id	sid	tid	token	lemma	pos
cana-2204	1	1	communications	communication	NOUN
cana-2204	1	2	on	on	ADP
cana-2204	1	3	applied	apply	VERB
cana-2204	1	4	nonlinear	nonlinear	ADJ
cana-2204	1	5	analysis	analysis	NOUN
cana-2204	1	6	issn	issn	NOUN
cana-2204	1	7	:	:	PUNCT
cana-2204	1	8	1074	1074	NUM
cana-2204	1	9	-	-	PUNCT
cana-2204	1	10	133x	133x	NUM
cana-2204	1	11	vol	vol	NOUN
cana-2204	1	12	32	32	NUM
cana-2204	1	13	no	no	NOUN
cana-2204	1	14	.	.	PUNCT
cana-2204	2	1	1s	1s	NUM
cana-2204	2	2	(	(	PUNCT
cana-2204	2	3	2025	2025	NUM
cana-2204	2	4	)	)	PUNCT
cana-2204	2	5	399	399	NUM
cana-2204	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2204	2	7	a	a	DET
cana-2204	2	8	new	new	ADJ
cana-2204	2	9	type	type	NOUN
cana-2204	2	10	of	of	ADP
cana-2204	2	11	(	(	PUNCT
cana-2204	2	12	𝝐	𝝐	NOUN
cana-2204	2	13	)	)	PUNCT
cana-2204	2	14	−	−	PROPN
cana-2204	2	15	lorentzian	lorentzian	ADJ
cana-2204	2	16	para	para	NOUN
cana-2204	2	17	-	-	PUNCT
cana-2204	2	18	sasakian	sasakian	NOUN
cana-2204	2	19	manifolds	manifold	VERB
cana-2204	2	20	shadab	shadab	PROPN
cana-2204	2	21	ahmad	ahmad	PROPN
cana-2204	2	22	khan*1	khan*1	PROPN
cana-2204	2	23	,	,	PUNCT
cana-2204	2	24	toukeer	toukeer	PROPN
cana-2204	2	25	khan2	khan2	NOUN
cana-2204	2	26	,	,	PUNCT
cana-2204	2	27	mohd	mohd	PROPN
cana-2204	2	28	bilal3	bilal3	NOUN
cana-2204	2	29	,	,	PUNCT
cana-2204	2	30	anis	anis	PROPN
cana-2204	2	31	ahmad4	ahmad4	PROPN
cana-2204	2	32	1	1	NUM
cana-2204	2	33	assistant	assistant	NOUN
cana-2204	2	34	professor	professor	NOUN
cana-2204	2	35	,	,	PUNCT
cana-2204	2	36	department	department	NOUN
cana-2204	2	37	of	of	ADP
cana-2204	2	38	mathematics	mathematics	PROPN
cana-2204	2	39	&	&	CCONJ
cana-2204	2	40	statistics	statistic	NOUN
cana-2204	2	41	,	,	PUNCT
cana-2204	2	42	integral	integral	ADJ
cana-2204	2	43	university	university	NOUN
cana-2204	2	44	,	,	PUNCT
cana-2204	2	45	lucknow	lucknow	PROPN
cana-2204	2	46	,	,	PUNCT
cana-2204	2	47	india226026	india226026	PROPN
cana-2204	2	48	sakhan@iul.ac.in	sakhan@iul.ac.in	ADV
cana-2204	2	49	*	*	PUNCT
cana-2204	3	1	2associate	2associate	NUM
cana-2204	3	2	professor	professor	NOUN
cana-2204	3	3	,	,	PUNCT
cana-2204	3	4	department	department	NOUN
cana-2204	3	5	of	of	ADP
cana-2204	3	6	liberal	liberal	ADJ
cana-2204	3	7	education	education	NOUN
cana-2204	3	8	,	,	PUNCT
cana-2204	3	9	faculty	faculty	NOUN
cana-2204	3	10	of	of	ADP
cana-2204	3	11	science	science	NOUN
cana-2204	3	12	,	,	PUNCT
cana-2204	3	13	era	era	NOUN
cana-2204	3	14	university	university	NOUN
cana-2204	3	15	,	,	PUNCT
cana-2204	3	16	lucknow	lucknow	PROPN
cana-2204	3	17	,	,	PUNCT
cana-2204	3	18	india	india	PROPN
cana-2204	3	19	-226003	-226003	PROPN
cana-2204	4	1	3department	3department	NUM
cana-2204	4	2	of	of	ADP
cana-2204	4	3	mathematical	mathematical	ADJ
cana-2204	4	4	sciences	science	NOUN
cana-2204	4	5	,	,	PUNCT
cana-2204	4	6	faculty	faculty	NOUN
cana-2204	4	7	of	of	ADP
cana-2204	4	8	applied	apply	VERB
cana-2204	4	9	sciences	science	NOUN
cana-2204	4	10	,	,	PUNCT
cana-2204	4	11	umm	umm	INTJ
cana-2204	4	12	al	al	PROPN
cana-2204	4	13	qura	qura	PROPN
cana-2204	4	14	university	university	PROPN
cana-2204	4	15	,	,	PUNCT
cana-2204	4	16	makkah	makkah	PROPN
cana-2204	4	17	21955	21955	NUM
cana-2204	4	18	,	,	PUNCT
cana-2204	4	19	saudi	saudi	PROPN
cana-2204	4	20	arabia	arabia	PROPN
cana-2204	4	21	4research	4research	PROPN
cana-2204	4	22	scholar	scholar	NOUN
cana-2204	4	23	,	,	PUNCT
cana-2204	4	24	department	department	NOUN
cana-2204	4	25	of	of	ADP
cana-2204	4	26	mathematics	mathematics	PROPN
cana-2204	4	27	&	&	CCONJ
cana-2204	4	28	statistics	statistic	NOUN
cana-2204	4	29	,	,	PUNCT
cana-2204	4	30	integral	integral	ADJ
cana-2204	4	31	university	university	NOUN
cana-2204	4	32	,	,	PUNCT
cana-2204	4	33	lucknow	lucknow	PROPN
cana-2204	4	34	,	,	PUNCT
cana-2204	4	35	india-226026	india-226026	ADJ
cana-2204	4	36	article	article	NOUN
cana-2204	4	37	history	history	NOUN
cana-2204	4	38	:	:	PUNCT
cana-2204	4	39	received	receive	VERB
cana-2204	4	40	:	:	PUNCT
cana-2204	4	41	21	21	NUM
cana-2204	4	42	-	-	SYM
cana-2204	4	43	08	08	NUM
cana-2204	4	44	-	-	PUNCT
cana-2204	4	45	2024	2024	NUM
cana-2204	4	46	revised	revise	VERB
cana-2204	4	47	:	:	PUNCT
cana-2204	4	48	02	02	NUM
cana-2204	4	49	-	-	SYM
cana-2204	4	50	10	10	NUM
cana-2204	4	51	-	-	PUNCT
cana-2204	4	52	2024	2024	NUM
cana-2204	4	53	accepted	accept	VERB
cana-2204	4	54	:	:	PUNCT
cana-2204	4	55	20	20	NUM
cana-2204	4	56	-	-	SYM
cana-2204	4	57	10	10	NUM
cana-2204	4	58	-	-	PUNCT
cana-2204	4	59	2024	2024	NUM
cana-2204	4	60	abstract	abstract	NOUN
cana-2204	4	61	:	:	PUNCT
cana-2204	4	62	the	the	DET
cana-2204	4	63	current	current	ADJ
cana-2204	4	64	investigation	investigation	NOUN
cana-2204	4	65	commences	commence	VERB
cana-2204	4	66	by	by	ADP
cana-2204	4	67	introducing	introduce	VERB
cana-2204	4	68	a	a	DET
cana-2204	4	69	novel	novel	ADJ
cana-2204	4	70	category	category	NOUN
cana-2204	4	71	termed	term	VERB
cana-2204	4	72	(	(	PUNCT
cana-2204	4	73	ϵ	ϵ	X
cana-2204	4	74	)	)	PUNCT
cana-2204	4	75	−lorentzian	−lorentzian	ADJ
cana-2204	4	76	para	para	PROPN
cana-2204	4	77	-	-	PUNCT
cana-2204	4	78	sasakian	sasakian	ADJ
cana-2204	4	79	manifolds	manifold	NOUN
cana-2204	4	80	,	,	PUNCT
cana-2204	4	81	employing	employ	VERB
cana-2204	4	82	the	the	DET
cana-2204	4	83	generalized	generalize	VERB
cana-2204	4	84	symmetric	symmetric	ADJ
cana-2204	4	85	metric	metric	ADJ
cana-2204	4	86	connection	connection	NOUN
cana-2204	4	87	of	of	ADP
cana-2204	4	88	a	a	DET
cana-2204	4	89	specific	specific	ADJ
cana-2204	4	90	type(α	type(α	NOUN
cana-2204	4	91	,	,	PUNCT
cana-2204	4	92	β	β	NOUN
cana-2204	4	93	)	)	PUNCT
cana-2204	4	94	.	.	PUNCT
cana-2204	5	1	several	several	ADJ
cana-2204	5	2	fundamental	fundamental	ADJ
cana-2204	5	3	outcomes	outcome	NOUN
cana-2204	5	4	concerning	concern	VERB
cana-2204	5	5	with	with	ADP
cana-2204	5	6	these	these	DET
cana-2204	5	7	manifolds	manifold	NOUN
cana-2204	5	8	are	be	AUX
cana-2204	5	9	derived	derive	VERB
cana-2204	5	10	.	.	PUNCT
cana-2204	6	1	subsequently	subsequently	ADV
cana-2204	6	2	,	,	PUNCT
cana-2204	6	3	we	we	PRON
cana-2204	6	4	delve	delve	VERB
cana-2204	6	5	into	into	ADP
cana-2204	6	6	the	the	DET
cana-2204	6	7	examination	examination	NOUN
cana-2204	6	8	of	of	ADP
cana-2204	6	9	conformally	conformally	ADV
cana-2204	6	10	flat	flat	ADJ
cana-2204	6	11	and	and	CCONJ
cana-2204	6	12	weyl	weyl	VERB
cana-2204	6	13	-	-	PUNCT
cana-2204	6	14	semi	semi	ADV
cana-2204	6	15	-	-	ADJ
cana-2204	6	16	symmetric	symmetric	ADJ
cana-2204	6	17	(	(	PUNCT
cana-2204	6	18	ϵ	ϵ	NOUN
cana-2204	6	19	)	)	PUNCT
cana-2204	6	20	−	−	PROPN
cana-2204	6	21	lorentzian	lorentzian	ADJ
cana-2204	6	22	para	para	NOUN
cana-2204	6	23	-	-	PUNCT
cana-2204	6	24	sasakian	sasakian	ADJ
cana-2204	6	25	manifolds	manifold	NOUN
cana-2204	6	26	,	,	PUNCT
cana-2204	6	27	utilizing	utilize	VERB
cana-2204	6	28	the	the	DET
cana-2204	6	29	generalized	generalize	VERB
cana-2204	6	30	symmetric	symmetric	ADJ
cana-2204	6	31	metric	metric	ADJ
cana-2204	6	32	connection	connection	NOUN
cana-2204	6	33	of	of	ADP
cana-2204	6	34	the	the	DET
cana-2204	6	35	type(α	type(α	PROPN
cana-2204	6	36	,	,	PUNCT
cana-2204	6	37	β	β	NOUN
cana-2204	6	38	)	)	PUNCT
cana-2204	6	39	.	.	PUNCT
cana-2204	7	1	keywords	keyword	NOUN
cana-2204	7	2	:	:	PUNCT
cana-2204	7	3	(	(	PUNCT
cana-2204	7	4	ϵ	ϵ	X
cana-2204	7	5	)	)	PUNCT
cana-2204	7	6	−lorentzian	−lorentzian	ADJ
cana-2204	7	7	para	para	PROPN
cana-2204	7	8	-	-	PUNCT
cana-2204	7	9	sasakian	sasakian	ADJ
cana-2204	7	10	manifolds	manifold	NOUN
cana-2204	7	11	,	,	PUNCT
cana-2204	7	12	generalized	generalize	VERB
cana-2204	7	13	symmetric	symmetric	ADJ
cana-2204	7	14	metric	metric	ADJ
cana-2204	7	15	connection	connection	NOUN
cana-2204	7	16	of	of	ADP
cana-2204	7	17	the	the	DET
cana-2204	7	18	type(α	type(α	PROPN
cana-2204	7	19	,	,	PUNCT
cana-2204	7	20	β	β	NOUN
cana-2204	7	21	)	)	PUNCT
cana-2204	7	22	,	,	PUNCT
cana-2204	7	23	conformally	conformally	ADV
cana-2204	7	24	flat	flat	ADJ
cana-2204	7	25	,	,	PUNCT
cana-2204	7	26	η−einstein	η−einstein	ADJ
cana-2204	7	27	manifold	manifold	ADJ
cana-2204	7	28	weylsemisymmetric	weylsemisymmetric	ADJ
cana-2204	7	29	and	and	CCONJ
cana-2204	7	30	quasi	quasi	ADJ
cana-2204	7	31	-	-	ADJ
cana-2204	7	32	constant	constant	ADJ
cana-2204	7	33	curvature	curvature	NOUN
cana-2204	7	34	.	.	PUNCT
cana-2204	8	1	mathematics	mathematic	NOUN
cana-2204	8	2	subject	subject	ADJ
cana-2204	8	3	classification	classification	NOUN
cana-2204	8	4	:	:	PUNCT
cana-2204	8	5	2000	2000	NUM
cana-2204	8	6	.	.	PUNCT
cana-2204	9	1	53c15	53c15	NUM
cana-2204	9	2	,	,	PUNCT
cana-2204	9	3	53c25	53c25	NUM
cana-2204	9	4	,	,	PUNCT
cana-2204	9	5	53c40	53c40	NUM
cana-2204	9	6	.	.	X
cana-2204	10	1	1	1	NUM
cana-2204	10	2	.	.	X
cana-2204	10	3	introduction	introduction	NOUN
cana-2204	10	4	in	in	ADP
cana-2204	10	5	[	[	X
cana-2204	10	6	2	2	NUM
cana-2204	10	7	]	]	PUNCT
cana-2204	10	8	,	,	PUNCT
cana-2204	10	9	the	the	DET
cana-2204	10	10	authors	author	NOUN
cana-2204	10	11	introduced	introduce	VERB
cana-2204	10	12	and	and	CCONJ
cana-2204	10	13	studied	study	VERB
cana-2204	10	14	the	the	DET
cana-2204	10	15	notion	notion	NOUN
cana-2204	10	16	of	of	ADP
cana-2204	10	17	special	special	ADJ
cana-2204	10	18	conformally	conformally	ADV
cana-2204	10	19	flat	flat	ADJ
cana-2204	10	20	space	space	NOUN
cana-2204	10	21	.	.	PUNCT
cana-2204	11	1	bejancu	bejancu	PROPN
cana-2204	11	2	et	et	PROPN
cana-2204	11	3	.	.	PUNCT
cana-2204	12	1	al	al	PROPN
cana-2204	12	2	.	.	PUNCT
cana-2204	13	1	[	[	X
cana-2204	13	2	1	1	NUM
cana-2204	13	3	]	]	PUNCT
cana-2204	13	4	,	,	PUNCT
cana-2204	13	5	introduced	introduce	VERB
cana-2204	13	6	the	the	DET
cana-2204	13	7	concept	concept	NOUN
cana-2204	13	8	of	of	ADP
cana-2204	13	9	(	(	PUNCT
cana-2204	13	10	𝜖)-sasakian	𝜖)-sasakian	PROPN
cana-2204	13	11	manifolds	manifold	NOUN
cana-2204	13	12	.	.	PUNCT
cana-2204	14	1	also	also	ADV
cana-2204	14	2	,	,	PUNCT
cana-2204	14	3	xufeng	xufeng	PROPN
cana-2204	14	4	and	and	CCONJ
cana-2204	14	5	xiaoli	xiaoli	NOUN
cana-2204	15	1	[	[	X
cana-2204	15	2	4	4	X
cana-2204	15	3	]	]	PUNCT
cana-2204	15	4	showed	show	VERB
cana-2204	15	5	that	that	SCONJ
cana-2204	15	6	every	every	DET
cana-2204	15	7	(	(	PUNCT
cana-2204	15	8	𝜖	𝜖	NOUN
cana-2204	15	9	)	)	PUNCT
cana-2204	15	10	-sasakian	-sasakian	ADJ
cana-2204	15	11	manifold	manifold	NOUN
cana-2204	15	12	must	must	AUX
cana-2204	15	13	be	be	AUX
cana-2204	15	14	a	a	DET
cana-2204	15	15	real	real	ADJ
cana-2204	15	16	hypersurface	hypersurface	NOUN
cana-2204	15	17	of	of	ADP
cana-2204	15	18	some	some	DET
cana-2204	15	19	indefinite	indefinite	ADJ
cana-2204	15	20	kaehler	kaehler	NOUN
cana-2204	15	21	manifold	manifold	ADJ
cana-2204	15	22	.	.	PUNCT
cana-2204	16	1	t.	t.	PROPN
cana-2204	16	2	takahashi	takahashi	PROPN
cana-2204	16	3	introduced	introduce	VERB
cana-2204	16	4	almost	almost	ADV
cana-2204	16	5	contact	contact	NOUN
cana-2204	16	6	manifolds	manifold	NOUN
cana-2204	16	7	equipped	equip	VERB
cana-2204	16	8	with	with	ADP
cana-2204	16	9	associated	associate	VERB
cana-2204	16	10	indefinite	indefinite	ADJ
cana-2204	16	11	metrics	metric	NOUN
cana-2204	16	12	in	in	ADP
cana-2204	16	13	1969	1969	NUM
cana-2204	16	14	and	and	CCONJ
cana-2204	16	15	studied	study	VERB
cana-2204	16	16	sasakian	sasakian	ADJ
cana-2204	16	17	manifolds	manifold	NOUN
cana-2204	16	18	equipped	equip	VERB
cana-2204	16	19	with	with	ADP
cana-2204	16	20	an	an	DET
cana-2204	16	21	associated	associated	ADJ
cana-2204	16	22	indefinite	indefinite	ADJ
cana-2204	16	23	metric	metric	ADJ
cana-2204	16	24	.	.	PUNCT
cana-2204	17	1	since	since	SCONJ
cana-2204	17	2	the	the	DET
cana-2204	17	3	substantial	substantial	ADJ
cana-2204	17	4	role	role	NOUN
cana-2204	17	5	that	that	PRON
cana-2204	17	6	sasakian	sasakian	PROPN
cana-2204	17	7	manifolds	manifold	NOUN
cana-2204	17	8	with	with	ADP
cana-2204	17	9	indefinite	indefinite	ADJ
cana-2204	17	10	metrics	metric	NOUN
cana-2204	17	11	play	play	VERB
cana-2204	17	12	in	in	ADP
cana-2204	17	13	physics	physics	NOUN
cana-2204	18	1	[	[	X
cana-2204	18	2	5	5	NUM
cana-2204	18	3	]	]	PUNCT
cana-2204	18	4	,	,	PUNCT
cana-2204	18	5	our	our	PRON
cana-2204	18	6	inclination	inclination	NOUN
cana-2204	18	7	naturally	naturally	ADV
cana-2204	18	8	lies	lie	VERB
cana-2204	18	9	in	in	ADP
cana-2204	18	10	exploring	explore	VERB
cana-2204	18	11	diverse	diverse	ADJ
cana-2204	18	12	contact	contact	NOUN
cana-2204	18	13	manifolds	manifold	NOUN
cana-2204	18	14	with	with	ADP
cana-2204	18	15	indefinite	indefinite	ADJ
cana-2204	18	16	metrics	metric	NOUN
cana-2204	18	17	.	.	PUNCT
cana-2204	19	1	recently	recently	ADV
cana-2204	19	2	,	,	PUNCT
cana-2204	19	3	in	in	ADP
cana-2204	19	4	2009	2009	NUM
cana-2204	19	5	,	,	PUNCT
cana-2204	19	6	u.c	u.c	PROPN
cana-2204	19	7	.	.	PROPN
cana-2204	19	8	de	de	PROPN
cana-2204	19	9	&	&	CCONJ
cana-2204	19	10	sarkar	sarkar	PROPN
cana-2204	20	1	[	[	X
cana-2204	20	2	8	8	NUM
cana-2204	20	3	]	]	PUNCT
cana-2204	20	4	,	,	PUNCT
cana-2204	20	5	studied(𝜖)-kenmotsu	studied(𝜖)-kenmotsu	PROPN
cana-2204	20	6	manifolds	manifold	VERB
cana-2204	20	7	.	.	PUNCT
cana-2204	21	1	k.	k.	PROPN
cana-2204	21	2	matsumoto	matsumoto	PROPN
cana-2204	22	1	[	[	X
cana-2204	22	2	7	7	NUM
cana-2204	22	3	]	]	PUNCT
cana-2204	22	4	,	,	PUNCT
cana-2204	22	5	introduced	introduce	VERB
cana-2204	22	6	the	the	DET
cana-2204	22	7	notion	notion	NOUN
cana-2204	22	8	of	of	ADP
cana-2204	22	9	lorentzian	lorentzian	ADJ
cana-2204	22	10	parasasakian	parasasakian	PROPN
cana-2204	22	11	manifolds	manifolds	PROPN
cana-2204	22	12	.	.	PUNCT
cana-2204	22	13	i.	i.	PROPN
cana-2204	22	14	mihai	mihai	PROPN
cana-2204	22	15	and	and	CCONJ
cana-2204	22	16	r.	r.	PROPN
cana-2204	22	17	rosca	rosca	PROPN
cana-2204	23	1	[	[	X
cana-2204	23	2	9	9	NUM
cana-2204	23	3	]	]	PUNCT
cana-2204	23	4	,	,	PUNCT
cana-2204	23	5	defined	define	VERB
cana-2204	23	6	the	the	DET
cana-2204	23	7	same	same	ADJ
cana-2204	23	8	notion	notion	NOUN
cana-2204	23	9	independently	independently	ADV
cana-2204	23	10	and	and	CCONJ
cana-2204	23	11	several	several	ADJ
cana-2204	23	12	authors	author	NOUN
cana-2204	23	13	[	[	X
cana-2204	23	14	10	10	NUM
cana-2204	23	15	]	]	PUNCT
cana-2204	23	16	,	,	PUNCT
cana-2204	23	17	[	[	X
cana-2204	23	18	11	11	NUM
cana-2204	23	19	]	]	PUNCT
cana-2204	23	20	,	,	PUNCT
cana-2204	23	21	[	[	X
cana-2204	23	22	12	12	NUM
cana-2204	23	23	]	]	PUNCT
cana-2204	23	24	,	,	PUNCT
cana-2204	23	25	[	[	X
cana-2204	23	26	14	14	NUM
cana-2204	23	27	]	]	PUNCT
cana-2204	23	28	,	,	PUNCT
cana-2204	23	29	[	[	X
cana-2204	23	30	16	16	NUM
cana-2204	23	31	]	]	PUNCT
cana-2204	23	32	,	,	PUNCT
cana-2204	23	33	[	[	X
cana-2204	23	34	17	17	NUM
cana-2204	23	35	]	]	PUNCT
cana-2204	23	36	also	also	ADV
cana-2204	23	37	studied	study	VERB
cana-2204	23	38	different	different	ADJ
cana-2204	23	39	structures	structure	NOUN
cana-2204	23	40	.	.	PUNCT
cana-2204	24	1	a	a	DET
cana-2204	24	2	linear	linear	ADJ
cana-2204	24	3	connection	connection	NOUN
cana-2204	24	4	∇̅	∇̅	NOUN
cana-2204	24	5	on	on	ADP
cana-2204	24	6	a	a	DET
cana-2204	24	7	riemannian	riemannian	ADJ
cana-2204	24	8	manifold	manifold	ADJ
cana-2204	24	9	𝑀	𝑀	PROPN
cana-2204	24	10	is	be	AUX
cana-2204	24	11	suggested	suggest	VERB
cana-2204	24	12	to	to	PART
cana-2204	24	13	be	be	AUX
cana-2204	24	14	a	a	DET
cana-2204	24	15	generalized	generalized	ADJ
cana-2204	24	16	symmetric	symmetric	ADJ
cana-2204	24	17	connection	connection	NOUN
cana-2204	24	18	if	if	SCONJ
cana-2204	24	19	its	its	PRON
cana-2204	24	20	torsion	torsion	NOUN
cana-2204	24	21	tensor	tensor	NOUN
cana-2204	24	22	𝑇	𝑇	PROPN
cana-2204	24	23	is	be	AUX
cana-2204	24	24	defined	define	VERB
cana-2204	24	25	as	as	ADP
cana-2204	24	26	:	:	PUNCT
cana-2204	24	27	𝑇(𝑋	𝑇(𝑋	NOUN
cana-2204	24	28	,	,	PUNCT
cana-2204	24	29	𝑌	𝑌	PROPN
cana-2204	24	30	)	)	PUNCT
cana-2204	24	31	=	=	SYM
cana-2204	24	32	𝛼{𝑢(𝑌)𝑋	𝛼{𝑢(𝑌)𝑋	PROPN
cana-2204	24	33	−	−	PROPN
cana-2204	24	34	𝑢(𝑋)𝑌	𝑢(𝑋)𝑌	PROPN
cana-2204	24	35	}	}	PUNCT
cana-2204	24	36	+	+	NUM
cana-2204	24	37	𝛽{𝑢(𝑌)𝜙𝑋	𝛽{𝑢(𝑌)𝜙𝑋	NOUN
cana-2204	24	38	−	−	NOUN
cana-2204	24	39	𝑢(𝑋)𝜙𝑌	𝑢(𝑋)𝜙𝑌	NOUN
cana-2204	24	40	}	}	PUNCT
cana-2204	24	41	(	(	PUNCT
cana-2204	24	42	1.1	1.1	NUM
cana-2204	24	43	)	)	PUNCT
cana-2204	24	44	communications	communication	NOUN
cana-2204	24	45	on	on	ADP
cana-2204	24	46	applied	apply	VERB
cana-2204	24	47	nonlinear	nonlinear	ADJ
cana-2204	24	48	analysis	analysis	NOUN
cana-2204	24	49	issn	issn	NOUN
cana-2204	24	50	:	:	PUNCT
cana-2204	24	51	1074	1074	NUM
cana-2204	24	52	-	-	PUNCT
cana-2204	24	53	133x	133x	NUM
cana-2204	24	54	vol	vol	NOUN
cana-2204	24	55	32	32	NUM
cana-2204	25	1	no	no	NOUN
cana-2204	25	2	.	.	PUNCT
cana-2204	26	1	1s	1s	NUM
cana-2204	26	2	(	(	PUNCT
cana-2204	26	3	2025	2025	NUM
cana-2204	26	4	)	)	PUNCT
cana-2204	26	5	400	400	NUM
cana-2204	26	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2204	26	7	for	for	ADP
cana-2204	26	8	any	any	DET
cana-2204	26	9	vector	vector	NOUN
cana-2204	26	10	field	field	NOUN
cana-2204	26	11	𝑋	𝑋	NOUN
cana-2204	26	12	and	and	CCONJ
cana-2204	26	13	𝑌	𝑌	PROPN
cana-2204	26	14	on	on	ADP
cana-2204	26	15	𝑀	𝑀	PROPN
cana-2204	26	16	,	,	PUNCT
cana-2204	26	17	where	where	SCONJ
cana-2204	26	18	𝛼	𝛼	X
cana-2204	26	19	and	and	CCONJ
cana-2204	26	20	𝛽	𝛽	NOUN
cana-2204	26	21	are	be	AUX
cana-2204	26	22	constant	constant	ADJ
