id	sid	tid	token	lemma	pos
cana-3453	1	1	communications	communication	NOUN
cana-3453	1	2	on	on	ADP
cana-3453	1	3	applied	apply	VERB
cana-3453	1	4	nonlinear	nonlinear	ADJ
cana-3453	1	5	analysis	analysis	NOUN
cana-3453	1	6	issn	issn	NOUN
cana-3453	1	7	:	:	PUNCT
cana-3453	1	8	1074	1074	NUM
cana-3453	1	9	-	-	PUNCT
cana-3453	1	10	133x	133x	NUM
cana-3453	1	11	vol	vol	NOUN
cana-3453	1	12	32	32	NUM
cana-3453	1	13	no	no	NOUN
cana-3453	1	14	.	.	PUNCT
cana-3453	2	1	7s	7	NOUN
cana-3453	2	2	(	(	PUNCT
cana-3453	2	3	2025	2025	NUM
cana-3453	2	4	)	)	PUNCT
cana-3453	2	5	415	415	NUM
cana-3453	2	6	https://internationalpubls.com	https://internationalpubls.com	NUM
cana-3453	2	7	invariant	invariant	ADJ
cana-3453	2	8	submanifolds	submanifold	NOUN
cana-3453	2	9	of	of	ADP
cana-3453	2	10	generalized	generalized	ADJ
cana-3453	2	11	sasakian	sasakian	ADJ
cana-3453	2	12	-	-	PUNCT
cana-3453	2	13	space	space	NOUN
cana-3453	2	14	-	-	PUNCT
cana-3453	2	15	forms	form	NOUN
cana-3453	2	16	sagarika	sagarika	VERB
cana-3453	2	17	nandy	nandy	PROPN
cana-3453	2	18	*	*	PUNCT
cana-3453	2	19	*	*	PUNCT
cana-3453	2	20	department	department	PROPN
cana-3453	2	21	of	of	ADP
cana-3453	2	22	mathematics	mathematic	NOUN
cana-3453	2	23	,	,	PUNCT
cana-3453	2	24	deshabandhu	deshabandhu	NOUN
cana-3453	2	25	mahavidyalaya	mahavidyalaya	PROPN
cana-3453	2	26	,	,	PUNCT
cana-3453	2	27	chittaranjan	chittaranjan	PROPN
cana-3453	2	28	,	,	PUNCT
cana-3453	2	29	paschim	paschim	NOUN
cana-3453	2	30	bardhaman	bardhaman	NOUN
cana-3453	2	31	,	,	PUNCT
cana-3453	2	32	west	west	PROPN
cana-3453	2	33	bengal	bengal	PROPN
cana-3453	2	34	,	,	PUNCT
cana-3453	2	35	india	india	PROPN
cana-3453	2	36	e	e	PROPN
cana-3453	2	37	-	-	NOUN
cana-3453	2	38	mail	mail	NOUN
cana-3453	2	39	:	:	PUNCT
cana-3453	3	1	dbmsagarika@gmail.com	dbmsagarika@gmail.com	X
cana-3453	3	2	article	article	NOUN
cana-3453	3	3	history	history	NOUN
cana-3453	3	4	:	:	PUNCT
cana-3453	3	5	received	receive	VERB
cana-3453	3	6	:	:	PUNCT
cana-3453	3	7	26	26	NUM
cana-3453	3	8	-	-	SYM
cana-3453	3	9	10	10	NUM
cana-3453	3	10	-	-	PUNCT
cana-3453	3	11	2024	2024	NUM
cana-3453	3	12	revised:10	revised:10	NOUN
cana-3453	3	13	-	-	PUNCT
cana-3453	3	14	11	11	NUM
cana-3453	3	15	-	-	PUNCT
cana-3453	3	16	2024	2024	NUM
cana-3453	3	17	accepted:18	accepted:18	PROPN
cana-3453	3	18	-	-	PUNCT
cana-3453	3	19	12	12	NUM
cana-3453	3	20	-	-	PUNCT
cana-3453	3	21	2024	2024	NUM
cana-3453	3	22	abstract	abstract	NOUN
cana-3453	3	23	:	:	PUNCT
cana-3453	3	24	the	the	DET
cana-3453	3	25	new	new	ADJ
cana-3453	3	26	characterizations	characterization	NOUN
cana-3453	3	27	of	of	ADP
cana-3453	3	28	invariant	invariant	ADJ
cana-3453	3	29	submanifolds	submanifold	NOUN
cana-3453	3	30	of	of	ADP
cana-3453	3	31	generalized	generalized	ADJ
cana-3453	3	32	sasakian	sasakian	ADJ
cana-3453	3	33	-	-	PUNCT
cana-3453	3	34	space	space	NOUN
cana-3453	3	35	forms	form	NOUN
cana-3453	3	36	(	(	PUNCT
cana-3453	3	37	ssf	ssf	NOUN
cana-3453	3	38	)	)	PUNCT
cana-3453	3	39	in	in	ADP
cana-3453	3	40	terms	term	NOUN
cana-3453	3	41	of	of	ADP
cana-3453	3	42	their	their	PRON
cana-3453	3	43	behavior	behavior	NOUN
cana-3453	3	44	with	with	ADP
cana-3453	3	45	respect	respect	NOUN
cana-3453	3	46	to	to	ADP
cana-3453	3	47	the	the	DET
cana-3453	3	48	various	various	ADJ
cana-3453	3	49	curvature	curvature	NOUN
cana-3453	3	50	tensors	tensor	NOUN
cana-3453	3	51	are	be	AUX
cana-3453	3	52	obtained	obtain	VERB
cana-3453	3	53	in	in	ADP
cana-3453	3	54	this	this	DET
cana-3453	3	55	work	work	NOUN
cana-3453	3	56	.	.	PUNCT
cana-3453	4	1	by	by	ADP
cana-3453	4	2	examining	examine	VERB
cana-3453	4	3	the	the	DET
cana-3453	4	4	connections	connection	NOUN
cana-3453	4	5	between	between	ADP
cana-3453	4	6	these	these	DET
cana-3453	4	7	submanifolds	submanifold	NOUN
cana-3453	4	8	'	'	PART
cana-3453	4	9	second	second	ADJ
cana-3453	4	10	fundamental	fundamental	ADJ
cana-3453	4	11	form	form	NOUN
cana-3453	4	12	σ	σ	NOUN
cana-3453	4	13	and	and	CCONJ
cana-3453	4	14	certain	certain	ADJ
cana-3453	4	15	curvature	curvature	NOUN
cana-3453	4	16	tensors	tensor	NOUN
cana-3453	4	17	wi	wi	PROPN
cana-3453	4	18	with	with	ADP
cana-3453	4	19	i	i	PROPN
cana-3453	4	20	=	=	SYM
cana-3453	4	21	2	2	NUM
cana-3453	4	22	,	,	PUNCT
cana-3453	4	23	3	3	NUM
cana-3453	4	24	,	,	PUNCT
cana-3453	4	25	4	4	NUM
cana-3453	4	26	,	,	PUNCT
cana-3453	4	27	6	6	NUM
cana-3453	4	28	,	,	PUNCT
cana-3453	4	29	7	7	NUM
cana-3453	4	30	,	,	PUNCT
cana-3453	4	31	we	we	PRON
cana-3453	4	32	derive	derive	VERB
cana-3453	4	33	necessary	necessary	ADJ
cana-3453	4	34	and	and	CCONJ
cana-3453	4	35	sufficient	sufficient	ADJ
cana-3453	4	36	conditions	condition	NOUN
cana-3453	4	37	for	for	ADP
cana-3453	4	38	their	their	PRON
cana-3453	4	39	geodesicity	geodesicity	NOUN
cana-3453	4	40	.	.	PUNCT
cana-3453	5	1	we	we	PRON
cana-3453	5	2	show	show	VERB
cana-3453	5	3	that	that	SCONJ
cana-3453	5	4	total	total	ADJ
cana-3453	5	5	geodesicity	geodesicity	NOUN
cana-3453	5	6	corresponds	correspond	VERB
cana-3453	5	7	to	to	ADP
cana-3453	5	8	the	the	DET
cana-3453	5	9	claim	claim	NOUN
cana-3453	5	10	that	that	SCONJ
cana-3453	5	11	tensor	tensor	NOUN
cana-3453	5	12	products	product	NOUN
cana-3453	5	13	vanish	vanish	VERB
cana-3453	5	14	,	,	PUNCT
cana-3453	5	15	q(σ	q(σ	PROPN
cana-3453	5	16	,	,	PUNCT
cana-3453	5	17	wi	wi	PROPN
cana-3453	5	18	)	)	PUNCT
cana-3453	5	19	=	=	SYM
cana-3453	5	20	0	0	NUM
cana-3453	5	21	,	,	PUNCT
cana-3453	5	22	subject	subject	ADJ
cana-3453	5	23	to	to	ADP
cana-3453	5	24	different	different	ADJ
cana-3453	5	25	non	non	ADJ
cana-3453	5	26	-	-	ADJ
cana-3453	5	27	degeneracy	degeneracy	ADJ
cana-3453	5	28	conditions	condition	NOUN
cana-3453	5	29	for	for	ADP
cana-3453	5	30	each	each	DET
cana-3453	5	31	curvature	curvature	NOUN
cana-3453	5	32	tensor	tensor	NOUN
cana-3453	5	33	.	.	PUNCT
cana-3453	6	1	the	the	DET
cana-3453	6	2	essential	essential	ADJ
cana-3453	6	3	characterization	characterization	NOUN
cana-3453	6	4	comes	come	VERB
cana-3453	6	5	from	from	ADP
cana-3453	6	6	the	the	DET
cana-3453	6	7	w6	w6	PROPN
cana-3453	6	8	curvature	curvature	NOUN
cana-3453	6	9	tensor	tensor	NOUN
cana-3453	6	10	,	,	PUNCT
cana-3453	6	11	which	which	PRON
cana-3453	6	12	suffices	suffice	VERB
cana-3453	6	13	to	to	PART
cana-3453	6	14	fulfil	fulfil	VERB
cana-3453	6	15	2n(f1−f3)≠0	2n(f1−f3)≠0	NUM
cana-3453	6	16	,	,	PUNCT
cana-3453	6	17	and	and	CCONJ
cana-3453	6	18	all	all	DET
cana-3453	6	19	other	other	ADJ
cana-3453	6	20	tensors	tensor	NOUN
cana-3453	6	21	lead	lead	VERB
cana-3453	6	22	to	to	ADP
cana-3453	6	23	complementary	complementary	ADJ
cana-3453	6	24	constraints	constraint	NOUN
cana-3453	6	25	concerning	concern	VERB
cana-3453	6	26	the	the	DET
cana-3453	6	27	structural	structural	ADJ
cana-3453	6	28	functions	function	NOUN
cana-3453	6	29	f1	f1	NOUN
cana-3453	6	30	,	,	PUNCT
cana-3453	6	31	f2	f2	PROPN
cana-3453	6	32	,	,	PUNCT
cana-3453	6	33	and	and	CCONJ
cana-3453	6	34	f3	f3	ADJ
cana-3453	6	35	.	.	PUNCT
cana-3453	7	1	these	these	DET
cana-3453	7	2	findings	finding	NOUN
cana-3453	7	3	offer	offer	VERB
cana-3453	7	4	various	various	ADJ
cana-3453	7	5	avenues	avenue	NOUN
cana-3453	7	6	for	for	ADP
cana-3453	7	7	studying	study	VERB
cana-3453	7	8	the	the	DET
cana-3453	7	9	geometric	geometric	ADJ
cana-3453	7	10	nature	nature	NOUN
cana-3453	7	11	of	of	ADP
cana-3453	7	12	invariant	invariant	ADJ
cana-3453	7	13	submanifolds	submanifold	NOUN
cana-3453	7	14	and	and	CCONJ
cana-3453	7	15	enhance	enhance	VERB
cana-3453	7	16	our	our	PRON
cana-3453	7	17	insights	insight	NOUN
cana-3453	7	18	into	into	ADP
cana-3453	7	19	the	the	DET
cana-3453	7	20	behaviour	behaviour	NOUN
cana-3453	7	21	of	of	ADP
cana-3453	7	22	generalized	generalized	ADJ
cana-3453	7	23	sasakian	sasakian	ADJ
cana-3453	7	24	-	-	PUNCT
cana-3453	7	25	space	space	NOUN
cana-3453	7	26	-	-	PUNCT
cana-3453	7	27	forms	form	NOUN
cana-3453	7	28	.	.	PUNCT
cana-3453	8	1	keywords	keyword	NOUN
cana-3453	8	2	:	:	PUNCT
cana-3453	8	3	generalized	generalized	ADJ
cana-3453	8	4	sasakian	sasakian	ADJ
cana-3453	8	5	-	-	PUNCT
cana-3453	8	6	space	space	NOUN
cana-3453	8	7	-	-	PUNCT
cana-3453	8	8	forms	form	NOUN
cana-3453	8	9	;	;	PUNCT
cana-3453	8	10	invariant	invariant	ADJ
cana-3453	8	11	submanifold	submanifold	NOUN
cana-3453	8	12	;	;	PUNCT
cana-3453	8	13	geodesic	geodesic	NOUN
cana-3453	8	14	;	;	PUNCT
cana-3453	8	15	curvature	curvature	NOUN
cana-3453	8	16	tensors	tensor	NOUN
cana-3453	8	17	;	;	PUNCT
cana-3453	8	18	second	second	ADJ
cana-3453	8	19	fundamental	fundamental	ADJ
cana-3453	8	20	form	form	NOUN
cana-3453	8	21	.	.	PUNCT
cana-3453	9	1	msc2010	msc2010	NOUN
cana-3453	9	2	:	:	PUNCT
cana-3453	9	3	53c40	53c40	NUM
cana-3453	9	4	,	,	PUNCT
cana-3453	9	5	53c22	53c22	NUM
cana-3453	9	6	,	,	PUNCT
cana-3453	9	7	53d15	53d15	NUM
cana-3453	9	8	.	.	NOUN
cana-3453	10	1	1	1	NUM
cana-3453	10	2	.	.	X
cana-3453	10	3	introduction	introduction	NOUN
cana-3453	10	4	the	the	DET
cana-3453	10	5	differential	differential	ADJ
cana-3453	10	6	geometry	geometry	NOUN
cana-3453	10	7	and	and	CCONJ
cana-3453	10	8	submanifold	submanifold	NOUN
cana-3453	10	9	theory	theory	NOUN
cana-3453	10	10	have	have	AUX
cana-3453	10	11	been	be	AUX
cana-3453	10	12	productive	productive	ADJ
cana-3453	10	13	in	in	ADP
cana-3453	10	14	getting	get	VERB
cana-3453	10	15	deep	deep	ADJ
cana-3453	10	16	information	information	NOUN
cana-3453	10	17	about	about	ADP
cana-3453	10	18	the	the	DET
cana-3453	10	19	manifold	manifold	ADJ
cana-3453	10	20	structure	structure	NOUN
cana-3453	10	21	.	.	PUNCT
cana-3453	11	1	the	the	DET
cana-3453	11	2	invariant	invariant	ADJ
cana-3453	11	3	submanifolds	submanifold	NOUN
cana-3453	11	4	are	be	AUX
cana-3453	11	5	one	one	NUM
cana-3453	11	6	of	of	ADP
cana-3453	11	7	these	these	DET
cana-3453	11	8	geometric	geometric	ADJ
cana-3453	11	9	structures	structure	NOUN
cana-3453	11	10	that	that	PRON
cana-3453	11	11	emerged	emerge	VERB
cana-3453	11	12	among	among	ADP
cana-3453	11	13	many	many	ADJ
cana-3453	11	14	others	other	NOUN
cana-3453	11	15	and	and	CCONJ
cana-3453	11	16	have	have	AUX
cana-3453	11	17	been	be	AUX
cana-3453	11	18	studied	study	VERB
cana-3453	11	19	due	due	ADP
cana-3453	11	20	to	to	ADP
cana-3453	11	21	their	their	PRON
cana-3453	11	22	nature	nature	NOUN
cana-3453	11	23	of	of	ADP
cana-3453	11	24	being	be	AUX
cana-3453	11	25	interesting	interesting	ADJ
cana-3453	11	26	and	and	CCONJ
cana-3453	11	27	giving	give	VERB
cana-3453	11	28	both	both	DET
cana-3453	11	29	an	an	DET
cana-3453	11	30	intrinsic	intrinsic	ADJ
cana-3453	11	31	and	and	CCONJ
cana-3453	11	32	extrinsic	extrinsic	ADJ
cana-3453	11	33	geometry	geometry	NOUN
cana-3453	11	34	of	of	ADP
cana-3453	11	35	the	the	DET
cana-3453	11	36	ambient	ambient	NOUN
cana-3453	11	37	or	or	CCONJ
cana-3453	11	38	underlying	underlying	ADJ
cana-3453	11	39	space	space	NOUN
cana-3453	11	40	.	.	PUNCT
cana-3453	12	1	they	they	PRON
cana-3453	12	2	are	be	AUX
cana-3453	12	3	important	important	ADJ
cana-3453	12	4	in	in	ADP
cana-3453	12	5	studying	study	VERB
cana-3453	12	6	geometric	geometric	ADJ
cana-3453	12	7	properties	property	NOUN
cana-3453	12	8	while	while	SCONJ
cana-3453	12	9	keeping	keep	VERB
cana-3453	12	10	some	some	DET
cana-3453	12	11	structures	structure	NOUN
cana-3453	12	12	of	of	ADP
cana-3453	12	13	the	the	DET
cana-3453	12	14	ambient	ambient	NOUN
cana-3453	12	15	manifold	manifold	NOUN
cana-3453	12	16	.	.	PUNCT
cana-3453	13	1	since	since	SCONJ
cana-3453	13	2	its	its	PRON
cana-3453	13	3	early	early	ADJ
cana-3453	13	4	days	day	NOUN
cana-3453	13	5	,	,	PUNCT
cana-3453	13	6	the	the	DET
cana-3453	13	7	geometric	geometric	ADJ
cana-3453	13	8	theory	theory	NOUN
cana-3453	13	9	of	of	ADP
cana-3453	13	10	submanifolds	submanifold	NOUN
cana-3453	13	11	has	have	AUX
cana-3453	13	12	developed	develop	VERB
cana-3453	13	13	through	through	ADP
cana-3453	13	14	local	local	ADJ
cana-3453	13	15	differential	differential	NOUN
cana-3453	13	16	geometry	geometry	NOUN
cana-3453	13	17	and	and	CCONJ
cana-3453	13	18	global	global	ADJ
cana-3453	13	19	analytical	analytical	ADJ
cana-3453	13	20	methods	method	NOUN
cana-3453	13	21	.	.	PUNCT
cana-3453	14	1	in	in	ADP
cana-3453	14	2	this	this	DET
cana-3453	14	3	context	context	NOUN
cana-3453	14	4	,	,	PUNCT
cana-3453	14	5	generalized	generalized	ADJ
cana-3453	14	6	ssf	ssf	NOUN
cana-3453	14	7	play	play	VERB
cana-3453	14	8	an	an	DET
cana-3453	14	9	ever	ever	ADV
cana-3453	14	10	more	more	ADV
cana-3453	14	11	central	central	ADJ
cana-3453	14	12	role	role	NOUN
cana-3453	14	13	in	in	ADP
cana-3453	14	14	studying	study	VERB
cana-3453	14	15	how	how	SCONJ
cana-3453	14	16	contact	contact	NOUN
cana-3453	14	17	structures	structure	NOUN
cana-3453	14	18	affect	affect	VERB
cana-3453	14	19	differential	differential	ADJ
cana-3453	14	20	geometric	geometric	ADJ
cana-3453	14	21	properties	property	NOUN
cana-3453	14	22	.	.	PUNCT
cana-3453	15	1	these	these	DET
cana-3453	15	2	manifolds	manifold	NOUN
cana-3453	15	3	are	be	AUX
cana-3453	15	4	a	a	DET
cana-3453	15	5	natural	natural	ADJ
cana-3453	15	6	generalization	generalization	NOUN
cana-3453	15	7	of	of	ADP
cana-3453	15	8	classical	classical	ADJ
cana-3453	15	9	space	space	NOUN
cana-3453	15	10	forms	form	NOUN
cana-3453	15	11	while	while	SCONJ
cana-3453	15	12	still	still	ADV
cana-3453	15	13	preserving	preserve	VERB
cana-3453	15	14	important	important	ADJ
cana-3453	15	15	contact	contact	NOUN
cana-3453	15	16	geometric	geometric	ADJ
cana-3453	15	17	properties	property	NOUN
cana-3453	15	18	.	.	PUNCT
cana-3453	16	1	generalized	generalized	ADJ
cana-3453	16	2	sasaki	sasaki	NOUN
cana-3453	16	3	spaces	space	NOUN
cana-3453	16	4	,	,	PUNCT
cana-3453	16	5	which	which	PRON
cana-3453	16	6	,	,	PUNCT
cana-3453	16	7	as	as	ADP
cana-3453	16	8	manifolds	manifold	NOUN
cana-3453	16	9	,	,	PUNCT
cana-3453	16	10	exhibit	exhibit	VERB
cana-3453	16	11	a	a	DET
cana-3453	16	12	rich	rich	ADJ
cana-3453	16	13	interplay	interplay	NOUN
cana-3453	16	14	between	between	ADP
cana-3453	16	15	metric	metric	ADJ
cana-3453	16	16	properties	property	NOUN
cana-3453	16	17	and	and	CCONJ
cana-3453	16	18	contact	contact	NOUN
cana-3453	16	19	structure	structure	NOUN
cana-3453	16	20	,	,	PUNCT
cana-3453	16	21	can	can	AUX
cana-3453	16	22	thus	thus	ADV
cana-3453	16	23	be	be	AUX
cana-3453	16	24	seen	see	VERB
cana-3453	16	25	as	as	ADP
cana-3453	16	26	a	a	DET
cana-3453	16	27	natural	natural	ADJ
cana-3453	16	28	setup	setup	NOUN
cana-3453	16	29	for	for	ADP
cana-3453	16	30	these	these	DET
cana-3453	16	31	studies	study	NOUN
cana-3453	16	32	since	since	SCONJ
cana-3453	16	33	they	they	PRON
cana-3453	16	34	are	be	AUX
cana-3453	16	35	generalizations	generalization	NOUN
cana-3453	16	36	of	of	ADP
cana-3453	16	37	classical	classical	ADJ
cana-3453	16	38	sasakian	sasakian	ADJ
cana-3453	16	39	manifold	manifold	NOUN
cana-3453	16	40	,	,	PUNCT
cana-3453	16	41	which	which	PRON
cana-3453	16	42	serves	serve	VERB
cana-3453	16	43	as	as	ADP
cana-3453	16	44	the	the	DET
cana-3453	16	45	building	building	NOUN
cana-3453	16	46	blocks	block	NOUN
cana-3453	16	47	to	to	ADP
cana-3453	16	48	the	the	DET
cana-3453	16	49	study	study	NOUN
cana-3453	16	50	of	of	ADP
cana-3453	16	51	the	the	DET
cana-3453	16	52	submanifold	submanifold	NOUN
cana-3453	16	53	behaviour	behaviour	NOUN
cana-3453	16	54	.	.	PUNCT
cana-3453	17	1	generalized	generalized	ADJ
cana-3453	17	2	geometry	geometry	NOUN
cana-3453	17	3	and	and	CCONJ
cana-3453	17	4	sasakian	sasakian	ADJ
cana-3453	17	5	geometry	geometry	NOUN
cana-3453	17	6	have	have	AUX
cana-3453	17	7	made	make	VERB
cana-3453	17	8	great	great	ADJ
cana-3453	17	9	strides	stride	NOUN
cana-3453	17	10	in	in	ADP
cana-3453	17	11	the	the	DET
cana-3453	17	12	last	last	ADJ
cana-3453	17	13	few	few	ADJ
cana-3453	17	14	decades	decade	NOUN
cana-3453	17	15	.	.	PUNCT
cana-3453	18	1	note	note	VERB
cana-3453	18	2	that	that	SCONJ
cana-3453	18	3	a	a	DET
cana-3453	18	4	generalized	generalized	ADJ
cana-3453	18	5	sasakian	sasakian	ADJ
cana-3453	18	6	-	-	PUNCT
cana-3453	18	7	space	space	NOUN
cana-3453	18	8	-	-	PUNCT
cana-3453	18	9	form	form	NOUN
cana-3453	18	10	can	can	AUX
cana-3453	18	11	be	be	AUX
cana-3453	18	12	considered	consider	VERB
cana-3453	18	13	a	a	DET
cana-3453	18	14	natural	natural	ADJ
cana-3453	18	15	generalization	generalization	NOUN
cana-3453	18	16	of	of	ADP
cana-3453	18	17	metric	metric	ADJ
cana-3453	18	18	contact	contact	NOUN
cana-3453	18	19	structures	structure	NOUN
cana-3453	18	20	relevant	relevant	ADJ
cana-3453	18	21	to	to	ADP
cana-3453	18	22	classical	classical	ADJ
cana-3453	18	23	ones	one	NOUN
cana-3453	18	24	,	,	PUNCT
cana-3453	18	25	known	know	VERB
cana-3453	18	26	to	to	PART
cana-3453	18	27	be	be	AUX
cana-3453	18	28	contained	contain	VERB
cana-3453	18	29	in	in	ADP
cana-3453	18	30	the	the	DET
cana-3453	18	31	classical	classical	ADJ
cana-3453	18	32	space	space	NOUN
cana-3453	18	33	forms	form	NOUN
cana-3453	18	34	.	.	PUNCT
cana-3453	19	1	this	this	DET
cana-3453	19	2	communications	communication	NOUN
cana-3453	19	3	on	on	ADP
cana-3453	19	4	applied	apply	VERB
cana-3453	19	5	nonlinear	nonlinear	ADJ
cana-3453	19	6	analysis	analysis	NOUN
cana-3453	19	7	issn	issn	NOUN
cana-3453	19	8	:	:	PUNCT
cana-3453	19	9	1074	1074	NUM
cana-3453	19	10	-	-	PUNCT
cana-3453	19	11	133x	133x	NUM
cana-3453	19	12	vol	vol	NOUN
cana-3453	19	13	32	32	NUM
cana-3453	19	14	no	no	NOUN
cana-3453	19	15	.	.	PUNCT
cana-3453	20	1	7s	7	NOUN
cana-3453	20	2	(	(	PUNCT
cana-3453	20	3	2025	2025	NUM
cana-3453	20	4	)	)	PUNCT
cana-3453	20	5	416	416	NUM
cana-3453	20	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3453	20	7	generalization	generalization	NOUN
cana-3453	20	8	has	have	AUX
cana-3453	20	9	attracted	attract	VERB
cana-3453	20	10	considerable	considerable	ADJ
cana-3453	20	11	attention	attention	NOUN
cana-3453	20	12	from	from	ADP
cana-3453	20	13	researchers	researcher	NOUN
cana-3453	20	14	worldwide	worldwide	ADV
cana-3453	20	15	,	,	PUNCT
cana-3453	20	16	as	as	SCONJ
cana-3453	20	17	evidenced	evidence	VERB
cana-3453	20	18	by	by	ADP
cana-3453	20	19	numerous	numerous	ADJ
cana-3453	20	20	contributions	contribution	NOUN
cana-3453	20	21	[	[	X
cana-3453	20	22	2	2	NUM
cana-3453	20	23	,	,	PUNCT
cana-3453	20	24	3	3	NUM
cana-3453	20	25	,	,	PUNCT
cana-3453	20	26	4	4	NUM
cana-3453	20	27	,	,	PUNCT
cana-3453	20	28	5	5	NUM
cana-3453	20	29	,	,	PUNCT
cana-3453	20	30	12	12	NUM
cana-3453	20	31	,	,	PUNCT
cana-3453	20	32	14	14	NUM
cana-3453	20	33	,	,	PUNCT
cana-3453	20	34	15	15	NUM
cana-3453	20	35	,	,	PUNCT
cana-3453	20	36	16	16	NUM
cana-3453	20	37	,	,	PUNCT
cana-3453	20	38	18	18	NUM
cana-3453	20	39	,	,	PUNCT
cana-3453	20	40	20	20	NUM
cana-3453	20	41	,	,	PUNCT
cana-3453	20	42	22	22	NUM
cana-3453	20	43	]	]	PUNCT
cana-3453	20	44	.	.	PUNCT
cana-3453	21	1	this	this	PRON
cana-3453	21	2	is	be	AUX
cana-3453	21	3	a	a	DET
cana-3453	21	4	more	more	ADV
cana-3453	21	5	involved	involved	ADJ
cana-3453	21	6	way	way	NOUN
cana-3453	21	7	of	of	ADP
cana-3453	21	8	describing	describe	VERB
cana-3453	21	9	things	thing	NOUN
cana-3453	21	10	in	in	ADP
cana-3453	21	11	terms	term	NOUN
cana-3453	21	12	of	of	ADP
cana-3453	21	13	curvature	curvature	NOUN
cana-3453	21	14	tensors	tensor	NOUN
cana-3453	21	15	.	.	PUNCT
cana-3453	22	1	such	such	ADJ
cana-3453	22	2	tensors	tensor	NOUN
cana-3453	22	3	,	,	PUNCT
cana-3453	22	4	though	though	SCONJ
cana-3453	22	5	aligned	align	VERB
cana-3453	22	6	with	with	ADP
cana-3453	22	7	the	the	DET
cana-3453	22	8	usual	usual	ADJ
cana-3453	22	9	suspects	suspect	NOUN
cana-3453	22	10	from	from	ADP
cana-3453	22	11	classical	classical	ADJ
cana-3453	22	12	curvature	curvature	NOUN
cana-3453	22	13	measurements	measurement	NOUN
cana-3453	22	14	,	,	PUNCT
cana-3453	22	15	express	express	VERB
cana-3453	22	16	new	new	ADJ
cana-3453	22	17	aspects	aspect	NOUN
cana-3453	22	18	of	of	ADP
cana-3453	22	19	how	how	SCONJ
cana-3453	22	20	the	the	DET
cana-3453	22	21	contact	contact	NOUN
cana-3453	22	22	structure	structure	NOUN
cana-3453	22	23	affects	affect	VERB
cana-3453	22	24	geometric	geometric	ADJ
cana-3453	22	25	properties	property	NOUN
cana-3453	22	26	.	.	PUNCT
cana-3453	23	1	several	several	ADJ
cana-3453	23	2	of	of	ADP
cana-3453	23	3	these	these	DET
cana-3453	23	4	expressions	expression	NOUN
cana-3453	23	5	can	can	AUX
cana-3453	23	6	be	be	AUX
cana-3453	23	7	reconstructed	reconstruct	VERB
cana-3453	23	8	as	as	ADP
cana-3453	23	9	some	some	DET
cana-3453	23	10	curvature	curvature	NOUN
cana-3453	23	11	tensors	tensor	NOUN
cana-3453	23	12	with	with	ADP
cana-3453	23	13	the	the	DET
cana-3453	23	14	second	second	ADJ
cana-3453	23	15	fundamental	fundamental	ADJ
cana-3453	23	16	form	form	NOUN
cana-3453	23	17	from	from	ADP
cana-3453	23	18	different	different	ADJ
cana-3453	23	19	points	point	NOUN
cana-3453	23	20	of	of	ADP
cana-3453	23	21	view	view	NOUN
cana-3453	23	22	to	to	PART
cana-3453	23	23	examine	examine	VERB
cana-3453	23	24	the	the	DET
cana-3453	23	25	second	second	ADJ
cana-3453	23	26	fundamental	fundamental	ADJ
cana-3453	23	27	form	form	NOUN
cana-3453	23	28	and	and	CCONJ
cana-3453	23	29	geodesic	geodesic	NOUN
cana-3453	23	30	submanifolds	submanifold	NOUN
cana-3453	23	31	related	relate	VERB
cana-3453	23	32	with	with	ADP
cana-3453	23	33	each	each	DET
cana-3453	23	34	other	other	ADJ
cana-3453	23	35	via	via	ADP
cana-3453	23	36	these	these	DET
cana-3453	23	37	second	second	ADJ
cana-3453	23	38	fundamental	fundamental	ADJ
cana-3453	23	39	form	form	NOUN
cana-3453	23	40	relationships	relationship	NOUN
cana-3453	23	41	.	.	PUNCT
cana-3453	24	1	by	by	ADP
cana-3453	24	2	examining	examine	VERB
cana-3453	24	3	the	the	DET
cana-3453	24	4	interplay	interplay	NOUN
cana-3453	24	5	between	between	ADP
cana-3453	24	6	the	the	DET
cana-3453	24	7	respective	respective	ADJ
cana-3453	24	8	metric	metric	ADJ
cana-3453	24	9	tensors	tensor	NOUN
cana-3453	24	10	and	and	CCONJ
cana-3453	24	11	associated	associate	VERB
cana-3453	24	12	contact	contact	NOUN
cana-3453	24	13	structures	structure	NOUN
cana-3453	24	14	on	on	ADP
cana-3453	24	15	such	such	ADJ
cana-3453	24	16	spaces	space	NOUN
cana-3453	24	17	,	,	PUNCT
cana-3453	24	18	we	we	PRON
cana-3453	24	19	uncover	uncover	VERB
cana-3453	24	20	striking	striking	ADJ
cana-3453	24	21	similarities	similarity	NOUN
cana-3453	24	22	to	to	ADP
cana-3453	24	23	the	the	DET
cana-3453	24	24	structures	structure	NOUN
cana-3453	24	25	of	of	ADP
cana-3453	24	26	classical	classical	ADJ
cana-3453	24	27	differential	differential	NOUN
cana-3453	24	28	geometry	geometry	NOUN
cana-3453	24	29	.	.	PUNCT
cana-3453	25	1	these	these	DET
cana-3453	25	2	behaviors	behavior	NOUN
cana-3453	25	3	differ	differ	VERB
cana-3453	25	4	from	from	ADP
cana-3453	25	5	riemannian	riemannian	ADJ
cana-3453	25	6	manifolds	manifold	NOUN
cana-3453	25	7	with	with	ADP
cana-3453	25	8	constant	constant	ADJ
cana-3453	25	9	sectional	sectional	ADJ
cana-3453	25	10	curvature	curvature	NOUN
cana-3453	25	11	,	,	PUNCT
cana-3453	25	12	as	as	SCONJ
cana-3453	25	13	their	their	PRON
cana-3453	25	14	curvature	curvature	NOUN
cana-3453	25	15	conditions	condition	NOUN
cana-3453	25	16	are	be	AUX
cana-3453	25	17	determined	determine	VERB
cana-3453	25	18	by	by	ADP
cana-3453	25	19	the	the	DET
cana-3453	25	20	functions	function	NOUN
cana-3453	25	21	f₁	f₁	NOUN
cana-3453	25	22	,	,	PUNCT
cana-3453	25	23	f₂	f₂	NOUN
cana-3453	25	24	and	and	CCONJ
cana-3453	25	25	f₃.	f₃.	NOUN
cana-3453	25	26	this	this	DET
cana-3453	25	27	modification	modification	NOUN
cana-3453	25	28	brings	bring	VERB
cana-3453	25	29	in	in	ADP
cana-3453	25	30	nontrivial	nontrivial	ADJ
cana-3453	25	31	geometric	geometric	ADJ
cana-3453	25	32	features	feature	NOUN
cana-3453	25	33	affecting	affect	VERB
cana-3453	25	34	local	local	ADJ
cana-3453	25	35	and	and	CCONJ
cana-3453	25	36	global	global	ADJ
cana-3453	25	37	properties	property	NOUN
cana-3453	25	38	of	of	ADP
cana-3453	25	39	invariant	invariant	ADJ
cana-3453	25	40	submanifolds	submanifold	NOUN
cana-3453	25	41	.	.	PUNCT
cana-3453	26	1	these	these	DET
cana-3453	26	2	functions	function	NOUN
cana-3453	26	3	are	be	AUX
cana-3453	26	4	related	relate	VERB
cana-3453	26	5	by	by	ADP
cana-3453	26	6	how	how	SCONJ
cana-3453	26	7	contact	contact	NOUN
cana-3453	26	8	structures	structure	NOUN
cana-3453	26	9	generalize	generalize	VERB
cana-3453	26	10	classical	classical	ADJ
cana-3453	26	11	curvature	curvature	NOUN
cana-3453	26	12	phenomena	phenomenon	NOUN
cana-3453	26	13	which	which	PRON
cana-3453	26	14	then	then	ADV
cana-3453	26	15	gives	give	VERB
cana-3453	26	16	rise	rise	NOUN
cana-3453	26	17	to	to	ADP
cana-3453	26	18	geometric	geometric	ADJ
cana-3453	26	19	phenomena	phenomenon	NOUN
cana-3453	26	20	only	only	ADV
cana-3453	26	21	found	find	VERB
cana-3453	26	22	in	in	ADP
cana-3453	26	23	contact	contact	NOUN
cana-3453	26	24	metric	metric	ADJ
cana-3453	26	25	geometry	geometry	NOUN
cana-3453	26	26	.	.	PUNCT
cana-3453	27	1	the	the	DET
cana-3453	27	2	primary	primary	ADJ
cana-3453	27	3	characteristic	characteristic	NOUN
cana-3453	27	4	that	that	PRON
cana-3453	27	5	characterizes	characterize	VERB
cana-3453	27	6	a	a	DET
cana-3453	27	7	generalized	generalized	ADJ
cana-3453	27	8	sasakian	sasakian	ADJ
cana-3453	27	9	-	-	PUNCT
cana-3453	27	10	space	space	NOUN
cana-3453	27	11	-	-	PUNCT
cana-3453	27	12	form	form	NOUN
cana-3453	27	13	is	be	AUX
cana-3453	27	14	its	its	PRON
cana-3453	27	15	curvature	curvature	NOUN
cana-3453	27	16	structure	structure	NOUN
cana-3453	27	17	.	.	PUNCT
cana-3453	28	1	if	if	SCONJ
cana-3453	28	2	m	m	NOUN
cana-3453	28	3	is	be	AUX
cana-3453	28	4	a	a	DET
cana-3453	28	5	nearly	nearly	ADV
cana-3453	28	6	contact	contact	NOUN
cana-3453	28	7	metric	metric	ADJ
cana-3453	28	8	manifold	manifold	ADJ
cana-3453	28	9	,	,	PUNCT
cana-3453	28	10	then	then	ADV
cana-3453	28	11	f1	f1	NOUN
cana-3453	28	12	,	,	PUNCT
cana-3453	28	13	f2	f2	PROPN
cana-3453	28	14	,	,	PUNCT
cana-3453	28	15	and	and	CCONJ
cana-3453	28	16	f3	f3	PROPN
cana-3453	28	17	are	be	AUX
cana-3453	28	18	its	its	PRON
cana-3453	28	19	three	three	NUM
cana-3453	28	20	differentiable	differentiable	ADJ
cana-3453	28	21	functions	function	NOUN
cana-3453	28	22	.	.	PUNCT
cana-3453	29	1	given	give	VERB
cana-3453	29	2	the	the	DET
cana-3453	29	3	following	follow	VERB
cana-3453	29	4	form	form	NOUN
cana-3453	29	5	's	's	PART
cana-3453	29	6	curvature	curvature	NOUN
cana-3453	29	7	tensor	tensor	NOUN
cana-3453	29	8	r	r	NOUN
cana-3453	29	9	:	:	PUNCT
cana-3453	29	10	𝑹(𝑿	𝑹(𝑿	NOUN
cana-3453	29	11	,	,	PUNCT
cana-3453	29	12	𝒀)𝒁	𝒀)𝒁	NOUN
cana-3453	29	13	=	=	SYM
cana-3453	29	14	𝒇𝟏{𝒈(𝒀	𝒇𝟏{𝒈(𝒀	PROPN
cana-3453	29	15	,	,	PUNCT
cana-3453	29	16	𝒁)𝑿	𝒁)𝑿	ADV
cana-3453	29	17	−	−	NOUN
cana-3453	29	18	𝒈(𝑿	𝒈(𝑿	NOUN
cana-3453	29	19	,	,	PUNCT
cana-3453	29	20	𝒁)𝒀	𝒁)𝒀	NOUN
cana-3453	29	21	}	}	PUNCT
cana-3453	29	22	+	+	CCONJ
cana-3453	29	23	𝒇𝟐{𝒈(𝑿	𝒇𝟐{𝒈(𝑿	PROPN
cana-3453	29	24	,	,	PUNCT
cana-3453	29	25	𝝓𝒁)𝝓𝒀	𝝓𝒁)𝝓𝒀	NOUN
cana-3453	29	26	−	−	NOUN
cana-3453	29	27	𝒈(𝒀	𝒈(𝒀	NOUN
cana-3453	29	28	,	,	PUNCT
cana-3453	29	29	𝝓𝒁)𝝓𝑿	𝝓𝒁)𝝓𝑿	PROPN
cana-3453	29	30	+	+	X
cana-3453	29	31	𝟐𝒈(𝑿	𝟐𝒈(𝑿	PROPN
cana-3453	29	32	,	,	PUNCT
cana-3453	29	33	𝝓𝒀)𝝓𝒁	𝝓𝒀)𝝓𝒁	NOUN
cana-3453	29	34	}	}	PUNCT
cana-3453	29	35	+	+	CCONJ
cana-3453	29	36	𝒇𝟑{𝜼(𝑿)𝜼(𝒁)𝒀	𝒇𝟑{𝜼(𝑿)𝜼(𝒁)𝒀	PRON
cana-3453	29	37	−	−	PROPN
cana-3453	29	38	𝜼(𝒀)𝜼(𝒁)𝑿	𝜼(𝒀)𝜼(𝒁)𝑿	ADJ
cana-3453	29	39	+	+	CCONJ
cana-3453	29	40	𝒈(𝑿	𝒈(𝑿	PROPN
cana-3453	29	41	,	,	PUNCT
cana-3453	29	42	𝒁)𝜼(𝒀)𝝃	𝒁)𝜼(𝒀)𝝃	VERB
cana-3453	29	43	−	−	NOUN
cana-3453	29	44	𝒈(𝒀	𝒈(𝒀	NOUN
cana-3453	29	45	,	,	PUNCT
cana-3453	29	46	𝒁)𝜼(𝑿)𝝃	𝒁)𝜼(𝑿)𝝃	PUNCT
cana-3453	29	47	}	}	PUNCT
cana-3453	29	48	(	(	PUNCT
cana-3453	29	49	1.1	1.1	NUM
cana-3453	29	50	)	)	PUNCT
cana-3453	29	51	for	for	ADP
cana-3453	29	52	any	any	DET
cana-3453	29	53	vector	vector	NOUN
cana-3453	29	54	fields	field	NOUN
cana-3453	29	55	x	x	X
cana-3453	29	56	,	,	PUNCT
cana-3453	29	57	y	y	PROPN
cana-3453	29	58	and	and	CCONJ
cana-3453	29	59	z	z	PROPN
cana-3453	29	60	on	on	ADP
cana-3453	29	61	m	m	PROPN
cana-3453	29	62	,	,	PUNCT
cana-3453	29	63	the	the	DET
cana-3453	29	64	manifold	manifold	NOUN
cana-3453	29	65	with	with	ADP
cana-3453	29	66	the	the	DET
cana-3453	29	67	structure	structure	NOUN
cana-3453	29	68	of	of	ADP
cana-3453	29	69	dimension	dimension	NOUN
cana-3453	29	70	(	(	PUNCT
cana-3453	29	71	2n+1),n>1	2n+1),n>1	NUM
cana-3453	29	72	is	be	AUX
cana-3453	29	73	called	call	VERB
cana-3453	29	74	a	a	DET
cana-3453	29	75	generalized	generalized	ADJ
cana-3453	29	76	sasakianspace	sasakianspace	NOUN
cana-3453	29	77	-	-	PUNCT
cana-3453	29	78	form	form	NOUN
cana-3453	29	79	if	if	SCONJ
cana-3453	29	80	the	the	DET
cana-3453	29	81	following	follow	VERB
cana-3453	29	82	condition	condition	NOUN
cana-3453	29	83	holds	hold	VERB
cana-3453	29	84	:	:	PUNCT
cana-3453	29	85	the	the	DET
cana-3453	29	86	condition	condition	NOUN
cana-3453	29	87	n>1	n>1	PROPN
cana-3453	29	88	will	will	AUX
cana-3453	29	89	be	be	AUX
cana-3453	29	90	held	hold	VERB
cana-3453	29	91	in	in	ADP
cana-3453	29	92	the	the	DET
cana-3453	29	93	rest	rest	NOUN
cana-3453	29	94	of	of	ADP
cana-3453	29	95	this	this	DET
cana-3453	29	96	paper	paper	NOUN
cana-3453	29	97	.	.	PUNCT
cana-3453	30	1	these	these	DET
cana-3453	30	2	structures	structure	NOUN
cana-3453	30	3	exhibit	exhibit	VERB
cana-3453	30	4	particularly	particularly	ADV
cana-3453	30	5	interesting	interesting	ADJ
cana-3453	30	6	properties	property	NOUN
cana-3453	30	7	when	when	SCONJ
cana-3453	30	8	the	the	DET
cana-3453	30	9	functions	function	NOUN
cana-3453	30	10	take	take	VERB
cana-3453	30	11	specific	specific	ADJ
cana-3453	30	12	values	value	NOUN
cana-3453	30	13	.	.	PUNCT
cana-3453	31	1	for	for	ADP
cana-3453	31	2	instance	instance	NOUN
cana-3453	31	3	,	,	PUNCT
cana-3453	31	4	when	when	SCONJ
cana-3453	31	5	𝑓1	𝑓1	PROPN
cana-3453	31	6	=	=	PROPN
cana-3453	31	7	𝑐+3	𝑐+3	NUM
cana-3453	31	8	4	4	NUM
cana-3453	31	9	,	,	PUNCT
cana-3453	31	10	𝑓2	𝑓2	ADJ
cana-3453	31	11	=	=	SYM
cana-3453	31	12	𝑓3	𝑓3	NOUN
cana-3453	31	13	=	=	SYM
cana-3453	31	14	𝑐−1	𝑐−1	NOUN
cana-3453	31	15	4	4	NUM
cana-3453	31	16	,	,	PUNCT
cana-3453	31	17	a	a	DET
cana-3453	31	18	generalized	generalize	VERB
cana-3453	31	19	sasakian	sasakian	ADJ
cana-3453	31	20	-	-	PUNCT
cana-3453	31	21	space	space	NOUN
cana-3453	31	22	-	-	PUNCT
cana-3453	31	23	form	form	NOUN
cana-3453	31	24	endowed	endow	VERB
cana-3453	31	25	with	with	ADP
cana-3453	31	26	a	a	DET
cana-3453	31	27	sasakian	sasakian	ADJ
cana-3453	31	28	structure	structure	NOUN
cana-3453	31	29	reduces	reduce	VERB
cana-3453	31	30	to	to	ADP
cana-3453	31	31	a	a	DET
cana-3453	31	32	classical	classical	ADJ
cana-3453	31	33	sasakian	sasakian	ADJ
cana-3453	31	34	-	-	PUNCT
cana-3453	31	35	space	space	NOUN
cana-3453	31	36	-	-	PUNCT
cana-3453	31	37	form	form	NOUN
cana-3453	31	38	.	.	PUNCT
cana-3453	32	1	hui	hui	PROPN
cana-3453	32	2	and	and	CCONJ
cana-3453	32	3	sarkar	sarkar	PROPN
cana-3453	32	4	[	[	X
cana-3453	32	5	18	18	NUM
cana-3453	32	6	]	]	PUNCT
cana-3453	32	7	recently	recently	ADV
cana-3453	32	8	achieved	achieve	VERB
cana-3453	32	9	advances	advance	NOUN
cana-3453	32	10	in	in	ADP
cana-3453	32	11	this	this	DET
cana-3453	32	12	respect	respect	NOUN
cana-3453	32	13	and	and	CCONJ
cana-3453	32	14	analyzed	analyze	VERB
cana-3453	32	15	w2	w2	NOUN
cana-3453	32	16	-	-	PUNCT
cana-3453	32	17	flat	flat	ADJ
cana-3453	32	18	generalized	generalize	VERB
cana-3453	32	19	sasakian	sasakian	ADJ
cana-3453	32	20	-	-	PUNCT
cana-3453	32	21	space	space	NOUN
cana-3453	32	22	-	-	PUNCT
cana-3453	32	23	forms	form	NOUN
cana-3453	32	24	in	in	ADP
cana-3453	32	25	detail	detail	NOUN
cana-3453	32	26	.	.	PUNCT
cana-3453	33	1	based	base	VERB
cana-3453	33	2	on	on	ADP
cana-3453	33	3	their	their	PRON
cana-3453	33	4	research	research	NOUN
cana-3453	33	5	,	,	PUNCT
cana-3453	33	6	they	they	PRON
cana-3453	33	7	provide	provide	VERB
cana-3453	33	8	necessary	necessary	ADJ
cana-3453	33	9	and	and	CCONJ
cana-3453	33	10	sufficient	sufficient	ADJ
cana-3453	33	11	conditions	condition	NOUN
cana-3453	33	12	for	for	ADP
cana-3453	33	13	w2	w2	NOUN
cana-3453	33	14	-	-	PUNCT
cana-3453	33	15	flatness	flatness	NOUN
cana-3453	33	16	and	and	CCONJ
cana-3453	33	17	discuss	discuss	VERB
cana-3453	33	18	the	the	DET
cana-3453	33	19	consequences	consequence	NOUN
cana-3453	33	20	of	of	ADP
cana-3453	33	21	conditions	condition	NOUN
cana-3453	33	22	like	like	ADP
cana-3453	33	23	w2	w2	NOUN
cana-3453	33	24	.	.	PUNCT
cana-3453	34	1	s=0	s=0	PROPN
cana-3453	34	2	.	.	PUNCT
cana-3453	35	1	they	they	PRON
cana-3453	35	2	showed	show	VERB
cana-3453	35	3	that	that	SCONJ
cana-3453	35	4	satisfying	satisfy	VERB
cana-3453	35	5	w2	w2	NOUN
cana-3453	35	6	are	be	AUX
cana-3453	35	7	generalized	generalize	VERB
cana-3453	35	8	sasakian	sasakian	ADJ
cana-3453	35	9	-	-	PUNCT
cana-3453	35	10	space	space	NOUN
cana-3453	35	11	-	-	PUNCT
cana-3453	35	12	forms	form	NOUN
cana-3453	35	13	.	.	PUNCT
cana-3453	36	1	r=0	r=0	PROPN
cana-3453	36	2	(	(	PUNCT
cana-3453	36	3	either	either	CCONJ
cana-3453	36	4	tent	tent	VERB
cana-3453	36	5	with	with	ADP
cana-3453	36	6	w2	w2	NOUN
cana-3453	36	7	-	-	PUNCT
cana-3453	36	8	flatness	flatness	NOUN
cana-3453	36	9	or	or	CCONJ
cana-3453	36	10	certain	certain	ADJ
cana-3453	36	11	curvature	curvature	NOUN
cana-3453	36	12	tensor	tensor	NOUN
cana-3453	36	13	properties	property	NOUN
cana-3453	36	14	)	)	PUNCT
cana-3453	36	15	.	.	PUNCT
cana-3453	37	1	curvature	curvature	NOUN
cana-3453	37	2	tensor	tensor	NOUN
cana-3453	37	3	has	have	AUX
cana-3453	37	4	also	also	ADV
cana-3453	37	5	been	be	AUX
cana-3453	37	6	largely	largely	ADV
cana-3453	37	7	established	establish	VERB
cana-3453	37	8	.	.	PUNCT
cana-3453	38	1	tripathi	tripathi	NOUN
cana-3453	38	2	et	et	PROPN
cana-3453	38	3	al	al	PROPN
cana-3453	38	4	.	.	PUNCT
cana-3453	39	1	the	the	DET
cana-3453	39	2	τ	τ	NOUN
cana-3453	39	3	-	-	PUNCT
cana-3453	39	4	curvature	curvature	NOUN
cana-3453	39	5	tensor	tensor	NOUN
cana-3453	39	6	was	be	AUX
cana-3453	39	7	then	then	ADV
cana-3453	39	8	introduced	introduce	VERB
cana-3453	39	9	in	in	ADP
cana-3453	39	10	[	[	X
cana-3453	39	11	1	1	NUM
cana-3453	39	12	]	]	PUNCT
cana-3453	39	13	and	and	CCONJ
cana-3453	39	14	later	later	ADV
cana-3453	39	15	generalized	generalize	VERB
cana-3453	39	16	to	to	ADP
cana-3453	39	17	(	(	PUNCT
cana-3453	39	18	k	k	X
cana-3453	39	19	,	,	PUNCT
cana-3453	39	20	μ)-contact	μ)-contact	NOUN
cana-3453	39	21	manifolds	manifold	NOUN
cana-3453	39	22	in	in	ADP
cana-3453	39	23	[	[	X
cana-3453	39	24	30	30	NUM
cana-3453	39	25	]	]	PUNCT
cana-3453	39	26	,	,	PUNCT
cana-3453	39	27	which	which	PRON
cana-3453	39	28	developed	develop	VERB
cana-3453	39	29	its	its	PRON
cana-3453	39	30	idea	idea	NOUN
cana-3453	39	31	further	far	ADV
cana-3453	39	32	.	.	PUNCT
cana-3453	40	1	this	this	DET
cana-3453	40	2	paper	paper	NOUN
cana-3453	40	3	presents	present	VERB
cana-3453	40	4	new	new	ADJ
cana-3453	40	5	results	result	NOUN
cana-3453	40	6	on	on	ADP
cana-3453	40	7	invariant	invariant	ADJ
cana-3453	40	8	submanifolds	submanifold	NOUN
cana-3453	40	9	of	of	ADP
cana-3453	40	10	generalized	generalized	ADJ
cana-3453	40	11	ssf	ssf	NOUN
cana-3453	40	12	.	.	PUNCT
cana-3453	41	1	this	this	PRON
cana-3453	41	2	leads	lead	VERB
cana-3453	41	3	us	we	PRON
cana-3453	41	4	to	to	PART
cana-3453	41	5	investigate	investigate	VERB
cana-3453	41	6	the	the	DET
cana-3453	41	7	necessary	necessary	ADJ
cana-3453	41	8	and	and	CCONJ
cana-3453	41	9	sufficient	sufficient	ADJ
cana-3453	41	10	conditions	condition	NOUN
cana-3453	41	11	on	on	ADP
cana-3453	41	12	the	the	DET
cana-3453	41	13	data	datum	NOUN
cana-3453	41	14	that	that	PRON
cana-3453	41	15	guarantee	guarantee	VERB
cana-3453	41	16	that	that	SCONJ
cana-3453	41	17	these	these	DET
cana-3453	41	18	submanifolds	submanifold	NOUN
cana-3453	41	19	are	be	AUX
cana-3453	41	20	totally	totally	ADV
cana-3453	41	21	geodesic	geodesic	ADJ
cana-3453	41	22	;	;	PUNCT
cana-3453	41	23	we	we	PRON
cana-3453	41	24	will	will	AUX
cana-3453	41	25	obtain	obtain	VERB
cana-3453	41	26	some	some	PRON
cana-3453	41	27	of	of	ADP
cana-3453	41	28	these	these	DET
cana-3453	41	29	conditions	condition	NOUN
cana-3453	41	30	concerning	concern	VERB
cana-3453	41	31	different	different	ADJ
cana-3453	41	32	geometric	geometric	ADJ
cana-3453	41	33	assumptions	assumption	NOUN
cana-3453	41	34	.	.	PUNCT
cana-3453	42	1	we	we	PRON
cana-3453	42	2	study	study	VERB
cana-3453	42	3	the	the	DET
cana-3453	42	4	situations	situation	NOUN
cana-3453	42	5	when	when	SCONJ
cana-3453	42	6	the	the	DET
cana-3453	42	7	second	second	ADJ
cana-3453	42	8	fundamental	fundamental	ADJ
cana-3453	42	9	form	form	NOUN
cana-3453	42	10	σ	σ	NOUN
cana-3453	42	11	satisfies	satisfie	NOUN
cana-3453	42	12	certain	certain	ADJ
cana-3453	42	13	relations	relation	NOUN
cana-3453	42	14	(	(	PUNCT
cana-3453	42	15	wi=2,3,4,6,7	wi=2,3,4,6,7	NOUN
cana-3453	42	16	)	)	PUNCT
cana-3453	42	17	,	,	PUNCT
cana-3453	42	18	in	in	ADP
cana-3453	42	19	terms	term	NOUN
cana-3453	42	20	of	of	ADP
cana-3453	42	21	the	the	DET
cana-3453	42	22	curvature	curvature	NOUN
cana-3453	42	23	tensors	tensor	NOUN
cana-3453	42	24	q(σ	q(σ	ADJ
cana-3453	42	25	,	,	PUNCT
cana-3453	42	26	w2)=0	w2)=0	NOUN
cana-3453	42	27	,	,	PUNCT
cana-3453	42	28	q(σ	q(σ	NUM
cana-3453	42	29	,	,	PUNCT
cana-3453	42	30	w3)=0	w3)=0	PROPN
cana-3453	42	31	,	,	PUNCT
cana-3453	42	32	q(σ	q(σ	ADJ
cana-3453	42	33	,	,	PUNCT
cana-3453	42	34	w4)=0	w4)=0	ADJ
cana-3453	42	35	,	,	PUNCT
cana-3453	42	36	q(σ	q(σ	NUM
cana-3453	42	37	,	,	PUNCT
cana-3453	42	38	w6)=	w6)=	NUM
cana-3453	42	39	and	and	CCONJ
cana-3453	42	40	q(σ	q(σ	ADJ
cana-3453	42	41	,	,	PUNCT
cana-3453	42	42	w7)=0	w7)=0	NOUN
cana-3453	42	43	.	.	PUNCT
cana-3453	43	1	communications	communication	NOUN
cana-3453	43	2	on	on	ADP
cana-3453	43	3	applied	apply	VERB
cana-3453	43	4	nonlinear	nonlinear	ADJ
cana-3453	43	5	analysis	analysis	NOUN
cana-3453	43	6	issn	issn	NOUN
cana-3453	43	7	:	:	PUNCT
cana-3453	43	8	1074	1074	NUM
cana-3453	43	9	-	-	PUNCT
cana-3453	43	10	133x	133x	NUM
cana-3453	43	11	vol	vol	NOUN
cana-3453	43	12	32	32	NUM
cana-3453	43	13	no	no	NOUN
cana-3453	43	14	.	.	PUNCT
cana-3453	44	1	7s	7	NOUN
cana-3453	44	2	(	(	PUNCT
cana-3453	44	3	2025	2025	NUM
cana-3453	44	4	)	)	PUNCT
cana-3453	44	5	417	417	NUM
cana-3453	44	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3453	44	7	the	the	DET
cana-3453	44	8	following	follow	VERB
cana-3453	44	9	sections	section	NOUN
cana-3453	44	10	are	be	AUX
cana-3453	44	11	organized	organize	VERB
cana-3453	44	12	:	:	PUNCT
cana-3453	44	13	section	section	NOUN
cana-3453	44	14	2	2	NUM
cana-3453	44	15	establishes	establish	VERB
cana-3453	44	16	preliminary	preliminary	ADJ
cana-3453	44	17	definitions	definition	NOUN
cana-3453	44	18	and	and	CCONJ
cana-3453	44	19	fundamental	fundamental	ADJ
cana-3453	44	20	properties	property	NOUN
cana-3453	44	21	of	of	ADP
cana-3453	44	22	generalized	generalized	ADJ
cana-3453	44	23	sasakian	sasakian	ADJ
cana-3453	44	24	space	space	NOUN
cana-3453	44	25	forms	form	NOUN
cana-3453	44	26	and	and	CCONJ
cana-3453	44	27	their	their	PRON
cana-3453	44	28	associated	associated	ADJ
cana-3453	44	29	curvature	curvature	NOUN
cana-3453	44	30	tensors	tensor	NOUN
cana-3453	44	31	.	.	PUNCT
cana-3453	45	1	section	section	NOUN
cana-3453	45	2	3	3	NUM
cana-3453	45	3	introduces	introduce	VERB
cana-3453	45	4	key	key	ADJ
cana-3453	45	5	concepts	concept	NOUN
cana-3453	45	6	related	relate	VERB
cana-3453	45	7	to	to	ADP
cana-3453	45	8	invariant	invariant	ADJ
cana-3453	45	9	submanifolds	submanifold	NOUN
cana-3453	45	10	.	.	PUNCT
cana-3453	46	1	the	the	DET
cana-3453	46	2	main	main	ADJ
cana-3453	46	3	results	result	NOUN
cana-3453	46	4	and	and	CCONJ
cana-3453	46	5	their	their	PRON
cana-3453	46	6	proofs	proof	NOUN
cana-3453	46	7	are	be	AUX
cana-3453	46	8	presented	present	VERB
cana-3453	46	9	in	in	ADP
cana-3453	46	10	sections	section	NOUN
cana-3453	46	11	4	4	NUM
cana-3453	46	12	through	through	ADP
cana-3453	46	13	8	8	NUM
cana-3453	46	14	,	,	PUNCT
cana-3453	46	15	each	each	PRON
cana-3453	46	16	addressing	address	VERB
cana-3453	46	17	specific	specific	ADJ
cana-3453	46	18	relationships	relationship	NOUN
cana-3453	46	19	between	between	ADP
cana-3453	46	20	the	the	DET
cana-3453	46	21	second	second	ADJ
cana-3453	46	22	fundamental	fundamental	ADJ
cana-3453	46	23	form	form	NOUN
cana-3453	46	24	and	and	CCONJ
cana-3453	46	25	various	various	ADJ
cana-3453	46	26	curvature	curvature	NOUN
cana-3453	46	27	tensors	tensor	NOUN
cana-3453	46	28	.	.	PUNCT
cana-3453	47	1	2	2	X
cana-3453	47	2	.	.	X
cana-3453	47	3	preliminaries	preliminary	NOUN
cana-3453	47	4	the	the	DET
cana-3453	47	5	structural	structural	ADJ
cana-3453	47	6	foundations	foundation	NOUN
cana-3453	47	7	of	of	ADP
cana-3453	47	8	generalized	generalized	ADJ
cana-3453	47	9	ssf	ssf	NOUN
cana-3453	47	10	emerge	emerge	VERB
cana-3453	47	11	from	from	ADP
cana-3453	47	12	the	the	DET
cana-3453	47	13	interplay	interplay	NOUN
cana-3453	47	14	between	between	ADP
cana-3453	47	15	metric	metric	ADJ
cana-3453	47	16	and	and	CCONJ
cana-3453	47	17	contact	contact	NOUN
cana-3453	47	18	properties	property	NOUN
cana-3453	47	19	.	.	PUNCT
cana-3453	48	1	these	these	DET
cana-3453	48	2	manifolds	manifold	NOUN
cana-3453	48	3	represent	represent	VERB
cana-3453	48	4	a	a	DET
cana-3453	48	5	natural	natural	ADJ
cana-3453	48	6	extension	extension	NOUN
cana-3453	48	7	of	of	ADP
cana-3453	48	8	constant	constant	ADJ
cana-3453	48	9	curvature	curvature	NOUN
cana-3453	48	10	spaces	space	NOUN
cana-3453	48	11	in	in	ADP
cana-3453	48	12	contact	contact	NOUN
cana-3453	48	13	geometry	geometry	NOUN
cana-3453	48	14	,	,	PUNCT
cana-3453	48	15	much	much	ADV
cana-3453	48	16	as	as	SCONJ
cana-3453	48	17	kählerian	kählerian	ADJ
cana-3453	48	18	-	-	PUNCT
cana-3453	48	19	space	space	NOUN
cana-3453	48	20	-	-	PUNCT
cana-3453	48	21	forms	form	NOUN
cana-3453	48	22	extend	extend	VERB
cana-3453	48	23	constant	constant	ADJ
cana-3453	48	24	curvature	curvature	NOUN
cana-3453	48	25	spaces	space	NOUN
cana-3453	48	26	in	in	ADP
cana-3453	48	27	complex	complex	ADJ
cana-3453	48	28	geometry	geometry	NOUN
cana-3453	48	29	.	.	PUNCT
cana-3453	49	1	the	the	DET
cana-3453	49	2	defining	define	VERB
cana-3453	49	3	tensors	tensor	NOUN
cana-3453	49	4	φ	φ	NUM
cana-3453	49	5	,	,	PUNCT
cana-3453	49	6	ξ	ξ	PROPN
cana-3453	49	7	,	,	PUNCT
cana-3453	49	8	and	and	CCONJ
cana-3453	49	9	η	η	PROPN
cana-3453	49	10	work	work	NOUN
cana-3453	49	11	together	together	ADV
cana-3453	49	12	to	to	PART
cana-3453	49	13	create	create	VERB
cana-3453	49	14	a	a	DET
cana-3453	49	15	rich	rich	ADJ
cana-3453	49	16	geometric	geometric	ADJ
cana-3453	49	17	structure	structure	NOUN
cana-3453	49	18	that	that	PRON
cana-3453	49	19	bridges	bridge	VERB
cana-3453	49	20	riemannian	riemannian	NOUN
cana-3453	49	21	and	and	CCONJ
cana-3453	49	22	contact	contact	NOUN
cana-3453	49	23	geometry	geometry	NOUN
cana-3453	49	24	.	.	PUNCT
cana-3453	50	1	these	these	DET
cana-3453	50	2	tensors	tensor	NOUN
cana-3453	50	3	are	be	AUX
cana-3453	50	4	more	more	ADJ
cana-3453	50	5	than	than	ADP
cana-3453	50	6	algebraic	algebraic	ADJ
cana-3453	50	7	relations	relation	NOUN
cana-3453	50	8	;	;	PUNCT
cana-3453	50	9	they	they	PRON
cana-3453	50	10	carry	carry	VERB
cana-3453	50	11	important	important	ADJ
cana-3453	50	12	geometrical	geometrical	ADJ
cana-3453	50	13	information	information	NOUN
cana-3453	50	14	about	about	ADP
cana-3453	50	15	how	how	SCONJ
cana-3453	50	16	the	the	DET
cana-3453	50	17	local	local	ADJ
cana-3453	50	18	and	and	CCONJ
cana-3453	50	19	global	global	ADJ
cana-3453	50	20	behaviour	behaviour	NOUN
cana-3453	50	21	of	of	ADP
cana-3453	50	22	the	the	DET
cana-3453	50	23	manifold	manifold	NOUN
cana-3453	50	24	is	be	AUX
cana-3453	50	25	constructed	construct	VERB
cana-3453	50	26	.	.	PUNCT
cana-3453	51	1	in	in	ADP
cana-3453	51	2	particular	particular	ADJ
cana-3453	51	3	,	,	PUNCT
cana-3453	51	4	we	we	PRON
cana-3453	51	5	are	be	AUX
cana-3453	51	6	interested	interested	ADJ
cana-3453	51	7	in	in	ADP
cana-3453	51	8	their	their	PRON
cana-3453	51	9	interaction	interaction	NOUN
cana-3453	51	10	with	with	ADP
cana-3453	51	11	the	the	DET
cana-3453	51	12	levi	levi	PROPN
cana-3453	51	13	-	-	PUNCT
cana-3453	51	14	civita	civita	PROPN
cana-3453	51	15	connection	connection	NOUN
cana-3453	51	16	,	,	PUNCT
cana-3453	51	17	which	which	PRON
cana-3453	51	18	gives	give	VERB
cana-3453	51	19	rise	rise	NOUN
cana-3453	51	20	to	to	ADP
cana-3453	51	21	several	several	ADJ
cana-3453	51	22	compatibility	compatibility	NOUN
cana-3453	51	23	conditions	condition	NOUN
cana-3453	51	24	that	that	PRON
cana-3453	51	25	ultimately	ultimately	ADV
cana-3453	51	26	define	define	VERB
cana-3453	51	27	the	the	DET
cana-3453	51	28	geometry	geometry	NOUN
cana-3453	51	29	in	in	ADP
cana-3453	51	30	question	question	NOUN
cana-3453	51	31	.	.	PUNCT
cana-3453	52	1	though	though	SCONJ
cana-3453	52	2	these	these	DET
cana-3453	52	3	conditions	condition	NOUN
cana-3453	52	4	sound	sound	VERB
cana-3453	52	5	somewhat	somewhat	ADV
cana-3453	52	6	technical	technical	ADJ
cana-3453	52	7	,	,	PUNCT
cana-3453	52	8	they	they	PRON
cana-3453	52	9	have	have	VERB
cana-3453	52	10	far	far	ADV
cana-3453	52	11	-	-	PUNCT
cana-3453	52	12	reaching	reach	VERB
cana-3453	52	13	geometric	geometric	ADJ
cana-3453	52	14	implications	implication	NOUN
cana-3453	52	15	:	:	PUNCT
cana-3453	52	16	they	they	PRON
cana-3453	52	17	tell	tell	VERB
cana-3453	52	18	us	we	PRON
cana-3453	52	19	how	how	SCONJ
cana-3453	52	20	we	we	PRON
cana-3453	52	21	can	can	AUX
cana-3453	52	22	"	"	PUNCT
cana-3453	52	23	transport	transport	VERB
cana-3453	52	24	"	"	PUNCT
cana-3453	52	25	vectors	vector	NOUN
cana-3453	52	26	along	along	ADP
cana-3453	52	27	curves	curve	NOUN
cana-3453	52	28	in	in	ADP
cana-3453	52	29	the	the	DET
cana-3453	52	30	manifold	manifold	NOUN
cana-3453	52	31	.	.	PUNCT
cana-3453	53	1	the	the	DET
cana-3453	53	2	algebraic	algebraic	ADJ
cana-3453	53	3	aspects	aspect	NOUN
cana-3453	53	4	of	of	ADP
cana-3453	53	5	almost	almost	ADV
cana-3453	53	6	contact	contact	NOUN
cana-3453	53	7	metric	metric	ADJ
cana-3453	53	8	structures	structure	NOUN
cana-3453	53	9	emerge	emerge	VERB
cana-3453	53	10	naturally	naturally	ADV
cana-3453	53	11	from	from	ADP
cana-3453	53	12	geometric	geometric	ADJ
cana-3453	53	13	considerations	consideration	NOUN
cana-3453	53	14	.	.	PUNCT
cana-3453	54	1	the	the	DET
cana-3453	54	2	present	present	ADJ
cana-3453	54	3	structures	structure	NOUN
cana-3453	54	4	arise	arise	VERB
cana-3453	54	5	as	as	ADP
cana-3453	54	6	formal	formal	ADJ
cana-3453	54	7	generalizations	generalization	NOUN
cana-3453	54	8	and	and	CCONJ
cana-3453	54	9	as	as	ADP
cana-3453	54	10	a	a	DET
cana-3453	54	11	natural	natural	ADJ
cana-3453	54	12	outcome	outcome	NOUN
cana-3453	54	13	of	of	ADP
cana-3453	54	14	extending	extend	VERB
cana-3453	54	15	kählerian	kählerian	ADJ
cana-3453	54	16	geometry	geometry	NOUN
cana-3453	54	17	to	to	ADP
cana-3453	54	18	odd	odd	ADV
cana-3453	54	19	-	-	PUNCT
cana-3453	54	20	dimensional	dimensional	ADJ
cana-3453	54	21	manifolds	manifold	NOUN
cana-3453	54	22	.	.	PUNCT
cana-3453	55	1	thus	thus	ADV
cana-3453	55	2	,	,	PUNCT
cana-3453	55	3	the	the	DET
cana-3453	55	4	interrelationships	interrelationship	NOUN
cana-3453	55	5	between	between	ADP
cana-3453	55	6	these	these	DET
cana-3453	55	7	structural	structural	ADJ
cana-3453	55	8	tensors	tensor	NOUN
cana-3453	55	9	φ	φ	NUM
cana-3453	55	10	,	,	PUNCT
cana-3453	55	11	ξ	ξ	PROPN
cana-3453	55	12	,	,	PUNCT
cana-3453	55	13	and	and	CCONJ
cana-3453	55	14	η	η	PROPN
cana-3453	55	15	reveal	reveal	VERB
cana-3453	55	16	basic	basic	ADJ
cana-3453	55	17	geometric	geometric	ADJ
cana-3453	55	18	attributes	attribute	NOUN
cana-3453	55	19	peculiar	peculiar	ADJ
cana-3453	55	20	to	to	ADP
cana-3453	55	21	this	this	DET
cana-3453	55	22	geometry	geometry	NOUN
cana-3453	55	23	(	(	PUNCT
cana-3453	55	24	amongst	amongst	ADP
cana-3453	55	25	all	all	DET
cana-3453	55	26	geometric	geometric	ADJ
cana-3453	55	27	structures	structure	NOUN
cana-3453	55	28	)	)	PUNCT
cana-3453	55	29	.	.	PUNCT
cana-3453	56	1	these	these	DET
cana-3453	56	2	relations	relation	NOUN
cana-3453	56	3	provide	provide	VERB
cana-3453	56	4	key	key	ADJ
cana-3453	56	5	information	information	NOUN
cana-3453	56	6	on	on	ADP
cana-3453	56	7	how	how	SCONJ
cana-3453	56	8	generalized	generalized	ADJ
cana-3453	56	9	sasakian	sasakian	ADJ
cana-3453	56	10	space	space	NOUN
cana-3453	56	11	forms	form	NOUN
cana-3453	56	12	differ	differ	VERB
cana-3453	56	13	from	from	ADP
cana-3453	56	14	classical	classical	ADJ
cana-3453	56	15	ones	one	NOUN
cana-3453	56	16	.	.	PUNCT
cana-3453	57	1	for	for	ADP
cana-3453	57	2	this	this	DET
cana-3453	57	3	investigation	investigation	NOUN
cana-3453	57	4	,	,	PUNCT
cana-3453	57	5	we	we	PRON
cana-3453	57	6	rely	rely	VERB
cana-3453	57	7	on	on	ADP
cana-3453	57	8	the	the	DET
cana-3453	57	9	geometric	geometric	ADJ
cana-3453	57	10	structure	structure	NOUN
cana-3453	57	11	of	of	ADP
cana-3453	57	12	riemannian	riemannian	NOUN
cana-3453	57	13	manifolds	manifold	NOUN
cana-3453	57	14	with	with	ADP
cana-3453	57	15	certain	certain	ADJ
cana-3453	57	16	contact	contact	NOUN
cana-3453	57	17	metric	metric	ADJ
cana-3453	57	18	properties	property	NOUN
cana-3453	57	19	.	.	PUNCT
cana-3453	58	1	let	let	VERB
cana-3453	58	2	m	m	PRON
cana-3453	58	3	be	be	AUX
cana-3453	58	4	a	a	DET
cana-3453	58	5	(	(	PUNCT
cana-3453	58	6	2n	2n	NUM
cana-3453	58	7	+	+	CCONJ
cana-3453	58	8	1)-dimensional	1)-dimensional	NUM
cana-3453	58	9	riemannian	riemannian	NOUN
cana-3453	58	10	manifold	manifold	NOUN
cana-3453	58	11	with	with	ADP
cana-3453	58	12	n	n	PROPN
cana-3453	58	13	>	>	X
cana-3453	58	14	1	1	X
cana-3453	58	15	.	.	PUNCT
cana-3453	59	1	let	let	VERB
cana-3453	59	2	m	m	PRON
cana-3453	59	3	be	be	AUX
cana-3453	59	4	an	an	DET
cana-3453	59	5	almost	almost	ADV
cana-3453	59	6	contact	contact	NOUN
cana-3453	59	7	metric	metric	ADJ
cana-3453	59	8	manifold	manifold	NOUN
cana-3453	59	9	provided	provide	VERB
cana-3453	59	10	with	with	ADP
cana-3453	59	11	three	three	NUM
cana-3453	59	12	important	important	ADJ
cana-3453	59	13	geometric	geometric	ADJ
cana-3453	59	14	objects	object	NOUN
cana-3453	59	15	:	:	PUNCT
cana-3453	60	1	1	1	X
cana-3453	60	2	.	.	X
cana-3453	61	1	a	a	DET
cana-3453	61	2	(	(	PUNCT
cana-3453	61	3	1,1)-tensor	1,1)-tensor	NUM
cana-3453	61	4	field	field	NOUN
cana-3453	61	5	denoted	denote	VERB
cana-3453	61	6	by	by	ADP
cana-3453	61	7	φ	φ	PROPN
cana-3453	61	8	2	2	NUM
cana-3453	61	9	.	.	PUNCT
cana-3453	62	1	a	a	DET
cana-3453	62	2	characteristic	characteristic	ADJ
cana-3453	62	3	vector	vector	NOUN
cana-3453	62	4	field	field	NOUN
cana-3453	62	5	ξ	ξ	PROPN
cana-3453	62	6	(	(	PUNCT
cana-3453	62	7	known	know	VERB
cana-3453	62	8	as	as	ADP
cana-3453	62	9	the	the	DET
cana-3453	62	10	structure	structure	NOUN
cana-3453	62	11	vector	vector	NOUN
cana-3453	62	12	field	field	NOUN
cana-3453	62	13	)	)	PUNCT
cana-3453	62	14	3	3	X
cana-3453	62	15	.	.	X
cana-3453	63	1	a	a	DET
cana-3453	63	2	1	1	NUM
cana-3453	63	3	-	-	PUNCT
cana-3453	63	4	form	form	NOUN
cana-3453	63	5	η	η	NOUN
cana-3453	63	6	these	these	DET
cana-3453	63	7	structures	structure	NOUN
cana-3453	63	8	exist	exist	VERB
cana-3453	63	9	on	on	ADP
cana-3453	63	10	𝑀2𝑛+1(𝑓1	𝑀2𝑛+1(𝑓1	NUM
cana-3453	63	11	,	,	PUNCT
cana-3453	63	12	𝑓2	𝑓2	ADJ
cana-3453	63	13	,	,	PUNCT
cana-3453	63	14	𝑓3	𝑓3	NOUN
cana-3453	63	15	)	)	PUNCT
cana-3453	63	16	and	and	CCONJ
cana-3453	63	17	satisfy	satisfy	VERB
cana-3453	63	18	the	the	DET
cana-3453	63	19	following	follow	VERB
cana-3453	63	20	fundamental	fundamental	ADJ
cana-3453	63	21	relationships	relationship	NOUN
cana-3453	63	22	:	:	PUNCT
cana-3453	63	23	𝝓𝟐(𝑿	𝝓𝟐(𝑿	NUM
cana-3453	63	24	)	)	PUNCT
cana-3453	63	25	=	=	PUNCT
cana-3453	63	26	−𝑿	−𝑿	PROPN
cana-3453	63	27	+	+	PUNCT
cana-3453	63	28	𝜼(𝑿)𝝃	𝜼(𝑿)𝝃	NUM
cana-3453	63	29	,	,	PUNCT
cana-3453	63	30	𝝓𝝃	𝝓𝝃	ADP
cana-3453	63	31	=	=	SYM
cana-3453	63	32	𝟎	𝟎	PROPN
cana-3453	63	33	,	,	PUNCT
cana-3453	63	34	(	(	PUNCT
cana-3453	63	35	2.1	2.1	NUM
cana-3453	63	36	)	)	PUNCT
cana-3453	63	37	𝜼(𝝃	𝜼(𝝃	NOUN
cana-3453	63	38	)	)	PUNCT
cana-3453	64	1	=	=	SYM
cana-3453	64	2	𝟏	𝟏	NUM
cana-3453	64	3	,	,	PUNCT
cana-3453	64	4	𝒈(𝑿	𝒈(𝑿	NOUN
cana-3453	64	5	,	,	PUNCT
cana-3453	64	6	𝝃	𝝃	NOUN
cana-3453	64	7	)	)	PUNCT
cana-3453	64	8	=	=	SYM
cana-3453	64	9	𝜼(𝑿	𝜼(𝑿	PROPN
cana-3453	64	10	)	)	PUNCT
cana-3453	64	11	,	,	PUNCT
cana-3453	64	12	𝜼(𝝓𝑿	𝜼(𝝓𝑿	X
cana-3453	64	13	)	)	PUNCT
cana-3453	64	14	=	=	SYM
cana-3453	65	1	𝟎	𝟎	PROPN
cana-3453	65	2	,	,	PUNCT
cana-3453	65	3	(	(	PUNCT
cana-3453	65	4	2.2	2.2	NUM
cana-3453	65	5	)	)	PUNCT
cana-3453	65	6	𝒈(𝝓𝑿	𝒈(𝝓𝑿	PROPN
cana-3453	65	7	,	,	PUNCT
cana-3453	65	8	𝝓𝒀	𝝓𝒀	NOUN
cana-3453	65	9	)	)	PUNCT
cana-3453	65	10	=	=	SYM
cana-3453	66	1	𝒈(𝑿	𝒈(𝑿	NOUN
cana-3453	66	2	,	,	PUNCT
cana-3453	66	3	𝒀	𝒀	NOUN
cana-3453	66	4	)	)	PUNCT
cana-3453	66	5	−	−	PROPN
cana-3453	66	6	𝜼(𝑿)𝜼(𝒀	𝜼(𝑿)𝜼(𝒀	NOUN
cana-3453	66	7	)	)	PUNCT
cana-3453	66	8	,	,	PUNCT
cana-3453	66	9	(	(	PUNCT
cana-3453	66	10	2.3	2.3	NUM
cana-3453	66	11	)	)	PUNCT
cana-3453	66	12	𝒈(𝝓𝑿	𝒈(𝝓𝑿	PROPN
cana-3453	66	13	,	,	PUNCT
cana-3453	66	14	𝒀	𝒀	NOUN
cana-3453	66	15	)	)	PUNCT
cana-3453	66	16	=	=	SYM
cana-3453	66	17	−𝒈(𝑿	−𝒈(𝑿	PROPN
cana-3453	66	18	,	,	PUNCT
cana-3453	66	19	𝝓𝒀	𝝓𝒀	NOUN
cana-3453	66	20	)	)	PUNCT
cana-3453	66	21	,	,	PUNCT
cana-3453	66	22	(	(	PUNCT
cana-3453	66	23	2.4	2.4	NUM
cana-3453	66	24	)	)	PUNCT
cana-3453	66	25	(	(	PUNCT
cana-3453	66	26	𝛁𝑿𝜼)(𝒀	𝛁𝑿𝜼)(𝒀	NOUN
cana-3453	66	27	)	)	PUNCT
cana-3453	66	28	=	=	SYM
cana-3453	66	29	𝒈(𝛁𝑿𝝃	𝒈(𝛁𝑿𝝃	NOUN
cana-3453	66	30	,	,	PUNCT
cana-3453	66	31	𝒀	𝒀	NOUN
cana-3453	66	32	)	)	PUNCT
cana-3453	66	33	.	.	PUNCT
cana-3453	67	1	(	(	PUNCT
cana-3453	67	2	2.5	2.5	NUM
cana-3453	67	3	)	)	PUNCT
cana-3453	67	4	equations	equation	NOUN
cana-3453	67	5	(	(	PUNCT
cana-3453	67	6	2.1)-(2.5	2.1)-(2.5	X
cana-3453	67	7	)	)	PUNCT
cana-3453	67	8	form	form	NOUN
cana-3453	67	9	all	all	DET
cana-3453	67	10	essential	essential	ADJ
cana-3453	67	11	algebraic	algebraic	ADJ
cana-3453	67	12	substratum	substratum	NOUN
cana-3453	67	13	of	of	ADP
cana-3453	67	14	almost	almost	ADV
cana-3453	67	15	contact	contact	VERB
cana-3453	67	16	metric	metric	ADJ
cana-3453	67	17	structures	structure	NOUN
cana-3453	67	18	.	.	PUNCT
cana-3453	68	1	the	the	DET
cana-3453	68	2	relationships	relationship	NOUN
cana-3453	68	3	you	you	PRON
cana-3453	68	4	describe	describe	VERB
cana-3453	68	5	show	show	VERB
cana-3453	68	6	how	how	SCONJ
cana-3453	68	7	the	the	DET
cana-3453	68	8	tensor	tensor	NOUN
cana-3453	68	9	field	field	NOUN
cana-3453	68	10	φ	φ	PROPN
cana-3453	68	11	interacts	interact	VERB
cana-3453	68	12	with	with	ADP
cana-3453	68	13	the	the	DET
cana-3453	68	14	metric	metric	ADJ
cana-3453	68	15	structure	structure	NOUN
cana-3453	68	16	g	g	PROPN
cana-3453	68	17	and	and	CCONJ
cana-3453	68	18	the	the	DET
cana-3453	68	19	characteristic	characteristic	ADJ
cana-3453	68	20	vector	vector	NOUN
cana-3453	68	21	field	field	NOUN
cana-3453	68	22	ξ	ξ	PROPN
cana-3453	68	23	.	.	PUNCT
cana-3453	69	1	in	in	ADP
cana-3453	69	2	particular	particular	ADJ
cana-3453	69	3	,	,	PUNCT
cana-3453	69	4	these	these	DET
cana-3453	69	5	equations	equation	NOUN
cana-3453	69	6	together	together	ADV
cana-3453	69	7	guarantee	guarantee	VERB
cana-3453	69	8	that	that	SCONJ
cana-3453	69	9	the	the	DET
cana-3453	69	10	metric	metric	NOUN
cana-3453	69	11	and	and	CCONJ
cana-3453	69	12	communications	communication	NOUN
cana-3453	69	13	on	on	ADP
cana-3453	69	14	applied	apply	VERB
cana-3453	69	15	nonlinear	nonlinear	ADJ
cana-3453	69	16	analysis	analysis	NOUN
cana-3453	69	17	issn	issn	NOUN
cana-3453	69	18	:	:	PUNCT
cana-3453	69	19	1074	1074	NUM
cana-3453	69	20	-	-	PUNCT
cana-3453	69	21	133x	133x	NUM
cana-3453	69	22	vol	vol	NOUN
cana-3453	69	23	32	32	NUM
cana-3453	69	24	no	no	NOUN
cana-3453	69	25	.	.	PUNCT
cana-3453	70	1	7s	7	NOUN
cana-3453	70	2	(	(	PUNCT
cana-3453	70	3	2025	2025	NUM
cana-3453	70	4	)	)	PUNCT
cana-3453	70	5	418	418	NUM
cana-3453	70	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3453	70	7	contact	contact	NOUN
cana-3453	70	8	properties	property	NOUN
cana-3453	70	9	are	be	AUX
cana-3453	70	10	compatible	compatible	ADJ
cana-3453	70	11	.	.	PUNCT
cana-3453	71	1	their	their	PRON
cana-3453	71	2	algebraic	algebraic	ADJ
cana-3453	71	3	form	form	NOUN
cana-3453	71	4	belies	belie	VERB
cana-3453	71	5	a	a	DET
cana-3453	71	6	wealth	wealth	NOUN
cana-3453	71	7	of	of	ADP
cana-3453	71	8	geometric	geometric	ADJ
cana-3453	71	9	implications	implication	NOUN
cana-3453	71	10	,	,	PUNCT
cana-3453	71	11	from	from	ADP
cana-3453	71	12	the	the	DET
cana-3453	71	13	measurement	measurement	NOUN
cana-3453	71	14	of	of	ADP
cana-3453	71	15	local	local	ADJ
cana-3453	71	16	angles	angle	NOUN
cana-3453	71	17	to	to	ADP
cana-3453	71	18	global	global	ADJ
cana-3453	71	19	topological	topological	ADJ
cana-3453	71	20	properties	property	NOUN
cana-3453	71	21	.	.	PUNCT
cana-3453	72	1	for	for	ADP
cana-3453	72	2	a	a	DET
cana-3453	72	3	generalized	generalized	ADJ
cana-3453	72	4	sasakian	sasakian	ADJ
cana-3453	72	5	-	-	PUNCT
cana-3453	72	6	space	space	NOUN
cana-3453	72	7	-	-	PUNCT
cana-3453	72	8	form	form	NOUN
cana-3453	72	9	,	,	PUNCT
cana-3453	72	10	𝑀2𝑛+1(𝑓1	𝑀2𝑛+1(𝑓1	NUM
cana-3453	72	11	,	,	PUNCT
cana-3453	72	12	𝑓2	𝑓2	ADJ
cana-3453	72	13	,	,	PUNCT
cana-3453	72	14	𝑓3	𝑓3	NOUN
cana-3453	72	15	)	)	PUNCT
cana-3453	72	16	,	,	PUNCT
cana-3453	72	17	with	with	ADP
cana-3453	72	18	n	n	PROPN
cana-3453	72	19	>	>	SYM
cana-3453	72	20	1	1	NUM
cana-3453	72	21	,	,	PUNCT
cana-3453	72	22	several	several	ADJ
cana-3453	72	23	main	main	ADJ
cana-3453	72	24	geometric	geometric	ADJ
cana-3453	72	25	properties	property	NOUN
cana-3453	72	26	arise	arise	VERB
cana-3453	72	27	.	.	PUNCT
cana-3453	73	1	the	the	DET
cana-3453	73	2	properties	property	NOUN
cana-3453	73	3	obtained	obtain	VERB
cana-3453	73	4	above	above	ADV
cana-3453	73	5	from	from	ADP
cana-3453	73	6	equation	equation	NOUN
cana-3453	73	7	(	(	PUNCT
cana-3453	73	8	1.1	1.1	NUM
cana-3453	73	9	)	)	PUNCT
cana-3453	73	10	can	can	AUX
cana-3453	73	11	be	be	AUX
cana-3453	73	12	expressed	express	VERB
cana-3453	73	13	through	through	ADP
cana-3453	73	14	a	a	DET
cana-3453	73	15	series	series	NOUN
cana-3453	73	16	comprising	comprise	VERB
cana-3453	73	17	fundamental	fundamental	ADJ
cana-3453	73	18	relationships	relationship	NOUN
cana-3453	73	19	:	:	PUNCT
cana-3453	73	20	(	(	PUNCT
cana-3453	73	21	𝛁𝑿𝛟)(𝒀	𝛁𝑿𝛟)(𝒀	NOUN
cana-3453	73	22	)	)	PUNCT
cana-3453	73	23	=	=	SYM
cana-3453	73	24	(	(	PUNCT
cana-3453	73	25	𝒇𝟏	𝒇𝟏	NOUN
cana-3453	73	26	−	−	PROPN
cana-3453	73	27	𝒇𝟑)[𝒈(𝑿	𝒇𝟑)[𝒈(𝑿	PROPN
cana-3453	73	28	,	,	PUNCT
cana-3453	73	29	𝒀)𝝃	𝒀)𝝃	PUNCT
cana-3453	73	30	−	−	NOUN
cana-3453	74	1	𝜼(𝒀)𝑿	𝜼(𝒀)𝑿	VERB
cana-3453	74	2	]	]	X
cana-3453	74	3	,	,	PUNCT
cana-3453	74	4	(	(	PUNCT
cana-3453	74	5	2.6	2.6	NUM
cana-3453	74	6	)	)	PUNCT
cana-3453	74	7	𝛁𝑿𝝃	𝛁𝑿𝝃	NOUN
cana-3453	74	8	=	=	PROPN
cana-3453	74	9	−(𝒇𝟏	−(𝒇𝟏	ADP
cana-3453	74	10	−	−	PROPN
cana-3453	74	11	𝒇𝟑)𝝓𝑿	𝒇𝟑)𝝓𝑿	NOUN
cana-3453	74	12	(	(	PUNCT
cana-3453	74	13	2.7	2.7	NUM
cana-3453	74	14	)	)	PUNCT
cana-3453	74	15	the	the	DET
cana-3453	74	16	differential	differential	ADJ
cana-3453	74	17	equations	equation	NOUN
cana-3453	74	18	(	(	PUNCT
cana-3453	74	19	2.6	2.6	NUM
cana-3453	74	20	)	)	PUNCT
cana-3453	74	21	and	and	CCONJ
cana-3453	74	22	(	(	PUNCT
cana-3453	74	23	2.7	2.7	NUM
cana-3453	74	24	)	)	PUNCT
cana-3453	74	25	give	give	VERB
cana-3453	74	26	the	the	DET
cana-3453	74	27	contact	contact	NOUN
cana-3453	74	28	structure	structure	NOUN
cana-3453	74	29	's	's	PART
cana-3453	74	30	parallel	parallel	ADJ
cana-3453	74	31	behaviour	behaviour	NOUN
cana-3453	74	32	.	.	PUNCT
cana-3453	75	1	as	as	SCONJ
cana-3453	75	2	demonstrated	demonstrate	VERB
cana-3453	75	3	in	in	ADP
cana-3453	75	4	the	the	DET
cana-3453	75	5	equations	equation	NOUN
cana-3453	75	6	above	above	ADV
cana-3453	75	7	,	,	PUNCT
cana-3453	75	8	the	the	DET
cana-3453	75	9	covariant	covariant	ADJ
cana-3453	75	10	derivatives	derivative	NOUN
cana-3453	75	11	of	of	ADP
cana-3453	75	12	φ	φ	PROPN
cana-3453	75	13	and	and	CCONJ
cana-3453	75	14	ξ	ξ	PROPN
cana-3453	75	15	are	be	AUX
cana-3453	75	16	governed	govern	VERB
cana-3453	75	17	by	by	ADP
cana-3453	75	18	the	the	DET
cana-3453	75	19	functions	function	NOUN
cana-3453	75	20	f₁	f₁	NOUN
cana-3453	75	21	and	and	CCONJ
cana-3453	75	22	f₃	f₃	NOUN
cana-3453	75	23	,	,	PUNCT
cana-3453	75	24	and	and	CCONJ
cana-3453	75	25	they	they	PRON
cana-3453	75	26	point	point	VERB
cana-3453	75	27	out	out	ADP
cana-3453	75	28	that	that	SCONJ
cana-3453	75	29	generalized	generalized	ADJ
cana-3453	75	30	ssf	ssf	NOUN
cana-3453	75	31	have	have	VERB
cana-3453	75	32	an	an	DET
cana-3453	75	33	explicitly	explicitly	ADV
cana-3453	75	34	prescribed	prescribe	VERB
cana-3453	75	35	connection	connection	NOUN
cana-3453	75	36	.	.	PUNCT
cana-3453	76	1	this	this	DET
cana-3453	76	2	behaviour	behaviour	NOUN
cana-3453	76	3	sets	set	VERB
cana-3453	76	4	these	these	DET
cana-3453	76	5	spaces	space	NOUN
cana-3453	76	6	apart	apart	ADV
cana-3453	76	7	from	from	ADP
cana-3453	76	8	other	other	ADJ
cana-3453	76	9	contact	contact	NOUN
cana-3453	76	10	metric	metric	ADJ
cana-3453	76	11	manifolds	manifold	NOUN
cana-3453	76	12	and	and	CCONJ
cana-3453	76	13	has	have	VERB
cana-3453	76	14	deep	deep	ADJ
cana-3453	76	15	consequences	consequence	NOUN
cana-3453	76	16	for	for	ADP
cana-3453	76	17	geodesic	geodesic	ADJ
cana-3453	76	18	behaviour	behaviour	NOUN
cana-3453	76	19	.	.	PUNCT
cana-3453	77	1	𝑸𝑿	𝑸𝑿	PROPN
cana-3453	77	2	=	=	PUNCT
cana-3453	77	3	(	(	PUNCT
cana-3453	77	4	𝟐𝒏𝒇𝟏	𝟐𝒏𝒇𝟏	X
cana-3453	77	5	+	+	CCONJ
cana-3453	77	6	𝟑𝒇𝟐	𝟑𝒇𝟐	NUM
cana-3453	77	7	−	−	NOUN
cana-3453	77	8	𝒇𝟑)𝑿	𝒇𝟑)𝑿	NOUN
cana-3453	77	9	−	−	NOUN
cana-3453	77	10	{	{	PUNCT
cana-3453	77	11	𝟑𝒇𝟐	𝟑𝒇𝟐	PUNCT
cana-3453	77	12	+	+	CCONJ
cana-3453	77	13	(	(	PUNCT
cana-3453	77	14	𝟐𝒏	𝟐𝒏	ADV
cana-3453	77	15	−	−	PROPN
cana-3453	77	16	𝟏)𝒇𝟑}𝜼(𝑿)𝜼(𝒀	𝟏)𝒇𝟑}𝜼(𝑿)𝜼(𝒀	NOUN
cana-3453	77	17	)	)	PUNCT
cana-3453	77	18	,	,	PUNCT
cana-3453	77	19	(	(	PUNCT
cana-3453	77	20	2.8	2.8	NUM
cana-3453	77	21	)	)	PUNCT
cana-3453	77	22	𝑺(𝑿	𝑺(𝑿	NUM
cana-3453	77	23	,	,	PUNCT
cana-3453	77	24	𝒀	𝒀	NOUN
cana-3453	77	25	)	)	PUNCT
cana-3453	77	26	=	=	PUNCT
cana-3453	77	27	(	(	PUNCT
cana-3453	77	28	𝟐𝒏𝒇𝟏	𝟐𝒏𝒇𝟏	X
cana-3453	77	29	+	+	CCONJ
cana-3453	77	30	𝟑𝒇𝟐	𝟑𝒇𝟐	X
cana-3453	77	31	−	−	PROPN
cana-3453	77	32	𝒇𝟑)𝒈(𝑿	𝒇𝟑)𝒈(𝑿	NOUN
cana-3453	77	33	,	,	PUNCT
cana-3453	77	34	𝒀	𝒀	NOUN
cana-3453	77	35	)	)	PUNCT
cana-3453	77	36	−	−	NOUN
cana-3453	77	37	{	{	PUNCT
cana-3453	77	38	𝟑𝒇𝟐	𝟑𝒇𝟐	PUNCT
cana-3453	77	39	+	+	CCONJ
cana-3453	77	40	(	(	PUNCT
cana-3453	77	41	𝟐𝒏	𝟐𝒏	ADV
cana-3453	77	42	−	−	PROPN
cana-3453	77	43	𝟏)𝒇𝟑}𝜼(𝑿)𝜼(𝒀	𝟏)𝒇𝟑}𝜼(𝑿)𝜼(𝒀	NOUN
cana-3453	77	44	)	)	PUNCT
cana-3453	77	45	,	,	PUNCT
cana-3453	77	46	(	(	PUNCT
cana-3453	77	47	2.9	2.9	NUM
cana-3453	77	48	)	)	PUNCT
cana-3453	77	49	𝒓	𝒓	NOUN
cana-3453	77	50	=	=	PUNCT
cana-3453	77	51	𝟐𝒏(𝟐𝒏	𝟐𝒏(𝟐𝒏	PROPN
cana-3453	78	1	+	+	CCONJ
cana-3453	78	2	𝟏)𝒇𝟏	𝟏)𝒇𝟏	PROPN
cana-3453	78	3	+	+	CCONJ
cana-3453	78	4	𝟔𝒏𝒇𝟐	𝟔𝒏𝒇𝟐	NUM
cana-3453	78	5	−	−	NOUN
cana-3453	78	6	𝟒𝒏𝒇𝟑	𝟒𝒏𝒇𝟑	ADV
cana-3453	78	7	,	,	PUNCT
cana-3453	78	8	(	(	PUNCT
cana-3453	78	9	2.10	2.10	NUM
cana-3453	78	10	)	)	PUNCT
cana-3453	78	11	the	the	DET
cana-3453	78	12	curvature	curvature	NOUN
cana-3453	78	13	properties	property	NOUN
cana-3453	78	14	of	of	ADP
cana-3453	78	15	these	these	DET
cana-3453	78	16	manifolds	manifold	NOUN
cana-3453	78	17	are	be	AUX
cana-3453	78	18	characterized	characterize	VERB
cana-3453	78	19	by	by	ADP
cana-3453	78	20	:	:	PUNCT
cana-3453	78	21	𝒓	𝒓	PROPN
cana-3453	78	22	=	=	PUNCT
cana-3453	78	23	𝟐𝒏(𝟐𝒏	𝟐𝒏(𝟐𝒏	PROPN
cana-3453	79	1	+	+	CCONJ
cana-3453	79	2	𝟏)𝒇𝟏	𝟏)𝒇𝟏	PROPN
cana-3453	79	3	+	+	CCONJ
cana-3453	79	4	𝟔𝒏𝒇𝟐	𝟔𝒏𝒇𝟐	NUM
cana-3453	79	5	−	−	NOUN
cana-3453	79	6	𝟒𝒏𝒇𝟑	𝟒𝒏𝒇𝟑	ADV
cana-3453	79	7	,	,	PUNCT
cana-3453	79	8	(	(	PUNCT
cana-3453	79	9	2.10	2.10	NUM
cana-3453	79	10	)	)	PUNCT
cana-3453	79	11	the	the	DET
cana-3453	79	12	ricci	ricci	PROPN
cana-3453	79	13	curvature	curvature	PROPN
cana-3453	79	14	equations	equation	NOUN
cana-3453	79	15	(	(	PUNCT
cana-3453	79	16	2.8)-(2.10	2.8)-(2.10	NUM
cana-3453	79	17	)	)	PUNCT
cana-3453	79	18	indicate	indicate	VERB
cana-3453	79	19	how	how	SCONJ
cana-3453	79	20	the	the	DET
cana-3453	79	21	scalar	scalar	ADJ
cana-3453	79	22	curvature	curvature	NOUN
cana-3453	79	23	properties	property	NOUN
cana-3453	79	24	of	of	ADP
cana-3453	79	25	the	the	DET
cana-3453	79	26	manifold	manifold	NOUN
cana-3453	79	27	are	be	AUX
cana-3453	79	28	affected	affect	VERB
cana-3453	79	29	by	by	ADP
cana-3453	79	30	the	the	DET
cana-3453	79	31	three	three	NUM
cana-3453	79	32	defining	define	VERB
cana-3453	79	33	functions	function	NOUN
cana-3453	79	34	f₁	f₁	NOUN
cana-3453	79	35	,	,	PUNCT
cana-3453	79	36	f₂	f₂	NOUN
cana-3453	79	37	,	,	PUNCT
cana-3453	79	38	and	and	CCONJ
cana-3453	79	39	f₃.	f₃.	AUX
cana-3453	79	40	our	our	PRON
cana-3453	79	41	results	result	NOUN
cana-3453	79	42	show	show	VERB
cana-3453	79	43	that	that	SCONJ
cana-3453	79	44	generalized	generalized	ADJ
cana-3453	79	45	ssf	ssf	NOUN
cana-3453	79	46	have	have	VERB
cana-3453	79	47	a	a	DET
cana-3453	79	48	rich	rich	ADJ
cana-3453	79	49	curvature	curvature	NOUN
cana-3453	79	50	structure	structure	NOUN
cana-3453	79	51	that	that	PRON
cana-3453	79	52	is	be	AUX
cana-3453	79	53	not	not	PART
cana-3453	79	54	limited	limit	VERB
cana-3453	79	55	to	to	ADP
cana-3453	79	56	the	the	DET
cana-3453	79	57	case	case	NOUN
cana-3453	79	58	of	of	ADP
cana-3453	79	59	constant	constant	ADJ
cana-3453	79	60	sectional	sectional	ADJ
cana-3453	79	61	curvature	curvature	NOUN
cana-3453	79	62	.	.	PUNCT
cana-3453	80	1	in	in	ADP
cana-3453	80	2	particular	particular	ADJ
cana-3453	80	3	,	,	PUNCT
cana-3453	80	4	the	the	DET
cana-3453	80	5	expression	expression	NOUN
cana-3453	80	6	for	for	ADP
cana-3453	80	7	the	the	DET
cana-3453	80	8	scalar	scalar	ADJ
cana-3453	80	9	curvature	curvature	NOUN
cana-3453	80	10	reveals	reveal	VERB
cana-3453	80	11	how	how	SCONJ
cana-3453	80	12	the	the	DET
cana-3453	80	13	contact	contact	NOUN
cana-3453	80	14	structure	structure	NOUN
cana-3453	80	15	enters	enter	VERB
cana-3453	80	16	the	the	DET
cana-3453	80	17	overall	overall	ADJ
cana-3453	80	18	geometry	geometry	NOUN
cana-3453	80	19	through	through	ADP
cana-3453	80	20	certain	certain	ADJ
cana-3453	80	21	combinations	combination	NOUN
cana-3453	80	22	of	of	ADP
cana-3453	80	23	these	these	DET
cana-3453	80	24	functions	function	NOUN
cana-3453	80	25	.	.	PUNCT
cana-3453	81	1	𝑹(𝑿	𝑹(𝑿	NOUN
cana-3453	81	2	,	,	PUNCT
cana-3453	81	3	𝒀)𝝃	𝒀)𝝃	PUNCT
cana-3453	81	4	=	=	PUNCT
cana-3453	82	1	(	(	PUNCT
cana-3453	82	2	𝒇𝟏	𝒇𝟏	NOUN
cana-3453	82	3	−	−	PROPN
cana-3453	82	4	𝒇𝟑){𝜼(𝒀)𝑿	𝒇𝟑){𝜼(𝒀)𝑿	PROPN
cana-3453	82	5	−	−	PROPN
cana-3453	82	6	𝜼(𝑿)𝒀	𝜼(𝑿)𝒀	NOUN
cana-3453	82	7	}	}	PUNCT
cana-3453	82	8	,	,	PUNCT
cana-3453	82	9	(	(	PUNCT
cana-3453	82	10	2.11	2.11	NUM
cana-3453	82	11	)	)	PUNCT
cana-3453	82	12	𝑹(𝝃	𝑹(𝝃	NOUN
cana-3453	82	13	,	,	PUNCT
cana-3453	82	14	𝑿)𝒀	𝑿)𝒀	NOUN
cana-3453	82	15	=	=	SYM
cana-3453	82	16	(	(	PUNCT
cana-3453	82	17	𝒇𝟏	𝒇𝟏	NOUN
cana-3453	82	18	−	−	PROPN
cana-3453	82	19	𝒇𝟑){𝒈(𝑿	𝒇𝟑){𝒈(𝑿	NUM
cana-3453	82	20	,	,	PUNCT
cana-3453	82	21	𝒀)𝝃	𝒀)𝝃	PUNCT
cana-3453	82	22	−	−	NOUN
cana-3453	82	23	𝜼(𝒀)𝑿	𝜼(𝒀)𝑿	VERB
cana-3453	82	24	}	}	PUNCT
cana-3453	82	25	,	,	PUNCT
cana-3453	82	26	(	(	PUNCT
cana-3453	82	27	2.12	2.12	NUM
cana-3453	82	28	)	)	PUNCT
cana-3453	82	29	𝜼(𝑹(𝑿	𝜼(𝑹(𝑿	NOUN
cana-3453	82	30	,	,	PUNCT
cana-3453	82	31	𝒀)𝒁	𝒀)𝒁	ADJ
cana-3453	82	32	)	)	PUNCT
cana-3453	82	33	=	=	PUNCT
cana-3453	82	34	(	(	PUNCT
cana-3453	82	35	𝒇𝟏	𝒇𝟏	NOUN
cana-3453	82	36	−	−	PROPN
cana-3453	82	37	𝒇𝟑){𝒈(𝒀	𝒇𝟑){𝒈(𝒀	NUM
cana-3453	82	38	,	,	PUNCT
cana-3453	82	39	𝒁)𝜼(𝑿	𝒁)𝜼(𝑿	NUM
cana-3453	82	40	)	)	PUNCT
cana-3453	82	41	−	−	PROPN
cana-3453	83	1	𝒈(𝑿	𝒈(𝑿	NOUN
cana-3453	83	2	,	,	PUNCT
cana-3453	83	3	𝒁)𝜼(𝒀	𝒁)𝜼(𝒀	PROPN
cana-3453	83	4	)	)	PUNCT
cana-3453	83	5	}	}	PUNCT
cana-3453	83	6	,	,	PUNCT
cana-3453	83	7	(	(	PUNCT
cana-3453	83	8	2.13	2.13	NUM
cana-3453	83	9	)	)	PUNCT
cana-3453	83	10	the	the	DET
cana-3453	83	11	curvature	curvature	NOUN
cana-3453	83	12	relations	relation	NOUN
cana-3453	83	13	(	(	PUNCT
cana-3453	83	14	2.11	2.11	NUM
cana-3453	83	15	)	)	PUNCT
cana-3453	83	16	−	−	PROPN
cana-3453	83	17	(	(	PUNCT
cana-3453	83	18	2.13	2.13	NUM
cana-3453	83	19	)	)	PUNCT
cana-3453	83	20	contain	contain	VERB
cana-3453	83	21	important	important	ADJ
cana-3453	83	22	information	information	NOUN
cana-3453	83	23	about	about	ADP
cana-3453	83	24	the	the	DET
cana-3453	83	25	sectional	sectional	ADJ
cana-3453	83	26	curvature	curvature	NOUN
cana-3453	83	27	directions	direction	NOUN
cana-3453	83	28	and	and	CCONJ
cana-3453	83	29	have	have	VERB
cana-3453	83	30	the	the	DET
cana-3453	83	31	characteristic	characteristic	ADJ
cana-3453	83	32	vector	vector	NOUN
cana-3453	83	33	field	field	NOUN
cana-3453	83	34	ξ	ξ	PROPN
cana-3453	83	35	.	.	PUNCT
cana-3453	84	1	the	the	DET
cana-3453	84	2	definition	definition	NOUN
cana-3453	84	3	of	of	ADP
cana-3453	84	4	f₁	f₁	ADJ
cana-3453	84	5	and	and	CCONJ
cana-3453	84	6	f₃	f₃	NOUN
cana-3453	84	7	reveals	reveal	VERB
cana-3453	84	8	the	the	DET
cana-3453	84	9	dynamical	dynamical	ADJ
cana-3453	84	10	mechanism	mechanism	NOUN
cana-3453	84	11	whereby	whereby	SCONJ
cana-3453	84	12	the	the	DET
cana-3453	84	13	difference	difference	NOUN
cana-3453	84	14	f₁-f₃	f₁-f₃	NOUN
cana-3453	84	15	governs	govern	VERB
cana-3453	84	16	curvature	curvature	NOUN
cana-3453	84	17	behaviour	behaviour	NOUN
cana-3453	84	18	along	along	ADP
cana-3453	84	19	the	the	DET
cana-3453	84	20	direction	direction	NOUN
cana-3453	84	21	of	of	ADP
cana-3453	84	22	the	the	DET
cana-3453	84	23	contact	contact	NOUN
cana-3453	84	24	distribution	distribution	NOUN
cana-3453	84	25	.	.	PUNCT
cana-3453	85	1	this	this	DET
cana-3453	85	2	geometric	geometric	ADJ
cana-3453	85	3	interpretation	interpretation	NOUN
cana-3453	85	4	will	will	AUX
cana-3453	85	5	help	help	VERB
cana-3453	85	6	us	we	PRON
cana-3453	85	7	understand	understand	VERB
cana-3453	85	8	why	why	SCONJ
cana-3453	85	9	this	this	DET
cana-3453	85	10	difference	difference	NOUN
cana-3453	85	11	occurs	occur	VERB
cana-3453	85	12	so	so	ADV
cana-3453	85	13	often	often	ADV
cana-3453	85	14	in	in	ADP
cana-3453	85	15	non	non	ADJ
cana-3453	85	16	-	-	ADJ
cana-3453	85	17	degeneracy	degeneracy	ADJ
cana-3453	85	18	conditions	condition	NOUN
cana-3453	85	19	for	for	ADP
cana-3453	85	20	the	the	DET
cana-3453	85	21	rest	rest	NOUN
cana-3453	85	22	of	of	ADP
cana-3453	85	23	our	our	PRON
cana-3453	85	24	analysis	analysis	NOUN
cana-3453	85	25	.	.	PUNCT
cana-3453	86	1	furthermore	furthermore	ADV
cana-3453	86	2	,	,	PUNCT
cana-3453	86	3	the	the	DET
cana-3453	86	4	following	follow	VERB
cana-3453	86	5	relations	relation	NOUN
cana-3453	86	6	based	base	VERB
cana-3453	86	7	on	on	ADP
cana-3453	86	8	ricci	ricci	PROPN
cana-3453	86	9	curvature	curvature	PROPN
cana-3453	86	10	tensor	tensor	NOUN
cana-3453	86	11	are	be	AUX
cana-3453	86	12	:	:	PUNCT
cana-3453	86	13	𝑺(𝑿	𝑺(𝑿	NUM
cana-3453	86	14	,	,	PUNCT
cana-3453	86	15	𝝃	𝝃	NOUN
cana-3453	86	16	)	)	PUNCT
cana-3453	86	17	=	=	PUNCT
cana-3453	87	1	𝟐𝒏(𝒇𝟏	𝟐𝒏(𝒇𝟏	ADJ
cana-3453	88	1	−	−	NUM
cana-3453	88	2	𝒇𝟑)𝜼(𝑿	𝒇𝟑)𝜼(𝑿	PROPN
cana-3453	88	3	)	)	PUNCT
cana-3453	88	4	,	,	PUNCT
cana-3453	88	5	(	(	PUNCT
cana-3453	88	6	2.14	2.14	NUM
cana-3453	88	7	)	)	PUNCT
cana-3453	88	8	𝑺(𝝃	𝑺(𝝃	NOUN
cana-3453	88	9	,	,	PUNCT
cana-3453	88	10	𝝃	𝝃	NOUN
cana-3453	88	11	)	)	PUNCT
cana-3453	88	12	=	=	PUNCT
cana-3453	88	13	𝟐𝒏(𝒇𝟏	𝟐𝒏(𝒇𝟏	ADJ
cana-3453	88	14	−	−	NUM
cana-3453	88	15	𝒇𝟑	𝒇𝟑	PROPN
cana-3453	88	16	)	)	PUNCT
cana-3453	88	17	,	,	PUNCT
cana-3453	88	18	(	(	PUNCT
cana-3453	88	19	2.15	2.15	NUM
cana-3453	88	20	)	)	PUNCT
cana-3453	88	21	𝑸𝝃	𝑸𝝃	PROPN
cana-3453	88	22	=	=	PUNCT
cana-3453	88	23	𝟐𝒏(𝒇𝟏	𝟐𝒏(𝒇𝟏	ADJ
cana-3453	88	24	−	−	PROPN
cana-3453	88	25	𝒇𝟑)𝝃	𝒇𝟑)𝝃	NOUN
cana-3453	88	26	,	,	PUNCT
cana-3453	88	27	(	(	PUNCT
cana-3453	88	28	2.16	2.16	NUM
cana-3453	88	29	)	)	PUNCT
cana-3453	88	30	communications	communication	NOUN
cana-3453	88	31	on	on	ADP
cana-3453	88	32	applied	apply	VERB
cana-3453	88	33	nonlinear	nonlinear	ADJ
cana-3453	88	34	analysis	analysis	NOUN
cana-3453	88	35	issn	issn	NOUN
cana-3453	88	36	:	:	PUNCT
cana-3453	88	37	1074	1074	NUM
cana-3453	88	38	-	-	PUNCT
cana-3453	88	39	133x	133x	NUM
cana-3453	88	40	vol	vol	NOUN
cana-3453	88	41	32	32	NUM
cana-3453	88	42	no	no	NOUN
cana-3453	88	43	.	.	PUNCT
cana-3453	89	1	7s	7	NOUN
cana-3453	89	2	(	(	PUNCT
cana-3453	89	3	2025	2025	NUM
cana-3453	89	4	)	)	PUNCT
cana-3453	89	5	419	419	NUM
cana-3453	89	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3453	89	7	the	the	DET
cana-3453	89	8	special	special	ADJ
cana-3453	89	9	properties	property	NOUN
cana-3453	89	10	of	of	ADP
cana-3453	89	11	ricci	ricci	PROPN
cana-3453	89	12	curvature	curvature	PROPN
cana-3453	89	13	,	,	PUNCT
cana-3453	89	14	whose	whose	DET
cana-3453	89	15	type	type	NOUN
cana-3453	89	16	is	be	AUX
cana-3453	89	17	(	(	PUNCT
cana-3453	89	18	2.14)--(2.16	2.14)--(2.16	NUM
cana-3453	89	19	)	)	PUNCT
cana-3453	89	20	in	in	ADP
cana-3453	89	21	a	a	DET
cana-3453	89	22	sense	sense	NOUN
cana-3453	89	23	along	along	ADP
cana-3453	89	24	the	the	DET
cana-3453	89	25	characteristic	characteristic	ADJ
cana-3453	89	26	direction	direction	NOUN
cana-3453	89	27	ξ	ξ	NOUN
cana-3453	89	28	,	,	PUNCT
cana-3453	89	29	reveal	reveal	VERB
cana-3453	89	30	some	some	DET
cana-3453	89	31	geometric	geometric	ADJ
cana-3453	89	32	behaviour	behaviour	NOUN
cana-3453	89	33	of	of	ADP
cana-3453	89	34	these	these	DET
cana-3453	89	35	spaces	space	NOUN
cana-3453	89	36	.	.	PUNCT
cana-3453	90	1	one	one	PRON
cana-3453	90	2	can	can	AUX
cana-3453	90	3	see	see	VERB
cana-3453	90	4	that	that	SCONJ
cana-3453	90	5	the	the	DET
cana-3453	90	6	ricci	ricci	PROPN
cana-3453	90	7	curvature	curvature	NOUN
cana-3453	90	8	in	in	ADP
cana-3453	90	9	directions	direction	NOUN
cana-3453	90	10	joined	join	VERB
cana-3453	90	11	with	with	ADP
cana-3453	90	12	ξ	ξ	PROPN
cana-3453	90	13	mainly	mainly	ADV
cana-3453	90	14	depends	depend	VERB
cana-3453	90	15	on	on	ADP
cana-3453	90	16	f₁	f₁	ADJ
cana-3453	90	17	f₃	f₃	PROPN
cana-3453	90	18	,	,	PUNCT
cana-3453	90	19	which	which	PRON
cana-3453	90	20	tells	tell	VERB
cana-3453	90	21	us	we	PRON
cana-3453	90	22	that	that	SCONJ
cana-3453	90	23	seeing	see	VERB
cana-3453	90	24	the	the	DET
cana-3453	90	25	way	way	NOUN
cana-3453	90	26	the	the	DET
cana-3453	90	27	contact	contact	NOUN
cana-3453	90	28	structure	structure	NOUN
cana-3453	90	29	behaves	behave	VERB
cana-3453	90	30	in	in	ADP
cana-3453	90	31	the	the	DET
cana-3453	90	32	presence	presence	NOUN
cana-3453	90	33	of	of	ADP
cana-3453	90	34	curvature	curvature	NOUN
cana-3453	90	35	is	be	AUX
cana-3453	90	36	somehow	somehow	ADV
cana-3453	90	37	an	an	DET
cana-3453	90	38	integral	integral	ADJ
cana-3453	90	39	eye	eye	NOUN
cana-3453	90	40	of	of	ADP
cana-3453	90	41	the	the	DET
cana-3453	90	42	discrepancy	discrepancy	NOUN
cana-3453	90	43	between	between	ADP
cana-3453	90	44	contact	contact	NOUN
cana-3453	90	45	hypothetically	hypothetically	ADV
cana-3453	90	46	rigid	rigid	ADJ
cana-3453	90	47	structure	structure	NOUN
cana-3453	90	48	based	base	VERB
cana-3453	90	49	on	on	ADP
cana-3453	90	50	the	the	DET
cana-3453	90	51	curvature	curvature	NOUN
cana-3453	90	52	expansion	expansion	NOUN
cana-3453	90	53	.	.	PUNCT
cana-3453	91	1	this	this	PRON
cana-3453	91	2	has	have	VERB
cana-3453	91	3	the	the	DET
cana-3453	91	4	simple	simple	ADJ
cana-3453	91	5	implication	implication	NOUN
cana-3453	91	6	that	that	SCONJ
cana-3453	91	7	the	the	DET
cana-3453	91	8	various	various	ADJ
cana-3453	91	9	curvature	curvature	NOUN
cana-3453	91	10	tensors	tensor	NOUN
cana-3453	91	11	wi	wi	PROPN
cana-3453	91	12	(	(	PUNCT
cana-3453	91	13	i	i	NOUN
cana-3453	91	14	=	=	NOUN
cana-3453	91	15	2	2	NUM
cana-3453	91	16	,	,	PUNCT
cana-3453	91	17	3	3	NUM
cana-3453	91	18	,	,	PUNCT
cana-3453	91	19	4	4	NUM
cana-3453	91	20	,	,	PUNCT
cana-3453	91	21	6	6	NUM
cana-3453	91	22	,	,	PUNCT
cana-3453	91	23	7	7	NUM
cana-3453	91	24	)	)	PUNCT
cana-3453	91	25	behave	behave	VERB
cana-3453	91	26	as	as	SCONJ
cana-3453	91	27	follows	follow	VERB