cana-2204	26	23	functions	function	NOUN
cana-2204	26	24	on	on	ADP
cana-2204	26	25	𝑀[13	𝑀[13	PROPN
cana-2204	26	26	]	]	PUNCT
cana-2204	26	27	,	,	PUNCT
cana-2204	26	28	𝜙	𝜙	PROPN
cana-2204	26	29	can	can	AUX
cana-2204	26	30	be	be	AUX
cana-2204	26	31	viewed	view	VERB
cana-2204	26	32	as	as	ADP
cana-2204	26	33	tensor	tensor	NOUN
cana-2204	26	34	of	of	ADP
cana-2204	26	35	type	type	NOUN
cana-2204	26	36	(	(	PUNCT
cana-2204	26	37	1	1	NUM
cana-2204	26	38	,	,	PUNCT
cana-2204	26	39	1	1	NUM
cana-2204	26	40	)	)	PUNCT
cana-2204	26	41	and	and	CCONJ
cana-2204	26	42	𝑢	𝑢	PROPN
cana-2204	26	43	is	be	AUX
cana-2204	26	44	regarded	regard	VERB
cana-2204	26	45	as	as	ADP
cana-2204	26	46	a	a	DET
cana-2204	26	47	1	1	NUM
cana-2204	26	48	-	-	PUNCT
cana-2204	26	49	form	form	NOUN
cana-2204	26	50	connected	connect	VERB
cana-2204	26	51	with	with	ADP
cana-2204	26	52	the	the	DET
cana-2204	26	53	vector	vector	NOUN
cana-2204	26	54	field	field	NOUN
cana-2204	26	55	which	which	PRON
cana-2204	26	56	has	have	VERB
cana-2204	26	57	a	a	DET
cana-2204	26	58	non	non	ADJ
cana-2204	26	59	-	-	ADJ
cana-2204	26	60	vanishing	vanishing	ADJ
cana-2204	26	61	smooth	smooth	ADJ
cana-2204	26	62	non	non	ADJ
cana-2204	26	63	-	-	ADJ
cana-2204	26	64	null	null	ADJ
cana-2204	26	65	unit	unit	NOUN
cana-2204	26	66	.	.	PUNCT
cana-2204	27	1	a	a	DET
cana-2204	27	2	linear	linear	ADJ
cana-2204	27	3	metric	metric	ADJ
cana-2204	27	4	connection	connection	NOUN
cana-2204	27	5	satisfying	satisfy	VERB
cana-2204	27	6	the	the	DET
cana-2204	27	7	equation	equation	NOUN
cana-2204	27	8	(	(	PUNCT
cana-2204	27	9	1.1	1.1	NUM
cana-2204	27	10	)	)	PUNCT
cana-2204	27	11	is	be	AUX
cana-2204	27	12	called	call	VERB
cana-2204	27	13	generalized	generalized	ADJ
cana-2204	27	14	symmetric	symmetric	ADJ
cana-2204	27	15	metric	metric	ADJ
cana-2204	27	16	connection	connection	NOUN
cana-2204	27	17	of	of	ADP
cana-2204	27	18	type(𝛼	type(𝛼	PROPN
cana-2204	27	19	,	,	PUNCT
cana-2204	27	20	𝛽	𝛽	NOUN
cana-2204	27	21	)	)	PUNCT
cana-2204	27	22	.	.	PUNCT
cana-2204	28	1	moreover	moreover	ADV
cana-2204	28	2	,	,	PUNCT
cana-2204	28	3	the	the	DET
cana-2204	28	4	connection	connection	NOUN
cana-2204	28	5	∇̅	∇̅	PROPN
cana-2204	28	6	is	be	AUX
cana-2204	28	7	said	say	VERB
cana-2204	28	8	to	to	PART
cana-2204	28	9	be	be	AUX
cana-2204	28	10	metric	metric	ADJ
cana-2204	28	11	connection	connection	NOUN
cana-2204	28	12	if	if	SCONJ
cana-2204	28	13	∇̅𝑔	∇̅𝑔	NOUN
cana-2204	28	14	=	=	NOUN
cana-2204	28	15	0	0	NUM
cana-2204	28	16	,	,	PUNCT
cana-2204	28	17	with	with	ADP
cana-2204	28	18	𝑔	𝑔	PROPN
cana-2204	28	19	as	as	ADP
cana-2204	28	20	metric	metric	ADJ
cana-2204	28	21	tensor	tensor	NOUN
cana-2204	28	22	[	[	X
cana-2204	28	23	15	15	NUM
cana-2204	28	24	]	]	PUNCT
cana-2204	28	25	.	.	PUNCT
cana-2204	29	1	this	this	DET
cana-2204	29	2	paper	paper	NOUN
cana-2204	29	3	is	be	AUX
cana-2204	29	4	organized	organize	VERB
cana-2204	29	5	as	as	SCONJ
cana-2204	29	6	follows	follow	VERB
cana-2204	29	7	:	:	PUNCT
cana-2204	29	8	section	section	NOUN
cana-2204	29	9	i	i	PRON
cana-2204	29	10	,	,	PUNCT
cana-2204	29	11	is	be	AUX
cana-2204	29	12	introductory	introductory	ADJ
cana-2204	29	13	.	.	PUNCT
cana-2204	30	1	section	section	PROPN
cana-2204	30	2	ii	ii	PROPN
cana-2204	30	3	,	,	PUNCT
cana-2204	30	4	is	be	AUX
cana-2204	30	5	devoted	devote	VERB
cana-2204	30	6	to	to	ADP
cana-2204	30	7	preliminaries	preliminary	NOUN
cana-2204	30	8	.	.	PUNCT
cana-2204	31	1	in	in	ADP
cana-2204	31	2	section	section	PROPN
cana-2204	31	3	iii	iii	PROPN
cana-2204	31	4	,	,	PUNCT
cana-2204	31	5	we	we	PRON
cana-2204	31	6	define	define	VERB
cana-2204	31	7	(	(	PUNCT
cana-2204	31	8	𝜖	𝜖	NOUN
cana-2204	31	9	)	)	PUNCT
cana-2204	31	10	−lorentzian	−lorentzian	ADJ
cana-2204	31	11	para	para	NOUN
cana-2204	31	12	-	-	PUNCT
cana-2204	31	13	sasakian	sasakian	NOUN
cana-2204	31	14	manifold	manifold	NOUN
cana-2204	31	15	with	with	ADP
cana-2204	31	16	generalized	generalized	ADJ
cana-2204	31	17	symmetric	symmetric	ADJ
cana-2204	31	18	metric	metric	ADJ
cana-2204	31	19	connection	connection	NOUN
cana-2204	31	20	.	.	PUNCT
cana-2204	32	1	we	we	PRON
cana-2204	32	2	also	also	ADV
cana-2204	32	3	give	give	VERB
cana-2204	32	4	some	some	DET
cana-2204	32	5	basic	basic	ADJ
cana-2204	32	6	results	result	NOUN
cana-2204	32	7	of	of	ADP
cana-2204	32	8	suchtype	suchtype	NOUN
cana-2204	32	9	of	of	ADP
cana-2204	32	10	manifold	manifold	NOUN
cana-2204	32	11	in	in	ADP
cana-2204	32	12	the	the	DET
cana-2204	32	13	same	same	ADJ
cana-2204	32	14	section	section	NOUN
cana-2204	32	15	.	.	PUNCT
cana-2204	33	1	in	in	ADP
cana-2204	33	2	section	section	NOUN
cana-2204	33	3	iv	iv	NUM
cana-2204	33	4	,	,	PUNCT
cana-2204	33	5	we	we	PRON
cana-2204	33	6	have	have	AUX
cana-2204	33	7	studied	study	VERB
cana-2204	33	8	conformally	conformally	ADV
cana-2204	33	9	flat	flat	ADJ
cana-2204	33	10	(	(	PUNCT
cana-2204	33	11	𝜖	𝜖	NOUN
cana-2204	33	12	)	)	PUNCT
cana-2204	33	13	−lorentzian	−lorentzian	ADJ
cana-2204	33	14	para	para	PROPN
cana-2204	33	15	sasakian	sasakian	PROPN
cana-2204	33	16	manifold	manifold	NOUN
cana-2204	33	17	with	with	ADP
cana-2204	33	18	generalized	generalized	ADJ
cana-2204	33	19	symmetric	symmetric	ADJ
cana-2204	33	20	metric	metric	ADJ
cana-2204	33	21	connection	connection	NOUN
cana-2204	33	22	.	.	PUNCT
cana-2204	34	1	in	in	ADP
cana-2204	34	2	section	section	NOUN
cana-2204	34	3	v	v	NOUN
cana-2204	34	4	,	,	PUNCT
cana-2204	34	5	we	we	PRON
cana-2204	34	6	consider	consider	VERB
cana-2204	34	7	weyl	weyl	VERB
cana-2204	34	8	-	-	PUNCT
cana-2204	34	9	semi	semi	ADV
cana-2204	34	10	-	-	ADJ
cana-2204	34	11	symmetric	symmetric	ADJ
cana-2204	34	12	(	(	PUNCT
cana-2204	34	13	𝜖	𝜖	NOUN
cana-2204	34	14	)	)	PUNCT
cana-2204	34	15	−	−	PROPN
cana-2204	34	16	lorentzian	lorentzian	ADJ
cana-2204	34	17	para	para	NOUN
cana-2204	34	18	-	-	PUNCT
cana-2204	34	19	sasakian	sasakian	NOUN
cana-2204	34	20	manifold	manifold	NOUN
cana-2204	34	21	.	.	PUNCT
cana-2204	35	1	2	2	X
cana-2204	35	2	.	.	NUM
cana-2204	35	3	preliminaries	preliminary	NOUN
cana-2204	35	4	an	an	DET
cana-2204	35	5	n	n	ADV
cana-2204	35	6	-	-	PUNCT
cana-2204	35	7	dimensional	dimensional	ADJ
cana-2204	35	8	differential	differential	NOUN
cana-2204	35	9	manifold	manifold	NOUN
cana-2204	35	10	is	be	AUX
cana-2204	35	11	called	call	VERB
cana-2204	35	12	an	an	DET
cana-2204	35	13	(	(	PUNCT
cana-2204	35	14	𝜖	𝜖	NOUN
cana-2204	35	15	)	)	PUNCT
cana-2204	35	16	−	−	PROPN
cana-2204	35	17	lorentzian	lorentzian	ADJ
cana-2204	35	18	para	para	NOUN
cana-2204	35	19	-	-	PUNCT
cana-2204	35	20	sasakian	sasakian	ADJ
cana-2204	35	21	manifold	manifold	NOUN
cana-2204	35	22	i.e.	i.e.	X
cana-2204	35	23	(	(	PUNCT
cana-2204	35	24	𝜖)-lp	𝜖)-lp	ADJ
cana-2204	35	25	sasakian	sasakian	ADJ
cana-2204	35	26	manifold	manifold	NOUN
cana-2204	35	27	,	,	PUNCT
cana-2204	35	28	if	if	SCONJ
cana-2204	35	29	it	it	PRON
cana-2204	35	30	admits	admit	VERB
cana-2204	35	31	a	a	DET
cana-2204	35	32	(	(	PUNCT
cana-2204	35	33	1,1	1,1	NUM
cana-2204	35	34	)	)	PUNCT
cana-2204	35	35	tensor	tensor	NOUN
cana-2204	35	36	field	field	NOUN
cana-2204	35	37	𝜙	𝜙	PROPN
cana-2204	35	38	,	,	PUNCT
cana-2204	35	39	a	a	DET
cana-2204	35	40	contravariant	contravariant	ADJ
cana-2204	35	41	vector	vector	NOUN
cana-2204	35	42	field	field	NOUN
cana-2204	35	43	𝜉	𝜉	PROPN
cana-2204	35	44	,	,	PUNCT
cana-2204	35	45	a	a	DET
cana-2204	35	46	1form	1form	NUM
cana-2204	35	47	𝜂	𝜂	NOUN
cana-2204	35	48	and	and	CCONJ
cana-2204	35	49	a	a	DET
cana-2204	35	50	lorentzian	lorentzian	ADJ
cana-2204	35	51	metric	metric	NOUN
cana-2204	35	52	𝑔	𝑔	PROPN
cana-2204	35	53	which	which	PRON
cana-2204	35	54	satisfies	satisfy	VERB
cana-2204	35	55	𝜙2𝑋	𝜙2𝑋	PRON
cana-2204	35	56	=	=	SYM
cana-2204	35	57	𝑋	𝑋	PROPN
cana-2204	35	58	+	+	X
cana-2204	35	59	𝜂(𝑋)𝜉	𝜂(𝑋)𝜉	NUM
cana-2204	35	60	,	,	PUNCT
cana-2204	35	61	𝜂(𝜉	𝜂(𝜉	NOUN
cana-2204	35	62	)	)	PUNCT
cana-2204	35	63	=	=	SYM
cana-2204	35	64	−1	−1	NOUN
cana-2204	35	65	(	(	PUNCT
cana-2204	35	66	2.1	2.1	NUM
cana-2204	35	67	)	)	PUNCT
cana-2204	35	68	𝑔(𝜉	𝑔(𝜉	PROPN
cana-2204	35	69	,	,	PUNCT
cana-2204	35	70	𝜉	𝜉	NOUN
cana-2204	35	71	)	)	PUNCT
cana-2204	35	72	=	=	SYM
cana-2204	35	73	−𝜖	−𝜖	PROPN
cana-2204	35	74	,	,	PUNCT
cana-2204	35	75	𝜂(𝑋	𝜂(𝑋	ADV
cana-2204	35	76	)	)	PUNCT
cana-2204	35	77	=	=	SYM
cana-2204	35	78	𝜖𝑔(𝑋	𝜖𝑔(𝑋	PROPN
cana-2204	35	79	,	,	PUNCT
cana-2204	35	80	𝜉	𝜉	NOUN
cana-2204	35	81	)	)	PUNCT
cana-2204	35	82	,	,	PUNCT
cana-2204	35	83	𝜙𝜉	𝜙𝜉	X
cana-2204	35	84	=	=	SYM
cana-2204	35	85	0	0	NUM
cana-2204	35	86	,	,	PUNCT
cana-2204	35	87	𝜂(𝜙𝑋	𝜂(𝜙𝑋	PROPN
cana-2204	35	88	)	)	PUNCT
cana-2204	35	89	=	=	SYM
cana-2204	35	90	0	0	NUM
cana-2204	35	91	(	(	PUNCT
cana-2204	35	92	2.2	2.2	NUM
cana-2204	35	93	)	)	PUNCT
cana-2204	35	94	𝑔(𝜙𝑋	𝑔(𝜙𝑋	PROPN
cana-2204	35	95	,	,	PUNCT
cana-2204	35	96	𝜙𝑌	𝜙𝑌	NUM
cana-2204	35	97	)	)	PUNCT
cana-2204	35	98	=	=	SYM
cana-2204	35	99	𝑔(𝑋	𝑔(𝑋	PROPN
cana-2204	35	100	,	,	PUNCT
cana-2204	35	101	𝑌	𝑌	PROPN
cana-2204	35	102	)	)	PUNCT
cana-2204	36	1	+	+	CCONJ
cana-2204	36	2	𝜖	𝜖	X
cana-2204	36	3	𝜂(𝑋	𝜂(𝑋	ADV
cana-2204	36	4	)	)	PUNCT
cana-2204	36	5	𝜂(𝑌	𝜂(𝑌	PROPN
cana-2204	36	6	)	)	PUNCT
cana-2204	36	7	(	(	PUNCT
cana-2204	36	8	2.3	2.3	NUM
cana-2204	36	9	)	)	PUNCT
cana-2204	36	10	(	(	PUNCT
cana-2204	36	11	∇x𝜙)𝑌	∇x𝜙)𝑌	NOUN
cana-2204	36	12	=	=	SYM
cana-2204	36	13	𝑔(𝑋	𝑔(𝑋	PROPN
cana-2204	36	14	,	,	PUNCT
cana-2204	36	15	𝑌)𝜉	𝑌)𝜉	PUNCT
cana-2204	37	1	+	+	CCONJ
cana-2204	38	1	𝜖𝜂(𝑌)𝑋	𝜖𝜂(𝑌)𝑋	X
cana-2204	38	2	+	+	NUM
cana-2204	38	3	2𝜖𝜂(𝑋)𝜂(𝑌)𝜉	2𝜖𝜂(𝑋)𝜂(𝑌)𝜉	NUM
cana-2204	38	4	(	(	PUNCT
cana-2204	38	5	2.4	2.4	NUM
cana-2204	38	6	)	)	PUNCT
cana-2204	38	7	∇𝑋𝜉	∇𝑋𝜉	NOUN
cana-2204	38	8	=	=	SYM
cana-2204	38	9	𝜖𝜙𝑋	𝜖𝜙𝑋	NOUN
cana-2204	38	10	(	(	PUNCT
cana-2204	38	11	2.5	2.5	NUM
cana-2204	38	12	)	)	PUNCT
cana-2204	38	13	(	(	PUNCT
cana-2204	38	14	∇𝑋𝜂)𝑋	∇𝑋𝜂)𝑋	X
cana-2204	38	15	=	=	SYM
cana-2204	38	16	𝑔(𝜙𝑋	𝑔(𝜙𝑋	PROPN
cana-2204	38	17	,	,	PUNCT
cana-2204	38	18	𝑌	𝑌	PROPN
cana-2204	38	19	)	)	PUNCT
cana-2204	38	20	(	(	PUNCT
cana-2204	38	21	2.6	2.6	NUM
cana-2204	38	22	)	)	PUNCT
cana-2204	38	23	for	for	ADP
cana-2204	38	24	arbitrary	arbitrary	ADJ
cana-2204	38	25	vector	vector	NOUN
cana-2204	38	26	field	field	NOUN
cana-2204	38	27	𝑋	𝑋	NOUN
cana-2204	38	28	and	and	CCONJ
cana-2204	38	29	𝑌	𝑌	PROPN
cana-2204	38	30	;	;	PUNCT
cana-2204	38	31	where	where	SCONJ
cana-2204	38	32	∇	∇	PROPN
cana-2204	38	33	denotes	denote	VERB
cana-2204	38	34	the	the	DET
cana-2204	38	35	operator	operator	NOUN
cana-2204	38	36	of	of	ADP
cana-2204	38	37	covariant	covariant	ADJ
cana-2204	38	38	differentiation	differentiation	NOUN
cana-2204	38	39	with	with	ADP
cana-2204	38	40	respect	respect	NOUN
cana-2204	38	41	to	to	ADP
cana-2204	38	42	the	the	DET
cana-2204	38	43	metric	metric	NOUN
cana-2204	39	1	[	[	X
cana-2204	39	2	7	7	NUM
cana-2204	39	3	]	]	PUNCT
cana-2204	39	4	,	,	PUNCT
cana-2204	39	5	[	[	X
cana-2204	39	6	8	8	NUM
cana-2204	39	7	]	]	PUNCT
cana-2204	39	8	on	on	ADP
cana-2204	39	9	an	an	DET
cana-2204	39	10	ndimensional	ndimensional	ADJ
cana-2204	39	11	(	(	PUNCT
cana-2204	39	12	𝜖	𝜖	NOUN
cana-2204	39	13	)	)	PUNCT
cana-2204	39	14	−	−	PROPN
cana-2204	39	15	lorentzian	lorentzian	ADJ
cana-2204	39	16	para	para	NOUN
cana-2204	39	17	-	-	PUNCT
cana-2204	39	18	sasakian	sasakian	NOUN
cana-2204	39	19	manifold	manifold	NOUN
cana-2204	39	20	with	with	ADP
cana-2204	39	21	structure	structure	NOUN
cana-2204	39	22	(	(	PUNCT
cana-2204	39	23	𝜙	𝜙	NOUN
cana-2204	39	24	,	,	PUNCT
cana-2204	39	25	𝜉	𝜉	X
cana-2204	39	26	,	,	PUNCT
cana-2204	39	27	𝜂	𝜂	NOUN
cana-2204	39	28	,	,	PUNCT
cana-2204	39	29	𝑔	𝑔	NOUN
cana-2204	39	30	)	)	PUNCT
cana-2204	39	31	,	,	PUNCT
cana-2204	39	32	the	the	DET
cana-2204	39	33	following	follow	VERB
cana-2204	39	34	results	result	NOUN
cana-2204	39	35	hold	hold	VERB
cana-2204	39	36	[	[	X
cana-2204	39	37	8	8	NUM
cana-2204	39	38	]	]	PUNCT
cana-2204	39	39	.	.	PUNCT
cana-2204	40	1	𝑅(𝑋	𝑅(𝑋	PROPN
cana-2204	40	2	,	,	PUNCT
cana-2204	40	3	𝑌)𝜉	𝑌)𝜉	PUNCT
cana-2204	41	1	=	=	SYM
cana-2204	41	2	𝜂(𝑌)𝑋	𝜂(𝑌)𝑋	NOUN
cana-2204	41	3	−	−	PROPN
cana-2204	41	4	𝜂(𝑋)𝑌	𝜂(𝑋)𝑌	NOUN
cana-2204	41	5	(	(	PUNCT
cana-2204	41	6	2.7	2.7	NUM
cana-2204	41	7	)	)	PUNCT
cana-2204	41	8	𝑅(𝜉	𝑅(𝜉	PROPN
cana-2204	41	9	,	,	PUNCT
cana-2204	41	10	𝑋)𝜉	𝑋)𝜉	ADV
cana-2204	41	11	=	=	SYM
cana-2204	41	12	𝜖𝑔(𝑋	𝜖𝑔(𝑋	PROPN
cana-2204	41	13	,	,	PUNCT
cana-2204	41	14	𝑌)𝜉	𝑌)𝜉	PUNCT
cana-2204	42	1	−	−	PROPN
cana-2204	43	1	𝜂(𝑋)𝑌	𝜂(𝑋)𝑌	INTJ
cana-2204	43	2	(	(	PUNCT
cana-2204	43	3	2.8	2.8	NUM
cana-2204	43	4	)	)	PUNCT
cana-2204	43	5	𝑔(𝑅(𝑋	𝑔(𝑅(𝑋	PROPN
cana-2204	43	6	,	,	PUNCT
cana-2204	43	7	𝑌)𝑍	𝑌)𝑍	PROPN
cana-2204	43	8	,	,	PUNCT
cana-2204	43	9	𝜉	𝜉	X
cana-2204	43	10	)	)	PUNCT
cana-2204	43	11	=	=	SYM
cana-2204	43	12	𝑔(𝑌	𝑔(𝑌	NOUN
cana-2204	43	13	,	,	PUNCT
cana-2204	43	14	𝑍)𝜂(𝑋	𝑍)𝜂(𝑋	ADJ
cana-2204	43	15	)	)	PUNCT
cana-2204	43	16	−	−	PRON
cana-2204	43	17	𝑔(𝑋	𝑔(𝑋	PROPN
cana-2204	43	18	,	,	PUNCT
cana-2204	43	19	𝑍)𝜂(𝑌	𝑍)𝜂(𝑌	X
cana-2204	43	20	)	)	PUNCT
cana-2204	43	21	(	(	PUNCT
cana-2204	43	22	2.9	2.9	NUM
cana-2204	43	23	)	)	PUNCT
cana-2204	43	24	𝑆(𝜙𝑋	𝑆(𝜙𝑋	PROPN
cana-2204	43	25	,	,	PUNCT
cana-2204	43	26	𝜙𝑌	𝜙𝑌	NUM
cana-2204	43	27	)	)	PUNCT
cana-2204	43	28	=	=	SYM
cana-2204	43	29	𝑆(𝑋	𝑆(𝑋	PROPN
cana-2204	43	30	,	,	PUNCT
cana-2204	43	31	𝑌	𝑌	PROPN
cana-2204	43	32	)	)	PUNCT
cana-2204	43	33	+	+	CCONJ
cana-2204	43	34	(	(	PUNCT
cana-2204	43	35	𝑛	𝑛	DET
cana-2204	43	36	−	−	PROPN
cana-2204	43	37	1)𝜂(𝑋)𝜂(𝑌	1)𝜂(𝑋)𝜂(𝑌	NOUN
cana-2204	43	38	)	)	PUNCT
cana-2204	43	39	(	(	PUNCT
cana-2204	43	40	2.10	2.10	NUM
cana-2204	43	41	)	)	PUNCT
cana-2204	43	42	𝑆(𝑋	𝑆(𝑋	PROPN
cana-2204	43	43	,	,	PUNCT
cana-2204	43	44	𝜉	𝜉	NOUN
cana-2204	43	45	)	)	PUNCT
cana-2204	43	46	=	=	SYM
cana-2204	43	47	(	(	PUNCT
cana-2204	43	48	𝑛	𝑛	DET
cana-2204	43	49	−	−	PROPN
cana-2204	43	50	1)𝜂(𝑌	1)𝜂(𝑌	NUM
cana-2204	43	51	)	)	PUNCT
cana-2204	43	52	(	(	PUNCT
cana-2204	43	53	2.11	2.11	NUM
cana-2204	43	54	)	)	PUNCT
cana-2204	43	55	𝑄𝑋	𝑄𝑋	NOUN
cana-2204	43	56	=	=	SYM
cana-2204	43	57	𝜖(𝑛	𝜖(𝑛	PROPN
cana-2204	43	58	−	−	PROPN
cana-2204	43	59	1)𝜉	1)𝜉	NUM
cana-2204	43	60	(	(	PUNCT
cana-2204	43	61	2.12	2.12	NUM
cana-2204	43	62	)	)	PUNCT
cana-2204	43	63	for	for	ADP
cana-2204	43	64	any	any	DET
cana-2204	43	65	vector	vector	NOUN
cana-2204	43	66	fields	field	NOUN
cana-2204	43	67	𝑋	𝑋	PROPN
cana-2204	43	68	,	,	PUNCT
cana-2204	43	69	𝑌	𝑌	PROPN
cana-2204	43	70	and	and	CCONJ
cana-2204	43	71	𝑍	𝑍	NOUN
cana-2204	43	72	;	;	PUNCT
cana-2204	43	73	where	where	SCONJ
cana-2204	43	74	𝑅	𝑅	PROPN
cana-2204	43	75	is	be	AUX
cana-2204	43	76	the	the	DET
cana-2204	43	77	riemannian	riemannian	ADJ
cana-2204	43	78	curvature	curvature	NOUN
cana-2204	43	79	tensor	tensor	NOUN
cana-2204	43	80	,	,	PUNCT
cana-2204	43	81	𝑆	𝑆	PROPN
cana-2204	43	82	is	be	AUX
cana-2204	43	83	the	the	DET
cana-2204	43	84	ricci	ricci	PROPN
cana-2204	43	85	tensor	tensor	NOUN
cana-2204	43	86	and	and	CCONJ
cana-2204	43	87	𝑄	𝑄	PROPN
cana-2204	43	88	is	be	AUX
cana-2204	43	89	the	the	DET
cana-2204	43	90	ricci	ricci	NOUN
cana-2204	43	91	operator	operator	NOUN
cana-2204	43	92	given	give	VERB
cana-2204	43	93	by	by	ADP
cana-2204	43	94	𝑔(𝑄𝑋	𝑔(𝑄𝑋	PROPN
cana-2204	43	95	,	,	PUNCT
cana-2204	43	96	𝑌	𝑌	PROPN
cana-2204	43	97	)	)	PUNCT
cana-2204	43	98	=	=	SYM
cana-2204	43	99	𝑆(𝑋	𝑆(𝑋	PROPN
cana-2204	43	100	,	,	PUNCT
cana-2204	43	101	𝑌	𝑌	PROPN
cana-2204	43	102	)	)	PUNCT
cana-2204	43	103	.	.	PUNCT
cana-2204	44	1	communications	communication	NOUN
cana-2204	44	2	on	on	ADP
cana-2204	44	3	applied	apply	VERB
cana-2204	44	4	nonlinear	nonlinear	ADJ
cana-2204	44	5	analysis	analysis	NOUN
cana-2204	44	6	issn	issn	NOUN
cana-2204	44	7	:	:	PUNCT
cana-2204	44	8	1074	1074	NUM
cana-2204	44	9	-	-	PUNCT
cana-2204	44	10	133x	133x	NUM
cana-2204	44	11	vol	vol	NOUN
cana-2204	44	12	32	32	NUM
cana-2204	44	13	no	no	NOUN
cana-2204	44	14	.	.	PUNCT
cana-2204	45	1	1s	1s	NUM
cana-2204	45	2	(	(	PUNCT
cana-2204	45	3	2025	2025	NUM
cana-2204	45	4	)	)	PUNCT
cana-2204	45	5	401	401	NUM
cana-2204	45	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2204	46	1	we	we	PRON
cana-2204	46	2	note	note	VERB
cana-2204	46	3	that	that	SCONJ
cana-2204	46	4	,	,	PUNCT
cana-2204	46	5	if	if	SCONJ
cana-2204	46	6	𝜖	𝜖	X
cana-2204	46	7	=	=	SYM
cana-2204	46	8	1and	1and	NUM
cana-2204	46	9	the	the	DET
cana-2204	46	10	structure	structure	NOUN
cana-2204	46	11	vector	vector	NOUN
cana-2204	46	12	field	field	NOUN
cana-2204	46	13	𝜉	𝜉	NOUN
cana-2204	46	14	is	be	AUX
cana-2204	46	15	space	space	NOUN
cana-2204	46	16	like	like	ADP
cana-2204	46	17	,	,	PUNCT
cana-2204	46	18	then	then	ADV
cana-2204	46	19	an	an	DET
cana-2204	46	20	(	(	PUNCT
cana-2204	46	21	𝜖)-lp	𝜖)-lp	ADJ
cana-2204	46	22	sasakian	sasakian	ADJ
cana-2204	46	23	manifold	manifold	NOUN
cana-2204	46	24	is	be	AUX
cana-2204	46	25	a	a	DET
cana-2204	46	26	usual	usual	ADJ
cana-2204	46	27	lp	lp	ADJ
cana-2204	46	28	-	-	ADJ
cana-2204	46	29	sasakian	sasakian	ADJ
cana-2204	46	30	manifold	manifold	NOUN
cana-2204	46	31	.	.	PUNCT
cana-2204	47	1	an	an	DET
cana-2204	47	2	(	(	PUNCT
cana-2204	47	3	𝜖)-lp	𝜖)-lp	ADJ
cana-2204	47	4	sasakian	sasakian	ADJ
cana-2204	47	5	manifold	manifold	NOUN
cana-2204	47	6	is	be	AUX
cana-2204	47	7	said	say	VERB
cana-2204	47	8	to	to	PART
cana-2204	47	9	be	be	AUX
cana-2204	47	10	einstein	einstein	PROPN
cana-2204	47	11	manifold	manifold	ADJ
cana-2204	47	12	if	if	SCONJ
cana-2204	47	13	its	its	PRON
cana-2204	47	14	ricci	ricci	PROPN
cana-2204	47	15	tensor	tensor	NOUN
cana-2204	47	16	𝑆	𝑆	PROPN