cana-3453	91	28	concerning	concern	VERB
cana-3453	91	29	ξ	ξ	PROPN
cana-3453	91	30	:	:	PUNCT
cana-3453	91	31	𝑾𝟐(𝑿	𝑾𝟐(𝑿	PROPN
cana-3453	91	32	,	,	PUNCT
cana-3453	91	33	𝒀)𝝃	𝒀)𝝃	PUNCT
cana-3453	91	34	=	=	PUNCT
cana-3453	92	1	(	(	PUNCT
cana-3453	92	2	𝒇𝟏	𝒇𝟏	NOUN
cana-3453	92	3	−	−	PROPN
cana-3453	92	4	𝒇𝟑)[𝜼(𝒀)𝑿	𝒇𝟑)[𝜼(𝒀)𝑿	ADP
cana-3453	92	5	−	−	PROPN
cana-3453	92	6	𝜼(𝑿)𝒀	𝜼(𝑿)𝒀	NOUN
cana-3453	92	7	]	]	PUNCT
cana-3453	93	1	+	+	CCONJ
cana-3453	93	2	𝟏	𝟏	NUM
cana-3453	93	3	𝟐𝒏	𝟐𝒏	NOUN
cana-3453	93	4	[	[	X
cana-3453	93	5	𝟐𝒏(𝒇𝟏	𝟐𝒏(𝒇𝟏	ADJ
cana-3453	93	6	−	−	PUNCT
cana-3453	93	7	𝒇𝟑)𝜼(𝑿)𝝃	𝒇𝟑)𝜼(𝑿)𝝃	NOUN
cana-3453	93	8	−	−	PROPN
cana-3453	94	1	𝑸𝑿	𝑸𝑿	PROPN
cana-3453	94	2	]	]	PUNCT
cana-3453	94	3	,	,	PUNCT
cana-3453	94	4	(	(	PUNCT
cana-3453	94	5	2.17	2.17	NUM
cana-3453	94	6	)	)	PUNCT
cana-3453	94	7	𝑾𝟑(𝑿	𝑾𝟑(𝑿	NOUN
cana-3453	94	8	,	,	PUNCT
cana-3453	94	9	𝒀)𝝃	𝒀)𝝃	PUNCT
cana-3453	94	10	=	=	PUNCT
cana-3453	95	1	(	(	PUNCT
cana-3453	95	2	𝒇𝟏	𝒇𝟏	NOUN
cana-3453	95	3	−	−	PROPN
cana-3453	95	4	𝒇𝟑)[𝜼(𝒀)𝑿	𝒇𝟑)[𝜼(𝒀)𝑿	ADP
cana-3453	95	5	−	−	PROPN
cana-3453	95	6	𝜼(𝑿)𝒀	𝜼(𝑿)𝒀	NOUN
cana-3453	95	7	]	]	PUNCT
cana-3453	95	8	−	−	PROPN
cana-3453	95	9	𝟏	𝟏	NUM
cana-3453	95	10	𝟐𝒏	𝟐𝒏	PROPN
cana-3453	96	1	[	[	X
cana-3453	96	2	𝟐𝒏(𝒇𝟏	𝟐𝒏(𝒇𝟏	ADJ
cana-3453	96	3	−	−	PUNCT
cana-3453	96	4	𝒇𝟑)𝜼(𝑿)𝒀	𝒇𝟑)𝜼(𝑿)𝒀	NOUN
cana-3453	96	5	−	−	ADJ
cana-3453	96	6	𝜼(𝒀)𝑸𝑿	𝜼(𝒀)𝑸𝑿	NOUN
cana-3453	96	7	]	]	X
cana-3453	96	8	,	,	PUNCT
cana-3453	96	9	(	(	PUNCT
cana-3453	96	10	2.18	2.18	NUM
cana-3453	96	11	)	)	PUNCT
cana-3453	96	12	𝑾𝟒(𝑿	𝑾𝟒(𝑿	PROPN
cana-3453	96	13	,	,	PUNCT
cana-3453	96	14	𝒀)𝝃	𝒀)𝝃	PUNCT
cana-3453	96	15	=	=	PUNCT
cana-3453	96	16	(	(	PUNCT
cana-3453	96	17	𝒇𝟏	𝒇𝟏	NOUN
cana-3453	96	18	−	−	PROPN
cana-3453	96	19	𝒇𝟑)[𝜼(𝒀)𝑿	𝒇𝟑)[𝜼(𝒀)𝑿	ADP
cana-3453	96	20	−	−	PROPN
cana-3453	96	21	𝜼(𝑿)𝒀	𝜼(𝑿)𝒀	NOUN
cana-3453	96	22	]	]	PUNCT
cana-3453	97	1	+	+	CCONJ
cana-3453	97	2	𝟏	𝟏	NUM
cana-3453	97	3	𝟐𝒏	𝟐𝒏	NOUN
cana-3453	98	1	[	[	X
cana-3453	98	2	𝜼(𝑿)𝑸𝒀	𝜼(𝑿)𝑸𝒀	NOUN
cana-3453	98	3	−	−	PROPN
cana-3453	98	4	𝒈(𝑿	𝒈(𝑿	PROPN
cana-3453	98	5	,	,	PUNCT
cana-3453	98	6	𝒀)𝝃	𝒀)𝝃	NOUN
cana-3453	98	7	]	]	PUNCT
cana-3453	98	8	,	,	PUNCT
cana-3453	98	9	(	(	PUNCT
cana-3453	98	10	2.19	2.19	NUM
cana-3453	98	11	)	)	PUNCT
cana-3453	98	12	𝑾𝟔(𝑿	𝑾𝟔(𝑿	NOUN
cana-3453	98	13	,	,	PUNCT
cana-3453	98	14	𝒀)𝝃	𝒀)𝝃	PUNCT
cana-3453	98	15	=	=	PUNCT
cana-3453	99	1	(	(	PUNCT
cana-3453	99	2	𝒇𝟏	𝒇𝟏	NOUN
cana-3453	99	3	−	−	PROPN
cana-3453	99	4	𝒇𝟑)[𝜼(𝒀)𝑿	𝒇𝟑)[𝜼(𝒀)𝑿	ADP
cana-3453	99	5	−	−	PROPN
cana-3453	99	6	𝜼(𝑿)𝒀	𝜼(𝑿)𝒀	NOUN
cana-3453	99	7	]	]	PUNCT
cana-3453	99	8	−	−	PROPN
cana-3453	100	1	(	(	PUNCT
cana-3453	100	2	𝒇𝟏	𝒇𝟏	NOUN
cana-3453	100	3	−	−	PROPN
cana-3453	100	4	𝒇𝟑)[𝜼(𝒀)𝑿	𝒇𝟑)[𝜼(𝒀)𝑿	PRON
cana-3453	100	5	−	−	PROPN
cana-3453	100	6	𝒈(𝑿	𝒈(𝑿	NOUN
cana-3453	100	7	,	,	PUNCT
cana-3453	100	8	𝒀)𝝃	𝒀)𝝃	NOUN
cana-3453	100	9	]	]	PUNCT
cana-3453	100	10	,	,	PUNCT
cana-3453	100	11	(	(	PUNCT
cana-3453	100	12	2.20	2.20	NUM
cana-3453	100	13	)	)	PUNCT
cana-3453	100	14	𝑾𝟕(𝑿	𝑾𝟕(𝑿	PROPN
cana-3453	100	15	,	,	PUNCT
cana-3453	100	16	𝒀)𝝃	𝒀)𝝃	PUNCT
cana-3453	100	17	=	=	PUNCT
cana-3453	101	1	(	(	PUNCT
cana-3453	101	2	𝒇𝟏	𝒇𝟏	NOUN
cana-3453	101	3	−	−	PROPN
cana-3453	101	4	𝒇𝟑)[𝜼(𝒀)𝑿	𝒇𝟑)[𝜼(𝒀)𝑿	ADP
cana-3453	101	5	−	−	PROPN
cana-3453	101	6	𝜼(𝑿)𝒀	𝜼(𝑿)𝒀	NOUN
cana-3453	101	7	]	]	PUNCT
cana-3453	101	8	−	−	PROPN
cana-3453	101	9	𝟏	𝟏	NUM
cana-3453	101	10	𝟐𝒏	𝟐𝒏	PROPN
cana-3453	102	1	[	[	X
cana-3453	102	2	𝟐𝒏(𝒇𝟏	𝟐𝒏(𝒇𝟏	ADJ
cana-3453	102	3	−	−	PRON
cana-3453	102	4	𝒇𝟑)𝜼(𝒀)𝑿	𝒇𝟑)𝜼(𝒀)𝑿	PROPN
cana-3453	102	5	−	−	PROPN
cana-3453	102	6	𝜼(𝒀)𝑸𝑿	𝜼(𝒀)𝑸𝑿	NOUN
cana-3453	102	7	]	]	PUNCT
cana-3453	102	8	.	.	PUNCT
cana-3453	103	1	(	(	PUNCT
cana-3453	103	2	2.21	2.21	NUM
cana-3453	103	3	)	)	PUNCT
cana-3453	103	4	these	these	DET
cana-3453	103	5	relationships	relationship	NOUN
cana-3453	103	6	constitute	constitute	VERB
cana-3453	103	7	the	the	DET
cana-3453	103	8	mathematical	mathematical	ADJ
cana-3453	103	9	basis	basis	NOUN
cana-3453	103	10	for	for	ADP
cana-3453	103	11	the	the	DET
cana-3453	103	12	later	later	ADJ
cana-3453	103	13	study	study	NOUN
cana-3453	103	14	of	of	ADP
cana-3453	103	15	invariant	invariant	ADJ
cana-3453	103	16	submanifolds	submanifold	NOUN
cana-3453	103	17	of	of	ADP
cana-3453	103	18	generalized	generalized	ADJ
cana-3453	103	19	sasakian	sasakian	ADJ
cana-3453	103	20	-	-	PUNCT
cana-3453	103	21	space	space	NOUN
cana-3453	103	22	-	-	PUNCT
cana-3453	103	23	forms	form	NOUN
cana-3453	103	24	.	.	PUNCT
cana-3453	104	1	the	the	DET
cana-3453	104	2	wi	wi	PROPN
cana-3453	104	3	curvature	curvature	NOUN
cana-3453	104	4	tensor	tensor	NOUN
cana-3453	104	5	equations	equation	NOUN
cana-3453	104	6	show	show	VERB
cana-3453	104	7	increasingly	increasingly	ADV
cana-3453	104	8	advanced	advanced	ADJ
cana-3453	104	9	dualities	duality	NOUN
cana-3453	104	10	between	between	ADP
cana-3453	104	11	metric	metric	ADJ
cana-3453	104	12	structure	structure	NOUN
cana-3453	104	13	,	,	PUNCT
cana-3453	104	14	contact	contact	NOUN
cana-3453	104	15	properties	property	NOUN
cana-3453	104	16	and	and	CCONJ
cana-3453	104	17	curvature	curvature	NOUN
cana-3453	104	18	.	.	PUNCT
cana-3453	105	1	these	these	DET
cana-3453	105	2	tensors	tensor	NOUN
cana-3453	105	3	reflect	reflect	VERB
cana-3453	105	4	certain	certain	ADJ
cana-3453	105	5	geometry	geometry	NOUN
cana-3453	105	6	properties	property	NOUN
cana-3453	105	7	that	that	PRON
cana-3453	105	8	can	can	AUX
cana-3453	105	9	be	be	AUX
cana-3453	105	10	appreciated	appreciate	VERB
cana-3453	105	11	,	,	PUNCT
cana-3453	105	12	and	and	CCONJ
cana-3453	105	13	they	they	PRON
cana-3453	105	14	inherit	inherit	VERB
cana-3453	105	15	some	some	DET
cana-3453	105	16	basic	basic	ADJ
cana-3453	105	17	properties	property	NOUN
cana-3453	105	18	from	from	ADP
cana-3453	105	19	the	the	DET
cana-3453	105	20	riemann	riemann	PROPN
cana-3453	105	21	curvature	curvature	PROPN
cana-3453	105	22	tensor	tensor	NOUN
cana-3453	105	23	.	.	PUNCT
cana-3453	106	1	these	these	DET
cana-3453	106	2	relationships	relationship	NOUN
cana-3453	106	3	will	will	AUX
cana-3453	106	4	be	be	AUX
cana-3453	106	5	very	very	ADV
cana-3453	106	6	important	important	ADJ
cana-3453	106	7	in	in	ADP
cana-3453	106	8	our	our	PRON
cana-3453	106	9	later	later	ADJ
cana-3453	106	10	analysis	analysis	NOUN
cana-3453	106	11	of	of	ADP
cana-3453	106	12	totally	totally	ADV
cana-3453	106	13	geodesic	geodesic	ADJ
cana-3453	106	14	invariant	invariant	ADJ
cana-3453	106	15	submanifolds	submanifold	NOUN
cana-3453	106	16	.	.	PUNCT
cana-3453	107	1	3	3	X
cana-3453	107	2	.	.	X
cana-3453	107	3	invariant	invariant	ADJ
cana-3453	107	4	submanifolds	submanifold	NOUN
cana-3453	107	5	of	of	ADP
cana-3453	107	6	generalized	generalized	ADJ
cana-3453	107	7	sasakian	sasakian	ADJ
cana-3453	107	8	-	-	PUNCT
cana-3453	107	9	space	space	NOUN
cana-3453	107	10	-	-	PUNCT
cana-3453	107	11	forms	form	NOUN
cana-3453	107	12	dual	dual	ADJ
cana-3453	107	13	to	to	ADP
cana-3453	107	14	commonness	commonness	NOUN
cana-3453	107	15	,	,	PUNCT
cana-3453	107	16	the	the	DET
cana-3453	107	17	structure	structure	NOUN
cana-3453	107	18	is	be	AUX
cana-3453	107	19	generally	generally	ADV
cana-3453	107	20	inherited	inherit	VERB
cana-3453	107	21	or	or	CCONJ
cana-3453	107	22	induced	induce	VERB
cana-3453	107	23	from	from	ADP
cana-3453	107	24	the	the	DET
cana-3453	107	25	ambient	ambient	ADJ
cana-3453	107	26	space	space	NOUN
cana-3453	107	27	,	,	PUNCT
cana-3453	107	28	and	and	CCONJ
cana-3453	107	29	the	the	DET
cana-3453	107	30	study	study	NOUN
cana-3453	107	31	of	of	ADP
cana-3453	107	32	invariant	invariant	ADJ
cana-3453	107	33	submanifolds	submanifold	NOUN
cana-3453	107	34	in	in	ADP
cana-3453	107	35	generalized	generalized	ADJ
cana-3453	107	36	sasakian	sasakian	ADJ
cana-3453	107	37	-	-	PUNCT
cana-3453	107	38	space	space	NOUN
cana-3453	107	39	-	-	PUNCT
cana-3453	107	40	forms	form	NOUN
cana-3453	107	41	establishes	establish	VERB
cana-3453	107	42	a	a	DET
cana-3453	107	43	balancing	balance	VERB
cana-3453	107	44	act	act	NOUN
cana-3453	107	45	between	between	ADP
cana-3453	107	46	the	the	DET
cana-3453	107	47	two	two	NUM
cana-3453	107	48	.	.	PUNCT
cana-3453	108	1	if	if	SCONJ
cana-3453	108	2	a	a	DET
cana-3453	108	3	submanifold	submanifold	NOUN
cana-3453	108	4	is	be	AUX
cana-3453	108	5	φ	φ	NOUN
cana-3453	108	6	-	-	ADJ
cana-3453	108	7	invariant	invariant	ADJ
cana-3453	108	8	,	,	PUNCT
cana-3453	108	9	it	it	PRON
cana-3453	108	10	inherits	inherit	VERB
cana-3453	108	11	contact	contact	NOUN
cana-3453	108	12	structures	structure	NOUN
cana-3453	108	13	and	and	CCONJ
cana-3453	108	14	metric	metric	ADJ
cana-3453	108	15	properties	property	NOUN
cana-3453	108	16	from	from	ADP
cana-3453	108	17	its	its	PRON
cana-3453	108	18	surroundings	surrounding	NOUN
cana-3453	108	19	.	.	PUNCT
cana-3453	109	1	with	with	ADP
cana-3453	109	2	this	this	DET
cana-3453	109	3	inheritance	inheritance	NOUN
cana-3453	109	4	comes	come	VERB
cana-3453	109	5	a	a	DET
cana-3453	109	6	refined	refined	ADJ
cana-3453	109	7	interaction	interaction	NOUN
cana-3453	109	8	between	between	ADP
cana-3453	109	9	the	the	DET
cana-3453	109	10	second	second	ADJ
cana-3453	109	11	fundamental	fundamental	ADJ
cana-3453	109	12	form	form	NOUN
cana-3453	109	13	and	and	CCONJ
cana-3453	109	14	the	the	DET
cana-3453	109	15	ambient	ambient	ADJ
cana-3453	109	16	space	space	NOUN
cana-3453	109	17	's	's	PART
cana-3453	109	18	fundamental	fundamental	ADJ
cana-3453	109	19	tensors	tensor	NOUN
cana-3453	109	20	.	.	PUNCT
cana-3453	110	1	when	when	SCONJ
cana-3453	110	2	the	the	DET
cana-3453	110	3	second	second	ADJ
cana-3453	110	4	fundamental	fundamental	ADJ
cana-3453	110	5	form	form	NOUN
cana-3453	110	6	σ	σ	NOUN
cana-3453	110	7	is	be	AUX
cana-3453	110	8	considered	consider	VERB
cana-3453	110	9	concerning	concern	VERB
cana-3453	110	10	it	it	PRON
cana-3453	110	11	,	,	PUNCT
cana-3453	110	12	it	it	PRON
cana-3453	110	13	becomes	becomes	AUX
cana-3453	110	14	especially	especially	ADV
cana-3453	110	15	telling	tell	VERB
cana-3453	110	16	to	to	PART
cana-3453	110	17	study	study	VERB
cana-3453	110	18	the	the	DET
cana-3453	110	19	structure	structure	NOUN
cana-3453	110	20	vector	vector	NOUN
cana-3453	110	21	field	field	NOUN
cana-3453	110	22	ξ	ξ	PROPN
cana-3453	110	23	.	.	PUNCT
cana-3453	111	1	the	the	DET
cana-3453	111	2	nonvanishing	nonvanishing	NOUN
cana-3453	111	3	of	of	ADP
cana-3453	111	4	σ(x	σ(x	PROPN
cana-3453	111	5	,	,	PUNCT
cana-3453	111	6	ξ	ξ	X
cana-3453	111	7	)	)	PUNCT
cana-3453	111	8	is	be	AUX
cana-3453	111	9	thus	thus	ADV
cana-3453	111	10	not	not	PART
cana-3453	111	11	a	a	DET
cana-3453	111	12	mere	mere	ADJ
cana-3453	111	13	technical	technical	ADJ
cana-3453	111	14	condition	condition	NOUN
cana-3453	111	15	but	but	CCONJ
cana-3453	111	16	rather	rather	ADV
cana-3453	111	17	encodes	encodes	DET
cana-3453	111	18	deep	deep	ADJ
cana-3453	111	19	geometric	geometric	ADJ
cana-3453	111	20	information	information	NOUN
cana-3453	111	21	about	about	ADP
cana-3453	111	22	the	the	DET
cana-3453	111	23	submanifold	submanifold	NOUN
cana-3453	111	24	's	's	PART
cana-3453	111	25	embedding	embed	VERB
cana-3453	111	26	into	into	ADP
cana-3453	111	27	the	the	DET
cana-3453	111	28	ambient	ambient	ADJ
cana-3453	111	29	manifold	manifold	NOUN
cana-3453	111	30	.	.	PUNCT
cana-3453	112	1	this	this	DET
cana-3453	112	2	relation	relation	NOUN
cana-3453	112	3	is	be	AUX
cana-3453	112	4	essential	essential	ADJ
cana-3453	112	5	to	to	ADP
cana-3453	112	6	understanding	understand	VERB
cana-3453	112	7	the	the	DET
cana-3453	112	8	nature	nature	NOUN
cana-3453	112	9	of	of	ADP
cana-3453	112	10	the	the	DET
cana-3453	112	11	submanifold	submanifold	NOUN
cana-3453	112	12	's	's	PART
cana-3453	112	13	geodesics	geodesic	NOUN
cana-3453	112	14	and	and	CCONJ
cana-3453	112	15	its	its	PRON
cana-3453	112	16	difference	difference	NOUN
cana-3453	112	17	from	from	ADP
cana-3453	112	18	the	the	DET
cana-3453	112	19	geodesics	geodesic	NOUN
cana-3453	112	20	of	of	ADP
cana-3453	112	21	an	an	DET
cana-3453	112	22	ambient	ambient	ADJ
cana-3453	112	23	space	space	NOUN
cana-3453	112	24	.	.	PUNCT
cana-3453	113	1	suppose	suppose	VERB
cana-3453	113	2	we	we	PRON
cana-3453	113	3	have	have	VERB
cana-3453	113	4	a	a	DET
cana-3453	113	5	submanifold	submanifold	NOUN
cana-3453	113	6	n	n	NOUN
cana-3453	113	7	of	of	ADP
cana-3453	113	8	a	a	DET
cana-3453	113	9	generalized	generalized	ADJ
cana-3453	113	10	sasakian	sasakian	ADJ
cana-3453	113	11	-	-	PUNCT
cana-3453	113	12	space	space	NOUN
cana-3453	113	13	-	-	PUNCT
cana-3453	113	14	form	form	NOUN
cana-3453	113	15	.	.	PUNCT
cana-3453	114	1	𝑀2𝑛+1(𝑓1	𝑀2𝑛+1(𝑓1	NUM
cana-3453	114	2	,	,	PUNCT
cana-3453	114	3	𝑓2	𝑓2	ADJ
cana-3453	114	4	,	,	PUNCT
cana-3453	114	5	𝑓3	𝑓3	NOUN
cana-3453	114	6	)	)	PUNCT
cana-3453	114	7	.	.	PUNCT
cana-3453	115	1	in	in	ADP
cana-3453	115	2	order	order	NOUN
cana-3453	115	3	to	to	PART
cana-3453	115	4	understand	understand	VERB
cana-3453	115	5	the	the	DET
cana-3453	115	6	geometry	geometry	NOUN
cana-3453	115	7	of	of	ADP
cana-3453	115	8	such	such	DET
cana-3453	115	9	an	an	DET
cana-3453	115	10	embedding	embedding	NOUN
cana-3453	115	11	,	,	PUNCT
cana-3453	115	12	one	one	PRON
cana-3453	115	13	should	should	AUX
cana-3453	115	14	investigate	investigate	VERB
cana-3453	115	15	relations	relation	NOUN
cana-3453	115	16	between	between	ADP
cana-3453	115	17	the	the	DET
cana-3453	115	18	connections	connection	NOUN
cana-3453	115	19	and	and	CCONJ
cana-3453	115	20	fundamental	fundamental	ADJ
cana-3453	115	21	forms	form	NOUN
cana-3453	115	22	of	of	ADP
cana-3453	115	23	the	the	DET
cana-3453	115	24	submanifold	submanifold	NOUN
cana-3453	115	25	and	and	CCONJ
cana-3453	115	26	the	the	DET
cana-3453	115	27	ambient	ambient	ADJ
cana-3453	115	28	space	space	NOUN
cana-3453	115	29	.	.	PUNCT
cana-3453	116	1	a	a	DET
cana-3453	116	2	couple	couple	NOUN
cana-3453	116	3	of	of	ADP
cana-3453	116	4	the	the	DET
cana-3453	116	5	canonical	canonical	ADJ
cana-3453	116	6	bundles	bundle	NOUN
cana-3453	116	7	that	that	PRON
cana-3453	116	8	we	we	PRON
cana-3453	116	9	consider	consider	VERB
cana-3453	116	10	in	in	ADP
cana-3453	116	11	this	this	DET
cana-3453	116	12	submanifold	submanifold	NOUN
cana-3453	116	13	n	n	AUX
cana-3453	116	14	are	be	AUX
cana-3453	116	15	:	:	PUNCT
cana-3453	116	16	1	1	X
cana-3453	116	17	.	.	PUNCT
cana-3453	117	1	the	the	DET
cana-3453	117	2	tangent	tangent	NOUN
cana-3453	117	3	bundle	bundle	PROPN
cana-3453	117	4	tn	tn	PROPN
cana-3453	117	5	2	2	NUM
cana-3453	117	6	.	.	PUNCT
cana-3453	118	1	the	the	DET
cana-3453	118	2	normal	normal	ADJ
cana-3453	118	3	bundle	bundle	NOUN
cana-3453	118	4	n⊥n	n⊥n	NOUN
cana-3453	118	5	communications	communication	NOUN
cana-3453	118	6	on	on	ADP
cana-3453	118	7	applied	apply	VERB
cana-3453	118	8	nonlinear	nonlinear	ADJ
cana-3453	118	9	analysis	analysis	NOUN
cana-3453	118	10	issn	issn	NOUN
cana-3453	118	11	:	:	PUNCT
cana-3453	118	12	1074	1074	NUM
cana-3453	118	13	-	-	PUNCT
cana-3453	118	14	133x	133x	NUM
cana-3453	118	15	vol	vol	NOUN
cana-3453	118	16	32	32	NUM
cana-3453	118	17	no	no	NOUN
cana-3453	118	18	.	.	PUNCT
cana-3453	119	1	7s	7	NOUN
cana-3453	119	2	(	(	PUNCT
cana-3453	119	3	2025	2025	NUM
cana-3453	119	4	)	)	PUNCT
cana-3453	119	5	420	420	NUM
cana-3453	119	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3453	119	7	restricting	restrict	VERB
cana-3453	119	8	the	the	DET
cana-3453	119	9	ambient	ambient	ADJ
cana-3453	119	10	geometric	geometric	ADJ
cana-3453	119	11	structures	structure	NOUN
cana-3453	119	12	to	to	ADP
cana-3453	119	13	the	the	DET
cana-3453	119	14	submanifold	submanifold	NOUN
cana-3453	119	15	is	be	AUX
cana-3453	119	16	required	require	VERB
cana-3453	119	17	to	to	PART
cana-3453	119	18	study	study	VERB
cana-3453	119	19	submanifolds	submanifold	NOUN
cana-3453	119	20	in	in	ADP
cana-3453	119	21	generalized	generalized	ADJ
cana-3453	119	22	sasakian	sasakian	ADJ
cana-3453	119	23	-	-	PUNCT
cana-3453	119	24	space	space	NOUN
cana-3453	119	25	-	-	PUNCT
cana-3453	119	26	forms	form	NOUN
cana-3453	119	27	.	.	PUNCT
cana-3453	120	1	the	the	DET
cana-3453	120	2	behaviour	behaviour	NOUN
cana-3453	120	3	of	of	ADP
cana-3453	120	4	invariant	invariant	ADJ
cana-3453	120	5	submanifolds	submanifold	NOUN
cana-3453	120	6	concerning	concern	VERB
cana-3453	120	7	the	the	DET
cana-3453	120	8	inheritance	inheritance	NOUN
cana-3453	120	9	of	of	ADP
cana-3453	120	10	geometric	geometric	ADJ
cana-3453	120	11	properties	property	NOUN
cana-3453	120	12	is	be	AUX
cana-3453	120	13	quite	quite	ADV
cana-3453	120	14	different	different	ADJ
cana-3453	120	15	than	than	ADP
cana-3453	120	16	in	in	ADP
cana-3453	120	17	other	other	ADJ
cana-3453	120	18	geometric	geometric	ADJ
cana-3453	120	19	settings	setting	NOUN
cana-3453	120	20	.	.	PUNCT
cana-3453	121	1	understanding	understand	VERB
cana-3453	121	2	the	the	DET
cana-3453	121	3	relationship	relationship	NOUN
cana-3453	121	4	between	between	ADP
cana-3453	121	5	ambient	ambient	ADJ
cana-3453	121	6	and	and	CCONJ
cana-3453	121	7	induced	induce	VERB
cana-3453	121	8	structures	structure	NOUN
cana-3453	121	9	yields	yield	NOUN
cana-3453	121	10	crucial	crucial	ADJ
cana-3453	121	11	insight	insight	NOUN
cana-3453	121	12	into	into	ADP
cana-3453	121	13	how	how	SCONJ
cana-3453	121	14	contact	contact	NOUN
cana-3453	121	15	geometry	geometry	NOUN
cana-3453	121	16	affects	affect	VERB
cana-3453	121	17	submanifold	submanifold	NOUN
cana-3453	121	18	behaviour	behaviour	NOUN
cana-3453	121	19	.	.	PUNCT
cana-3453	122	1	these	these	DET
cana-3453	122	2	relations	relation	NOUN
cana-3453	122	3	and	and	CCONJ
cana-3453	122	4	manifestations	manifestation	NOUN
cana-3453	122	5	are	be	AUX
cana-3453	122	6	local	local	ADJ
cana-3453	122	7	(	(	PUNCT
cana-3453	122	8	involving	involve	VERB
cana-3453	122	9	second	second	ADJ
cana-3453	122	10	fundamental	fundamental	ADJ
cana-3453	122	11	forms	form	NOUN
cana-3453	122	12	,	,	PUNCT
cana-3453	122	13	etc	etc	X
cana-3453	122	14	.	.	X
cana-3453	122	15	)	)	PUNCT
cana-3453	123	1	and	and	CCONJ
cana-3453	123	2	global	global	ADJ
cana-3453	123	3	(	(	PUNCT
cana-3453	123	4	involving	involve	VERB
cana-3453	123	5	,	,	PUNCT
cana-3453	123	6	e.g.	e.g.	ADV
cana-3453	123	7	curvature	curvature	NOUN
cana-3453	123	8	properties	property	NOUN
cana-3453	123	9	)	)	PUNCT
cana-3453	123	10	.	.	PUNCT
cana-3453	124	1	the	the	DET
cana-3453	124	2	relationship	relationship	NOUN
cana-3453	124	3	between	between	ADP
cana-3453	124	4	these	these	DET
cana-3453	124	5	structures	structure	NOUN
cana-3453	124	6	is	be	AUX
cana-3453	124	7	governed	govern	VERB
cana-3453	124	8	by	by	ADP
cana-3453	124	9	the	the	DET
cana-3453	124	10	so	so	ADV
cana-3453	124	11	-	-	PUNCT
cana-3453	124	12	called	call	VERB
cana-3453	124	13	gauss	gauss	ADJ
cana-3453	124	14	-	-	PUNCT
cana-3453	124	15	weingarten	weingarten	ADJ
cana-3453	124	16	formulae	formulae	NOUN
cana-3453	124	17	,	,	PUNCT
cana-3453	124	18	which	which	PRON
cana-3453	124	19	are	be	AUX
cana-3453	124	20	the	the	DET
cana-3453	124	21	main	main	ADJ
cana-3453	124	22	equations	equation	NOUN
cana-3453	124	23	describing	describe	VERB
cana-3453	124	24	the	the	DET
cana-3453	124	25	geometry	geometry	NOUN
cana-3453	124	26	of	of	ADP
cana-3453	124	27	the	the	DET
cana-3453	124	28	embedding	embed	VERB
cana-3453	124	29	:	:	PUNCT
cana-3453	124	30	�	�	NOUN
cana-3453	124	31	̅	̅	NOUN
cana-3453	124	32	�	�	NOUN
cana-3453	124	33	𝑿𝒀	𝑿𝒀	NOUN
cana-3453	124	34	=	=	PUNCT
cana-3453	124	35	𝛁𝑿𝒀	𝛁𝑿𝒀	NOUN
cana-3453	124	36	+	+	NUM
cana-3453	124	37	𝝈(𝑿	𝝈(𝑿	NOUN
cana-3453	124	38	,	,	PUNCT
cana-3453	124	39	𝒀	𝒀	NOUN
cana-3453	124	40	)	)	PUNCT
cana-3453	124	41	(	(	PUNCT
cana-3453	124	42	3.1	3.1	NUM
cana-3453	124	43	)	)	PUNCT
cana-3453	124	44	�	�	NOUN
cana-3453	124	45	̅	̅	NOUN
cana-3453	124	46	�	�	NOUN
cana-3453	124	47	𝑿𝑽	𝑿𝑽	NOUN
cana-3453	124	48	=	=	SYM
cana-3453	125	1	−𝐀𝑽𝑿	−𝐀𝑽𝑿	PROPN
cana-3453	125	2	+	+	CCONJ
cana-3453	125	3	𝛁𝑿	𝛁𝑿	NOUN
cana-3453	125	4	⊥𝑽	⊥𝑽	ADJ
cana-3453	125	5	(	(	PUNCT
cana-3453	125	6	3.2	3.2	NUM
cana-3453	125	7	)	)	PUNCT
cana-3453	125	8	these	these	DET
cana-3453	125	9	equations	equation	NOUN
cana-3453	125	10	hold	hold	VERB
cana-3453	125	11	for	for	ADP
cana-3453	125	12	all	all	DET
cana-3453	125	13	vector	vector	NOUN
cana-3453	125	14	fields	field	NOUN
cana-3453	125	15	x	x	X
cana-3453	125	16	,	,	PUNCT
cana-3453	125	17	y∈γ(tn	y∈γ(tn	PROPN
cana-3453	125	18	)	)	PUNCT
cana-3453	125	19	and	and	CCONJ
cana-3453	125	20	v∈γ(t⊥n	v∈γ(t⊥n	NUM
cana-3453	125	21	)	)	PUNCT
cana-3453	125	22	,	,	PUNCT
cana-3453	125	23	where	where	SCONJ
cana-3453	125	24	:	:	PUNCT
cana-3453	125	25	•	•	NUM
cana-3453	125	26	∇⊥	∇⊥	NOUN
cana-3453	125	27	represents	represent	VERB
cana-3453	125	28	the	the	DET
cana-3453	125	29	connection	connection	NOUN
cana-3453	125	30	in	in	ADP
cana-3453	125	31	the	the	DET
cana-3453	125	32	normal	normal	ADJ
cana-3453	125	33	bundle	bundle	NOUN
cana-3453	125	34	•	•	PROPN
cana-3453	125	35	σ	σ	PROPN
cana-3453	125	36	denotes	denote	NOUN
cana-3453	125	37	the	the	DET
cana-3453	125	38	second	second	ADJ
cana-3453	125	39	fundamental	fundamental	ADJ
cana-3453	125	40	form	form	NOUN
cana-3453	125	41	•	•	ADP
cana-3453	125	42	av	av	PROPN
cana-3453	125	43	is	be	AUX
cana-3453	125	44	the	the	DET
cana-3453	125	45	shape	shape	NOUN
cana-3453	125	46	operator	operator	NOUN
cana-3453	125	47	corresponding	correspond	VERB
cana-3453	125	48	to	to	ADP
cana-3453	125	49	the	the	DET
cana-3453	125	50	normal	normal	ADJ
cana-3453	125	51	vector	vector	NOUN
cana-3453	125	52	field	field	NOUN
cana-3453	125	53	v	v	ADP
cana-3453	125	54	the	the	DET
cana-3453	125	55	shape	shape	NOUN
cana-3453	125	56	operator	operator	NOUN
cana-3453	125	57	and	and	CCONJ
cana-3453	125	58	the	the	DET
cana-3453	125	59	second	second	ADJ
cana-3453	125	60	fundamental	fundamental	ADJ
cana-3453	125	61	form	form	NOUN
cana-3453	125	62	are	be	AUX
cana-3453	125	63	intrinsically	intrinsically	ADV
cana-3453	125	64	connected	connect	VERB
cana-3453	125	65	through	through	ADP
cana-3453	125	66	a	a	DET
cana-3453	125	67	relationship	relationship	NOUN
cana-3453	125	68	that	that	PRON
cana-3453	125	69	characterizes	characterize	VERB
cana-3453	125	70	their	their	PRON
cana-3453	125	71	geometric	geometric	ADJ
cana-3453	125	72	interaction	interaction	NOUN
cana-3453	125	73	:	:	PUNCT
cana-3453	126	1	g(σ(x	g(σ(x	PROPN
cana-3453	126	2	,	,	PUNCT
cana-3453	126	3	y),v)=g(avx	y),v)=g(avx	NOUN
cana-3453	126	4	,	,	PUNCT
cana-3453	126	5	y	y	NOUN
cana-3453	126	6	)	)	PUNCT
cana-3453	126	7	this	this	DET
cana-3453	126	8	relationship	relationship	NOUN
cana-3453	126	9	plays	play	VERB
cana-3453	126	10	a	a	DET
cana-3453	126	11	crucial	crucial	ADJ
cana-3453	126	12	role	role	NOUN
cana-3453	126	13	in	in	ADP
cana-3453	126	14	understanding	understand	VERB
cana-3453	126	15	the	the	DET
cana-3453	126	16	geometric	geometric	ADJ
cana-3453	126	17	properties	property	NOUN
cana-3453	126	18	of	of	ADP
cana-3453	126	19	the	the	DET
cana-3453	126	20	submanifold	submanifold	NOUN
cana-3453	126	21	.	.	PUNCT
cana-3453	127	1	a	a	DET
cana-3453	127	2	particularly	particularly	ADV
cana-3453	127	3	important	important	ADJ
cana-3453	127	4	case	case	NOUN
cana-3453	127	5	arises	arise	VERB
cana-3453	127	6	when	when	SCONJ
cana-3453	127	7	examining	examine	VERB
cana-3453	127	8	the	the	DET
cana-3453	127	9	behaviour	behaviour	NOUN
cana-3453	127	10	of	of	ADP
cana-3453	127	11	the	the	DET
cana-3453	127	12	second	second	ADJ
cana-3453	127	13	fundamental	fundamental	ADJ
cana-3453	127	14	form	form	NOUN
cana-3453	127	15	concerning	concern	VERB
cana-3453	127	16	the	the	DET
cana-3453	127	17	structure	structure	NOUN
cana-3453	127	18	vector	vector	NOUN
cana-3453	127	19	field	field	NOUN
cana-3453	127	20	ξ	ξ	X
cana-3453	127	21	:	:	PUNCT
cana-3453	127	22	σ(x	σ(x	PROPN
cana-3453	127	23	,	,	PUNCT
cana-3453	127	24	ξ)=0	ξ)=0	PROPN
cana-3453	127	25	(	(	PUNCT
cana-3453	127	26	3.3	3.3	NUM
cana-3453	127	27	)	)	PUNCT
cana-3453	127	28	it	it	PRON
cana-3453	127	29	plays	play	VERB
cana-3453	127	30	an	an	DET
cana-3453	127	31	important	important	ADJ
cana-3453	127	32	role	role	NOUN
cana-3453	127	33	in	in	ADP
cana-3453	127	34	the	the	DET
cana-3453	127	35	geometry	geometry	NOUN
cana-3453	127	36	of	of	ADP
cana-3453	127	37	the	the	DET
cana-3453	127	38	submanifold	submanifold	NOUN
cana-3453	127	39	and	and	CCONJ
cana-3453	127	40	will	will	AUX
cana-3453	127	41	be	be	AUX
cana-3453	127	42	helpful	helpful	ADJ
cana-3453	127	43	for	for	ADP
cana-3453	127	44	our	our	PRON
cana-3453	127	45	analysis	analysis	NOUN
cana-3453	127	46	in	in	ADP
cana-3453	127	47	the	the	DET
cana-3453	127	48	next	next	ADJ
cana-3453	127	49	sections	section	NOUN
cana-3453	127	50	.	.	PUNCT
cana-3453	128	1	the	the	DET
cana-3453	128	2	notion	notion	NOUN
cana-3453	128	3	of	of	ADP
cana-3453	128	4	a	a	DET
cana-3453	128	5	geodesic	geodesic	ADJ
cana-3453	128	6	submanifold	submanifold	NOUN
cana-3453	128	7	appears	appear	VERB
cana-3453	128	8	in	in	ADP
cana-3453	128	9	the	the	DET
cana-3453	128	10	context	context	NOUN
cana-3453	128	11	where	where	SCONJ
cana-3453	128	12	the	the	DET
cana-3453	128	13	second	second	ADJ
cana-3453	128	14	fundamental	fundamental	ADJ
cana-3453	128	15	form	form	NOUN
cana-3453	128	16	vanishes	vanish	VERB
cana-3453	128	17	identically	identically	ADV
cana-3453	128	18	.	.	PUNCT
cana-3453	129	1	this	this	PRON
cana-3453	129	2	is	be	AUX
cana-3453	129	3	the	the	DET
cana-3453	129	4	most	most	ADV
cana-3453	129	5	natural	natural	ADJ
cana-3453	129	6	embedding	embedding	NOUN
cana-3453	129	7	of	of	ADP
cana-3453	129	8	a	a	DET
cana-3453	129	9	manifold	manifold	NOUN
cana-3453	129	10	into	into	ADP
cana-3453	129	11	another	another	PRON
cana-3453	129	12	,	,	PUNCT
cana-3453	129	13	preserving	preserve	VERB
cana-3453	129	14	geodesics	geodesic	NOUN
cana-3453	129	15	and	and	CCONJ
cana-3453	129	16	minimizing	minimize	VERB
cana-3453	129	17	the	the	DET
cana-3453	129	18	amount	amount	NOUN
cana-3453	129	19	of	of	ADP
cana-3453	129	20	geometric	geometric	ADJ
cana-3453	129	21	distortion	distortion	NOUN
cana-3453	129	22	.	.	PUNCT
cana-3453	130	1	the	the	DET
cana-3453	130	2	relationships	relationship	NOUN
cana-3453	130	3	we	we	PRON
cana-3453	130	4	have	have	AUX
cana-3453	130	5	established	establish	VERB
cana-3453	130	6	in	in	ADP
cana-3453	130	7	this	this	DET
cana-3453	130	8	section	section	NOUN
cana-3453	130	9	provide	provide	VERB
cana-3453	130	10	an	an	DET
cana-3453	130	11	essential	essential	ADJ
cana-3453	130	12	background	background	NOUN
cana-3453	130	13	for	for	ADP
cana-3453	130	14	our	our	PRON
cana-3453	130	15	study	study	NOUN
cana-3453	130	16	of	of	ADP
cana-3453	130	17	under	under	ADP
cana-3453	130	18	which	which	PRON
cana-3453	130	19	conditions	condition	NOUN
cana-3453	130	20	invariant	invariant	ADJ
cana-3453	130	21	submanifolds	submanifold	NOUN
cana-3453	130	22	of	of	ADP
cana-3453	130	23	generalized	generalized	ADJ
cana-3453	130	24	sasakian	sasakian	ADJ
cana-3453	130	25	-	-	PUNCT
cana-3453	130	26	space	space	NOUN
cana-3453	130	27	-	-	PUNCT
cana-3453	130	28	forms	form	NOUN
cana-3453	130	29	are	be	AUX
cana-3453	130	30	geodesic	geodesic	ADJ
cana-3453	130	31	.	.	PUNCT
cana-3453	131	1	we	we	PRON
cana-3453	131	2	will	will	AUX
cana-3453	131	3	follow	follow	VERB
cana-3453	131	4	with	with	ADP
cana-3453	131	5	conditions	condition	NOUN
cana-3453	131	6	satisfying	satisfy	VERB
cana-3453	131	7	these	these	PRON
cana-3453	131	8	to	to	PART
cana-3453	131	9	determine	determine	VERB
cana-3453	131	10	how	how	SCONJ
cana-3453	131	11	they	they	PRON
cana-3453	131	12	relate	relate	VERB
cana-3453	131	13	to	to	ADP
cana-3453	131	14	different	different	ADJ
cana-3453	131	15	curvature	curvature	NOUN
cana-3453	131	16	tensors	tensor	NOUN
cana-3453	131	17	,	,	PUNCT
cana-3453	131	18	which	which	PRON
cana-3453	131	19	describe	describe	VERB
cana-3453	131	20	the	the	DET
cana-3453	131	21	geometric	geometric	ADJ
cana-3453	131	22	characteristics	characteristic	NOUN
cana-3453	131	23	of	of	ADP
cana-3453	131	24	these	these	DET
cana-3453	131	25	submanifolds	submanifold	NOUN
cana-3453	131	26	.	.	PUNCT
cana-3453	132	1	this	this	DET
cana-3453	132	2	intriguing	intriguing	ADJ
cana-3453	132	3	connection	connection	NOUN
cana-3453	132	4	of	of	ADP
cana-3453	132	5	curvature	curvature	NOUN
cana-3453	132	6	tensors	tensor	NOUN
cana-3453	132	7	and	and	CCONJ
cana-3453	132	8	invariant	invariant	ADJ
cana-3453	132	9	submanifolds	submanifold	NOUN
cana-3453	132	10	involves	involve	VERB
cana-3453	132	11	many	many	ADJ
cana-3453	132	12	sophisticated	sophisticated	ADJ
cana-3453	132	13	geometric	geometric	ADJ
cana-3453	132	14	aspects	aspect	NOUN
cana-3453	132	15	that	that	PRON
cana-3453	132	16	beg	beg	VERB
cana-3453	132	17	special	special	ADJ
cana-3453	132	18	attention	attention	NOUN
cana-3453	132	19	.	.	PUNCT
cana-3453	133	1	this	this	DET
cana-3453	133	2	question	question	NOUN
cana-3453	133	3	,	,	PUNCT
cana-3453	133	4	the	the	DET
cana-3453	133	5	interaction	interaction	NOUN
cana-3453	133	6	of	of	ADP
cana-3453	133	7	these	these	DET
cana-3453	133	8	two	two	NUM
cana-3453	133	9	types	type	NOUN
cana-3453	133	10	of	of	ADP
cana-3453	133	11	tensors	tensor	NOUN
cana-3453	133	12	with	with	ADP
cana-3453	133	13	the	the	DET
cana-3453	133	14	second	second	ADJ
cana-3453	133	15	fundamental	fundamental	ADJ
cana-3453	133	16	form	form	NOUN
cana-3453	133	17	,	,	PUNCT
cana-3453	133	18	is	be	AUX
cana-3453	133	19	not	not	PART
cana-3453	133	20	just	just	ADV
cana-3453	133	21	a	a	DET
cana-3453	133	22	technical	technical	ADJ
cana-3453	133	23	question	question	NOUN
cana-3453	133	24	,	,	PUNCT
cana-3453	133	25	but	but	CCONJ
cana-3453	133	26	rather	rather	ADV
cana-3453	133	27	tells	tell	VERB
cana-3453	133	28	us	we	PRON
cana-3453	133	29	a	a	DET
cana-3453	133	30	lot	lot	NOUN
cana-3453	133	31	about	about	ADP
cana-3453	133	32	the	the	DET
cana-3453	133	33	structure	structure	NOUN
cana-3453	133	34	of	of	ADP
cana-3453	133	35	the	the	DET
cana-3453	133	36	ambient	ambient	ADJ
cana-3453	133	37	space	space	NOUN
cana-3453	133	38	and	and	CCONJ
cana-3453	133	39	submanifolds	submanifold	NOUN
cana-3453	133	40	.	.	PUNCT
cana-3453	134	1	these	these	DET
cana-3453	134	2	entanglements	entanglement	NOUN
cana-3453	134	3	are	be	AUX
cana-3453	134	4	especially	especially	ADV
cana-3453	134	5	relevant	relevant	ADJ
cana-3453	134	6	™	™	NOUN
cana-3453	134	7	when	when	SCONJ
cana-3453	134	8	studying	study	VERB
cana-3453	134	9	conditions	condition	NOUN
cana-3453	134	10	for	for	ADP
cana-3453	134	11	total	total	ADJ
cana-3453	134	12	geodesicity	geodesicity	NOUN
cana-3453	134	13	,	,	PUNCT
cana-3453	134	14	since	since	SCONJ
cana-3453	134	15	they	they	PRON
cana-3453	134	16	provide	provide	VERB
cana-3453	134	17	insight	insight	NOUN
cana-3453	134	18	into	into	ADP
cana-3453	134	19	how	how	SCONJ
cana-3453	134	20	the	the	DET
cana-3453	134	21	geometry	geometry	NOUN
cana-3453	134	22	of	of	ADP
cana-3453	134	23	the	the	DET
cana-3453	134	24	submanifold	submanifold	NOUN
cana-3453	134	25	fits	fit	VERB
cana-3453	134	26	with	with	ADP
cana-3453	134	27	that	that	PRON
cana-3453	134	28	of	of	ADP
cana-3453	134	29	the	the	DET
cana-3453	134	30	ambient	ambient	ADJ
cana-3453	134	31	space	space	NOUN
cana-3453	134	32	.	.	PUNCT
cana-3453	135	1	communications	communication	NOUN
cana-3453	135	2	on	on	ADP
cana-3453	135	3	applied	apply	VERB
cana-3453	135	4	nonlinear	nonlinear	ADJ
cana-3453	135	5	analysis	analysis	NOUN
cana-3453	135	6	issn	issn	NOUN
cana-3453	135	7	:	:	PUNCT
cana-3453	135	8	1074	1074	NUM
cana-3453	135	9	-	-	PUNCT
cana-3453	135	10	133x	133x	NUM
cana-3453	135	11	vol	vol	NOUN
cana-3453	135	12	32	32	NUM
cana-3453	135	13	no	no	NOUN
cana-3453	135	14	.	.	PUNCT
cana-3453	136	1	7s	7	NOUN
cana-3453	136	2	(	(	PUNCT
cana-3453	136	3	2025	2025	NUM
cana-3453	136	4	)	)	PUNCT
cana-3453	136	5	421	421	NUM
cana-3453	136	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3453	137	1	this	this	PRON
cana-3453	137	2	includes	include	VERB
cana-3453	137	3	the	the	DET
cana-3453	137	4	study	study	NOUN
cana-3453	137	5	of	of	ADP
cana-3453	137	6	other	other	ADJ
cana-3453	137	7	curvature	curvature	NOUN
cana-3453	137	8	tensors	tensor	NOUN
cana-3453	137	9	wi	wi	PROPN
cana-3453	137	10	,	,	PUNCT
cana-3453	137	11	which	which	PRON
cana-3453	137	12	gives	give	VERB
cana-3453	137	13	different	different	ADJ
cana-3453	137	14	point	point	NOUN
cana-3453	137	15	of	of	ADP
cana-3453	137	16	views	view	NOUN
cana-3453	137	17	of	of	ADP
cana-3453	137	18	the	the	DET
cana-3453	137	19	same	same	ADJ
cana-3453	137	20	geometric	geometric	ADJ
cana-3453	137	21	situation	situation	NOUN
cana-3453	137	22	.	.	PUNCT
cana-3453	138	1	each	each	DET
cana-3453	138	2	tensor	tensor	NOUN
cana-3453	138	3	encodes	encode	VERB
cana-3453	138	4	a	a	DET
cana-3453	138	5	different	different	ADJ
cana-3453	138	6	facet	facet	NOUN
cana-3453	138	7	of	of	ADP
cana-3453	138	8	the	the	DET
cana-3453	138	9	manifold	manifold	NOUN
cana-3453	138	10	's	's	PART
cana-3453	138	11	geometry	geometry	NOUN
cana-3453	138	12	,	,	PUNCT
cana-3453	138	13	akin	akin	ADJ
cana-3453	138	14	to	to	ADP
cana-3453	138	15	how	how	SCONJ
cana-3453	138	16	observing	observe	VERB
cana-3453	138	17	a	a	DET
cana-3453	138	18	complex	complex	ADJ
cana-3453	138	19	sculpture	sculpture	NOUN
cana-3453	138	20	from	from	ADP
cana-3453	138	21	various	various	ADJ
cana-3453	138	22	vantage	vantage	NOUN
cana-3453	138	23	points	point	NOUN
cana-3453	138	24	unveils	unveil	VERB
cana-3453	138	25	different	different	ADJ
cana-3453	138	26	characteristics	characteristic	NOUN
cana-3453	138	27	of	of	ADP
cana-3453	138	28	its	its	PRON
cana-3453	138	29	shape	shape	NOUN
cana-3453	138	30	.	.	PUNCT
cana-3453	139	1	this	this	DET
cana-3453	139	2	multiplicity	multiplicity	NOUN
cana-3453	139	3	of	of	ADP
cana-3453	139	4	perspectives	perspective	NOUN
cana-3453	139	5	is	be	AUX
cana-3453	139	6	crucial	crucial	ADJ
cana-3453	139	7	for	for	ADP
cana-3453	139	8	constructing	construct	VERB
cana-3453	139	9	a	a	DET
cana-3453	139	10	thorough	thorough	ADJ
cana-3453	139	11	understanding	understanding	NOUN
cana-3453	139	12	of	of	ADP
cana-3453	139	13	the	the	DET
cana-3453	139	14	geometric	geometric	ADJ
cana-3453	139	15	structures	structure	NOUN
cana-3453	139	16	involved	involve	VERB
cana-3453	139	17	.	.	PUNCT
cana-3453	140	1	4	4	X
cana-3453	140	2	.	.	X
cana-3453	140	3	geometry	geometry	NOUN
cana-3453	140	4	of	of	ADP
cana-3453	140	5	invariant	invariant	ADJ
cana-3453	140	6	submanifolds	submanifold	NOUN
cana-3453	140	7	satisfying	satisfy	VERB
cana-3453	140	8	the	the	DET
cana-3453	140	9	w2	w2	NOUN
cana-3453	140	10	curvature	curvature	NOUN
cana-3453	140	11	condition	condition	NOUN
cana-3453	140	12	robust	robust	ADJ
cana-3453	140	13	,	,	PUNCT
cana-3453	140	14	the	the	DET
cana-3453	140	15	curvature	curvature	NOUN
cana-3453	140	16	for	for	ADP
cana-3453	140	17	the	the	DET
cana-3453	140	18	w₂	w₂	PROPN
cana-3453	140	19	is	be	AUX
cana-3453	140	20	a	a	DET
cana-3453	140	21	powerful	powerful	ADJ
cana-3453	140	22	refinement	refinement	NOUN
cana-3453	140	23	of	of	ADP
cana-3453	140	24	the	the	DET
cana-3453	140	25	curvature	curvature	NOUN
cana-3453	140	26	for	for	ADP
cana-3453	140	27	the	the	DET
cana-3453	140	28	riemann	riemann	PROPN
cana-3453	140	29	.	.	PUNCT
cana-3453	141	1	this	this	DET
cana-3453	141	2	twstory	twstory	NOUN
cana-3453	141	3	has	have	VERB
cana-3453	141	4	some	some	DET
cana-3453	141	5	subtle	subtle	ADJ
cana-3453	141	6	geometric	geometric	ADJ
cana-3453	141	7	features	feature	NOUN
cana-3453	141	8	which	which	PRON
cana-3453	141	9	are	be	AUX
cana-3453	141	10	hidden	hide	VERB
cana-3453	141	11	from	from	ADP
cana-3453	141	12	the	the	DET
cana-3453	141	13	purely	purely	ADV
cana-3453	141	14	classical	classical	ADJ
cana-3453	141	15	curvature	curvature	NOUN
cana-3453	141	16	tensor	tensor	NOUN
cana-3453	141	17	through	through	ADP
cana-3453	141	18	the	the	DET
cana-3453	141	19	consideration	consideration	NOUN
cana-3453	141	20	of	of	ADP
cana-3453	141	21	its	its	PRON
cana-3453	141	22	interaction	interaction	NOUN
cana-3453	141	23	with	with	ADP
cana-3453	141	24	invariant	invariant	ADJ
cana-3453	141	25	submanifolds	submanifold	NOUN
cana-3453	141	26	.	.	PUNCT
cana-3453	142	1	conditions	condition	NOUN
cana-3453	142	2	like	like	ADP
cana-3453	142	3	q(σ	q(σ	PROPN
cana-3453	142	4	,	,	PUNCT
cana-3453	142	5	w₂	w₂	PROPN
cana-3453	142	6	)	)	PUNCT
cana-3453	142	7	=	=	SYM
cana-3453	142	8	0	0	NUM
cana-3453	142	9	go	go	VERB
cana-3453	142	10	beyond	beyond	ADP
cana-3453	142	11	purely	purely	ADV
cana-3453	142	12	algebraic	algebraic	ADJ
cana-3453	142	13	constraints	constraint	NOUN
cana-3453	142	14	and	and	CCONJ
cana-3453	142	15	give	give	VERB
cana-3453	142	16	some	some	DET
cana-3453	142	17	information	information	NOUN
cana-3453	142	18	about	about	ADP
cana-3453	142	19	the	the	DET
cana-3453	142	20	curvature	curvature	NOUN
cana-3453	142	21	of	of	ADP
cana-3453	142	22	the	the	DET
cana-3453	142	23	submanifold	submanifold	NOUN
cana-3453	142	24	in	in	ADP
cana-3453	142	25	the	the	DET
cana-3453	142	26	ambient	ambient	ADJ
cana-3453	142	27	space	space	NOUN
cana-3453	142	28	.	.	PUNCT
cana-3453	143	1	in	in	ADP
cana-3453	143	2	this	this	DET
cana-3453	143	3	case	case	NOUN
cana-3453	143	4	,	,	PUNCT
cana-3453	143	5	the	the	DET
cana-3453	143	6	non	non	ADJ
cana-3453	143	7	-	-	ADJ
cana-3453	143	8	degeneracy	degeneracy	ADJ
cana-3453	143	9	condition	condition	NOUN
cana-3453	143	10	that	that	PRON
cana-3453	143	11	involves	involve	VERB
cana-3453	143	12	f₂	f₂	NOUN
cana-3453	143	13	and	and	CCONJ
cana-3453	143	14	f₃	f₃	NOUN
cana-3453	143	15	is	be	AUX
cana-3453	143	16	not	not	PART
cana-3453	143	17	an	an	DET
cana-3453	143	18	arbitrary	arbitrary	ADJ
cana-3453	143	19	technical	technical	ADJ
cana-3453	143	20	necessity	necessity	NOUN
cana-3453	143	21	but	but	CCONJ
cana-3453	143	22	a	a	DET
cana-3453	143	23	condition	condition	NOUN
cana-3453	143	24	that	that	PRON
cana-3453	143	25	arises	arise	VERB
cana-3453	143	26	naturally	naturally	ADV
cana-3453	143	27	because	because	SCONJ
cana-3453	143	28	we	we	PRON
cana-3453	143	29	are	be	AUX
cana-3453	143	30	working	work	VERB
cana-3453	143	31	on	on	ADP
cana-3453	143	32	generalized	generalized	ADJ
cana-3453	143	33	ssf	ssf	NOUN
cana-3453	143	34	.	.	PUNCT
cana-3453	144	1	here	here	ADV
cana-3453	144	2	,	,	PUNCT
cana-3453	144	3	we	we	PRON
cana-3453	144	4	explore	explore	VERB
cana-3453	144	5	how	how	SCONJ
cana-3453	144	6	the	the	DET
cana-3453	144	7	geometry	geometry	NOUN
cana-3453	144	8	of	of	ADP
cana-3453	144	9	the	the	DET
cana-3453	144	10	invariant	invariant	ADJ
cana-3453	144	11	submanifolds	submanifold	NOUN
cana-3453	144	12	is	be	AUX
cana-3453	144	13	governed	govern	VERB
cana-3453	144	14	by	by	ADP
cana-3453	144	15	the	the	DET
cana-3453	144	16	interaction	interaction	NOUN
cana-3453	144	17	of	of	ADP
cana-3453	144	18	the	the	DET
cana-3453	144	19	corresponding	corresponding	ADJ
cana-3453	144	20	second	second	ADJ
cana-3453	144	21	fundamental	fundamental	ADJ
cana-3453	144	22	form	form	NOUN
cana-3453	144	23	σ	σ	NOUN
cana-3453	144	24	with	with	ADP
cana-3453	144	25	a	a	DET
cana-3453	144	26	certain	certain	ADJ
cana-3453	144	27	curvature	curvature	NOUN
cana-3453	144	28	tensor	tensor	NOUN
cana-3453	144	29	w2	w2	NOUN
cana-3453	144	30	.	.	PUNCT
cana-3453	145	1	the	the	DET
cana-3453	145	2	present	present	ADJ
cana-3453	145	3	study	study	NOUN
cana-3453	145	4	addresses	address	NOUN
cana-3453	145	5	when	when	SCONJ
cana-3453	145	6	those	those	DET
cana-3453	145	7	submanifolds	submanifold	NOUN
cana-3453	145	8	become	become	VERB
cana-3453	145	9	geodesic	geodesic	ADJ
cana-3453	145	10	under	under	ADP
cana-3453	145	11	certain	certain	ADJ
cana-3453	145	12	curvature	curvature	NOUN
cana-3453	145	13	conditions	condition	NOUN
cana-3453	145	14	.	.	PUNCT
cana-3453	146	1	analyzing	analyze	VERB
cana-3453	146	2	tensor	tensor	NOUN
cana-3453	146	3	derivations	derivation	NOUN
cana-3453	146	4	is	be	AUX
cana-3453	146	5	needed	need	VERB
cana-3453	146	6	to	to	PART
cana-3453	146	7	study	study	VERB
cana-3453	146	8	the	the	DET
cana-3453	146	9	w₂	w₂	PROPN
cana-3453	146	10	curvature	curvature	NOUN
cana-3453	146	11	tensor	tensor	NOUN
cana-3453	146	12	behaviour	behaviour	NOUN
cana-3453	146	13	under	under	ADP
cana-3453	146	14	invariant	invariant	ADJ
cana-3453	146	15	submanifolds	submanifold	NOUN
cana-3453	146	16	.	.	PUNCT
cana-3453	147	1	types	type	NOUN
cana-3453	147	2	of	of	ADP
cana-3453	147	3	the	the	DET
cana-3453	147	4	operator	operator	NOUN
cana-3453	147	5	q(σ	q(σ	NOUN
cana-3453	147	6	,	,	PUNCT
cana-3453	147	7	w₂)—which	w₂)—which	PRON
cana-3453	147	8	implements	implement	VERB
cana-3453	147	9	the	the	DET
cana-3453	147	10	second	second	ADJ
cana-3453	147	11	fundamental	fundamental	ADJ
cana-3453	147	12	form	form	NOUN
cana-3453	147	13	on	on	ADP
cana-3453	147	14	the	the	DET
cana-3453	147	15	image	image	NOUN
cana-3453	147	16	of	of	ADP
cana-3453	147	17	the	the	DET
cana-3453	147	18	submanifold	submanifold	NOUN
cana-3453	147	19	under	under	ADP
cana-3453	147	20	the	the	DET
cana-3453	147	21	second	second	ADJ
cana-3453	147	22	fundamental	fundamental	ADJ
cana-3453	147	23	map	map	NOUN
cana-3453	147	24	,	,	PUNCT
cana-3453	147	25	and	and	CCONJ
cana-3453	147	26	σ	σ	PROPN
cana-3453	147	27	is	be	AUX
cana-3453	147	28	intrinsic	intrinsic	ADJ
cana-3453	147	29	information	information	NOUN
cana-3453	147	30	by	by	ADP
cana-3453	147	31	how	how	SCONJ
cana-3453	147	32	the	the	DET
cana-3453	147	33	submanifold	submanifold	NOUN
cana-3453	147	34	curves	curve	VERB
cana-3453	147	35	inside	inside	ADP
cana-3453	147	36	the	the	DET
cana-3453	147	37	ambient	ambient	ADJ
cana-3453	147	38	space	space	NOUN
cana-3453	147	39	.	.	PUNCT
cana-3453	148	1	this	this	DET
cana-3453	148	2	relation	relation	NOUN
cana-3453	148	3	goes	go	VERB
cana-3453	148	4	deeper	deep	ADJ
cana-3453	148	5	and	and	CCONJ
cana-3453	148	6	presents	present	VERB
cana-3453	148	7	essential	essential	ADJ
cana-3453	148	8	geometric	geometric	ADJ
cana-3453	148	9	features	feature	NOUN
cana-3453	148	10	of	of	ADP
cana-3453	148	11	totally	totally	ADV
cana-3453	148	12	geodesic	geodesic	ADJ
cana-3453	148	13	submanifolds	submanifold	NOUN
cana-3453	148	14	beyond	beyond	ADP
cana-3453	148	15	their	their	PRON
cana-3453	148	16	simple	simple	ADJ
cana-3453	148	17	algebraic	algebraic	ADJ
cana-3453	148	18	properties	property	NOUN
cana-3453	148	19	.	.	PUNCT
cana-3453	149	1	theorem	theorem	VERB
cana-3453	149	2	4.1	4.1	NUM
cana-3453	149	3	.	.	PUNCT
cana-3453	150	1	let	let	VERB
cana-3453	150	2	m	m	PRON
cana-3453	150	3	be	be	AUX
cana-3453	150	4	a	a	DET
cana-3453	150	5	generalized	generalized	ADJ
cana-3453	150	6	sasakian	sasakian	ADJ
cana-3453	150	7	-	-	PUNCT
cana-3453	150	8	space	space	NOUN
cana-3453	150	9	-	-	PUNCT
cana-3453	150	10	form	form	NOUN
cana-3453	150	11	and	and	CCONJ
cana-3453	150	12	n	n	PRON
cana-3453	150	13	be	be	VERB
cana-3453	150	14	an	an	DET
cana-3453	150	15	invariant	invariant	ADJ
cana-3453	150	16	submanifold	submanifold	NOUN
cana-3453	150	17	in	in	ADP
cana-3453	150	18	m.	m.	NOUN
cana-3453	150	19	the	the	DET
cana-3453	150	20	submanifold	submanifold	NOUN
cana-3453	150	21	n	n	AUX
cana-3453	150	22	is	be	AUX
cana-3453	150	23	geodesic	geodesic	ADJ
cana-3453	150	24	in	in	ADP
cana-3453	150	25	m	m	PROPN
cana-3453	150	26	if	if	SCONJ
cana-3453	151	1	and	and	CCONJ
cana-3453	151	2	only	only	ADV
cana-3453	151	3	if	if	SCONJ
cana-3453	151	4	q(σ	q(σ	NUM
cana-3453	151	5	,	,	PUNCT
cana-3453	151	6	w2)=0	w2)=0	PROPN
cana-3453	151	7	,	,	PUNCT
cana-3453	151	8	with	with	ADP
cana-3453	151	9	the	the	DET
cana-3453	151	10	non	non	ADJ
cana-3453	151	11	-	-	ADJ
cana-3453	151	12	degeneracy	degeneracy	ADJ
cana-3453	151	13	condition	condition	NOUN
cana-3453	151	14	(	(	PUNCT
cana-3453	151	15	2n−1){3f2+(2n−1)f3}≠0	2n−1){3f2+(2n−1)f3}≠0	NOUN
cana-3453	151	16	.	.	PUNCT
cana-3453	151	17	proof	proof	NOUN
cana-3453	151	18	.	.	PUNCT
cana-3453	152	1	to	to	PART
cana-3453	152	2	handle	handle	VERB
cana-3453	152	3	this	this	PRON
cana-3453	152	4	systematically	systematically	ADV
cana-3453	152	5	,	,	PUNCT
cana-3453	152	6	recall	recall	VERB
cana-3453	152	7	that	that	SCONJ
cana-3453	152	8	we	we	PRON
cana-3453	152	9	should	should	AUX
cana-3453	152	10	consider	consider	VERB
cana-3453	152	11	invariant	invariant	ADJ
cana-3453	152	12	submanifolds	submanifold	NOUN
cana-3453	152	13	n	n	PRON
cana-3453	152	14	such	such	ADJ
cana-3453	152	15	that	that	SCONJ
cana-3453	152	16	q(σ	q(σ	ADJ
cana-3453	152	17	,	,	PUNCT
cana-3453	152	18	w2)=0	w2)=0	PROPN
cana-3453	152	19	.	.	PUNCT
cana-3453	153	1	this	this	PRON
cana-3453	153	2	is	be	AUX
cana-3453	153	3	equivalent	equivalent	ADJ
cana-3453	153	4	to	to	ADP
cana-3453	153	5	the	the	DET
cana-3453	153	6	condition	condition	NOUN
cana-3453	153	7	:	:	PUNCT
cana-3453	153	8	q(σ	q(σ	ADJ
cana-3453	153	9	,	,	PUNCT
cana-3453	153	10	w2)=q(σ	w2)=q(σ	PROPN
cana-3453	153	11	,	,	PUNCT
cana-3453	153	12	w2)(x	w2)(x	PROPN
cana-3453	153	13	,	,	PUNCT
cana-3453	153	14	y	y	PROPN
cana-3453	153	15	,	,	PUNCT
cana-3453	153	16	z;u	z;u	PROPN
cana-3453	153	17	,	,	PUNCT
cana-3453	153	18	v)=((u∧σv).w2)(x	v)=((u∧σv).w2)(x	NOUN
cana-3453	153	19	,	,	PUNCT
cana-3453	153	20	y)z=0	y)z=0	PROPN
cana-3453	153	21	this	this	PRON
cana-3453	153	22	expands	expand	VERB
cana-3453	153	23	to	to	ADP
cana-3453	153	24	:	:	PUNCT
cana-3453	153	25	−𝑾𝟐(𝑿	−𝑾𝟐(𝑿	NOUN
cana-3453	153	26	,	,	PUNCT
cana-3453	153	27	(	(	PUNCT
cana-3453	153	28	𝑼	𝑼	PROPN
cana-3453	153	29	∧𝝈	∧𝝈	NUM
cana-3453	153	30	𝑽)𝒀)𝒁	𝑽)𝒀)𝒁	VERB
cana-3453	153	31	−	−	PROPN
cana-3453	153	32	𝑾𝟐(𝑿	𝑾𝟐(𝑿	PROPN
cana-3453	153	33	,	,	PUNCT
cana-3453	153	34	𝒀)(𝑼	𝒀)(𝑼	ADP
cana-3453	153	35	∧𝝈	∧𝝈	NUM
cana-3453	153	36	𝑽)𝒁	𝑽)𝒁	NOUN
cana-3453	153	37	−	−	PROPN
cana-3453	153	38	𝐖𝟐(𝐗	𝐖𝟐(𝐗	PROPN
cana-3453	153	39	,	,	PUNCT
cana-3453	153	40	𝐘)(𝐔	𝐘)(𝐔	PUNCT
cana-3453	153	41	∧	∧	NOUN
cana-3453	153	42	𝛔𝐕)𝐙	𝛔𝐕)𝐙	NOUN
cana-3453	153	43	=	=	SYM
cana-3453	153	44	𝟎	𝟎	X
cana-3453	153	45	(	(	PUNCT
cana-3453	153	46	𝟒.	𝟒.	X
cana-3453	153	47	𝟏	𝟏	X
cana-3453	153	48	)	)	PUNCT
cana-3453	153	49	where	where	SCONJ
cana-3453	153	50	the	the	DET
cana-3453	153	51	operator	operator	NOUN
cana-3453	153	52	u∧σv	u∧σv	NOUN
cana-3453	153	53	is	be	AUX
cana-3453	153	54	defined	define	VERB
cana-3453	153	55	by	by	ADP
cana-3453	153	56	:	:	PUNCT
cana-3453	153	57	(	(	PUNCT
cana-3453	153	58	u∧σv)p	u∧σv)p	NOUN
cana-3453	153	59	=	=	ADJ
cana-3453	153	60	σ(v	σ(v	NOUN
cana-3453	153	61	,	,	PUNCT
cana-3453	153	62	p)u−σ(u	p)u−σ(u	NOUN
cana-3453	153	63	,	,	PUNCT
cana-3453	153	64	p)v	p)v	NOUN
cana-3453	153	65	(	(	PUNCT
cana-3453	153	66	4.2	4.2	NUM
cana-3453	153	67	)	)	PUNCT
cana-3453	153	68	substituting	substitute	VERB
cana-3453	153	69	(	(	PUNCT
cana-3453	153	70	4.2	4.2	NUM
cana-3453	153	71	)	)	PUNCT
cana-3453	153	72	into	into	ADP
cana-3453	153	73	(	(	PUNCT
cana-3453	153	74	4.1	4.1	NUM
cana-3453	153	75	)	)	PUNCT
cana-3453	153	76	yields	yield	NOUN
cana-3453	153	77	:	:	PUNCT
cana-3453	153	78	communications	communication	NOUN
cana-3453	153	79	on	on	ADP
cana-3453	153	80	applied	apply	VERB
cana-3453	153	81	nonlinear	nonlinear	ADJ
cana-3453	153	82	analysis	analysis	NOUN
cana-3453	153	83	issn	issn	NOUN
cana-3453	153	84	:	:	PUNCT
cana-3453	153	85	1074	1074	NUM
cana-3453	153	86	-	-	PUNCT
cana-3453	153	87	133x	133x	NUM
cana-3453	153	88	vol	vol	NOUN
cana-3453	153	89	32	32	NUM
cana-3453	153	90	no	no	NOUN
cana-3453	153	91	.	.	PUNCT
cana-3453	154	1	7s	7	NOUN
cana-3453	154	2	(	(	PUNCT
cana-3453	154	3	2025	2025	NUM
cana-3453	154	4	)	)	PUNCT
cana-3453	154	5	422	422	NUM
cana-3453	155	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3453	155	2	−𝝈(𝑽	−𝝈(𝑽	PROPN
cana-3453	155	3	,	,	PUNCT
cana-3453	155	4	𝑿)𝑾𝟐(𝑼	𝑿)𝑾𝟐(𝑼	NOUN
cana-3453	155	5	,	,	PUNCT
cana-3453	155	6	𝒀)𝒁	𝒀)𝒁	VERB
cana-3453	155	7	+	+	CCONJ
cana-3453	155	8	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	155	9	,	,	PUNCT
cana-3453	155	10	𝑿)𝑾𝟐(𝑽	𝑿)𝑾𝟐(𝑽	NOUN
cana-3453	155	11	,	,	PUNCT
cana-3453	155	12	𝒀)𝒁	𝒀)𝒁	VERB
cana-3453	155	13	−	−	PROPN
cana-3453	155	14	𝝈(𝑽	𝝈(𝑽	PROPN
cana-3453	155	15	,	,	PUNCT
cana-3453	155	16	𝒀)𝑾𝟐(𝑿	𝒀)𝑾𝟐(𝑿	NOUN
cana-3453	155	17	,	,	PUNCT
cana-3453	155	18	𝑼)𝒁	𝑼)𝒁	NOUN
cana-3453	155	19	+	+	CCONJ
cana-3453	155	20	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	155	21	,	,	PUNCT
cana-3453	155	22	𝒀)𝑾𝟐(𝑿	𝒀)𝑾𝟐(𝑿	NOUN
cana-3453	155	23	,	,	PUNCT
cana-3453	155	24	𝑽)𝒁	𝑽)𝒁	NOUN
cana-3453	155	25	−	−	PROPN
cana-3453	155	26	𝝈(𝑽	𝝈(𝑽	NOUN
cana-3453	155	27	,	,	PUNCT
cana-3453	155	28	𝒁)𝑾𝟐(𝑿	𝒁)𝑾𝟐(𝑿	NOUN
cana-3453	155	29	,	,	PUNCT
cana-3453	155	30	𝒀)𝑼	𝒀)𝑼	NOUN
cana-3453	155	31	+	+	CCONJ
cana-3453	155	32	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	155	33	,	,	PUNCT
cana-3453	155	34	𝒁)𝑾𝟐(𝑿	𝒁)𝑾𝟐(𝑿	NOUN
cana-3453	155	35	,	,	PUNCT
cana-3453	155	36	𝒀)𝑽	𝒀)𝑽	NOUN
cana-3453	155	37	=	=	SYM
cana-3453	155	38	𝟎.	𝟎.	X
cana-3453	155	39	(	(	PUNCT
cana-3453	155	40	4.3	4.3	NUM
cana-3453	155	41	)	)	PUNCT
cana-3453	155	42	using	use	VERB
cana-3453	155	43	the	the	DET
cana-3453	155	44	algebraic	algebraic	ADJ
cana-3453	155	45	expansion	expansion	NOUN
cana-3453	155	46	of	of	ADP
cana-3453	155	47	the	the	DET
cana-3453	155	48	tensor	tensor	NOUN
cana-3453	155	49	derivative	derivative	NOUN
cana-3453	155	50	,	,	PUNCT
cana-3453	155	51	we	we	PRON
cana-3453	155	52	get	get	VERB
cana-3453	155	53	multiple	multiple	ADJ
cana-3453	155	54	terms	term	NOUN
cana-3453	155	55	that	that	PRON
cana-3453	155	56	depend	depend	VERB
cana-3453	155	57	on	on	ADP
cana-3453	155	58	second	second	ADJ
cana-3453	155	59	fundamental	fundamental	ADJ
cana-3453	155	60	form	form	NOUN
cana-3453	155	61	and	and	CCONJ
cana-3453	155	62	components	component	NOUN
cana-3453	155	63	of	of	ADP
cana-3453	155	64	w₂	w₂	PROPN
cana-3453	155	65	curvature	curvature	NOUN
cana-3453	155	66	.	.	PUNCT
cana-3453	156	1	thus	thus	ADV
cana-3453	156	2	each	each	DET
cana-3453	156	3	term	term	NOUN
cana-3453	156	4	provides	provide	VERB
cana-3453	156	5	different	different	ADJ
cana-3453	156	6	geometric	geometric	ADJ
cana-3453	156	7	data	datum	NOUN
cana-3453	156	8	of	of	ADP
cana-3453	156	9	how	how	SCONJ
cana-3453	156	10	the	the	DET
cana-3453	156	11	submanifold	submanifold	NOUN
cana-3453	156	12	folds	fold	VERB
cana-3453	156	13	into	into	ADP
cana-3453	156	14	the	the	DET
cana-3453	156	15	ambient	ambient	ADJ
cana-3453	156	16	space	space	NOUN
cana-3453	156	17	.	.	PUNCT
cana-3453	157	1	a	a	DET
cana-3453	157	2	more	more	ADV
cana-3453	157	3	informative	informative	ADJ
cana-3453	157	4	case	case	NOUN
cana-3453	157	5	happens	happen	VERB
cana-3453	157	6	when	when	SCONJ
cana-3453	157	7	we	we	PRON
cana-3453	157	8	compute	compute	VERB
cana-3453	157	9	the	the	DET
cana-3453	157	10	inner	inner	ADJ
cana-3453	157	11	product	product	NOUN
cana-3453	157	12	between	between	ADP
cana-3453	157	13	σ	σ	PROPN
cana-3453	157	14	and	and	CCONJ
cana-3453	157	15	w₂	w₂	PROPN
cana-3453	157	16	along	along	ADP
cana-3453	157	17	the	the	DET
cana-3453	157	18	characteristic	characteristic	ADJ
cana-3453	157	19	direction	direction	NOUN
cana-3453	157	20	ξ	ξ	PROPN
cana-3453	157	21	.	.	PUNCT
cana-3453	158	1	when	when	SCONJ
cana-3453	158	2	we	we	PRON
cana-3453	158	3	set	set	VERB
cana-3453	158	4	z	z	NOUN
cana-3453	158	5	=	=	PROPN
cana-3453	158	6	v	v	NOUN
cana-3453	158	7	=	=	SYM
cana-3453	158	8	ξ	ξ	X
cana-3453	158	9	in	in	ADP
cana-3453	158	10	(	(	PUNCT
cana-3453	158	11	4.3	4.3	NUM
cana-3453	158	12	)	)	PUNCT
cana-3453	158	13	and	and	CCONJ
cana-3453	158	14	apply	apply	VERB
cana-3453	158	15	condition	condition	NOUN
cana-3453	158	16	(	(	PUNCT
cana-3453	158	17	3.3	3.3	NUM
cana-3453	158	18	)	)	PUNCT
cana-3453	158	19	,	,	PUNCT
cana-3453	158	20	we	we	PRON
cana-3453	158	21	obtain	obtain	VERB
cana-3453	158	22	:	:	PUNCT
cana-3453	158	23	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	158	24	,	,	PUNCT
cana-3453	158	25	𝑿)𝑾𝟐(𝝃	𝑿)𝑾𝟐(𝝃	NOUN
cana-3453	158	26	,	,	PUNCT
cana-3453	158	27	𝒀)𝝃	𝒀)𝝃	PUNCT
cana-3453	159	1	+	+	CCONJ
cana-3453	159	2	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	159	3	,	,	PUNCT
cana-3453	159	4	𝒀)𝑾𝟐(𝑿	𝒀)𝑾𝟐(𝑿	NOUN
cana-3453	159	5	,	,	PUNCT
cana-3453	159	6	𝝃)𝝃	𝝃)𝝃	X
cana-3453	159	7	=	=	SYM
cana-3453	159	8	𝟎	𝟎	X
cana-3453	159	9	(	(	PUNCT
cana-3453	159	10	4.4	4.4	NUM
cana-3453	159	11	)	)	PUNCT
cana-3453	159	12	applying	apply	VERB
cana-3453	159	13	equation	equation	NOUN
cana-3453	159	14	(	(	PUNCT
cana-3453	159	15	2.17	2.17	NUM
cana-3453	159	16	)	)	PUNCT
cana-3453	159	17	to	to	ADP
cana-3453	159	18	(	(	PUNCT
cana-3453	159	19	4.4	4.4	NUM
cana-3453	159	20	):	):	PUNCT
cana-3453	159	21	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	159	22	,	,	PUNCT
cana-3453	159	23	𝑿)[(𝒇𝟏	𝑿)[(𝒇𝟏	VERB
cana-3453	159	24	−	−	PROPN
cana-3453	159	25	𝒇𝟑){𝜼(𝒀)𝝃	𝒇𝟑){𝜼(𝒀)𝝃	PUNCT
cana-3453	159	26	−	−	PROPN
cana-3453	159	27	𝒀	𝒀	NOUN
cana-3453	159	28	}	}	PUNCT
cana-3453	159	29	+	+	CCONJ
cana-3453	159	30	𝟏	𝟏	NUM
cana-3453	159	31	𝟐𝒏	𝟐𝒏	NOUN
cana-3453	159	32	{	{	PUNCT
cana-3453	159	33	𝑸𝒀	𝑸𝒀	PROPN
cana-3453	159	34	−	−	PROPN
cana-3453	159	35	𝟐𝒏(𝒇𝟏	𝟐𝒏(𝒇𝟏	ADJ
cana-3453	159	36	−	−	PROPN
cana-3453	159	37	𝒇𝟑)𝜼(𝒀)𝝃	𝒇𝟑)𝜼(𝒀)𝝃	NUM
cana-3453	159	38	}	}	PUNCT
cana-3453	159	39	]	]	PUNCT
cana-3453	160	1	+	+	CCONJ
cana-3453	160	2	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	160	3	,	,	PUNCT
cana-3453	160	4	𝒀)[(𝒇𝟏	𝒀)[(𝒇𝟏	VERB
cana-3453	160	5	−	−	NOUN
cana-3453	160	6	𝒇𝟑){𝑿	𝒇𝟑){𝑿	NOUN
cana-3453	160	7	−	−	PROPN
cana-3453	160	8	𝜼(𝑿)𝝃	𝜼(𝑿)𝝃	CCONJ
cana-3453	160	9	}	}	PUNCT
cana-3453	160	10	+	+	CCONJ
cana-3453	160	11	𝟏	𝟏	NUM
cana-3453	160	12	𝟐𝒏	𝟐𝒏	NOUN
cana-3453	160	13	{	{	PUNCT
cana-3453	160	14	𝟐𝒏(𝒇𝟏	𝟐𝒏(𝒇𝟏	ADJ
cana-3453	160	15	−	−	PUNCT
cana-3453	160	16	𝒇𝟑)𝜼(𝑿)𝝃	𝒇𝟑)𝜼(𝑿)𝝃	NOUN
cana-3453	160	17	−	−	PROPN
cana-3453	160	18	𝑸𝑿	𝑸𝑿	PROPN
cana-3453	160	19	}	}	PUNCT
cana-3453	160	20	]	]	PUNCT
cana-3453	160	21	=	=	SYM
cana-3453	160	22	𝟎	𝟎	X
cana-3453	160	23	(	(	PUNCT
cana-3453	160	24	4.5	4.5	NUM
cana-3453	160	25	)	)	PUNCT
cana-3453	160	26	it	it	PRON
cana-3453	160	27	is	be	AUX
cana-3453	160	28	geometrically	geometrically	ADV
cana-3453	160	29	remarkable	remarkable	ADJ
cana-3453	160	30	that	that	SCONJ
cana-3453	160	31	these	these	DET
cana-3453	160	32	equations	equation	NOUN
cana-3453	160	33	involve	involve	VERB
cana-3453	160	34	mixed	mixed	ADJ
cana-3453	160	35	terms	term	NOUN
cana-3453	160	36	of	of	ADP
cana-3453	160	37	both	both	CCONJ
cana-3453	160	38	the	the	DET
cana-3453	160	39	metric	metric	ADJ
cana-3453	160	40	tensor	tensor	NOUN
cana-3453	160	41	g	g	NOUN
cana-3453	160	42	and	and	CCONJ
cana-3453	160	43	the	the	DET
cana-3453	160	44	structure	structure	NOUN
cana-3453	160	45	tensor	tensor	NOUN
cana-3453	160	46	φ	φ	PROPN
cana-3453	160	47	.	.	PUNCT
cana-3453	161	1	when	when	SCONJ
cana-3453	161	2	we	we	PRON
cana-3453	161	3	turn	turn	VERB
cana-3453	161	4	to	to	ADP
cana-3453	161	5	the	the	DET
cana-3453	161	6	generalized	generalize	VERB
cana-3453	161	7	sasakian	sasakian	ADJ
cana-3453	161	8	-	-	PUNCT
cana-3453	161	9	space	space	NOUN
cana-3453	161	10	-	-	PUNCT
cana-3453	161	11	forms	form	NOUN
cana-3453	161	12	for	for	ADP
cana-3453	161	13	which	which	PRON
cana-3453	161	14	we	we	PRON
cana-3453	161	15	can	can	AUX
cana-3453	161	16	form	form	VERB
cana-3453	161	17	a	a	DET
cana-3453	161	18	metric	metric	NOUN
cana-3453	161	19	and	and	CCONJ
cana-3453	161	20	a	a	DET
cana-3453	161	21	contact	contact	NOUN
cana-3453	161	22	structure	structure	NOUN
cana-3453	161	23	,	,	PUNCT
cana-3453	161	24	these	these	DET
cana-3453	161	25	combinations	combination	NOUN
cana-3453	161	26	are	be	AUX
cana-3453	161	27	a	a	DET
cana-3453	161	28	reflection	reflection	NOUN
cana-3453	161	29	of	of	ADP
cana-3453	161	30	that	that	DET
cana-3453	161	31	dual	dual	ADJ
cana-3453	161	32	structure	structure	NOUN
cana-3453	161	33	.	.	PUNCT
cana-3453	162	1	these	these	DET
cana-3453	162	2	two	two	NUM
cana-3453	162	3	structures	structure	NOUN
cana-3453	162	4	interact	interact	VERB
cana-3453	162	5	through	through	ADP
cana-3453	162	6	the	the	DET
cana-3453	162	7	second	second	ADJ
cana-3453	162	8	fundamental	fundamental	ADJ
cana-3453	162	9	form	form	NOUN
cana-3453	162	10	,	,	PUNCT
cana-3453	162	11	governed	govern	VERB
cana-3453	162	12	projectively	projectively	ADV
cana-3453	162	13	by	by	ADP
cana-3453	162	14	the	the	DET
cana-3453	162	15	wi	wi	PROPN
cana-3453	162	16	tensors	tensor	NOUN
cana-3453	162	17	and	and	CCONJ
cana-3453	162	18	thus	thus	ADV
cana-3453	162	19	intertwining	intertwine	VERB
cana-3453	162	20	within	within	ADP
cana-3453	162	21	the	the	DET
cana-3453	162	22	ambient	ambient	ADJ
cana-3453	162	23	geometry	geometry	NOUN
cana-3453	162	24	.	.	PUNCT
cana-3453	163	1	above	above	ADV
cana-3453	163	2	,	,	PUNCT
cana-3453	163	3	each	each	PRON
cana-3453	163	4	of	of	ADP
cana-3453	163	5	the	the	DET
cana-3453	163	6	terms	term	NOUN
cana-3453	163	7	in	in	ADP
cana-3453	163	8	these	these	DET
cana-3453	163	9	equations	equation	NOUN
cana-3453	163	10	have	have	VERB
cana-3453	163	11	some	some	DET
cana-3453	163	12	specific	specific	ADJ
cana-3453	163	13	geometric	geometric	ADJ
cana-3453	163	14	meaning	meaning	NOUN
cana-3453	163	15	the	the	DET
cana-3453	163	16	metric	metric	ADJ
cana-3453	163	17	terms	term	NOUN
cana-3453	163	18	are	be	AUX
cana-3453	163	19	about	about	ADP
cana-3453	163	20	how	how	SCONJ
cana-3453	163	21	the	the	DET
cana-3453	163	22	structure	structure	NOUN
cana-3453	163	23	preserves	preserve	VERB
cana-3453	163	24	distances	distance	NOUN
cana-3453	163	25	,	,	PUNCT
cana-3453	163	26	whereas	whereas	SCONJ
cana-3453	163	27	the	the	DET
cana-3453	163	28	φ	φ	VERB
cana-3453	163	29	-	-	PUNCT
cana-3453	163	30	dependent	dependent	ADJ
cana-3453	163	31	terms	term	NOUN
cana-3453	163	32	store	store	VERB
cana-3453	163	33	information	information	NOUN
cana-3453	163	34	about	about	ADP
cana-3453	163	35	how	how	SCONJ
cana-3453	163	36	the	the	DET
cana-3453	163	37	contact	contact	NOUN
cana-3453	163	38	structure	structure	NOUN
cana-3453	163	39	twists	twist	VERB
cana-3453	163	40	the	the	DET
cana-3453	163	41	embedding	embed	VERB
cana-3453	163	42	.	.	PUNCT
cana-3453	164	1	taking	take	VERB
cana-3453	164	2	the	the	DET
cana-3453	164	3	inner	inner	ADJ
cana-3453	164	4	product	product	NOUN
cana-3453	164	5	with	with	ADP
cana-3453	164	6	w	w	PROPN
cana-3453	164	7	:	:	PUNCT
cana-3453	164	8	𝝈(𝑼	𝝈(𝑼	PROPN
cana-3453	164	9	,	,	PUNCT
cana-3453	164	10	𝑿	𝑿	PROPN
cana-3453	164	11	)	)	PUNCT
cana-3453	165	1	[	[	X
cana-3453	165	2	(	(	PUNCT
cana-3453	165	3	𝒇𝟏	𝒇𝟏	NOUN
cana-3453	165	4	−	−	PROPN
cana-3453	165	5	𝒇𝟑){𝜼(𝒀)𝜼(𝑾	𝒇𝟑){𝜼(𝒀)𝜼(𝑾	NOUN
cana-3453	165	6	)	)	PUNCT
cana-3453	165	7	−	−	NOUN
cana-3453	165	8	𝒈(𝒀	𝒈(𝒀	NOUN
cana-3453	165	9	,	,	PUNCT
cana-3453	165	10	𝑾	𝑾	ADJ
cana-3453	165	11	)	)	PUNCT
cana-3453	165	12	}	}	PUNCT
cana-3453	166	1	+	+	CCONJ
cana-3453	166	2	𝟏	𝟏	NUM
cana-3453	166	3	𝟐𝒏	𝟐𝒏	NOUN
cana-3453	166	4	{	{	PUNCT
cana-3453	166	5	𝒈(𝑸𝒀	𝒈(𝑸𝒀	NOUN
cana-3453	166	6	,	,	PUNCT
cana-3453	166	7	𝑾	𝑾	ADJ
cana-3453	166	8	)	)	PUNCT
cana-3453	166	9	−	−	PROPN
cana-3453	166	10	𝟐𝒏(𝒇𝟏	𝟐𝒏(𝒇𝟏	ADJ
cana-3453	167	1	−	−	PROPN
cana-3453	167	2	𝒇𝟑)𝜼(𝒀)𝜼(𝑾	𝒇𝟑)𝜼(𝒀)𝜼(𝑾	NUM
cana-3453	167	3	)	)	PUNCT
cana-3453	167	4	}	}	PUNCT
cana-3453	167	5	]	]	PUNCT
cana-3453	168	1	+	+	CCONJ
cana-3453	168	2	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	168	3	,	,	PUNCT
cana-3453	168	4	𝒀	𝒀	NOUN
cana-3453	168	5	)	)	PUNCT
cana-3453	169	1	[	[	X
cana-3453	169	2	(	(	PUNCT
cana-3453	169	3	𝒇𝟏	𝒇𝟏	NOUN
cana-3453	169	4	−	−	PROPN
cana-3453	169	5	𝒇𝟑){𝒈(𝑿	𝒇𝟑){𝒈(𝑿	NUM
cana-3453	169	6	,	,	PUNCT
cana-3453	169	7	𝑾	𝑾	ADJ
cana-3453	169	8	)	)	PUNCT
cana-3453	169	9	−	−	PROPN
cana-3453	169	10	𝜼(𝑿)𝜼(𝑾	𝜼(𝑿)𝜼(𝑾	NOUN
cana-3453	169	11	)	)	PUNCT
cana-3453	169	12	}	}	PUNCT
cana-3453	170	1	+	+	CCONJ
cana-3453	170	2	𝟏	𝟏	NUM
cana-3453	170	3	𝟐𝒏	𝟐𝒏	NOUN
cana-3453	170	4	{	{	PUNCT
cana-3453	170	5	𝟐𝒏(𝒇𝟏	𝟐𝒏(𝒇𝟏	ADJ
cana-3453	170	6	−	−	PROPN
cana-3453	170	7	𝒇𝟑)𝜼(𝑿)𝜼(𝑾	𝒇𝟑)𝜼(𝑿)𝜼(𝑾	PROPN
cana-3453	170	8	)	)	PUNCT
cana-3453	170	9	−	−	ADP
cana-3453	170	10	𝒈(𝑸𝑿	𝒈(𝑸𝑿	PROPN
cana-3453	170	11	,	,	PUNCT
cana-3453	170	12	𝑾	𝑾	ADJ
cana-3453	170	13	)	)	PUNCT
cana-3453	170	14	}	}	PUNCT
cana-3453	170	15	]	]	PUNCT
cana-3453	170	16	=	=	SYM
cana-3453	170	17	𝟎	𝟎	X
cana-3453	170	18	(	(	PUNCT
cana-3453	170	19	4.6)0	4.6)0	NUM
cana-3453	170	20	contracting	contract	VERB
cana-3453	170	21	y	y	PROPN
cana-3453	170	22	and	and	CCONJ
cana-3453	170	23	w	w	PROPN
cana-3453	170	24	,	,	PUNCT
cana-3453	170	25	we	we	PRON
cana-3453	170	26	get	get	VERB
cana-3453	170	27	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	170	28	,	,	PUNCT
cana-3453	170	29	𝑿)(𝟐𝒏	𝑿)(𝟐𝒏	ADJ
cana-3453	170	30	−	−	NOUN
cana-3453	170	31	𝟏){𝟑𝒇𝟐	𝟏){𝟑𝒇𝟐	NOUN
cana-3453	170	32	+	+	CCONJ
cana-3453	170	33	(	(	PUNCT
cana-3453	170	34	𝟐𝒏	𝟐𝒏	PROPN
cana-3453	170	35	−	−	PROPN
cana-3453	170	36	𝟏)𝒇𝟑	𝟏)𝒇𝟑	NOUN
cana-3453	170	37	}	}	PUNCT
cana-3453	170	38	=	=	SYM
cana-3453	170	39	𝟎	𝟎	X
cana-3453	170	40	(	(	PUNCT
cana-3453	170	41	4.7	4.7	NUM
cana-3453	170	42	)	)	PUNCT
cana-3453	170	43	given	give	VERB
cana-3453	170	44	our	our	PRON
cana-3453	170	45	non	non	ADJ
cana-3453	170	46	-	-	ADJ
cana-3453	170	47	degeneracy	degeneracy	ADJ
cana-3453	170	48	condition	condition	NOUN
cana-3453	170	49	(	(	PUNCT
cana-3453	170	50	2𝑛	2𝑛	PROPN
cana-3453	170	51	−	−	PROPN
cana-3453	170	52	1){3𝑓2	1){3𝑓2	PROPN
cana-3453	170	53	+	+	CCONJ
cana-3453	170	54	(	(	PUNCT
cana-3453	170	55	2𝑛	2𝑛	PROPN
cana-3453	170	56	−	−	PROPN
cana-3453	170	57	1)𝑓3	1)𝑓3	PROPN
cana-3453	170	58	}	}	PUNCT
cana-3453	170	59	≠	≠	PROPN
cana-3453	170	60	0	0	NUM
cana-3453	170	61	.	.	PUNCT
cana-3453	171	1	we	we	PRON
cana-3453	171	2	conclude	conclude	VERB
cana-3453	171	3	:	:	PUNCT
cana-3453	171	4	σ(u	σ(u	NOUN
cana-3453	171	5	,	,	PUNCT
cana-3453	171	6	x)=0	x)=0	PROPN
cana-3453	171	7	it	it	PRON
cana-3453	171	8	follows	follow	VERB
cana-3453	171	9	that	that	SCONJ
cana-3453	171	10	n	n	PRON
cana-3453	171	11	is	be	AUX
cana-3453	171	12	geodesic	geodesic	ADJ
cana-3453	171	13	as	as	ADP
cana-3453	171	14	a	a	DET
cana-3453	171	15	submanifold	submanifold	NOUN
cana-3453	171	16	.	.	PUNCT
cana-3453	172	1	the	the	DET
cana-3453	172	2	converse	converse	NOUN
cana-3453	172	3	follows	follow	VERB
cana-3453	172	4	immediately	immediately	ADV
cana-3453	172	5	because	because	SCONJ
cana-3453	172	6	a	a	DET
cana-3453	172	7	geodesic	geodesic	NOUN
cana-3453	172	8	submanifold	submanifold	NOUN
cana-3453	172	9	trivially	trivially	ADV
cana-3453	172	10	satisfies	satisfy	VERB
cana-3453	172	11	q(σ	q(σ	NUM
cana-3453	172	12	,	,	PUNCT
cana-3453	172	13	w2)=0	w2)=0	PROPN
cana-3453	172	14	.	.	PUNCT
cana-3453	173	1	this	this	DET
cana-3453	173	2	theorem	theorem	NOUN
cana-3453	173	3	gives	give	VERB
cana-3453	173	4	us	we	PRON
cana-3453	173	5	a	a	DET
cana-3453	173	6	very	very	ADV
cana-3453	173	7	clear	clear	ADJ
cana-3453	173	8	characterization	characterization	NOUN
cana-3453	173	9	of	of	ADP
cana-3453	173	10	totally	totally	ADV
cana-3453	173	11	geodesic	geodesic	ADJ
cana-3453	173	12	invariant	invariant	ADJ
cana-3453	173	13	submanifolds	submanifold	NOUN
cana-3453	173	14	utilizing	utilize	VERB
cana-3453	173	15	the	the	DET
cana-3453	173	16	behaviour	behaviour	NOUN
cana-3453	173	17	of	of	ADP
cana-3453	173	18	their	their	PRON
cana-3453	173	19	second	second	ADJ
cana-3453	173	20	fundamental	fundamental	ADJ
cana-3453	173	21	form	form	NOUN
cana-3453	173	22	concerning	concern	VERB
cana-3453	173	23	the	the	DET
cana-3453	173	24	w2	w2	NOUN
cana-3453	173	25	curvature	curvature	NOUN
cana-3453	173	26	tensor	tensor	NOUN
cana-3453	173	27	.	.	PUNCT
cana-3453	174	1	the	the	DET
cana-3453	174	2	appearance	appearance	NOUN
cana-3453	174	3	of	of	ADP
cana-3453	174	4	the	the	DET
cana-3453	174	5	non	non	ADJ
cana-3453	174	6	-	-	ADJ
cana-3453	174	7	degeneracy	degeneracy	ADJ
cana-3453	174	8	condition	condition	NOUN
cana-3453	174	9	(	(	PUNCT
cana-3453	174	10	2n	2n	NUM
cana-3453	174	11	−	−	PROPN
cana-3453	174	12	1){3f₂	1){3f₂	PROPN
cana-3453	174	13	+	+	CCONJ
cana-3453	174	14	(	(	PUNCT
cana-3453	174	15	2n	2n	NUM
cana-3453	174	16	−	−	NOUN
cana-3453	174	17	1)f₃	1)f₃	NUM
cana-3453	174	18	}	}	PUNCT
cana-3453	174	19	≠	≠	PROPN
cana-3453	174	20	0	0	NUM
cana-3453	174	21	encodes	encode	NOUN
cana-3453	174	22	important	important	ADJ
cana-3453	174	23	geometric	geometric	ADJ
cana-3453	174	24	properties	property	NOUN
cana-3453	174	25	of	of	ADP
cana-3453	174	26	the	the	DET
cana-3453	174	27	ambient	ambient	ADJ
cana-3453	174	28	space	space	NOUN
cana-3453	174	29	.	.	PUNCT
cana-3453	175	1	this	this	DET
cana-3453	175	2	condition	condition	NOUN
cana-3453	175	3	guarantees	guarantee	VERB
cana-3453	175	4	that	that	SCONJ
cana-3453	175	5	the	the	DET
cana-3453	175	6	correlation	correlation	NOUN
cana-3453	175	7	between	between	ADP
cana-3453	175	8	w2	w2	NOUN
cana-3453	175	9	and	and	CCONJ
cana-3453	175	10	σ	σ	PROPN
cana-3453	175	11	encodes	encode	NOUN
cana-3453	175	12	significant	significant	ADJ
cana-3453	175	13	geometric	geometric	ADJ
cana-3453	175	14	information	information	NOUN
cana-3453	175	15	and	and	CCONJ
cana-3453	175	16	does	do	AUX
cana-3453	175	17	not	not	PART
cana-3453	175	18	collapse	collapse	VERB
cana-3453	175	19	into	into	ADP
cana-3453	175	20	degenerate	degenerate	ADJ
cana-3453	175	21	cases	case	NOUN
cana-3453	175	22	.	.	PUNCT
cana-3453	176	1	therefore	therefore	ADV
cana-3453	176	2	,	,	PUNCT
cana-3453	176	3	this	this	DET
cana-3453	176	4	condition	condition	NOUN
cana-3453	176	5	with	with	ADP
cana-3453	176	6	the	the	DET
cana-3453	176	7	f₂	f₂	NOUN
cana-3453	176	8	,	,	PUNCT
cana-3453	176	9	f₃	f₃	ADJ
cana-3453	176	10	pair	pair	NOUN
cana-3453	176	11	is	be	AUX
cana-3453	176	12	also	also	ADV
cana-3453	176	13	elementary	elementary	ADJ
cana-3453	176	14	regarding	regard	VERB
cana-3453	176	15	the	the	DET
cana-3453	176	16	contact	contact	NOUN
cana-3453	176	17	structure	structure	NOUN
cana-3453	176	18	.	.	PUNCT
cana-3453	177	1	communications	communication	NOUN
cana-3453	177	2	on	on	ADP
cana-3453	177	3	applied	apply	VERB
cana-3453	177	4	nonlinear	nonlinear	ADJ
cana-3453	177	5	analysis	analysis	NOUN
cana-3453	177	6	issn	issn	NOUN
cana-3453	177	7	:	:	PUNCT
cana-3453	177	8	1074	1074	NUM
cana-3453	177	9	-	-	PUNCT
cana-3453	177	10	133x	133x	NUM
cana-3453	177	11	vol	vol	NOUN
cana-3453	177	12	32	32	NUM
cana-3453	177	13	no	no	NOUN
cana-3453	177	14	.	.	PUNCT
cana-3453	178	1	7s	7	NOUN
cana-3453	178	2	(	(	PUNCT
cana-3453	178	3	2025	2025	NUM
cana-3453	178	4	)	)	PUNCT
cana-3453	178	5	423	423	NUM
cana-3453	178	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3453	178	7	5	5	NUM
cana-3453	178	8	.	.	PUNCT
cana-3453	178	9	characterization	characterization	NOUN
cana-3453	178	10	of	of	ADP
cana-3453	178	11	invariant	invariant	ADJ
cana-3453	178	12	submanifolds	submanifold	NOUN
cana-3453	178	13	through	through	ADP
cana-3453	178	14	w3	w3	PROPN
cana-3453	178	15	curvature	curvature	NOUN
cana-3453	178	16	interaction	interaction	NOUN
cana-3453	178	17	here	here	ADV
cana-3453	178	18	,	,	PUNCT
cana-3453	178	19	we	we	PRON
cana-3453	178	20	will	will	AUX
cana-3453	178	21	see	see	VERB
cana-3453	178	22	how	how	SCONJ
cana-3453	178	23	the	the	DET
cana-3453	178	24	geometry	geometry	NOUN
cana-3453	178	25	of	of	ADP
cana-3453	178	26	invariant	invariant	ADJ
cana-3453	178	27	submanifolds	submanifold	NOUN
cana-3453	178	28	and	and	CCONJ
cana-3453	178	29	the	the	DET
cana-3453	178	30	second	second	ADJ
cana-3453	178	31	fundamental	fundamental	ADJ
cana-3453	178	32	form	form	NOUN
cana-3453	178	33	encode	encode	ADJ
cana-3453	178	34	information	information	NOUN
cana-3453	178	35	about	about	ADP
cana-3453	178	36	the	the	DET
cana-3453	178	37	w3	w3	PROPN
cana-3453	178	38	curvature	curvature	PROPN
cana-3453	178	39	tensor	tensor	NOUN
cana-3453	178	40	.	.	PUNCT
cana-3453	179	1	we	we	PRON
cana-3453	179	2	give	give	VERB
cana-3453	179	3	conditions	condition	NOUN
cana-3453	179	4	under	under	ADP
cana-3453	179	5	which	which	PRON
cana-3453	179	6	such	such	ADJ
cana-3453	179	7	submanifolds	submanifold	NOUN
cana-3453	179	8	are	be	AUX
cana-3453	179	9	geodesic	geodesic	ADJ
cana-3453	179	10	.	.	PUNCT
cana-3453	180	1	specifically	specifically	ADV
cana-3453	180	2	,	,	PUNCT
cana-3453	180	3	the	the	DET
cana-3453	180	4	w₃	w₃	PROPN
cana-3453	180	5	curvature	curvature	NOUN
cana-3453	180	6	tensor	tensor	NOUN
cana-3453	180	7	has	have	VERB
cana-3453	180	8	some	some	DET
cana-3453	180	9	unique	unique	ADJ
cana-3453	180	10	properties	property	NOUN
cana-3453	180	11	different	different	ADJ
cana-3453	180	12	from	from	ADP
cana-3453	180	13	other	other	ADJ
cana-3453	180	14	generalized	generalized	ADJ
cana-3453	180	15	curvature	curvature	NOUN
cana-3453	180	16	tensors	tensor	NOUN
cana-3453	180	17	in	in	ADP
cana-3453	180	18	contact	contact	NOUN
cana-3453	180	19	metric	metric	ADJ
cana-3453	180	20	geometry	geometry	NOUN
cana-3453	180	21	.	.	PUNCT
cana-3453	181	1	w₃	w₃	NOUN
cana-3453	181	2	derives	derive	VERB
cana-3453	181	3	from	from	ADP
cana-3453	181	4	the	the	DET
cana-3453	181	5	contact	contact	NOUN
cana-3453	181	6	structure	structure	NOUN
cana-3453	181	7	,	,	PUNCT
cana-3453	181	8	a	a	DET
cana-3453	181	9	type	type	NOUN
cana-3453	181	10	of	of	ADP
cana-3453	181	11	riemannian	riemannian	ADJ
cana-3453	181	12	structure	structure	NOUN
cana-3453	181	13	lacking	lack	VERB
cana-3453	181	14	local	local	ADJ
cana-3453	181	15	equivalences	equivalence	NOUN
cana-3453	181	16	to	to	ADP
cana-3453	181	17	basic	basic	ADJ
cana-3453	181	18	geometric	geometric	ADJ
cana-3453	181	19	invariants	invariant	NOUN
cana-3453	181	20	such	such	ADJ
cana-3453	181	21	as	as	ADP
cana-3453	181	22	the	the	DET
cana-3453	181	23	riemann	riemann	PROPN
cana-3453	181	24	curvature	curvature	PROPN
cana-3453	181	25	tensor	tensor	NOUN
cana-3453	181	26	.	.	PUNCT
cana-3453	182	1	the	the	DET
cana-3453	182	2	way	way	NOUN
cana-3453	182	3	this	this	DET
cana-3453	182	4	tensor	tensor	NOUN
cana-3453	182	5	acts	act	VERB
cana-3453	182	6	along	along	ADP
cana-3453	182	7	the	the	DET
cana-3453	182	8	characteristic	characteristic	ADJ
cana-3453	182	9	direction	direction	NOUN
cana-3453	182	10	ξ	ξ	PROPN
cana-3453	182	11	encodes	encode	NOUN
cana-3453	182	12	essential	essential	ADJ
cana-3453	182	13	contact	contact	NOUN
cana-3453	182	14	geometric	geometric	ADJ
cana-3453	182	15	information	information	NOUN
cana-3453	182	16	about	about	ADP
cana-3453	182	17	the	the	DET
cana-3453	182	18	manifold	manifold	NOUN
cana-3453	182	19	.	.	PUNCT
cana-3453	183	1	these	these	DET
cana-3453	183	2	results	result	NOUN
cana-3453	183	3	hold	hold	VERB
cana-3453	183	4	important	important	ADJ
cana-3453	183	5	geometric	geometric	ADJ
cana-3453	183	6	significance	significance	NOUN
cana-3453	183	7	in	in	ADP
cana-3453	183	8	understanding	understand	VERB
cana-3453	183	9	how	how	SCONJ
cana-3453	183	10	the	the	DET
cana-3453	183	11	contact	contact	NOUN
cana-3453	183	12	structure	structure	NOUN
cana-3453	183	13	constrains	constrain	VERB
cana-3453	183	14	the	the	DET
cana-3453	183	15	possible	possible	ADJ
cana-3453	183	16	configurations	configuration	NOUN
cana-3453	183	17	of	of	ADP
cana-3453	183	18	totally	totally	ADV
cana-3453	183	19	geodesic	geodesic	ADJ
cana-3453	183	20	submanifolds	submanifold	NOUN
cana-3453	183	21	through	through	ADP
cana-3453	183	22	the	the	DET
cana-3453	183	23	interactions	interaction	NOUN
cana-3453	183	24	between	between	ADP
cana-3453	183	25	w₃	w₃	NOUN
cana-3453	183	26	and	and	CCONJ
cana-3453	183	27	invariant	invariant	ADJ
cana-3453	183	28	submanifolds	submanifold	NOUN
cana-3453	183	29	.	.	PUNCT
cana-3453	184	1	this	this	PRON
cana-3453	184	2	is	be	AUX
cana-3453	184	3	not	not	PART
cana-3453	184	4	just	just	ADV
cana-3453	184	5	algebraically	algebraically	ADV
cana-3453	184	6	compatible	compatible	ADJ
cana-3453	184	7	;	;	PUNCT
cana-3453	184	8	the	the	DET
cana-3453	184	9	relation	relation	NOUN
cana-3453	184	10	of	of	ADP
cana-3453	184	11	w₃	w₃	NOUN
cana-3453	184	12	to	to	ADP
cana-3453	184	13	the	the	DET
cana-3453	184	14	second	second	ADJ
cana-3453	184	15	fundamental	fundamental	ADJ
cana-3453	184	16	form	form	NOUN
cana-3453	184	17	has	have	VERB
cana-3453	184	18	more	more	ADJ
cana-3453	184	19	geometric	geometric	ADJ
cana-3453	184	20	consequences	consequence	NOUN
cana-3453	184	21	,	,	PUNCT
cana-3453	184	22	namely	namely	ADV
cana-3453	184	23	,	,	PUNCT
cana-3453	184	24	concerning	concern	VERB
cana-3453	184	25	how	how	SCONJ
cana-3453	184	26	the	the	DET
cana-3453	184	27	position	position	NOUN
cana-3453	184	28	of	of	ADP
cana-3453	184	29	the	the	DET
cana-3453	184	30	submanifold	submanifold	NOUN
cana-3453	184	31	is	be	AUX
cana-3453	184	32	related	relate	VERB
cana-3453	184	33	to	to	ADP
cana-3453	184	34	the	the	DET
cana-3453	184	35	contact	contact	NOUN
cana-3453	184	36	structure	structure	NOUN
cana-3453	184	37	of	of	ADP
cana-3453	184	38	the	the	DET
cana-3453	184	39	ambient	ambient	ADJ
cana-3453	184	40	space	space	NOUN
cana-3453	184	41	.	.	PUNCT
cana-3453	185	1	the	the	DET
cana-3453	185	2	w₃	w₃	PROPN
cana-3453	185	3	curvature	curvature	NOUN
cana-3453	185	4	tensor	tensor	NOUN
cana-3453	185	5	gives	give	VERB
cana-3453	185	6	an	an	DET
cana-3453	185	7	alternative	alternative	ADJ
cana-3453	185	8	viewpoint	viewpoint	NOUN
cana-3453	185	9	on	on	ADP
cana-3453	185	10	totally	totally	ADV
cana-3453	185	11	geodesic	geodesic	ADJ
cana-3453	185	12	submanifolds	submanifold	NOUN
cana-3453	185	13	via	via	ADP
cana-3453	185	14	its	its	PRON
cana-3453	185	15	special	special	ADJ
cana-3453	185	16	action	action	NOUN
cana-3453	185	17	on	on	ADP
cana-3453	185	18	the	the	DET
cana-3453	185	19	contact	contact	NOUN
cana-3453	185	20	structure	structure	NOUN
cana-3453	185	21	.	.	PUNCT
cana-3453	186	1	in	in	ADP
cana-3453	186	2	contrast	contrast	NOUN
cana-3453	186	3	to	to	ADP
cana-3453	186	4	w₂	w₂	PROPN
cana-3453	186	5	,	,	PUNCT
cana-3453	186	6	this	this	DET
cana-3453	186	7	tensor	tensor	NOUN
cana-3453	186	8	includes	include	VERB
cana-3453	186	9	terms	term	NOUN
cana-3453	186	10	involving	involve	VERB
cana-3453	186	11	generalizations	generalization	NOUN
cana-3453	186	12	of	of	ADP
cana-3453	186	13	how	how	SCONJ
cana-3453	186	14	the	the	DET
cana-3453	186	15	submanifold	submanifold	NOUN
cana-3453	186	16	sits	sit	VERB
cana-3453	186	17	in	in	ADP
cana-3453	186	18	the	the	DET
cana-3453	186	19	ambient	ambient	ADJ
cana-3453	186	20	space	space	NOUN
cana-3453	186	21	geometry	geometry	NOUN
cana-3453	186	22	.	.	PUNCT
cana-3453	187	1	theorem	theorem	VERB
cana-3453	187	2	5.1	5.1	NUM
cana-3453	187	3	.	.	PUNCT
cana-3453	188	1	let	let	VERB
cana-3453	188	2	n	n	PRON
cana-3453	188	3	be	be	AUX
cana-3453	188	4	an	an	DET
cana-3453	188	5	invariant	invariant	ADJ
cana-3453	188	6	submanifold	submanifold	NOUN
cana-3453	188	7	of	of	ADP
cana-3453	188	8	a	a	DET
cana-3453	188	9	generalized	generalized	ADJ
cana-3453	188	10	sasakian	sasakian	ADJ
cana-3453	188	11	-	-	PUNCT
cana-3453	188	12	space	space	NOUN
cana-3453	188	13	-	-	PUNCT
cana-3453	188	14	form	form	NOUN
cana-3453	188	15	m.	m.	NOUN
cana-3453	188	16	the	the	DET
cana-3453	188	17	submanifold	submanifold	NOUN
cana-3453	188	18	n	n	PRON
cana-3453	188	19	becomes	become	VERB
cana-3453	188	20	totally	totally	ADV
cana-3453	188	21	geodesic	geodesic	ADJ
cana-3453	188	22	if	if	SCONJ
cana-3453	188	23	and	and	CCONJ
cana-3453	188	24	only	only	ADV
cana-3453	188	25	if	if	SCONJ
cana-3453	188	26	q(σ	q(σ	NUM
cana-3453	188	27	,	,	PUNCT
cana-3453	188	28	w3)=0	w3)=0	NUM
cana-3453	188	29	,	,	PUNCT
cana-3453	188	30	provided	provide	VERB
cana-3453	188	31	that	that	SCONJ
cana-3453	188	32	that	that	SCONJ
cana-3453	188	33	{	{	PUNCT
cana-3453	188	34	𝟒𝒏(𝟏	𝟒𝒏(𝟏	X
cana-3453	188	35	−	−	NOUN
cana-3453	188	36	𝟐𝒏)𝒇𝟏	𝟐𝒏)𝒇𝟏	PROPN
cana-3453	189	1	+	+	PUNCT
cana-3453	189	2	𝟑𝒇𝟐	𝟑𝒇𝟐	PUNCT
cana-3453	189	3	+	+	CCONJ
cana-3453	189	4	(	(	PUNCT
cana-3453	189	5	𝟒𝒏	𝟒𝒏	ADJ
cana-3453	189	6	+	+	PROPN
cana-3453	189	7	𝟏)(𝟐𝒏	𝟏)(𝟐𝒏	PROPN
cana-3453	189	8	−	−	PROPN
cana-3453	189	9	𝟏)𝒇𝟑	𝟏)𝒇𝟑	PROPN
cana-3453	189	10	}	}	PUNCT
cana-3453	189	11	≠	≠	PROPN
cana-3453	189	12	𝟎.	𝟎.	PUNCT
cana-3453	189	13	proof	proof	NOUN
cana-3453	189	14	.	.	PUNCT
cana-3453	190	1	consider	consider	VERB
cana-3453	190	2	an	an	DET
cana-3453	190	3	invariant	invariant	ADJ
cana-3453	190	4	submanifold	submanifold	NOUN
cana-3453	190	5	n	n	ADP
cana-3453	190	6	satisfying	satisfy	VERB