cana-2204	47	17	is	be	AUX
cana-2204	47	18	of	of	ADP
cana-2204	47	19	the	the	DET
cana-2204	47	20	form	form	NOUN
cana-2204	47	21	𝑆(𝑋	𝑆(𝑋	PROPN
cana-2204	47	22	,	,	PUNCT
cana-2204	47	23	𝑌	𝑌	PROPN
cana-2204	47	24	)	)	PUNCT
cana-2204	47	25	=	=	SYM
cana-2204	47	26	𝜆𝑔(𝑋	𝜆𝑔(𝑋	PROPN
cana-2204	47	27	,	,	PUNCT
cana-2204	47	28	𝑌	𝑌	PROPN
cana-2204	47	29	)	)	PUNCT
cana-2204	47	30	where	where	SCONJ
cana-2204	47	31	𝜆	𝜆	NOUN
cana-2204	47	32	is	be	AUX
cana-2204	47	33	a	a	DET
cana-2204	47	34	constant	constant	ADJ
cana-2204	47	35	.	.	PUNCT
cana-2204	48	1	definition	definition	NOUN
cana-2204	48	2	2.1.an	2.1.an	NUM
cana-2204	48	3	(	(	PUNCT
cana-2204	48	4	𝜖)-lp	𝜖)-lp	PROPN
cana-2204	48	5	sasakian	sasakian	ADJ
cana-2204	48	6	manifold	manifold	NOUN
cana-2204	48	7	will	will	AUX
cana-2204	48	8	be	be	AUX
cana-2204	48	9	called	call	VERB
cana-2204	48	10	a	a	DET
cana-2204	48	11	manifold	manifold	NOUN
cana-2204	48	12	of	of	ADP
cana-2204	48	13	quasi	quasi	ADJ
cana-2204	48	14	-	-	ADJ
cana-2204	48	15	constant	constant	ADJ
cana-2204	48	16	curvature	curvature	NOUN
cana-2204	48	17	if	if	SCONJ
cana-2204	48	18	the	the	DET
cana-2204	48	19	curvature	curvature	NOUN
cana-2204	48	20	tensor	tensor	NOUN
cana-2204	48	21	�	�	PROPN
cana-2204	48	22	̃	̃	PROPN
cana-2204	48	23	�	�	PROPN
cana-2204	48	24	of	of	ADP
cana-2204	48	25	type	type	NOUN
cana-2204	48	26	(	(	PUNCT
cana-2204	48	27	0,4	0,4	NOUN
cana-2204	48	28	)	)	PUNCT
cana-2204	48	29	satisfies	satisfy	VERB
cana-2204	48	30	the	the	DET
cana-2204	48	31	condition	condition	NOUN
cana-2204	48	32	�	�	PROPN
cana-2204	48	33	̃	̃	PROPN
cana-2204	48	34	�	�	NOUN
cana-2204	48	35	(𝑋	(𝑋	NUM
cana-2204	48	36	,	,	PUNCT
cana-2204	48	37	𝑌	𝑌	PROPN
cana-2204	48	38	,	,	PUNCT
cana-2204	48	39	𝑍,𝑊	𝑍,𝑊	ADJ
cana-2204	48	40	)	)	PUNCT
cana-2204	48	41	=	=	SYM
cana-2204	48	42	𝑎[𝑔(𝑌	𝑎[𝑔(𝑌	X
cana-2204	48	43	,	,	PUNCT
cana-2204	48	44	𝑍)𝑔(𝑋,𝑊	𝑍)𝑔(𝑋,𝑊	NUM
cana-2204	48	45	)	)	PUNCT
cana-2204	48	46	−	−	PRON
cana-2204	48	47	𝑔(𝑋	𝑔(𝑋	PROPN
cana-2204	48	48	,	,	PUNCT
cana-2204	48	49	𝑍)𝑔(𝑌,𝑊	𝑍)𝑔(𝑌,𝑊	X
cana-2204	48	50	)	)	PUNCT
cana-2204	48	51	]	]	PUNCT
cana-2204	49	1	+	+	CCONJ
cana-2204	49	2	𝑏[𝑇(𝑌)𝑇(𝑍)𝑔(𝑋,𝑊	𝑏[𝑇(𝑌)𝑇(𝑍)𝑔(𝑋,𝑊	NUM
cana-2204	49	3	)	)	PUNCT
cana-2204	49	4	−𝑇(𝑋)𝑇(𝑍)𝑔(𝑌,𝑊	−𝑇(𝑋)𝑇(𝑍)𝑔(𝑌,𝑊	PROPN
cana-2204	49	5	)	)	PUNCT
cana-2204	49	6	+	+	NUM
cana-2204	49	7	𝑇(𝑋)𝑇(𝑊)𝑔(𝑌	𝑇(𝑋)𝑇(𝑊)𝑔(𝑌	NOUN
cana-2204	49	8	,	,	PUNCT
cana-2204	49	9	𝑍	𝑍	NOUN
cana-2204	49	10	)	)	PUNCT
cana-2204	49	11	−	−	PROPN
cana-2204	49	12	𝑇(𝑌)𝑇(𝑊)}𝑔(𝑋	𝑇(𝑌)𝑇(𝑊)}𝑔(𝑋	PROPN
cana-2204	49	13	,	,	PUNCT
cana-2204	49	14	𝑍	𝑍	PROPN
cana-2204	49	15	)	)	PUNCT
cana-2204	49	16	}	}	PUNCT
cana-2204	49	17	(	(	PUNCT
cana-2204	49	18	2.13	2.13	NUM
cana-2204	49	19	)	)	PUNCT
cana-2204	49	20	where	where	SCONJ
cana-2204	49	21	�	�	PROPN
cana-2204	49	22	̃	̃	PROPN
cana-2204	49	23	�	�	NOUN
cana-2204	49	24	(𝑋	(𝑋	NUM
cana-2204	49	25	,	,	PUNCT
cana-2204	49	26	𝑌	𝑌	PROPN
cana-2204	49	27	,	,	PUNCT
cana-2204	49	28	𝑍,𝑊	𝑍,𝑊	ADJ
cana-2204	49	29	)	)	PUNCT
cana-2204	49	30	=	=	SYM
cana-2204	49	31	𝑔(𝑅(𝑋	𝑔(𝑅(𝑋	PROPN
cana-2204	49	32	,	,	PUNCT
cana-2204	49	33	𝑌)𝑍,𝑊	𝑌)𝑍,𝑊	ADJ
cana-2204	49	34	)	)	PUNCT
cana-2204	49	35	𝑅	𝑅	PROPN
cana-2204	49	36	is	be	AUX
cana-2204	49	37	the	the	DET
cana-2204	49	38	curvature	curvature	NOUN
cana-2204	49	39	tensor	tensor	NOUN
cana-2204	49	40	of	of	ADP
cana-2204	49	41	type	type	NOUN
cana-2204	49	42	(	(	PUNCT
cana-2204	49	43	1,3);𝑎	1,3);𝑎	NOUN
cana-2204	49	44	,	,	PUNCT
cana-2204	49	45	𝑏	𝑏	PROPN
cana-2204	49	46	are	be	AUX
cana-2204	49	47	scalar	scalar	ADJ
cana-2204	49	48	functions	function	NOUN
cana-2204	49	49	and	and	CCONJ
cana-2204	49	50	𝜌	𝜌	NOUN
cana-2204	49	51	is	be	AUX
cana-2204	49	52	a	a	DET
cana-2204	49	53	unit	unit	NOUN
cana-2204	49	54	vector	vector	NOUN
cana-2204	49	55	field	field	NOUN
cana-2204	49	56	defined	define	VERB
cana-2204	49	57	by	by	ADP
cana-2204	49	58	𝑔(𝑋	𝑔(𝑋	PROPN
cana-2204	49	59	,	,	PUNCT
cana-2204	49	60	𝜌	𝜌	ADP
cana-2204	49	61	)	)	PUNCT
cana-2204	49	62	=	=	SYM
cana-2204	49	63	𝑇(𝑋	𝑇(𝑋	NOUN
cana-2204	49	64	)	)	PUNCT
cana-2204	49	65	(	(	PUNCT
cana-2204	49	66	2.14	2.14	NUM
cana-2204	49	67	)	)	PUNCT
cana-2204	49	68	the	the	DET
cana-2204	49	69	notion	notion	NOUN
cana-2204	49	70	of	of	ADP
cana-2204	49	71	quasi	quasi	ADJ
cana-2204	49	72	-	-	ADJ
cana-2204	49	73	constant	constant	ADJ
cana-2204	49	74	curvature	curvature	NOUN
cana-2204	49	75	for	for	ADP
cana-2204	49	76	riemannian	riemannian	ADJ
cana-2204	49	77	manifolds	manifold	NOUN
cana-2204	49	78	was	be	AUX
cana-2204	49	79	given	give	VERB
cana-2204	49	80	by	by	ADP
cana-2204	49	81	chen	chen	PROPN
cana-2204	49	82	and	and	CCONJ
cana-2204	49	83	yano	yano	PROPN
cana-2204	50	1	[	[	X
cana-2204	50	2	2	2	NUM
cana-2204	50	3	]	]	PUNCT
cana-2204	50	4	.	.	PUNCT
cana-2204	51	1	definition	definition	NOUN
cana-2204	51	2	2.2.an	2.2.an	NUM
cana-2204	51	3	(	(	PUNCT
cana-2204	51	4	𝜖)-lp	𝜖)-lp	PROPN
cana-2204	51	5	sasakian	sasakian	ADJ
cana-2204	51	6	manifold	manifold	NOUN
cana-2204	51	7	will	will	AUX
cana-2204	51	8	be	be	AUX
cana-2204	51	9	called	call	VERB
cana-2204	51	10	𝜂-einstein	𝜂-einstein	PROPN
cana-2204	51	11	manifold	manifold	ADJ
cana-2204	51	12	if	if	SCONJ
cana-2204	51	13	the	the	DET
cana-2204	51	14	ricci	ricci	PROPN
cana-2204	51	15	tensor	tensor	NOUN
cana-2204	51	16	𝑆	𝑆	PROPN
cana-2204	51	17	of	of	ADP
cana-2204	51	18	type	type	NOUN
cana-2204	51	19	(	(	PUNCT
cana-2204	51	20	0	0	NUM
cana-2204	51	21	,	,	PUNCT
cana-2204	51	22	2	2	NUM
cana-2204	51	23	)	)	PUNCT
cana-2204	51	24	satisfies	satisfie	NOUN
cana-2204	51	25	[	[	X
cana-2204	51	26	2	2	NUM
cana-2204	51	27	]	]	X
cana-2204	51	28	𝑆(𝑋	𝑆(𝑋	PROPN
cana-2204	51	29	,	,	PUNCT
cana-2204	51	30	𝑌	𝑌	PROPN
cana-2204	51	31	)	)	PUNCT
cana-2204	51	32	=	=	SYM
cana-2204	51	33	𝑎𝑔(𝑋	𝑎𝑔(𝑋	PROPN
cana-2204	51	34	,	,	PUNCT
cana-2204	51	35	𝑌	𝑌	PROPN
cana-2204	51	36	)	)	PUNCT
cana-2204	51	37	+	+	NUM
cana-2204	51	38	𝑏𝜂(𝑋)𝜂(𝑌	𝑏𝜂(𝑋)𝜂(𝑌	NOUN
cana-2204	51	39	)	)	PUNCT
cana-2204	51	40	where	where	SCONJ
cana-2204	51	41	𝑎	𝑎	NOUN
cana-2204	51	42	and	and	CCONJ
cana-2204	51	43	𝑏	𝑏	NOUN
cana-2204	51	44	are	be	AUX
cana-2204	51	45	scalar	scalar	ADJ
cana-2204	51	46	functions	function	NOUN
cana-2204	51	47	.	.	PUNCT
cana-2204	52	1	definition	definition	NOUN
cana-2204	52	2	2.3.a	2.3.a	PROPN
cana-2204	52	3	type	type	NOUN
cana-2204	52	4	of	of	ADP
cana-2204	52	5	riemannian	riemannian	NOUN
cana-2204	52	6	manifold	manifold	NOUN
cana-2204	52	7	whose	whose	DET
cana-2204	52	8	curvature	curvature	NOUN
cana-2204	52	9	tensor	tensor	NOUN
cana-2204	52	10	�	�	PROPN
cana-2204	52	11	̃	̃	PROPN
cana-2204	52	12	�	�	PROPN
cana-2204	52	13	of	of	ADP
cana-2204	52	14	type	type	NOUN
cana-2204	52	15	(	(	PUNCT
cana-2204	52	16	0	0	NUM
cana-2204	52	17	,	,	PUNCT
cana-2204	52	18	4	4	NUM
cana-2204	52	19	)	)	PUNCT
cana-2204	52	20	satisfies	satisfy	VERB
cana-2204	52	21	the	the	DET
cana-2204	52	22	condition	condition	NOUN
cana-2204	52	23	�	�	PROPN
cana-2204	52	24	̃	̃	PROPN
cana-2204	52	25	�	�	NOUN
cana-2204	52	26	(𝑋	(𝑋	NUM
cana-2204	52	27	,	,	PUNCT
cana-2204	52	28	𝑌	𝑌	PROPN
cana-2204	52	29	,	,	PUNCT
cana-2204	52	30	𝑍,𝑊	𝑍,𝑊	ADJ
cana-2204	52	31	)	)	PUNCT
cana-2204	52	32	=	=	SYM
cana-2204	52	33	𝐹(𝑌	𝐹(𝑌	NOUN
cana-2204	52	34	,	,	PUNCT
cana-2204	52	35	𝑍)𝐹(𝑋,𝑊	𝑍)𝐹(𝑋,𝑊	NUM
cana-2204	52	36	)	)	PUNCT
cana-2204	52	37	−	−	ADP
cana-2204	52	38	𝐹(𝑋	𝐹(𝑋	PROPN
cana-2204	52	39	,	,	PUNCT
cana-2204	52	40	𝑍)𝐹(𝑌,𝑊	𝑍)𝐹(𝑌,𝑊	PROPN
cana-2204	52	41	)	)	PUNCT
cana-2204	52	42	(	(	PUNCT
cana-2204	52	43	2.15	2.15	NUM
cana-2204	52	44	)	)	PUNCT
cana-2204	52	45	is	be	AUX
cana-2204	52	46	called	call	VERB
cana-2204	52	47	a	a	DET
cana-2204	52	48	special	special	ADJ
cana-2204	52	49	manifold	manifold	NOUN
cana-2204	52	50	with	with	ADP
cana-2204	52	51	the	the	DET
cana-2204	52	52	associate	associate	ADJ
cana-2204	52	53	symmetric	symmetric	ADJ
cana-2204	52	54	tensor	tensor	NOUN
cana-2204	52	55	𝐹	𝐹	PROPN
cana-2204	52	56	of	of	ADP
cana-2204	52	57	type	type	NOUN
cana-2204	52	58	(	(	PUNCT
cana-2204	52	59	0,2	0,2	NUM
cana-2204	52	60	)	)	PUNCT
cana-2204	52	61	and	and	CCONJ
cana-2204	52	62	is	be	AUX
cana-2204	52	63	denoted	denote	VERB
cana-2204	52	64	by	by	ADP
cana-2204	52	65	𝜓(𝐹)𝑛.	𝜓(𝐹)𝑛.	NOUN
cana-2204	52	66	in	in	ADP
cana-2204	52	67	1956	1956	NUM
cana-2204	52	68	,	,	PUNCT
cana-2204	53	1	s.	s.	PROPN
cana-2204	53	2	s.	s.	PROPN
cana-2204	53	3	chern	chern	PROPN
cana-2204	54	1	[	[	X
cana-2204	54	2	3	3	NUM
cana-2204	54	3	]	]	PUNCT
cana-2204	54	4	studied	study	VERB
cana-2204	54	5	such	such	ADJ
cana-2204	54	6	type	type	NOUN
cana-2204	54	7	of	of	ADP
cana-2204	54	8	manifolds	manifold	NOUN
cana-2204	54	9	.	.	PUNCT
cana-2204	55	1	these	these	DET
cana-2204	55	2	manifolds	manifold	NOUN
cana-2204	55	3	are	be	AUX
cana-2204	55	4	important	important	ADJ
cana-2204	55	5	for	for	ADP
cana-2204	55	6	the	the	DET
cana-2204	55	7	following	follow	VERB
cana-2204	55	8	reasons	reason	NOUN
cana-2204	55	9	.	.	PUNCT
cana-2204	56	1	firstly	firstly	ADV
cana-2204	56	2	,	,	PUNCT
cana-2204	56	3	for	for	ADP
cana-2204	56	4	possessing	possess	VERB
cana-2204	56	5	some	some	DET
cana-2204	56	6	remarkable	remarkable	ADJ
cana-2204	56	7	properties	property	NOUN
cana-2204	56	8	related	relate	VERB
cana-2204	56	9	to	to	PART
cana-2204	56	10	curvature	curvature	VERB
cana-2204	56	11	and	and	CCONJ
cana-2204	56	12	characteristic	characteristic	ADJ
cana-2204	56	13	classes	class	NOUN
cana-2204	56	14	and	and	CCONJ
cana-2204	56	15	secondly	secondly	ADV
cana-2204	56	16	,	,	PUNCT
cana-2204	56	17	for	for	ADP
cana-2204	56	18	containing	contain	VERB
cana-2204	56	19	a	a	DET
cana-2204	56	20	manifold	manifold	NOUN
cana-2204	56	21	of	of	ADP
cana-2204	56	22	quasi	quasi	ADJ
cana-2204	56	23	-	-	ADJ
cana-2204	56	24	constant	constant	ADJ
cana-2204	56	25	curvature	curvature	NOUN
cana-2204	56	26	[	[	X
cana-2204	56	27	2	2	NUM
cana-2204	56	28	]	]	PUNCT
cana-2204	56	29	.	.	PUNCT
cana-2204	57	1	definition	definition	NOUN
cana-2204	57	2	2.4.an	2.4.an	NUM
cana-2204	57	3	(	(	PUNCT
cana-2204	57	4	𝜖)-lp	𝜖)-lp	ADJ
cana-2204	57	5	sasakian	sasakian	ADJ
cana-2204	57	6	manifold	manifold	NOUN
cana-2204	57	7	will	will	AUX
cana-2204	57	8	be	be	AUX
cana-2204	57	9	called	call	VERB
cana-2204	57	10	weyl	weyl	VERB
cana-2204	57	11	-	-	PUNCT
cana-2204	57	12	semi	semi	ADV
cana-2204	57	13	-	-	ADJ
cana-2204	57	14	symmetric	symmetric	ADJ
cana-2204	57	15	if	if	SCONJ
cana-2204	57	16	it	it	PRON
cana-2204	57	17	satisfies	satisfy	VERB
cana-2204	57	18	(	(	PUNCT
cana-2204	57	19	𝑅.	𝑅.	X
cana-2204	57	20	(	(	PUNCT
cana-2204	57	21	𝑋	𝑋	PROPN
cana-2204	57	22	,	,	PUNCT
cana-2204	57	23	𝑌	𝑌	PROPN
cana-2204	57	24	)	)	PUNCT
cana-2204	57	25	.	.	PUNCT
cana-2204	58	1	𝐶)(𝑌	𝐶)(𝑌	ADJ
cana-2204	58	2	,	,	PUNCT
cana-2204	58	3	𝑍)𝑊	𝑍)𝑊	NOUN
cana-2204	58	4	=	=	NOUN
cana-2204	58	5	0	0	NUM
cana-2204	58	6	where	where	SCONJ
cana-2204	58	7	𝑅(𝑋	𝑅(𝑋	NOUN
cana-2204	58	8	,	,	PUNCT
cana-2204	58	9	𝑌	𝑌	PROPN
cana-2204	58	10	)	)	PUNCT
cana-2204	58	11	denotes	denote	VERB
cana-2204	58	12	the	the	DET
cana-2204	58	13	curvature	curvature	NOUN
cana-2204	58	14	operator	operator	NOUN
cana-2204	58	15	and	and	CCONJ
cana-2204	58	16	𝐶(𝑌	𝐶(𝑌	PROPN
cana-2204	58	17	,	,	PUNCT
cana-2204	58	18	𝑍)𝑊	𝑍)𝑊	NOUN
cana-2204	58	19	is	be	AUX
cana-2204	58	20	the	the	DET
cana-2204	58	21	weyl	weyl	VERB
cana-2204	58	22	-	-	PUNCT
cana-2204	58	23	conformal	conformal	ADJ
cana-2204	58	24	curvature	curvature	NOUN
cana-2204	58	25	tensor	tensor	NOUN
cana-2204	58	26	.	.	PUNCT
cana-2204	59	1	3	3	X
cana-2204	59	2	.	.	X
cana-2204	59	3	on	on	ADP
cana-2204	59	4	(	(	PUNCT
cana-2204	59	5	𝝐	𝝐	NOUN
cana-2204	59	6	)	)	PUNCT
cana-2204	59	7	−lorentzian	−lorentzian	ADJ
cana-2204	59	8	para	para	NOUN
cana-2204	59	9	-	-	PUNCT
cana-2204	59	10	sasakian	sasakian	NOUN
cana-2204	59	11	manifold	manifold	NOUN
cana-2204	59	12	with	with	ADP
cana-2204	59	13	parallelized	parallelize	VERB
cana-2204	59	14	generalized	generalize	VERB
cana-2204	59	15	symmetric	symmetric	ADJ
cana-2204	59	16	metric	metric	ADJ
cana-2204	59	17	connection	connection	NOUN
cana-2204	59	18	theorem	theorem	VERB
cana-2204	59	19	3.1	3.1	NUM
cana-2204	59	20	.	.	PUNCT
cana-2204	60	1	for	for	ADP
cana-2204	60	2	an	an	DET
cana-2204	60	3	(	(	PUNCT
cana-2204	60	4	𝜖	𝜖	NOUN
cana-2204	60	5	)	)	PUNCT
cana-2204	60	6	−lorentzian	−lorentzian	ADJ
cana-2204	60	7	para	para	NOUN
cana-2204	60	8	-	-	PUNCT
cana-2204	60	9	sasakian	sasakian	NOUN
cana-2204	60	10	manifold	manifold	NOUN
cana-2204	60	11	,	,	PUNCT
cana-2204	60	12	the	the	DET
cana-2204	60	13	generalized	generalize	VERB
cana-2204	60	14	symmetric	symmetric	ADJ
cana-2204	60	15	metric	metric	ADJ
cana-2204	60	16	connection	connection	NOUN
cana-2204	60	17	∇̅	∇̅	PROPN
cana-2204	60	18	of	of	ADP
cana-2204	60	19	type	type	NOUN
cana-2204	60	20	(	(	PUNCT
cana-2204	60	21	α	α	NOUN
cana-2204	60	22	,	,	PUNCT
cana-2204	60	23	β	β	NOUN
cana-2204	60	24	)	)	PUNCT
cana-2204	60	25	is	be	AUX
cana-2204	60	26	given	give	VERB
cana-2204	60	27	by	by	ADP
cana-2204	60	28	∇̅𝑋𝑌	∇̅𝑋𝑌	ADJ
cana-2204	61	1	=	=	PUNCT
cana-2204	61	2	∇𝑋𝑌	∇𝑋𝑌	PROPN
cana-2204	62	1	+	+	CCONJ
cana-2204	62	2	𝛼{𝜂(𝑌)𝑋	𝛼{𝜂(𝑌)𝑋	NOUN
cana-2204	62	3	−	−	NOUN
cana-2204	62	4	𝜖𝑔(𝑋	𝜖𝑔(𝑋	PROPN
cana-2204	62	5	,	,	PUNCT
cana-2204	62	6	𝑌)𝜉	𝑌)𝜉	CCONJ
cana-2204	62	7	}	}	PUNCT
cana-2204	63	1	+	+	CCONJ
cana-2204	63	2	𝛽{𝜂(𝑌)𝜙𝑋	𝛽{𝜂(𝑌)𝜙𝑋	NOUN
cana-2204	63	3	−	−	NOUN
cana-2204	63	4	𝜖𝑔(𝜙𝑋	𝜖𝑔(𝜙𝑋	PROPN
cana-2204	63	5	,	,	PUNCT
cana-2204	63	6	𝑌)𝜉	𝑌)𝜉	CCONJ
cana-2204	63	7	}	}	PUNCT
cana-2204	63	8	(	(	PUNCT
cana-2204	63	9	3.1	3.1	NUM
cana-2204	63	10	)	)	PUNCT
cana-2204	63	11	communications	communication	NOUN
cana-2204	63	12	on	on	ADP
cana-2204	63	13	applied	apply	VERB
cana-2204	63	14	nonlinear	nonlinear	ADJ
cana-2204	63	15	analysis	analysis	NOUN
cana-2204	63	16	issn	issn	NOUN
cana-2204	63	17	:	:	PUNCT
cana-2204	63	18	1074	1074	NUM
cana-2204	63	19	-	-	PUNCT
cana-2204	63	20	133x	133x	NUM
cana-2204	63	21	vol	vol	NOUN
cana-2204	63	22	32	32	NUM
cana-2204	63	23	no	no	NOUN
cana-2204	63	24	.	.	PUNCT
cana-2204	64	1	1s	1s	NUM
cana-2204	64	2	(	(	PUNCT
cana-2204	64	3	2025	2025	NUM
cana-2204	64	4	)	)	PUNCT
cana-2204	64	5	402	402	NUM
cana-2204	65	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-2204	65	2	proof	proof	NOUN
cana-2204	65	3	:	:	PUNCT
cana-2204	65	4	the	the	DET
cana-2204	65	5	relation	relation	NOUN
cana-2204	65	6	between	between	ADP
cana-2204	65	7	a	a	DET
cana-2204	65	8	linear	linear	ADJ
cana-2204	65	9	connection	connection	NOUN
cana-2204	65	10	∇̅	∇̅	PROPN
cana-2204	65	11	and	and	CCONJ
cana-2204	65	12	levi	levi	PROPN
cana-2204	65	13	-	-	PUNCT
cana-2204	65	14	civita	civita	PROPN
cana-2204	65	15	connection	connection	NOUN
cana-2204	65	16	∇	∇	X
cana-2204	65	17	is	be	AUX
cana-2204	65	18	given	give	VERB
cana-2204	65	19	by	by	ADP
cana-2204	65	20	∇̅𝑋𝑌	∇̅𝑋𝑌	ADJ
cana-2204	66	1	=	=	PUNCT
cana-2204	66	2	∇𝑋𝑌	∇𝑋𝑌	PROPN
cana-2204	67	1	+	+	CCONJ
cana-2204	67	2	𝐻(𝑋	𝐻(𝑋	NOUN
cana-2204	67	3	,	,	PUNCT
cana-2204	67	4	𝑌	𝑌	PROPN
cana-2204	67	5	)	)	PUNCT
cana-2204	67	6	(	(	PUNCT
cana-2204	67	7	3.2	3.2	NUM
cana-2204	67	8	)	)	PUNCT
cana-2204	67	9	for	for	ADP
cana-2204	67	10	all	all	DET
cana-2204	67	11	vector	vector	NOUN
cana-2204	67	12	field	field	NOUN
cana-2204	67	13	𝑋	𝑋	NOUN
cana-2204	67	14	and	and	CCONJ
cana-2204	67	15	𝑌.	𝑌.	PROPN
cana-2204	67	16	the	the	DET
cana-2204	67	17	following	follow	VERB
cana-2204	67	18	equation	equation	NOUN
cana-2204	67	19	is	be	AUX
cana-2204	67	20	such	such	ADJ
cana-2204	67	21	that	that	SCONJ
cana-2204	67	22	∇̅	∇̅	PROPN
cana-2204	67	23	is	be	AUX
cana-2204	67	24	a	a	DET
cana-2204	67	25	generalized	generalize	VERB
cana-2204	67	26	symmetric	symmetric	ADJ
cana-2204	67	27	metric	metric	ADJ
cana-2204	67	28	connection	connection	NOUN
cana-2204	67	29	of∇.	of∇.	NUM
cana-2204	67	30	in	in	ADP
cana-2204	67	31	which	which	PRON
cana-2204	67	32	𝐻	𝐻	PROPN
cana-2204	67	33	is	be	AUX
cana-2204	67	34	viewed	view	VERB
cana-2204	67	35	as	as	ADP
cana-2204	67	36	a	a	DET
cana-2204	67	37	tensor	tensor	NOUN
cana-2204	67	38	of	of	ADP
cana-2204	67	39	type	type	NOUN
cana-2204	67	40	(	(	PUNCT
cana-2204	67	41	1	1	NUM
cana-2204	67	42	,	,	PUNCT
cana-2204	67	43	2	2	NUM
cana-2204	67	44	)	)	PUNCT
cana-2204	67	45	,	,	PUNCT
cana-2204	67	46	given	give	VERB
cana-2204	67	47	by	by	ADP
cana-2204	67	48	𝐻(𝑋	𝐻(𝑋	NOUN
cana-2204	67	49	,	,	PUNCT
cana-2204	67	50	𝑌	𝑌	PROPN
cana-2204	67	51	)	)	PUNCT
cana-2204	67	52	=	=	SYM
cana-2204	67	53	1	1	NUM
cana-2204	67	54	2	2	NUM
cana-2204	67	55	[	[	X
cana-2204	67	56	𝑇(𝑋	𝑇(𝑋	NOUN
cana-2204	67	57	,	,	PUNCT
cana-2204	67	58	𝑌	𝑌	PROPN
cana-2204	67	59	)	)	PUNCT
cana-2204	67	60	+	+	CCONJ
cana-2204	67	61	𝑇	𝑇	PROPN
cana-2204	67	62	′(𝑋	′(𝑋	NOUN
cana-2204	67	63	,	,	PUNCT
cana-2204	67	64	𝑌	𝑌	PROPN
cana-2204	67	65	)	)	PUNCT
cana-2204	67	66	+	+	CCONJ
cana-2204	67	67	𝑇	𝑇	PROPN
cana-2204	67	68	′(𝑌	′(𝑌	NOUN
cana-2204	67	69	,	,	PUNCT
cana-2204	67	70	𝑋	𝑋	PROPN
cana-2204	67	71	)	)	PUNCT
cana-2204	67	72	(	(	PUNCT
cana-2204	67	73	3.3	3.3	NUM
cana-2204	67	74	)	)	PUNCT
cana-2204	67	75	where	where	SCONJ
cana-2204	67	76	t	t	PROPN
cana-2204	67	77	is	be	AUX
cana-2204	67	78	viewed	view	VERB
cana-2204	67	79	as	as	ADP
cana-2204	67	80	the	the	DET
cana-2204	67	81	torsion	torsion	NOUN
cana-2204	67	82	tensor	tensor	NOUN
cana-2204	67	83	of	of	ADP
cana-2204	67	84	∇̅	∇̅	PROPN
cana-2204	67	85	and	and	CCONJ
cana-2204	67	86	𝑔(𝑇	𝑔(𝑇	NOUN
cana-2204	67	87	′(𝑋	′(𝑋	NOUN
cana-2204	67	88	,	,	PUNCT
cana-2204	67	89	𝑌),𝑊	𝑌),𝑊	PROPN
cana-2204	67	90	)	)	PUNCT
cana-2204	67	91	=	=	SYM
cana-2204	67	92	𝑔(𝑇(𝑊,𝑋	𝑔(𝑇(𝑊,𝑋	NOUN
cana-2204	67	93	)	)	PUNCT
cana-2204	67	94	,	,	PUNCT
cana-2204	67	95	𝑌	𝑌	PROPN
cana-2204	67	96	)	)	PUNCT
cana-2204	67	97	(	(	PUNCT
cana-2204	67	98	3.4	3.4	NUM
cana-2204	67	99	)	)	PUNCT
cana-2204	67	100	replacing	replace	VERB
cana-2204	67	101	𝑊	𝑊	PROPN
cana-2204	67	102	by𝜉	by𝜉	NOUN
cana-2204	67	103	,	,	PUNCT
cana-2204	67	104	using	use	VERB
cana-2204	67	105	(	(	PUNCT
cana-2204	67	106	1.1	1.1	NUM
cana-2204	67	107	)	)	PUNCT
cana-2204	67	108	,	,	PUNCT
cana-2204	67	109	(	(	PUNCT
cana-2204	67	110	2.1	2.1	NUM
cana-2204	67	111	)	)	PUNCT
cana-2204	67	112	,	,	PUNCT
cana-2204	67	113	(	(	PUNCT
cana-2204	67	114	2.2	2.2	NUM
cana-2204	67	115	)	)	PUNCT
cana-2204	67	116	and	and	CCONJ
cana-2204	67	117	(	(	PUNCT
cana-2204	67	118	2.3	2.3	NUM
cana-2204	67	119	)	)	PUNCT
cana-2204	67	120	in	in	ADP
cana-2204	67	121	(	(	PUNCT
cana-2204	67	122	3.4	3.4	NUM
cana-2204	67	123	)	)	PUNCT
cana-2204	67	124	,	,	PUNCT
cana-2204	67	125	we	we	PRON
cana-2204	67	126	have	have	VERB
cana-2204	67	127	𝑔(𝑇	𝑔(𝑇	NOUN
cana-2204	67	128	′(𝑋	′(𝑋	NOUN
cana-2204	67	129	,	,	PUNCT
cana-2204	67	130	𝑌	𝑌	PROPN
cana-2204	67	131	)	)	PUNCT
cana-2204	67	132	,	,	PUNCT
cana-2204	67	133	𝜉	𝜉	X
cana-2204	67	134	)	)	PUNCT
cana-2204	67	135	=	=	SYM
cana-2204	67	136	𝛼𝜂(𝑋)𝑔(𝑌	𝛼𝜂(𝑋)𝑔(𝑌	NOUN
cana-2204	67	137	,	,	PUNCT
cana-2204	67	138	𝜉	𝜉	NOUN
cana-2204	67	139	)	)	PUNCT
cana-2204	67	140	−	−	ADP
cana-2204	67	141	𝜖𝛼𝑔(𝑋	𝜖𝛼𝑔(𝑋	PROPN
cana-2204	67	142	,	,	PUNCT
cana-2204	67	143	𝑌)𝑔(𝜉	𝑌)𝑔(𝜉	PROPN
cana-2204	67	144	,	,	PUNCT
cana-2204	67	145	𝜉	𝜉	X
cana-2204	67	146	)	)	PUNCT
cana-2204	68	1	+	+	CCONJ
cana-2204	68	2	𝛽𝜂(𝑋)𝑔(𝜙𝑌	𝛽𝜂(𝑋)𝑔(𝜙𝑌	PROPN
cana-2204	68	3	,	,	PUNCT
cana-2204	68	4	𝜉	𝜉	NOUN
cana-2204	68	5	)	)	PUNCT
cana-2204	68	6	−	−	NOUN
cana-2204	68	7	𝜖𝛽𝑔(𝜙𝑋	𝜖𝛽𝑔(𝜙𝑋	NOUN
cana-2204	68	8	,	,	PUNCT
cana-2204	68	9	𝑌)𝑔(𝜉	𝑌)𝑔(𝜉	PROPN
cana-2204	68	10	,	,	PUNCT
cana-2204	68	11	𝜉	𝜉	NOUN
cana-2204	68	12	)	)	PUNCT
cana-2204	68	13	𝑇	𝑇	NOUN
cana-2204	68	14	′(𝑋	′(𝑋	NOUN
cana-2204	68	15	,	,	PUNCT
cana-2204	68	16	𝑌	𝑌	PROPN
cana-2204	68	17	)	)	PUNCT
cana-2204	68	18	=	=	SYM
cana-2204	69	1	𝛼𝜂(𝑋)𝑌	𝛼𝜂(𝑋)𝑌	PROPN
cana-2204	69	2	−	−	PROPN
cana-2204	69	3	𝜖𝛼𝑔(𝑋	𝜖𝛼𝑔(𝑋	PROPN
cana-2204	69	4	,	,	PUNCT
cana-2204	69	5	𝑌)𝜉	𝑌)𝜉	PUNCT
cana-2204	70	1	+	+	CCONJ
cana-2204	70	2	𝛽𝜂(𝑋)𝜙𝑌	𝛽𝜂(𝑋)𝜙𝑌	PROPN
cana-2204	70	3	−	−	NOUN
cana-2204	70	4	𝜖𝛽𝑔(𝜙𝑋	𝜖𝛽𝑔(𝜙𝑋	NOUN
cana-2204	70	5	,	,	PUNCT
cana-2204	70	6	𝑌)𝜉	𝑌)𝜉	X
cana-2204	70	7	(	(	PUNCT
cana-2204	70	8	3.5	3.5	X
cana-2204	70	9	)	)	PUNCT
cana-2204	70	10	𝑇	𝑇	PROPN
cana-2204	70	11	′(𝑌	′(𝑌	NOUN
cana-2204	70	12	,	,	PUNCT
cana-2204	70	13	𝑋	𝑋	NOUN
cana-2204	70	14	)	)	PUNCT
cana-2204	70	15	=	=	SYM
cana-2204	70	16	𝛼𝜂(𝑌)𝑋	𝛼𝜂(𝑌)𝑋	ADJ
cana-2204	71	1	−	−	PROPN
cana-2204	71	2	𝜖𝛼𝑔(𝑋	𝜖𝛼𝑔(𝑋	PROPN
cana-2204	71	3	,	,	PUNCT
cana-2204	71	4	𝑌)𝜉	𝑌)𝜉	PUNCT
cana-2204	72	1	+	+	CCONJ
cana-2204	72	2	𝛽𝜂(𝑌)𝜙𝑋	𝛽𝜂(𝑌)𝜙𝑋	NOUN
cana-2204	72	3	−	−	NOUN
cana-2204	72	4	𝜖𝛽𝑔(𝑋	𝜖𝛽𝑔(𝑋	NUM
cana-2204	72	5	,	,	PUNCT
cana-2204	72	6	𝜙𝑌)𝜉	𝜙𝑌)𝜉	PROPN
cana-2204	72	7	(	(	PUNCT
cana-2204	72	8	3.6	3.6	NUM
cana-2204	72	9	)	)	PUNCT
cana-2204	72	10	using	use	VERB
cana-2204	72	11	equations	equation	NOUN
cana-2204	72	12	(	(	PUNCT
cana-2204	72	13	1.1	1.1	NUM
cana-2204	72	14	)	)	PUNCT
cana-2204	72	15	,	,	PUNCT
cana-2204	72	16	(	(	PUNCT
cana-2204	72	17	2.2	2.2	NUM
cana-2204	72	18	)	)	PUNCT
cana-2204	72	19	,	,	PUNCT
cana-2204	72	20	(	(	PUNCT
cana-2204	72	21	3.5	3.5	NUM
cana-2204	72	22	)	)	PUNCT
cana-2204	72	23	and	and	CCONJ
cana-2204	72	24	(	(	PUNCT
cana-2204	72	25	3.6	3.6	NUM
cana-2204	72	26	)	)	PUNCT
cana-2204	72	27	in	in	ADP
cana-2204	72	28	(	(	PUNCT
cana-2204	72	29	3.3	3.3	NUM
cana-2204	72	30	)	)	PUNCT
cana-2204	72	31	,	,	PUNCT
cana-2204	72	32	we	we	PRON
cana-2204	72	33	obtained	obtain	VERB
cana-2204	72	34	𝐻(𝑋	𝐻(𝑋	NOUN
cana-2204	72	35	,	,	PUNCT
cana-2204	72	36	𝑌	𝑌	PROPN
cana-2204	72	37	)	)	PUNCT
cana-2204	72	38	=	=	SYM
cana-2204	72	39	𝛼{𝜂(𝑌)𝑋	𝛼{𝜂(𝑌)𝑋	NOUN
cana-2204	72	40	−	−	NOUN
cana-2204	72	41	𝜖𝑔(𝑋	𝜖𝑔(𝑋	PROPN
cana-2204	72	42	,	,	PUNCT
cana-2204	72	43	𝑌)𝜉	𝑌)𝜉	CCONJ
cana-2204	72	44	}	}	PUNCT
cana-2204	72	45	+	+	CCONJ
cana-2204	72	46	𝛽{𝜂(𝑌)𝜙𝑋	𝛽{𝜂(𝑌)𝜙𝑋	NOUN
cana-2204	72	47	−	−	NOUN
cana-2204	72	48	𝜖𝑔(𝜙𝑋	𝜖𝑔(𝜙𝑋	PROPN
cana-2204	72	49	,	,	PUNCT
cana-2204	72	50	𝑌)𝜉	𝑌)𝜉	CCONJ
cana-2204	72	51	}	}	PUNCT
cana-2204	72	52	(	(	PUNCT
cana-2204	72	53	3.7	3.7	NUM
cana-2204	72	54	)	)	PUNCT
cana-2204	72	55	using	use	VERB
cana-2204	72	56	above	above	ADV
cana-2204	72	57	in	in	ADP
cana-2204	72	58	(	(	PUNCT
cana-2204	72	59	3.2	3.2	NUM
cana-2204	72	60	)	)	PUNCT
cana-2204	72	61	proves	prove	VERB
cana-2204	72	62	to	to	ADP
cana-2204	72	63	our	our	PRON
cana-2204	72	64	assertion	assertion	NOUN
cana-2204	72	65	.	.	PUNCT
cana-2204	73	1	now	now	ADV
cana-2204	73	2	,	,	PUNCT
cana-2204	73	3	substituting	substitute	VERB
cana-2204	73	4	𝑌	𝑌	PROPN
cana-2204	73	5	=	=	SYM
cana-2204	73	6	𝜉	𝜉	NOUN
cana-2204	73	7	in	in	ADP
cana-2204	73	8	equation	equation	NOUN
cana-2204	73	9	(	(	PUNCT
cana-2204	73	10	3.1	3.1	NUM
cana-2204	73	11	)	)	PUNCT
cana-2204	73	12	,	,	PUNCT
cana-2204	73	13	we	we	PRON
cana-2204	73	14	obtained	obtain	VERB
cana-2204	73	15	∇̅𝑋𝜉	∇̅𝑋𝜉	PROPN
cana-2204	73	16	=	=	PUNCT
cana-2204	73	17	∇𝑋𝜉	∇𝑋𝜉	NOUN
cana-2204	73	18	−	−	NOUN
cana-2204	73	19	𝛼{𝑋	𝛼{𝑋	PROPN
cana-2204	73	20	+	+	PUNCT
cana-2204	74	1	𝜂(𝑋)𝜉	𝜂(𝑋)𝜉	NUM
cana-2204	74	2	}	}	PUNCT
cana-2204	74	3	−	−	PUNCT
cana-2204	74	4	𝛽(𝜙𝑋	𝛽(𝜙𝑋	NOUN
cana-2204	74	5	)	)	PUNCT
cana-2204	74	6	if	if	SCONJ
cana-2204	74	7	the	the	DET
cana-2204	74	8	vector	vector	NOUN
cana-2204	74	9	field	field	NOUN
cana-2204	74	10	𝜉	𝜉	AUX
cana-2204	74	11	representing	represent	VERB
cana-2204	74	12	a	a	DET
cana-2204	74	13	unit	unit	NOUN
cana-2204	74	14	of	of	ADP
cana-2204	74	15	time	time	NOUN
cana-2204	74	16	like	like	INTJ
cana-2204	74	17	is	be	AUX
cana-2204	74	18	aligned	align	VERB
cana-2204	74	19	in	in	ADP
cana-2204	74	20	parallel	parallel	NOUN
cana-2204	74	21	according	accord	VERB
cana-2204	74	22	to	to	ADP
cana-2204	74	23	a	a	DET
cana-2204	74	24	generalized	generalize	VERB
cana-2204	74	25	symmetric	symmetric	ADJ
cana-2204	74	26	metric	metric	ADJ
cana-2204	74	27	connection	connection	NOUN
cana-2204	74	28	,	,	PUNCT
cana-2204	74	29	that	that	PRON
cana-2204	74	30	is	be	AUX
cana-2204	74	31	∇̅𝑋𝜉	∇̅𝑋𝜉	NOUN
cana-2204	74	32	=	=	SYM
cana-2204	74	33	0	0	NUM
cana-2204	74	34	,	,	PUNCT
cana-2204	74	35	we	we	PRON
cana-2204	74	36	have	have	VERB
cana-2204	74	37	∇𝑋𝜉	∇𝑋𝜉	NOUN
cana-2204	74	38	=	=	SYM
cana-2204	74	39	𝛼{𝑋	𝛼{𝑋	PROPN
cana-2204	74	40	+	+	CCONJ
cana-2204	74	41	𝜂(𝑋)𝜉	𝜂(𝑋)𝜉	X
cana-2204	74	42	}	}	PUNCT
cana-2204	74	43	+	+	CCONJ
cana-2204	74	44	𝛽(𝜙𝑋	𝛽(𝜙𝑋	NOUN
cana-2204	74	45	)	)	PUNCT
cana-2204	74	46	(	(	PUNCT
cana-2204	74	47	3.8	3.8	NUM
cana-2204	74	48	)	)	PUNCT
cana-2204	74	49	then	then	ADV
cana-2204	74	50	∇̅	∇̅	NOUN
cana-2204	74	51	is	be	AUX
cana-2204	74	52	called	call	VERB
cana-2204	74	53	generalized	generalized	ADJ
cana-2204	74	54	symmetric	symmetric	ADJ
cana-2204	74	55	metric	metric	NOUN
cana-2204	74	56	𝜉	𝜉	NOUN
cana-2204	74	57	connection	connection	NOUN
cana-2204	74	58	.	.	PUNCT
cana-2204	75	1	using	use	VERB
cana-2204	75	2	equation	equation	NOUN
cana-2204	75	3	(	(	PUNCT
cana-2204	75	4	2.2	2.2	NUM
cana-2204	75	5	)	)	PUNCT
cana-2204	75	6	and	and	CCONJ
cana-2204	75	7	(	(	PUNCT
cana-2204	75	8	2.3	2.3	NUM
cana-2204	75	9	)	)	PUNCT
cana-2204	75	10	and	and	CCONJ
cana-2204	75	11	replacing	replace	VERB
cana-2204	75	12	𝑌	𝑌	PROPN
cana-2204	75	13	by	by	ADP
cana-2204	75	14	𝜙𝑌	𝜙𝑌	ADJ
cana-2204	75	15	in	in	ADP
cana-2204	75	16	equation	equation	NOUN
cana-2204	75	17	(	(	PUNCT
cana-2204	75	18	3.1	3.1	NUM
cana-2204	75	19	)	)	PUNCT
cana-2204	75	20	,	,	PUNCT
cana-2204	75	21	we	we	PRON
cana-2204	75	22	have	have	VERB
cana-2204	75	23	∇̅𝑋𝜙𝑌	∇̅𝑋𝜙𝑌	PROPN
cana-2204	75	24	=	=	X
cana-2204	75	25	∇𝑋𝜙𝑌	∇𝑋𝜙𝑌	ADJ
cana-2204	75	26	−	−	PROPN
cana-2204	75	27	𝜖𝛼𝑔(𝑋	𝜖𝛼𝑔(𝑋	PROPN
cana-2204	75	28	,	,	PUNCT
cana-2204	75	29	𝜙𝑌)𝜉	𝜙𝑌)𝜉	PROPN
cana-2204	75	30	−	−	PROPN
cana-2204	75	31	𝜖𝛽{𝑔(𝑋	𝜖𝛽{𝑔(𝑋	NUM
cana-2204	75	32	,	,	PUNCT
cana-2204	75	33	𝑌	𝑌	PROPN
cana-2204	75	34	)	)	PUNCT
cana-2204	75	35	+	+	NUM
cana-2204	75	36	𝜖	𝜖	X
cana-2204	75	37	𝜂(𝑋	𝜂(𝑋	ADV
cana-2204	75	38	)	)	PUNCT
cana-2204	75	39	𝜂(𝑌)}𝜉	𝜂(𝑌)}𝜉	VERB
cana-2204	75	40	using	use	VERB
cana-2204	75	41	covariant	covariant	ADJ
cana-2204	75	42	differentiation	differentiation	NOUN
cana-2204	75	43	in	in	ADP
cana-2204	75	44	above	above	ADV
cana-2204	75	45	,	,	PUNCT
cana-2204	75	46	we	we	PRON
cana-2204	75	47	have	have	VERB
cana-2204	75	48	(	(	PUNCT
cana-2204	75	49	∇̅𝑋𝜙)𝑌	∇̅𝑋𝜙)𝑌	PROPN
cana-2204	75	50	=	=	PUNCT
cana-2204	75	51	(	(	PUNCT
cana-2204	75	52	∇𝑋𝜙)𝑌	∇𝑋𝜙)𝑌	PROPN
cana-2204	75	53	−	−	NOUN
cana-2204	75	54	𝛼{𝜂(𝑌)𝜙𝑋	𝛼{𝜂(𝑌)𝜙𝑋	NOUN
cana-2204	75	55	+	+	CCONJ
cana-2204	75	56	𝜖𝑔(𝑋	𝜖𝑔(𝑋	PROPN
cana-2204	75	57	,	,	PUNCT
cana-2204	75	58	𝜙𝑌)𝜉	𝜙𝑌)𝜉	PROPN
cana-2204	75	59	}	}	PUNCT
cana-2204	75	60	−	−	ADP
cana-2204	75	61	𝛽{𝜂(𝑌)𝑋	𝛽{𝜂(𝑌)𝑋	PROPN
cana-2204	75	62	+	+	CCONJ
cana-2204	75	63	𝜖𝑔(𝑋	𝜖𝑔(𝑋	PROPN
cana-2204	75	64	,	,	PUNCT
cana-2204	75	65	𝑌)𝜉	𝑌)𝜉	PUNCT
cana-2204	76	1	+	+	CCONJ
cana-2204	76	2	𝜂(𝑋)𝜂(𝑌)𝜉	𝜂(𝑋)𝜂(𝑌)𝜉	NOUN
cana-2204	76	3	}	}	PUNCT
cana-2204	76	4	(	(	PUNCT
cana-2204	76	5	3.9	3.9	NUM
cana-2204	76	6	)	)	PUNCT
cana-2204	76	7	now	now	ADV
cana-2204	76	8	,	,	PUNCT
cana-2204	76	9	as	as	SCONJ
cana-2204	76	10	we	we	PRON
cana-2204	76	11	know	know	VERB
cana-2204	76	12	that	that	SCONJ
cana-2204	76	13	(	(	PUNCT
cana-2204	76	14	∇̅𝑋𝜂)𝑌	∇̅𝑋𝜂)𝑌	X
cana-2204	76	15	=	=	PUNCT
cana-2204	76	16	∇̅𝑋𝜂𝑌	∇̅𝑋𝜂𝑌	PROPN
cana-2204	76	17	+	+	CCONJ
cana-2204	76	18	𝜂	𝜂	PROPN
cana-2204	76	19	(	(	PUNCT
cana-2204	76	20	∇̅𝑋𝑌	∇̅𝑋𝑌	ADV
cana-2204	76	21	)	)	PUNCT
cana-2204	76	22	using	use	VERB
cana-2204	76	23	equation	equation	NOUN
cana-2204	76	24	(	(	PUNCT
cana-2204	76	25	3.1	3.1	NUM
cana-2204	76	26	)	)	PUNCT
cana-2204	76	27	in	in	ADP
cana-2204	76	28	above	above	ADV
cana-2204	76	29	,	,	PUNCT
cana-2204	76	30	we	we	PRON
cana-2204	76	31	obtained	obtain	VERB
cana-2204	76	32	(	(	PUNCT
cana-2204	76	33	∇̅𝑋𝜂)𝑌	∇̅𝑋𝜂)𝑌	X
cana-2204	76	34	=	=	SYM
cana-2204	76	35	(	(	PUNCT
cana-2204	76	36	∇𝑋𝜂)𝑌	∇𝑋𝜂)𝑌	PROPN
cana-2204	76	37	−	−	PROPN
cana-2204	76	38	𝛼{𝜖𝑔(𝑋	𝛼{𝜖𝑔(𝑋	NOUN
cana-2204	76	39	,	,	PUNCT
cana-2204	76	40	𝑌	𝑌	PROPN
cana-2204	76	41	)	)	PUNCT
cana-2204	76	42	+	+	SYM
cana-2204	77	1	𝜂(𝑋)𝜂(𝑌	𝜂(𝑋)𝜂(𝑌	NOUN
cana-2204	77	2	)	)	PUNCT
cana-2204	77	3	}	}	PUNCT
cana-2204	77	4	−	−	ADP
cana-2204	77	5	𝜖𝛽𝑔(𝜙𝑋	𝜖𝛽𝑔(𝜙𝑋	NOUN
cana-2204	77	6	,	,	PUNCT
cana-2204	77	7	𝑌	𝑌	PROPN
cana-2204	77	8	)	)	PUNCT
cana-2204	77	9	(	(	PUNCT
cana-2204	77	10	3.10	3.10	NUM
cana-2204	77	11	)	)	PUNCT
cana-2204	77	12	for	for	ADP
cana-2204	77	13	any	any	DET
cana-2204	77	14	vector	vector	NOUN
cana-2204	77	15	field	field	NOUN
cana-2204	77	16	𝑋	𝑋	NOUN
cana-2204	77	17	and	and	CCONJ
cana-2204	77	18	𝑌	𝑌	PROPN
cana-2204	77	19	on	on	ADP
cana-2204	77	20	𝑀.	𝑀.	PROPN
cana-2204	77	21	now	now	ADV
cana-2204	77	22	,	,	PUNCT
cana-2204	77	23	we	we	PRON
cana-2204	77	24	suppose	suppose	VERB
cana-2204	77	25	that	that	SCONJ
cana-2204	77	26	communications	communication	NOUN
cana-2204	77	27	on	on	ADP
cana-2204	77	28	applied	apply	VERB
cana-2204	77	29	nonlinear	nonlinear	ADJ
cana-2204	77	30	analysis	analysis	NOUN
cana-2204	77	31	issn	issn	NOUN
cana-2204	77	32	:	:	PUNCT
cana-2204	77	33	1074	1074	NUM
cana-2204	77	34	-	-	PUNCT
cana-2204	77	35	133x	133x	NUM
cana-2204	77	36	vol	vol	NOUN
cana-2204	77	37	32	32	NUM
cana-2204	77	38	no	no	NOUN
cana-2204	77	39	.	.	PUNCT
cana-2204	78	1	1s	1s	NUM
cana-2204	78	2	(	(	PUNCT
cana-2204	78	3	2025	2025	NUM
cana-2204	78	4	)	)	PUNCT
cana-2204	78	5	403	403	NUM
cana-2204	78	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2204	78	7	(	(	PUNCT
cana-2204	78	8	∇̅𝑋𝜙)𝑌	∇̅𝑋𝜙)𝑌	PROPN
cana-2204	78	9	=	=	SYM
cana-2204	78	10	0	0	NUM
cana-2204	78	11	and	and	CCONJ
cana-2204	78	12	(	(	PUNCT
cana-2204	78	13	∇̅𝑋𝜂)𝑌	∇̅𝑋𝜂)𝑌	X
cana-2204	78	14	=	=	SYM
cana-2204	78	15	0	0	X
cana-2204	78	16	.	.	PUNCT
cana-2204	79	1	then	then	ADV
cana-2204	79	2	the	the	DET
cana-2204	79	3	equation	equation	NOUN
cana-2204	79	4	(	(	PUNCT
cana-2204	79	5	3.9	3.9	NUM
cana-2204	79	6	)	)	PUNCT
cana-2204	79	7	and	and	CCONJ
cana-2204	79	8	(	(	PUNCT
cana-2204	79	9	3.10	3.10	NUM
cana-2204	79	10	)	)	PUNCT
cana-2204	79	11	will	will	AUX
cana-2204	79	12	be	be	AUX
cana-2204	79	13	as	as	SCONJ
cana-2204	79	14	follows	follow	VERB
cana-2204	79	15	(	(	PUNCT
cana-2204	79	16	∇𝑋𝜙)𝑌	∇𝑋𝜙)𝑌	PROPN
cana-2204	79	17	=	=	SYM
cana-2204	79	18	𝛼{𝜂(𝑌)𝜙𝑋	𝛼{𝜂(𝑌)𝜙𝑋	PROPN
cana-2204	79	19	+	+	CCONJ
cana-2204	79	20	𝜖𝑔(𝑋	𝜖𝑔(𝑋	PROPN
cana-2204	79	21	,	,	PUNCT
cana-2204	79	22	𝜙𝑌)𝜉	𝜙𝑌)𝜉	PROPN
cana-2204	79	23	}	}	PUNCT
cana-2204	79	24	+	+	CCONJ
cana-2204	79	25	𝛽{𝜂(𝑌	𝛽{𝜂(𝑌	X
cana-2204	79	26	)	)	PUNCT
cana-2204	79	27	+	+	CCONJ
cana-2204	79	28	𝜖𝑔(𝑋	𝜖𝑔(𝑋	PROPN
cana-2204	79	29	,	,	PUNCT
cana-2204	79	30	𝑌)𝜉	𝑌)𝜉	PUNCT
cana-2204	80	1	+	+	CCONJ
cana-2204	81	1	2𝜂(𝑋	2𝜂(𝑋	NUM
cana-2204	81	2	)	)	PUNCT
cana-2204	81	3	𝜂(𝑌)𝜉	𝜂(𝑌)𝜉	NUM
cana-2204	81	4	}	}	PUNCT
cana-2204	81	5	(	(	PUNCT
cana-2204	81	6	3.11	3.11	NUM
cana-2204	81	7	)	)	PUNCT
cana-2204	81	8	and	and	CCONJ
cana-2204	81	9	(	(	PUNCT
cana-2204	81	10	∇𝑋𝜂)𝑌	∇𝑋𝜂)𝑌	PROPN
cana-2204	81	11	=	=	SYM
cana-2204	81	12	𝛼{𝜖𝑔(𝑋	𝛼{𝜖𝑔(𝑋	PROPN
cana-2204	81	13	,	,	PUNCT
cana-2204	81	14	𝑌	𝑌	PROPN
cana-2204	81	15	)	)	PUNCT
cana-2204	82	1	+	+	SYM
cana-2204	82	2	𝜂(𝑋)𝜂(𝑌	𝜂(𝑋)𝜂(𝑌	NOUN
cana-2204	82	3	)	)	PUNCT
cana-2204	82	4	}	}	PUNCT
cana-2204	83	1	+	+	CCONJ
cana-2204	83	2	𝜖𝛽𝑔(𝜙𝑋	𝜖𝛽𝑔(𝜙𝑋	NOUN
cana-2204	83	3	,	,	PUNCT
cana-2204	83	4	𝑌	𝑌	PROPN
cana-2204	83	5	)	)	PUNCT
cana-2204	83	6	(	(	PUNCT
cana-2204	83	7	3.12	3.12	NUM
cana-2204	83	8	)	)	PUNCT
cana-2204	83	9	a	a	DET
cana-2204	83	10	linear	linear	ADJ
cana-2204	83	11	connection	connection	NOUN
cana-2204	83	12	∇̅	∇̅	ADJ
cana-2204	83	13	satisfying	satisfy	VERB
cana-2204	83	14	equation	equation	NOUN
cana-2204	83	15	(	(	PUNCT
cana-2204	83	16	3.11	3.11	NUM
cana-2204	83	17	)	)	PUNCT
cana-2204	83	18	and	and	CCONJ
cana-2204	83	19	(	(	PUNCT
cana-2204	83	20	3.12	3.12	NUM
cana-2204	83	21	)	)	PUNCT
cana-2204	83	22	is	be	AUX
cana-2204	83	23	called	call	VERB
cana-2204	83	24	𝜙	𝜙	DET
cana-2204	83	25	−parallel	−parallel	ADV
cana-2204	83	26	generalized	generalize	VERB
cana-2204	83	27	symmetric	symmetric	ADJ
cana-2204	83	28	connection	connection	NOUN
cana-2204	83	29	and	and	CCONJ
cana-2204	83	30	𝜂	𝜂	DET
cana-2204	83	31	−parallel	−parallel	ADV
cana-2204	83	32	generalized	generalize	VERB
cana-2204	83	33	symmetric	symmetric	ADJ
cana-2204	83	34	connection	connection	NOUN
cana-2204	83	35	respectively	respectively	ADV
cana-2204	83	36	.	.	PUNCT
cana-2204	84	1	definition	definition	NOUN
cana-2204	84	2	3.2.let	3.2.let	PROPN
cana-2204	84	3	𝑀	𝑀	PROPN
cana-2204	84	4	be	be	VERB
cana-2204	84	5	a	a	DET
cana-2204	84	6	(	(	PUNCT
cana-2204	84	7	𝜖	𝜖	NOUN
cana-2204	84	8	)	)	PUNCT
cana-2204	84	9	−lorentzian	−lorentzian	ADJ
cana-2204	84	10	para	para	NOUN
cana-2204	84	11	-	-	PUNCT
cana-2204	84	12	sasakianmanifold	sasakianmanifold	ADJ
cana-2204	84	13	.	.	PUNCT
cana-2204	85	1	if	if	SCONJ
cana-2204	85	2	𝑀	𝑀	PROPN
cana-2204	85	3	satisfies	satisfy	VERB
cana-2204	85	4	the	the	DET
cana-2204	85	5	equations	equation	NOUN
cana-2204	85	6	(	(	PUNCT
cana-2204	85	7	3.8	3.8	NUM
cana-2204	85	8	)	)	PUNCT
cana-2204	85	9	,	,	PUNCT
cana-2204	85	10	(	(	PUNCT
cana-2204	85	11	3.11	3.11	NUM
cana-2204	85	12	)	)	PUNCT
cana-2204	85	13	and	and	CCONJ
cana-2204	85	14	(	(	PUNCT
cana-2204	85	15	3.12	3.12	NUM
cana-2204	85	16	)	)	PUNCT
cana-2204	85	17	,	,	PUNCT
cana-2204	85	18	then	then	ADV
cana-2204	85	19	𝑀	𝑀	PROPN
cana-2204	85	20	is	be	AUX
cana-2204	85	21	called	call	VERB
cana-2204	85	22	(	(	PUNCT
cana-2204	85	23	𝜖	𝜖	NOUN
cana-2204	85	24	)	)	PUNCT
cana-2204	85	25	−lorentzian	−lorentzian	ADJ
cana-2204	85	26	para	para	NOUN
cana-2204	85	27	-	-	PUNCT
cana-2204	85	28	sasakian	sasakian	NOUN
cana-2204	85	29	manifold	manifold	NOUN
cana-2204	85	30	with	with	ADP
cana-2204	85	31	parallelized	parallelize	VERB
cana-2204	85	32	generalized	generalize	VERB
cana-2204	85	33	symmetric	symmetric	ADJ
cana-2204	85	34	metric	metric	ADJ
cana-2204	85	35	connection	connection	NOUN
cana-2204	85	36	,	,	PUNCT
cana-2204	85	37	this	this	PRON
cana-2204	85	38	means	mean	VERB
cana-2204	85	39	that	that	SCONJ
cana-2204	85	40	the	the	DET
cana-2204	85	41	connection	connection	NOUN
cana-2204	85	42	∇̅	∇̅	NOUN
cana-2204	85	43	is	be	AUX
cana-2204	85	44	generalized	generalize	VERB
cana-2204	85	45	symmetric	symmetric	ADJ
cana-2204	85	46	metric	metric	ADJ
cana-2204	85	47	𝜉	𝜉	PROPN
cana-2204	85	48	connection	connection	NOUN
cana-2204	85	49	,	,	PUNCT
cana-2204	85	50	𝜙	𝜙	NOUN
cana-2204	85	51	−parallel	−parallel	ADV
cana-2204	85	52	generalized	generalize	VERB
cana-2204	85	53	symmetric	symmetric	ADJ
cana-2204	85	54	connection	connection	NOUN
cana-2204	85	55	and	and	CCONJ
cana-2204	85	56	𝜂	𝜂	DET
cana-2204	85	57	−parallel	−parallel	ADV
cana-2204	85	58	generalized	generalize	VERB
cana-2204	85	59	symmetric	symmetric	ADJ
cana-2204	85	60	connection	connection	NOUN
cana-2204	85	61	proposition	proposition	NOUN
cana-2204	85	62	3.3	3.3	NUM
cana-2204	85	63	.	.	PUNCT
cana-2204	86	1	let	let	VERB
cana-2204	86	2	𝑀	𝑀	PRON
cana-2204	86	3	be	be	AUX
cana-2204	86	4	a	a	DET
cana-2204	86	5	(	(	PUNCT
cana-2204	86	6	𝜖	𝜖	NOUN
cana-2204	86	7	)	)	PUNCT
cana-2204	86	8	−lorentzian	−lorentzian	ADJ
cana-2204	86	9	para	para	NOUN