cana-3453	190	7	q(σ	q(σ	NUM
cana-3453	190	8	,	,	PUNCT
cana-3453	190	9	w3)=0	w3)=0	PROPN
cana-3453	190	10	.	.	PUNCT
cana-3453	191	1	this	this	DET
cana-3453	191	2	condition	condition	NOUN
cana-3453	191	3	can	can	AUX
cana-3453	191	4	be	be	AUX
cana-3453	191	5	expressed	express	VERB
cana-3453	191	6	through	through	ADP
cana-3453	191	7	the	the	DET
cana-3453	191	8	derivation	derivation	NOUN
cana-3453	191	9	:	:	PUNCT
cana-3453	192	1	𝑸(𝝈	𝑸(𝝈	ADJ
cana-3453	192	2	,	,	PUNCT
cana-3453	192	3	𝑾𝟑)(𝑿	𝑾𝟑)(𝑿	ADJ
cana-3453	192	4	,	,	PUNCT
cana-3453	192	5	𝒀	𝒀	PROPN
cana-3453	192	6	,	,	PUNCT
cana-3453	192	7	𝒁	𝒁	PROPN
cana-3453	192	8	;	;	PUNCT
cana-3453	192	9	𝑼	𝑼	ADJ
cana-3453	192	10	,	,	PUNCT
cana-3453	192	11	𝑽	𝑽	PROPN
cana-3453	192	12	)	)	PUNCT
cana-3453	192	13	=	=	SYM
cana-3453	192	14	(	(	PUNCT
cana-3453	192	15	(	(	PUNCT
cana-3453	192	16	𝑼	𝑼	PROPN
cana-3453	192	17	∧𝝈	∧𝝈	NUM
cana-3453	192	18	𝑽	𝑽	PROPN
cana-3453	192	19	)	)	PUNCT
cana-3453	192	20	.	.	PUNCT
cana-3453	193	1	𝑾𝟑)𝑿	𝑾𝟑)𝑿	X
cana-3453	193	2	,	,	PUNCT
cana-3453	193	3	𝒀)𝒁	𝒀)𝒁	NOUN
cana-3453	193	4	=	=	SYM
cana-3453	193	5	𝟎	𝟎	NUM
cana-3453	193	6	which	which	PRON
cana-3453	193	7	expands	expand	VERB
cana-3453	193	8	to	to	ADP
cana-3453	193	9	:	:	PUNCT
cana-3453	193	10	=	=	SYM
cana-3453	193	11	−	−	PROPN
cana-3453	193	12	𝑾𝟑((𝑼	𝑾𝟑((𝑼	VERB
cana-3453	194	1	∧𝝈	∧𝝈	SYM
cana-3453	194	2	𝑽)𝑿	𝑽)𝑿	PROPN
cana-3453	194	3	,	,	PUNCT
cana-3453	194	4	𝒀)𝒁	𝒀)𝒁	VERB
cana-3453	194	5	−	−	PROPN
cana-3453	194	6	𝑾𝟑(𝑿	𝑾𝟑(𝑿	PROPN
cana-3453	194	7	,	,	PUNCT
cana-3453	194	8	(	(	PUNCT
cana-3453	194	9	𝑼	𝑼	PROPN
cana-3453	194	10	∧𝝈	∧𝝈	NUM
cana-3453	194	11	𝑽)𝒀)𝒁	𝑽)𝒀)𝒁	VERB
cana-3453	194	12	−	−	PROPN
cana-3453	194	13	𝑾𝟑(𝑿	𝑾𝟑(𝑿	NOUN
cana-3453	194	14	,	,	PUNCT
cana-3453	194	15	𝒀)(𝑼	𝒀)(𝑼	ADP
cana-3453	194	16	∧𝝈	∧𝝈	NUM
cana-3453	194	17	𝑽)𝒁	𝑽)𝒁	NOUN
cana-3453	194	18	)	)	PUNCT
cana-3453	194	19	(	(	PUNCT
cana-3453	194	20	5.1	5.1	NUM
cana-3453	194	21	)	)	PUNCT
cana-3453	194	22	applying	apply	VERB
cana-3453	194	23	the	the	DET
cana-3453	194	24	definition	definition	NOUN
cana-3453	194	25	of	of	ADP
cana-3453	194	26	the	the	DET
cana-3453	194	27	operator	operator	NOUN
cana-3453	194	28	u∧σ	u∧σ	VERB
cana-3453	194	29	to	to	ADP
cana-3453	194	30	equation	equation	NOUN
cana-3453	194	31	(	(	PUNCT
cana-3453	194	32	5.1	5.1	NUM
cana-3453	194	33	)	)	PUNCT
cana-3453	194	34	yields	yield	NOUN
cana-3453	194	35	:	:	PUNCT
cana-3453	194	36	−𝝈(𝑽	−𝝈(𝑽	NOUN
cana-3453	194	37	,	,	PUNCT
cana-3453	194	38	𝑿)𝑾𝟑(𝑼	𝑿)𝑾𝟑(𝑼	NOUN
cana-3453	194	39	,	,	PUNCT
cana-3453	194	40	𝒀)𝒁	𝒀)𝒁	VERB
cana-3453	194	41	+	+	CCONJ
cana-3453	194	42	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	194	43	,	,	PUNCT
cana-3453	194	44	𝑿)𝑾𝟑(𝑽	𝑿)𝑾𝟑(𝑽	PROPN
cana-3453	194	45	,	,	PUNCT
cana-3453	194	46	𝒀)𝒁	𝒀)𝒁	VERB
cana-3453	194	47	−	−	PROPN
cana-3453	194	48	𝝈(𝑽	𝝈(𝑽	PROPN
cana-3453	194	49	,	,	PUNCT
cana-3453	194	50	𝒀)𝑾𝟑(𝑿	𝒀)𝑾𝟑(𝑿	NOUN
cana-3453	194	51	,	,	PUNCT
cana-3453	194	52	𝑼)𝒁	𝑼)𝒁	NOUN
cana-3453	194	53	+	+	CCONJ
cana-3453	194	54	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	194	55	,	,	PUNCT
cana-3453	194	56	𝒀)𝑾𝟑(𝑿	𝒀)𝑾𝟑(𝑿	NOUN
cana-3453	194	57	,	,	PUNCT
cana-3453	194	58	𝑽)𝒁	𝑽)𝒁	VERB
cana-3453	194	59	−	−	PROPN
cana-3453	194	60	𝝈(𝑽	𝝈(𝑽	NOUN
cana-3453	194	61	,	,	PUNCT
cana-3453	194	62	𝒁)𝑾𝟑(𝑿	𝒁)𝑾𝟑(𝑿	NOUN
cana-3453	194	63	,	,	PUNCT
cana-3453	194	64	𝒀)𝑼	𝒀)𝑼	NOUN
cana-3453	194	65	+	+	CCONJ
cana-3453	194	66	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	194	67	,	,	PUNCT
cana-3453	194	68	𝒁)𝑾𝟑(𝑿	𝒁)𝑾𝟑(𝑿	NOUN
cana-3453	194	69	,	,	PUNCT
cana-3453	194	70	𝒀)𝑽	𝒀)𝑽	NOUN
cana-3453	194	71	=	=	SYM
cana-3453	194	72	𝟎	𝟎	X
cana-3453	194	73	(	(	PUNCT
cana-3453	194	74	5.2	5.2	NUM
cana-3453	194	75	)	)	PUNCT
cana-3453	194	76	the	the	DET
cana-3453	194	77	tensor	tensor	NOUN
cana-3453	194	78	w₃	w₃	NOUN
cana-3453	194	79	shows	show	VERB
cana-3453	194	80	features	feature	NOUN
cana-3453	194	81	that	that	PRON
cana-3453	194	82	are	be	AUX
cana-3453	194	83	different	different	ADJ
cana-3453	194	84	from	from	ADP
cana-3453	194	85	the	the	DET
cana-3453	194	86	one	one	NOUN
cana-3453	194	87	shown	show	VERB
cana-3453	194	88	by	by	ADP
cana-3453	194	89	w₂	w₂	PROPN
cana-3453	194	90	when	when	SCONJ
cana-3453	194	91	we	we	PRON
cana-3453	194	92	analyze	analyze	VERB
cana-3453	194	93	the	the	DET
cana-3453	194	94	behaviour	behaviour	NOUN
cana-3453	194	95	along	along	ADP
cana-3453	194	96	the	the	DET
cana-3453	194	97	characteristic	characteristic	ADJ
cana-3453	194	98	direction	direction	NOUN
cana-3453	194	99	ξ	ξ	PROPN
cana-3453	194	100	.	.	PUNCT
cana-3453	195	1	as	as	SCONJ
cana-3453	195	2	the	the	DET
cana-3453	195	3	structural	structural	ADJ
cana-3453	195	4	equations	equation	NOUN
cana-3453	195	5	show	show	VERB
cana-3453	195	6	,	,	PUNCT
cana-3453	195	7	contact	contact	NOUN
cana-3453	195	8	metric	metric	ADJ
cana-3453	195	9	properties	property	NOUN
cana-3453	195	10	affect	affect	VERB
cana-3453	195	11	the	the	DET
cana-3453	195	12	geometric	geometric	ADJ
cana-3453	195	13	correspondence	correspondence	NOUN
cana-3453	195	14	concerning	concern	VERB
cana-3453	195	15	the	the	DET
cana-3453	195	16	submanifold	submanifold	NOUN
cana-3453	195	17	and	and	CCONJ
cana-3453	195	18	ambient	ambient	ADJ
cana-3453	195	19	space	space	NOUN
cana-3453	195	20	.	.	PUNCT
cana-3453	196	1	inclusion	inclusion	NOUN
cana-3453	196	2	of	of	ADP
cana-3453	196	3	z	z	PROPN
cana-3453	196	4	=	=	PROPN
cana-3453	196	5	v	v	NOUN
cana-3453	196	6	=	=	SYM
cana-3453	196	7	ξ	ξ	X
cana-3453	196	8	in	in	ADP
cana-3453	196	9	(	(	PUNCT
cana-3453	196	10	5.2	5.2	NUM
cana-3453	196	11	)	)	PUNCT
cana-3453	196	12	and	and	CCONJ
cana-3453	196	13	dependency	dependency	NOUN
cana-3453	196	14	using	use	VERB
cana-3453	196	15	condition	condition	NOUN
cana-3453	196	16	(	(	PUNCT
cana-3453	196	17	3.3	3.3	NUM
cana-3453	196	18	):	):	PUNCT
cana-3453	196	19	𝝈(𝑼	𝝈(𝑼	PROPN
cana-3453	196	20	,	,	PUNCT
cana-3453	196	21	𝑿)𝑾𝟑(𝝃	𝑿)𝑾𝟑(𝝃	NOUN
cana-3453	196	22	,	,	PUNCT
cana-3453	196	23	𝒀)𝝃	𝒀)𝝃	PUNCT
cana-3453	197	1	+	+	CCONJ
cana-3453	197	2	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	197	3	,	,	PUNCT
cana-3453	197	4	𝒀)𝑾𝟑(𝑿	𝒀)𝑾𝟑(𝑿	NOUN
cana-3453	197	5	,	,	PUNCT
cana-3453	197	6	𝝃)𝝃	𝝃)𝝃	X
cana-3453	197	7	=	=	SYM
cana-3453	197	8	𝟎	𝟎	X
cana-3453	197	9	(	(	PUNCT
cana-3453	197	10	5.3	5.3	NUM
cana-3453	197	11	)	)	PUNCT
cana-3453	197	12	substituting	substitute	VERB
cana-3453	197	13	equation	equation	NOUN
cana-3453	197	14	(	(	PUNCT
cana-3453	197	15	2.18	2.18	NUM
cana-3453	197	16	):	):	PUNCT
cana-3453	197	17	communications	communication	NOUN
cana-3453	197	18	on	on	ADP
cana-3453	197	19	applied	apply	VERB
cana-3453	197	20	nonlinear	nonlinear	ADJ
cana-3453	197	21	analysis	analysis	NOUN
cana-3453	197	22	issn	issn	NOUN
cana-3453	197	23	:	:	PUNCT
cana-3453	197	24	1074	1074	NUM
cana-3453	197	25	-	-	PUNCT
cana-3453	197	26	133x	133x	NUM
cana-3453	197	27	vol	vol	NOUN
cana-3453	197	28	32	32	NUM
cana-3453	197	29	no	no	NOUN
cana-3453	197	30	.	.	PUNCT
cana-3453	198	1	7s	7	NOUN
cana-3453	198	2	(	(	PUNCT
cana-3453	198	3	2025	2025	NUM
cana-3453	198	4	)	)	PUNCT
cana-3453	198	5	424	424	NUM
cana-3453	198	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3453	198	7	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	198	8	,	,	PUNCT
cana-3453	198	9	𝑿)[𝟐(𝒇𝟏	𝑿)[𝟐(𝒇𝟏	PUNCT
cana-3453	198	10	−	−	PROPN
cana-3453	198	11	𝒇𝟑){𝜼(𝒀)𝝃	𝒇𝟑){𝜼(𝒀)𝝃	NUM
cana-3453	198	12	−	−	PROPN
cana-3453	198	13	𝒀	𝒀	NOUN
cana-3453	198	14	}	}	PUNCT
cana-3453	198	15	]	]	PUNCT
cana-3453	199	1	+	+	CCONJ
cana-3453	199	2	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	199	3	,	,	PUNCT
cana-3453	199	4	𝒀	𝒀	NOUN
cana-3453	199	5	)	)	PUNCT
cana-3453	200	1	[	[	X
cana-3453	200	2	(	(	PUNCT
cana-3453	200	3	𝒇𝟏	𝒇𝟏	NOUN
cana-3453	200	4	−	−	PROPN
cana-3453	200	5	𝒇𝟑){𝑿	𝒇𝟑){𝑿	PROPN
cana-3453	200	6	−	−	PROPN
cana-3453	200	7	𝜼(𝑿)𝝃	𝜼(𝑿)𝝃	CCONJ
cana-3453	200	8	}	}	PUNCT
cana-3453	200	9	−	−	PROPN
cana-3453	200	10	𝟏	𝟏	NUM
cana-3453	200	11	𝟐𝒏	𝟐𝒏	NOUN
cana-3453	200	12	{	{	PUNCT
cana-3453	200	13	𝟐𝒏(𝒇𝟏	𝟐𝒏(𝒇𝟏	ADJ
cana-3453	200	14	−	−	PUNCT
cana-3453	200	15	𝒇𝟑)𝜼(𝑿)𝝃	𝒇𝟑)𝜼(𝑿)𝝃	NOUN
cana-3453	200	16	−	−	PROPN
cana-3453	200	17	𝑸𝑿	𝑸𝑿	PROPN
cana-3453	200	18	}	}	PUNCT
cana-3453	200	19	]	]	PUNCT
cana-3453	200	20	=	=	SYM
cana-3453	200	21	𝟎	𝟎	X
cana-3453	200	22	(	(	PUNCT
cana-3453	200	23	5.4	5.4	NUM
cana-3453	200	24	)	)	PUNCT
cana-3453	200	25	taking	take	VERB
cana-3453	200	26	the	the	DET
cana-3453	200	27	inner	inner	ADJ
cana-3453	200	28	product	product	NOUN
cana-3453	200	29	with	with	ADP
cana-3453	200	30	w	w	PROPN
cana-3453	200	31	:	:	PUNCT
cana-3453	200	32	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	200	33	,	,	PUNCT
cana-3453	200	34	𝑿)[𝟐(𝒇𝟏	𝑿)[𝟐(𝒇𝟏	PUNCT
cana-3453	200	35	−	−	PROPN
cana-3453	200	36	𝒇𝟑){𝜼(𝒀)𝜼(𝑾	𝒇𝟑){𝜼(𝒀)𝜼(𝑾	SYM
cana-3453	200	37	)	)	PUNCT
cana-3453	200	38	−	−	NOUN
cana-3453	200	39	𝒈(𝒀	𝒈(𝒀	NOUN
cana-3453	200	40	,	,	PUNCT
cana-3453	200	41	𝑾	𝑾	ADJ
cana-3453	200	42	)	)	PUNCT
cana-3453	200	43	}	}	PUNCT
cana-3453	200	44	]	]	PUNCT
cana-3453	201	1	+	+	CCONJ
cana-3453	201	2	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	201	3	,	,	PUNCT
cana-3453	201	4	𝒀	𝒀	NOUN
cana-3453	201	5	)	)	PUNCT
cana-3453	202	1	[	[	X
cana-3453	202	2	(	(	PUNCT
cana-3453	202	3	𝒇𝟏	𝒇𝟏	NOUN
cana-3453	202	4	−	−	PROPN
cana-3453	202	5	𝒇𝟑){𝒈(𝑿	𝒇𝟑){𝒈(𝑿	NUM
cana-3453	202	6	,	,	PUNCT
cana-3453	202	7	𝑾	𝑾	ADJ
cana-3453	202	8	)	)	PUNCT
cana-3453	202	9	−	−	PROPN
cana-3453	202	10	𝜼(𝑿)𝜼(𝑾	𝜼(𝑿)𝜼(𝑾	NOUN
cana-3453	202	11	)	)	PUNCT
cana-3453	202	12	}	}	PUNCT
cana-3453	202	13	−	−	ADP
cana-3453	202	14	𝟏	𝟏	NUM
cana-3453	202	15	𝟐𝒏	𝟐𝒏	NOUN
cana-3453	202	16	{	{	PUNCT
cana-3453	202	17	𝟐𝒏(𝒇𝟏	𝟐𝒏(𝒇𝟏	ADJ
cana-3453	202	18	−	−	PROPN
cana-3453	202	19	𝒇𝟑)𝜼(𝑿)𝜼(𝑾	𝒇𝟑)𝜼(𝑿)𝜼(𝑾	PROPN
cana-3453	202	20	)	)	PUNCT
cana-3453	202	21	−	−	ADP
cana-3453	202	22	𝒈(𝑸𝑿	𝒈(𝑸𝑿	PROPN
cana-3453	202	23	,	,	PUNCT
cana-3453	202	24	𝑾	𝑾	ADJ
cana-3453	202	25	)	)	PUNCT
cana-3453	202	26	}	}	PUNCT
cana-3453	202	27	]	]	PUNCT
cana-3453	202	28	=	=	SYM
cana-3453	202	29	𝟎	𝟎	X
cana-3453	202	30	(	(	PUNCT
cana-3453	202	31	5.5	5.5	NUM
cana-3453	202	32	)	)	PUNCT
cana-3453	202	33	contracting	contract	VERB
cana-3453	202	34	y	y	PROPN
cana-3453	202	35	and	and	CCONJ
cana-3453	202	36	w	w	PROPN
cana-3453	202	37	leads	lead	VERB
cana-3453	202	38	to	to	ADP
cana-3453	202	39	:	:	PUNCT
cana-3453	202	40	𝝈(𝑼	𝝈(𝑼	PROPN
cana-3453	202	41	,	,	PUNCT
cana-3453	202	42	𝑿){𝟒𝒏(𝟏	𝑿){𝟒𝒏(𝟏	PROPN
cana-3453	202	43	−	−	PROPN
cana-3453	202	44	𝟐𝒏)𝒇𝟏	𝟐𝒏)𝒇𝟏	PROPN
cana-3453	203	1	+	+	PUNCT
cana-3453	203	2	𝟑𝒇𝟐	𝟑𝒇𝟐	PUNCT
cana-3453	203	3	+	+	CCONJ
cana-3453	203	4	(	(	PUNCT
cana-3453	203	5	𝟒𝒏	𝟒𝒏	ADJ
cana-3453	203	6	+	+	PROPN
cana-3453	203	7	𝟏)(𝟐𝒏	𝟏)(𝟐𝒏	PROPN
cana-3453	204	1	−	−	PROPN
cana-3453	204	2	𝟏)𝒇𝟑	𝟏)𝒇𝟑	NOUN
cana-3453	204	3	}	}	PUNCT
cana-3453	204	4	=	=	SYM
cana-3453	204	5	𝟎.	𝟎.	X
cana-3453	204	6	(	(	PUNCT
cana-3453	204	7	5.6	5.6	NUM
cana-3453	204	8	)	)	PUNCT
cana-3453	204	9	under	under	ADP
cana-3453	204	10	our	our	PRON
cana-3453	204	11	non	non	ADJ
cana-3453	204	12	-	-	ADJ
cana-3453	204	13	degeneracy	degeneracy	ADJ
cana-3453	204	14	condition	condition	NOUN
cana-3453	204	15	{	{	PUNCT
cana-3453	204	16	4n(1−2n)f1	4n(1−2n)f1	NUM
cana-3453	204	17	+	+	NOUN
cana-3453	204	18	3f2+(4n+1)(2n−1)f3}≠0	3f2+(4n+1)(2n−1)f3}≠0	NUM
cana-3453	204	19	,	,	PUNCT
cana-3453	204	20	we	we	PRON
cana-3453	204	21	conclude	conclude	VERB
cana-3453	204	22	:	:	PUNCT
cana-3453	205	1	σ(u	σ(u	NOUN
cana-3453	205	2	,	,	PUNCT
cana-3453	205	3	x)=0	x)=0	PROPN
cana-3453	205	4	this	this	PRON
cana-3453	205	5	shows	show	VERB
cana-3453	205	6	that	that	SCONJ
cana-3453	205	7	n	n	PRON
cana-3453	205	8	is	be	AUX
cana-3453	205	9	geodesic	geodesic	ADJ
cana-3453	205	10	.	.	PUNCT
cana-3453	206	1	the	the	DET
cana-3453	206	2	other	other	ADJ
cana-3453	206	3	direction	direction	NOUN
cana-3453	206	4	is	be	AUX
cana-3453	206	5	also	also	ADV
cana-3453	206	6	straightforward	straightforward	ADJ
cana-3453	206	7	because	because	SCONJ
cana-3453	206	8	geodesic	geodesic	ADJ
cana-3453	206	9	submanifolds	submanifold	NOUN
cana-3453	206	10	are	be	AUX
cana-3453	206	11	automatically	automatically	ADV
cana-3453	206	12	of	of	ADP
cana-3453	206	13	class	class	NOUN
cana-3453	206	14	q	q	PUNCT
cana-3453	206	15	by	by	ADP
cana-3453	206	16	definition	definition	NOUN
cana-3453	206	17	(	(	PUNCT
cana-3453	206	18	q(σ	q(σ	X
cana-3453	206	19	,	,	PUNCT
cana-3453	206	20	w3)=0	w3)=0	PROPN
cana-3453	206	21	)	)	PUNCT
cana-3453	206	22	.	.	PUNCT
cana-3453	207	1	the	the	DET
cana-3453	207	2	theorem	theorem	NOUN
cana-3453	207	3	gives	give	VERB
cana-3453	207	4	a	a	DET
cana-3453	207	5	manifold	manifold	ADJ
cana-3453	207	6	-	-	PUNCT
cana-3453	207	7	analytical	analytical	ADJ
cana-3453	207	8	property	property	NOUN
cana-3453	207	9	of	of	ADP
cana-3453	207	10	these	these	DET
cana-3453	207	11	invariant	invariant	ADJ
cana-3453	207	12	submanifolds	submanifold	NOUN
cana-3453	207	13	as	as	ADP
cana-3453	207	14	totally	totally	ADV
cana-3453	207	15	geodesical	geodesical	ADJ
cana-3453	207	16	,	,	PUNCT
cana-3453	207	17	thus	thus	ADV
cana-3453	207	18	characterizing	characterize	VERB
cana-3453	207	19	completely	completely	ADV
cana-3453	207	20	different	different	ADJ
cana-3453	207	21	invariant	invariant	ADJ
cana-3453	207	22	submanifolds	submanifold	NOUN
cana-3453	207	23	concerning	concern	VERB
cana-3453	207	24	their	their	PRON
cana-3453	207	25	interaction	interaction	NOUN
cana-3453	207	26	with	with	ADP
cana-3453	207	27	the	the	DET
cana-3453	207	28	w3	w3	PROPN
cana-3453	207	29	curvature	curvature	PROPN
cana-3453	207	30	tensor	tensor	NOUN
cana-3453	207	31	,	,	PUNCT
cana-3453	207	32	adding	add	VERB
cana-3453	207	33	another	another	DET
cana-3453	207	34	piece	piece	NOUN
cana-3453	207	35	of	of	ADP
cana-3453	207	36	the	the	DET
cana-3453	207	37	puzzle	puzzle	NOUN
cana-3453	207	38	to	to	ADP
cana-3453	207	39	our	our	PRON
cana-3453	207	40	existing	exist	VERB
cana-3453	207	41	results	result	NOUN
cana-3453	207	42	with	with	ADP
cana-3453	207	43	w2	w2	NOUN
cana-3453	207	44	.	.	PUNCT
cana-3453	208	1	in	in	ADP
cana-3453	208	2	w₂	w₂	PROPN
cana-3453	208	3	,	,	PUNCT
cana-3453	208	4	the	the	DET
cana-3453	208	5	corresponding	corresponding	ADJ
cana-3453	208	6	non	non	ADJ
cana-3453	208	7	-	-	ADJ
cana-3453	208	8	degeneracy	degeneracy	ADJ
cana-3453	208	9	condition	condition	NOUN
cana-3453	208	10	is	be	AUX
cana-3453	208	11	direct	direct	ADJ
cana-3453	208	12	and	and	CCONJ
cana-3453	208	13	simple	simple	ADJ
cana-3453	208	14	,	,	PUNCT
cana-3453	208	15	whereas	whereas	SCONJ
cana-3453	208	16	,	,	PUNCT
cana-3453	208	17	for	for	ADP
cana-3453	208	18	w₃	w₃	NOUN
cana-3453	208	19	,	,	PUNCT
cana-3453	208	20	we	we	PRON
cana-3453	208	21	will	will	AUX
cana-3453	208	22	see	see	VERB
cana-3453	208	23	that	that	SCONJ
cana-3453	208	24	it	it	PRON
cana-3453	208	25	is	be	AUX
cana-3453	208	26	much	much	ADV
cana-3453	208	27	more	more	ADV
cana-3453	208	28	interesting	interesting	ADJ
cana-3453	208	29	and	and	CCONJ
cana-3453	208	30	stranger	strange	ADJ
cana-3453	208	31	and	and	CCONJ
cana-3453	208	32	is	be	AUX
cana-3453	208	33	the	the	DET
cana-3453	208	34	way	way	NOUN
cana-3453	208	35	that	that	PRON
cana-3453	208	36	we	we	PRON
cana-3453	208	37	can	can	AUX
cana-3453	208	38	solve	solve	VERB
cana-3453	208	39	w₃	w₃	NOUN
cana-3453	208	40	and	and	CCONJ
cana-3453	208	41	allows	allow	VERB
cana-3453	208	42	us	we	PRON
cana-3453	208	43	to	to	ADP
cana-3453	208	44	a	a	DET
cana-3453	208	45	determinant	determinant	ADJ
cana-3453	208	46	function	function	NOUN
cana-3453	208	47	on	on	ADP
cana-3453	208	48	f₁	f₁	ADJ
cana-3453	208	49	,	,	PUNCT
cana-3453	208	50	f₂	f₂	NOUN
cana-3453	208	51	and	and	CCONJ
cana-3453	208	52	f₃.	f₃.	NOUN
cana-3453	208	53	this	this	PRON
cana-3453	208	54	allows	allow	VERB
cana-3453	208	55	the	the	DET
cana-3453	208	56	geometrical	geometrical	ADJ
cana-3453	208	57	information	information	NOUN
cana-3453	208	58	contained	contain	VERB
cana-3453	208	59	in	in	ADP
cana-3453	208	60	w₃	w₃	NOUN
cana-3453	208	61	to	to	PART
cana-3453	208	62	have	have	AUX
cana-3453	208	63	a	a	DET
cana-3453	208	64	sensible	sensible	ADJ
cana-3453	208	65	meaning	meaning	NOUN
cana-3453	208	66	,	,	PUNCT
cana-3453	208	67	previously	previously	ADV
cana-3453	208	68	used	use	VERB
cana-3453	208	69	to	to	PART
cana-3453	208	70	characterize	characterize	VERB
cana-3453	208	71	geodesic	geodesic	ADJ
cana-3453	208	72	submanifolds	submanifold	NOUN
cana-3453	208	73	.	.	PUNCT
cana-3453	209	1	the	the	DET
cana-3453	209	2	basic	basic	ADJ
cana-3453	209	3	framework	framework	NOUN
cana-3453	209	4	from	from	ADP
cana-3453	209	5	w2	w2	NOUN
cana-3453	209	6	to	to	ADP
cana-3453	209	7	w4	w4	PROPN
cana-3453	209	8	shows	show	VERB
cana-3453	209	9	an	an	DET
cana-3453	209	10	improvement	improvement	NOUN
cana-3453	209	11	of	of	ADP
cana-3453	209	12	geometry	geometry	NOUN
cana-3453	209	13	interlacing	interlace	VERB
cana-3453	209	14	results	result	NOUN
cana-3453	209	15	in	in	ADP
cana-3453	209	16	generalized	generalized	ADJ
cana-3453	209	17	sasakian	sasakian	ADJ
cana-3453	209	18	-	-	PUNCT
cana-3453	209	19	space	space	NOUN
cana-3453	209	20	-	-	PUNCT
cana-3453	209	21	forms	form	NOUN
cana-3453	209	22	.	.	PUNCT
cana-3453	210	1	the	the	DET
cana-3453	210	2	higher	high	ADJ
cana-3453	210	3	tensor	tensor	NOUN
cana-3453	210	4	is	be	AUX
cana-3453	210	5	built	build	VERB
cana-3453	210	6	on	on	ADP
cana-3453	210	7	the	the	DET
cana-3453	210	8	new	new	ADJ
cana-3453	210	9	information	information	NOUN
cana-3453	210	10	gained	gain	VERB
cana-3453	210	11	from	from	ADP
cana-3453	210	12	the	the	DET
cana-3453	210	13	previous	previous	ADJ
cana-3453	210	14	one	one	NOUN
cana-3453	210	15	and	and	CCONJ
cana-3453	210	16	explores	explore	VERB
cana-3453	210	17	the	the	DET
cana-3453	210	18	new	new	ADJ
cana-3453	210	19	geometric	geometric	ADJ
cana-3453	210	20	nature	nature	NOUN
cana-3453	210	21	.	.	PUNCT
cana-3453	211	1	this	this	DET
cana-3453	211	2	progressive	progressive	ADJ
cana-3453	211	3	technique	technique	NOUN
cana-3453	211	4	of	of	ADP
cana-3453	211	5	knowing	know	VERB
cana-3453	211	6	the	the	DET
cana-3453	211	7	geometry	geometry	NOUN
cana-3453	211	8	creates	create	VERB
cana-3453	211	9	more	more	ADV
cana-3453	211	10	and	and	CCONJ
cana-3453	211	11	more	more	ADJ
cana-3453	211	12	refined	refined	ADJ
cana-3453	211	13	instruments	instrument	NOUN
cana-3453	211	14	for	for	ADP
cana-3453	211	15	studying	study	VERB
cana-3453	211	16	the	the	DET
cana-3453	211	17	submanifold	submanifold	NOUN
cana-3453	211	18	behaviour	behaviour	NOUN
cana-3453	211	19	.	.	PUNCT
cana-3453	212	1	instead	instead	ADV
cana-3453	212	2	,	,	PUNCT
cana-3453	212	3	the	the	DET
cana-3453	212	4	varying	vary	VERB
cana-3453	212	5	complexities	complexity	NOUN
cana-3453	212	6	of	of	ADP
cana-3453	212	7	non	non	ADJ
cana-3453	212	8	-	-	ADJ
cana-3453	212	9	degeneracy	degeneracy	ADJ
cana-3453	212	10	conditions	condition	NOUN
cana-3453	212	11	associated	associate	VERB
cana-3453	212	12	to	to	ADP
cana-3453	212	13	each	each	DET
cana-3453	212	14	tensor	tensor	NOUN
cana-3453	212	15	carry	carry	VERB
cana-3453	212	16	the	the	DET
cana-3453	212	17	implication	implication	NOUN
cana-3453	212	18	(	(	PUNCT
cana-3453	212	19	which	which	PRON
cana-3453	212	20	is	be	AUX
cana-3453	212	21	already	already	ADV
cana-3453	212	22	known	know	VERB
cana-3453	212	23	to	to	PART
cana-3453	212	24	hold	hold	VERB
cana-3453	212	25	)	)	PUNCT
cana-3453	212	26	that	that	SCONJ
cana-3453	212	27	nature	nature	NOUN
cana-3453	212	28	has	have	AUX
cana-3453	212	29	a	a	DET
cana-3453	212	30	preference	preference	NOUN
cana-3453	212	31	for	for	ADP
cana-3453	212	32	several	several	ADJ
cana-3453	212	33	complementary	complementary	ADJ
cana-3453	212	34	descriptions	description	NOUN
cana-3453	212	35	of	of	ADP
cana-3453	212	36	fundamental	fundamental	ADJ
cana-3453	212	37	geometric	geometric	ADJ
cana-3453	212	38	features	feature	NOUN
cana-3453	212	39	.	.	PUNCT
cana-3453	213	1	6	6	X
cana-3453	213	2	.	.	X
cana-3453	213	3	invariant	invariant	ADJ
cana-3453	213	4	submanifolds	submanifold	NOUN
cana-3453	213	5	and	and	CCONJ
cana-3453	213	6	the	the	DET
cana-3453	213	7	w4	w4	NOUN
cana-3453	213	8	curvature	curvature	NOUN
cana-3453	213	9	relationship	relationship	NOUN
cana-3453	213	10	in	in	ADP
cana-3453	213	11	this	this	DET
cana-3453	213	12	sense	sense	NOUN
cana-3453	213	13	,	,	PUNCT
cana-3453	213	14	this	this	DET
cana-3453	213	15	section	section	NOUN
cana-3453	213	16	presents	present	VERB
cana-3453	213	17	a	a	DET
cana-3453	213	18	new	new	ADJ
cana-3453	213	19	characterization	characterization	NOUN
cana-3453	213	20	of	of	ADP
cana-3453	213	21	totally	totally	ADV
cana-3453	213	22	geodesic	geodesic	ADJ
cana-3453	213	23	submanifolds	submanifold	NOUN
cana-3453	213	24	in	in	ADP
cana-3453	213	25	terms	term	NOUN
cana-3453	213	26	of	of	ADP
cana-3453	213	27	their	their	PRON
cana-3453	213	28	relations	relation	NOUN
cana-3453	213	29	with	with	ADP
cana-3453	213	30	the	the	DET
cana-3453	213	31	w4	w4	NOUN
cana-3453	213	32	curvature	curvature	NOUN
cana-3453	213	33	tensor	tensor	NOUN
cana-3453	213	34	.	.	PUNCT
cana-3453	214	1	we	we	PRON
cana-3453	214	2	set	set	VERB
cana-3453	214	3	out	out	ADP
cana-3453	214	4	necessary	necessary	ADJ
cana-3453	214	5	and	and	CCONJ
cana-3453	214	6	sufficient	sufficient	ADJ
cana-3453	214	7	conditions	condition	NOUN
cana-3453	214	8	under	under	ADP
cana-3453	214	9	which	which	PRON
cana-3453	214	10	this	this	DET
cana-3453	214	11	interaction	interaction	NOUN
cana-3453	214	12	determines	determine	VERB
cana-3453	214	13	the	the	DET
cana-3453	214	14	geometric	geometric	ADJ
cana-3453	214	15	structure	structure	NOUN
cana-3453	214	16	of	of	ADP
cana-3453	214	17	the	the	DET
cana-3453	214	18	submanifold	submanifold	NOUN
cana-3453	214	19	.	.	PUNCT
cana-3453	215	1	more	more	ADV
cana-3453	215	2	specifically	specifically	ADV
cana-3453	215	3	,	,	PUNCT
cana-3453	215	4	while	while	SCONJ
cana-3453	215	5	natural	natural	ADJ
cana-3453	215	6	numbers	number	NOUN
cana-3453	215	7	up	up	ADP
cana-3453	215	8	to	to	PART
cana-3453	215	9	4	4	NUM
cana-3453	215	10	are	be	AUX
cana-3453	215	11	certain	certain	ADJ
cana-3453	215	12	types	type	NOUN
cana-3453	215	13	of	of	ADP
cana-3453	215	14	w₄	w₄	NOUN
cana-3453	215	15	curvature	curvature	NOUN
cana-3453	215	16	tensor	tensor	NOUN
cana-3453	215	17	and	and	CCONJ
cana-3453	215	18	while	while	SCONJ
cana-3453	215	19	w₂	w₂	PROPN
cana-3453	215	20	and	and	CCONJ
cana-3453	215	21	w₃	w₃	PROPN
cana-3453	215	22	have	have	VERB
cana-3453	215	23	properties	property	NOUN
cana-3453	215	24	that	that	PRON
cana-3453	215	25	extend	extend	VERB
cana-3453	215	26	to	to	ADP
cana-3453	215	27	w₄	w₄	NOUN
cana-3453	215	28	,	,	PUNCT
cana-3453	215	29	there	there	PRON
cana-3453	215	30	are	be	VERB
cana-3453	215	31	properties	property	NOUN
cana-3453	215	32	exhibited	exhibit	VERB
cana-3453	215	33	exclusively	exclusively	ADV
cana-3453	215	34	by	by	ADP
cana-3453	215	35	the	the	DET
cana-3453	215	36	w₄	w₄	PROPN
cana-3453	215	37	curvature	curvature	NOUN
cana-3453	215	38	tensor	tensor	NOUN
cana-3453	215	39	itself	itself	PRON
cana-3453	215	40	,	,	PUNCT
cana-3453	215	41	properties	property	NOUN
cana-3453	215	42	that	that	PRON
cana-3453	215	43	have	have	VERB
cana-3453	215	44	to	to	PART
cana-3453	215	45	do	do	VERB
cana-3453	215	46	with	with	ADP
cana-3453	215	47	how	how	SCONJ
cana-3453	215	48	it	it	PRON
cana-3453	215	49	interacts	interact	VERB
cana-3453	215	50	with	with	ADP
cana-3453	215	51	the	the	DET
cana-3453	215	52	contact	contact	NOUN
cana-3453	215	53	structure	structure	NOUN
cana-3453	215	54	.	.	PUNCT
cana-3453	216	1	it	it	PRON
cana-3453	216	2	encodes	encode	VERB
cana-3453	216	3	geometric	geometric	ADJ
cana-3453	216	4	data	datum	NOUN
cana-3453	216	5	relevant	relevant	ADJ
cana-3453	216	6	to	to	ADP
cana-3453	216	7	studying	study	VERB
cana-3453	216	8	the	the	DET
cana-3453	216	9	interaction	interaction	NOUN
cana-3453	216	10	between	between	ADP
cana-3453	216	11	a	a	DET
cana-3453	216	12	contact	contact	NOUN
cana-3453	216	13	metric	metric	ADJ
cana-3453	216	14	structure	structure	NOUN
cana-3453	216	15	on	on	ADP
cana-3453	216	16	the	the	DET
cana-3453	216	17	ambient	ambient	ADJ
cana-3453	216	18	space	space	NOUN
cana-3453	216	19	and	and	CCONJ
cana-3453	216	20	the	the	DET
cana-3453	216	21	invariant	invariant	ADJ
cana-3453	216	22	submanifolds	submanifold	NOUN
cana-3453	216	23	.	.	PUNCT
cana-3453	217	1	the	the	DET
cana-3453	217	2	result	result	NOUN
cana-3453	217	3	w₄	w₄	NOUN
cana-3453	217	4	encompasses	encompass	VERB
cana-3453	217	5	terms	term	NOUN
cana-3453	217	6	with	with	ADP
cana-3453	217	7	metric	metric	ADJ
cana-3453	217	8	tensor	tensor	NOUN
cana-3453	217	9	communications	communication	NOUN
cana-3453	217	10	on	on	ADP
cana-3453	217	11	applied	apply	VERB
cana-3453	217	12	nonlinear	nonlinear	ADJ
cana-3453	217	13	analysis	analysis	NOUN
cana-3453	217	14	issn	issn	NOUN
cana-3453	217	15	:	:	PUNCT
cana-3453	217	16	1074	1074	NUM
cana-3453	217	17	-	-	PUNCT
cana-3453	217	18	133x	133x	NUM
cana-3453	217	19	vol	vol	NOUN
cana-3453	217	20	32	32	NUM
cana-3453	217	21	no	no	NOUN
cana-3453	217	22	.	.	PUNCT
cana-3453	218	1	7s	7	NOUN
cana-3453	218	2	(	(	PUNCT
cana-3453	218	3	2025	2025	NUM
cana-3453	218	4	)	)	PUNCT
cana-3453	218	5	425	425	NUM
cana-3453	218	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3453	218	7	and	and	CCONJ
cana-3453	218	8	structure	structure	NOUN
cana-3453	218	9	tensor	tensor	NOUN
cana-3453	218	10	φ	φ	PROPN
cana-3453	218	11	,	,	PUNCT
cana-3453	218	12	allowing	allow	VERB
cana-3453	218	13	a	a	DET
cana-3453	218	14	balanced	balanced	ADJ
cana-3453	218	15	view	view	NOUN
cana-3453	218	16	of	of	ADP
cana-3453	218	17	the	the	DET
cana-3453	218	18	manifold	manifold	NOUN
cana-3453	218	19	's	's	PART
cana-3453	218	20	geometric	geometric	ADJ
cana-3453	218	21	and	and	CCONJ
cana-3453	218	22	contact	contact	NOUN
cana-3453	218	23	aspects	aspect	NOUN
cana-3453	218	24	.	.	PUNCT
cana-3453	219	1	such	such	DET
cana-3453	219	2	a	a	DET
cana-3453	219	3	balance	balance	NOUN
cana-3453	219	4	contributes	contribute	VERB
cana-3453	219	5	a	a	DET
cana-3453	219	6	lot	lot	NOUN
cana-3453	219	7	when	when	SCONJ
cana-3453	219	8	looking	look	VERB
cana-3453	219	9	at	at	ADP
cana-3453	219	10	totally	totally	ADV
cana-3453	219	11	geodesic	geodesic	ADJ
cana-3453	219	12	submanifolds	submanifold	NOUN
cana-3453	219	13	,	,	PUNCT
cana-3453	219	14	as	as	SCONJ
cana-3453	219	15	it	it	PRON
cana-3453	219	16	demonstrates	demonstrate	VERB
cana-3453	219	17	how	how	SCONJ
cana-3453	219	18	the	the	DET
cana-3453	219	19	contact	contact	NOUN
cana-3453	219	20	structure	structure	NOUN
cana-3453	219	21	affects	affect	VERB
cana-3453	219	22	the	the	DET
cana-3453	219	23	alignment	alignment	NOUN
cana-3453	219	24	of	of	ADP
cana-3453	219	25	geodesics	geodesic	NOUN
cana-3453	219	26	in	in	ADP
cana-3453	219	27	the	the	DET
cana-3453	219	28	submanifold	submanifold	NOUN
cana-3453	219	29	with	with	ADP
cana-3453	219	30	those	those	PRON
cana-3453	219	31	in	in	ADP
cana-3453	219	32	the	the	DET
cana-3453	219	33	ambient	ambient	ADJ
cana-3453	219	34	space	space	NOUN
cana-3453	219	35	.	.	PUNCT
cana-3453	220	1	the	the	DET
cana-3453	220	2	novelty	novelty	NOUN
cana-3453	220	3	here	here	ADV
cana-3453	220	4	lies	lie	VERB
cana-3453	220	5	in	in	ADP
cana-3453	220	6	the	the	DET
cana-3453	220	7	different	different	ADJ
cana-3453	220	8	geometric	geometric	ADJ
cana-3453	220	9	interactions	interaction	NOUN
cana-3453	220	10	of	of	ADP
cana-3453	220	11	the	the	DET
cana-3453	220	12	w₄	w₄	NOUN
cana-3453	220	13	curvature	curvature	NOUN
cana-3453	220	14	tensor	tensor	NOUN
cana-3453	220	15	underneath	underneath	ADP
cana-3453	220	16	the	the	DET
cana-3453	220	17	underlying	underlie	VERB
cana-3453	220	18	metric	metric	ADJ
cana-3453	220	19	and	and	CCONJ
cana-3453	220	20	contact	contact	NOUN
cana-3453	220	21	structures	structure	NOUN
cana-3453	220	22	.	.	PUNCT
cana-3453	221	1	behaviour	behaviour	NOUN
cana-3453	221	2	along	along	ADP
cana-3453	221	3	the	the	DET
cana-3453	221	4	characteristic	characteristic	ADJ
cana-3453	221	5	direction	direction	NOUN
cana-3453	221	6	illuminates	illuminate	VERB
cana-3453	221	7	complementary	complementary	ADJ
cana-3453	221	8	properties	property	NOUN
cana-3453	221	9	relative	relative	ADJ
cana-3453	221	10	to	to	ADP
cana-3453	221	11	what	what	PRON
cana-3453	221	12	we	we	PRON
cana-3453	221	13	learned	learn	VERB
cana-3453	221	14	from	from	ADP
cana-3453	221	15	w₂	w₂	PROPN
cana-3453	221	16	and	and	CCONJ
cana-3453	221	17	w₃.	w₃.	PROPN
cana-3453	221	18	theorem	theorem	VERB
cana-3453	221	19	6.1	6.1	NUM
cana-3453	221	20	.	.	PUNCT
cana-3453	222	1	let	let	VERB
cana-3453	222	2	m	m	PRON
cana-3453	222	3	be	be	AUX
cana-3453	222	4	a	a	DET
cana-3453	222	5	generalized	generalized	ADJ
cana-3453	222	6	sasakian	sasakian	ADJ
cana-3453	222	7	-	-	PUNCT
cana-3453	222	8	space	space	NOUN
cana-3453	222	9	-	-	PUNCT
cana-3453	222	10	form	form	NOUN
cana-3453	222	11	and	and	CCONJ
cana-3453	222	12	n∈mbe	n∈mbe	VERB
cana-3453	222	13	an	an	DET
cana-3453	222	14	invariant	invariant	ADJ
cana-3453	222	15	submanifold	submanifold	NOUN
cana-3453	222	16	.	.	PUNCT
cana-3453	223	1	the	the	DET
cana-3453	223	2	following	follow	VERB
cana-3453	223	3	conditions	condition	NOUN
cana-3453	223	4	are	be	AUX
cana-3453	223	5	equivalent	equivalent	ADJ
cana-3453	223	6	:	:	PUNCT
cana-3453	223	7	1	1	X
cana-3453	223	8	.	.	X
cana-3453	224	1	n	n	PRON
cana-3453	224	2	is	be	AUX
cana-3453	224	3	totally	totally	ADV
cana-3453	224	4	geodesic	geodesic	ADJ
cana-3453	224	5	2	2	NUM
cana-3453	224	6	.	.	X
cana-3453	224	7	q(σ	q(σ	NUM
cana-3453	224	8	,	,	PUNCT
cana-3453	224	9	w4)=0	w4)=0	PUNCT
cana-3453	224	10	this	this	DET
cana-3453	224	11	equivalence	equivalence	NOUN
cana-3453	224	12	holds	hold	VERB
cana-3453	224	13	under	under	ADP
cana-3453	224	14	the	the	DET
cana-3453	224	15	condition	condition	NOUN
cana-3453	224	16	{	{	PUNCT
cana-3453	224	17	f1	f1	NOUN
cana-3453	224	18	+	+	PROPN
cana-3453	224	19	3f2	3f2	NUM
cana-3453	224	20	+	+	ADJ
cana-3453	224	21	2(n−1)f3}≠0	2(n−1)f3}≠0	PROPN
cana-3453	224	22	proof	proof	NOUN
cana-3453	224	23	.	.	PUNCT
cana-3453	225	1	let	let	VERB
cana-3453	225	2	m	m	PRON
cana-3453	225	3	be	be	AUX
cana-3453	225	4	a	a	DET
cana-3453	225	5	generalized	generalized	ADJ
cana-3453	225	6	sasakian	sasakian	ADJ
cana-3453	225	7	-	-	PUNCT
cana-3453	225	8	space	space	NOUN
cana-3453	225	9	-	-	PUNCT
cana-3453	225	10	form	form	NOUN
cana-3453	225	11	,	,	PUNCT
cana-3453	225	12	and	and	CCONJ
cana-3453	225	13	let	let	VERB
cana-3453	225	14	n	n	PRON
cana-3453	225	15	be	be	AUX
cana-3453	225	16	an	an	DET
cana-3453	225	17	invariant	invariant	ADJ
cana-3453	225	18	submanifold	submanifold	NOUN
cana-3453	225	19	of	of	ADP
cana-3453	225	20	m	m	PROPN
cana-3453	225	21	,	,	PUNCT
cana-3453	225	22	then	then	ADV
cana-3453	225	23	the	the	DET
cana-3453	225	24	following	following	ADJ
cana-3453	225	25	statements	statement	NOUN
cana-3453	225	26	are	be	AUX
cana-3453	225	27	equivalent	equivalent	ADJ
cana-3453	225	28	:	:	PUNCT
cana-3453	225	29	𝑸(𝝈	𝑸(𝝈	X
cana-3453	225	30	,	,	PUNCT
cana-3453	225	31	𝑾𝟒	𝑾𝟒	NOUN
cana-3453	225	32	)	)	PUNCT
cana-3453	225	33	=	=	SYM
cana-3453	226	1	𝑸(𝝈	𝑸(𝝈	ADJ
cana-3453	226	2	,	,	PUNCT
cana-3453	226	3	𝑾𝟒)(𝑿	𝑾𝟒)(𝑿	NOUN
cana-3453	226	4	,	,	PUNCT
cana-3453	226	5	𝒀	𝒀	PROPN
cana-3453	226	6	,	,	PUNCT
cana-3453	226	7	𝒁	𝒁	PROPN
cana-3453	226	8	;	;	PUNCT
cana-3453	226	9	𝑼	𝑼	ADJ
cana-3453	226	10	,	,	PUNCT
cana-3453	226	11	𝑽	𝑽	PROPN
cana-3453	226	12	)	)	PUNCT
cana-3453	226	13	=	=	SYM
cana-3453	226	14	(	(	PUNCT
cana-3453	226	15	(	(	PUNCT
cana-3453	226	16	𝑼	𝑼	PROPN
cana-3453	226	17	∧𝝈	∧𝝈	NUM
cana-3453	226	18	𝑽	𝑽	PROPN
cana-3453	226	19	)	)	PUNCT
cana-3453	226	20	.	.	PUNCT
cana-3453	227	1	𝑾𝟒)(𝑿	𝑾𝟒)(𝑿	PROPN
cana-3453	227	2	,	,	PUNCT
cana-3453	227	3	𝒀)𝒁	𝒀)𝒁	VERB
cana-3453	227	4	=	=	SYM
cana-3453	227	5	𝟎	𝟎	PUNCT
cana-3453	227	6	this	this	PRON
cana-3453	227	7	expands	expand	VERB
cana-3453	227	8	to	to	ADP
cana-3453	227	9	:	:	PUNCT
cana-3453	227	10	−𝑾𝟒((𝑼	−𝑾𝟒((𝑼	PROPN
cana-3453	227	11	∧𝝈	∧𝝈	NUM
cana-3453	227	12	𝑽)𝑿	𝑽)𝑿	PROPN
cana-3453	227	13	,	,	PUNCT
cana-3453	227	14	𝒀)𝒁	𝒀)𝒁	VERB
cana-3453	227	15	−	−	PROPN
cana-3453	227	16	𝑾𝟒(𝑿	𝑾𝟒(𝑿	PROPN
cana-3453	227	17	,	,	PUNCT
cana-3453	227	18	(	(	PUNCT
cana-3453	227	19	𝑼	𝑼	PROPN
cana-3453	227	20	∧𝝈	∧𝝈	NUM
cana-3453	227	21	𝑽)𝒀)𝒁	𝑽)𝒀)𝒁	VERB
cana-3453	227	22	−	−	PROPN
cana-3453	227	23	𝑾𝟒(𝑿	𝑾𝟒(𝑿	PROPN
cana-3453	227	24	,	,	PUNCT
cana-3453	227	25	𝒀)(𝑼	𝒀)(𝑼	ADP
cana-3453	227	26	∧𝝈	∧𝝈	NUM
cana-3453	227	27	𝑽)𝒁	𝑽)𝒁	NOUN
cana-3453	227	28	(	(	PUNCT
cana-3453	227	29	6.1	6.1	NUM
cana-3453	227	30	)	)	PUNCT
cana-3453	227	31	applying	apply	VERB
cana-3453	227	32	the	the	DET
cana-3453	227	33	fundamental	fundamental	ADJ
cana-3453	227	34	relationship	relationship	NOUN
cana-3453	227	35	:	:	PUNCT
cana-3453	227	36	−𝝈(𝑽	−𝝈(𝑽	PROPN
cana-3453	227	37	,	,	PUNCT
cana-3453	227	38	𝑿)𝑾𝟒(𝑼	𝑿)𝑾𝟒(𝑼	PROPN
cana-3453	227	39	,	,	PUNCT
cana-3453	227	40	𝒀)𝒁	𝒀)𝒁	VERB
cana-3453	227	41	+	+	CCONJ
cana-3453	227	42	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	227	43	,	,	PUNCT
cana-3453	227	44	𝑿)𝑾𝟒(𝑽	𝑿)𝑾𝟒(𝑽	ADV
cana-3453	227	45	,	,	PUNCT
cana-3453	227	46	𝒀)𝒁	𝒀)𝒁	VERB
cana-3453	227	47	−	−	PROPN
cana-3453	227	48	𝝈(𝑽	𝝈(𝑽	PROPN
cana-3453	227	49	,	,	PUNCT
cana-3453	227	50	𝒀)𝑾𝟒(𝑿	𝒀)𝑾𝟒(𝑿	ADJ
cana-3453	227	51	,	,	PUNCT
cana-3453	227	52	𝑼)𝒁	𝑼)𝒁	NOUN
cana-3453	227	53	+	+	CCONJ
cana-3453	227	54	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	227	55	,	,	PUNCT
cana-3453	227	56	𝒀)𝑾𝟒(𝑿	𝒀)𝑾𝟒(𝑿	NOUN
cana-3453	227	57	,	,	PUNCT
cana-3453	227	58	𝑽)𝒁	𝑽)𝒁	NOUN
cana-3453	227	59	−	−	PROPN
cana-3453	227	60	𝝈(𝑽	𝝈(𝑽	PROPN
cana-3453	227	61	,	,	PUNCT
cana-3453	227	62	𝒁)𝑾𝟒(𝑿	𝒁)𝑾𝟒(𝑿	NOUN
cana-3453	227	63	,	,	PUNCT
cana-3453	227	64	𝒀)𝑼	𝒀)𝑼	NOUN
cana-3453	227	65	+	+	CCONJ
cana-3453	227	66	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	227	67	,	,	PUNCT
cana-3453	227	68	𝒁)𝑾𝟒(𝑿	𝒁)𝑾𝟒(𝑿	NOUN
cana-3453	227	69	,	,	PUNCT
cana-3453	227	70	𝒀)𝑽	𝒀)𝑽	NOUN
cana-3453	227	71	=	=	SYM
cana-3453	227	72	𝟎	𝟎	X
cana-3453	227	73	(	(	PUNCT
cana-3453	227	74	6.2	6.2	NUM
cana-3453	227	75	)	)	PUNCT
cana-3453	227	76	setting	set	VERB
cana-3453	227	77	z	z	NOUN
cana-3453	227	78	=	=	NOUN
cana-3453	227	79	v	v	NOUN
cana-3453	227	80	=	=	SYM
cana-3453	227	81	ξ	ξ	X
cana-3453	227	82	and	and	CCONJ
cana-3453	227	83	utilizing	utilize	VERB
cana-3453	227	84	(	(	PUNCT
cana-3453	227	85	3.3	3.3	NUM
cana-3453	227	86	):	):	PUNCT
cana-3453	227	87	𝝈(𝑼	𝝈(𝑼	PROPN
cana-3453	227	88	,	,	PUNCT
cana-3453	227	89	𝑿)𝑾𝟒(𝝃	𝑿)𝑾𝟒(𝝃	ADJ
cana-3453	227	90	,	,	PUNCT
cana-3453	227	91	𝒀)𝝃	𝒀)𝝃	PUNCT
cana-3453	228	1	+	+	CCONJ
cana-3453	228	2	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	228	3	,	,	PUNCT
cana-3453	228	4	𝒀)𝑾𝟒(𝑿	𝒀)𝑾𝟒(𝑿	ADJ
cana-3453	228	5	,	,	PUNCT
cana-3453	228	6	𝝃)𝝃	𝝃)𝝃	X
cana-3453	228	7	=	=	SYM
cana-3453	228	8	𝟎	𝟎	X
cana-3453	228	9	(	(	PUNCT
cana-3453	228	10	6.3	6.3	NUM
cana-3453	228	11	)	)	PUNCT
cana-3453	228	12	the	the	DET
cana-3453	228	13	internal	internal	ADJ
cana-3453	228	14	structure	structure	NOUN
cana-3453	228	15	of	of	ADP
cana-3453	228	16	w₄	w₄	NOUN
cana-3453	228	17	gives	give	VERB
cana-3453	228	18	rise	rise	NOUN
cana-3453	228	19	to	to	ADP
cana-3453	228	20	relationships	relationship	NOUN
cana-3453	228	21	between	between	ADP
cana-3453	228	22	σ	σ	PROPN
cana-3453	228	23	and	and	CCONJ
cana-3453	228	24	the	the	DET
cana-3453	228	25	ambient	ambient	ADJ
cana-3453	228	26	geometry	geometry	NOUN
cana-3453	228	27	that	that	PRON
cana-3453	228	28	are	be	AUX
cana-3453	228	29	very	very	ADV
cana-3453	228	30	different	different	ADJ
cana-3453	228	31	from	from	ADP
cana-3453	228	32	those	those	PRON
cana-3453	228	33	seen	see	VERB
cana-3453	228	34	with	with	ADP
cana-3453	228	35	earlier	early	ADJ
cana-3453	228	36	tensors	tensor	NOUN
cana-3453	228	37	.	.	PUNCT
cana-3453	229	1	these	these	DET
cana-3453	229	2	connections	connection	NOUN
cana-3453	229	3	offer	offer	VERB
cana-3453	229	4	additional	additional	ADJ
cana-3453	229	5	geometric	geometric	ADJ
cana-3453	229	6	insight	insight	NOUN
cana-3453	229	7	into	into	ADP
cana-3453	229	8	the	the	DET
cana-3453	229	9	nature	nature	NOUN
cana-3453	229	10	of	of	ADP
cana-3453	229	11	totally	totally	ADV
cana-3453	229	12	geodesic	geodesic	ADJ
cana-3453	229	13	submanifolds	submanifold	NOUN
cana-3453	229	14	.	.	PUNCT
cana-3453	230	1	substituting	substitute	VERB
cana-3453	230	2	equation	equation	NOUN
cana-3453	230	3	(	(	PUNCT
cana-3453	230	4	2.19	2.19	NUM
cana-3453	230	5	):	):	PUNCT
cana-3453	230	6	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	230	7	,	,	PUNCT
cana-3453	230	8	𝑿)[(𝒇𝟏	𝑿)[(𝒇𝟏	VERB
cana-3453	230	9	−	−	PROPN
cana-3453	230	10	𝒇𝟑){𝜼(𝒀)𝝃	𝒇𝟑){𝜼(𝒀)𝝃	PUNCT
cana-3453	230	11	−	−	PROPN
cana-3453	230	12	𝒀	𝒀	NOUN
cana-3453	230	13	}	}	PUNCT
cana-3453	230	14	+	+	CCONJ
cana-3453	230	15	𝟏	𝟏	NUM
cana-3453	230	16	𝟐𝒏	𝟐𝒏	NOUN
cana-3453	230	17	{	{	PUNCT
cana-3453	230	18	𝑸𝒀	𝑸𝒀	PROPN
cana-3453	230	19	−	−	PROPN
cana-3453	230	20	𝟐𝒏(𝒇𝟏	𝟐𝒏(𝒇𝟏	ADJ
cana-3453	230	21	−	−	PROPN
cana-3453	230	22	𝒇𝟑)𝜼(𝒀)𝝃	𝒇𝟑)𝜼(𝒀)𝝃	NUM
cana-3453	230	23	]	]	PUNCT
cana-3453	231	1	+	+	NOUN
cana-3453	231	2	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	231	3	,	,	PUNCT
cana-3453	231	4	𝒀)(𝒇𝟏	𝒀)(𝒇𝟏	ADJ
cana-3453	231	5	−	−	NOUN
cana-3453	231	6	𝒇𝟑){𝑿	𝒇𝟑){𝑿	SYM
cana-3453	231	7	−	−	PROPN
cana-3453	231	8	𝜼(𝑿)𝝃	𝜼(𝑿)𝝃	CCONJ
cana-3453	231	9	}	}	PUNCT
cana-3453	231	10	=	=	SYM
cana-3453	231	11	𝟎	𝟎	X
cana-3453	231	12	(	(	PUNCT
cana-3453	231	13	6.4	6.4	NUM
cana-3453	231	14	)	)	PUNCT
cana-3453	231	15	taking	take	VERB
cana-3453	231	16	the	the	DET
cana-3453	231	17	inner	inner	ADJ
cana-3453	231	18	product	product	NOUN
cana-3453	231	19	with	with	ADP
cana-3453	231	20	w	w	PROPN
cana-3453	231	21	:	:	PUNCT
cana-3453	231	22	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	231	23	,	,	PUNCT
cana-3453	231	24	𝑿)[(𝒇𝟏	𝑿)[(𝒇𝟏	VERB
cana-3453	231	25	−	−	PROPN
cana-3453	231	26	𝒇𝟑){𝜼(𝒀)𝜼(𝑾	𝒇𝟑){𝜼(𝒀)𝜼(𝑾	NOUN
cana-3453	231	27	)	)	PUNCT
cana-3453	231	28	−	−	NOUN
cana-3453	232	1	𝒈(𝒀	𝒈(𝒀	NOUN
cana-3453	232	2	,	,	PUNCT
cana-3453	232	3	𝑾	𝑾	ADJ
cana-3453	232	4	)	)	PUNCT
cana-3453	232	5	}	}	PUNCT
cana-3453	233	1	+	+	CCONJ
cana-3453	233	2	𝟏	𝟏	NUM
cana-3453	233	3	𝟐𝒏	𝟐𝒏	NOUN
cana-3453	233	4	{	{	PUNCT
cana-3453	233	5	𝒈(𝑸𝒀	𝒈(𝑸𝒀	NOUN
cana-3453	233	6	,	,	PUNCT
cana-3453	233	7	𝑾	𝑾	ADJ
cana-3453	233	8	)	)	PUNCT
cana-3453	233	9	−	−	PROPN
cana-3453	233	10	𝟐𝒏(𝒇𝟏	𝟐𝒏(𝒇𝟏	ADJ
cana-3453	234	1	−	−	PROPN
cana-3453	234	2	𝒇𝟑)𝜼(𝒀)𝜼(𝑾	𝒇𝟑)𝜼(𝒀)𝜼(𝑾	NUM
cana-3453	234	3	)	)	PUNCT
cana-3453	234	4	]	]	PUNCT
cana-3453	235	1	+	+	NOUN
cana-3453	235	2	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	235	3	,	,	PUNCT
cana-3453	235	4	𝒀)(𝒇𝟏	𝒀)(𝒇𝟏	ADJ
cana-3453	235	5	−	−	NOUN
cana-3453	235	6	𝒇𝟑){𝒈(𝑿	𝒇𝟑){𝒈(𝑿	NUM
cana-3453	235	7	,	,	PUNCT
cana-3453	235	8	𝑾	𝑾	ADJ
cana-3453	235	9	)	)	PUNCT
cana-3453	235	10	−	−	PROPN
cana-3453	235	11	𝜼(𝑿)𝜼(𝑾	𝜼(𝑿)𝜼(𝑾	NOUN
cana-3453	235	12	)	)	PUNCT
cana-3453	235	13	}	}	PUNCT
cana-3453	235	14	=	=	SYM
cana-3453	235	15	𝟎	𝟎	X
cana-3453	235	16	(	(	PUNCT
cana-3453	235	17	6.5	6.5	NUM
cana-3453	235	18	)	)	PUNCT
cana-3453	235	19	contracting	contract	VERB
cana-3453	235	20	y	y	PROPN
cana-3453	235	21	and	and	CCONJ
cana-3453	235	22	w	w	PROPN
cana-3453	235	23	leads	lead	VERB
cana-3453	235	24	to	to	ADP
cana-3453	235	25	:	:	PUNCT
cana-3453	235	26	communications	communication	NOUN
cana-3453	235	27	on	on	ADP
cana-3453	235	28	applied	apply	VERB
cana-3453	235	29	nonlinear	nonlinear	ADJ
cana-3453	235	30	analysis	analysis	NOUN
cana-3453	235	31	issn	issn	NOUN
cana-3453	235	32	:	:	PUNCT
cana-3453	235	33	1074	1074	NUM
cana-3453	235	34	-	-	PUNCT
cana-3453	235	35	133x	133x	NUM
cana-3453	235	36	vol	vol	NOUN
cana-3453	235	37	32	32	NUM
cana-3453	235	38	no	no	NOUN
cana-3453	235	39	.	.	PUNCT
cana-3453	236	1	7s	7	NOUN
cana-3453	236	2	(	(	PUNCT
cana-3453	236	3	2025	2025	NUM
cana-3453	236	4	)	)	PUNCT
cana-3453	236	5	426	426	NUM
cana-3453	236	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3453	236	7	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	236	8	,	,	PUNCT
cana-3453	236	9	𝑿){𝒇𝟏	𝑿){𝒇𝟏	NOUN
cana-3453	237	1	+	+	X
cana-3453	237	2	𝟑𝒇𝟐	𝟑𝒇𝟐	PUNCT
cana-3453	238	1	+	+	PUNCT
cana-3453	238	2	𝟐(𝒏	𝟐(𝒏	NOUN
cana-3453	238	3	−	−	PROPN
cana-3453	239	1	𝟏)𝒇𝟑	𝟏)𝒇𝟑	NOUN
cana-3453	239	2	}	}	PUNCT
cana-3453	239	3	=	=	PUNCT
cana-3453	239	4	𝟎.	𝟎.	X
cana-3453	239	5	(	(	PUNCT
cana-3453	239	6	6.6	6.6	NUM
cana-3453	239	7	)	)	PUNCT
cana-3453	239	8	given	give	VERB
cana-3453	239	9	our	our	PRON
cana-3453	239	10	non	non	ADJ
cana-3453	239	11	-	-	ADJ
cana-3453	239	12	degeneracy	degeneracy	ADJ
cana-3453	239	13	condition	condition	NOUN
cana-3453	239	14	{	{	PUNCT
cana-3453	239	15	f1	f1	PROPN
cana-3453	239	16	+	+	PROPN
cana-3453	239	17	3f2	3f2	NUM
cana-3453	239	18	+	+	NOUN
cana-3453	239	19	2(n−1)f3}≠0	2(n−1)f3}≠0	NUM
cana-3453	239	20	,	,	PUNCT
cana-3453	239	21	we	we	PRON
cana-3453	239	22	conclude	conclude	VERB
cana-3453	239	23	:	:	PUNCT
cana-3453	240	1	σ(u	σ(u	NOUN
cana-3453	240	2	,	,	PUNCT
cana-3453	240	3	x)=0	x)=0	PROPN
cana-3453	240	4	this	this	PRON
cana-3453	240	5	shows	show	VERB
cana-3453	240	6	that	that	SCONJ
cana-3453	240	7	n	n	PRON
cana-3453	240	8	is	be	AUX
cana-3453	240	9	completely	completely	ADV
cana-3453	240	10	geodesic	geodesic	ADJ
cana-3453	240	11	.	.	PUNCT
cana-3453	241	1	the	the	DET
cana-3453	241	2	converse	converse	NOUN
cana-3453	241	3	is	be	AUX
cana-3453	241	4	simple	simple	ADJ
cana-3453	241	5	:	:	PUNCT
cana-3453	241	6	if	if	SCONJ
cana-3453	241	7	n	n	PRON
cana-3453	241	8	is	be	AUX
cana-3453	241	9	totally	totally	ADV
cana-3453	241	10	geodesic	geodesic	ADJ
cana-3453	241	11	,	,	PUNCT
cana-3453	241	12	we	we	PRON
cana-3453	241	13	have	have	VERB
cana-3453	241	14	σ=0	σ=0	NOUN
cana-3453	241	15	identically	identically	ADV
cana-3453	241	16	,	,	PUNCT
cana-3453	241	17	hence	hence	ADV
cana-3453	241	18	q(σ	q(σ	NUM
cana-3453	241	19	,	,	PUNCT
cana-3453	241	20	w4)=0	w4)=0	ADJ
cana-3453	241	21	.	.	PUNCT
cana-3453	242	1	this	this	DET
cana-3453	242	2	theorem	theorem	NOUN
cana-3453	242	3	establishes	establish	VERB
cana-3453	242	4	a	a	DET
cana-3453	242	5	strong	strong	ADJ
cana-3453	242	6	relationship	relationship	NOUN
cana-3453	242	7	between	between	ADP
cana-3453	242	8	the	the	DET
cana-3453	242	9	w4	w4	NOUN
cana-3453	242	10	curvature	curvature	NOUN
cana-3453	242	11	tensor	tensor	NOUN
cana-3453	242	12	and	and	CCONJ
cana-3453	242	13	the	the	DET
cana-3453	242	14	geometric	geometric	ADJ
cana-3453	242	15	properties	property	NOUN
cana-3453	242	16	of	of	ADP
cana-3453	242	17	invariant	invariant	ADJ
cana-3453	242	18	submanifolds	submanifold	NOUN
cana-3453	242	19	,	,	PUNCT
cana-3453	242	20	leading	lead	VERB
cana-3453	242	21	to	to	ADP
cana-3453	242	22	a	a	DET
cana-3453	242	23	novel	novel	ADJ
cana-3453	242	24	criterion	criterion	NOUN
cana-3453	242	25	for	for	ADP
cana-3453	242	26	total	total	ADJ
cana-3453	242	27	geodesicity	geodesicity	NOUN
cana-3453	242	28	that	that	PRON
cana-3453	242	29	enhances	enhance	VERB
cana-3453	242	30	our	our	PRON
cana-3453	242	31	previous	previous	ADJ
cana-3453	242	32	descriptions	description	NOUN
cana-3453	242	33	in	in	ADP
cana-3453	242	34	terms	term	NOUN
cana-3453	242	35	of	of	ADP
cana-3453	242	36	w2	w2	NOUN
cana-3453	242	37	and	and	CCONJ
cana-3453	242	38	w3	w3	PROPN
cana-3453	242	39	.	.	PUNCT
cana-3453	243	1	7	7	X
cana-3453	243	2	.	.	X
cana-3453	243	3	total	total	ADJ
cana-3453	243	4	geodesicity	geodesicity	NOUN
cana-3453	243	5	through	through	ADP
cana-3453	243	6	w6	w6	PROPN
cana-3453	243	7	curvature	curvature	NOUN
cana-3453	243	8	interactions	interaction	NOUN
cana-3453	243	9	in	in	ADP
cana-3453	243	10	this	this	DET
cana-3453	243	11	section	section	NOUN
cana-3453	243	12	,	,	PUNCT
cana-3453	243	13	we	we	PRON
cana-3453	243	14	study	study	VERB
cana-3453	243	15	a	a	DET
cana-3453	243	16	different	different	ADJ
cana-3453	243	17	definition	definition	NOUN
cana-3453	243	18	of	of	ADP
cana-3453	243	19	totally	totally	ADV
cana-3453	243	20	geodesic	geodesic	ADJ
cana-3453	243	21	submanifolds	submanifold	NOUN
cana-3453	243	22	.	.	PUNCT
cana-3453	244	1	we	we	PRON
cana-3453	244	2	analyze	analyze	VERB
cana-3453	244	3	geodesic	geodesic	ADJ
cana-3453	244	4	submanifolds	submanifold	NOUN
cana-3453	244	5	regarding	regard	VERB
cana-3453	244	6	their	their	PRON
cana-3453	244	7	interaction	interaction	NOUN
cana-3453	244	8	with	with	ADP
cana-3453	244	9	the	the	DET
cana-3453	244	10	w6	w6	PROPN
cana-3453	244	11	curvature	curvature	NOUN
cana-3453	244	12	tensor	tensor	NOUN
cana-3453	244	13	.	.	PUNCT
cana-3453	245	1	the	the	DET
cana-3453	245	2	conditions	condition	NOUN
cana-3453	245	3	we	we	PRON
cana-3453	245	4	obtain	obtain	VERB
cana-3453	245	5	afford	afford	VERB
cana-3453	245	6	a	a	DET
cana-3453	245	7	simpler	simple	ADJ
cana-3453	245	8	test	test	NOUN
cana-3453	245	9	than	than	ADP
cana-3453	245	10	those	those	PRON
cana-3453	245	11	of	of	ADP
cana-3453	245	12	earlier	early	ADJ
cana-3453	245	13	sections	section	NOUN
cana-3453	245	14	,	,	PUNCT
cana-3453	245	15	underscoring	underscore	VERB
cana-3453	245	16	the	the	DET
cana-3453	245	17	uniqueness	uniqueness	NOUN
cana-3453	245	18	of	of	ADP
cana-3453	245	19	the	the	DET
cana-3453	245	20	w6	w6	PROPN
cana-3453	245	21	tensor	tensor	NOUN
cana-3453	245	22	.	.	PUNCT
cana-3453	245	23	of	of	ADP
cana-3453	245	24	all	all	DET
cana-3453	245	25	the	the	DET
cana-3453	245	26	curvature	curvature	NOUN
cana-3453	245	27	tensors	tensor	NOUN
cana-3453	245	28	considered	consider	VERB
cana-3453	245	29	here	here	ADV
cana-3453	245	30	,	,	PUNCT
cana-3453	245	31	w₆	w₆	NOUN
cana-3453	245	32	has	have	VERB
cana-3453	245	33	strikingly	strikingly	ADV
cana-3453	245	34	simple	simple	ADJ
cana-3453	245	35	properties	property	NOUN
cana-3453	245	36	in	in	ADP
cana-3453	245	37	terms	term	NOUN
cana-3453	245	38	of	of	ADP
cana-3453	245	39	its	its	PRON
cana-3453	245	40	interaction	interaction	NOUN
cana-3453	245	41	with	with	ADP
cana-3453	245	42	invariant	invariant	ADJ
cana-3453	245	43	submanifolds	submanifold	NOUN
cana-3453	245	44	.	.	PUNCT
cana-3453	246	1	the	the	DET
cana-3453	246	2	simplification	simplification	NOUN
cana-3453	246	3	of	of	ADP
cana-3453	246	4	conditions	condition	NOUN
cana-3453	246	5	on	on	ADP
cana-3453	246	6	w₆	w₆	NOUN
cana-3453	246	7	at	at	ADP
cana-3453	246	8	the	the	DET
cana-3453	246	9	level	level	NOUN
cana-3453	246	10	of	of	ADP
cana-3453	246	11	w₄	w₄	NOUN
cana-3453	246	12	is	be	AUX
cana-3453	246	13	not	not	PART
cana-3453	246	14	an	an	DET
cana-3453	246	15	accident	accident	NOUN
cana-3453	246	16	but	but	CCONJ
cana-3453	246	17	reflects	reflect	VERB
cana-3453	246	18	underlying	underlie	VERB
cana-3453	246	19	geometric	geometric	ADJ
cana-3453	246	20	information	information	NOUN
cana-3453	246	21	about	about	ADP
cana-3453	246	22	generalized	generalized	ADJ
cana-3453	246	23	ssf	ssf	NOUN
cana-3453	246	24	.	.	PUNCT
cana-3453	247	1	the	the	DET
cana-3453	247	2	action	action	NOUN
cana-3453	247	3	of	of	ADP
cana-3453	247	4	this	this	DET
cana-3453	247	5	tensor	tensor	NOUN
cana-3453	247	6	offers	offer	VERB
cana-3453	247	7	a	a	DET
cana-3453	247	8	view	view	NOUN
cana-3453	247	9	of	of	ADP
cana-3453	247	10	the	the	DET
cana-3453	247	11	impact	impact	NOUN
cana-3453	247	12	of	of	ADP
cana-3453	247	13	the	the	DET
cana-3453	247	14	contact	contact	NOUN
cana-3453	247	15	structure	structure	NOUN
cana-3453	247	16	on	on	ADP
cana-3453	247	17	the	the	DET
cana-3453	247	18	geometry	geometry	NOUN
cana-3453	247	19	of	of	ADP
cana-3453	247	20	invariant	invariant	ADJ
cana-3453	247	21	submanifolds	submanifold	NOUN
cana-3453	247	22	.	.	PUNCT
cana-3453	248	1	the	the	DET
cana-3453	248	2	second	second	ADJ
cana-3453	248	3	fundamental	fundamental	ADJ
cana-3453	248	4	form	form	NOUN
cana-3453	248	5	and	and	CCONJ
cana-3453	248	6	its	its	PRON
cana-3453	248	7	geometric	geometric	ADJ
cana-3453	248	8	integration	integration	NOUN
cana-3453	248	9	in	in	ADP
cana-3453	248	10	w₂	w₂	PROPN
cana-3453	248	11	is	be	AUX
cana-3453	248	12	found	find	VERB
cana-3453	248	13	in	in	ADP
cana-3453	248	14	w₆	w₆	NOUN
cana-3453	248	15	,	,	PUNCT
cana-3453	248	16	a	a	DET
cana-3453	248	17	primary	primary	ADJ
cana-3453	248	18	source	source	NOUN
cana-3453	248	19	that	that	PRON
cana-3453	248	20	keeps	keep	VERB
cana-3453	248	21	necessary	necessary	ADJ
cana-3453	248	22	information	information	NOUN
cana-3453	248	23	about	about	ADP
cana-3453	248	24	the	the	DET
cana-3453	248	25	submanifold	submanifold	NOUN
cana-3453	248	26	,	,	PUNCT
cana-3453	248	27	including	include	VERB
cana-3453	248	28	its	its	PRON
cana-3453	248	29	contact	contact	NOUN
cana-3453	248	30	metric	metric	ADJ
cana-3453	248	31	properties	property	NOUN
cana-3453	248	32	with	with	ADP
cana-3453	248	33	that	that	PRON
cana-3453	248	34	of	of	ADP
cana-3453	248	35	the	the	DET
cana-3453	248	36	ambient	ambient	ADJ
cana-3453	248	37	space	space	NOUN
cana-3453	248	38	.	.	PUNCT
cana-3453	249	1	the	the	DET
cana-3453	249	2	non	non	ADJ
cana-3453	249	3	-	-	ADJ
cana-3453	249	4	degeneracy	degeneracy	ADJ
cana-3453	249	5	condition	condition	NOUN
cana-3453	249	6	associated	associate	VERB
cana-3453	249	7	with	with	ADP
cana-3453	249	8	w₆	w₆	NOUN
cana-3453	249	9	is	be	AUX
cana-3453	249	10	relatively	relatively	ADV
cana-3453	249	11	simple	simple	ADJ
cana-3453	249	12	,	,	PUNCT
cana-3453	249	13	which	which	PRON
cana-3453	249	14	could	could	AUX
cana-3453	249	15	lead	lead	VERB
cana-3453	249	16	to	to	ADP
cana-3453	249	17	the	the	DET
cana-3453	249	18	conclusion	conclusion	NOUN
cana-3453	249	19	that	that	SCONJ
cana-3453	249	20	this	this	DET
cana-3453	249	21	tensor	tensor	NOUN
cana-3453	249	22	is	be	AUX
cana-3453	249	23	particularly	particularly	ADV
cana-3453	249	24	suitable	suitable	ADJ
cana-3453	249	25	for	for	ADP
cana-3453	249	26	practical	practical	ADJ
cana-3453	249	27	applications	application	NOUN
cana-3453	249	28	of	of	ADP
cana-3453	249	29	contact	contact	NOUN
cana-3453	249	30	geometry	geometry	NOUN
cana-3453	249	31	.	.	PUNCT
cana-3453	250	1	no	no	DET
cana-3453	250	2	curvature	curvature	NOUN
cana-3453	250	3	tensor	tensor	NOUN
cana-3453	250	4	studied	study	VERB
cana-3453	250	5	to	to	ADP
cana-3453	250	6	date	date	NOUN
cana-3453	250	7	interacts	interact	VERB
cana-3453	250	8	with	with	ADP
cana-3453	250	9	invariant	invariant	ADJ
cana-3453	250	10	submanifolds	submanifold	NOUN
cana-3453	250	11	like	like	ADP
cana-3453	250	12	the	the	DET
cana-3453	250	13	w₆	w₆	NOUN
cana-3453	250	14	tensor	tensor	NOUN
cana-3453	250	15	;	;	PUNCT
cana-3453	250	16	it	it	PRON
cana-3453	250	17	is	be	AUX
cana-3453	250	18	simply	simply	ADV
cana-3453	250	19	an	an	DET
cana-3453	250	20	elegant	elegant	ADJ
cana-3453	250	21	animal	animal	NOUN
cana-3453	250	22	.	.	PUNCT
cana-3453	251	1	such	such	ADJ
cana-3453	251	2	decomposition	decomposition	NOUN
cana-3453	251	3	manifests	manifest	NOUN
cana-3453	251	4	in	in	ADP
cana-3453	251	5	a	a	DET
cana-3453	251	6	simple	simple	ADJ
cana-3453	251	7	algebraic	algebraic	ADJ
cana-3453	251	8	structure	structure	NOUN
cana-3453	251	9	,	,	PUNCT
cana-3453	251	10	providing	provide	VERB
cana-3453	251	11	clearer	clear	ADJ
cana-3453	251	12	geometric	geometric	ADJ
cana-3453	251	13	interpretations	interpretation	NOUN
cana-3453	251	14	while	while	SCONJ
cana-3453	251	15	encoding	encode	VERB
cana-3453	251	16	essential	essential	ADJ
cana-3453	251	17	information	information	NOUN
cana-3453	251	18	about	about	ADP
cana-3453	251	19	the	the	DET
cana-3453	251	20	geodesic	geodesic	ADJ
cana-3453	251	21	properties	property	NOUN
cana-3453	251	22	of	of	ADP
cana-3453	251	23	the	the	DET
cana-3453	251	24	manifold	manifold	NOUN
cana-3453	251	25	.	.	PUNCT
cana-3453	252	1	theorem	theorem	VERB
cana-3453	252	2	7.1	7.1	NUM
cana-3453	252	3	.	.	PUNCT
cana-3453	253	1	suppose	suppose	VERB
cana-3453	253	2	2n(f1−f3)≠0	2n(f1−f3)≠0	NUM
cana-3453	253	3	,	,	PUNCT
cana-3453	253	4	then	then	ADV
cana-3453	253	5	an	an	DET
cana-3453	253	6	invariant	invariant	ADJ
cana-3453	253	7	submanifold	submanifold	NOUN
cana-3453	253	8	n	n	PROPN
cana-3453	253	9	of	of	ADP
cana-3453	253	10	the	the	DET
cana-3453	253	11	generalized	generalized	ADJ
cana-3453	253	12	sasakianspace	sasakianspace	NOUN
cana-3453	253	13	-	-	PUNCT
cana-3453	253	14	form	form	NOUN
cana-3453	253	15	m	m	NOUN
cana-3453	253	16	is	be	AUX
cana-3453	253	17	geodesic	geodesic	ADJ
cana-3453	254	1	if	if	SCONJ
cana-3453	254	2	and	and	CCONJ
cana-3453	254	3	only	only	ADV
cana-3453	254	4	if	if	SCONJ
cana-3453	254	5	q(σ	q(σ	NUM
cana-3453	254	6	,	,	PUNCT
cana-3453	254	7	w6)=0	w6)=0	NOUN
cana-3453	254	8	.	.	PUNCT
cana-3453	255	1	proof	proof	NOUN
cana-3453	255	2	.	.	PUNCT
cana-3453	256	1	consider	consider	VERB
cana-3453	256	2	an	an	DET
cana-3453	256	3	invariant	invariant	ADJ
cana-3453	256	4	submanifold	submanifold	NOUN
cana-3453	256	5	n	n	ADP
cana-3453	256	6	where	where	SCONJ
cana-3453	256	7	q(σ	q(σ	ADJ
cana-3453	256	8	,	,	PUNCT
cana-3453	256	9	w6)=0	w6)=0	PROPN
cana-3453	256	10	.	.	PUNCT
cana-3453	257	1	we	we	PRON
cana-3453	257	2	can	can	AUX
cana-3453	257	3	express	express	VERB
cana-3453	257	4	this	this	DET
cana-3453	257	5	condition	condition	NOUN
cana-3453	257	6	through	through	ADP
cana-3453	257	7	:	:	PUNCT
cana-3453	257	8	𝑸(𝝈	𝑸(𝝈	NUM
cana-3453	257	9	,	,	PUNCT
cana-3453	257	10	𝑾𝟔	𝑾𝟔	PROPN
cana-3453	257	11	)	)	PUNCT
cana-3453	257	12	=	=	SYM
cana-3453	258	1	𝑸(𝝈	𝑸(𝝈	ADJ
cana-3453	258	2	,	,	PUNCT
cana-3453	258	3	𝑾𝟔)(𝑿	𝑾𝟔)(𝑿	PROPN
cana-3453	258	4	,	,	PUNCT
cana-3453	258	5	𝒀	𝒀	PROPN
cana-3453	258	6	,	,	PUNCT
cana-3453	258	7	𝒁	𝒁	PROPN
cana-3453	258	8	;	;	PUNCT
cana-3453	258	9	𝑼	𝑼	ADJ
cana-3453	258	10	,	,	PUNCT
cana-3453	258	11	𝑽	𝑽	PROPN
cana-3453	258	12	)	)	PUNCT
cana-3453	258	13	=	=	SYM
cana-3453	258	14	(	(	PUNCT
cana-3453	258	15	(	(	PUNCT
cana-3453	258	16	𝑼	𝑼	PROPN
cana-3453	258	17	∧𝝈	∧𝝈	NUM
cana-3453	258	18	𝑽	𝑽	PROPN
cana-3453	258	19	)	)	PUNCT
cana-3453	258	20	.	.	PUNCT
cana-3453	259	1	𝑾𝟔)(𝑿	𝑾𝟔)(𝑿	PROPN
cana-3453	259	2	,	,	PUNCT
cana-3453	259	3	𝒀)𝒁	𝒀)𝒁	VERB
cana-3453	259	4	=	=	SYM
cana-3453	259	5	𝟎	𝟎	NUM
cana-3453	259	6	this	this	PRON
cana-3453	259	7	leads	lead	VERB
cana-3453	259	8	to	to	ADP
cana-3453	259	9	:	:	PUNCT
cana-3453	259	10	−𝑾𝟔((𝑼	−𝑾𝟔((𝑼	PROPN
cana-3453	259	11	∧𝝈	∧𝝈	NUM
cana-3453	259	12	𝑽)𝑿	𝑽)𝑿	PROPN
cana-3453	259	13	,	,	PUNCT
cana-3453	259	14	𝒀)𝒁	𝒀)𝒁	VERB
cana-3453	259	15	−	−	PROPN
cana-3453	259	16	𝑾𝟔(𝑿	𝑾𝟔(𝑿	NOUN
cana-3453	259	17	,	,	PUNCT
cana-3453	259	18	(	(	PUNCT
cana-3453	259	19	𝑼	𝑼	PROPN
cana-3453	259	20	∧𝝈	∧𝝈	NUM
cana-3453	259	21	𝑽)𝒀)𝒁	𝑽)𝒀)𝒁	VERB
cana-3453	259	22	−	−	NOUN
cana-3453	259	23	𝑾𝟔(𝑿	𝑾𝟔(𝑿	NOUN
cana-3453	259	24	,	,	PUNCT
cana-3453	259	25	𝒀)(𝑼	𝒀)(𝑼	SYM
cana-3453	259	26	∧𝝈	∧𝝈	NUM
cana-3453	259	27	𝑽)𝒁	𝑽)𝒁	NOUN
cana-3453	259	28	(	(	PUNCT
cana-3453	259	29	7.1	7.1	NUM
cana-3453	259	30	)	)	PUNCT
cana-3453	259	31	applying	apply	VERB
cana-3453	259	32	the	the	DET
cana-3453	259	33	structural	structural	ADJ
cana-3453	259	34	operator	operator	NOUN
cana-3453	259	35	and	and	CCONJ
cana-3453	259	36	expanding	expand	VERB
cana-3453	259	37	:	:	PUNCT
cana-3453	259	38	−𝝈(𝑽	−𝝈(𝑽	PROPN
cana-3453	259	39	,	,	PUNCT
cana-3453	259	40	𝑿)𝑾𝟔(𝑼	𝑿)𝑾𝟔(𝑼	NOUN
cana-3453	259	41	,	,	PUNCT
cana-3453	259	42	𝒀)𝒁	𝒀)𝒁	VERB
cana-3453	259	43	+	+	CCONJ
cana-3453	259	44	𝝈(𝑼	𝝈(𝑼	PROPN
cana-3453	259	45	,	,	PUNCT
cana-3453	259	46	𝑿)𝑾𝟔(𝑽	𝑿)𝑾𝟔(𝑽	NOUN
cana-3453	259	47	,	,	PUNCT
cana-3453	259	48	𝒀)𝒁	𝒀)𝒁	VERB
cana-3453	259	49	−	−	PROPN
cana-3453	259	50	𝝈(𝑽	𝝈(𝑽	PROPN
cana-3453	259	51	,	,	PUNCT
cana-3453	259	52	𝒀)𝑾𝟔(𝑿	𝒀)𝑾𝟔(𝑿	ADV
cana-3453	259	53	,	,	PUNCT
cana-3453	259	54	𝑼)𝒁	𝑼)𝒁	NOUN
cana-3453	259	55	+	+	CCONJ
cana-3453	260	1	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	260	2	,	,	PUNCT
cana-3453	260	3	𝒀)𝑾𝟔(𝑿	𝒀)𝑾𝟔(𝑿	ADV
cana-3453	260	4	,	,	PUNCT
cana-3453	260	5	𝑽)𝒁	𝑽)𝒁	VERB
cana-3453	260	6	−	−	PROPN
cana-3453	260	7	𝝈(𝑽	𝝈(𝑽	PROPN
cana-3453	260	8	,	,	PUNCT
cana-3453	260	9	𝒁)𝑾𝟔(𝑿	𝒁)𝑾𝟔(𝑿	PROPN
cana-3453	260	10	,	,	PUNCT
cana-3453	260	11	𝒀)𝑼	𝒀)𝑼	X
cana-3453	260	12	+	+	CCONJ
cana-3453	260	13	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	260	14	,	,	PUNCT
cana-3453	260	15	𝒁)𝑾𝟔(𝑿	𝒁)𝑾𝟔(𝑿	ADV
cana-3453	260	16	,	,	PUNCT
cana-3453	260	17	𝒀)𝑽	𝒀)𝑽	NOUN
cana-3453	260	18	=	=	SYM
cana-3453	260	19	𝟎	𝟎	X
cana-3453	260	20	(	(	PUNCT
cana-3453	260	21	7.2	7.2	NUM
cana-3453	260	22	)	)	PUNCT
cana-3453	260	23	setting	set	VERB
cana-3453	260	24	z	z	NOUN
cana-3453	260	25	=	=	NOUN
cana-3453	260	26	v	v	NOUN
cana-3453	260	27	=	=	SYM
cana-3453	260	28	ξ	ξ	PROPN
cana-3453	260	29	and	and	CCONJ
cana-3453	260	30	using	use	VERB
cana-3453	260	31	the	the	DET
cana-3453	260	32	fundamental	fundamental	ADJ
cana-3453	260	33	condition	condition	NOUN
cana-3453	260	34	(	(	PUNCT
cana-3453	260	35	3.3	3.3	NUM
cana-3453	260	36	):	):	PUNCT
cana-3453	260	37	communications	communication	NOUN
cana-3453	260	38	on	on	ADP
cana-3453	260	39	applied	apply	VERB
cana-3453	260	40	nonlinear	nonlinear	ADJ
cana-3453	260	41	analysis	analysis	NOUN
cana-3453	260	42	issn	issn	NOUN
cana-3453	260	43	:	:	PUNCT
cana-3453	260	44	1074	1074	NUM
cana-3453	260	45	-	-	PUNCT
cana-3453	260	46	133x	133x	NUM
cana-3453	260	47	vol	vol	NOUN
cana-3453	260	48	32	32	NUM
cana-3453	260	49	no	no	NOUN
cana-3453	260	50	.	.	PUNCT
cana-3453	261	1	7s	7	NOUN
cana-3453	261	2	(	(	PUNCT
cana-3453	261	3	2025	2025	NUM
cana-3453	261	4	)	)	PUNCT
cana-3453	261	5	427	427	NUM
cana-3453	261	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3453	261	7	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	261	8	,	,	PUNCT
cana-3453	261	9	𝑿)𝑾𝟔(𝝃	𝑿)𝑾𝟔(𝝃	INTJ
cana-3453	261	10	,	,	PUNCT
cana-3453	261	11	𝒀)𝝃	𝒀)𝝃	PUNCT
cana-3453	262	1	+	+	CCONJ
cana-3453	262	2	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	262	3	,	,	PUNCT
cana-3453	262	4	𝒀)𝑾𝟔(𝑿	𝒀)𝑾𝟔(𝑿	ADV
cana-3453	262	5	,	,	PUNCT
cana-3453	262	6	𝝃)𝝃	𝝃)𝝃	X
cana-3453	262	7	=	=	SYM
cana-3453	262	8	𝟎	𝟎	X
cana-3453	262	9	(	(	PUNCT
cana-3453	262	10	7.3	7.3	NUM
cana-3453	262	11	)	)	PUNCT
cana-3453	262	12	this	this	DET
cana-3453	262	13	simple	simple	ADJ
cana-3453	262	14	nature	nature	NOUN
cana-3453	262	15	of	of	ADP
cana-3453	262	16	w6	w6	PROPN
cana-3453	262	17	points	point	NOUN
cana-3453	262	18	suggests	suggest	VERB
cana-3453	262	19	that	that	SCONJ
cana-3453	262	20	this	this	DET
cana-3453	262	21	tensor	tensor	NOUN
cana-3453	262	22	plays	play	VERB
cana-3453	262	23	a	a	DET
cana-3453	262	24	special	special	ADJ
cana-3453	262	25	role	role	NOUN
cana-3453	262	26	in	in	ADP
cana-3453	262	27	the	the	DET
cana-3453	262	28	submanifoldgeometry	submanifoldgeometry	NOUN
cana-3453	262	29	description	description	NOUN
cana-3453	262	30	.	.	PUNCT
cana-3453	263	1	in	in	ADP
cana-3453	263	2	contrast	contrast	NOUN
cana-3453	263	3	,	,	PUNCT
cana-3453	263	4	while	while	SCONJ
cana-3453	263	5	the	the	DET
cana-3453	263	6	characterization	characterization	NOUN
cana-3453	263	7	of	of	ADP
cana-3453	263	8	other	other	ADJ
cana-3453	263	9	tensors	tensor	NOUN
cana-3453	263	10	requires	require	VERB
cana-3453	263	11	non	non	ADJ
cana-3453	263	12	-	-	ADJ
cana-3453	263	13	degeneracy	degeneracy	ADJ
cana-3453	263	14	conditions	condition	NOUN
cana-3453	263	15	of	of	ADP
cana-3453	263	16	vast	vast	ADJ
cana-3453	263	17	complexity	complexity	NOUN
cana-3453	263	18	,	,	PUNCT
cana-3453	263	19	w6	w6	PROPN
cana-3453	263	20	enjoys	enjoy	VERB
cana-3453	263	21	an	an	DET
cana-3453	263	22	exquisite	exquisite	ADJ
cana-3453	263	23	simplicity	simplicity	NOUN
cana-3453	263	24	of	of	ADP
cana-3453	263	25	requirements	requirement	NOUN
cana-3453	263	26	.	.	PUNCT
cana-3453	264	1	this	this	DET
cana-3453	264	2	reduction	reduction	NOUN
cana-3453	264	3	means	mean	VERB
cana-3453	264	4	that	that	SCONJ
cana-3453	264	5	w6	w6	PROPN
cana-3453	264	6	might	might	AUX
cana-3453	264	7	be	be	AUX
cana-3453	264	8	especially	especially	ADV
cana-3453	264	9	useful	useful	ADJ
cana-3453	264	10	for	for	ADP
cana-3453	264	11	practical	practical	ADJ
cana-3453	264	12	implementations	implementation	NOUN
cana-3453	264	13	and	and	CCONJ
cana-3453	264	14	theoretical	theoretical	ADJ
cana-3453	264	15	analyses	analysis	NOUN
cana-3453	264	16	that	that	PRON
cana-3453	264	17	require	require	VERB
cana-3453	264	18	reducing	reduce	VERB
cana-3453	264	19	computational	computational	ADJ
cana-3453	264	20	complexity	complexity	NOUN
cana-3453	264	21	,	,	PUNCT
cana-3453	264	22	while	while	SCONJ
cana-3453	264	23	preserving	preserve	VERB
cana-3453	264	24	geometric	geometric	ADJ
cana-3453	264	25	insight	insight	NOUN
cana-3453	264	26	.	.	PUNCT
cana-3453	265	1	substituting	substitute	VERB
cana-3453	265	2	equation	equation	NOUN
cana-3453	265	3	(	(	PUNCT
cana-3453	265	4	2.20	2.20	NUM
cana-3453	265	5	):	):	PUNCT
cana-3453	265	6	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	265	7	,	,	PUNCT
cana-3453	265	8	𝑿)[(𝒇𝟏	𝑿)[(𝒇𝟏	VERB
cana-3453	265	9	−	−	PROPN
cana-3453	265	10	𝒇𝟑){𝜼(𝒀)𝝃	𝒇𝟑){𝜼(𝒀)𝝃	PUNCT
cana-3453	265	11	−	−	PROPN
cana-3453	266	1	𝒀	𝒀	NOUN
cana-3453	266	2	}	}	PUNCT
cana-3453	266	3	]	]	PUNCT
cana-3453	266	4	=	=	SYM
cana-3453	266	5	𝟎	𝟎	X
cana-3453	266	6	(	(	PUNCT
cana-3453	266	7	7.4	7.4	NUM
cana-3453	266	8	)	)	PUNCT
cana-3453	266	9	taking	take	VERB
cana-3453	266	10	the	the	DET
cana-3453	266	11	inner	inner	ADJ
cana-3453	266	12	product	product	NOUN
cana-3453	266	13	with	with	ADP
cana-3453	266	14	w	w	PROPN
cana-3453	266	15	:	:	PUNCT
cana-3453	266	16	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	266	17	,	,	PUNCT
cana-3453	266	18	𝑿)[(𝒇𝟏	𝑿)[(𝒇𝟏	VERB
cana-3453	266	19	−	−	PROPN
cana-3453	266	20	𝒇𝟑){𝜼(𝒀)𝜼(𝑾	𝒇𝟑){𝜼(𝒀)𝜼(𝑾	NOUN
cana-3453	266	21	)	)	PUNCT
cana-3453	267	1	−	−	NOUN
cana-3453	267	2	𝒈(𝒀	𝒈(𝒀	NOUN
cana-3453	267	3	,	,	PUNCT
cana-3453	267	4	𝑾	𝑾	ADJ
cana-3453	267	5	)	)	PUNCT
cana-3453	267	6	}	}	PUNCT
cana-3453	267	7	]	]	PUNCT
cana-3453	267	8	=	=	SYM
cana-3453	267	9	𝟎	𝟎	X
cana-3453	267	10	(	(	PUNCT
cana-3453	267	11	7.5	7.5	NUM
cana-3453	267	12	)	)	PUNCT
cana-3453	267	13	contracting	contract	VERB
cana-3453	267	14	y	y	PROPN
cana-3453	267	15	and	and	CCONJ
cana-3453	267	16	w	w	PROPN
cana-3453	267	17	yields	yield	NOUN
cana-3453	267	18	:	:	PUNCT
cana-3453	267	19	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	267	20	,	,	PUNCT
cana-3453	267	21	𝑿){𝟐𝒏(𝒇𝟏	𝑿){𝟐𝒏(𝒇𝟏	VERB
cana-3453	267	22	−	−	PROPN
cana-3453	267	23	𝒇𝟑	𝒇𝟑	NOUN
cana-3453	267	24	)	)	PUNCT
cana-3453	267	25	}	}	PUNCT
cana-3453	268	1	=	=	SYM
cana-3453	268	2	𝟎.	𝟎.	X
cana-3453	268	3	(	(	PUNCT
cana-3453	268	4	7.6	7.6	NUM
cana-3453	268	5	)	)	PUNCT
cana-3453	268	6	under	under	ADP
cana-3453	268	7	our	our	PRON
cana-3453	268	8	non	non	ADJ
cana-3453	268	9	-	-	ADJ
cana-3453	268	10	degeneracy	degeneracy	ADJ
cana-3453	268	11	condition	condition	NOUN
cana-3453	268	12	2n(f1−f3)≠0	2n(f1−f3)≠0	NUM
cana-3453	268	13	,	,	PUNCT
cana-3453	268	14	we	we	PRON
cana-3453	268	15	obtain	obtain	VERB
cana-3453	268	16	:	:	PUNCT
cana-3453	268	17	σ(u	σ(u	NOUN
cana-3453	268	18	,	,	PUNCT
cana-3453	268	19	x)=0	x)=0	PROPN
cana-3453	268	20	it	it	PRON
cana-3453	268	21	follows	follow	VERB
cana-3453	268	22	that	that	SCONJ
cana-3453	268	23	n	n	PRON
cana-3453	268	24	is	be	AUX
cana-3453	268	25	geodesic	geodesic	ADJ
cana-3453	268	26	.	.	PUNCT
cana-3453	269	1	conversely	conversely	ADV
cana-3453	269	2	,	,	PUNCT
cana-3453	269	3	any	any	DET
cana-3453	269	4	geodesic	geodesic	NOUN
cana-3453	269	5	submanifold	submanifold	NOUN
cana-3453	269	6	satisfies	satisfy	VERB
cana-3453	269	7	q(σ	q(σ	NUM
cana-3453	269	8	,	,	PUNCT
cana-3453	269	9	w6)=0	w6)=0	VERB
cana-3453	269	10	by	by	ADP
cana-3453	269	11	definition	definition	NOUN
cana-3453	269	12	;	;	PUNCT
cana-3453	269	13	thus	thus	ADV
cana-3453	269	14	,	,	PUNCT
cana-3453	269	15	the	the	DET
cana-3453	269	16	converse	converse	NOUN
cana-3453	269	17	holds	hold	VERB
cana-3453	269	18	trivially	trivially	ADV
cana-3453	269	19	.	.	PUNCT
cana-3453	270	1	as	as	ADP
cana-3453	270	2	such	such	ADJ
cana-3453	270	3	,	,	PUNCT
cana-3453	270	4	this	this	DET
cana-3453	270	5	theorem	theorem	NOUN
cana-3453	270	6	gives	give	VERB
cana-3453	270	7	a	a	DET
cana-3453	270	8	remarkably	remarkably	ADV
cana-3453	270	9	beautiful	beautiful	ADJ
cana-3453	270	10	characterization	characterization	NOUN
cana-3453	270	11	of	of	ADP
cana-3453	270	12	totally	totally	ADV
cana-3453	270	13	geodesic	geodesic	ADJ
cana-3453	270	14	invariant	invariant	ADJ
cana-3453	270	15	submanifolds	submanifold	NOUN
cana-3453	270	16	in	in	ADP
cana-3453	270	17	terms	term	NOUN
cana-3453	270	18	of	of	ADP
cana-3453	270	19	the	the	DET
cana-3453	270	20	w6	w6	PROPN
cana-3453	270	21	curvature	curvature	NOUN
cana-3453	270	22	tensor	tensor	NOUN
cana-3453	270	23	.	.	PUNCT
cana-3453	271	1	the	the	DET
cana-3453	271	2	geometrical	geometrical	ADJ
cana-3453	271	3	structure	structure	NOUN
cana-3453	271	4	of	of	ADP
cana-3453	271	5	these	these	DET
cana-3453	271	6	fold	fold	ADJ
cana-3453	271	7	submanifolds	submanifold	NOUN
cana-3453	271	8	might	might	AUX
cana-3453	271	9	be	be	AUX
cana-3453	271	10	analyzed	analyze	VERB
cana-3453	271	11	particularly	particularly	ADV
cana-3453	271	12	well	well	ADV
cana-3453	271	13	within	within	ADP
cana-3453	271	14	the	the	DET
cana-3453	271	15	family	family	NOUN
cana-3453	271	16	of	of	ADP
cana-3453	271	17	orbits	orbit	NOUN
cana-3453	271	18	of	of	ADP
cana-3453	271	19	the	the	DET
cana-3453	271	20	w6	w6	PROPN
cana-3453	271	21	representation	representation	NOUN
cana-3453	271	22	thanks	thank	NOUN
cana-3453	271	23	to	to	ADP
cana-3453	271	24	the	the	DET
cana-3453	271	25	non	non	ADJ
cana-3453	271	26	-	-	ADJ
cana-3453	271	27	degeneracy	degeneracy	ADJ
cana-3453	271	28	condition	condition	NOUN
cana-3453	271	29	2n(f1−f3)≠0	2n(f1−f3)≠0	NUM
cana-3453	271	30	itself	itself	PRON
cana-3453	271	31	being	be	AUX
cana-3453	271	32	extremely	extremely	ADV
cana-3453	271	33	easy	easy	ADJ
cana-3453	271	34	to	to	PART
cana-3453	271	35	check	check	VERB
cana-3453	271	36	.	.	PUNCT
cana-3453	272	1	so	so	ADV
cana-3453	272	2	,	,	PUNCT
cana-3453	272	3	the	the	DET
cana-3453	272	4	very	very	ADV
cana-3453	272	5	simple	simple	ADJ
cana-3453	272	6	non	non	ADJ
cana-3453	272	7	-	-	ADJ
cana-3453	272	8	degeneracy	degeneracy	ADJ
cana-3453	272	9	condition	condition	NOUN
cana-3453	272	10	2n(f₁	2n(f₁	NUM
cana-3453	272	11	−	−	PROPN
cana-3453	272	12	f₃	f₃	PROPN
cana-3453	272	13	)	)	PUNCT
cana-3453	272	14	≠	≠	PROPN
cana-3453	272	15	0	0	NUM
cana-3453	272	16	indicates	indicate	VERB
cana-3453	272	17	that	that	SCONJ
cana-3453	272	18	w₆	w₆	NOUN
cana-3453	272	19	encodes	encode	VERB
cana-3453	272	20	fundamental	fundamental	ADJ
cana-3453	272	21	geometric	geometric	ADJ
cana-3453	272	22	properties	property	NOUN
cana-3453	272	23	more	more	ADV
cana-3453	272	24	directly	directly	ADV
cana-3453	272	25	than	than	ADP
cana-3453	272	26	any	any	DET
cana-3453	272	27	other	other	ADJ
cana-3453	272	28	curvature	curvature	NOUN
cana-3453	272	29	tensors	tensor	NOUN
cana-3453	272	30	.	.	PUNCT
cana-3453	273	1	this	this	DET
cana-3453	273	2	simplification	simplification	NOUN
cana-3453	273	3	is	be	AUX
cana-3453	273	4	not	not	PART
cana-3453	273	5	just	just	ADV
cana-3453	273	6	algebraic	algebraic	ADJ
cana-3453	273	7	but	but	CCONJ
cana-3453	273	8	is	be	AUX
cana-3453	273	9	rooted	root	VERB
cana-3453	273	10	in	in	ADP
cana-3453	273	11	the	the	DET
cana-3453	273	12	basic	basic	ADJ
cana-3453	273	13	geometric	geometric	ADJ
cana-3453	273	14	properties	property	NOUN
cana-3453	273	15	.	.	PUNCT
cana-3453	274	1	8	8	NUM
cana-3453	274	2	.	.	X
cana-3453	274	3	characterization	characterization	NOUN
cana-3453	274	4	through	through	ADP
cana-3453	274	5	w7	w7	ADJ
cana-3453	274	6	curvature	curvature	NOUN
cana-3453	274	7	interactions	interaction	NOUN
cana-3453	274	8	we	we	PRON
cana-3453	274	9	conclude	conclude	VERB
cana-3453	274	10	this	this	DET
cana-3453	274	11	section	section	NOUN
cana-3453	274	12	with	with	ADP
cana-3453	274	13	our	our	PRON
cana-3453	274	14	main	main	ADJ
cana-3453	274	15	characterization	characterization	NOUN
cana-3453	274	16	of	of	ADP
cana-3453	274	17	totally	totally	ADV
cana-3453	274	18	geodesic	geodesic	ADJ
cana-3453	274	19	submanifolds	submanifold	NOUN
cana-3453	274	20	in	in	ADP
cana-3453	274	21	terms	term	NOUN
cana-3453	274	22	of	of	ADP
cana-3453	274	23	the	the	DET
cana-3453	274	24	w7	w7	ADJ
cana-3453	274	25	curvature	curvature	NOUN
cana-3453	274	26	tensor	tensor	NOUN
cana-3453	274	27	.	.	PUNCT
cana-3453	275	1	the	the	DET
cana-3453	275	2	relations	relation	NOUN
cana-3453	275	3	we	we	PRON
cana-3453	275	4	obtain	obtain	VERB
cana-3453	275	5	scale	scale	NOUN
cana-3453	275	6	our	our	PRON
cana-3453	275	7	thorough	thorough	ADJ
cana-3453	275	8	study	study	NOUN
cana-3453	275	9	between	between	ADP
cana-3453	275	10	relations	relation	NOUN
cana-3453	275	11	of	of	ADP
cana-3453	275	12	curvature	curvature	NOUN
cana-3453	275	13	tensors	tensor	NOUN
cana-3453	275	14	in	in	ADP
cana-3453	275	15	generalized	generalized	ADJ
cana-3453	275	16	ssf	ssf	NOUN
cana-3453	275	17	.	.	PUNCT
cana-3453	276	1	our	our	PRON
cana-3453	276	2	work	work	NOUN
cana-3453	276	3	,	,	PUNCT
cana-3453	276	4	as	as	SCONJ
cana-3453	276	5	the	the	DET
cana-3453	276	6	previously	previously	ADV
cana-3453	276	7	selected	select	VERB
cana-3453	276	8	paragraphs	paragraph	NOUN
cana-3453	276	9	and	and	CCONJ
cana-3453	276	10	sentence	sentence	NOUN
cana-3453	276	11	describes	describe	NOUN
cana-3453	276	12	,	,	PUNCT
cana-3453	276	13	is	be	AUX
cana-3453	276	14	an	an	DET
cana-3453	276	15	extension	extension	NOUN
cana-3453	276	16	of	of	ADP
cana-3453	276	17	the	the	DET
cana-3453	276	18	aforementioned	aforementioned	ADJ
cana-3453	276	19	one	one	NOUN
cana-3453	276	20	with	with	ADP
cana-3453	276	21	the	the	DET
cana-3453	276	22	introduction	introduction	NOUN
cana-3453	276	23	of	of	ADP
cana-3453	276	24	the	the	DET
cana-3453	276	25	w₇	w₇	NOUN
cana-3453	276	26	curvature	curvature	NOUN
cana-3453	276	27	tensor	tensor	NOUN
cana-3453	276	28	to	to	PART
cana-3453	276	29	fully	fully	ADV
cana-3453	276	30	complete	complete	VERB
cana-3453	276	31	our	our	PRON
cana-3453	276	32	knowledge	knowledge	NOUN
cana-3453	276	33	about	about	ADP
cana-3453	276	34	the	the	DET
cana-3453	276	35	problem	problem	NOUN
cana-3453	276	36	in	in	ADP
cana-3453	276	37	the	the	DET
cana-3453	276	38	context	context	NOUN
cana-3453	276	39	of	of	ADP
cana-3453	276	40	generalized	generalized	ADJ
cana-3453	276	41	sasakian	sasakian	ADJ
cana-3453	276	42	-	-	PUNCT
cana-3453	276	43	social	social	ADJ
cana-3453	276	44	forms	form	NOUN
cana-3453	276	45	.	.	PUNCT
cana-3453	277	1	this	this	DET
cana-3453	277	2	tensor	tensor	NOUN
cana-3453	277	3	has	have	VERB
cana-3453	277	4	some	some	DET
cana-3453	277	5	elements	element	NOUN
cana-3453	277	6	found	find	VERB
cana-3453	277	7	in	in	ADP
cana-3453	277	8	previously	previously	ADV
cana-3453	277	9	explored	explore	VERB
cana-3453	277	10	curvature	curvature	NOUN
cana-3453	277	11	tensors	tensor	NOUN
cana-3453	277	12	but	but	CCONJ
cana-3453	277	13	also	also	ADV
cana-3453	277	14	has	have	VERB
cana-3453	277	15	some	some	DET
cana-3453	277	16	peculiarities	peculiarity	NOUN
cana-3453	277	17	.	.	PUNCT
cana-3453	278	1	particularly	particularly	ADV
cana-3453	278	2	,	,	PUNCT
cana-3453	278	3	the	the	DET
cana-3453	278	4	nature	nature	NOUN
cana-3453	278	5	of	of	ADP
cana-3453	278	6	w₇	w₇	NOUN
cana-3453	278	7	in	in	ADP
cana-3453	278	8	the	the	DET
cana-3453	278	9	characteristic	characteristic	ADJ
cana-3453	278	10	direction	direction	NOUN
cana-3453	278	11	is	be	AUX
cana-3453	278	12	key	key	ADJ
cana-3453	278	13	to	to	ADP
cana-3453	278	14	understanding	understand	VERB
cana-3453	278	15	the	the	DET
cana-3453	278	16	contact	contact	NOUN
cana-3453	278	17	structure	structure	NOUN
cana-3453	278	18	's	's	PART
cana-3453	278	19	influence	influence	NOUN
cana-3453	278	20	over	over	ADP
cana-3453	278	21	the	the	DET
cana-3453	278	22	geometries	geometry	NOUN
cana-3453	278	23	of	of	ADP
cana-3453	278	24	invariant	invariant	ADJ
cana-3453	278	25	submanifolds	submanifold	NOUN
cana-3453	278	26	.	.	PUNCT
cana-3453	279	1	the	the	DET
cana-3453	279	2	relationship	relationship	NOUN
cana-3453	279	3	between	between	ADP
cana-3453	279	4	w₇	w₇	PROPN
cana-3453	279	5	and	and	CCONJ
cana-3453	279	6	the	the	DET
cana-3453	279	7	second	second	ADJ
cana-3453	279	8	fundamental	fundamental	ADJ
cana-3453	279	9	form	form	NOUN
cana-3453	279	10	has	have	AUX
cana-3453	279	11	shown	show	VERB
cana-3453	279	12	subtle	subtle	ADJ
cana-3453	279	13	geometric	geometric	ADJ
cana-3453	279	14	properties	property	NOUN
cana-3453	279	15	,	,	PUNCT
cana-3453	279	16	which	which	PRON
cana-3453	279	17	hold	hold	VERB
cana-3453	279	18	in	in	ADP
cana-3453	279	19	addition	addition	NOUN
cana-3453	279	20	to	to	ADP
cana-3453	279	21	the	the	DET
cana-3453	279	22	victory	victory	NOUN
cana-3453	279	23	from	from	ADP
cana-3453	279	24	another	another	DET
cana-3453	279	25	curvature	curvature	NOUN
cana-3453	279	26	tensor	tensor	NOUN
cana-3453	279	27	.	.	PUNCT
cana-3453	280	1	moreover	moreover	ADV
cana-3453	280	2	,	,	PUNCT
cana-3453	280	3	while	while	SCONJ
cana-3453	280	4	the	the	DET
cana-3453	280	5	non	non	ADJ
cana-3453	280	6	-	-	ADJ
cana-3453	280	7	degeneracy	degeneracy	ADJ
cana-3453	280	8	condition	condition	NOUN
cana-3453	280	9	for	for	ADP
cana-3453	280	10	w₇	w₇	NOUN
cana-3453	280	11	is	be	AUX
cana-3453	280	12	far	far	ADV
cana-3453	280	13	more	more	ADV
cana-3453	280	14	complicated	complicated	ADJ
cana-3453	280	15	than	than	ADP
cana-3453	280	16	that	that	PRON
cana-3453	280	17	of	of	ADP
cana-3453	280	18	w₆	w₆	NOUN
cana-3453	280	19	,	,	PUNCT
cana-3453	280	20	it	it	PRON
cana-3453	280	21	then	then	ADV
cana-3453	280	22	picks	pick	VERB
cana-3453	280	23	up	up	ADP
cana-3453	280	24	essential	essential	ADJ
cana-3453	280	25	geometric	geometric	ADJ
cana-3453	280	26	information	information	NOUN
cana-3453	280	27	that	that	PRON
cana-3453	280	28	other	other	ADJ
cana-3453	280	29	curvature	curvature	NOUN
cana-3453	280	30	tensors	tensor	NOUN
cana-3453	280	31	alone	alone	ADV
cana-3453	280	32	can	can	AUX
cana-3453	280	33	not	not	PART
cana-3453	280	34	see	see	VERB
cana-3453	280	35	.	.	PUNCT
cana-3453	281	1	communications	communication	NOUN
cana-3453	281	2	on	on	ADP
cana-3453	281	3	applied	apply	VERB
cana-3453	281	4	nonlinear	nonlinear	ADJ
cana-3453	281	5	analysis	analysis	NOUN
cana-3453	281	6	issn	issn	NOUN
cana-3453	281	7	:	:	PUNCT
cana-3453	281	8	1074	1074	NUM
cana-3453	281	9	-	-	PUNCT
cana-3453	281	10	133x	133x	NUM
cana-3453	281	11	vol	vol	NOUN
cana-3453	281	12	32	32	NUM
cana-3453	281	13	no	no	NOUN
cana-3453	281	14	.	.	PUNCT
cana-3453	282	1	7s	7	NOUN
cana-3453	282	2	(	(	PUNCT
cana-3453	282	3	2025	2025	NUM
cana-3453	282	4	)	)	PUNCT
cana-3453	282	5	428	428	NUM
cana-3453	283	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3453	283	2	there	there	PRON
cana-3453	283	3	is	be	VERB
cana-3453	283	4	a	a	DET
cana-3453	283	5	complete	complete	ADJ
cana-3453	283	6	geometric	geometric	ADJ
cana-3453	283	7	characterization	characterization	NOUN
cana-3453	283	8	of	of	ADP
cana-3453	283	9	totally	totally	ADV
cana-3453	283	10	geodesic	geodesic	ADJ
cana-3453	283	11	properties	property	NOUN
cana-3453	283	12	with	with	ADP
cana-3453	283	13	the	the	DET
cana-3453	283	14	last	last	ADJ
cana-3453	283	15	curvature	curvature	NOUN
cana-3453	283	16	tensor	tensor	NOUN
cana-3453	283	17	w₇.	w₇.	PUNCT
cana-3453	283	18	unlike	unlike	ADP
cana-3453	283	19	previous	previous	ADJ
cana-3453	283	20	tensors	tensor	NOUN
cana-3453	283	21	,	,	PUNCT
cana-3453	283	22	its	its	PRON
cana-3453	283	23	structure	structure	NOUN
cana-3453	283	24	combines	combine	VERB
cana-3453	283	25	elements	element	NOUN
cana-3453	283	26	seen	see	VERB
cana-3453	283	27	in	in	ADP
cana-3453	283	28	previous	previous	ADJ
cana-3453	283	29	tensors	tensor	NOUN
cana-3453	283	30	but	but	CCONJ
cana-3453	283	31	with	with	ADP
cana-3453	283	32	distinct	distinct	ADJ
cana-3453	283	33	features	feature	NOUN
cana-3453	283	34	that	that	PRON
cana-3453	283	35	capture	capture	VERB
cana-3453	283	36	different	different	ADJ
cana-3453	283	37	geometric	geometric	ADJ
cana-3453	283	38	information	information	NOUN
cana-3453	283	39	.	.	PUNCT
cana-3453	284	1	theorem	theorem	VERB
cana-3453	284	2	8.1	8.1	NUM
cana-3453	284	3	.	.	PUNCT
cana-3453	285	1	for	for	ADP
cana-3453	285	2	an	an	DET