cana-2204	86	10	-	-	PUNCT
cana-2204	86	11	sasakian	sasakian	NOUN
cana-2204	86	12	manifold	manifold	NOUN
cana-2204	86	13	with	with	ADP
cana-2204	86	14	parallelized	parallelize	VERB
cana-2204	86	15	generalized	generalize	VERB
cana-2204	86	16	symmetric	symmetric	ADJ
cana-2204	86	17	metric	metric	ADJ
cana-2204	86	18	connection	connection	NOUN
cana-2204	86	19	then	then	ADV
cana-2204	86	20	following	follow	VERB
cana-2204	86	21	relation	relation	NOUN
cana-2204	86	22	holds	hold	VERB
cana-2204	86	23	(	(	PUNCT
cana-2204	86	24	∇̅𝑋𝜙)𝑌	∇̅𝑋𝜙)𝑌	PROPN
cana-2204	86	25	=	=	SYM
cana-2204	86	26	(	(	PUNCT
cana-2204	86	27	𝜖	𝜖	PROPN
cana-2204	86	28	−	−	PROPN
cana-2204	86	29	𝛽	𝛽	NOUN
cana-2204	86	30	)	)	PUNCT
cana-2204	86	31	𝜂(𝑌)𝑋	𝜂(𝑌)𝑋	NOUN
cana-2204	86	32	+	+	CCONJ
cana-2204	86	33	2(𝜖	2(𝜖	NUM
cana-2204	86	34	+	+	CCONJ
cana-2204	86	35	1)𝜂(𝑋	1)𝜂(𝑋	NUM
cana-2204	86	36	)	)	PUNCT
cana-2204	86	37	𝜂(𝑌)𝜉	𝜂(𝑌)𝜉	VERB
cana-2204	87	1	+	+	CCONJ
cana-2204	87	2	(	(	PUNCT
cana-2204	87	3	1−	1−	NUM
cana-2204	87	4	ϵ𝛽)𝑔(𝑋	ϵ𝛽)𝑔(𝑋	PROPN
cana-2204	87	5	,	,	PUNCT
cana-2204	87	6	𝑌)𝜉	𝑌)𝜉	PUNCT
cana-2204	88	1	−	−	PROPN
cana-2204	88	2	𝛼𝜂(𝑌)𝜙𝑋	𝛼𝜂(𝑌)𝜙𝑋	NOUN
cana-2204	88	3	−	−	PROPN
cana-2204	88	4	ϵα𝑔(𝑋	ϵα𝑔(𝑋	NUM
cana-2204	88	5	,	,	PUNCT
cana-2204	88	6	𝜙𝑌)𝜉	𝜙𝑌)𝜉	PROPN
cana-2204	88	7	(	(	PUNCT
cana-2204	88	8	3.13	3.13	NUM
cana-2204	88	9	)	)	PUNCT
cana-2204	88	10	for	for	ADP
cana-2204	88	11	all	all	DET
cana-2204	88	12	vector	vector	NOUN
cana-2204	88	13	field	field	NOUN
cana-2204	88	14	𝑋	𝑋	NOUN
cana-2204	88	15	and	and	CCONJ
cana-2204	88	16	𝑌	𝑌	PROPN
cana-2204	88	17	on	on	ADP
cana-2204	88	18	𝑀.	𝑀.	NOUN
cana-2204	88	19	proposition	proposition	NOUN
cana-2204	88	20	3.4	3.4	NUM
cana-2204	88	21	.	.	PUNCT
cana-2204	89	1	in	in	ADP
cana-2204	89	2	a	a	DET
cana-2204	89	3	(	(	PUNCT
cana-2204	89	4	𝜖	𝜖	NOUN
cana-2204	89	5	)	)	PUNCT
cana-2204	89	6	−	−	PROPN
cana-2204	89	7	lorentzian	lorentzian	ADJ
cana-2204	89	8	para	para	NOUN
cana-2204	89	9	-	-	PUNCT
cana-2204	89	10	sasakian	sasakian	NOUN
cana-2204	89	11	manifold	manifold	NOUN
cana-2204	89	12	with	with	ADP
cana-2204	89	13	parallelized	parallelize	VERB
cana-2204	89	14	generalized	generalize	VERB
cana-2204	89	15	symmetric	symmetric	ADJ
cana-2204	89	16	metric	metric	ADJ
cana-2204	89	17	connection	connection	NOUN
cana-2204	89	18	,	,	PUNCT
cana-2204	89	19	curvature	curvature	NOUN
cana-2204	89	20	tensor	tensor	NOUN
cana-2204	89	21	𝑅	𝑅	PROPN
cana-2204	89	22	and	and	CCONJ
cana-2204	89	23	ricci	ricci	PROPN
cana-2204	89	24	tensor	tensor	NOUN
cana-2204	89	25	𝑆	𝑆	PROPN
cana-2204	89	26	and	and	CCONJ
cana-2204	89	27	ricci	ricci	PROPN
cana-2204	89	28	operator	operator	NOUN
cana-2204	89	29	𝑄	𝑄	PROPN
cana-2204	89	30	has	have	VERB
cana-2204	89	31	the	the	DET
cana-2204	89	32	following	follow	VERB
cana-2204	89	33	relations	relation	NOUN
cana-2204	89	34	𝑅(𝑋	𝑅(𝑋	NOUN
cana-2204	89	35	,	,	PUNCT
cana-2204	89	36	𝑌)𝜉	𝑌)𝜉	PUNCT
cana-2204	90	1	=	=	SYM
cana-2204	90	2	(	(	PUNCT
cana-2204	90	3	𝛼2	𝛼2	PROPN
cana-2204	90	4	+	+	CCONJ
cana-2204	90	5	𝛽2){𝜂(𝑌)𝑋	𝛽2){𝜂(𝑌)𝑋	VERB
cana-2204	90	6	−	−	ADJ
cana-2204	90	7	𝜂(𝑋)𝑌	𝜂(𝑋)𝑌	NOUN
cana-2204	90	8	}	}	PUNCT
cana-2204	90	9	+	+	NUM
cana-2204	90	10	2𝛼𝛽{𝜂(𝑌)𝜙𝑋	2𝛼𝛽{𝜂(𝑌)𝜙𝑋	NUM
cana-2204	90	11	−	−	NOUN
cana-2204	90	12	𝜂(𝑋)𝜙𝑌	𝜂(𝑋)𝜙𝑌	ADJ
cana-2204	90	13	}	}	PUNCT
cana-2204	90	14	+	+	CCONJ
cana-2204	90	15	(	(	PUNCT
cana-2204	90	16	𝑋𝛽)𝜙𝑌	𝑋𝛽)𝜙𝑌	PROPN
cana-2204	90	17	−	−	PROPN
cana-2204	90	18	(	(	PUNCT
cana-2204	90	19	𝑌𝛽)𝜙𝑋	𝑌𝛽)𝜙𝑋	NOUN
cana-2204	90	20	+	+	ADJ
cana-2204	90	21	(	(	PUNCT
cana-2204	90	22	𝑋𝛼)𝜙2𝑌	𝑋𝛼)𝜙2𝑌	ADJ
cana-2204	90	23	−	−	PROPN
cana-2204	90	24	(	(	PUNCT
cana-2204	90	25	𝑌𝛼)𝜙2𝑋	𝑌𝛼)𝜙2𝑋	PROPN
cana-2204	90	26	(	(	PUNCT
cana-2204	90	27	3.14	3.14	NUM
cana-2204	90	28	)	)	PUNCT
cana-2204	90	29	𝜂(𝑅(𝑋	𝜂(𝑅(𝑋	PROPN
cana-2204	90	30	,	,	PUNCT
cana-2204	90	31	𝑌)𝑍	𝑌)𝑍	X
cana-2204	90	32	=	=	PUNCT
cana-2204	90	33	𝜖(𝛼2	𝜖(𝛼2	PROPN
cana-2204	90	34	+	+	CCONJ
cana-2204	90	35	𝛽2){𝜂(𝑋)𝑔(𝑌	𝛽2){𝜂(𝑋)𝑔(𝑌	PROPN
cana-2204	90	36	,	,	PUNCT
cana-2204	90	37	𝑍	𝑍	PROPN
cana-2204	90	38	)	)	PUNCT
cana-2204	90	39	−	−	NOUN
cana-2204	90	40	𝜂(𝑌)𝑔(𝑋	𝜂(𝑌)𝑔(𝑋	NOUN
cana-2204	90	41	,	,	PUNCT
cana-2204	90	42	𝑍	𝑍	PROPN
cana-2204	90	43	)	)	PUNCT
cana-2204	90	44	}	}	PUNCT
cana-2204	90	45	(	(	PUNCT
cana-2204	90	46	3.15	3.15	NUM
cana-2204	90	47	)	)	PUNCT
cana-2204	90	48	𝑅(𝜉	𝑅(𝜉	PROPN
cana-2204	90	49	,	,	PUNCT
cana-2204	90	50	𝑋)𝑌	𝑋)𝑌	X
cana-2204	90	51	=	=	SYM
cana-2204	90	52	(	(	PUNCT
cana-2204	90	53	𝛼2	𝛼2	PROPN
cana-2204	90	54	+	+	CCONJ
cana-2204	90	55	𝛽2){𝜖𝑔(𝑋	𝛽2){𝜖𝑔(𝑋	PROPN
cana-2204	90	56	,	,	PUNCT
cana-2204	90	57	𝑌)𝜉	𝑌)𝜉	PUNCT
cana-2204	91	1	−	−	PROPN
cana-2204	92	1	𝜂(𝑌)𝑋	𝜂(𝑌)𝑋	NOUN
cana-2204	92	2	}	}	PUNCT
cana-2204	92	3	+	+	CCONJ
cana-2204	92	4	{	{	PUNCT
cana-2204	92	5	𝜉𝛽	𝜉𝛽	ADJ
cana-2204	92	6	−	−	PROPN
cana-2204	92	7	2𝛼𝛽𝜂(𝑌)}𝜙𝑋	2𝛼𝛽𝜂(𝑌)}𝜙𝑋	NUM
cana-2204	92	8	+	+	CCONJ
cana-2204	92	9	(	(	PUNCT
cana-2204	92	10	𝜉𝛼)𝑋	𝜉𝛼)𝑋	PROPN
cana-2204	92	11	+	+	PROPN
cana-2204	92	12	𝜖𝛼𝜉𝑔(𝑋	𝜖𝛼𝜉𝑔(𝑋	PROPN
cana-2204	92	13	,	,	PUNCT
cana-2204	92	14	𝑌)𝑌	𝑌)𝑌	NOUN
cana-2204	92	15	−	−	PROPN
cana-2204	92	16	(	(	PUNCT
cana-2204	92	17	𝑋𝛼)𝜉	𝑋𝛼)𝜉	NOUN
cana-2204	92	18	−	−	PROPN
cana-2204	92	19	(	(	PUNCT
cana-2204	92	20	𝑋𝛼)𝜂(𝑌)𝑌	𝑋𝛼)𝜂(𝑌)𝑌	ADJ
cana-2204	92	21	(	(	PUNCT
cana-2204	92	22	3.16	3.16	NUM
cana-2204	92	23	)	)	PUNCT
cana-2204	92	24	𝑅(𝜉	𝑅(𝜉	PROPN
cana-2204	92	25	,	,	PUNCT
cana-2204	92	26	𝑋)𝜉	𝑋)𝜉	ADV
cana-2204	92	27	=	=	SYM
cana-2204	92	28	(	(	PUNCT
cana-2204	92	29	𝛼2	𝛼2	PROPN
cana-2204	92	30	+	+	CCONJ
cana-2204	92	31	𝛽2	𝛽2	PROPN
cana-2204	92	32	+	+	CCONJ
cana-2204	92	33	𝜉𝛼)𝜙2𝑋	𝜉𝛼)𝜙2𝑋	PROPN
cana-2204	92	34	+	+	CCONJ
cana-2204	92	35	(	(	PUNCT
cana-2204	92	36	2𝛼𝛽	2𝛼𝛽	NOUN
cana-2204	92	37	+	+	CCONJ
cana-2204	92	38	𝜉𝛽)𝜙𝑋	𝜉𝛽)𝜙𝑋	PROPN
cana-2204	92	39	(	(	PUNCT
cana-2204	92	40	3.17	3.17	NUM
cana-2204	92	41	)	)	PUNCT
cana-2204	92	42	𝑆(𝑋	𝑆(𝑋	PROPN
cana-2204	92	43	,	,	PUNCT
cana-2204	92	44	𝜉	𝜉	NOUN
cana-2204	92	45	)	)	PUNCT
cana-2204	92	46	=	=	SYM
cana-2204	92	47	(	(	PUNCT
cana-2204	92	48	𝑛	𝑛	DET
cana-2204	92	49	−	−	PROPN
cana-2204	92	50	1)(𝛼2	1)(𝛼2	NUM
cana-2204	92	51	+	+	CCONJ
cana-2204	92	52	𝛽2)𝜂(𝑋	𝛽2)𝜂(𝑋	NOUN
cana-2204	92	53	)	)	PUNCT
cana-2204	92	54	(	(	PUNCT
cana-2204	92	55	3.18	3.18	NUM
cana-2204	92	56	)	)	PUNCT
cana-2204	93	1	𝑄𝜉	𝑄𝜉	PROPN
cana-2204	93	2	=	=	PUNCT
cana-2204	93	3	𝜖(𝑛	𝜖(𝑛	PROPN
cana-2204	93	4	−	−	PROPN
cana-2204	93	5	1)(𝛼2	1)(𝛼2	NUM
cana-2204	93	6	+	+	CCONJ
cana-2204	93	7	𝛽2)𝜉	𝛽2)𝜉	ADJ
cana-2204	93	8	(	(	PUNCT
cana-2204	93	9	3.19	3.19	NUM
cana-2204	93	10	)	)	PUNCT
cana-2204	93	11	for	for	ADP
cana-2204	93	12	any	any	DET
cana-2204	93	13	𝑋	𝑋	PROPN
cana-2204	93	14	,	,	PUNCT
cana-2204	93	15	𝑌	𝑌	PROPN
cana-2204	93	16	,	,	PUNCT
cana-2204	93	17	𝑍	𝑍	PROPN
cana-2204	93	18	∈	∈	PROPN
cana-2204	93	19	𝜒(𝑀	𝜒(𝑀	PROPN
cana-2204	93	20	)	)	PUNCT
cana-2204	93	21	.	.	PUNCT
cana-2204	94	1	proof	proof	NOUN
cana-2204	94	2	:	:	PUNCT
cana-2204	94	3	as	as	SCONJ
cana-2204	94	4	we	we	PRON
cana-2204	94	5	know	know	VERB
cana-2204	94	6	that	that	PRON
cana-2204	94	7	,	,	PUNCT
cana-2204	94	8	curvature	curvature	NOUN
cana-2204	94	9	tensor	tensor	NOUN
cana-2204	94	10	is	be	AUX
cana-2204	94	11	𝑅(𝑋	𝑅(𝑋	NOUN
cana-2204	94	12	,	,	PUNCT
cana-2204	94	13	𝑌)𝜉	𝑌)𝜉	PUNCT
cana-2204	95	1	=	=	PUNCT
cana-2204	95	2	∇𝑋∇𝑌𝜉	∇𝑋∇𝑌𝜉	PROPN
cana-2204	95	3	−	−	NOUN
cana-2204	95	4	∇𝑌∇𝑋𝜉	∇𝑌∇𝑋𝜉	ADJ
cana-2204	95	5	−	−	PROPN
cana-2204	95	6	∇[𝑋,𝑌]𝜉	∇[𝑋,𝑌]𝜉	NOUN
cana-2204	95	7	(	(	PUNCT
cana-2204	95	8	3.20	3.20	NUM
cana-2204	95	9	)	)	PUNCT
cana-2204	95	10	communications	communication	NOUN
cana-2204	95	11	on	on	ADP
cana-2204	95	12	applied	apply	VERB
cana-2204	95	13	nonlinear	nonlinear	ADJ
cana-2204	95	14	analysis	analysis	NOUN
cana-2204	95	15	issn	issn	NOUN
cana-2204	95	16	:	:	PUNCT
cana-2204	95	17	1074	1074	NUM
cana-2204	95	18	-	-	PUNCT
cana-2204	95	19	133x	133x	NUM
cana-2204	95	20	vol	vol	NOUN
cana-2204	95	21	32	32	NUM
cana-2204	96	1	no	no	NOUN
cana-2204	96	2	.	.	PUNCT
cana-2204	97	1	1s	1s	NUM
cana-2204	97	2	(	(	PUNCT
cana-2204	97	3	2025	2025	NUM
cana-2204	97	4	)	)	PUNCT
cana-2204	97	5	404	404	NUM
cana-2204	97	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2204	97	7	using	use	VERB
cana-2204	97	8	equation	equation	NOUN
cana-2204	97	9	(	(	PUNCT
cana-2204	97	10	3.8	3.8	NUM
cana-2204	97	11	)	)	PUNCT
cana-2204	97	12	and	and	CCONJ
cana-2204	97	13	(	(	PUNCT
cana-2204	97	14	3.12	3.12	NUM
cana-2204	97	15	)	)	PUNCT
cana-2204	97	16	,	,	PUNCT
cana-2204	97	17	we	we	PRON
cana-2204	97	18	obtained	obtain	VERB
cana-2204	97	19	∇𝑋∇𝑌𝜉	∇𝑋∇𝑌𝜉	NOUN
cana-2204	97	20	=	=	PUNCT
cana-2204	97	21	(	(	PUNCT
cana-2204	97	22	𝑋𝛼)(𝜙	𝑋𝛼)(𝜙	NOUN
cana-2204	97	23	2𝑌	2𝑌	NOUN
cana-2204	97	24	)	)	PUNCT
cana-2204	98	1	+	+	CCONJ
cana-2204	98	2	𝛼(∇𝑋𝑌	𝛼(∇𝑋𝑌	NUM
cana-2204	98	3	)	)	PUNCT
cana-2204	99	1	+	+	CCONJ
cana-2204	99	2	𝜖𝛼	𝜖𝛼	ADP
cana-2204	99	3	2𝑔(𝑋	2𝑔(𝑋	NUM
cana-2204	99	4	,	,	PUNCT
cana-2204	99	5	𝑌	𝑌	PROPN
cana-2204	99	6	+	+	CCONJ
cana-2204	99	7	𝛼2𝜂(𝑋)𝜂(𝑌)𝜉	𝛼2𝜂(𝑋)𝜂(𝑌)𝜉	PROPN
cana-2204	99	8	+	+	CCONJ
cana-2204	99	9	𝜖𝛼𝛽𝑔(𝜙𝑋	𝜖𝛼𝛽𝑔(𝜙𝑋	NOUN
cana-2204	99	10	,	,	PUNCT
cana-2204	99	11	𝑌)𝜉	𝑌)𝜉	PUNCT
cana-2204	100	1	+	+	CCONJ
cana-2204	100	2	𝛼2𝜂(𝑌)𝑋	𝛼2𝜂(𝑌)𝑋	NOUN
cana-2204	100	3	+	+	CCONJ
cana-2204	100	4	𝛼2𝜂(𝑋)𝜂(𝑌)𝜉	𝛼2𝜂(𝑋)𝜂(𝑌)𝜉	PROPN
cana-2204	100	5	+	+	NUM
cana-2204	100	6	𝛼𝛽𝜂(𝑌)𝜙𝑋	𝛼𝛽𝜂(𝑌)𝜙𝑋	X
cana-2204	100	7	+	+	CCONJ
cana-2204	100	8	(	(	PUNCT
cana-2204	100	9	𝑋𝛽)(𝜙𝑌	𝑋𝛽)(𝜙𝑌	NOUN
cana-2204	100	10	)	)	PUNCT
cana-2204	100	11	+	+	CCONJ
cana-2204	100	12	𝛽(∇𝑋𝜙𝑌	𝛽(∇𝑋𝜙𝑌	ADJ
cana-2204	100	13	)	)	PUNCT
cana-2204	100	14	(	(	PUNCT
cana-2204	100	15	3.21	3.21	NUM
cana-2204	100	16	)	)	PUNCT
cana-2204	100	17	similarly	similarly	ADV
cana-2204	100	18	,	,	PUNCT
cana-2204	100	19	∇𝑌∇𝑋𝜉	∇𝑌∇𝑋𝜉	NOUN
cana-2204	100	20	=	=	SYM
cana-2204	100	21	(	(	PUNCT
cana-2204	100	22	𝑌𝛼)(𝜙	𝑌𝛼)(𝜙	NOUN
cana-2204	100	23	2𝑋	2𝑋	PROPN
cana-2204	100	24	)	)	PUNCT
cana-2204	100	25	+	+	NUM
cana-2204	100	26	𝛼(∇𝑌𝑋	𝛼(∇𝑌𝑋	NOUN
cana-2204	100	27	)	)	PUNCT
cana-2204	101	1	+	+	CCONJ
cana-2204	101	2	𝜖𝛼	𝜖𝛼	ADP
cana-2204	101	3	2𝑔(𝑋	2𝑔(𝑋	NUM
cana-2204	101	4	,	,	PUNCT
cana-2204	101	5	𝑌)𝜉	𝑌)𝜉	PUNCT
cana-2204	102	1	+	+	CCONJ
cana-2204	102	2	𝛼2𝜂(𝑋)𝜂(𝑌)𝜉	𝛼2𝜂(𝑋)𝜂(𝑌)𝜉	PROPN
cana-2204	102	3	+	+	CCONJ
cana-2204	102	4	𝜖𝛼𝛽𝑔(𝑋	𝜖𝛼𝛽𝑔(𝑋	PROPN
cana-2204	102	5	,	,	PUNCT
cana-2204	102	6	𝜙𝑌)𝜉	𝜙𝑌)𝜉	PROPN
cana-2204	102	7	+	+	PROPN
cana-2204	102	8	𝛼2𝜂(𝑋)𝑌	𝛼2𝜂(𝑋)𝑌	ADJ
cana-2204	102	9	+	+	CCONJ
cana-2204	102	10	𝛼2𝜂(𝑋)𝜂(𝑌)𝜉	𝛼2𝜂(𝑋)𝜂(𝑌)𝜉	PROPN
cana-2204	102	11	+	+	CCONJ
cana-2204	102	12	𝛼𝛽𝜂(𝑋)𝜙𝑌	𝛼𝛽𝜂(𝑋)𝜙𝑌	NOUN
cana-2204	102	13	+	+	CCONJ
cana-2204	102	14	(	(	PUNCT
cana-2204	102	15	𝑌𝛽)(𝜙𝑋	𝑌𝛽)(𝜙𝑋	PROPN
cana-2204	102	16	)	)	PUNCT
cana-2204	102	17	+	+	CCONJ
cana-2204	102	18	𝛽(∇𝑌𝜙𝑋	𝛽(∇𝑌𝜙𝑋	ADJ
cana-2204	102	19	)	)	PUNCT
cana-2204	102	20	(	(	PUNCT
cana-2204	102	21	3.22	3.22	NUM
cana-2204	102	22	)	)	PUNCT
cana-2204	102	23	using	use	VERB
cana-2204	102	24	equation	equation	NOUN
cana-2204	102	25	(	(	PUNCT
cana-2204	102	26	3.8	3.8	NUM
cana-2204	102	27	)	)	PUNCT
cana-2204	102	28	and	and	CCONJ
cana-2204	102	29	(	(	PUNCT
cana-2204	102	30	3.11	3.11	NUM
cana-2204	102	31	)	)	PUNCT
cana-2204	102	32	,	,	PUNCT
cana-2204	102	33	we	we	PRON
cana-2204	102	34	can	can	AUX
cana-2204	102	35	write	write	VERB
cana-2204	102	36	as	as	ADP
cana-2204	102	37	𝛻[𝑋,𝑌]𝜉	𝛻[𝑋,𝑌]𝜉	PROPN
cana-2204	102	38	=	=	SYM
cana-2204	102	39	𝛼{[𝑋	𝛼{[𝑋	PROPN
cana-2204	102	40	,	,	PUNCT
cana-2204	102	41	𝑌	𝑌	PROPN
cana-2204	102	42	]	]	X
cana-2204	102	43	+	+	CCONJ
cana-2204	102	44	𝜂([𝑋	𝜂([𝑋	ADJ
cana-2204	102	45	,	,	PUNCT
cana-2204	102	46	𝑌])𝜉	𝑌])𝜉	NUM
cana-2204	102	47	}	}	PUNCT
cana-2204	102	48	+	+	NUM
cana-2204	102	49	𝛽𝜙[𝑋	𝛽𝜙[𝑋	NOUN
cana-2204	102	50	,	,	PUNCT
cana-2204	102	51	𝑌	𝑌	PROPN
cana-2204	102	52	]	]	PUNCT
cana-2204	102	53	∇[𝑋,𝑌]𝜉	∇[𝑋,𝑌]𝜉	NOUN
cana-2204	102	54	=	=	SYM
cana-2204	102	55	𝛼(∇𝑋𝑌	𝛼(∇𝑋𝑌	PROPN
cana-2204	102	56	)	)	PUNCT
cana-2204	102	57	−	−	PROPN
cana-2204	102	58	𝛼(∇𝑌𝑋	𝛼(∇𝑌𝑋	NOUN
cana-2204	102	59	)	)	PUNCT
cana-2204	103	1	+	+	CCONJ
cana-2204	103	2	𝛽∇𝑋𝜙𝑌	𝛽∇𝑋𝜙𝑌	PROPN
cana-2204	103	3	−	−	ADP
cana-2204	103	4	𝛽∇𝑌𝜙𝑋	𝛽∇𝑌𝜙𝑋	ADJ
cana-2204	103	5	−	−	PROPN
cana-2204	103	6	𝛼𝛽𝜂(𝑌)𝜙𝑋	𝛼𝛽𝜂(𝑌)𝜙𝑋	NOUN
cana-2204	103	7	+	+	CCONJ
cana-2204	103	8	𝛼𝛽𝜂(𝑋)𝜙𝑌	𝛼𝛽𝜂(𝑋)𝜙𝑌	PROPN
cana-2204	103	9	+	+	NOUN
cana-2204	103	10	𝛽2𝜂(𝑋)𝑌	𝛽2𝜂(𝑋)𝑌	ADJ
cana-2204	103	11	−	−	NOUN
cana-2204	103	12	𝛽2𝜂(𝑌)𝑋	𝛽2𝜂(𝑌)𝑋	NOUN
cana-2204	103	13	(	(	PUNCT
cana-2204	103	14	3.23	3.23	NUM
cana-2204	103	15	)	)	PUNCT
cana-2204	103	16	using	use	VERB
cana-2204	103	17	equation	equation	NOUN
cana-2204	103	18	(	(	PUNCT
cana-2204	103	19	3.20	3.20	NUM
cana-2204	103	20	)	)	PUNCT
cana-2204	103	21	,	,	PUNCT
cana-2204	103	22	(	(	PUNCT
cana-2204	103	23	3.21	3.21	NUM
cana-2204	103	24	)	)	PUNCT
cana-2204	103	25	,	,	PUNCT
cana-2204	103	26	(	(	PUNCT
cana-2204	103	27	3.22	3.22	NUM
cana-2204	103	28	)	)	PUNCT
cana-2204	103	29	and	and	CCONJ
cana-2204	103	30	(	(	PUNCT
cana-2204	103	31	3.23	3.23	NUM
cana-2204	103	32	)	)	PUNCT
cana-2204	103	33	,	,	PUNCT
cana-2204	103	34	we	we	PRON
cana-2204	103	35	obtained	obtain	VERB
cana-2204	103	36	𝑅(𝑋	𝑅(𝑋	NOUN
cana-2204	103	37	,	,	PUNCT
cana-2204	103	38	𝑌)𝜉	𝑌)𝜉	PUNCT
cana-2204	104	1	=	=	SYM
cana-2204	104	2	(	(	PUNCT
cana-2204	104	3	𝛼2	𝛼2	PROPN
cana-2204	104	4	+	+	CCONJ
cana-2204	104	5	𝛽2){𝜂(𝑌)𝑋	𝛽2){𝜂(𝑌)𝑋	VERB
cana-2204	104	6	−	−	NOUN
cana-2204	104	7	𝜂(𝑋)𝑌	𝜂(𝑋)𝑌	NOUN
cana-2204	104	8	+	+	CCONJ
cana-2204	104	9	2𝛼𝛽{𝜂(𝑌)𝜙𝑋	2𝛼𝛽{𝜂(𝑌)𝜙𝑋	NUM
cana-2204	104	10	−	−	NOUN
cana-2204	104	11	𝜂(𝑋)𝜙𝑌	𝜂(𝑋)𝜙𝑌	ADJ
cana-2204	104	12	}	}	PUNCT
cana-2204	104	13	+	+	CCONJ
cana-2204	104	14	(	(	PUNCT
cana-2204	104	15	𝑋𝛽)(𝜙𝑌	𝑋𝛽)(𝜙𝑌	NOUN
cana-2204	104	16	)	)	PUNCT
cana-2204	104	17	−(𝑌𝛽)(𝜙𝑋	−(𝑌𝛽)(𝜙𝑋	NOUN
cana-2204	104	18	)	)	PUNCT
cana-2204	105	1	+	+	CCONJ
cana-2204	105	2	(	(	PUNCT
cana-2204	105	3	𝑋𝛼)(𝜙2𝑌	𝑋𝛼)(𝜙2𝑌	ADJ
cana-2204	105	4	)	)	PUNCT
cana-2204	105	5	−	−	PROPN
cana-2204	105	6	(	(	PUNCT
cana-2204	105	7	𝑌𝛼)(𝜙2𝑋	𝑌𝛼)(𝜙2𝑋	PROPN
cana-2204	105	8	)	)	PUNCT
cana-2204	105	9	(	(	PUNCT
cana-2204	105	10	3.24	3.24	NUM
cana-2204	105	11	)	)	PUNCT
cana-2204	105	12	interchanging	interchanging	ADJ
cana-2204	105	13	𝑍	𝑍	NOUN
cana-2204	105	14	and	and	CCONJ
cana-2204	105	15	𝜉	𝜉	X
cana-2204	105	16	in	in	ADP
cana-2204	105	17	(	(	PUNCT
cana-2204	105	18	3.14	3.14	NUM
cana-2204	105	19	)	)	PUNCT
cana-2204	105	20	,	,	PUNCT
cana-2204	105	21	we	we	PRON
cana-2204	105	22	have	have	AUX
cana-2204	105	23	𝑔(𝑅(𝑋	𝑔(𝑅(𝑋	PROPN
cana-2204	105	24	,	,	PUNCT
cana-2204	105	25	𝑌)𝑍	𝑌)𝑍	PROPN
cana-2204	105	26	,	,	PUNCT
cana-2204	105	27	𝜉	𝜉	X
cana-2204	105	28	)	)	PUNCT
cana-2204	105	29	=	=	SYM
cana-2204	105	30	(	(	PUNCT
cana-2204	105	31	𝛼2	𝛼2	PROPN
cana-2204	105	32	+	+	CCONJ
cana-2204	105	33	𝛽2){𝜖𝑔(𝑌	𝛽2){𝜖𝑔(𝑌	PROPN
cana-2204	105	34	,	,	PUNCT
cana-2204	105	35	𝑍)𝑔(𝑋	𝑍)𝑔(𝑋	NUM
cana-2204	105	36	,	,	PUNCT
cana-2204	105	37	𝜉	𝜉	NOUN
cana-2204	105	38	)	)	PUNCT
cana-2204	105	39	−	−	ADP
cana-2204	105	40	𝜖𝑔(𝑋	𝜖𝑔(𝑋	PROPN
cana-2204	105	41	,	,	PUNCT
cana-2204	105	42	𝑍)𝑔(𝑌	𝑍)𝑔(𝑌	X
cana-2204	105	43	,	,	PUNCT
cana-2204	105	44	𝜉	𝜉	NOUN
cana-2204	105	45	)	)	PUNCT
cana-2204	105	46	}	}	PUNCT
cana-2204	105	47	+2𝛼𝛽{𝜖𝑔(𝑌	+2𝛼𝛽{𝜖𝑔(𝑌	PROPN
cana-2204	105	48	,	,	PUNCT
cana-2204	105	49	𝑍)𝑔(𝜙𝑋	𝑍)𝑔(𝜙𝑋	PROPN
cana-2204	105	50	,	,	PUNCT
cana-2204	105	51	𝜉	𝜉	NOUN
cana-2204	105	52	)	)	PUNCT
cana-2204	105	53	−	−	NOUN
cana-2204	105	54	𝜖𝑔(𝑋	𝜖𝑔(𝑋	PROPN
cana-2204	105	55	,	,	PUNCT
cana-2204	105	56	𝑍)𝑔(𝜙𝑌	𝑍)𝑔(𝜙𝑌	PROPN
cana-2204	105	57	,	,	PUNCT
cana-2204	105	58	𝜉	𝜉	NOUN
cana-2204	105	59	)	)	PUNCT
cana-2204	105	60	}	}	PUNCT
cana-2204	106	1	+	+	CCONJ
cana-2204	106	2	(	(	PUNCT
cana-2204	106	3	𝑋𝛽)𝑔(𝜙𝑌	𝑋𝛽)𝑔(𝜙𝑌	NUM
cana-2204	106	4	,	,	PUNCT
cana-2204	106	5	𝜉	𝜉	NOUN
cana-2204	106	6	)	)	PUNCT
cana-2204	106	7	−(𝑌𝛽)𝑔(𝜙𝑋	−(𝑌𝛽)𝑔(𝜙𝑋	PUNCT
cana-2204	106	8	,	,	PUNCT
cana-2204	106	9	𝜉	𝜉	X
cana-2204	106	10	)	)	PUNCT
cana-2204	106	11	+	+	CCONJ
cana-2204	106	12	(	(	PUNCT
cana-2204	106	13	𝑋𝛼)𝑔(𝜙2𝑌	𝑋𝛼)𝑔(𝜙2𝑌	ADJ
cana-2204	106	14	,	,	PUNCT
cana-2204	106	15	𝜉	𝜉	NOUN
cana-2204	106	16	)	)	PUNCT
cana-2204	106	17	−	−	PROPN
cana-2204	106	18	(	(	PUNCT
cana-2204	106	19	𝑌𝛼)𝑔(𝜙2𝑋	𝑌𝛼)𝑔(𝜙2𝑋	PROPN
cana-2204	106	20	,	,	PUNCT
cana-2204	106	21	𝜉	𝜉	NOUN
cana-2204	106	22	)	)	PUNCT
cana-2204	106	23	𝜂(𝑅(𝑋	𝜂(𝑅(𝑋	PROPN
cana-2204	106	24	,	,	PUNCT
cana-2204	106	25	𝑌)𝑍	𝑌)𝑍	PROPN
cana-2204	106	26	)	)	PUNCT
cana-2204	107	1	=	=	PRON
cana-2204	107	2	𝜖(𝛼2	𝜖(𝛼2	NOUN
cana-2204	107	3	+	+	CCONJ
cana-2204	107	4	𝛽2){𝜂(𝑋)𝑔(𝑌	𝛽2){𝜂(𝑋)𝑔(𝑌	PROPN
cana-2204	107	5	,	,	PUNCT
cana-2204	107	6	𝑍	𝑍	PROPN
cana-2204	107	7	)	)	PUNCT
cana-2204	107	8	−	−	NOUN
cana-2204	107	9	𝜂(𝑌)𝑔(𝑋	𝜂(𝑌)𝑔(𝑋	NOUN
cana-2204	107	10	,	,	PUNCT
cana-2204	107	11	𝑍	𝑍	PROPN
cana-2204	107	12	)	)	PUNCT
cana-2204	107	13	}	}	PUNCT
cana-2204	107	14	(	(	PUNCT
cana-2204	107	15	3.25	3.25	NUM
cana-2204	107	16	)	)	PUNCT
cana-2204	107	17	from	from	ADP
cana-2204	107	18	equation	equation	NOUN
cana-2204	107	19	(	(	PUNCT
cana-2204	107	20	3.14	3.14	NUM
cana-2204	107	21	)	)	PUNCT
cana-2204	107	22	,	,	PUNCT
cana-2204	107	23	we	we	PRON
cana-2204	107	24	have	have	VERB
cana-2204	107	25	𝑅(𝜉	𝑅(𝜉	NOUN
cana-2204	107	26	,	,	PUNCT
cana-2204	107	27	𝑋)𝑌	𝑋)𝑌	X
cana-2204	107	28	=	=	SYM
cana-2204	107	29	(	(	PUNCT
cana-2204	107	30	𝛼2	𝛼2	PROPN
cana-2204	107	31	+	+	CCONJ
cana-2204	107	32	𝛽2){𝜖𝑔(𝑋	𝛽2){𝜖𝑔(𝑋	PROPN
cana-2204	107	33	,	,	PUNCT
cana-2204	107	34	𝑌)𝜉	𝑌)𝜉	PUNCT
cana-2204	108	1	−	−	PROPN
cana-2204	109	1	𝜂(𝑌)𝑋	𝜂(𝑌)𝑋	NOUN
cana-2204	109	2	}	}	PUNCT
cana-2204	109	3	+	+	CCONJ
cana-2204	109	4	{	{	PUNCT
cana-2204	109	5	𝜉𝛽	𝜉𝛽	ADJ
cana-2204	109	6	−	−	PROPN
cana-2204	109	7	2𝛼𝛽𝜂(𝑌)}𝜙𝑋	2𝛼𝛽𝜂(𝑌)}𝜙𝑋	NUM
cana-2204	109	8	+	+	CCONJ