cana-3453	285	3	invariant	invariant	ADJ
cana-3453	285	4	submanifold	submanifold	NOUN
cana-3453	285	5	n	n	PROPN
cana-3453	285	6	of	of	ADP
cana-3453	285	7	a	a	DET
cana-3453	285	8	generalized	generalized	ADJ
cana-3453	285	9	sasakian	sasakian	ADJ
cana-3453	285	10	-	-	PUNCT
cana-3453	285	11	space	space	NOUN
cana-3453	285	12	-	-	PUNCT
cana-3453	285	13	form	form	NOUN
cana-3453	285	14	m	m	NOUN
cana-3453	285	15	,	,	PUNCT
cana-3453	285	16	n	n	PRON
cana-3453	285	17	is	be	AUX
cana-3453	285	18	geodesic	geodesic	ADJ
cana-3453	285	19	if	if	SCONJ
cana-3453	285	20	and	and	CCONJ
cana-3453	285	21	only	only	ADV
cana-3453	285	22	if	if	SCONJ
cana-3453	285	23	(	(	PUNCT
cana-3453	285	24	𝜎	𝜎	INTJ
cana-3453	285	25	,	,	PUNCT
cana-3453	285	26	𝑊7	𝑊7	PROPN
cana-3453	285	27	)	)	PUNCT
cana-3453	286	1	=	=	PUNCT
cana-3453	286	2	0	0	NUM
cana-3453	286	3	,	,	PUNCT
cana-3453	286	4	provided	provide	VERB
cana-3453	286	5	that	that	SCONJ
cana-3453	286	6	{	{	PUNCT
cana-3453	286	7	2𝑛(1	2𝑛(1	NOUN
cana-3453	286	8	−	−	NOUN
cana-3453	286	9	2𝑛)𝑓1	2𝑛)𝑓1	NUM
cana-3453	286	10	+	+	CCONJ
cana-3453	286	11	3𝑓2	3𝑓2	NUM
cana-3453	286	12	−	−	PROPN
cana-3453	286	13	(	(	PUNCT
cana-3453	286	14	2𝑛	2𝑛	PROPN
cana-3453	286	15	+	+	CCONJ
cana-3453	286	16	1)(2𝑛	1)(2𝑛	NUM
cana-3453	286	17	−	−	PROPN
cana-3453	286	18	1)𝑓3	1)𝑓3	NUM
cana-3453	286	19	}	}	PUNCT
cana-3453	286	20	≠	≠	PROPN
cana-3453	286	21	0	0	NUM
cana-3453	286	22	.	.	PUNCT
cana-3453	287	1	proof	proof	NOUN
cana-3453	287	2	.	.	PUNCT
cana-3453	288	1	consider	consider	VERB
cana-3453	288	2	an	an	DET
cana-3453	288	3	invariant	invariant	ADJ
cana-3453	288	4	submanifold	submanifold	NOUN
cana-3453	288	5	n	n	ADP
cana-3453	288	6	satisfying	satisfy	VERB
cana-3453	288	7	q(σ	q(σ	NUM
cana-3453	288	8	,	,	PUNCT
cana-3453	288	9	w7)=0	w7)=0	NOUN
cana-3453	288	10	.	.	PUNCT
cana-3453	289	1	we	we	PRON
cana-3453	289	2	begin	begin	VERB
cana-3453	289	3	by	by	ADP
cana-3453	289	4	expressing	express	VERB
cana-3453	289	5	:	:	PUNCT
cana-3453	289	6	𝑸(𝝈	𝑸(𝝈	ADJ
cana-3453	289	7	,	,	PUNCT
cana-3453	289	8	𝑾𝟕	𝑾𝟕	ADJ
cana-3453	289	9	)	)	PUNCT
cana-3453	289	10	=	=	PUNCT
cana-3453	290	1	𝑸(𝝈	𝑸(𝝈	ADJ
cana-3453	290	2	,	,	PUNCT
cana-3453	290	3	𝑾𝟕)(𝑿	𝑾𝟕)(𝑿	ADJ
cana-3453	290	4	,	,	PUNCT
cana-3453	290	5	𝒀	𝒀	PROPN
cana-3453	290	6	,	,	PUNCT
cana-3453	290	7	𝒁	𝒁	PROPN
cana-3453	290	8	;	;	PUNCT
cana-3453	290	9	𝑼	𝑼	ADJ
cana-3453	290	10	,	,	PUNCT
cana-3453	290	11	𝑽	𝑽	PROPN
cana-3453	290	12	)	)	PUNCT
cana-3453	290	13	=	=	SYM
cana-3453	290	14	(	(	PUNCT
cana-3453	290	15	(	(	PUNCT
cana-3453	290	16	𝑼	𝑼	PROPN
cana-3453	290	17	∧𝝈	∧𝝈	NUM
cana-3453	290	18	𝑽	𝑽	PROPN
cana-3453	290	19	)	)	PUNCT
cana-3453	290	20	.	.	PUNCT
cana-3453	291	1	𝑾𝟕)(𝑿	𝑾𝟕)(𝑿	ADJ
cana-3453	291	2	,	,	PUNCT
cana-3453	291	3	𝒀)𝒁	𝒀)𝒁	VERB
cana-3453	291	4	=	=	SYM
cana-3453	291	5	𝟎	𝟎	PUNCT
cana-3453	291	6	this	this	PRON
cana-3453	291	7	expands	expand	VERB
cana-3453	291	8	to	to	ADP
cana-3453	291	9	:	:	PUNCT
cana-3453	291	10	−𝑾𝟕((𝑼	−𝑾𝟕((𝑼	ADP
cana-3453	291	11	∧𝝈	∧𝝈	NUM
cana-3453	291	12	𝑽)𝑿	𝑽)𝑿	PROPN
cana-3453	291	13	,	,	PUNCT
cana-3453	291	14	𝒀)𝒁	𝒀)𝒁	VERB
cana-3453	291	15	−	−	PROPN
cana-3453	291	16	𝑾𝟕(𝑿	𝑾𝟕(𝑿	PROPN
cana-3453	291	17	,	,	PUNCT
cana-3453	291	18	(	(	PUNCT
cana-3453	291	19	𝑼	𝑼	PROPN
cana-3453	291	20	∧𝝈	∧𝝈	NUM
cana-3453	291	21	𝑽)𝒀)𝒁	𝑽)𝒀)𝒁	VERB
cana-3453	291	22	−	−	NOUN
cana-3453	291	23	𝑾𝟕(𝑿	𝑾𝟕(𝑿	PROPN
cana-3453	291	24	,	,	PUNCT
cana-3453	291	25	𝒀)(𝑼	𝒀)(𝑼	ADP
cana-3453	291	26	∧𝝈	∧𝝈	NUM
cana-3453	291	27	𝑽)𝒁	𝑽)𝒁	NOUN
cana-3453	291	28	(	(	PUNCT
cana-3453	291	29	8.1	8.1	NUM
cana-3453	291	30	)	)	PUNCT
cana-3453	291	31	using	use	VERB
cana-3453	291	32	the	the	DET
cana-3453	291	33	fundamental	fundamental	ADJ
cana-3453	291	34	operator	operator	NOUN
cana-3453	291	35	relationship	relationship	NOUN
cana-3453	291	36	:	:	PUNCT
cana-3453	291	37	−𝝈(𝑽	−𝝈(𝑽	PROPN
cana-3453	291	38	,	,	PUNCT
cana-3453	291	39	𝑿)𝑾𝟕(𝑼	𝑿)𝑾𝟕(𝑼	NOUN
cana-3453	291	40	,	,	PUNCT
cana-3453	291	41	𝒀)𝒁	𝒀)𝒁	VERB
cana-3453	291	42	+	+	CCONJ
cana-3453	291	43	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	291	44	,	,	PUNCT
cana-3453	291	45	𝑿)𝑾𝟕(𝑽	𝑿)𝑾𝟕(𝑽	ADV
cana-3453	291	46	,	,	PUNCT
cana-3453	291	47	𝒀)𝒁	𝒀)𝒁	VERB
cana-3453	291	48	−	−	PROPN
cana-3453	291	49	𝝈(𝑽	𝝈(𝑽	NUM
cana-3453	291	50	,	,	PUNCT
cana-3453	291	51	𝒀)𝑾𝟕(𝑿	𝒀)𝑾𝟕(𝑿	NOUN
cana-3453	291	52	,	,	PUNCT
cana-3453	291	53	𝑼)𝒁	𝑼)𝒁	NOUN
cana-3453	291	54	+	+	CCONJ
cana-3453	291	55	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	291	56	,	,	PUNCT
cana-3453	291	57	𝒀)𝑾𝟕(𝑿	𝒀)𝑾𝟕(𝑿	NOUN
cana-3453	291	58	,	,	PUNCT
cana-3453	291	59	𝑽)𝒁	𝑽)𝒁	VERB
cana-3453	291	60	−	−	PROPN
cana-3453	291	61	𝝈(𝑽	𝝈(𝑽	NUM
cana-3453	291	62	,	,	PUNCT
cana-3453	291	63	𝒁)𝑾𝟕(𝑿	𝒁)𝑾𝟕(𝑿	X
cana-3453	291	64	,	,	PUNCT
cana-3453	291	65	𝒀)𝑼	𝒀)𝑼	NOUN
cana-3453	291	66	+	+	CCONJ
cana-3453	291	67	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	291	68	,	,	PUNCT
cana-3453	291	69	𝒁)𝑾𝟕(𝑿	𝒁)𝑾𝟕(𝑿	X
cana-3453	291	70	,	,	PUNCT
cana-3453	291	71	𝒀)𝑽	𝒀)𝑽	NOUN
cana-3453	291	72	=	=	SYM
cana-3453	291	73	𝟎	𝟎	X
cana-3453	291	74	(	(	PUNCT
cana-3453	291	75	8.2	8.2	NUM
cana-3453	291	76	)	)	PUNCT
cana-3453	291	77	setting	set	VERB
cana-3453	291	78	z	z	NOUN
cana-3453	291	79	=	=	NOUN
cana-3453	291	80	v	v	NOUN
cana-3453	291	81	=	=	SYM
cana-3453	291	82	ξ	ξ	X
cana-3453	291	83	and	and	CCONJ
cana-3453	291	84	applying	apply	VERB
cana-3453	291	85	condition	condition	NOUN
cana-3453	291	86	(	(	PUNCT
cana-3453	291	87	3.3	3.3	NUM
cana-3453	291	88	):	):	PUNCT
cana-3453	291	89	𝝈(𝑼	𝝈(𝑼	PROPN
cana-3453	291	90	,	,	PUNCT
cana-3453	291	91	𝑿)𝑾𝟕(𝝃	𝑿)𝑾𝟕(𝝃	NOUN
cana-3453	291	92	,	,	PUNCT
cana-3453	291	93	𝒀)𝝃	𝒀)𝝃	PUNCT
cana-3453	292	1	+	+	CCONJ
cana-3453	292	2	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	292	3	,	,	PUNCT
cana-3453	292	4	𝒀)𝑾𝟕(𝑿	𝒀)𝑾𝟕(𝑿	NOUN
cana-3453	292	5	,	,	PUNCT
cana-3453	292	6	𝝃)𝝃	𝝃)𝝃	X
cana-3453	292	7	=	=	SYM
cana-3453	292	8	𝟎	𝟎	X
cana-3453	292	9	(	(	PUNCT
cana-3453	292	10	8.3	8.3	NUM
cana-3453	292	11	)	)	PUNCT
cana-3453	292	12	the	the	DET
cana-3453	292	13	behaviour	behaviour	NOUN
cana-3453	292	14	of	of	ADP
cana-3453	292	15	w₇	w₇	PROPN
cana-3453	292	16	along	along	ADP
cana-3453	292	17	the	the	DET
cana-3453	292	18	characteristic	characteristic	ADJ
cana-3453	292	19	direction	direction	NOUN
cana-3453	292	20	reveals	reveal	VERB
cana-3453	292	21	subtle	subtle	ADJ
cana-3453	292	22	geometric	geometric	ADJ
cana-3453	292	23	properties	property	NOUN
cana-3453	292	24	that	that	PRON
cana-3453	292	25	complement	complement	VERB
cana-3453	292	26	our	our	PRON
cana-3453	292	27	understanding	understanding	NOUN
cana-3453	292	28	of	of	ADP
cana-3453	292	29	other	other	ADJ
cana-3453	292	30	curvature	curvature	NOUN
cana-3453	292	31	tensors	tensor	NOUN
cana-3453	292	32	.	.	PUNCT
cana-3453	293	1	these	these	DET
cana-3453	293	2	properties	property	NOUN
cana-3453	293	3	prove	prove	VERB
cana-3453	293	4	essential	essential	ADJ
cana-3453	293	5	in	in	ADP
cana-3453	293	6	establishing	establish	VERB
cana-3453	293	7	a	a	DET
cana-3453	293	8	comprehensive	comprehensive	ADJ
cana-3453	293	9	characterization	characterization	NOUN
cana-3453	293	10	of	of	ADP
cana-3453	293	11	totally	totally	ADV
cana-3453	293	12	geodesic	geodesic	ADJ
cana-3453	293	13	submanifolds	submanifold	NOUN
cana-3453	293	14	.	.	PUNCT
cana-3453	294	1	substituting	substitute	VERB
cana-3453	294	2	equation	equation	NOUN
cana-3453	294	3	(	(	PUNCT
cana-3453	294	4	2.21	2.21	NUM
cana-3453	294	5	):	):	PUNCT
cana-3453	294	6	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	294	7	,	,	PUNCT
cana-3453	294	8	𝑿)[(𝒇𝟏	𝑿)[(𝒇𝟏	VERB
cana-3453	294	9	−	−	PROPN
cana-3453	294	10	𝒇𝟑){𝜼(𝒀)𝝃	𝒇𝟑){𝜼(𝒀)𝝃	PUNCT
cana-3453	294	11	−	−	PROPN
cana-3453	295	1	𝒀	𝒀	NOUN
cana-3453	295	2	}	}	PUNCT
cana-3453	295	3	]	]	PUNCT
cana-3453	296	1	+	+	CCONJ
cana-3453	296	2	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	296	3	,	,	PUNCT
cana-3453	296	4	𝒀)[𝑸𝒀	𝒀)[𝑸𝒀	PROPN
cana-3453	296	5	−	−	PROPN
cana-3453	296	6	𝟐𝒏(𝒇𝟏	𝟐𝒏(𝒇𝟏	ADJ
cana-3453	296	7	−	−	PUNCT
cana-3453	296	8	𝒇𝟑)𝜼(𝑿)𝝃	𝒇𝟑)𝜼(𝑿)𝝃	NOUN
cana-3453	296	9	]	]	X
cana-3453	296	10	=	=	SYM
cana-3453	296	11	𝟎	𝟎	X
cana-3453	296	12	(	(	PUNCT
cana-3453	296	13	8.4	8.4	NUM
cana-3453	296	14	)	)	PUNCT
cana-3453	296	15	taking	take	VERB
cana-3453	296	16	the	the	DET
cana-3453	296	17	inner	inner	ADJ
cana-3453	296	18	product	product	NOUN
cana-3453	296	19	with	with	ADP
cana-3453	296	20	w	w	PROPN
cana-3453	296	21	:	:	PUNCT
cana-3453	296	22	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	296	23	,	,	PUNCT
cana-3453	296	24	𝑿)[(𝒇𝟏	𝑿)[(𝒇𝟏	VERB
cana-3453	296	25	−	−	PROPN
cana-3453	296	26	𝒇𝟑){𝜼(𝒀)𝜼(𝑾	𝒇𝟑){𝜼(𝒀)𝜼(𝑾	NOUN
cana-3453	296	27	)	)	PUNCT
cana-3453	296	28	−	−	NOUN
cana-3453	297	1	𝒈(𝒀	𝒈(𝒀	NOUN
cana-3453	297	2	,	,	PUNCT
cana-3453	297	3	𝑾	𝑾	ADJ
cana-3453	297	4	)	)	PUNCT
cana-3453	297	5	}	}	PUNCT
cana-3453	297	6	]	]	PUNCT
cana-3453	298	1	+	+	CCONJ
cana-3453	298	2	𝝈(𝑼	𝝈(𝑼	NOUN
cana-3453	298	3	,	,	PUNCT
cana-3453	298	4	𝒀)[𝒈(𝑸𝒀	𝒀)[𝒈(𝑸𝒀	NOUN
cana-3453	298	5	,	,	PUNCT
cana-3453	298	6	𝑾	𝑾	ADJ
cana-3453	298	7	)	)	PUNCT
cana-3453	298	8	−	−	PROPN
cana-3453	298	9	𝟐𝒏(𝒇𝟏	𝟐𝒏(𝒇𝟏	ADJ
cana-3453	298	10	−	−	PROPN
cana-3453	298	11	𝒇𝟑)𝜼(𝑿)𝜼(𝑾	𝒇𝟑)𝜼(𝑿)𝜼(𝑾	PROPN
cana-3453	298	12	)	)	PUNCT
cana-3453	298	13	]	]	PUNCT
cana-3453	298	14	=	=	SYM
cana-3453	298	15	𝟎	𝟎	X
cana-3453	298	16	(	(	PUNCT
cana-3453	298	17	8.5	8.5	NUM
cana-3453	298	18	)	)	PUNCT
cana-3453	298	19	contracting	contract	VERB
cana-3453	298	20	y	y	PROPN
cana-3453	298	21	and	and	CCONJ
cana-3453	298	22	w	w	PROPN
cana-3453	298	23	:	:	PUNCT
cana-3453	298	24	𝝈(𝑼	𝝈(𝑼	PROPN
cana-3453	298	25	,	,	PUNCT
cana-3453	298	26	𝑿){𝟐𝒏(𝟏	𝑿){𝟐𝒏(𝟏	NOUN
cana-3453	298	27	−	−	PROPN
cana-3453	298	28	𝟐𝒏)𝒇𝟏	𝟐𝒏)𝒇𝟏	PROPN
cana-3453	299	1	+	+	PUNCT
cana-3453	299	2	𝟑𝒇𝟐	𝟑𝒇𝟐	X
cana-3453	299	3	−	−	PROPN
cana-3453	300	1	(	(	PUNCT
cana-3453	300	2	𝟐𝒏	𝟐𝒏	PROPN
cana-3453	300	3	+	+	NUM
cana-3453	300	4	𝟏)(𝟐𝒏	𝟏)(𝟐𝒏	PROPN
cana-3453	300	5	−	−	PROPN
cana-3453	301	1	𝟏)𝒇𝟑	𝟏)𝒇𝟑	NOUN
cana-3453	301	2	}	}	PUNCT
cana-3453	301	3	=	=	SYM
cana-3453	301	4	𝟎.	𝟎.	X
cana-3453	301	5	(	(	PUNCT
cana-3453	301	6	8.6	8.6	NUM
cana-3453	301	7	)	)	PUNCT
cana-3453	301	8	given	give	VERB
cana-3453	301	9	our	our	PRON
cana-3453	301	10	non	non	ADJ
cana-3453	301	11	-	-	ADJ
cana-3453	301	12	degeneracy	degeneracy	ADJ
cana-3453	301	13	condition	condition	NOUN
cana-3453	301	14	{	{	PUNCT
cana-3453	301	15	2𝑛(1	2𝑛(1	NUM
cana-3453	301	16	−	−	NOUN
cana-3453	301	17	2𝑛)𝑓1	2𝑛)𝑓1	NUM
cana-3453	301	18	+	+	CCONJ
cana-3453	301	19	3𝑓2	3𝑓2	NUM
cana-3453	301	20	−	−	PROPN
cana-3453	301	21	(	(	PUNCT
cana-3453	301	22	2𝑛	2𝑛	PROPN
cana-3453	301	23	+	+	CCONJ
cana-3453	301	24	1)(2𝑛	1)(2𝑛	NUM
cana-3453	301	25	−	−	PROPN
cana-3453	301	26	1)𝑓3	1)𝑓3	NUM
cana-3453	301	27	}	}	PUNCT
cana-3453	301	28	≠	≠	PROPN
cana-3453	301	29	0	0	NUM
cana-3453	301	30	.	.	PUNCT
cana-3453	302	1	we	we	PRON
cana-3453	302	2	conclude	conclude	VERB
cana-3453	302	3	:	:	PUNCT
cana-3453	302	4	σ(u	σ(u	NOUN
cana-3453	302	5	,	,	PUNCT
cana-3453	302	6	x)=0	x)=0	PROPN
cana-3453	302	7	this	this	PRON
cana-3453	302	8	establishes	establish	VERB
cana-3453	302	9	that	that	SCONJ
cana-3453	302	10	n	n	NOUN
cana-3453	302	11	is	be	AUX
cana-3453	302	12	geodesic	geodesic	ADJ
cana-3453	302	13	.	.	PUNCT
cana-3453	303	1	the	the	DET
cana-3453	303	2	converse	converse	NOUN
cana-3453	303	3	follows	follow	VERB
cana-3453	303	4	naturally	naturally	ADV
cana-3453	303	5	as	as	SCONJ
cana-3453	303	6	any	any	DET
cana-3453	303	7	geodesic	geodesic	NOUN
cana-3453	303	8	submanifold	submanifold	NOUN
cana-3453	303	9	satisfies	satisfy	VERB
cana-3453	303	10	q(σ	q(σ	NUM
cana-3453	303	11	,	,	PUNCT
cana-3453	303	12	w7)=0	w7)=0	NOUN
cana-3453	303	13	.	.	PUNCT
cana-3453	304	1	the	the	DET
cana-3453	304	2	non	non	ADJ
cana-3453	304	3	-	-	ADJ
cana-3453	304	4	degeneracy	degeneracy	ADJ
cana-3453	304	5	condition	condition	NOUN
cana-3453	304	6	derived	derive	VERB
cana-3453	304	7	in	in	ADP
cana-3453	304	8	proposition	proposition	NOUN
cana-3453	304	9	w₇	w₇	NOUN
cana-3453	304	10	illuminates	illuminate	VERB
cana-3453	304	11	how	how	SCONJ
cana-3453	304	12	the	the	DET
cana-3453	304	13	contact	contact	NOUN
cana-3453	304	14	structure	structure	NOUN
cana-3453	304	15	(	(	PUNCT
cana-3453	304	16	defined	define	VERB
cana-3453	304	17	by	by	ADP
cana-3453	304	18	a	a	DET
cana-3453	304	19	one	one	NUM
cana-3453	304	20	-	-	PUNCT
cana-3453	304	21	form	form	NOUN
cana-3453	304	22	η	η	NOUN
cana-3453	304	23	)	)	PUNCT
cana-3453	304	24	offers	offer	VERB
cana-3453	304	25	intertwining	intertwine	VERB
cana-3453	304	26	metric	metric	ADJ
cana-3453	304	27	or	or	CCONJ
cana-3453	304	28	geometric	geometric	ADJ
cana-3453	304	29	properties	property	NOUN
cana-3453	304	30	interlacing	interlace	VERB
cana-3453	304	31	that	that	PRON
cana-3453	304	32	guarantee	guarantee	VERB
cana-3453	304	33	the	the	DET
cana-3453	304	34	most	most	ADV
cana-3453	304	35	interesting	interesting	ADJ
cana-3453	304	36	characterization	characterization	NOUN
cana-3453	304	37	by	by	ADP
cana-3453	304	38	geometry	geometry	NOUN
cana-3453	304	39	.	.	PUNCT
cana-3453	305	1	although	although	SCONJ
cana-3453	305	2	w₆	w₆	NOUN
cana-3453	305	3	is	be	AUX
cana-3453	305	4	more	more	ADV
cana-3453	305	5	complicated	complicated	ADJ
cana-3453	305	6	than	than	ADP
cana-3453	305	7	this	this	PRON
cana-3453	305	8	,	,	PUNCT
cana-3453	305	9	this	this	DET
cana-3453	305	10	condition	condition	NOUN
cana-3453	305	11	encodes	encode	VERB
cana-3453	305	12	some	some	DET
cana-3453	305	13	important	important	ADJ
cana-3453	305	14	features	feature	NOUN
cana-3453	305	15	that	that	PRON
cana-3453	305	16	are	be	AUX
cana-3453	305	17	not	not	PART
cana-3453	305	18	easily	easily	ADV
cana-3453	305	19	seen	see	VERB
cana-3453	305	20	through	through	ADP
cana-3453	305	21	other	other	ADJ
cana-3453	305	22	tensors	tensor	NOUN
cana-3453	305	23	.	.	PUNCT
cana-3453	306	1	this	this	DET
cana-3453	306	2	last	last	ADJ
cana-3453	306	3	theory	theory	NOUN
cana-3453	306	4	finalizes	finalize	VERB
cana-3453	306	5	our	our	PRON
cana-3453	306	6	study	study	NOUN
cana-3453	306	7	concerning	concern	VERB
cana-3453	306	8	the	the	DET
cana-3453	306	9	relations	relation	NOUN
cana-3453	306	10	between	between	ADP
cana-3453	306	11	curvature	curvature	NOUN
cana-3453	306	12	tensors	tensor	NOUN
cana-3453	306	13	and	and	CCONJ
cana-3453	306	14	geodesic	geodesic	ADJ
cana-3453	306	15	invariant	invariant	ADJ
cana-3453	306	16	submanifolds	submanifold	NOUN
cana-3453	306	17	of	of	ADP
cana-3453	306	18	generalized	generalized	ADJ
cana-3453	306	19	sasakian	sasakian	ADJ
cana-3453	306	20	-	-	PUNCT
cana-3453	306	21	space	space	NOUN
cana-3453	306	22	-	-	PUNCT
cana-3453	306	23	forms	form	NOUN
cana-3453	306	24	.	.	PUNCT
cana-3453	307	1	along	along	ADP
cana-3453	307	2	with	with	ADP
cana-3453	307	3	our	our	PRON
cana-3453	307	4	previous	previous	ADJ
cana-3453	307	5	results	result	NOUN
cana-3453	307	6	,	,	PUNCT
cana-3453	307	7	we	we	PRON
cana-3453	307	8	communications	communication	VERB
cana-3453	307	9	on	on	ADP
cana-3453	307	10	applied	apply	VERB
cana-3453	307	11	nonlinear	nonlinear	ADJ
cana-3453	307	12	analysis	analysis	NOUN
cana-3453	307	13	issn	issn	NOUN
cana-3453	307	14	:	:	PUNCT
cana-3453	307	15	1074	1074	NUM
cana-3453	307	16	-	-	PUNCT
cana-3453	307	17	133x	133x	NUM
cana-3453	307	18	vol	vol	NOUN
cana-3453	307	19	32	32	NUM
cana-3453	307	20	no	no	NOUN
cana-3453	307	21	.	.	PUNCT
cana-3453	308	1	7s	7	NOUN
cana-3453	308	2	(	(	PUNCT
cana-3453	308	3	2025	2025	NUM
cana-3453	308	4	)	)	PUNCT
cana-3453	308	5	429	429	NUM
cana-3453	308	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3453	308	7	succinctly	succinctly	ADV
cana-3453	308	8	delineate	delineate	VERB
cana-3453	308	9	five	five	NUM
cana-3453	308	10	different	different	ADJ
cana-3453	308	11	yet	yet	CCONJ
cana-3453	308	12	interrelated	interrelated	ADJ
cana-3453	308	13	characterizations	characterization	NOUN
cana-3453	308	14	of	of	ADP
cana-3453	308	15	totally	totally	ADV
cana-3453	308	16	geodesic	geodesic	ADJ
cana-3453	308	17	submanifolds	submanifold	NOUN
cana-3453	308	18	via	via	ADP
cana-3453	308	19	their	their	PRON
cana-3453	308	20	interactions	interaction	NOUN
cana-3453	308	21	with	with	ADP
cana-3453	308	22	the	the	DET
cana-3453	308	23	curvature	curvature	NOUN
cana-3453	308	24	tensors	tensor	NOUN
cana-3453	308	25	w2	w2	NOUN
cana-3453	308	26	to	to	AUX
cana-3453	308	27	w7	w7	PROPN
cana-3453	308	28	.	.	PUNCT
cana-3453	309	1	despite	despite	SCONJ
cana-3453	309	2	its	its	PRON
cana-3453	309	3	apparent	apparent	ADJ
cana-3453	309	4	simplicity	simplicity	NOUN
cana-3453	309	5	,	,	PUNCT
cana-3453	309	6	the	the	DET
cana-3453	309	7	previously	previously	ADV
cana-3453	309	8	defined	define	VERB
cana-3453	309	9	invariants	invariant	NOUN
cana-3453	309	10	provide	provide	VERB
cana-3453	309	11	different	different	ADJ
cana-3453	309	12	characteristic	characteristic	ADJ
cana-3453	309	13	views	view	NOUN
cana-3453	309	14	of	of	ADP
cana-3453	309	15	the	the	DET
cana-3453	309	16	geometric	geometric	ADJ
cana-3453	309	17	structure	structure	NOUN
cana-3453	309	18	of	of	ADP
cana-3453	309	19	invariant	invariant	ADJ
cana-3453	309	20	submanifolds	submanifold	NOUN
cana-3453	309	21	,	,	PUNCT
cana-3453	309	22	as	as	SCONJ
cana-3453	309	23	we	we	PRON
cana-3453	309	24	will	will	AUX
cana-3453	309	25	see	see	VERB
cana-3453	309	26	in	in	ADP
cana-3453	309	27	some	some	DET
cana-3453	309	28	detail	detail	NOUN
cana-3453	309	29	:	:	PUNCT
cana-3453	309	30	the	the	DET
cana-3453	309	31	advantages	advantage	NOUN
cana-3453	309	32	of	of	ADP
cana-3453	309	33	each	each	PRON
cana-3453	309	34	are	be	AUX
cana-3453	309	35	its	its	PRON
cana-3453	309	36	computational	computational	ADJ
cana-3453	309	37	simplicity	simplicity	NOUN
cana-3453	309	38	in	in	ADP
cana-3453	309	39	settings	setting	NOUN
cana-3453	309	40	of	of	ADP
cana-3453	309	41	interest	interest	NOUN
cana-3453	309	42	and	and	CCONJ
cana-3453	309	43	its	its	PRON
cana-3453	309	44	geometrical	geometrical	ADJ
cana-3453	309	45	insight	insight	NOUN
cana-3453	309	46	into	into	ADP
cana-3453	309	47	the	the	DET
cana-3453	309	48	nature	nature	NOUN
cana-3453	309	49	of	of	ADP
cana-3453	309	50	the	the	DET
cana-3453	309	51	adjoint	adjoint	PROPN
cana-3453	309	52	configuration	configuration	NOUN
cana-3453	309	53	.	.	PUNCT
cana-3453	310	1	the	the	DET
cana-3453	310	2	different	different	ADJ
cana-3453	310	3	non	non	ADJ
cana-3453	310	4	-	-	ADJ
cana-3453	310	5	degeneracy	degeneracy	ADJ
cana-3453	310	6	conditions	condition	NOUN
cana-3453	310	7	we	we	PRON
cana-3453	310	8	obtained	obtain	VERB
cana-3453	310	9	indicate	indicate	VERB
cana-3453	310	10	that	that	SCONJ
cana-3453	310	11	the	the	DET
cana-3453	310	12	analysis	analysis	NOUN
cana-3453	310	13	in	in	ADP
cana-3453	310	14	a	a	DET
cana-3453	310	15	given	give	VERB
cana-3453	310	16	space	space	NOUN
cana-3453	310	17	form	form	NOUN
cana-3453	310	18	will	will	AUX
cana-3453	310	19	require	require	VERB
cana-3453	310	20	different	different	ADJ
cana-3453	310	21	curvature	curvature	NOUN
cana-3453	310	22	tensors	tensor	NOUN
cana-3453	310	23	depending	depend	VERB
cana-3453	310	24	on	on	ADP
cana-3453	310	25	the	the	DET
cana-3453	310	26	new	new	ADJ
cana-3453	310	27	functions	function	NOUN
cana-3453	310	28	f1	f1	NOUN
cana-3453	310	29	,	,	PUNCT
cana-3453	310	30	f2	f2	PROPN
cana-3453	310	31	,	,	PUNCT
cana-3453	310	32	and	and	CCONJ
cana-3453	310	33	f3	f3	ADJ
cana-3453	310	34	for	for	ADP
cana-3453	310	35	each	each	DET
cana-3453	310	36	wave	wave	NOUN
cana-3453	310	37	source	source	NOUN
cana-3453	310	38	.	.	PUNCT
cana-3453	311	1	the	the	DET
cana-3453	311	2	study	study	NOUN
cana-3453	311	3	of	of	ADP
cana-3453	311	4	these	these	DET
cana-3453	311	5	five	five	NUM
cana-3453	311	6	curvature	curvature	NOUN
cana-3453	311	7	tensors	tensor	NOUN
cana-3453	311	8	provides	provide	VERB
cana-3453	311	9	a	a	DET
cana-3453	311	10	holistic	holistic	ADJ
cana-3453	311	11	view	view	NOUN
cana-3453	311	12	of	of	ADP
cana-3453	311	13	how	how	SCONJ
cana-3453	311	14	the	the	DET
cana-3453	311	15	contact	contact	NOUN
cana-3453	311	16	metric	metric	ADJ
cana-3453	311	17	structures	structure	NOUN
cana-3453	311	18	affect	affect	VERB
cana-3453	311	19	the	the	DET
cana-3453	311	20	geometry	geometry	NOUN
cana-3453	311	21	of	of	ADP
cana-3453	311	22	invariant	invariant	ADJ
cana-3453	311	23	submanifolds	submanifold	NOUN
cana-3453	311	24	.	.	PUNCT
cana-3453	312	1	while	while	SCONJ
cana-3453	312	2	each	each	DET
cana-3453	312	3	tensor	tensor	NOUN
cana-3453	312	4	is	be	AUX
cana-3453	312	5	novel	novel	ADJ
cana-3453	312	6	in	in	ADP
cana-3453	312	7	its	its	PRON
cana-3453	312	8	own	own	ADJ
cana-3453	312	9	right	right	NOUN
cana-3453	312	10	,	,	PUNCT
cana-3453	312	11	all	all	DET
cana-3453	312	12	three	three	NUM
cana-3453	312	13	provide	provide	VERB
cana-3453	312	14	a	a	DET
cana-3453	312	15	framework	framework	NOUN
cana-3453	312	16	for	for	ADP
cana-3453	312	17	understanding	understand	VERB
cana-3453	312	18	geodesic	geodesic	ADJ
cana-3453	312	19	submanifolds	submanifold	NOUN
cana-3453	312	20	of	of	ADP
cana-3453	312	21	generalized	generalized	ADJ
cana-3453	312	22	ssf	ssf	NOUN
cana-3453	312	23	.	.	PUNCT
cana-3453	313	1	non	non	ADJ
cana-3453	313	2	-	-	ADJ
cana-3453	313	3	degeneracy	degeneracy	ADJ
cana-3453	313	4	conditions	condition	NOUN
cana-3453	313	5	vary	vary	VERB
cana-3453	313	6	in	in	ADP
cana-3453	313	7	complexity	complexity	NOUN
cana-3453	313	8	,	,	PUNCT
cana-3453	313	9	so	so	SCONJ
cana-3453	313	10	that	that	SCONJ
cana-3453	313	11	different	different	ADJ
cana-3453	313	12	tensors	tensor	NOUN
cana-3453	313	13	may	may	AUX
cana-3453	313	14	be	be	AUX
cana-3453	313	15	better	well	ADV
cana-3453	313	16	suited	suit	VERB
cana-3453	313	17	for	for	ADP
cana-3453	313	18	analysis	analysis	NOUN
cana-3453	313	19	in	in	ADP
cana-3453	313	20	different	different	ADJ
cana-3453	313	21	geometries	geometry	NOUN
cana-3453	313	22	.	.	PUNCT
cana-3453	314	1	the	the	DET
cana-3453	314	2	implications	implication	NOUN
cana-3453	314	3	of	of	ADP
cana-3453	314	4	this	this	DET
cana-3453	314	5	hierarchic	hierarchic	ADJ
cana-3453	314	6	relation	relation	NOUN
cana-3453	314	7	between	between	ADP
cana-3453	314	8	curvature	curvature	NOUN
cana-3453	314	9	tensors	tensor	NOUN
cana-3453	314	10	and	and	CCONJ
cana-3453	314	11	geometric	geometric	ADJ
cana-3453	314	12	characteristics	characteristic	NOUN
cana-3453	314	13	in	in	ADP
cana-3453	314	14	a	a	DET
cana-3453	314	15	submanifold	submanifold	NOUN
cana-3453	314	16	lead	lead	NOUN
cana-3453	314	17	to	to	ADP
cana-3453	314	18	further	further	ADJ
cana-3453	314	19	research	research	NOUN
cana-3453	314	20	in	in	ADP
cana-3453	314	21	the	the	DET
cana-3453	314	22	context	context	NOUN
cana-3453	314	23	of	of	ADP
cana-3453	314	24	broader	broad	ADJ
cana-3453	314	25	classes	class	NOUN
cana-3453	314	26	within	within	ADP
cana-3453	314	27	contact	contact	NOUN
cana-3453	314	28	metric	metric	ADJ
cana-3453	314	29	geometry	geometry	NOUN
cana-3453	314	30	.	.	PUNCT
cana-3453	315	1	summary	summary	NOUN
cana-3453	315	2	of	of	ADP
cana-3453	315	3	key	key	ADJ
cana-3453	315	4	results	result	NOUN
cana-3453	315	5	our	our	PRON
cana-3453	315	6	investigation	investigation	NOUN
cana-3453	315	7	has	have	AUX
cana-3453	315	8	revealed	reveal	VERB
cana-3453	315	9	five	five	NUM
cana-3453	315	10	distinct	distinct	ADJ
cana-3453	315	11	characterizations	characterization	NOUN
cana-3453	315	12	of	of	ADP
cana-3453	315	13	totally	totally	ADV
cana-3453	315	14	geodesic	geodesic	ADJ
cana-3453	315	15	invariant	invariant	ADJ
cana-3453	315	16	submanifolds	submanifold	NOUN
cana-3453	315	17	in	in	ADP
cana-3453	315	18	generalized	generalized	ADJ
cana-3453	315	19	sasakian	sasakian	ADJ
cana-3453	315	20	-	-	PUNCT
cana-3453	315	21	space	space	NOUN
cana-3453	315	22	-	-	PUNCT
cana-3453	315	23	forms	form	NOUN
cana-3453	315	24	,	,	PUNCT
cana-3453	315	25	each	each	PRON
cana-3453	315	26	involving	involve	VERB
cana-3453	315	27	different	different	ADJ
cana-3453	315	28	curvature	curvature	NOUN
cana-3453	315	29	tensors	tensor	NOUN
cana-3453	315	30	:	:	PUNCT
cana-3453	315	31	1	1	X
cana-3453	315	32	.	.	PUNCT
cana-3453	315	33	w2	w2	NOUN
cana-3453	315	34	characterization	characterization	NOUN
cana-3453	315	35	:	:	PUNCT
cana-3453	315	36	total	total	ADJ
cana-3453	315	37	geodesicity	geodesicity	NOUN
cana-3453	315	38	is	be	AUX
cana-3453	315	39	equivalent	equivalent	ADJ
cana-3453	315	40	to	to	ADP
cana-3453	315	41	q(σ	q(σ	NUM
cana-3453	315	42	,	,	PUNCT
cana-3453	315	43	w2)=0	w2)=0	PROPN
cana-3453	315	44	when	when	SCONJ
cana-3453	315	45	(	(	PUNCT
cana-3453	315	46	2n−1){3f2+(2n−1)f3}≠0	2n−1){3f2+(2n−1)f3}≠0	NOUN
cana-3453	315	47	2	2	NUM
cana-3453	315	48	.	.	PUNCT
cana-3453	315	49	w3	w3	NOUN
cana-3453	315	50	characterization	characterization	NOUN
cana-3453	315	51	:	:	PUNCT
cana-3453	315	52	the	the	DET
cana-3453	315	53	condition	condition	NOUN
cana-3453	315	54	q(σ	q(σ	NOUN
cana-3453	315	55	,	,	PUNCT
cana-3453	315	56	w3)=0	w3)=0	NUM
cana-3453	315	57	ensures	ensure	VERB
cana-3453	315	58	total	total	ADJ
cana-3453	315	59	geodesicity	geodesicity	NOUN
cana-3453	315	60	under	under	ADP
cana-3453	315	61	{	{	PUNCT
cana-3453	315	62	4n(1−2n)f1	4n(1−2n)f1	NUM
cana-3453	315	63	+	+	NOUN
cana-3453	315	64	3f2+(4n+1)(2n−1)f3}≠0	3f2+(4n+1)(2n−1)f3}≠0	PROPN
cana-3453	315	65	3	3	NUM
cana-3453	315	66	.	.	PUNCT
cana-3453	315	67	w4	w4	ADJ
cana-3453	315	68	characterization	characterization	NOUN
cana-3453	315	69	:	:	PUNCT
cana-3453	315	70	total	total	ADJ
cana-3453	315	71	geodesicity	geodesicity	NOUN
cana-3453	315	72	is	be	AUX
cana-3453	315	73	characterized	characterize	VERB
cana-3453	315	74	by	by	ADP
cana-3453	315	75	q(σ	q(σ	NUM
cana-3453	315	76	,	,	PUNCT
cana-3453	315	77	w4)=0	w4)=0	ADJ
cana-3453	315	78	when	when	SCONJ
cana-3453	315	79	{	{	PUNCT
cana-3453	315	80	f1	f1	NOUN
cana-3453	315	81	+	+	NOUN
cana-3453	315	82	3f2	3f2	NUM
cana-3453	315	83	+	+	ADJ
cana-3453	315	84	2(n−1)f3}≠0	2(n−1)f3}≠0	NUM
cana-3453	315	85	4	4	NUM
cana-3453	315	86	.	.	PUNCT
cana-3453	315	87	w6	w6	ADJ
cana-3453	315	88	characterization	characterization	NOUN
cana-3453	315	89	:	:	PUNCT
cana-3453	315	90	the	the	DET
cana-3453	315	91	simplest	simple	ADJ
cana-3453	315	92	condition	condition	NOUN
cana-3453	315	93	appears	appear	VERB
cana-3453	315	94	with	with	ADP
cana-3453	315	95	q(σ	q(σ	NUM
cana-3453	315	96	,	,	PUNCT
cana-3453	315	97	w6)=0	w6)=0	NOUN
cana-3453	315	98	under	under	ADP
cana-3453	315	99	2n(f1−f3)≠0	2n(f1−f3)≠0	NUM
cana-3453	315	100	5	5	NUM
cana-3453	315	101	.	.	PUNCT
cana-3453	315	102	w7	w7	ADJ
cana-3453	315	103	characterization	characterization	NOUN
cana-3453	315	104	:	:	PUNCT
cana-3453	315	105	the	the	DET
cana-3453	315	106	final	final	ADJ
cana-3453	315	107	characterization	characterization	NOUN
cana-3453	315	108	uses	use	VERB
cana-3453	315	109	q(σ	q(σ	NUM
cana-3453	315	110	,	,	PUNCT
cana-3453	315	111	w7)=0	w7)=0	VERB
cana-3453	315	112	with	with	ADP
cana-3453	315	113	{	{	PUNCT
cana-3453	315	114	2n(1−2n)f1	2n(1−2n)f1	NUM
cana-3453	315	115	+	+	NOUN
cana-3453	315	116	3f2−(2n+1)(2n−1)f3}≠0	3f2−(2n+1)(2n−1)f3}≠0	NUM
cana-3453	315	117	these	these	DET
cana-3453	315	118	five	five	NUM
cana-3453	315	119	curvature	curvature	NOUN
cana-3453	315	120	tensors	tensor	NOUN
cana-3453	315	121	represent	represent	VERB
cana-3453	315	122	more	more	ADJ
cana-3453	315	123	than	than	ADP
cana-3453	315	124	their	their	PRON
cana-3453	315	125	individual	individual	ADJ
cana-3453	315	126	characterizations	characterization	NOUN
cana-3453	315	127	;	;	PUNCT
cana-3453	315	128	their	their	PRON
cana-3453	315	129	total	total	ADJ
cana-3453	315	130	impact	impact	NOUN
cana-3453	315	131	is	be	AUX
cana-3453	315	132	summed	sum	VERB
cana-3453	315	133	with	with	ADP
cana-3453	315	134	these	these	DET
cana-3453	315	135	five	five	NUM
cana-3453	315	136	.	.	PUNCT
cana-3453	316	1	they	they	PRON
cana-3453	316	2	offer	offer	VERB
cana-3453	316	3	a	a	DET
cana-3453	316	4	complete	complete	ADJ
cana-3453	316	5	account	account	NOUN
cana-3453	316	6	of	of	ADP
cana-3453	316	7	the	the	DET
cana-3453	316	8	geometry	geometry	NOUN
cana-3453	316	9	of	of	ADP
cana-3453	316	10	generalized	generalized	ADJ
cana-3453	316	11	sasakianspace	sasakianspace	NOUN
cana-3453	316	12	-	-	PUNCT
cana-3453	316	13	forms	form	NOUN
cana-3453	316	14	.	.	PUNCT
cana-3453	317	1	that	that	SCONJ
cana-3453	317	2	the	the	DET
cana-3453	317	3	conditions	condition	NOUN
cana-3453	317	4	upon	upon	SCONJ
cana-3453	317	5	which	which	PRON
cana-3453	317	6	each	each	DET
cana-3453	317	7	tensor	tensor	NOUN
cana-3453	317	8	gives	give	VERB
cana-3453	317	9	its	its	PRON
cana-3453	317	10	characterization	characterization	NOUN
cana-3453	317	11	are	be	AUX
cana-3453	317	12	so	so	ADV
cana-3453	317	13	diverse	diverse	ADJ
cana-3453	317	14	suggests	suggest	VERB
cana-3453	317	15	that	that	SCONJ
cana-3453	317	16	these	these	DET
cana-3453	317	17	spaces	space	NOUN
cana-3453	317	18	carry	carry	VERB
cana-3453	317	19	a	a	DET
cana-3453	317	20	rich	rich	ADJ
cana-3453	317	21	geometric	geometric	ADJ
cana-3453	317	22	structure	structure	NOUN
cana-3453	317	23	,	,	PUNCT
cana-3453	317	24	one	one	NUM
cana-3453	317	25	that	that	PRON
cana-3453	317	26	certainly	certainly	ADV
cana-3453	317	27	can	can	AUX
cana-3453	317	28	not	not	PART
cana-3453	317	29	be	be	AUX
cana-3453	317	30	ascertained	ascertain	VERB
cana-3453	317	31	through	through	ADP
cana-3453	317	32	only	only	ADV
cana-3453	317	33	one	one	NUM
cana-3453	317	34	perspective	perspective	NOUN
cana-3453	317	35	.	.	PUNCT
cana-3453	318	1	on	on	ADP
cana-3453	318	2	one	one	NUM
cana-3453	318	3	hand	hand	NOUN
cana-3453	318	4	,	,	PUNCT
cana-3453	318	5	this	this	DET
cana-3453	318	6	variety	variety	NOUN
cana-3453	318	7	of	of	ADP
cana-3453	318	8	descriptions	description	NOUN
cana-3453	318	9	extends	extend	VERB
cana-3453	318	10	our	our	PRON
cana-3453	318	11	insight	insight	NOUN
cana-3453	318	12	and	and	CCONJ
cana-3453	318	13	on	on	ADP
cana-3453	318	14	the	the	DET
cana-3453	318	15	other	other	ADJ
cana-3453	318	16	hand	hand	NOUN
cana-3453	318	17	,	,	PUNCT
cana-3453	318	18	it	it	PRON
cana-3453	318	19	offers	offer	VERB
cana-3453	318	20	technical	technical	ADJ
cana-3453	318	21	versatility	versatility	NOUN
cana-3453	318	22	to	to	PART
cana-3453	318	23	study	study	VERB
cana-3453	318	24	particular	particular	ADJ
cana-3453	318	25	geometrical	geometrical	ADJ
cana-3453	318	26	contexts	contexts	NOUN
cana-3453	318	27	.	.	PUNCT
cana-3453	319	1	conclusion	conclusion	NOUN
cana-3453	319	2	the	the	DET
cana-3453	319	3	current	current	ADJ
cana-3453	319	4	investigation	investigation	NOUN
cana-3453	319	5	has	have	AUX
cana-3453	319	6	provided	provide	VERB
cana-3453	319	7	a	a	DET
cana-3453	319	8	general	general	ADJ
cana-3453	319	9	framework	framework	NOUN
cana-3453	319	10	to	to	PART
cana-3453	319	11	understand	understand	VERB
cana-3453	319	12	the	the	DET
cana-3453	319	13	geodesic	geodesic	ADJ
cana-3453	319	14	invariant	invariant	ADJ
cana-3453	319	15	submanifolds	submanifold	NOUN
cana-3453	319	16	of	of	ADP
cana-3453	319	17	generalized	generalized	ADJ
cana-3453	319	18	sasakian	sasakian	ADJ
cana-3453	319	19	-	-	PUNCT
cana-3453	319	20	space	space	NOUN
cana-3453	319	21	-	-	PUNCT
cana-3453	319	22	forms	form	NOUN
cana-3453	319	23	via	via	ADP
cana-3453	319	24	their	their	PRON
cana-3453	319	25	relationships	relationship	NOUN
cana-3453	319	26	with	with	ADP
cana-3453	319	27	different	different	ADJ
cana-3453	319	28	curvature	curvature	NOUN
cana-3453	319	29	tensors	tensor	NOUN
cana-3453	319	30	.	.	PUNCT
cana-3453	320	1	we	we	PRON
cana-3453	320	2	show	show	VERB
cana-3453	320	3	that	that	SCONJ
cana-3453	320	4	the	the	DET
cana-3453	320	5	geometry	geometry	NOUN
cana-3453	320	6	of	of	ADP
cana-3453	320	7	these	these	DET
cana-3453	320	8	submanifolds	submanifold	NOUN
cana-3453	320	9	can	can	AUX
cana-3453	320	10	be	be	AUX
cana-3453	320	11	described	describe	VERB
cana-3453	320	12	in	in	ADP
cana-3453	320	13	several	several	ADJ
cana-3453	320	14	equivalent	equivalent	ADJ
cana-3453	320	15	ways	way	NOUN
cana-3453	320	16	,	,	PUNCT
cana-3453	320	17	with	with	ADP
cana-3453	320	18	each	each	DET
cana-3453	320	19	one	one	NOUN
cana-3453	320	20	highlighting	highlight	VERB
cana-3453	320	21	some	some	DET
cana-3453	320	22	characteristic	characteristic	ADJ
cana-3453	320	23	aspect	aspect	NOUN
cana-3453	320	24	of	of	ADP
cana-3453	320	25	their	their	PRON
cana-3453	320	26	geometry	geometry	NOUN
cana-3453	320	27	.	.	PUNCT
cana-3453	321	1	communications	communication	NOUN
cana-3453	321	2	on	on	ADP
cana-3453	321	3	applied	apply	VERB
cana-3453	321	4	nonlinear	nonlinear	ADJ
cana-3453	321	5	analysis	analysis	NOUN
cana-3453	321	6	issn	issn	NOUN
cana-3453	321	7	:	:	PUNCT
cana-3453	321	8	1074	1074	NUM
cana-3453	321	9	-	-	PUNCT
cana-3453	321	10	133x	133x	NUM
cana-3453	321	11	vol	vol	NOUN
cana-3453	321	12	32	32	NUM
cana-3453	321	13	no	no	NOUN
cana-3453	321	14	.	.	PUNCT
cana-3453	322	1	7s	7	NOUN
cana-3453	322	2	(	(	PUNCT
cana-3453	322	3	2025	2025	NUM
cana-3453	322	4	)	)	PUNCT
cana-3453	322	5	430	430	NUM
cana-3453	322	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3453	322	7	the	the	DET
cana-3453	322	8	dictates	dictate	NOUN
cana-3453	322	9	of	of	ADP
cana-3453	322	10	the	the	DET
cana-3453	322	11	second	second	ADJ
cana-3453	322	12	fundamental	fundamental	ADJ
cana-3453	322	13	form	form	NOUN
cana-3453	322	14	σ	σ	NUM
cana-3453	322	15	relate	relate	VERB
cana-3453	322	16	to	to	ADP
cana-3453	322	17	the	the	DET
cana-3453	322	18	curvature	curvature	NOUN
cana-3453	322	19	tensors	tensor	NOUN
cana-3453	322	20	wi	wi	PROPN
cana-3453	322	21	(	(	PUNCT
cana-3453	322	22	i	i	NOUN
cana-3453	322	23	=	=	NOUN
cana-3453	322	24	2	2	NUM
cana-3453	322	25	,	,	PUNCT
cana-3453	322	26	3	3	NUM
cana-3453	322	27	,	,	PUNCT
cana-3453	322	28	4	4	NUM
cana-3453	322	29	,	,	PUNCT
cana-3453	322	30	6	6	NUM
cana-3453	322	31	,	,	PUNCT
cana-3453	322	32	7	7	NUM
cana-3453	322	33	)	)	PUNCT
cana-3453	322	34	and	and	CCONJ
cana-3453	322	35	show	show	VERB
cana-3453	322	36	different	different	ADJ
cana-3453	322	37	aspects	aspect	NOUN
cana-3453	322	38	of	of	ADP
cana-3453	322	39	the	the	DET
cana-3453	322	40	same	same	ADJ
cana-3453	322	41	geometric	geometric	ADJ
cana-3453	322	42	property	property	NOUN
cana-3453	322	43	total	total	NOUN
cana-3453	322	44	geodesicity	geodesicity	NOUN
cana-3453	322	45	.	.	PUNCT
cana-3453	323	1	of	of	ADP
cana-3453	323	2	the	the	DET
cana-3453	323	3	various	various	ADJ
cana-3453	323	4	characterizations	characterization	NOUN
cana-3453	323	5	,	,	PUNCT
cana-3453	323	6	the	the	DET
cana-3453	323	7	w6	w6	PROPN
cana-3453	323	8	curvature	curvature	NOUN
cana-3453	323	9	tensor	tensor	NOUN
cana-3453	323	10	produces	produce	VERB
cana-3453	323	11	the	the	DET
cana-3453	323	12	neatest	neat	ADJ
cana-3453	323	13	condition	condition	NOUN
cana-3453	323	14	,	,	PUNCT
cana-3453	323	15	with	with	SCONJ
cana-3453	323	16	minimal	minimal	ADJ
cana-3453	323	17	nondegeneracy	nondegeneracy	NOUN
cana-3453	323	18	condition	condition	NOUN
cana-3453	323	19	required	require	VERB
cana-3453	323	20	,	,	PUNCT
cana-3453	323	21	and	and	CCONJ
cana-3453	323	22	thus	thus	ADV
cana-3453	323	23	seems	seem	VERB
cana-3453	323	24	especially	especially	ADV
cana-3453	323	25	suited	suit	VERB
cana-3453	323	26	to	to	ADP
cana-3453	323	27	practical	practical	ADJ
cana-3453	323	28	applications	application	NOUN
cana-3453	323	29	.	.	PUNCT
cana-3453	324	1	these	these	DET
cana-3453	324	2	results	result	NOUN
cana-3453	324	3	generalize	generalize	VERB
cana-3453	324	4	known	know	VERB
cana-3453	324	5	facts	fact	NOUN
cana-3453	324	6	about	about	ADP
cana-3453	324	7	generalized	generalized	ADJ
cana-3453	324	8	sasakian	sasakian	ADJ
cana-3453	324	9	space	space	NOUN
cana-3453	324	10	forms	form	NOUN
cana-3453	324	11	and	and	CCONJ
cana-3453	324	12	introduce	introduce	VERB
cana-3453	324	13	new	new	ADJ
cana-3453	324	14	tools	tool	NOUN
cana-3453	324	15	to	to	PART
cana-3453	324	16	study	study	VERB
cana-3453	324	17	their	their	PRON
cana-3453	324	18	submanifolds	submanifold	NOUN
cana-3453	324	19	.	.	PUNCT
cana-3453	325	1	geometers	geometer	NOUN
cana-3453	325	2	now	now	ADV
cana-3453	325	3	have	have	VERB
cana-3453	325	4	a	a	DET
cana-3453	325	5	range	range	NOUN
cana-3453	325	6	of	of	ADP
cana-3453	325	7	available	available	ADJ
cana-3453	325	8	conditions	condition	NOUN
cana-3453	325	9	that	that	SCONJ
cana-3453	325	10	they	they	PRON
cana-3453	325	11	can	can	AUX
cana-3453	325	12	use	use	VERB
cana-3453	325	13	as	as	ADP
cana-3453	325	14	the	the	DET
cana-3453	325	15	most	most	ADV
cana-3453	325	16	suitable	suitable	ADJ
cana-3453	325	17	characterization	characterization	NOUN
cana-3453	325	18	according	accord	VERB
cana-3453	325	19	to	to	ADP
cana-3453	325	20	the	the	DET
cana-3453	325	21	setting	setting	NOUN
cana-3453	325	22	of	of	ADP
cana-3453	325	23	their	their	PRON
cana-3453	325	24	study	study	NOUN
cana-3453	325	25	.	.	PUNCT
cana-3453	326	1	in	in	ADP
cana-3453	326	2	addition	addition	NOUN
cana-3453	326	3	,	,	PUNCT
cana-3453	326	4	the	the	DET
cana-3453	326	5	distinct	distinct	ADJ
cana-3453	326	6	nondegenerate	nondegenerate	NOUN
cana-3453	326	7	conditions	condition	NOUN
cana-3453	326	8	associated	associate	VERB
cana-3453	326	9	with	with	ADP
cana-3453	326	10	these	these	DET
cana-3453	326	11	different	different	ADJ
cana-3453	326	12	characterizations	characterization	NOUN
cana-3453	326	13	suggest	suggest	VERB
cana-3453	326	14	that	that	SCONJ
cana-3453	326	15	the	the	DET
cana-3453	326	16	results	result	NOUN
cana-3453	326	17	may	may	AUX
cana-3453	326	18	be	be	AUX
cana-3453	326	19	complementary	complementary	ADJ
cana-3453	326	20	and	and	CCONJ
cana-3453	326	21	potentially	potentially	ADV
cana-3453	326	22	useful	useful	ADJ
cana-3453	326	23	in	in	ADP
cana-3453	326	24	different	different	ADJ
cana-3453	326	25	regions	region	NOUN
cana-3453	326	26	of	of	ADP
cana-3453	326	27	the	the	DET
cana-3453	326	28	manifold	manifold	NOUN
cana-3453	326	29	for	for	ADP
cana-3453	326	30	varying	vary	VERB
cana-3453	326	31	f1	f1	NOUN
cana-3453	326	32	,	,	PUNCT
cana-3453	326	33	f2	f2	PROPN