cana-2204	109	9	(	(	PUNCT
cana-2204	109	10	𝜉𝛼)𝑋	𝜉𝛼)𝑋	PROPN
cana-2204	109	11	+	+	PROPN
cana-2204	109	12	𝜖𝛼𝜉𝑔(𝑋	𝜖𝛼𝜉𝑔(𝑋	PROPN
cana-2204	109	13	,	,	PUNCT
cana-2204	109	14	𝑌)𝑌	𝑌)𝑌	NOUN
cana-2204	109	15	−	−	PROPN
cana-2204	109	16	(	(	PUNCT
cana-2204	109	17	𝑋𝛼)𝜉	𝑋𝛼)𝜉	NOUN
cana-2204	109	18	−	−	PROPN
cana-2204	109	19	(	(	PUNCT
cana-2204	109	20	𝑋𝛼)𝜂(𝑌)𝑌	𝑋𝛼)𝜂(𝑌)𝑌	ADJ
cana-2204	109	21	(	(	PUNCT
cana-2204	109	22	3.26	3.26	NUM
cana-2204	109	23	)	)	PUNCT
cana-2204	109	24	putting	put	VERB
cana-2204	109	25	𝑌for	𝑌for	PROPN
cana-2204	109	26	𝜉	𝜉	NOUN
cana-2204	109	27	in	in	ADP
cana-2204	109	28	above	above	ADP
cana-2204	109	29	equation	equation	NOUN
cana-2204	109	30	,	,	PUNCT
cana-2204	109	31	we	we	PRON
cana-2204	109	32	have	have	VERB
cana-2204	109	33	𝑅(𝜉	𝑅(𝜉	NOUN
cana-2204	109	34	,	,	PUNCT
cana-2204	109	35	𝑋)𝜉	𝑋)𝜉	ADV
cana-2204	109	36	=	=	SYM
cana-2204	109	37	(	(	PUNCT
cana-2204	109	38	𝛼2	𝛼2	PROPN
cana-2204	109	39	+	+	CCONJ
cana-2204	109	40	𝛽2){𝑋	𝛽2){𝑋	NUM
cana-2204	109	41	+	+	CCONJ
cana-2204	109	42	𝜂(𝑋)𝜉	𝜂(𝑋)𝜉	NUM
cana-2204	109	43	}	}	PUNCT
cana-2204	109	44	+	+	CCONJ
cana-2204	109	45	{	{	PUNCT
cana-2204	109	46	𝜉𝛽	𝜉𝛽	ADJ
cana-2204	109	47	+	+	PROPN
cana-2204	109	48	2𝛼𝛽}𝜙𝑋	2𝛼𝛽}𝜙𝑋	NUM
cana-2204	109	49	+	+	PUNCT
cana-2204	109	50	𝜉𝛼{𝑋	𝜉𝛼{𝑋	NOUN
cana-2204	109	51	+	+	CCONJ
cana-2204	109	52	𝜂(𝑋)𝜉	𝜂(𝑋)𝜉	X
cana-2204	109	53	}	}	PUNCT
cana-2204	109	54	𝑅(𝜉	𝑅(𝜉	PART
cana-2204	109	55	,	,	PUNCT
cana-2204	109	56	𝑋)𝜉	𝑋)𝜉	ADV
cana-2204	109	57	=	=	SYM
cana-2204	109	58	(	(	PUNCT
cana-2204	109	59	𝛼2	𝛼2	PROPN
cana-2204	109	60	+	+	CCONJ
cana-2204	109	61	𝛽2	𝛽2	PROPN
cana-2204	109	62	+	+	CCONJ
cana-2204	109	63	𝜉𝛼)𝜙2𝑋	𝜉𝛼)𝜙2𝑋	PROPN
cana-2204	109	64	+	+	CCONJ
cana-2204	109	65	(	(	PUNCT
cana-2204	109	66	𝜉𝛽	𝜉𝛽	ADJ
cana-2204	109	67	+	+	PROPN
cana-2204	109	68	2𝛼𝛽)𝜙𝑋	2𝛼𝛽)𝜙𝑋	NUM
cana-2204	109	69	(	(	PUNCT
cana-2204	109	70	3.27	3.27	NUM
cana-2204	109	71	)	)	PUNCT
cana-2204	109	72	now	now	ADV
cana-2204	109	73	,	,	PUNCT
cana-2204	109	74	from	from	ADP
cana-2204	109	75	equation	equation	NOUN
cana-2204	109	76	(	(	PUNCT
cana-2204	109	77	3.15	3.15	NUM
cana-2204	109	78	)	)	PUNCT
cana-2204	109	79	,	,	PUNCT
cana-2204	109	80	we	we	PRON
cana-2204	109	81	have	have	VERB
cana-2204	109	82	𝑔(𝑅(𝑋	𝑔(𝑅(𝑋	PROPN
cana-2204	109	83	,	,	PUNCT
cana-2204	109	84	𝑌)𝑍	𝑌)𝑍	PROPN
cana-2204	109	85	,	,	PUNCT
cana-2204	109	86	𝜉	𝜉	X
cana-2204	109	87	)	)	PUNCT
cana-2204	109	88	=	=	SYM
cana-2204	109	89	(	(	PUNCT
cana-2204	109	90	𝛼2	𝛼2	PROPN
cana-2204	109	91	+	+	CCONJ
cana-2204	109	92	𝛽2){𝜖𝑔(𝑋	𝛽2){𝜖𝑔(𝑋	PROPN
cana-2204	109	93	,	,	PUNCT
cana-2204	109	94	𝜉)𝑔(𝑌	𝜉)𝑔(𝑌	PROPN
cana-2204	109	95	,	,	PUNCT
cana-2204	109	96	𝑍	𝑍	NOUN
cana-2204	109	97	)	)	PUNCT
cana-2204	109	98	−	−	NOUN
cana-2204	109	99	𝜖𝑔(𝑌	𝜖𝑔(𝑌	NOUN
cana-2204	109	100	,	,	PUNCT
cana-2204	109	101	𝜉)𝑔(𝑋	𝜉)𝑔(𝑋	PROPN
cana-2204	109	102	,	,	PUNCT
cana-2204	109	103	𝑍	𝑍	NOUN
cana-2204	109	104	)	)	PUNCT
cana-2204	109	105	}	}	PUNCT
cana-2204	109	106	putting	put	VERB
cana-2204	109	107	𝑌	𝑌	PROPN
cana-2204	109	108	=	=	SYM
cana-2204	109	109	𝑍	𝑍	PROPN
cana-2204	109	110	=	=	PUNCT
cana-2204	109	111	𝑒𝑖	𝑒𝑖	NOUN
cana-2204	109	112	,	,	PUNCT
cana-2204	109	113	where	where	SCONJ
cana-2204	109	114	𝑒𝑖	𝑒𝑖	NOUN
cana-2204	109	115	is	be	AUX
cana-2204	109	116	an	an	DET
cana-2204	109	117	orthonormal	orthonormal	ADJ
cana-2204	109	118	basis	basis	NOUN
cana-2204	109	119	of	of	ADP
cana-2204	109	120	the	the	DET
cana-2204	109	121	tangent	tangent	ADJ
cana-2204	109	122	space	space	NOUN
cana-2204	109	123	at	at	ADP
cana-2204	109	124	each	each	DET
cana-2204	109	125	point	point	NOUN
cana-2204	109	126	of	of	ADP
cana-2204	109	127	the	the	DET
cana-2204	109	128	manifold	manifold	NOUN
cana-2204	109	129	and	and	CCONJ
cana-2204	109	130	taking	take	VERB
cana-2204	109	131	summation	summation	NOUN
cana-2204	109	132	over	over	ADP
cana-2204	109	133	𝑖	𝑖	PROPN
cana-2204	109	134	and	and	CCONJ
cana-2204	109	135	𝑖	𝑖	NOUN
cana-2204	109	136	=	=	SYM
cana-2204	109	137	1,2	1,2	NUM
cana-2204	109	138	,	,	PUNCT
cana-2204	109	139	.	.	PUNCT
cana-2204	109	140	.	.	PUNCT
cana-2204	109	141	.	.	PUNCT
cana-2204	109	142	.	.	PUNCT
cana-2204	110	1	.	.	PUNCT
cana-2204	111	1	,	,	PUNCT
cana-2204	111	2	𝑛	𝑛	PRON
cana-2204	111	3	then	then	ADV
cana-2204	111	4	,	,	PUNCT
cana-2204	111	5	we	we	PRON
cana-2204	111	6	get	get	VERB
cana-2204	111	7	𝑔(𝑅(𝑋	𝑔(𝑅(𝑋	NOUN
cana-2204	111	8	,	,	PUNCT
cana-2204	111	9	𝑒𝑖)𝑒𝑖	𝑒𝑖)𝑒𝑖	NOUN
cana-2204	111	10	,	,	PUNCT
cana-2204	111	11	𝜉	𝜉	NOUN
cana-2204	111	12	)	)	PUNCT
cana-2204	111	13	=	=	SYM
cana-2204	111	14	(	(	PUNCT
cana-2204	111	15	𝛼	𝛼	NOUN
cana-2204	111	16	2	2	NUM
cana-2204	111	17	+	+	SYM
cana-2204	111	18	𝛽2){𝜖𝑔(𝑋	𝛽2){𝜖𝑔(𝑋	PROPN
cana-2204	111	19	,	,	PUNCT
cana-2204	111	20	𝜉)𝑔(𝑒𝑖	𝜉)𝑔(𝑒𝑖	X
cana-2204	111	21	,	,	PUNCT
cana-2204	111	22	𝑒𝑖	𝑒𝑖	NOUN
cana-2204	111	23	)	)	PUNCT
cana-2204	111	24	−	−	PROPN
cana-2204	111	25	𝜖𝑔(𝑒𝑖	𝜖𝑔(𝑒𝑖	PROPN
cana-2204	111	26	,	,	PUNCT
cana-2204	111	27	𝜉)𝑔(𝑋	𝜉)𝑔(𝑋	PROPN
cana-2204	111	28	,	,	PUNCT
cana-2204	111	29	𝑒𝑖	𝑒𝑖	NOUN
cana-2204	111	30	)	)	PUNCT
cana-2204	111	31	}	}	PUNCT
cana-2204	111	32	𝑆(𝑋	𝑆(𝑋	PROPN
cana-2204	111	33	,	,	PUNCT
cana-2204	111	34	𝜉	𝜉	NOUN
cana-2204	111	35	)	)	PUNCT
cana-2204	111	36	=	=	SYM
cana-2204	111	37	(	(	PUNCT
cana-2204	111	38	𝑛	𝑛	PRON
cana-2204	112	1	−	−	PROPN
cana-2204	112	2	1)(𝛼2	1)(𝛼2	NUM
cana-2204	112	3	+	+	CCONJ
cana-2204	112	4	𝛽2)𝜂(𝑋	𝛽2)𝜂(𝑋	NOUN
cana-2204	112	5	)	)	PUNCT
cana-2204	112	6	(	(	PUNCT
cana-2204	112	7	3.28	3.28	NUM
cana-2204	112	8	)	)	PUNCT
cana-2204	112	9	communications	communication	NOUN
cana-2204	112	10	on	on	ADP
cana-2204	112	11	applied	apply	VERB
cana-2204	112	12	nonlinear	nonlinear	ADJ
cana-2204	112	13	analysis	analysis	NOUN
cana-2204	112	14	issn	issn	NOUN
cana-2204	112	15	:	:	PUNCT
cana-2204	112	16	1074	1074	NUM
cana-2204	112	17	-	-	PUNCT
cana-2204	112	18	133x	133x	NUM
cana-2204	112	19	vol	vol	NOUN
cana-2204	112	20	32	32	NUM
cana-2204	113	1	no	no	NOUN
cana-2204	113	2	.	.	PUNCT
cana-2204	114	1	1s	1s	NUM
cana-2204	114	2	(	(	PUNCT
cana-2204	114	3	2025	2025	NUM
cana-2204	114	4	)	)	PUNCT
cana-2204	114	5	405	405	NUM
cana-2204	114	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2204	114	7	using	use	VERB
cana-2204	114	8	equation	equation	NOUN
cana-2204	114	9	(	(	PUNCT
cana-2204	114	10	3.28	3.28	NUM
cana-2204	114	11	)	)	PUNCT
cana-2204	114	12	,	,	PUNCT
cana-2204	114	13	we	we	PRON
cana-2204	114	14	have	have	VERB
cana-2204	114	15	𝑄𝜉	𝑄𝜉	PROPN
cana-2204	114	16	=	=	SYM
cana-2204	114	17	𝜖(𝑛	𝜖(𝑛	PROPN
cana-2204	114	18	−	−	PROPN
cana-2204	114	19	1)(𝛼2	1)(𝛼2	NUM
cana-2204	115	1	+	+	CCONJ
cana-2204	115	2	𝛽2)𝜉	𝛽2)𝜉	ADJ
cana-2204	115	3	(	(	PUNCT
cana-2204	115	4	3.29	3.29	NUM
cana-2204	115	5	)	)	PUNCT
cana-2204	115	6	in	in	ADP
cana-2204	115	7	a	a	DET
cana-2204	115	8	3	3	NUM
cana-2204	115	9	-	-	PUNCT
cana-2204	115	10	dimensional	dimensional	ADJ
cana-2204	115	11	manifold	manifold	NOUN
cana-2204	115	12	,	,	PUNCT
cana-2204	115	13	the	the	DET
cana-2204	115	14	curvature	curvature	NOUN
cana-2204	115	15	tensor	tensor	NOUN
cana-2204	115	16	is	be	AUX
cana-2204	115	17	given	give	VERB
cana-2204	115	18	by	by	ADP
cana-2204	115	19	𝑅(𝑋	𝑅(𝑋	NOUN
cana-2204	115	20	,	,	PUNCT
cana-2204	115	21	𝑌)𝑍	𝑌)𝑍	X
cana-2204	115	22	=	=	SYM
cana-2204	115	23	𝑆(𝑌	𝑆(𝑌	PROPN
cana-2204	115	24	,	,	PUNCT
cana-2204	115	25	𝑍)𝑋	𝑍)𝑋	ADJ
cana-2204	115	26	−	−	PROPN
cana-2204	115	27	𝑔(𝑋	𝑔(𝑋	PROPN
cana-2204	115	28	,	,	PUNCT
cana-2204	115	29	𝑍)𝑄𝑌	𝑍)𝑄𝑌	PROPN
cana-2204	115	30	+	+	CCONJ
cana-2204	115	31	𝑔(𝑌	𝑔(𝑌	NUM
cana-2204	115	32	,	,	PUNCT
cana-2204	115	33	𝑍)𝑄	𝑍)𝑄	NOUN
cana-2204	115	34	−	−	PROPN
cana-2204	115	35	𝑆(𝑋	𝑆(𝑋	PROPN
cana-2204	115	36	,	,	PUNCT
cana-2204	115	37	𝑍)𝑌	𝑍)𝑌	NOUN
cana-2204	116	1	−	−	PROPN
cana-2204	116	2	𝑟	𝑟	NOUN
cana-2204	116	3	2	2	NUM
cana-2204	116	4	{	{	PUNCT
cana-2204	116	5	𝑔(𝑌	𝑔(𝑌	PROPN
cana-2204	116	6	,	,	PUNCT
cana-2204	116	7	𝑍)𝑋	𝑍)𝑋	NOUN
cana-2204	116	8	−	−	PROPN
cana-2204	116	9	𝑔{𝑋	𝑔{𝑋	PROPN
cana-2204	116	10	,	,	PUNCT
cana-2204	116	11	𝑍)𝑌	𝑍)𝑌	PROPN
cana-2204	116	12	}	}	PUNCT
cana-2204	116	13	(	(	PUNCT
cana-2204	116	14	3.30	3.30	NUM
cana-2204	116	15	)	)	PUNCT
cana-2204	116	16	theorem	theorem	VERB
cana-2204	116	17	3.5	3.5	NUM
cana-2204	116	18	:	:	PUNCT
cana-2204	116	19	in	in	ADP
cana-2204	116	20	a	a	DET
cana-2204	116	21	3	3	NUM
cana-2204	116	22	dimensional	dimensional	ADJ
cana-2204	116	23	(	(	PUNCT
cana-2204	116	24	𝜖	𝜖	NOUN
cana-2204	116	25	)	)	PUNCT
cana-2204	116	26	−lorentzian	−lorentzian	ADJ
cana-2204	116	27	para	para	NOUN
cana-2204	116	28	-	-	PUNCT
cana-2204	116	29	sasakian	sasakian	NOUN
cana-2204	116	30	manifold	manifold	NOUN
cana-2204	116	31	with	with	ADP
cana-2204	116	32	parallelized	parallelize	VERB
cana-2204	116	33	generalized	generalize	VERB
cana-2204	116	34	symmetric	symmetric	ADJ
cana-2204	116	35	metric	metric	ADJ
cana-2204	116	36	connection	connection	NOUN
cana-2204	116	37	,	,	PUNCT
cana-2204	116	38	the	the	DET
cana-2204	116	39	ricci	ricci	NOUN
cana-2204	116	40	operator	operator	NOUN
cana-2204	116	41	𝑄	𝑄	PRON
cana-2204	116	42	is	be	AUX
cana-2204	116	43	given	give	VERB
cana-2204	116	44	by	by	ADP
cana-2204	116	45	𝑄𝑋	𝑄𝑋	PROPN
cana-2204	116	46	=	=	SYM
cana-2204	116	47	{	{	PUNCT
cana-2204	116	48	𝑟	𝑟	NOUN
cana-2204	116	49	2	2	NUM
cana-2204	116	50	−	−	NOUN
cana-2204	116	51	2𝜖(𝛼2	2𝜖(𝛼2	NUM
cana-2204	116	52	+	+	CCONJ
cana-2204	116	53	𝛽2	𝛽2	NOUN
cana-2204	116	54	)	)	PUNCT
cana-2204	117	1	+	+	CCONJ
cana-2204	117	2	𝜖𝜉𝛼}𝑋	𝜖𝜉𝛼}𝑋	ADJ
cana-2204	117	3	+	+	CCONJ
cana-2204	117	4	{	{	PUNCT
cana-2204	117	5	𝑟	𝑟	NOUN
cana-2204	117	6	2	2	NUM
cana-2204	117	7	−	−	NOUN
cana-2204	117	8	4𝜖(𝛼2	4𝜖(𝛼2	NOUN
cana-2204	117	9	+	+	NOUN
cana-2204	117	10	𝛽2	𝛽2	NOUN
cana-2204	117	11	)	)	PUNCT
cana-2204	118	1	+	+	CCONJ
cana-2204	118	2	𝜖𝜉𝛼}𝜂(𝑋)𝜉	𝜖𝜉𝛼}𝜂(𝑋)𝜉	PUNCT
cana-2204	118	3	+	+	CCONJ
cana-2204	118	4	𝜖{𝜉𝛽	𝜖{𝜉𝛽	PROPN
cana-2204	118	5	+	+	CCONJ
cana-2204	118	6	2𝛼𝛽	2𝛼𝛽	NOUN
cana-2204	118	7	}	}	PUNCT
cana-2204	118	8	(	(	PUNCT
cana-2204	118	9	3.31	3.31	NUM
cana-2204	118	10	)	)	PUNCT
cana-2204	118	11	for	for	ADP
cana-2204	118	12	any	any	DET
cana-2204	118	13	𝑋	𝑋	PROPN
cana-2204	118	14	,	,	PUNCT
cana-2204	118	15	𝑌	𝑌	PROPN
cana-2204	118	16	,	,	PUNCT
cana-2204	118	17	𝑍	𝑍	PROPN
cana-2204	118	18	∈	∈	PROPN
cana-2204	118	19	𝜒(𝑀	𝜒(𝑀	PROPN
cana-2204	118	20	)	)	PUNCT
cana-2204	118	21	.	.	PUNCT
cana-2204	119	1	proof	proof	NOUN
cana-2204	119	2	:	:	PUNCT
cana-2204	119	3	in	in	ADP
cana-2204	119	4	a	a	DET
cana-2204	119	5	3	3	NUM
cana-2204	119	6	-	-	PUNCT
cana-2204	119	7	dimensional	dimensional	ADJ
cana-2204	119	8	(	(	PUNCT
cana-2204	119	9	𝜖	𝜖	NOUN
cana-2204	119	10	)	)	PUNCT
cana-2204	119	11	−lorentzian	−lorentzian	ADJ
cana-2204	119	12	para	para	NOUN
cana-2204	119	13	-	-	PUNCT
cana-2204	119	14	sasakian	sasakian	NOUN
cana-2204	119	15	manifold	manifold	NOUN
cana-2204	119	16	with	with	ADP
cana-2204	119	17	parallelized	parallelize	VERB
cana-2204	119	18	generalized	generalize	VERB
cana-2204	119	19	symmetric	symmetric	ADJ
cana-2204	119	20	metric	metric	ADJ
cana-2204	119	21	connection	connection	NOUN
cana-2204	119	22	,	,	PUNCT
cana-2204	119	23	the	the	DET
cana-2204	119	24	curvature	curvature	NOUN
cana-2204	119	25	tensor	tensor	NOUN
cana-2204	119	26	is	be	AUX
cana-2204	119	27	𝑅(𝑋	𝑅(𝑋	NOUN
cana-2204	119	28	,	,	PUNCT
cana-2204	119	29	𝑌)𝜉	𝑌)𝜉	PUNCT
cana-2204	120	1	=	=	SYM
cana-2204	120	2	𝑆(𝑌	𝑆(𝑌	NOUN
cana-2204	120	3	,	,	PUNCT
cana-2204	120	4	𝜉)𝑋	𝜉)𝑋	ADJ
cana-2204	120	5	−	−	ADP
cana-2204	120	6	𝑆(𝑋	𝑆(𝑋	PROPN
cana-2204	120	7	,	,	PUNCT
cana-2204	120	8	𝜉)𝑌	𝜉)𝑌	VERB
cana-2204	120	9	−	−	PRON
cana-2204	120	10	𝑔(𝑋	𝑔(𝑋	NOUN
cana-2204	120	11	,	,	PUNCT
cana-2204	120	12	𝜉)𝑄𝑌	𝜉)𝑄𝑌	NOUN
cana-2204	120	13	+	+	CCONJ
cana-2204	120	14	𝑔(𝑌	𝑔(𝑌	NUM
cana-2204	120	15	,	,	PUNCT
cana-2204	120	16	𝜉)𝑄𝑋	𝜉)𝑄𝑋	ADP
cana-2204	120	17	−	−	PROPN
cana-2204	120	18	𝑟	𝑟	SYM
cana-2204	120	19	2	2	NUM
cana-2204	120	20	{	{	PUNCT
cana-2204	120	21	𝑔(𝑌	𝑔(𝑌	NUM
cana-2204	120	22	,	,	PUNCT
cana-2204	120	23	𝜉)𝑋	𝜉)𝑋	ADJ
cana-2204	120	24	−	−	ADP
cana-2204	120	25	𝑔(𝑋	𝑔(𝑋	NOUN
cana-2204	120	26	,	,	PUNCT
cana-2204	120	27	𝜉)𝑌	𝜉)𝑌	NOUN
cana-2204	120	28	}	}	PUNCT
cana-2204	120	29	using	use	VERB
cana-2204	120	30	equation	equation	NOUN
cana-2204	120	31	(	(	PUNCT
cana-2204	120	32	3.14	3.14	NUM
cana-2204	120	33	)	)	PUNCT
cana-2204	120	34	and	and	CCONJ
cana-2204	120	35	(	(	PUNCT
cana-2204	120	36	3.18	3.18	NUM
cana-2204	120	37	)	)	PUNCT
cana-2204	120	38	in	in	ADP
cana-2204	120	39	above	above	ADP
cana-2204	120	40	equation	equation	NOUN
cana-2204	120	41	,	,	PUNCT
cana-2204	120	42	then	then	ADV
cana-2204	120	43	we	we	PRON
cana-2204	120	44	have	have	VERB
cana-2204	120	45	{	{	PUNCT
cana-2204	120	46	r	r	NOUN
cana-2204	120	47	2𝜖	2𝜖	NUM
cana-2204	120	48	−	−	PROPN
cana-2204	120	49	2(α2	2(α2	NUM
cana-2204	120	50	+	+	CCONJ
cana-2204	120	51	β	β	PROPN
cana-2204	120	52	2	2	NUM
cana-2204	120	53	)	)	PUNCT
cana-2204	120	54	}	}	PUNCT
cana-2204	120	55	{	{	PUNCT
cana-2204	120	56	𝜂(𝑌)𝑋	𝜂(𝑌)𝑋	NOUN
cana-2204	120	57	−	−	PROPN
cana-2204	120	58	𝜂(𝑋)𝑌	𝜂(𝑋)𝑌	NOUN
cana-2204	120	59	}	}	PUNCT
cana-2204	120	60	+	+	NUM
cana-2204	120	61	2𝛼𝛽{𝜂(𝑌)𝜙𝑋	2𝛼𝛽{𝜂(𝑌)𝜙𝑋	NUM
cana-2204	120	62	−	−	NOUN
cana-2204	120	63	𝜂(𝑋)𝜙𝑌	𝜂(𝑋)𝜙𝑌	ADJ
cana-2204	120	64	}	}	PUNCT
cana-2204	120	65	+	+	PROPN
cana-2204	120	66	(	(	PUNCT
cana-2204	120	67	𝑋𝛽)𝜙𝑌	𝑋𝛽)𝜙𝑌	PROPN
cana-2204	120	68	−	−	PROPN
cana-2204	120	69	(	(	PUNCT
cana-2204	120	70	𝑌𝛽)𝜙𝑋	𝑌𝛽)𝜙𝑋	NOUN
cana-2204	120	71	+	+	CCONJ
cana-2204	120	72	(	(	PUNCT
cana-2204	120	73	𝑋𝛼)𝜙2𝑌	𝑋𝛼)𝜙2𝑌	ADJ
cana-2204	120	74	−	−	PROPN
cana-2204	120	75	(	(	PUNCT
cana-2204	120	76	𝑌𝛼)𝜙2𝑋	𝑌𝛼)𝜙2𝑋	NOUN
cana-2204	120	77	=	=	SYM
cana-2204	120	78	1	1	NUM
cana-2204	120	79	𝜖	𝜖	X
cana-2204	120	80	𝜂(𝑌)𝑄𝑋	𝜂(𝑌)𝑄𝑋	NOUN
cana-2204	120	81	−	−	PROPN
cana-2204	120	82	1	1	NUM
cana-2204	120	83	𝜖	𝜖	PRON
cana-2204	120	84	𝜂(𝑋)𝑄𝑌	𝜂(𝑋)𝑄𝑌	NOUN
cana-2204	120	85	putting	put	VERB
cana-2204	120	86	𝑌	𝑌	PROPN
cana-2204	120	87	=	=	PUNCT
cana-2204	120	88	𝜉	𝜉	NOUN
cana-2204	120	89	in	in	ADP
cana-2204	120	90	above	above	ADP
cana-2204	120	91	equation	equation	NOUN
cana-2204	120	92	,	,	PUNCT
cana-2204	120	93	we	we	PRON
cana-2204	120	94	obtained	obtain	VERB
cana-2204	120	95	𝑄𝑋	𝑄𝑋	PROPN
cana-2204	120	96	=	=	PRON
cana-2204	120	97	{	{	PUNCT
cana-2204	120	98	r	r	NOUN
cana-2204	120	99	2	2	NUM
cana-2204	120	100	−	−	NOUN
cana-2204	120	101	2𝜖(α2	2𝜖(α2	NUM
cana-2204	121	1	+	+	CCONJ
cana-2204	121	2	β	β	PROPN
cana-2204	121	3	2	2	NUM
cana-2204	121	4	)	)	PUNCT
cana-2204	121	5	+	+	NUM
cana-2204	121	6	𝜖𝜉𝛼}{𝑋	𝜖𝜉𝛼}{𝑋	NOUN
cana-2204	121	7	+	+	CCONJ
cana-2204	121	8	𝜂(𝑋)𝜉	𝜂(𝑋)𝜉	NUM
cana-2204	121	9	}	}	PUNCT
cana-2204	121	10	+	+	CCONJ
cana-2204	121	11	𝜖(2𝛼𝛽	𝜖(2𝛼𝛽	PROPN
cana-2204	121	12	+	+	NUM
cana-2204	121	13	𝜉𝛽))𝜙𝑋	𝜉𝛽))𝜙𝑋	NOUN
cana-2204	121	14	−	−	NOUN
cana-2204	121	15	2𝜖(α2	2𝜖(α2	NUM
cana-2204	122	1	+	+	CCONJ
cana-2204	122	2	β	β	X
cana-2204	122	3	2)𝜂(𝑋)𝜉	2)𝜂(𝑋)𝜉	NUM
cana-2204	122	4	thus	thus	ADV
cana-2204	122	5	,	,	PUNCT
cana-2204	122	6	from	from	ADP
cana-2204	122	7	above	above	ADP
cana-2204	122	8	we	we	PRON
cana-2204	122	9	have	have	VERB
cana-2204	122	10	result	result	NOUN
cana-2204	122	11	(	(	PUNCT
cana-2204	122	12	3.31	3.31	NUM
cana-2204	122	13	)	)	PUNCT
cana-2204	122	14	.	.	PUNCT
cana-2204	123	1	theorem	theorem	VERB
cana-2204	123	2	3.6	3.6	NUM
cana-2204	123	3	:	:	PUNCT
cana-2204	123	4	in	in	ADP
cana-2204	123	5	a	a	DET
cana-2204	123	6	3	3	NUM
cana-2204	123	7	dimensional	dimensional	ADJ
cana-2204	123	8	(	(	PUNCT
cana-2204	123	9	𝜖	𝜖	NOUN
cana-2204	123	10	)	)	PUNCT
cana-2204	123	11	−lorentzian	−lorentzian	ADJ
cana-2204	123	12	para	para	NOUN
cana-2204	123	13	-	-	PUNCT
cana-2204	123	14	sasakian	sasakian	NOUN
cana-2204	123	15	manifold	manifold	NOUN
cana-2204	123	16	with	with	ADP
cana-2204	123	17	parallelized	parallelize	VERB
cana-2204	123	18	generalized	generalize	VERB
cana-2204	123	19	symmetric	symmetric	ADJ
cana-2204	123	20	metric	metric	ADJ
cana-2204	123	21	connection	connection	NOUN
cana-2204	123	22	,	,	PUNCT
cana-2204	123	23	scalar	scalar	ADJ
cana-2204	123	24	and	and	CCONJ
cana-2204	123	25	ricci	ricci	PROPN
cana-2204	123	26	curvature	curvature	NOUN
cana-2204	123	27	are	be	AUX
cana-2204	123	28	given	give	VERB
cana-2204	123	29	by	by	ADP
cana-2204	123	30	the	the	DET
cana-2204	123	31	following	follow	VERB
cana-2204	123	32	expressions	expression	NOUN
cana-2204	123	33	𝑟	𝑟	X
cana-2204	123	34	=	=	SYM
cana-2204	123	35	2	2	NUM
cana-2204	123	36	(	(	PUNCT
cana-2204	123	37	1	1	NUM
cana-2204	123	38	+	+	NUM
cana-2204	123	39	8	8	NUM
cana-2204	123	40	𝜖−1	𝜖−1	PROPN
cana-2204	123	41	)	)	PUNCT
cana-2204	123	42	(	(	PUNCT
cana-2204	123	43	𝛼2	𝛼2	PROPN
cana-2204	123	44	+	+	SYM
cana-2204	123	45	𝛽2	𝛽2	NOUN
cana-2204	123	46	)	)	PUNCT
cana-2204	123	47	(	(	PUNCT
cana-2204	123	48	3.32	3.32	NUM
cana-2204	123	49	)	)	PUNCT
cana-2204	123	50	𝑆(𝑌	𝑆(𝑌	NOUN
cana-2204	123	51	,	,	PUNCT
cana-2204	123	52	𝑍	𝑍	PROPN
cana-2204	123	53	)	)	PUNCT
cana-2204	123	54	=	=	SYM
cana-2204	123	55	2(𝛼2	2(𝛼2	PROPN
cana-2204	123	56	+	+	CCONJ
cana-2204	123	57	𝛽2	𝛽2	NOUN
cana-2204	123	58	)	)	PUNCT
cana-2204	123	59	{	{	PUNCT
cana-2204	123	60	(	(	PUNCT
cana-2204	123	61	1	1	NUM
cana-2204	123	62	+	+	SYM
cana-2204	123	63	4	4	NUM
cana-2204	123	64	𝜖−1	𝜖−1	PROPN
cana-2204	123	65	)	)	PUNCT
cana-2204	123	66	𝑔(𝑌	𝑔(𝑌	PROPN
cana-2204	123	67	,	,	PUNCT
cana-2204	123	68	𝑍	𝑍	PROPN
cana-2204	123	69	)	)	PUNCT
cana-2204	123	70	+	+	CCONJ
cana-2204	123	71	2𝜂(𝑌)𝜂(𝑍	2𝜂(𝑌)𝜂(𝑍	NUM
cana-2204	123	72	)	)	PUNCT
cana-2204	123	73	}	}	PUNCT
cana-2204	123	74	(	(	PUNCT
cana-2204	123	75	3.33	3.33	NUM
cana-2204	123	76	)	)	PUNCT
cana-2204	123	77	for	for	ADP
cana-2204	123	78	any	any	DET
cana-2204	123	79	𝑋	𝑋	PROPN
cana-2204	123	80	,	,	PUNCT
cana-2204	123	81	𝑌	𝑌	PROPN
cana-2204	123	82	,	,	PUNCT
cana-2204	123	83	𝑍	𝑍	PROPN
cana-2204	123	84	∈	∈	PROPN
cana-2204	123	85	𝜒(𝑀	𝜒(𝑀	PROPN
cana-2204	123	86	)	)	PUNCT
cana-2204	123	87	.	.	PUNCT
cana-2204	124	1	proof	proof	NOUN
cana-2204	124	2	:	:	PUNCT
cana-2204	124	3	on	on	ADP
cana-2204	124	4	putting	put	VERB
cana-2204	124	5	𝜉	𝜉	PRON
cana-2204	124	6	in	in	ADP
cana-2204	124	7	place	place	NOUN
cana-2204	124	8	of	of	ADP
cana-2204	124	9	𝑋	𝑋	NOUN
cana-2204	124	10	in	in	ADP
cana-2204	124	11	equation	equation	NOUN
cana-2204	124	12	(	(	PUNCT
cana-2204	124	13	3.30	3.30	NUM
cana-2204	124	14	)	)	PUNCT
cana-2204	124	15	,	,	PUNCT
cana-2204	124	16	we	we	PRON
cana-2204	124	17	have	have	VERB
cana-2204	124	18	the	the	DET
cana-2204	124	19	equation	equation	NOUN
cana-2204	124	20	as	as	ADP
cana-2204	124	21	𝑅(𝜉	𝑅(𝜉	PROPN
cana-2204	124	22	,	,	PUNCT
cana-2204	124	23	𝑌)𝑍	𝑌)𝑍	X
cana-2204	124	24	=	=	SYM
cana-2204	124	25	𝑆(𝑌	𝑆(𝑌	PROPN
cana-2204	124	26	,	,	PUNCT
cana-2204	124	27	𝑍)𝜉	𝑍)𝜉	PROPN
cana-2204	124	28	−	−	PROPN
cana-2204	124	29	𝑔(𝜉	𝑔(𝜉	PROPN
cana-2204	124	30	,	,	PUNCT
cana-2204	124	31	𝑍)𝑄𝑌	𝑍)𝑄𝑌	PROPN
cana-2204	124	32	+	+	CCONJ
cana-2204	124	33	𝑔(𝑌	𝑔(𝑌	NUM
cana-2204	124	34	,	,	PUNCT
cana-2204	124	35	𝑍)𝑄𝜉	𝑍)𝑄𝜉	ADP
cana-2204	124	36	−	−	PROPN
cana-2204	124	37	𝑆(𝜉	𝑆(𝜉	PROPN
cana-2204	124	38	,	,	PUNCT
cana-2204	124	39	𝑍)𝑌	𝑍)𝑌	NOUN
cana-2204	124	40	−	−	PROPN
cana-2204	124	41	𝑟	𝑟	NOUN
cana-2204	124	42	2	2	NUM
cana-2204	124	43	{	{	PUNCT
cana-2204	124	44	𝑔(𝑌	𝑔(𝑌	PROPN
cana-2204	124	45	,	,	PUNCT
cana-2204	124	46	𝑍)𝜉	𝑍)𝜉	CCONJ
cana-2204	125	1	−	−	PROPN
cana-2204	125	2	𝑔{𝜉	𝑔{𝜉	PROPN
cana-2204	125	3	,	,	PUNCT
cana-2204	125	4	𝑍)𝑌	𝑍)𝑌	NOUN
cana-2204	125	5	}	}	PUNCT
cana-2204	125	6	𝜂(𝑅(𝜉	𝜂(𝑅(𝜉	PROPN
cana-2204	125	7	,	,	PUNCT
cana-2204	125	8	𝑌)𝑍	𝑌)𝑍	PROPN
cana-2204	125	9	)	)	PUNCT
cana-2204	125	10	=	=	PUNCT
cana-2204	125	11	−𝑆(𝑌	−𝑆(𝑌	PROPN
cana-2204	125	12	,	,	PUNCT
cana-2204	125	13	𝑍	𝑍	PROPN
cana-2204	125	14	)	)	PUNCT
cana-2204	125	15	−	−	PROPN
cana-2204	125	16	𝜂(𝑍)𝑆(𝑌	𝜂(𝑍)𝑆(𝑌	PROPN
cana-2204	125	17	,	,	PUNCT
cana-2204	125	18	𝜉	𝜉	NOUN
cana-2204	125	19	)	)	PUNCT
cana-2204	125	20	+	+	CCONJ
cana-2204	125	21	𝜖𝑆(𝜉	𝜖𝑆(𝜉	PROPN
cana-2204	125	22	,	,	PUNCT
cana-2204	125	23	𝜉)𝑔(𝑌	𝜉)𝑔(𝑌	PROPN
cana-2204	125	24	,	,	PUNCT
cana-2204	125	25	𝑍	𝑍	NOUN
cana-2204	125	26	)	)	PUNCT
cana-2204	125	27	−	−	NOUN
cana-2204	125	28	𝜂(𝑌)𝑆(𝜉	𝜂(𝑌)𝑆(𝜉	PROPN
cana-2204	125	29	,	,	PUNCT
cana-2204	125	30	𝑍	𝑍	PROPN
cana-2204	125	31	)	)	PUNCT
cana-2204	125	32	+	+	NUM
cana-2204	125	33	𝑟	𝑟	NOUN
cana-2204	125	34	2	2	NUM
cana-2204	125	35	{	{	PUNCT
cana-2204	125	36	𝑔(𝑌	𝑔(𝑌	NUM
cana-2204	125	37	,	,	PUNCT
cana-2204	125	38	𝑍	𝑍	PROPN
cana-2204	125	39	)	)	PUNCT
cana-2204	125	40	+	+	NUM
cana-2204	125	41	𝜖𝜂(𝑌)𝜂(𝑍	𝜖𝜂(𝑌)𝜂(𝑍	NOUN
cana-2204	125	42	)	)	PUNCT
cana-2204	125	43	}	}	PUNCT
cana-2204	125	44	communications	communication	NOUN
cana-2204	125	45	on	on	ADP
cana-2204	125	46	applied	apply	VERB
cana-2204	125	47	nonlinear	nonlinear	ADJ
cana-2204	125	48	analysis	analysis	NOUN
cana-2204	125	49	issn	issn	NOUN
cana-2204	125	50	:	:	PUNCT
cana-2204	125	51	1074	1074	NUM
cana-2204	125	52	-	-	PUNCT
cana-2204	125	53	133x	133x	NUM
cana-2204	125	54	vol	vol	NOUN
cana-2204	125	55	32	32	NUM
cana-2204	125	56	no	no	NOUN
cana-2204	125	57	.	.	PUNCT
cana-2204	126	1	1s	1s	NUM
cana-2204	126	2	(	(	PUNCT
cana-2204	126	3	2025	2025	NUM
cana-2204	126	4	)	)	PUNCT
cana-2204	126	5	406	406	NUM
cana-2204	126	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2204	126	7	using	use	VERB
cana-2204	126	8	equation	equation	NOUN
cana-2204	126	9	(	(	PUNCT
cana-2204	126	10	3.15	3.15	NUM
cana-2204	126	11	)	)	PUNCT
cana-2204	126	12	,	,	PUNCT
cana-2204	126	13	(	(	PUNCT
cana-2204	126	14	3.18	3.18	NUM
cana-2204	126	15	)	)	PUNCT
cana-2204	126	16	in	in	ADP
cana-2204	126	17	above	above	ADV
cana-2204	126	18	,	,	PUNCT
cana-2204	126	19	we	we	PRON
cana-2204	126	20	have	have	VERB
cana-2204	126	21	𝜖(𝛼2	𝜖(𝛼2	NOUN
cana-2204	126	22	+	+	CCONJ
cana-2204	126	23	𝛽2){𝜂(𝑋)𝑔(𝑌	𝛽2){𝜂(𝑋)𝑔(𝑌	PROPN
cana-2204	126	24	,	,	PUNCT
cana-2204	126	25	𝑍	𝑍	PROPN
cana-2204	126	26	)	)	PUNCT
cana-2204	127	1	−	−	NOUN
cana-2204	127	2	𝜂(𝑌)𝑔(𝑋	𝜂(𝑌)𝑔(𝑋	NOUN
cana-2204	127	3	,	,	PUNCT
cana-2204	127	4	𝑍	𝑍	PROPN
cana-2204	127	5	)	)	PUNCT
cana-2204	127	6	}	}	PUNCT
cana-2204	127	7	=	=	SYM
cana-2204	127	8	−𝑆(𝑌	−𝑆(𝑌	PROPN
cana-2204	127	9	,	,	PUNCT
cana-2204	127	10	𝑍	𝑍	PROPN
cana-2204	127	11	)	)	PUNCT
cana-2204	127	12	−	−	PROPN
cana-2204	127	13	𝜂(𝑍){2(𝛼2	𝜂(𝑍){2(𝛼2	NOUN
cana-2204	127	14	+	+	CCONJ
cana-2204	127	15	𝛽2)𝜂(𝑌	𝛽2)𝜂(𝑌	PROPN
cana-2204	127	16	)	)	PUNCT
cana-2204	127	17	}	}	PUNCT
cana-2204	128	1	+	+	VERB
cana-2204	128	2	𝜖{2𝜖(𝛼2	𝜖{2𝜖(𝛼2	PROPN
cana-2204	128	3	+	+	CCONJ
cana-2204	128	4	𝛽2)𝑔(𝜉	𝛽2)𝑔(𝜉	PROPN
cana-2204	128	5	,	,	PUNCT
cana-2204	128	6	𝜉)}𝑔(𝑌	𝜉)}𝑔(𝑌	PROPN
cana-2204	128	7	,	,	PUNCT
cana-2204	128	8	𝑍	𝑍	PROPN
cana-2204	128	9	)	)	PUNCT
cana-2204	128	10	−𝜂(𝑌){2(𝛼2	−𝜂(𝑌){2(𝛼2	NOUN
cana-2204	128	11	+	+	CCONJ
cana-2204	128	12	𝛽2)𝜂(𝑌	𝛽2)𝜂(𝑌	PROPN
cana-2204	128	13	)	)	PUNCT
cana-2204	128	14	}	}	PUNCT
cana-2204	128	15	+	+	NUM
cana-2204	128	16	𝑟	𝑟	SYM
cana-2204	128	17	2	2	NUM
cana-2204	128	18	{	{	PUNCT
cana-2204	128	19	𝑔(𝑌	𝑔(𝑌	NUM
cana-2204	128	20	,	,	PUNCT
cana-2204	128	21	𝑍	𝑍	PROPN
cana-2204	128	22	)	)	PUNCT
cana-2204	128	23	+	+	NUM
cana-2204	128	24	𝜖𝜂(𝑌)𝜂(𝑍	𝜖𝜂(𝑌)𝜂(𝑍	NOUN
cana-2204	128	25	)	)	PUNCT
cana-2204	128	26	}	}	PUNCT
cana-2204	128	27	𝑆(𝑌	𝑆(𝑌	VERB
cana-2204	128	28	,	,	PUNCT
cana-2204	128	29	𝑍	𝑍	PROPN
cana-2204	128	30	)	)	PUNCT
cana-2204	128	31	=	=	SYM
cana-2204	128	32	{	{	PUNCT
cana-2204	128	33	𝑟	𝑟	NOUN
cana-2204	128	34	2	2	NUM
cana-2204	128	35	−	−	NOUN
cana-2204	128	36	𝜖(𝛼2	𝜖(𝛼2	NOUN
cana-2204	128	37	+	+	CCONJ
cana-2204	128	38	𝛽2	𝛽2	NOUN
cana-2204	128	39	)	)	PUNCT
cana-2204	128	40	+	+	NUM
cana-2204	128	41	λ	λ	X
cana-2204	128	42	}	}	PUNCT
cana-2204	128	43	{	{	PUNCT
cana-2204	128	44	𝑔(𝑌	𝑔(𝑌	PROPN
cana-2204	128	45	,	,	PUNCT
cana-2204	128	46	𝑍	𝑍	PROPN
cana-2204	128	47	)	)	PUNCT
cana-2204	128	48	+	+	NUM
cana-2204	128	49	𝜖𝜂(𝑌)𝜂(𝑍	𝜖𝜂(𝑌)𝜂(𝑍	NOUN
cana-2204	128	50	)	)	PUNCT
cana-2204	128	51	}	}	PUNCT
cana-2204	128	52	−	−	ADP
cana-2204	128	53	λ𝑔(𝑌	λ𝑔(𝑌	NOUN
cana-2204	128	54	,	,	PUNCT
cana-2204	128	55	𝑍	𝑍	PROPN
cana-2204	128	56	)	)	PUNCT
cana-2204	128	57	−	−	PROPN
cana-2204	128	58	2𝜖λ𝜂(𝑌)𝜂(𝑌	2𝜖λ𝜂(𝑌)𝜂(𝑌	PROPN
cana-2204	128	59	)	)	PUNCT
cana-2204	128	60	(	(	PUNCT
cana-2204	128	61	3.34	3.34	NUM
cana-2204	128	62	)	)	PUNCT
cana-2204	128	63	where	where	SCONJ
cana-2204	128	64	λ	λ	X
cana-2204	128	65	=	=	NOUN
cana-2204	128	66	2ϵ(α2	2ϵ(α2	NUM
cana-2204	128	67	+	+	CCONJ
cana-2204	128	68	β	β	PROPN
cana-2204	128	69	2	2	NUM
cana-2204	128	70	)	)	PUNCT
cana-2204	128	71	,	,	PUNCT
cana-2204	128	72	now	now	ADV
cana-2204	128	73	taking	take	VERB
cana-2204	128	74	an	an	DET
cana-2204	128	75	orthonormal	orthonormal	ADJ
cana-2204	128	76	frame	frame	NOUN
cana-2204	128	77	filed	file	VERB
cana-2204	128	78	in	in	ADP
cana-2204	128	79	the	the	DET
cana-2204	128	80	above	above	ADJ
cana-2204	128	81	equation	equation	NOUN
cana-2204	128	82	over	over	ADP
cana-2204	128	83	𝑌	𝑌	PROPN
cana-2204	128	84	and	and	CCONJ
cana-2204	128	85	𝑍	𝑍	NOUN
cana-2204	128	86	,	,	PUNCT
cana-2204	128	87	we	we	PRON
cana-2204	128	88	have	have	VERB
cana-2204	128	89	the	the	DET
cana-2204	128	90	scalar	scalar	ADJ
cana-2204	128	91	curvature	curvature	NOUN
cana-2204	128	92	.	.	PUNCT
cana-2204	129	1	using	use	VERB
cana-2204	129	2	equations	equation	NOUN
cana-2204	129	3	(	(	PUNCT
cana-2204	129	4	3.32	3.32	NUM
cana-2204	129	5	)	)	PUNCT
cana-2204	129	6	and	and	CCONJ
cana-2204	129	7	(	(	PUNCT
cana-2204	129	8	3.34	3.34	NUM
cana-2204	129	9	)	)	PUNCT
cana-2204	129	10	,	,	PUNCT
cana-2204	129	11	the	the	DET
cana-2204	129	12	expression	expression	NOUN
cana-2204	129	13	of	of	ADP
cana-2204	129	14	the	the	DET
cana-2204	129	15	ricci	ricci	PROPN
cana-2204	129	16	tensor	tensor	NOUN
cana-2204	129	17	is	be	AUX
cana-2204	129	18	obtained	obtain	VERB
cana-2204	129	19	.	.	PUNCT
cana-2204	130	1	4	4	X
cana-2204	130	2	.	.	X
cana-2204	130	3	conformally	conformally	ADV
cana-2204	130	4	flat	flat	ADJ
cana-2204	130	5	(	(	PUNCT
cana-2204	130	6	𝝐	𝝐	NOUN
cana-2204	130	7	)	)	PUNCT
cana-2204	130	8	−lorentzian	−lorentzian	ADJ
cana-2204	130	9	para	para	NOUN
cana-2204	130	10	-	-	PUNCT
cana-2204	130	11	sasakian	sasakian	NOUN
cana-2204	130	12	manifold	manifold	NOUN
cana-2204	130	13	with	with	ADP
cana-2204	130	14	parallelized	parallelize	VERB
cana-2204	130	15	generalized	generalize	VERB
cana-2204	130	16	symmetric	symmetric	ADJ
cana-2204	130	17	metric	metric	ADJ
cana-2204	130	18	connection	connection	NOUN
cana-2204	130	19	the	the	DET
cana-2204	130	20	weyl	weyl	VERB
cana-2204	130	21	conformal	conformal	NOUN
cana-2204	130	22	curvature	curvature	NOUN
cana-2204	130	23	tensor	tensor	NOUN
cana-2204	130	24	𝐶	𝐶	PROPN
cana-2204	130	25	of	of	ADP
cana-2204	130	26	type	type	NOUN
cana-2204	130	27	(	(	PUNCT
cana-2204	130	28	1	1	NUM
cana-2204	130	29	,	,	PUNCT
cana-2204	130	30	3	3	NUM
cana-2204	130	31	)	)	PUNCT
cana-2204	130	32	of	of	ADP
cana-2204	130	33	an	an	DET
cana-2204	130	34	n	n	ADV
cana-2204	130	35	-	-	PUNCT
cana-2204	130	36	dimensional	dimensional	ADJ
cana-2204	130	37	riemannian	riemannian	ADJ
cana-2204	130	38	manifold	manifold	NOUN
cana-2204	130	39	is	be	AUX
cana-2204	130	40	given	give	VERB
cana-2204	130	41	by	by	ADP
cana-2204	130	42	𝐶(𝑋	𝐶(𝑋	PROPN
cana-2204	130	43	,	,	PUNCT
cana-2204	130	44	𝑌)𝑍	𝑌)𝑍	NOUN
cana-2204	130	45	=	=	PUNCT
cana-2204	130	46	𝑅(𝑋	𝑅(𝑋	PROPN
cana-2204	130	47	,	,	PUNCT
cana-2204	130	48	𝑌)𝑍	𝑌)𝑍	PROPN
cana-2204	130	49	−	−	PROPN
cana-2204	130	50	1	1	NUM
cana-2204	130	51	(	(	PUNCT
cana-2204	130	52	𝑛	𝑛	PRON
cana-2204	130	53	−	−	PROPN
cana-2204	130	54	2	2	NUM
cana-2204	130	55	)	)	PUNCT
cana-2204	131	1	[	[	X
cana-2204	131	2	𝑆(𝑌	𝑆(𝑌	NOUN
cana-2204	131	3	,	,	PUNCT
cana-2204	131	4	𝑍)𝑋	𝑍)𝑋	PROPN
cana-2204	131	5	−	−	PROPN
cana-2204	131	6	𝑆(𝑋	𝑆(𝑋	PROPN
cana-2204	131	7	,	,	PUNCT
cana-2204	131	8	𝑍)𝑌	𝑍)𝑌	NOUN
cana-2204	131	9	+	+	CCONJ
cana-2204	131	10	𝑔(𝑌	𝑔(𝑌	PROPN
cana-2204	131	11	,	,	PUNCT
cana-2204	131	12	𝑍)𝑄𝑋	𝑍)𝑄𝑋	PROPN
cana-2204	131	13	−	−	PROPN
cana-2204	131	14	𝑔(𝑋	𝑔(𝑋	PROPN
cana-2204	131	15	,	,	PUNCT
cana-2204	131	16	𝑍)𝑄𝑌	𝑍)𝑄𝑌	PROPN
cana-2204	131	17	]	]	X
cana-2204	131	18	+	+	NUM
cana-2204	131	19	𝑟	𝑟	NOUN
cana-2204	131	20	(	(	PUNCT
cana-2204	131	21	𝑛−1)(𝑛−2	𝑛−1)(𝑛−2	NOUN
cana-2204	131	22	)	)	PUNCT
cana-2204	132	1	[	[	X
cana-2204	132	2	𝑔(𝑌	𝑔(𝑌	NUM
cana-2204	132	3	,	,	PUNCT
cana-2204	132	4	𝑍)𝑋	𝑍)𝑋	NOUN
cana-2204	132	5	−	−	PROPN
cana-2204	132	6	𝑔(𝑋	𝑔(𝑋	PROPN
cana-2204	132	7	,	,	PUNCT
cana-2204	132	8	𝑍)𝑌	𝑍)𝑌	PROPN
cana-2204	132	9	]	]	X
cana-2204	132	10	(	(	PUNCT
cana-2204	132	11	4.1	4.1	NUM
cana-2204	132	12	)	)	PUNCT
cana-2204	132	13	where	where	SCONJ
cana-2204	132	14	𝑄	𝑄	PRON
cana-2204	132	15	is	be	AUX
cana-2204	132	16	the	the	DET
cana-2204	132	17	ricci	ricci	NOUN
cana-2204	132	18	operator	operator	NOUN
cana-2204	132	19	defined	define	VERB
cana-2204	132	20	by	by	ADP
cana-2204	132	21	g(qx	g(qx	PROPN
cana-2204	132	22	,	,	PUNCT
cana-2204	132	23	y	y	PROPN
cana-2204	132	24	)	)	PUNCT
cana-2204	132	25	=	=	SYM
cana-2204	132	26	s(x	s(x	NOUN
cana-2204	132	27	,	,	PUNCT
cana-2204	132	28	y)and	y)and	NOUN
cana-2204	132	29	𝑟	𝑟	NOUN
cana-2204	132	30	is	be	AUX
cana-2204	132	31	the	the	DET
cana-2204	132	32	scalar	scalar	ADJ
cana-2204	132	33	curvature	curvature	NOUN
cana-2204	132	34	.	.	PUNCT
cana-2204	133	1	let	let	VERB
cana-2204	133	2	us	we	PRON
cana-2204	133	3	suppose	suppose	VERB
cana-2204	133	4	that	that	SCONJ
cana-2204	133	5	the	the	DET
cana-2204	133	6	manifold	manifold	NOUN
cana-2204	133	7	is	be	AUX
cana-2204	133	8	conformally	conformally	ADV
cana-2204	133	9	flat	flat	ADJ
cana-2204	133	10	.	.	PUNCT
cana-2204	134	1	then	then	ADV
cana-2204	134	2	from	from	ADP
cana-2204	134	3	the	the	DET
cana-2204	134	4	above	above	ADJ
cana-2204	134	5	equation	equation	NOUN
cana-2204	134	6	,	,	PUNCT
cana-2204	134	7	we	we	PRON
cana-2204	134	8	have	have	VERB
cana-2204	134	9	𝑔(𝑅(𝑋	𝑔(𝑅(𝑋	NOUN
cana-2204	134	10	,	,	PUNCT
cana-2204	134	11	𝑌)𝑍,𝑊	𝑌)𝑍,𝑊	ADJ
cana-2204	134	12	)	)	PUNCT
cana-2204	134	13	=	=	SYM
cana-2204	134	14	1	1	NUM
cana-2204	134	15	(	(	PUNCT
cana-2204	134	16	𝑛	𝑛	PRON
cana-2204	134	17	−	−	PROPN
cana-2204	134	18	2	2	NUM
cana-2204	134	19	)	)	PUNCT
cana-2204	135	1	[	[	X
cana-2204	135	2	𝑆(𝑌	𝑆(𝑌	NOUN
cana-2204	135	3	,	,	PUNCT
cana-2204	135	4	𝑍)𝑔(𝑋,𝑊	𝑍)𝑔(𝑋,𝑊	ADJ
cana-2204	135	5	)	)	PUNCT
cana-2204	135	6	−	−	PROPN
cana-2204	135	7	𝑆(𝑋	𝑆(𝑋	PROPN
cana-2204	135	8	,	,	PUNCT
cana-2204	135	9	𝑍)𝑔(𝑌,𝑊	𝑍)𝑔(𝑌,𝑊	X
cana-2204	135	10	)	)	PUNCT
cana-2204	135	11	+	+	SYM
cana-2204	135	12	𝑔(𝑌	𝑔(𝑌	NUM
cana-2204	135	13	,	,	PUNCT
cana-2204	135	14	𝑍)𝑆(𝑋,𝑊	𝑍)𝑆(𝑋,𝑊	NUM
cana-2204	135	15	)	)	PUNCT
cana-2204	135	16	−𝑔(𝑋	−𝑔(𝑋	PROPN
cana-2204	135	17	,	,	PUNCT
cana-2204	135	18	𝑍)𝑆(𝑌,𝑊	𝑍)𝑆(𝑌,𝑊	PROPN
cana-2204	135	19	)	)	PUNCT
cana-2204	135	20	]	]	PUNCT
cana-2204	136	1	−	−	PROPN
cana-2204	136	2	𝑟	𝑟	X
cana-2204	136	3	(	(	PUNCT
cana-2204	136	4	𝑛−1)(𝑛−2	𝑛−1)(𝑛−2	NOUN
cana-2204	136	5	)	)	PUNCT
cana-2204	137	1	[	[	X
cana-2204	137	2	𝑔(𝑌	𝑔(𝑌	NUM
cana-2204	137	3	,	,	PUNCT
cana-2204	137	4	𝑍)𝑔(𝑋,𝑊	𝑍)𝑔(𝑋,𝑊	NUM
cana-2204	137	5	)	)	PUNCT
cana-2204	137	6	−	−	PRON
cana-2204	137	7	𝑔(𝑋	𝑔(𝑋	PROPN
cana-2204	137	8	,	,	PUNCT
cana-2204	137	9	𝑍)𝑔(𝑌,𝑊	𝑍)𝑔(𝑌,𝑊	X
cana-2204	137	10	)	)	PUNCT
cana-2204	137	11	]	]	PUNCT
cana-2204	137	12	(	(	PUNCT
cana-2204	137	13	4.2	4.2	X
cana-2204	137	14	)	)	PUNCT
cana-2204	137	15	putting	put	VERB
cana-2204	137	16	𝑊	𝑊	NOUN
cana-2204	137	17	=	=	SYM
cana-2204	137	18	𝜉	𝜉	NOUN
cana-2204	137	19	in	in	ADP
cana-2204	137	20	(	(	PUNCT
cana-2204	137	21	4.2	4.2	NUM
cana-2204	137	22	)	)	PUNCT
cana-2204	137	23	,	,	PUNCT
cana-2204	137	24	we	we	PRON
cana-2204	137	25	get	get	VERB
cana-2204	137	26	𝑔(𝑅(𝑋	𝑔(𝑅(𝑋	PROPN
cana-2204	137	27	,	,	PUNCT
cana-2204	137	28	𝑌)𝑍	𝑌)𝑍	PROPN
cana-2204	137	29	,	,	PUNCT
cana-2204	137	30	𝜉	𝜉	X
cana-2204	137	31	)	)	PUNCT
cana-2204	137	32	=	=	SYM
cana-2204	137	33	1	1	NUM
cana-2204	137	34	(	(	PUNCT
cana-2204	137	35	𝑛	𝑛	PRON
cana-2204	137	36	−	−	PROPN
cana-2204	137	37	2	2	NUM
cana-2204	137	38	)	)	PUNCT
cana-2204	138	1	[	[	X
cana-2204	138	2	𝑆(𝑌	𝑆(𝑌	NOUN
cana-2204	138	3	,	,	PUNCT
cana-2204	138	4	𝑍)𝑔(𝑋	𝑍)𝑔(𝑋	NUM
cana-2204	138	5	,	,	PUNCT
cana-2204	138	6	𝜉	𝜉	NOUN
cana-2204	138	7	)	)	PUNCT
cana-2204	138	8	−	−	PROPN
cana-2204	138	9	𝑆(𝑋	𝑆(𝑋	PROPN
cana-2204	138	10	,	,	PUNCT
cana-2204	138	11	𝑍)𝑔(𝑌	𝑍)𝑔(𝑌	NOUN
cana-2204	138	12	,	,	PUNCT
cana-2204	138	13	𝜉	𝜉	X
cana-2204	138	14	)	)	PUNCT
cana-2204	138	15	+	+	CCONJ
cana-2204	138	16	𝑔(𝑌	𝑔(𝑌	PROPN
cana-2204	138	17	,	,	PUNCT
cana-2204	138	18	𝑍)𝑆(𝑋	𝑍)𝑆(𝑋	PROPN
cana-2204	138	19	,	,	PUNCT
cana-2204	138	20	𝜉	𝜉	NOUN
cana-2204	138	21	)	)	PUNCT
cana-2204	138	22	−	−	NOUN
cana-2204	138	23	𝑔(𝑋	𝑔(𝑋	PROPN
cana-2204	138	24	,	,	PUNCT
cana-2204	138	25	𝑍)𝑆(𝑌	𝑍)𝑆(𝑌	NOUN
cana-2204	138	26	,	,	PUNCT
cana-2204	138	27	𝜉	𝜉	NOUN
cana-2204	138	28	)	)	PUNCT
cana-2204	138	29	]	]	PUNCT
cana-2204	139	1	−	−	PROPN
cana-2204	139	2	𝑟	𝑟	NOUN
cana-2204	139	3	(	(	PUNCT
cana-2204	139	4	𝑛	𝑛	PRON
cana-2204	139	5	−	−	NOUN
cana-2204	139	6	1)(𝑛	1)(𝑛	NUM
cana-2204	139	7	−	−	NOUN
cana-2204	139	8	2	2	NUM
cana-2204	139	9	)	)	PUNCT
cana-2204	140	1	[	[	X
cana-2204	140	2	𝑔(𝑌	𝑔(𝑌	NUM
cana-2204	140	3	,	,	PUNCT
cana-2204	140	4	𝑍)𝑔(𝑋	𝑍)𝑔(𝑋	NUM
cana-2204	140	5	,	,	PUNCT
cana-2204	140	6	𝜉	𝜉	NOUN
cana-2204	140	7	)	)	PUNCT
cana-2204	140	8	−	−	PRON
cana-2204	141	1	𝑔(𝑋	𝑔(𝑋	PROPN
cana-2204	141	2	,	,	PUNCT
cana-2204	141	3	𝑍)𝑔(𝑌	𝑍)𝑔(𝑌	X
cana-2204	141	4	,	,	PUNCT
cana-2204	141	5	𝜉	𝜉	NOUN
cana-2204	141	6	)	)	PUNCT
cana-2204	141	7	]	]	PUNCT
cana-2204	141	8	𝜖𝜂(𝑅(𝑋	𝜖𝜂(𝑅(𝑋	PROPN
cana-2204	141	9	,	,	PUNCT
cana-2204	141	10	𝑌)𝑍	𝑌)𝑍	NUM
cana-2204	141	11	)	)	PUNCT
cana-2204	141	12	=	=	SYM
cana-2204	141	13	1	1	NUM
cana-2204	141	14	(	(	PUNCT
cana-2204	141	15	𝑛	𝑛	PRON
cana-2204	141	16	−	−	PROPN
cana-2204	141	17	2	2	NUM
cana-2204	141	18	)	)	PUNCT
cana-2204	142	1	[	[	X
cana-2204	142	2	𝜖𝑆(𝑌	𝜖𝑆(𝑌	NOUN
cana-2204	142	3	,	,	PUNCT
cana-2204	142	4	𝑍)𝜂(𝑋	𝑍)𝜂(𝑋	ADJ
cana-2204	142	5	)	)	PUNCT
cana-2204	142	6	−	−	PROPN
cana-2204	142	7	𝜖𝑆(𝑋	𝜖𝑆(𝑋	PROPN
cana-2204	142	8	,	,	PUNCT
cana-2204	142	9	𝑍)𝜂(𝑌	𝑍)𝜂(𝑌	X
cana-2204	142	10	)	)	PUNCT
cana-2204	143	1	+	+	SYM
cana-2204	143	2	𝑔(𝑌	𝑔(𝑌	PROPN
cana-2204	143	3	,	,	PUNCT
cana-2204	143	4	𝑍)𝑆(𝑋	𝑍)𝑆(𝑋	PROPN
cana-2204	143	5	,	,	PUNCT
cana-2204	143	6	𝜉	𝜉	NOUN
cana-2204	143	7	)	)	PUNCT
cana-2204	143	8	−	−	NOUN
cana-2204	143	9	𝑔(𝑋	𝑔(𝑋	PROPN
cana-2204	143	10	,	,	PUNCT
cana-2204	143	11	𝑍)𝑆(𝑌	𝑍)𝑆(𝑌	NOUN
cana-2204	143	12	,	,	PUNCT
cana-2204	143	13	𝜉	𝜉	NOUN
cana-2204	143	14	)	)	PUNCT
cana-2204	143	15	]	]	PUNCT
cana-2204	144	1	−	−	PROPN
cana-2204	144	2	𝑟	𝑟	X
cana-2204	144	3	(	(	PUNCT
cana-2204	144	4	𝑛−1)(𝑛−2	𝑛−1)(𝑛−2	NOUN
cana-2204	144	5	)	)	PUNCT
cana-2204	145	1	[	[	X
cana-2204	145	2	𝜖𝑔(𝑌	𝜖𝑔(𝑌	NOUN
cana-2204	145	3	,	,	PUNCT
cana-2204	145	4	𝑍)𝜂(𝑋	𝑍)𝜂(𝑋	NUM
cana-2204	145	5	)	)	PUNCT
cana-2204	145	6	−	−	PRON
cana-2204	145	7	𝜖𝑔(𝑋	𝜖𝑔(𝑋	PROPN
cana-2204	145	8	,	,	PUNCT
cana-2204	145	9	𝑍)𝜂(𝑌	𝑍)𝜂(𝑌	PROPN
cana-2204	145	10	)	)	PUNCT
cana-2204	145	11	]	]	PUNCT
cana-2204	145	12	(	(	PUNCT
cana-2204	145	13	4.3	4.3	NUM
cana-2204	145	14	)	)	PUNCT
cana-2204	145	15	using	use	VERB