cana-3453	326	34	,	,	PUNCT
cana-3453	326	35	and	and	CCONJ
cana-3453	326	36	f3	f3	ADJ
cana-3453	326	37	values	value	NOUN
cana-3453	326	38	.	.	PUNCT
cana-3453	327	1	future	future	ADJ
cana-3453	327	2	research	research	NOUN
cana-3453	327	3	directions	direction	NOUN
cana-3453	327	4	could	could	AUX
cana-3453	327	5	focus	focus	VERB
cana-3453	327	6	on	on	ADP
cana-3453	327	7	studying	study	VERB
cana-3453	327	8	similar	similar	ADJ
cana-3453	327	9	characterizations	characterization	NOUN
cana-3453	327	10	for	for	ADP
cana-3453	327	11	other	other	ADJ
cana-3453	327	12	kinds	kind	NOUN
cana-3453	327	13	of	of	ADP
cana-3453	327	14	submanifolds	submanifold	NOUN
cana-3453	327	15	,	,	PUNCT
cana-3453	327	16	understanding	understand	VERB
cana-3453	327	17	the	the	DET
cana-3453	327	18	geometric	geometric	ADJ
cana-3453	327	19	significance	significance	NOUN
cana-3453	327	20	of	of	ADP
cana-3453	327	21	the	the	DET
cana-3453	327	22	different	different	ADJ
cana-3453	327	23	non	non	ADJ
cana-3453	327	24	-	-	ADJ
cana-3453	327	25	degeneracy	degeneracy	ADJ
cana-3453	327	26	conditions	condition	NOUN
cana-3453	327	27	,	,	PUNCT
cana-3453	327	28	and	and	CCONJ
cana-3453	327	29	finally	finally	ADV
cana-3453	327	30	,	,	PUNCT
cana-3453	327	31	considering	consider	VERB
cana-3453	327	32	how	how	SCONJ
cana-3453	327	33	these	these	DET
cana-3453	327	34	results	result	NOUN
cana-3453	327	35	can	can	AUX
cana-3453	327	36	be	be	AUX
cana-3453	327	37	generalized	generalize	VERB
cana-3453	327	38	to	to	ADP
cana-3453	327	39	other	other	ADJ
cana-3453	327	40	generalized	generalized	ADJ
cana-3453	327	41	geometrical	geometrical	ADJ
cana-3453	327	42	structures	structure	NOUN
cana-3453	327	43	.	.	PUNCT
cana-3453	328	1	moreover	moreover	ADV
cana-3453	328	2	,	,	PUNCT
cana-3453	328	3	the	the	DET
cana-3453	328	4	interplay	interplay	NOUN
cana-3453	328	5	of	of	ADP
cana-3453	328	6	these	these	DET
cana-3453	328	7	characterizations	characterization	NOUN
cana-3453	328	8	with	with	ADP
cana-3453	328	9	other	other	ADJ
cana-3453	328	10	geometric	geometric	ADJ
cana-3453	328	11	features	feature	NOUN
cana-3453	328	12	of	of	ADP
cana-3453	328	13	ssf	ssf	NOUN
cana-3453	328	14	could	could	AUX
cana-3453	328	15	also	also	ADV
cana-3453	328	16	lead	lead	VERB
cana-3453	328	17	to	to	ADP
cana-3453	328	18	profitable	profitable	ADJ
cana-3453	328	19	attributions	attribution	NOUN
cana-3453	328	20	.	.	PUNCT
cana-3453	329	1	our	our	PRON
cana-3453	329	2	results	result	NOUN
cana-3453	329	3	expand	expand	VERB
cana-3453	329	4	the	the	DET
cana-3453	329	5	lexicon	lexicon	NOUN
cana-3453	329	6	of	of	ADP
cana-3453	329	7	differential	differential	ADJ
cana-3453	329	8	geometry	geometry	NOUN
cana-3453	329	9	with	with	ADP
cana-3453	329	10	several	several	ADJ
cana-3453	329	11	equivalent	equivalent	ADJ
cana-3453	329	12	geometric	geometric	ADJ
cana-3453	329	13	conditions	condition	NOUN
cana-3453	329	14	for	for	ADP
cana-3453	329	15	a	a	DET
cana-3453	329	16	submanifold	submanifold	NOUN
cana-3453	329	17	to	to	PART
cana-3453	329	18	satisfy	satisfy	VERB
cana-3453	329	19	being	be	AUX
cana-3453	329	20	geodesic	geodesic	ADJ
cana-3453	329	21	,	,	PUNCT
cana-3453	329	22	each	each	PRON
cana-3453	329	23	of	of	ADP
cana-3453	329	24	which	which	PRON
cana-3453	329	25	sheds	shed	VERB
cana-3453	329	26	light	light	NOUN
cana-3453	329	27	on	on	ADP
cana-3453	329	28	the	the	DET
cana-3453	329	29	underlying	underlie	VERB
cana-3453	329	30	geometric	geometric	ADJ
cana-3453	329	31	structure	structure	NOUN
cana-3453	329	32	of	of	ADP
cana-3453	329	33	generalized	generalized	ADJ
cana-3453	329	34	ssf	ssf	NOUN
cana-3453	329	35	.	.	PUNCT
cana-3453	330	1	references	reference	NOUN
cana-3453	330	2	:	:	PUNCT
cana-3453	331	1	[	[	X
cana-3453	331	2	1	1	NUM
cana-3453	331	3	]	]	X
cana-3453	331	4	tripathi	tripathi	PROPN
cana-3453	331	5	,	,	PUNCT
cana-3453	331	6	m.	m.	NOUN
cana-3453	331	7	m.	m.	NOUN
cana-3453	331	8	and	and	CCONJ
cana-3453	331	9	gupta	gupta	PROPN
cana-3453	331	10	,	,	PUNCT
cana-3453	331	11	p.	p.	PROPN
cana-3453	331	12	,	,	PUNCT
cana-3453	331	13	t	t	PROPN
cana-3453	331	14	-	-	PUNCT
cana-3453	331	15	curvature	curvature	NOUN
cana-3453	331	16	tensor	tensor	NOUN
cana-3453	331	17	on	on	ADP
cana-3453	331	18	a	a	DET
cana-3453	331	19	semi	semi	ADJ
cana-3453	331	20	-	-	ADJ
cana-3453	331	21	riemannian	riemannian	ADJ
cana-3453	331	22	manifold	manifold	NOUN
cana-3453	331	23	,	,	PUNCT
cana-3453	331	24	j.adv	j.adv	PROPN
cana-3453	331	25	.	.	PUNCT
cana-3453	331	26	math	math	NOUN
cana-3453	331	27	,	,	PUNCT
cana-3453	331	28	stud	stud	NOUN
cana-3453	331	29	.	.	PUNCT
cana-3453	332	1	4	4	NUM
cana-3453	332	2	,	,	PUNCT
cana-3453	332	3	no.1	no.1	NUM
cana-3453	332	4	,	,	PUNCT
cana-3453	332	5	(	(	PUNCT
cana-3453	332	6	2011	2011	NUM
cana-3453	332	7	)	)	PUNCT
cana-3453	332	8	,	,	PUNCT
cana-3453	332	9	117	117	NUM
cana-3453	332	10	-	-	SYM
cana-3453	332	11	129	129	NUM
cana-3453	332	12	.	.	PUNCT
cana-3453	333	1	[	[	X
cana-3453	333	2	2	2	NUM
cana-3453	333	3	]	]	PUNCT
cana-3453	333	4	alegre	alegre	NOUN
cana-3453	333	5	,	,	PUNCT
cana-3453	333	6	p.	p.	PROPN
cana-3453	333	7	,	,	PUNCT
cana-3453	333	8	blair	blair	PROPN
cana-3453	333	9	,	,	PUNCT
cana-3453	333	10	d.	d.	PROPN
cana-3453	333	11	e.	e.	PROPN
cana-3453	333	12	and	and	CCONJ
cana-3453	333	13	carriazo	carriazo	PROPN
cana-3453	333	14	,	,	PUNCT
cana-3453	333	15	a.	a.	NOUN
cana-3453	333	16	,	,	PUNCT
cana-3453	333	17	generalized	generalize	VERB
cana-3453	333	18	sasakian	sasakian	ADJ
cana-3453	333	19	-	-	PUNCT
cana-3453	333	20	space	space	NOUN
cana-3453	333	21	-	-	PUNCT
cana-3453	333	22	forms	form	NOUN
cana-3453	333	23	,	,	PUNCT
cana-3453	333	24	israel	israel	PROPN
cana-3453	333	25	j.	j.	PROPN
cana-3453	333	26	math	math	PROPN
cana-3453	333	27	.	.	PUNCT
cana-3453	333	28	,	,	PUNCT
cana-3453	333	29	141	141	NUM
cana-3453	333	30	(	(	PUNCT
cana-3453	333	31	2004	2004	NUM
cana-3453	333	32	)	)	PUNCT
cana-3453	333	33	,	,	PUNCT
cana-3453	333	34	157–183	157–183	NUM
cana-3453	333	35	.	.	PUNCT
cana-3453	334	1	[	[	X
cana-3453	334	2	3	3	NUM
cana-3453	334	3	]	]	PUNCT
cana-3453	334	4	alegre	alegre	NOUN
cana-3453	334	5	,	,	PUNCT
cana-3453	334	6	p.	p.	NOUN
cana-3453	334	7	and	and	CCONJ
cana-3453	334	8	carriazo	carriazo	PROPN
cana-3453	334	9	,	,	PUNCT
cana-3453	334	10	a.	a.	NOUN
cana-3453	334	11	,	,	PUNCT
cana-3453	334	12	structures	structure	NOUN
cana-3453	334	13	on	on	ADP
cana-3453	334	14	generalized	generalized	ADJ
cana-3453	334	15	sasakian	sasakian	ADJ
cana-3453	334	16	-	-	PUNCT
cana-3453	334	17	space	space	NOUN
cana-3453	334	18	-	-	PUNCT
cana-3453	334	19	forms	form	NOUN
cana-3453	334	20	,	,	PUNCT
cana-3453	334	21	diff	diff	PROPN
cana-3453	334	22	.	.	PUNCT
cana-3453	335	1	geo	geo	PROPN
cana-3453	335	2	.	.	PROPN
cana-3453	336	1	and	and	CCONJ
cana-3453	336	2	its	its	PRON
cana-3453	336	3	application	application	NOUN
cana-3453	336	4	,	,	PUNCT
cana-3453	336	5	26	26	NUM
cana-3453	336	6	(	(	PUNCT
cana-3453	336	7	2008	2008	NUM
cana-3453	336	8	)	)	PUNCT
cana-3453	336	9	,	,	PUNCT
cana-3453	336	10	656–666	656–666	NUM
cana-3453	336	11	.	.	PUNCT
cana-3453	337	1	[	[	X
cana-3453	337	2	4	4	NUM
cana-3453	337	3	]	]	PUNCT
cana-3453	337	4	alegre	alegre	NOUN
cana-3453	337	5	,	,	PUNCT
cana-3453	337	6	p.	p.	NOUN
cana-3453	337	7	and	and	CCONJ
cana-3453	337	8	carriazo	carriazo	PROPN
cana-3453	337	9	,	,	PUNCT
cana-3453	337	10	a.	a.	NOUN
cana-3453	337	11	,	,	PUNCT
cana-3453	337	12	submanifolds	submanifold	NOUN
cana-3453	337	13	of	of	ADP
cana-3453	337	14	generalized	generalized	ADJ
cana-3453	337	15	sasakian	sasakian	ADJ
cana-3453	337	16	-	-	PUNCT
cana-3453	337	17	space	space	NOUN
cana-3453	337	18	-	-	PUNCT
cana-3453	337	19	forms	form	NOUN
cana-3453	337	20	,	,	PUNCT
cana-3453	337	21	taiwanese	taiwanese	ADJ
cana-3453	337	22	j.	j.	PROPN
cana-3453	337	23	math	math	PROPN
cana-3453	337	24	.	.	PUNCT
cana-3453	337	25	,	,	PUNCT
cana-3453	337	26	13	13	NUM
cana-3453	337	27	(	(	PUNCT
cana-3453	337	28	2009	2009	NUM
cana-3453	337	29	)	)	PUNCT
cana-3453	337	30	,	,	PUNCT
cana-3453	337	31	923	923	NUM
cana-3453	337	32	–	–	PUNCT
cana-3453	337	33	941	941	NUM
cana-3453	337	34	.	.	PUNCT
cana-3453	338	1	[	[	X
cana-3453	338	2	5	5	NUM
cana-3453	338	3	]	]	PUNCT
cana-3453	338	4	alegre	alegre	NOUN
cana-3453	338	5	,	,	PUNCT
cana-3453	338	6	p.	p.	NOUN
cana-3453	338	7	and	and	CCONJ
cana-3453	338	8	carriazo	carriazo	PROPN
cana-3453	338	9	,	,	PUNCT
cana-3453	338	10	a.	a.	NOUN
cana-3453	338	11	,	,	PUNCT
cana-3453	338	12	generalized	generalize	VERB
cana-3453	338	13	sasakian	sasakian	ADJ
cana-3453	338	14	-	-	PUNCT
cana-3453	338	15	space	space	NOUN
cana-3453	338	16	-	-	PUNCT
cana-3453	338	17	forms	form	NOUN
cana-3453	338	18	and	and	CCONJ
cana-3453	338	19	conformal	conformal	ADJ
cana-3453	338	20	changes	change	NOUN
cana-3453	338	21	of	of	ADP
cana-3453	338	22	the	the	DET
cana-3453	338	23	metric	metric	ADJ
cana-3453	338	24	,	,	PUNCT
cana-3453	338	25	results	result	NOUN
cana-3453	338	26	in	in	ADP
cana-3453	338	27	math	math	NOUN
cana-3453	338	28	.	.	PUNCT
cana-3453	338	29	,	,	PUNCT
cana-3453	338	30	59	59	NUM
cana-3453	338	31	(	(	PUNCT
cana-3453	338	32	2011	2011	NUM
cana-3453	338	33	)	)	PUNCT
cana-3453	338	34	,	,	PUNCT
cana-3453	338	35	485–493	485–493	NUM
cana-3453	338	36	,	,	PUNCT
cana-3453	338	37	doi	doi	X
cana-3453	338	38	10.1007	10.1007	NUM
cana-3453	338	39	/	/	SYM
cana-3453	338	40	s00025	s00025	PROPN
cana-3453	338	41	-	-	PUNCT
cana-3453	338	42	011	011	NUM
cana-3453	338	43	-	-	PUNCT
cana-3453	338	44	0115	0115	NUM
cana-3453	338	45	-	-	PUNCT
cana-3453	338	46	z.	z.	PROPN
cana-3453	339	1	[	[	X
cana-3453	339	2	6	6	NUM
cana-3453	339	3	]	]	PUNCT
cana-3453	339	4	atçeken	atçeken	VERB
cana-3453	339	5	,	,	PUNCT
cana-3453	339	6	m.	m.	NOUN
cana-3453	339	7	,	,	PUNCT
cana-3453	339	8	contact	contact	NOUN
cana-3453	339	9	cr	cr	NOUN
cana-3453	339	10	-	-	PUNCT
cana-3453	339	11	submanifolds	submanifold	NOUN
cana-3453	339	12	of	of	ADP
cana-3453	339	13	kenmotsu	kenmotsu	PROPN
cana-3453	339	14	manifolds	manifolds	PROPN
cana-3453	339	15	,	,	PUNCT
cana-3453	339	16	serdica	serdica	PROPN
cana-3453	339	17	math	math	PROPN
cana-3453	339	18	.	.	PUNCT
cana-3453	340	1	j.	j.	PROPN
cana-3453	340	2	,	,	PUNCT
cana-3453	340	3	37	37	NUM
cana-3453	340	4	(	(	PUNCT
cana-3453	340	5	2011	2011	NUM
cana-3453	340	6	)	)	PUNCT
cana-3453	340	7	,	,	PUNCT
cana-3453	340	8	67	67	NUM
cana-3453	340	9	-	-	SYM
cana-3453	340	10	78	78	NUM
cana-3453	340	11	.	.	PUNCT
cana-3453	341	1	[	[	X
cana-3453	341	2	7	7	NUM
cana-3453	341	3	]	]	PUNCT
cana-3453	341	4	atçeken	atçeken	VERB
cana-3453	341	5	,	,	PUNCT
cana-3453	341	6	m.	m.	NOUN
cana-3453	341	7	,	,	PUNCT
cana-3453	341	8	contact	contact	NOUN
cana-3453	341	9	cr	cr	NOUN
cana-3453	341	10	-	-	PUNCT
cana-3453	341	11	warped	warp	VERB
cana-3453	341	12	product	product	NOUN
cana-3453	341	13	submanifolds	submanifold	NOUN
cana-3453	341	14	in	in	ADP
cana-3453	341	15	cosymplectic	cosymplectic	ADJ
cana-3453	341	16	space	space	NOUN
cana-3453	341	17	forms	form	NOUN
cana-3453	341	18	,	,	PUNCT
cana-3453	341	19	collect	collect	NOUN
cana-3453	341	20	.	.	PUNCT
cana-3453	342	1	math	math	NOUN
cana-3453	342	2	.	.	PUNCT
cana-3453	343	1	(	(	PUNCT
cana-3453	343	2	2011	2011	NUM
cana-3453	343	3	)	)	PUNCT
cana-3453	343	4	62:17–26	62:17–26	NUM
cana-3453	343	5	,	,	PUNCT
cana-3453	343	6	doi	doi	NOUN
cana-3453	343	7	10.1007	10.1007	NUM
cana-3453	343	8	/	/	SYM
cana-3453	343	9	s13348	s13348	NOUN
cana-3453	343	10	-	-	PUNCT
cana-3453	343	11	010	010	NUM
cana-3453	343	12	-	-	PUNCT
cana-3453	343	13	0002	0002	NUM
cana-3453	343	14	-	-	PUNCT
cana-3453	343	15	z.	z.	NOUN
cana-3453	344	1	[	[	X
cana-3453	344	2	8	8	NUM
cana-3453	344	3	]	]	PUNCT
cana-3453	344	4	atçeken	atçeken	VERB
cana-3453	344	5	,	,	PUNCT
cana-3453	344	6	m.	m.	NOUN
cana-3453	344	7	and	and	CCONJ
cana-3453	344	8	dirik	dirik	PROPN
cana-3453	344	9	,	,	PUNCT
cana-3453	344	10	s.	s.	PROPN
cana-3453	344	11	,	,	PUNCT
cana-3453	344	12	on	on	ADP
cana-3453	344	13	contact	contact	NOUN
cana-3453	344	14	cr	cr	NOUN
cana-3453	344	15	-	-	PUNCT
cana-3453	344	16	submanifolds	submanifold	NOUN
cana-3453	344	17	of	of	ADP
cana-3453	344	18	kenmotsu	kenmotsu	PROPN
cana-3453	344	19	manifolds	manifolds	PROPN
cana-3453	344	20	,	,	PUNCT
cana-3453	344	21	acta	acta	PROPN
cana-3453	344	22	univ	univ	PROPN
cana-3453	344	23	.	.	PUNCT
cana-3453	345	1	sap	sap	PROPN
cana-3453	345	2	.	.	PUNCT
cana-3453	345	3	math	math	NOUN
cana-3453	345	4	.	.	PUNCT
cana-3453	346	1	4	4	NUM
cana-3453	346	2	(	(	PUNCT
cana-3453	346	3	2012	2012	NUM
cana-3453	346	4	)	)	PUNCT
cana-3453	346	5	,	,	PUNCT
cana-3453	346	6	182	182	NUM
cana-3453	346	7	-	-	SYM
cana-3453	346	8	198	198	NUM
cana-3453	346	9	.	.	PUNCT
cana-3453	347	1	acta	acta	PROPN
cana-3453	347	2	univ	univ	PROPN
cana-3453	347	3	.	.	PUNCT
cana-3453	348	1	sapientiae	sapientiae	PROPN
cana-3453	348	2	,	,	PUNCT
cana-3453	348	3	mathematica	mathematica	PROPN
cana-3453	348	4	,	,	PUNCT
cana-3453	348	5	4	4	NUM
cana-3453	348	6	,	,	PUNCT
cana-3453	348	7	2	2	NUM
cana-3453	348	8	(	(	PUNCT
cana-3453	348	9	2012	2012	NUM
cana-3453	348	10	)	)	PUNCT
cana-3453	348	11	182–198	182–198	NUM
cana-3453	348	12	.	.	PUNCT
cana-3453	349	1	[	[	X
cana-3453	349	2	9	9	NUM
cana-3453	349	3	]	]	SYM
cana-3453	349	4	bejancu	bejancu	NOUN
cana-3453	349	5	,	,	PUNCT
cana-3453	349	6	a.	a.	NOUN
cana-3453	349	7	,	,	PUNCT
cana-3453	349	8	geometry	geometry	NOUN
cana-3453	349	9	of	of	ADP
cana-3453	349	10	cr	cr	PROPN
cana-3453	349	11	-	-	PUNCT
cana-3453	349	12	submanifolds	submanifolds	PROPN
cana-3453	349	13	,	,	PUNCT
cana-3453	349	14	d.	d.	PROPN
cana-3453	349	15	reidel	reidel	PROPN
cana-3453	349	16	pub	pub	PROPN
cana-3453	349	17	.	.	PUNCT
cana-3453	349	18	co.	co.	PROPN
cana-3453	349	19	dordrecht	dordrecht	PROPN
cana-3453	349	20	,	,	PUNCT
cana-3453	349	21	holland	holland	PROPN
cana-3453	349	22	,	,	PUNCT
cana-3453	349	23	1986	1986	NUM
cana-3453	349	24	.	.	PUNCT
cana-3453	350	1	[	[	X
cana-3453	350	2	10	10	NUM
cana-3453	350	3	]	]	X
cana-3453	350	4	belkhelfa	belkhelfa	NOUN
cana-3453	350	5	,	,	PUNCT
cana-3453	350	6	m.	m.	NOUN
cana-3453	350	7	,	,	PUNCT
cana-3453	350	8	deszcz	deszcz	ADV
cana-3453	350	9	,	,	PUNCT
cana-3453	350	10	r.	r.	PROPN
cana-3453	350	11	and	and	CCONJ
cana-3453	350	12	verstraelen	verstraelen	PROPN
cana-3453	350	13	,	,	PUNCT
cana-3453	350	14	l.	l.	PROPN
cana-3453	350	15	,	,	PUNCT
cana-3453	350	16	symmetry	symmetry	NOUN
cana-3453	350	17	properties	property	NOUN
cana-3453	350	18	of	of	ADP
cana-3453	350	19	generalized	generalized	ADJ
cana-3453	350	20	sasakian	sasakian	ADJ
cana-3453	350	21	-	-	PUNCT
cana-3453	350	22	space	space	NOUN
cana-3453	350	23	-	-	PUNCT
cana-3453	350	24	forms	form	NOUN
cana-3453	350	25	,	,	PUNCT
cana-3453	350	26	soochow	soochow	PROPN
cana-3453	350	27	j.	j.	PROPN
cana-3453	350	28	math	math	PROPN
cana-3453	350	29	.	.	PUNCT
cana-3453	350	30	,	,	PUNCT
cana-3453	350	31	31	31	NUM
cana-3453	350	32	(	(	PUNCT
cana-3453	350	33	2005	2005	NUM
cana-3453	350	34	)	)	PUNCT
cana-3453	350	35	,	,	PUNCT
cana-3453	350	36	611–616	611–616	NUM
cana-3453	350	37	.	.	PUNCT
cana-3453	351	1	[	[	X
cana-3453	351	2	11	11	NUM
cana-3453	351	3	]	]	X
cana-3453	351	4	blair	blair	PROPN
cana-3453	351	5	,	,	PUNCT
cana-3453	351	6	d.	d.	PROPN
cana-3453	351	7	e.	e.	PROPN
cana-3453	351	8	,	,	PUNCT
cana-3453	351	9	contact	contact	NOUN
cana-3453	351	10	manifolds	manifold	NOUN
cana-3453	351	11	in	in	ADP
cana-3453	351	12	riemannian	riemannian	ADJ
cana-3453	351	13	geometry	geometry	NOUN
cana-3453	351	14	,	,	PUNCT
cana-3453	351	15	lecture	lecture	NOUN
cana-3453	351	16	notes	note	NOUN
cana-3453	351	17	in	in	ADP
cana-3453	351	18	math	math	NOUN
cana-3453	351	19	.	.	PUNCT
cana-3453	352	1	509	509	NUM
cana-3453	352	2	,	,	PUNCT
cana-3453	352	3	springer	springer	NOUN
cana-3453	352	4	-	-	PUNCT
cana-3453	352	5	verlag	verlag	PROPN
cana-3453	352	6	,	,	PUNCT
cana-3453	352	7	1976	1976	NUM
cana-3453	352	8	.	.	PUNCT
cana-3453	353	1	[	[	X
cana-3453	353	2	12	12	NUM
cana-3453	353	3	]	]	X
cana-3453	353	4	carriazo	carriazo	NOUN
cana-3453	353	5	,	,	PUNCT
cana-3453	353	6	a.	a.	NOUN
cana-3453	353	7	,	,	PUNCT
cana-3453	353	8	on	on	ADP
cana-3453	353	9	generalized	generalized	ADJ
cana-3453	353	10	sasakian	sasakian	ADJ
cana-3453	353	11	-	-	PUNCT
cana-3453	353	12	space	space	NOUN
cana-3453	353	13	-	-	PUNCT
cana-3453	353	14	forms	form	NOUN
cana-3453	353	15	,	,	PUNCT
cana-3453	353	16	proceedings	proceeding	NOUN
cana-3453	353	17	of	of	ADP
cana-3453	353	18	the	the	DET
cana-3453	353	19	ninth	ninth	ADJ
cana-3453	353	20	international	international	ADJ
cana-3453	353	21	workshop	workshop	NOUN
cana-3453	353	22	on	on	ADP
cana-3453	353	23	diff	diff	PROPN
cana-3453	353	24	.	.	PUNCT
cana-3453	354	1	geom	geom	PROPN
cana-3453	354	2	.	.	PROPN
cana-3453	354	3	,	,	PUNCT
cana-3453	354	4	9	9	NUM
cana-3453	354	5	(	(	PUNCT
cana-3453	354	6	2005	2005	NUM
cana-3453	354	7	)	)	PUNCT
cana-3453	354	8	,	,	PUNCT
cana-3453	354	9	31–39	31–39	NUM
cana-3453	354	10	.	.	PUNCT
cana-3453	355	1	[	[	X
cana-3453	355	2	13	13	NUM
cana-3453	355	3	]	]	X
cana-3453	355	4	chen	chen	PROPN
cana-3453	355	5	,	,	PUNCT
cana-3453	355	6	b.	b.	PROPN
cana-3453	355	7	y.	y.	PROPN
cana-3453	355	8	,	,	PUNCT
cana-3453	355	9	geometry	geometry	NOUN
cana-3453	355	10	of	of	ADP
cana-3453	355	11	slant	slant	ADJ
cana-3453	355	12	submanifolds	submanifold	NOUN
cana-3453	355	13	,	,	PUNCT
cana-3453	355	14	katholieke	katholieke	PROPN
cana-3453	355	15	universiteit	universiteit	PROPN
cana-3453	355	16	leuven	leuven	PROPN
cana-3453	355	17	,	,	PUNCT
cana-3453	355	18	1990	1990	NUM
cana-3453	355	19	.	.	PUNCT
cana-3453	356	1	[	[	X
cana-3453	356	2	14	14	NUM
cana-3453	356	3	]	]	PUNCT
cana-3453	356	4	cîrnu	cîrnu	NOUN
cana-3453	356	5	,	,	PUNCT
cana-3453	356	6	m.	m.	NOUN
cana-3453	356	7	,	,	PUNCT
cana-3453	356	8	cohomology	cohomology	NOUN
cana-3453	356	9	and	and	CCONJ
cana-3453	356	10	stability	stability	NOUN
cana-3453	356	11	of	of	ADP
cana-3453	356	12	generalized	generalized	ADJ
cana-3453	356	13	sasakian	sasakian	ADJ
cana-3453	356	14	-	-	PUNCT
cana-3453	356	15	space	space	NOUN
cana-3453	356	16	-	-	PUNCT
cana-3453	356	17	forms	form	NOUN
cana-3453	356	18	to	to	PART
cana-3453	356	19	appear	appear	VERB
cana-3453	356	20	in	in	ADP
cana-3453	356	21	bull	bull	NOUN
cana-3453	356	22	.	.	PUNCT
cana-3453	357	1	malaysian	malaysian	PROPN
cana-3453	357	2	mathematical	mathematical	PROPN
cana-3453	357	3	sciences	sciences	PROPN
cana-3453	357	4	society	society	NOUN
cana-3453	357	5	.	.	PUNCT
cana-3453	358	1	[	[	X
cana-3453	358	2	15	15	NUM
cana-3453	358	3	]	]	X
cana-3453	358	4	ghefari	ghefari	PROPN
cana-3453	358	5	,	,	PUNCT
cana-3453	358	6	r.	r.	PROPN
cana-3453	358	7	a.	a.	PROPN
cana-3453	358	8	,	,	PUNCT
cana-3453	358	9	solamy	solamy	PROPN
cana-3453	358	10	,	,	PUNCT
cana-3453	358	11	f.	f.	PROPN
cana-3453	358	12	r.	r.	PROPN
cana-3453	358	13	a.	a.	PROPN
cana-3453	358	14	and	and	CCONJ
cana-3453	358	15	shahid	shahid	PROPN
cana-3453	358	16	,	,	PUNCT
cana-3453	358	17	m.	m.	PROPN
cana-3453	358	18	h.	h.	PROPN
cana-3453	358	19	,	,	PUNCT
cana-3453	358	20	cr	cr	NOUN
cana-3453	358	21	-	-	PUNCT
cana-3453	358	22	submanifolds	submanifold	NOUN
cana-3453	358	23	of	of	ADP
cana-3453	358	24	generalized	generalized	ADJ
cana-3453	358	25	sasakian	sasakian	ADJ
cana-3453	358	26	-	-	PUNCT
cana-3453	358	27	space	space	NOUN
cana-3453	358	28	-	-	PUNCT
cana-3453	358	29	forms	form	NOUN
cana-3453	358	30	,	,	PUNCT
cana-3453	358	31	jp	jp	NOUN
cana-3453	358	32	j.	j.	PROPN
cana-3453	358	33	geom	geom	PROPN
cana-3453	358	34	.	.	PUNCT
cana-3453	359	1	and	and	CCONJ
cana-3453	359	2	topology	topology	NOUN
cana-3453	359	3	,	,	PUNCT
cana-3453	359	4	6	6	NUM
cana-3453	359	5	(	(	PUNCT
cana-3453	359	6	2006	2006	NUM
cana-3453	359	7	)	)	PUNCT
cana-3453	359	8	,	,	PUNCT
cana-3453	359	9	151–166	151–166	NUM
cana-3453	359	10	.	.	PUNCT
cana-3453	360	1	communications	communication	NOUN
cana-3453	360	2	on	on	ADP
cana-3453	360	3	applied	apply	VERB
cana-3453	360	4	nonlinear	nonlinear	ADJ
cana-3453	360	5	analysis	analysis	NOUN
cana-3453	360	6	issn	issn	NOUN
cana-3453	360	7	:	:	PUNCT
cana-3453	360	8	1074	1074	NUM
cana-3453	360	9	-	-	PUNCT
cana-3453	360	10	133x	133x	NUM
cana-3453	360	11	vol	vol	NOUN
cana-3453	360	12	32	32	NUM
cana-3453	360	13	no	no	NOUN
cana-3453	360	14	.	.	PUNCT
cana-3453	361	1	7s	7	NOUN
cana-3453	361	2	(	(	PUNCT
cana-3453	361	3	2025	2025	NUM
cana-3453	361	4	)	)	PUNCT
cana-3453	361	5	431	431	NUM
cana-3453	361	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3453	362	1	[	[	X
cana-3453	362	2	16	16	NUM
cana-3453	362	3	]	]	X
cana-3453	362	4	gherib	gherib	PROPN
cana-3453	362	5	,	,	PUNCT
cana-3453	362	6	f.	f.	PROPN
cana-3453	362	7	,	,	PUNCT
cana-3453	362	8	gorine	gorine	NOUN
cana-3453	362	9	,	,	PUNCT
cana-3453	362	10	m.	m.	NOUN
cana-3453	362	11	and	and	CCONJ
cana-3453	362	12	belkhelfa	belkhelfa	NOUN
cana-3453	362	13	,	,	PUNCT
cana-3453	362	14	m.	m.	NOUN
cana-3453	362	15	,	,	PUNCT
cana-3453	362	16	parallel	parallel	ADJ
cana-3453	362	17	and	and	CCONJ
cana-3453	362	18	semi	semi	ADJ
cana-3453	362	19	symmetry	symmetry	NOUN
cana-3453	362	20	of	of	ADP
cana-3453	362	21	some	some	DET
cana-3453	362	22	tensors	tensor	NOUN
cana-3453	362	23	in	in	ADP
cana-3453	362	24	generalized	generalized	ADJ
cana-3453	362	25	sasakian	sasakian	ADJ
cana-3453	362	26	-	-	PUNCT
cana-3453	362	27	spaceforms	spaceform	NOUN
cana-3453	362	28	,	,	PUNCT
cana-3453	362	29	bull	bull	NOUN
cana-3453	362	30	.	.	PUNCT
cana-3453	363	1	trans	trans	PROPN
cana-3453	363	2	.	.	PROPN
cana-3453	363	3	univ	univ	PROPN
cana-3453	363	4	.	.	PUNCT
cana-3453	363	5	brasov	brasov	PROPN
cana-3453	363	6	,	,	PUNCT
cana-3453	363	7	series	series	PROPN
cana-3453	363	8	iii	iii	PROPN
cana-3453	363	9	:	:	PUNCT
cana-3453	363	10	mathematics	mathematic	NOUN
cana-3453	363	11	,	,	PUNCT
cana-3453	363	12	informatics	informatic	NOUN
cana-3453	363	13	,	,	PUNCT
cana-3453	363	14	physics	physics	NOUN
cana-3453	363	15	,	,	PUNCT
cana-3453	363	16	1(50	1(50	NUM
cana-3453	363	17	)	)	PUNCT
cana-3453	363	18	(	(	PUNCT
cana-3453	363	19	2008	2008	NUM
cana-3453	363	20	)	)	PUNCT
cana-3453	363	21	,	,	PUNCT
cana-3453	363	22	139–148	139–148	NUM
cana-3453	363	23	.	.	PUNCT
cana-3453	364	1	[	[	X
cana-3453	364	2	17	17	NUM
cana-3453	364	3	]	]	X
cana-3453	364	4	hui	hui	PROPN
cana-3453	364	5	,	,	PUNCT
cana-3453	364	6	s.	s.	PROPN
cana-3453	364	7	k.	k.	PROPN
cana-3453	364	8	and	and	CCONJ
cana-3453	364	9	atçeken	atçeken	VERB
cana-3453	364	10	,	,	PUNCT
cana-3453	364	11	m.	m.	NOUN
cana-3453	364	12	,	,	PUNCT
cana-3453	364	13	contact	contact	NOUN
cana-3453	364	14	warped	warp	VERB
cana-3453	364	15	product	product	NOUN
cana-3453	364	16	semi	semi	ADJ
cana-3453	364	17	-	-	ADJ
cana-3453	364	18	slant	slant	ADJ
cana-3453	364	19	submanifolds	submanifold	NOUN
cana-3453	364	20	of	of	ADP
cana-3453	364	21	(	(	PUNCT
cana-3453	364	22	lcs)n	lcs)n	PROPN
cana-3453	364	23	manifolds	manifolds	PROPN
cana-3453	364	24	,	,	PUNCT
cana-3453	364	25	acta	acta	PROPN
cana-3453	364	26	univ	univ	PROPN
cana-3453	364	27	.	.	PUNCT
cana-3453	365	1	sapientiae	sapientiae	PROPN
cana-3453	365	2	math	math	PROPN
cana-3453	365	3	.	.	PUNCT
cana-3453	366	1	,	,	PUNCT
cana-3453	366	2	3	3	NUM
cana-3453	366	3	(	(	PUNCT
cana-3453	366	4	2011	2011	NUM
cana-3453	366	5	)	)	PUNCT
cana-3453	366	6	,	,	PUNCT
cana-3453	366	7	212–224	212–224	NUM
cana-3453	366	8	.	.	PUNCT
cana-3453	367	1	[	[	X
cana-3453	367	2	18	18	NUM
cana-3453	367	3	]	]	X
cana-3453	367	4	hui	hui	PROPN
cana-3453	367	5	,	,	PUNCT
cana-3453	367	6	s.	s.	PROPN
cana-3453	367	7	k.	k.	PROPN
cana-3453	367	8	and	and	CCONJ
cana-3453	367	9	sarkar	sarkar	PROPN
cana-3453	367	10	,	,	PUNCT
cana-3453	367	11	a.	a.	PROPN
cana-3453	367	12	,	,	PUNCT
cana-3453	367	13	on	on	ADP
cana-3453	367	14	the	the	DET
cana-3453	367	15	w2	w2	NOUN
cana-3453	367	16	-	-	PUNCT
cana-3453	367	17	curvature	curvature	NOUN
cana-3453	367	18	tensor	tensor	NOUN
cana-3453	367	19	of	of	ADP
cana-3453	367	20	generalized	generalized	ADJ
cana-3453	367	21	sasakian	sasakian	ADJ
cana-3453	367	22	-	-	PUNCT
cana-3453	367	23	space	space	NOUN
cana-3453	367	24	-	-	PUNCT
cana-3453	367	25	forms	form	NOUN
cana-3453	367	26	,	,	PUNCT
cana-3453	367	27	math	math	NOUN
cana-3453	367	28	.	.	PUNCT
cana-3453	368	1	pannonica	pannonica	PROPN
cana-3453	368	2	,	,	PUNCT
cana-3453	368	3	23	23	NUM
cana-3453	368	4	(	(	PUNCT
cana-3453	368	5	2012	2012	NUM
cana-3453	368	6	)	)	PUNCT
cana-3453	368	7	,	,	PUNCT
cana-3453	368	8	113–124	113–124	NUM
cana-3453	368	9	.	.	PUNCT
cana-3453	369	1	[	[	X
cana-3453	369	2	19	19	NUM
cana-3453	369	3	]	]	X
cana-3453	369	4	kentaro	kentaro	NOUN
cana-3453	369	5	,	,	PUNCT
cana-3453	369	6	y.	y.	PROPN
cana-3453	369	7	and	and	CCONJ
cana-3453	369	8	masahiro	masahiro	PROPN
cana-3453	369	9	,	,	PUNCT
cana-3453	369	10	k.	k.	PROPN
cana-3453	369	11	,	,	PUNCT
cana-3453	369	12	contact	contact	NOUN
cana-3453	369	13	cr	cr	PROPN
cana-3453	369	14	submanifolds	submanifolds	PROPN
cana-3453	369	15	,	,	PUNCT
cana-3453	369	16	kodai	kodai	PROPN
cana-3453	369	17	math	math	PROPN
cana-3453	369	18	.	.	PUNCT
cana-3453	370	1	j.	j.	PROPN
cana-3453	370	2	,	,	PUNCT
cana-3453	370	3	5	5	NUM
cana-3453	370	4	(	(	PUNCT
cana-3453	370	5	1982	1982	NUM
cana-3453	370	6	)	)	PUNCT
cana-3453	370	7	,	,	PUNCT
cana-3453	370	8	238	238	NUM
cana-3453	370	9	-	-	SYM
cana-3453	370	10	252	252	NUM
cana-3453	370	11	.	.	PUNCT
cana-3453	371	1	[	[	X
cana-3453	371	2	20	20	NUM
cana-3453	371	3	]	]	X
cana-3453	371	4	kim	kim	PROPN
cana-3453	371	5	,	,	PUNCT
cana-3453	371	6	u.	u.	PROPN
cana-3453	371	7	k.	k.	PROPN
cana-3453	371	8	,	,	PUNCT
cana-3453	371	9	conformally	conformally	ADV
cana-3453	371	10	flat	flat	ADJ
cana-3453	371	11	generalized	generalize	VERB
cana-3453	371	12	sasakian	sasakian	ADJ
cana-3453	371	13	-	-	PUNCT
cana-3453	371	14	space	space	NOUN
cana-3453	371	15	-	-	PUNCT
cana-3453	371	16	forms	form	NOUN
cana-3453	371	17	and	and	CCONJ
cana-3453	371	18	locally	locally	ADV
cana-3453	371	19	symmetric	symmetric	ADJ
cana-3453	371	20	generalized	generalize	VERB
cana-3453	371	21	sasakian	sasakian	ADJ
cana-3453	371	22	-	-	PUNCT
cana-3453	371	23	spaceforms	spaceform	NOUN
cana-3453	371	24	,	,	PUNCT
cana-3453	371	25	note	note	VERB
cana-3453	371	26	di	di	PROPN
cana-3453	371	27	matematica	matematica	PROPN
cana-3453	371	28	,	,	PUNCT
cana-3453	371	29	26	26	NUM
cana-3453	371	30	(	(	PUNCT
cana-3453	371	31	2006	2006	NUM
cana-3453	371	32	)	)	PUNCT
cana-3453	371	33	,	,	PUNCT
cana-3453	371	34	55–67	55–67	NUM
cana-3453	371	35	.	.	PUNCT
cana-3453	372	1	[	[	X
cana-3453	372	2	21	21	NUM
cana-3453	372	3	]	]	X
cana-3453	372	4	khan	khan	PROPN
cana-3453	372	5	,	,	PUNCT
cana-3453	372	6	v.	v.	PROPN
cana-3453	372	7	a.	a.	PROPN
cana-3453	372	8	,	,	PUNCT
cana-3453	372	9	khan	khan	PROPN
cana-3453	372	10	,	,	PUNCT
cana-3453	372	11	k.	k.	PROPN
cana-3453	372	12	a.	a.	PROPN
cana-3453	372	13	and	and	CCONJ
cana-3453	372	14	uddin	uddin	PROPN
cana-3453	372	15	,	,	PUNCT
cana-3453	372	16	s.	s.	PROPN
cana-3453	372	17	,	,	PUNCT
cana-3453	372	18	contact	contact	NOUN
cana-3453	372	19	cr	cr	NOUN
cana-3453	372	20	-	-	PUNCT
cana-3453	372	21	warped	warp	VERB
cana-3453	372	22	product	product	NOUN
cana-3453	372	23	submanifolds	submanifold	NOUN
cana-3453	372	24	of	of	ADP
cana-3453	372	25	kenmotsu	kenmotsu	PROPN
cana-3453	372	26	manifolds	manifolds	PROPN
cana-3453	372	27	,	,	PUNCT
cana-3453	372	28	thai	thai	PROPN
cana-3453	372	29	j.	j.	PROPN
cana-3453	372	30	math	math	PROPN
cana-3453	372	31	.	.	PUNCT
cana-3453	372	32	,	,	PUNCT
cana-3453	372	33	6	6	NUM
cana-3453	372	34	(	(	PUNCT
cana-3453	372	35	2008	2008	NUM
cana-3453	372	36	)	)	PUNCT
cana-3453	372	37	,	,	PUNCT
cana-3453	372	38	139	139	NUM
cana-3453	372	39	-	-	SYM
cana-3453	372	40	154	154	NUM
cana-3453	372	41	.	.	PUNCT
cana-3453	373	1	[	[	X
cana-3453	373	2	22	22	NUM
cana-3453	373	3	]	]	PUNCT
cana-3453	373	4	narain	narain	NOUN
cana-3453	373	5	,	,	PUNCT
cana-3453	373	6	d.	d.	PROPN
cana-3453	373	7	,	,	PUNCT
cana-3453	373	8	yadav	yadav	PROPN
cana-3453	373	9	,	,	PUNCT
cana-3453	373	10	s.	s.	PROPN
cana-3453	373	11	and	and	CCONJ
cana-3453	373	12	dwivedi	dwivedi	PROPN
cana-3453	373	13	,	,	PUNCT
cana-3453	373	14	p.	p.	PROPN
cana-3453	373	15	k.	k.	PROPN
cana-3453	373	16	,	,	PUNCT
cana-3453	373	17	on	on	ADP
cana-3453	373	18	generalized	generalized	ADJ
cana-3453	373	19	sasakian	sasakian	ADJ
cana-3453	373	20	-	-	PUNCT
cana-3453	373	21	space	space	NOUN
cana-3453	373	22	-	-	PUNCT
cana-3453	373	23	forms	form	NOUN
cana-3453	373	24	satisfying	satisfy	VERB
cana-3453	373	25	certain	certain	ADJ
cana-3453	373	26	conditions	condition	NOUN
cana-3453	373	27	,	,	PUNCT
cana-3453	373	28	int	int	NOUN
cana-3453	373	29	.	.	PUNCT
cana-3453	374	1	j.	j.	PROPN
cana-3453	374	2	math	math	PROPN
cana-3453	374	3	.	.	PUNCT
cana-3453	375	1	and	and	CCONJ
cana-3453	375	2	analysis	analysis	NOUN
cana-3453	375	3	,	,	PUNCT
cana-3453	375	4	3	3	NUM
cana-3453	375	5	(	(	PUNCT
cana-3453	375	6	2011	2011	NUM
cana-3453	375	7	)	)	PUNCT
cana-3453	375	8	,	,	PUNCT
cana-3453	375	9	1–12	1–12	NOUN
cana-3453	375	10	.	.	PUNCT
cana-3453	376	1	[	[	X
cana-3453	376	2	23	23	NUM
cana-3453	376	3	]	]	PUNCT
cana-3453	376	4	olteanu	olteanu	X
cana-3453	376	5	,	,	PUNCT
cana-3453	376	6	a.	a.	NOUN
cana-3453	376	7	,	,	PUNCT
cana-3453	376	8	legendrian	legendrian	ADJ
cana-3453	376	9	warped	warped	ADJ
cana-3453	376	10	product	product	NOUN
cana-3453	376	11	submanifolds	submanifold	NOUN
cana-3453	376	12	in	in	ADP
cana-3453	376	13	generalized	generalized	ADJ
cana-3453	376	14	sasakian	sasakian	ADJ
cana-3453	376	15	-	-	PUNCT
cana-3453	376	16	space	space	NOUN
cana-3453	376	17	forms	form	NOUN
cana-3453	376	18	,	,	PUNCT
cana-3453	376	19	acta	acta	PROPN
cana-3453	376	20	mathematica	mathematica	PROPN
cana-3453	376	21	academiae	academiae	PROPN
cana-3453	376	22	paedagogice	paedagogice	NOUN
cana-3453	376	23	nyiregyhaziensis	nyiregyhaziensis	NOUN
cana-3453	376	24	,	,	PUNCT
cana-3453	376	25	25	25	NUM
cana-3453	376	26	(	(	PUNCT
cana-3453	376	27	2009	2009	NUM
cana-3453	376	28	)	)	PUNCT
cana-3453	376	29	,	,	PUNCT
cana-3453	376	30	137–144	137–144	NUM
cana-3453	376	31	.	.	PUNCT
cana-3453	377	1	[	[	X
cana-3453	377	2	24	24	NUM
cana-3453	377	3	]	]	PUNCT
cana-3453	377	4	olteanu	olteanu	X
cana-3453	377	5	,	,	PUNCT
cana-3453	377	6	a.	a.	NOUN
cana-3453	377	7	,	,	PUNCT
cana-3453	377	8	a	a	DET
cana-3453	377	9	general	general	ADJ
cana-3453	377	10	inequality	inequality	NOUN
cana-3453	377	11	for	for	ADP
cana-3453	377	12	doubly	doubly	ADV
cana-3453	377	13	warped	warped	ADJ
cana-3453	377	14	product	product	NOUN
cana-3453	377	15	submanifolds	submanifold	NOUN
cana-3453	377	16	,	,	PUNCT
cana-3453	377	17	math	math	NOUN
cana-3453	377	18	.	.	PUNCT
cana-3453	378	1	j.	j.	PROPN
cana-3453	378	2	okayama	okayama	PROPN
cana-3453	378	3	univ	univ	PROPN
cana-3453	378	4	.	.	PROPN
cana-3453	378	5	,	,	PUNCT
cana-3453	378	6	52	52	NUM
cana-3453	378	7	(	(	PUNCT
cana-3453	378	8	2010	2010	NUM
cana-3453	378	9	)	)	PUNCT
cana-3453	378	10	,	,	PUNCT
cana-3453	378	11	133	133	NUM
cana-3453	378	12	–	–	SYM
cana-3453	378	13	142	142	NUM
cana-3453	378	14	.	.	PUNCT
cana-3453	379	1	[	[	X
cana-3453	379	2	25	25	NUM
cana-3453	379	3	]	]	PUNCT
cana-3453	379	4	shukla	shukla	NOUN
cana-3453	379	5	,	,	PUNCT
cana-3453	379	6	s.	s.	PROPN
cana-3453	379	7	s.	s.	PROPN
cana-3453	379	8	and	and	CCONJ
cana-3453	379	9	chaubey	chaubey	PROPN
cana-3453	379	10	,	,	PUNCT
cana-3453	379	11	p.	p.	PROPN
cana-3453	379	12	k.	k.	PROPN
cana-3453	379	13	,	,	PUNCT
cana-3453	379	14	on	on	ADP
cana-3453	379	15	invariant	invariant	ADJ
cana-3453	379	16	submanifolds	submanifold	NOUN
cana-3453	379	17	in	in	ADP
cana-3453	379	18	generalized	generalized	ADJ
cana-3453	379	19	sasakian	sasakian	ADJ
cana-3453	379	20	space	space	NOUN
cana-3453	379	21	forms	form	NOUN
cana-3453	379	22	,	,	PUNCT
cana-3453	379	23	j.	j.	PROPN
cana-3453	379	24	dynamical	dynamical	ADJ
cana-3453	379	25	systems	system	NOUN
cana-3453	379	26	and	and	CCONJ
cana-3453	379	27	geometric	geometric	ADJ
cana-3453	379	28	theories	theory	NOUN
cana-3453	379	29	,	,	PUNCT
cana-3453	379	30	8	8	NUM
cana-3453	379	31	(	(	PUNCT
cana-3453	379	32	2010	2010	NUM
cana-3453	379	33	)	)	PUNCT
cana-3453	379	34	,	,	PUNCT
cana-3453	379	35	173–188	173–188	NUM
cana-3453	379	36	,	,	PUNCT
cana-3453	379	37	doi	doi	NOUN
cana-3453	379	38	:	:	PUNCT
cana-3453	379	39	10.1080/1726037x.2010.10698583	10.1080/1726037x.2010.10698583	NOUN
cana-3453	379	40	.	.	PUNCT
cana-3453	380	1	[	[	X
cana-3453	380	2	26	26	NUM
cana-3453	380	3	]	]	PUNCT
cana-3453	380	4	sreenivasa	sreenivasa	NOUN
cana-3453	380	5	,	,	PUNCT
cana-3453	380	6	g.	g.	PROPN
cana-3453	380	7	t.	t.	PROPN
cana-3453	380	8	,	,	PUNCT
cana-3453	380	9	venkatesha	venkatesha	PROPN
cana-3453	380	10	and	and	CCONJ
cana-3453	380	11	bagewadi	bagewadi	NOUN
cana-3453	380	12	,	,	PUNCT
cana-3453	380	13	c.	c.	PROPN
cana-3453	380	14	s.	s.	PROPN
cana-3453	380	15	,	,	PUNCT
cana-3453	380	16	some	some	DET
cana-3453	380	17	results	result	NOUN
cana-3453	380	18	on	on	ADP
cana-3453	380	19	(	(	PUNCT
cana-3453	380	20	lcs)2n+1manifolds	lcs)2n+1manifold	NOUN
cana-3453	380	21	,	,	PUNCT
cana-3453	380	22	bull	bull	NOUN
cana-3453	380	23	.	.	PUNCT
cana-3453	380	24	math	math	NOUN
cana-3453	380	25	.	.	PUNCT
cana-3453	381	1	analysis	analysis	NOUN
cana-3453	381	2	and	and	CCONJ
cana-3453	381	3	appl	appl	NOUN
cana-3453	381	4	.	.	PROPN
cana-3453	381	5	,	,	PUNCT
cana-3453	381	6	1(3	1(3	NUM
cana-3453	381	7	)	)	PUNCT
cana-3453	381	8	(	(	PUNCT
cana-3453	381	9	2009	2009	NUM
cana-3453	381	10	)	)	PUNCT
cana-3453	381	11	,	,	PUNCT
cana-3453	381	12	64–70	64–70	NUM
cana-3453	381	13	.	.	PUNCT
cana-3453	382	1	[	[	X
cana-3453	382	2	27	27	NUM
cana-3453	382	3	]	]	X
cana-3453	382	4	yadav	yadav	PROPN
cana-3453	382	5	,	,	PUNCT
cana-3453	382	6	s.	s.	PROPN
cana-3453	382	7	,	,	PUNCT
cana-3453	382	8	suthar	suthar	VERB
cana-3453	382	9	,	,	PUNCT
cana-3453	382	10	d.	d.	PROPN
cana-3453	382	11	l.	l.	PROPN
cana-3453	382	12	and	and	CCONJ
cana-3453	382	13	srivastava	srivastava	PROPN
cana-3453	382	14	,	,	PUNCT
cana-3453	382	15	a.	a.	PROPN
cana-3453	382	16	k.	k.	PROPN
cana-3453	382	17	,	,	PUNCT
cana-3453	382	18	some	some	DET
cana-3453	382	19	results	result	NOUN
cana-3453	382	20	on	on	ADP
cana-3453	382	21	𝑀(𝑓1	𝑀(𝑓1	PROPN
cana-3453	382	22	,	,	PUNCT
cana-3453	382	23	𝑓2	𝑓2	NOUN
cana-3453	382	24	,	,	PUNCT
cana-3453	382	25	𝑓3)2𝑛+1manifolds	𝑓3)2𝑛+1manifold	NOUN
cana-3453	382	26	,	,	PUNCT
cana-3453	382	27	int	int	NOUN
cana-3453	382	28	.	.	PUNCT
cana-3453	383	1	j.	j.	PROPN
cana-3453	383	2	pure	pure	PROPN
cana-3453	383	3	and	and	CCONJ
cana-3453	383	4	appl	appl	PROPN
cana-3453	383	5	.	.	PROPN
cana-3453	383	6	math	math	PROPN
cana-3453	383	7	.	.	PUNCT
cana-3453	384	1	,	,	PUNCT
cana-3453	384	2	70	70	NUM
cana-3453	384	3	(	(	PUNCT
cana-3453	384	4	2011	2011	NUM
cana-3453	384	5	)	)	PUNCT
cana-3453	384	6	,	,	PUNCT
cana-3453	384	7	415–423	415–423	NUM
cana-3453	384	8	.	.	PUNCT
cana-3453	385	1	[	[	X
cana-3453	385	2	28	28	NUM
cana-3453	385	3	]	]	SYM
cana-3453	385	4	yano	yano	PROPN
cana-3453	385	5	,	,	PUNCT
cana-3453	385	6	k.	k.	PROPN
cana-3453	385	7	and	and	CCONJ
cana-3453	385	8	kon	kon	PROPN
cana-3453	385	9	,	,	PUNCT
cana-3453	385	10	m.	m.	NOUN
cana-3453	385	11	,	,	PUNCT
cana-3453	385	12	structures	structure	NOUN
cana-3453	385	13	on	on	ADP
cana-3453	385	14	manifolds	manifold	NOUN
cana-3453	385	15	,	,	PUNCT
cana-3453	385	16	world	world	NOUN
cana-3453	385	17	scientific	scientific	PROPN
cana-3453	385	18	publishing	publishing	PROPN
cana-3453	385	19	co.	co.	PROPN
cana-3453	385	20	,	,	PUNCT
cana-3453	385	21	singapore	singapore	PROPN
cana-3453	385	22	,	,	PUNCT
cana-3453	385	23	1984	1984	NUM
cana-3453	385	24	.	.	PUNCT
cana-3453	386	1	[	[	X
cana-3453	386	2	29	29	NUM
cana-3453	386	3	]	]	X
cana-3453	386	4	kumari	kumari	X
cana-3453	386	5	,	,	PUNCT
cana-3453	386	6	a.	a.	NOUN
cana-3453	386	7	and	and	CCONJ
cana-3453	386	8	chanyal	chanyal	ADJ
cana-3453	386	9	,	,	PUNCT
cana-3453	386	10	s.	s.	PROPN
cana-3453	386	11	k.	k.	PROPN
cana-3453	386	12	,	,	PUNCT
cana-3453	386	13	on	on	ADP
cana-3453	386	14	the	the	DET
cana-3453	386	15	t	t	PROPN
cana-3453	386	16	curvature	curvature	NOUN
cana-3453	386	17	tensor	tensor	NOUN
cana-3453	386	18	of	of	ADP
cana-3453	386	19	generalized	generalized	ADJ
cana-3453	386	20	sasakian	sasakian	ADJ
cana-3453	386	21	-	-	PUNCT
cana-3453	386	22	space	space	NOUN
cana-3453	386	23	-	-	PUNCT
cana-3453	386	24	forms	form	NOUN
cana-3453	386	25	,	,	PUNCT
cana-3453	386	26	iosr	iosr	ADJ
cana-3453	386	27	journal	journal	NOUN
cana-3453	386	28	of	of	ADP
cana-3453	386	29	mathematics	mathematic	NOUN
cana-3453	386	30	,	,	PUNCT
cana-3453	386	31	11(3	11(3	NUM
cana-3453	386	32	)	)	PUNCT
cana-3453	386	33	(	(	PUNCT
cana-3453	386	34	2015	2015	NUM
cana-3453	386	35	)	)	PUNCT
cana-3453	386	36	61	61	NUM
cana-3453	386	37	-	-	SYM
cana-3453	386	38	68	68	NUM
cana-3453	386	39	.	.	PUNCT
cana-3453	387	1	[	[	X
cana-3453	387	2	30	30	NUM
cana-3453	387	3	]	]	X
cana-3453	387	4	nagaraja	nagaraja	PROPN
cana-3453	387	5	,	,	PUNCT
cana-3453	387	6	h.	h.	PROPN
cana-3453	387	7	g.	g.	PROPN
cana-3453	387	8	and	and	CCONJ
cana-3453	387	9	somashekhara	somashekhara	PROPN
cana-3453	387	10	,	,	PUNCT
cana-3453	387	11	g.	g.	PROPN
cana-3453	387	12	,	,	PUNCT
cana-3453	387	13	τ	τ	NOUN
cana-3453	387	14	-	-	PUNCT
cana-3453	387	15	curvature	curvature	NOUN
cana-3453	387	16	tensor	tensor	NOUN
cana-3453	387	17	in	in	ADP
cana-3453	387	18	(	(	PUNCT
cana-3453	387	19	𝑘	𝑘	NOUN
cana-3453	387	20	,	,	PUNCT
cana-3453	387	21	𝜇)-contact	𝜇)-contact	PUNCT
cana-3453	387	22	manifolds	manifolds	PROPN
cana-3453	387	23	,	,	PUNCT
cana-3453	387	24	mathematica	mathematica	PROPN
cana-3453	387	25	aeterna	aeterna	PROPN
cana-3453	387	26	,	,	PUNCT
cana-3453	387	27	2	2	NUM
cana-3453	387	28	(	(	PUNCT
cana-3453	387	29	6	6	NUM
cana-3453	387	30	)	)	PUNCT
cana-3453	387	31	(	(	PUNCT
cana-3453	387	32	2012	2012	NUM
cana-3453	387	33	)	)	PUNCT
cana-3453	387	34	523	523	NUM
cana-3453	387	35	-	-	SYM
cana-3453	387	36	532	532	NUM
cana-3453	387	37	.	.	PUNCT