cana-2204	145	16	equation	equation	NOUN
cana-2204	145	17	(	(	PUNCT
cana-2204	145	18	3.15	3.15	NUM
cana-2204	145	19	)	)	PUNCT
cana-2204	145	20	,	,	PUNCT
cana-2204	145	21	(	(	PUNCT
cana-2204	145	22	3.18	3.18	NUM
cana-2204	145	23	)	)	PUNCT
cana-2204	145	24	and	and	CCONJ
cana-2204	145	25	(	(	PUNCT
cana-2204	145	26	4.3	4.3	NUM
cana-2204	145	27	)	)	PUNCT
cana-2204	145	28	,	,	PUNCT
cana-2204	145	29	we	we	PRON
cana-2204	145	30	obtained	obtain	VERB
cana-2204	145	31	(	(	PUNCT
cana-2204	145	32	𝛼2	𝛼2	PROPN
cana-2204	145	33	+	+	SYM
cana-2204	145	34	𝛽2){𝜂(𝑋)𝑔(𝑌	𝛽2){𝜂(𝑋)𝑔(𝑌	PROPN
cana-2204	145	35	,	,	PUNCT
cana-2204	145	36	𝑍	𝑍	PROPN
cana-2204	145	37	)	)	PUNCT
cana-2204	145	38	−	−	NOUN
cana-2204	145	39	𝜂(𝑌)𝑔(𝑋	𝜂(𝑌)𝑔(𝑋	NOUN
cana-2204	145	40	,	,	PUNCT
cana-2204	145	41	𝑍	𝑍	NOUN
cana-2204	145	42	)	)	PUNCT
cana-2204	145	43	}	}	PUNCT
cana-2204	145	44	=	=	SYM
cana-2204	145	45	𝜖	𝜖	X
cana-2204	145	46	(	(	PUNCT
cana-2204	145	47	𝑛	𝑛	PROPN
cana-2204	145	48	−	−	PROPN
cana-2204	145	49	2	2	NUM
cana-2204	145	50	)	)	PUNCT
cana-2204	146	1	[	[	X
cana-2204	146	2	𝑆(𝑌	𝑆(𝑌	NOUN
cana-2204	146	3	,	,	PUNCT
cana-2204	146	4	𝑍)𝜂(𝑋	𝑍)𝜂(𝑋	X
cana-2204	146	5	)	)	PUNCT
cana-2204	146	6	−	−	PROPN
cana-2204	146	7	𝑆(𝑋	𝑆(𝑋	PROPN
cana-2204	146	8	,	,	PUNCT
cana-2204	146	9	𝑍)𝜂(𝑌	𝑍)𝜂(𝑌	X
cana-2204	146	10	)	)	PUNCT
cana-2204	147	1	+	+	CCONJ
cana-2204	147	2	(	(	PUNCT
cana-2204	147	3	𝑛	𝑛	DET
cana-2204	147	4	−	−	PROPN
cana-2204	147	5	1)(𝛼2	1)(𝛼2	NUM
cana-2204	147	6	+	+	CCONJ
cana-2204	147	7	𝛽2)𝑔(𝑌	𝛽2)𝑔(𝑌	PROPN
cana-2204	147	8	,	,	PUNCT
cana-2204	147	9	𝑍)𝜂(𝑋	𝑍)𝜂(𝑋	ADJ
cana-2204	147	10	)	)	PUNCT
cana-2204	147	11	communications	communication	NOUN
cana-2204	147	12	on	on	ADP
cana-2204	147	13	applied	apply	VERB
cana-2204	147	14	nonlinear	nonlinear	ADJ
cana-2204	147	15	analysis	analysis	NOUN
cana-2204	147	16	issn	issn	NOUN
cana-2204	147	17	:	:	PUNCT
cana-2204	147	18	1074	1074	NUM
cana-2204	147	19	-	-	PUNCT
cana-2204	147	20	133x	133x	NUM
cana-2204	147	21	vol	vol	NOUN
cana-2204	147	22	32	32	NUM
cana-2204	147	23	no	no	NOUN
cana-2204	147	24	.	.	PUNCT
cana-2204	148	1	1s	1s	NUM
cana-2204	148	2	(	(	PUNCT
cana-2204	148	3	2025	2025	NUM
cana-2204	148	4	)	)	PUNCT
cana-2204	148	5	407	407	NUM
cana-2204	148	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2204	148	7	−(𝑛	−(𝑛	NOUN
cana-2204	148	8	−	−	PROPN
cana-2204	148	9	1)(𝛼2	1)(𝛼2	NUM
cana-2204	149	1	+	+	CCONJ
cana-2204	149	2	𝛽2)𝑔(𝑋	𝛽2)𝑔(𝑋	PROPN
cana-2204	149	3	,	,	PUNCT
cana-2204	149	4	𝑍)𝜂(𝑌	𝑍)𝜂(𝑌	X
cana-2204	149	5	)	)	PUNCT
cana-2204	149	6	]	]	PUNCT
cana-2204	150	1	−	−	PROPN
cana-2204	150	2	𝜖𝑟	𝜖𝑟	X
cana-2204	150	3	(	(	PUNCT
cana-2204	150	4	𝑛	𝑛	PRON
cana-2204	150	5	−	−	NOUN
cana-2204	150	6	1)(𝑛	1)(𝑛	NUM
cana-2204	150	7	−	−	NOUN
cana-2204	150	8	2	2	NUM
cana-2204	150	9	)	)	PUNCT
cana-2204	151	1	[	[	X
cana-2204	151	2	𝑔(𝑌	𝑔(𝑌	NUM
cana-2204	151	3	,	,	PUNCT
cana-2204	151	4	𝑍)𝜂(𝑋	𝑍)𝜂(𝑋	ADJ
cana-2204	151	5	)	)	PUNCT
cana-2204	151	6	−	−	PRON
cana-2204	151	7	𝑔(𝑋	𝑔(𝑋	PROPN
cana-2204	151	8	,	,	PUNCT
cana-2204	151	9	𝑍)𝜂(𝑌	𝑍)𝜂(𝑌	PROPN
cana-2204	151	10	)	)	PUNCT
cana-2204	151	11	]	]	PUNCT
cana-2204	152	1	𝑆(𝑌	𝑆(𝑌	NOUN
cana-2204	152	2	,	,	PUNCT
cana-2204	152	3	𝑍)𝜂(𝑋	𝑍)𝜂(𝑋	X
cana-2204	152	4	)	)	PUNCT
cana-2204	152	5	=	=	SYM
cana-2204	152	6	𝑆(𝑋	𝑆(𝑋	PROPN
cana-2204	152	7	,	,	PUNCT
cana-2204	152	8	𝑍)𝜂(𝑌	𝑍)𝜂(𝑌	X
cana-2204	152	9	)	)	PUNCT
cana-2204	152	10	+	+	CCONJ
cana-2204	152	11	{	{	PUNCT
cana-2204	152	12	𝑟	𝑟	X
cana-2204	152	13	(	(	PUNCT
cana-2204	152	14	𝑛−1	𝑛−1	NUM
cana-2204	152	15	)	)	PUNCT
cana-2204	152	16	−	−	NOUN
cana-2204	152	17	𝜖(𝛼2	𝜖(𝛼2	NOUN
cana-2204	152	18	+	+	SYM
cana-2204	152	19	𝛽2	𝛽2	NOUN
cana-2204	152	20	)	)	PUNCT
cana-2204	152	21	}	}	PUNCT
cana-2204	152	22	{	{	PUNCT
cana-2204	152	23	𝑔(𝑌	𝑔(𝑌	PROPN
cana-2204	152	24	,	,	PUNCT
cana-2204	152	25	𝑍)𝜂(𝑋	𝑍)𝜂(𝑋	ADJ
cana-2204	152	26	)	)	PUNCT
cana-2204	152	27	−	−	PRON
cana-2204	152	28	𝑔(𝑋	𝑔(𝑋	PROPN
cana-2204	152	29	,	,	PUNCT
cana-2204	152	30	𝑍)𝜂(𝑌	𝑍)𝜂(𝑌	PROPN
cana-2204	152	31	)	)	PUNCT
cana-2204	152	32	}	}	PUNCT
cana-2204	152	33	(	(	PUNCT
cana-2204	152	34	4.4	4.4	NUM
cana-2204	152	35	)	)	PUNCT
cana-2204	152	36	using	use	VERB
cana-2204	152	37	equation	equation	NOUN
cana-2204	152	38	(	(	PUNCT
cana-2204	152	39	3.18	3.18	NUM
cana-2204	152	40	)	)	PUNCT
cana-2204	152	41	and	and	CCONJ
cana-2204	152	42	replacing	replace	VERB
cana-2204	152	43	𝑋	𝑋	NOUN
cana-2204	152	44	=	=	PUNCT
cana-2204	152	45	𝜉	𝜉	NOUN
cana-2204	152	46	in	in	ADP
cana-2204	152	47	(	(	PUNCT
cana-2204	152	48	4.4	4.4	NUM
cana-2204	152	49	)	)	PUNCT
cana-2204	152	50	,	,	PUNCT
cana-2204	152	51	we	we	PRON
cana-2204	152	52	obtained	obtain	VERB
cana-2204	152	53	𝑆(𝑌	𝑆(𝑌	NOUN
cana-2204	152	54	,	,	PUNCT
cana-2204	152	55	𝑍	𝑍	PROPN
cana-2204	152	56	)	)	PUNCT
cana-2204	152	57	=	=	SYM
cana-2204	152	58	{	{	PUNCT
cana-2204	152	59	𝑟	𝑟	X
cana-2204	152	60	(	(	PUNCT
cana-2204	152	61	𝑛−1	𝑛−1	NUM
cana-2204	152	62	)	)	PUNCT
cana-2204	153	1	−	−	NOUN
cana-2204	153	2	𝜖(𝛼2	𝜖(𝛼2	NOUN
cana-2204	153	3	+	+	SYM
cana-2204	153	4	𝛽2	𝛽2	NOUN
cana-2204	153	5	)	)	PUNCT
cana-2204	153	6	}	}	PUNCT
cana-2204	153	7	𝑔(𝑌	𝑔(𝑌	NUM
cana-2204	153	8	,	,	PUNCT
cana-2204	153	9	𝑍	𝑍	PROPN
cana-2204	153	10	)	)	PUNCT
cana-2204	153	11	+	+	CCONJ
cana-2204	153	12	{	{	PUNCT
cana-2204	153	13	𝑟	𝑟	X
cana-2204	153	14	(	(	PUNCT
cana-2204	153	15	𝑛−1	𝑛−1	NUM
cana-2204	153	16	)	)	PUNCT
cana-2204	153	17	−	−	NOUN
cana-2204	153	18	𝜖𝑛(𝛼2	𝜖𝑛(𝛼2	ADP
cana-2204	153	19	+	+	CCONJ
cana-2204	153	20	𝛽2	𝛽2	NOUN
cana-2204	153	21	)	)	PUNCT
cana-2204	153	22	}	}	PUNCT
cana-2204	153	23	𝜂(𝑌)𝜂(𝑍	𝜂(𝑌)𝜂(𝑍	NOUN
cana-2204	153	24	)	)	PUNCT
cana-2204	153	25	(	(	PUNCT
cana-2204	153	26	4.5	4.5	NUM
cana-2204	153	27	)	)	PUNCT
cana-2204	153	28	hence	hence	ADV
cana-2204	153	29	,	,	PUNCT
cana-2204	153	30	we	we	PRON
cana-2204	153	31	can	can	AUX
cana-2204	153	32	state	state	VERB
cana-2204	153	33	the	the	DET
cana-2204	153	34	following	follow	VERB
cana-2204	153	35	using	use	VERB
cana-2204	153	36	definition	definition	NOUN
cana-2204	153	37	(	(	PUNCT
cana-2204	153	38	2.2	2.2	NUM
cana-2204	153	39	)	)	PUNCT
cana-2204	153	40	.	.	PUNCT
cana-2204	154	1	theorem	theorem	VERB
cana-2204	154	2	4.1.an	4.1.an	PRON
cana-2204	154	3	𝑛	𝑛	PROPN
cana-2204	155	1	−dimensional(𝑛	−dimensional(𝑛	NOUN
cana-2204	155	2	>	>	SYM
cana-2204	155	3	1	1	X
cana-2204	155	4	)	)	PUNCT
cana-2204	155	5	conformally	conformally	ADV
cana-2204	155	6	flat	flat	ADJ
cana-2204	155	7	(	(	PUNCT
cana-2204	155	8	𝜖	𝜖	NOUN
cana-2204	155	9	)	)	PUNCT
cana-2204	155	10	−	−	PROPN
cana-2204	155	11	lorentzian	lorentzian	ADJ
cana-2204	155	12	para	para	NOUN
cana-2204	155	13	-	-	PUNCT
cana-2204	155	14	sasakian	sasakian	NOUN
cana-2204	155	15	manifold	manifold	NOUN
cana-2204	155	16	with	with	ADP
cana-2204	155	17	parallelized	parallelize	VERB
cana-2204	155	18	generalized	generalize	VERB
cana-2204	155	19	symmetric	symmetric	ADJ
cana-2204	155	20	metric	metric	ADJ
cana-2204	155	21	connection	connection	NOUN
cana-2204	155	22	is	be	AUX
cana-2204	155	23	an	an	DET
cana-2204	155	24	𝜂	𝜂	NOUN
cana-2204	155	25	−einstein	−einstein	NOUN
cana-2204	155	26	manifold	manifold	NOUN
cana-2204	155	27	.	.	PUNCT
cana-2204	156	1	using	use	VERB
cana-2204	156	2	equation	equation	NOUN
cana-2204	156	3	(	(	PUNCT
cana-2204	156	4	4.5	4.5	NUM
cana-2204	156	5	)	)	PUNCT
cana-2204	156	6	in	in	ADP
cana-2204	156	7	(	(	PUNCT
cana-2204	156	8	4.2	4.2	NUM
cana-2204	156	9	)	)	PUNCT
cana-2204	156	10	,	,	PUNCT
cana-2204	156	11	we	we	PRON
cana-2204	156	12	get	get	VERB
cana-2204	156	13	𝑔(𝑅(𝑋	𝑔(𝑅(𝑋	NOUN
cana-2204	156	14	,	,	PUNCT
cana-2204	156	15	𝑌)𝑍,𝑊	𝑌)𝑍,𝑊	ADJ
cana-2204	156	16	)	)	PUNCT
cana-2204	156	17	=	=	SYM
cana-2204	156	18	1	1	NUM
cana-2204	156	19	(	(	PUNCT
cana-2204	156	20	𝑛	𝑛	PRON
cana-2204	156	21	−	−	PROPN
cana-2204	156	22	2	2	NUM
cana-2204	156	23	)	)	PUNCT
cana-2204	157	1	[	[	X
cana-2204	157	2	{	{	PUNCT
cana-2204	157	3	{	{	PUNCT
cana-2204	157	4	𝑟	𝑟	X
cana-2204	157	5	(	(	PUNCT
cana-2204	157	6	𝑛	𝑛	PRON
cana-2204	157	7	−	−	PROPN
cana-2204	157	8	1	1	NUM
cana-2204	157	9	)	)	PUNCT
cana-2204	157	10	−	−	NOUN
cana-2204	157	11	𝜖(𝛼2	𝜖(𝛼2	NOUN
cana-2204	157	12	+	+	CCONJ
cana-2204	157	13	𝛽2	𝛽2	NOUN
cana-2204	157	14	)	)	PUNCT
cana-2204	157	15	}	}	PUNCT
cana-2204	157	16	𝑔(𝑌	𝑔(𝑌	NUM
cana-2204	157	17	,	,	PUNCT
cana-2204	157	18	𝑍	𝑍	PROPN
cana-2204	157	19	)	)	PUNCT
cana-2204	157	20	+	+	CCONJ
cana-2204	157	21	{	{	PUNCT
cana-2204	157	22	𝑟	𝑟	NOUN
cana-2204	157	23	(	(	PUNCT
cana-2204	157	24	𝑛	𝑛	PRON
cana-2204	157	25	−	−	PROPN
cana-2204	157	26	1	1	NUM
cana-2204	157	27	)	)	PUNCT
cana-2204	157	28	−	−	NOUN
cana-2204	157	29	𝜖𝑛(𝛼2	𝜖𝑛(𝛼2	ADP
cana-2204	157	30	+	+	X
cana-2204	157	31	𝛽2	𝛽2	NOUN
cana-2204	157	32	)	)	PUNCT
cana-2204	157	33	}	}	PUNCT
cana-2204	157	34	𝜂(𝑌)𝜂(𝑍)}𝑔(𝑋,𝑊	𝜂(𝑌)𝜂(𝑍)}𝑔(𝑋,𝑊	PROPN
cana-2204	157	35	)	)	PUNCT
cana-2204	157	36	−	−	NOUN
cana-2204	157	37	{	{	PUNCT
cana-2204	157	38	{	{	PUNCT
cana-2204	157	39	𝑟	𝑟	X
cana-2204	157	40	(	(	PUNCT
cana-2204	157	41	𝑛	𝑛	PRON
cana-2204	157	42	−	−	PROPN
cana-2204	157	43	1	1	NUM
cana-2204	157	44	)	)	PUNCT
cana-2204	157	45	−	−	NOUN
cana-2204	157	46	𝜖(𝛼2	𝜖(𝛼2	NOUN
cana-2204	157	47	+	+	CCONJ
cana-2204	157	48	𝛽2	𝛽2	NOUN
cana-2204	157	49	)	)	PUNCT
cana-2204	157	50	}	}	PUNCT
cana-2204	157	51	𝑔(𝑋	𝑔(𝑋	PROPN
cana-2204	157	52	,	,	PUNCT
cana-2204	157	53	𝑍	𝑍	PROPN
cana-2204	157	54	)	)	PUNCT
cana-2204	157	55	+	+	CCONJ
cana-2204	157	56	{	{	PUNCT
cana-2204	157	57	𝑟	𝑟	NOUN
cana-2204	157	58	(	(	PUNCT
cana-2204	157	59	𝑛	𝑛	PRON
cana-2204	157	60	−	−	PROPN
cana-2204	157	61	1	1	NUM
cana-2204	157	62	)	)	PUNCT
cana-2204	157	63	−	−	NOUN
cana-2204	157	64	𝜖𝑛(𝛼2	𝜖𝑛(𝛼2	ADP
cana-2204	157	65	+	+	X
cana-2204	157	66	𝛽2	𝛽2	NOUN
cana-2204	157	67	)	)	PUNCT
cana-2204	157	68	}	}	PUNCT
cana-2204	157	69	𝜂(𝑋)𝜂(𝑍)}𝑔(𝑌,𝑊	𝜂(𝑋)𝜂(𝑍)}𝑔(𝑌,𝑊	NOUN
cana-2204	157	70	)	)	PUNCT
cana-2204	157	71	+	+	NOUN
cana-2204	157	72	{	{	PUNCT
cana-2204	157	73	{	{	PUNCT
cana-2204	157	74	𝑟	𝑟	X
cana-2204	157	75	(	(	PUNCT
cana-2204	157	76	𝑛	𝑛	PRON
cana-2204	157	77	−	−	PROPN
cana-2204	157	78	1	1	NUM
cana-2204	157	79	)	)	PUNCT
cana-2204	157	80	−	−	NOUN
cana-2204	157	81	𝜖(𝛼2	𝜖(𝛼2	NOUN
cana-2204	157	82	+	+	CCONJ
cana-2204	157	83	𝛽2	𝛽2	NOUN
cana-2204	157	84	)	)	PUNCT
cana-2204	157	85	}	}	PUNCT
cana-2204	157	86	𝑔(𝑋,𝑊	𝑔(𝑋,𝑊	NOUN
cana-2204	157	87	)	)	PUNCT
cana-2204	158	1	+	+	CCONJ
cana-2204	158	2	{	{	PUNCT
cana-2204	158	3	𝑟	𝑟	NOUN
cana-2204	158	4	(	(	PUNCT
cana-2204	158	5	𝑛	𝑛	PRON
cana-2204	158	6	−	−	PROPN
cana-2204	158	7	1	1	NUM
cana-2204	158	8	)	)	PUNCT
cana-2204	158	9	−	−	NOUN
cana-2204	158	10	𝜖𝑛(𝛼2	𝜖𝑛(𝛼2	ADP
cana-2204	158	11	+	+	CCONJ
cana-2204	158	12	𝛽2	𝛽2	NOUN
cana-2204	158	13	)	)	PUNCT
cana-2204	158	14	}	}	PUNCT
cana-2204	158	15	𝜂(𝑋)𝜂(𝑊)}𝑔(𝑌	𝜂(𝑋)𝜂(𝑊)}𝑔(𝑌	NOUN
cana-2204	158	16	,	,	PUNCT
cana-2204	158	17	𝑍	𝑍	PROPN
cana-2204	158	18	)	)	PUNCT
cana-2204	158	19	{	{	PUNCT
cana-2204	158	20	𝑟	𝑟	NOUN
cana-2204	158	21	(	(	PUNCT
cana-2204	158	22	𝑛	𝑛	PRON
cana-2204	158	23	−	−	PROPN
cana-2204	158	24	1	1	NUM
cana-2204	158	25	)	)	PUNCT
cana-2204	158	26	−	−	NOUN
cana-2204	158	27	𝜖(𝛼2	𝜖(𝛼2	NOUN
cana-2204	158	28	+	+	CCONJ
cana-2204	158	29	𝛽2	𝛽2	NOUN
cana-2204	158	30	)	)	PUNCT
cana-2204	158	31	}	}	PUNCT
cana-2204	158	32	𝑔(𝑌,𝑊	𝑔(𝑌,𝑊	NOUN
cana-2204	158	33	)	)	PUNCT
cana-2204	159	1	+	+	CCONJ
cana-2204	159	2	{	{	PUNCT
cana-2204	159	3	𝑟	𝑟	NOUN
cana-2204	159	4	(	(	PUNCT
cana-2204	159	5	𝑛	𝑛	PRON
cana-2204	159	6	−	−	PROPN
cana-2204	159	7	1	1	NUM
cana-2204	159	8	)	)	PUNCT
cana-2204	159	9	−	−	NOUN
cana-2204	159	10	𝜖𝑛(𝛼2	𝜖𝑛(𝛼2	ADP
cana-2204	159	11	+	+	X
cana-2204	159	12	𝛽2	𝛽2	NOUN
cana-2204	159	13	)	)	PUNCT
cana-2204	159	14	}	}	PUNCT
cana-2204	159	15	𝜂(𝑌)𝜂(𝑊)}𝑔(𝑋	𝜂(𝑌)𝜂(𝑊)}𝑔(𝑋	NUM
cana-2204	159	16	,	,	PUNCT
cana-2204	159	17	𝑍	𝑍	PROPN
cana-2204	159	18	)	)	PUNCT
cana-2204	159	19	]	]	PUNCT
cana-2204	159	20	−	−	PROPN
cana-2204	159	21	𝑟	𝑟	NOUN
cana-2204	159	22	(	(	PUNCT
cana-2204	159	23	𝑛	𝑛	PRON
cana-2204	159	24	−	−	NOUN
cana-2204	159	25	1)(𝑛	1)(𝑛	NUM
cana-2204	159	26	−	−	NOUN
cana-2204	159	27	2	2	NUM
cana-2204	159	28	)	)	PUNCT
cana-2204	160	1	[	[	X
cana-2204	160	2	𝑔(𝑌	𝑔(𝑌	NUM
cana-2204	160	3	,	,	PUNCT
cana-2204	160	4	𝑍)𝑔(𝑋,𝑊	𝑍)𝑔(𝑋,𝑊	NUM
cana-2204	160	5	)	)	PUNCT
cana-2204	160	6	−	−	PRON
cana-2204	160	7	𝑔(𝑋	𝑔(𝑋	PROPN
cana-2204	160	8	,	,	PUNCT
cana-2204	160	9	𝑍)𝑔(𝑌,𝑊	𝑍)𝑔(𝑌,𝑊	X
cana-2204	160	10	)	)	PUNCT
cana-2204	160	11	]	]	PUNCT
cana-2204	161	1	𝑔(𝑅(𝑋	𝑔(𝑅(𝑋	X
cana-2204	161	2	,	,	PUNCT
cana-2204	161	3	𝑌)𝑍,𝑊	𝑌)𝑍,𝑊	ADJ
cana-2204	161	4	)	)	PUNCT
cana-2204	161	5	=	=	SYM
cana-2204	161	6	{	{	PUNCT
cana-2204	161	7	𝑟	𝑟	X
cana-2204	161	8	−	−	X
cana-2204	161	9	2𝜖(𝑛	2𝜖(𝑛	NOUN
cana-2204	161	10	−	−	NOUN
cana-2204	161	11	1)(𝛼2	1)(𝛼2	NUM
cana-2204	161	12	+	+	NOUN
cana-2204	161	13	𝛽2	𝛽2	NOUN
cana-2204	161	14	)	)	PUNCT
cana-2204	161	15	(	(	PUNCT
cana-2204	161	16	𝑛	𝑛	PRON
cana-2204	161	17	−	−	NOUN
cana-2204	161	18	1)(𝑛	1)(𝑛	NUM
cana-2204	161	19	−	−	NOUN
cana-2204	161	20	2	2	NUM
cana-2204	161	21	)	)	PUNCT
cana-2204	161	22	}	}	PUNCT
cana-2204	161	23	{	{	PUNCT
cana-2204	161	24	𝑔(𝑌	𝑔(𝑌	PROPN
cana-2204	161	25	,	,	PUNCT
cana-2204	161	26	𝑍)𝑔(𝑋,𝑊	𝑍)𝑔(𝑋,𝑊	NUM
cana-2204	161	27	)	)	PUNCT
cana-2204	161	28	−	−	PRON
cana-2204	161	29	𝑔(𝑋	𝑔(𝑋	PROPN
cana-2204	161	30	,	,	PUNCT
cana-2204	161	31	𝑍)𝑔(𝑌,𝑊	𝑍)𝑔(𝑌,𝑊	X
cana-2204	161	32	)	)	PUNCT
cana-2204	161	33	}	}	PUNCT
cana-2204	162	1	+	+	CCONJ
cana-2204	162	2	{	{	PUNCT
cana-2204	162	3	𝑟	𝑟	NOUN
cana-2204	162	4	−	−	PROPN
cana-2204	162	5	𝜖𝑛(𝑛	𝜖𝑛(𝑛	NOUN
cana-2204	162	6	−	−	PROPN
cana-2204	162	7	1)(𝛼2	1)(𝛼2	NUM
cana-2204	162	8	+	+	SYM
cana-2204	162	9	𝛽2	𝛽2	NOUN
cana-2204	162	10	)	)	PUNCT
cana-2204	162	11	(	(	PUNCT
cana-2204	162	12	𝑛	𝑛	PRON
cana-2204	162	13	−	−	NOUN
cana-2204	162	14	1)(𝑛	1)(𝑛	NUM
cana-2204	162	15	−	−	NOUN
cana-2204	162	16	2	2	NUM
cana-2204	162	17	)	)	PUNCT
cana-2204	162	18	}	}	PUNCT
cana-2204	162	19	{	{	PUNCT
cana-2204	162	20	𝜂(𝑌)𝜂(𝑍)𝑔(𝑋,𝑊	𝜂(𝑌)𝜂(𝑍)𝑔(𝑋,𝑊	ADJ
cana-2204	162	21	)	)	PUNCT
cana-2204	162	22	−𝜂(𝑋)𝜂(𝑍)𝑔(𝑌,𝑊	−𝜂(𝑋)𝜂(𝑍)𝑔(𝑌,𝑊	NOUN
cana-2204	162	23	)	)	PUNCT
cana-2204	163	1	+	+	ADJ
cana-2204	163	2	𝜂(𝑋)𝜂(𝑊)𝑔(𝑌	𝜂(𝑋)𝜂(𝑊)𝑔(𝑌	ADJ
cana-2204	163	3	,	,	PUNCT
cana-2204	163	4	𝑍	𝑍	NOUN
cana-2204	163	5	)	)	PUNCT
cana-2204	163	6	−𝜂(𝑌)𝜂(𝑊)}𝑔(𝑋	−𝜂(𝑌)𝜂(𝑊)}𝑔(𝑋	PROPN
cana-2204	163	7	,	,	PUNCT
cana-2204	163	8	𝑍	𝑍	NOUN
cana-2204	163	9	)	)	PUNCT
cana-2204	163	10	}	}	PUNCT
cana-2204	163	11	in	in	ADP
cana-2204	163	12	view	view	NOUN
cana-2204	163	13	of	of	ADP
cana-2204	163	14	definition	definition	NOUN
cana-2204	163	15	(	(	PUNCT
cana-2204	163	16	2.1	2.1	NUM
cana-2204	163	17	)	)	PUNCT
cana-2204	163	18	and	and	CCONJ
cana-2204	163	19	above	above	ADP
cana-2204	163	20	relation	relation	NOUN
cana-2204	163	21	,	,	PUNCT
cana-2204	163	22	we	we	PRON
cana-2204	163	23	have	have	VERB
cana-2204	163	24	the	the	DET
cana-2204	163	25	following	following	NOUN
cana-2204	163	26	.	.	PUNCT
cana-2204	164	1	communications	communication	NOUN
cana-2204	164	2	on	on	ADP
cana-2204	164	3	applied	apply	VERB
cana-2204	164	4	nonlinear	nonlinear	ADJ
cana-2204	164	5	analysis	analysis	NOUN
cana-2204	164	6	issn	issn	NOUN
cana-2204	164	7	:	:	PUNCT
cana-2204	164	8	1074	1074	NUM
cana-2204	164	9	-	-	PUNCT
cana-2204	164	10	133x	133x	NUM
cana-2204	164	11	vol	vol	NOUN
cana-2204	164	12	32	32	NUM
cana-2204	164	13	no	no	NOUN
cana-2204	164	14	.	.	PUNCT
cana-2204	165	1	1s	1s	NUM
cana-2204	165	2	(	(	PUNCT
cana-2204	165	3	2025	2025	NUM
cana-2204	165	4	)	)	PUNCT
cana-2204	165	5	408	408	NUM
cana-2204	165	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2204	165	7	theorem	theorem	VERB
cana-2204	165	8	4.2.an	4.2.an	NUM
cana-2204	165	9	𝑛	𝑛	PRON
cana-2204	165	10	−dimensional	−dimensional	VERB
cana-2204	165	11	(	(	PUNCT
cana-2204	165	12	𝑛	𝑛	PART
cana-2204	165	13	>	>	SYM
cana-2204	165	14	1	1	NUM
cana-2204	165	15	)	)	PUNCT
cana-2204	165	16	conformally	conformally	ADV
cana-2204	165	17	flat	flat	ADJ
cana-2204	165	18	(	(	PUNCT
cana-2204	165	19	𝜖	𝜖	NOUN
cana-2204	165	20	)	)	PUNCT
cana-2204	165	21	−	−	PROPN
cana-2204	165	22	lorentzian	lorentzian	ADJ
cana-2204	165	23	para	para	NOUN
cana-2204	165	24	-	-	PUNCT
cana-2204	165	25	sasakian	sasakian	NOUN
cana-2204	165	26	manifold	manifold	NOUN
cana-2204	165	27	with	with	ADP
cana-2204	165	28	parallelized	parallelize	VERB
cana-2204	165	29	generalized	generalize	VERB
cana-2204	165	30	symmetric	symmetric	ADJ
cana-2204	165	31	metric	metric	ADJ
cana-2204	165	32	connection	connection	NOUN
cana-2204	165	33	is	be	AUX
cana-2204	165	34	of	of	ADP
cana-2204	165	35	quasi	quasi	ADJ
cana-2204	165	36	-	-	ADJ
cana-2204	165	37	constant	constant	ADJ
cana-2204	165	38	curvature	curvature	NOUN
cana-2204	165	39	.	.	PUNCT
cana-2204	166	1	it	it	PRON
cana-2204	166	2	is	be	AUX
cana-2204	166	3	already	already	ADV
cana-2204	166	4	proved	prove	VERB
cana-2204	166	5	that	that	SCONJ
cana-2204	166	6	𝜓(𝐹)𝑛	𝜓(𝐹)𝑛	PUNCT
cana-2204	166	7	contains	contain	VERB
cana-2204	166	8	a	a	DET
cana-2204	166	9	manifold	manifold	NOUN
cana-2204	166	10	of	of	ADP
cana-2204	166	11	quasi	quasi	ADJ
cana-2204	166	12	-	-	ADJ
cana-2204	166	13	constant	constant	ADJ
cana-2204	166	14	curvature	curvature	NOUN
cana-2204	166	15	as	as	ADP
cana-2204	166	16	a	a	DET
cana-2204	166	17	subclass	subclass	NOUN
cana-2204	166	18	.	.	PUNCT
cana-2204	167	1	let	let	VERB
cana-2204	167	2	us	we	PRON
cana-2204	167	3	suppose	suppose	VERB
cana-2204	168	1	𝐹(𝑋	𝐹(𝑋	PROPN
cana-2204	168	2	,	,	PUNCT
cana-2204	168	3	𝑌	𝑌	PROPN
cana-2204	168	4	)	)	PUNCT
cana-2204	168	5	=	=	SYM
cana-2204	168	6	𝑝𝑔(𝑋	𝑝𝑔(𝑋	PROPN
cana-2204	168	7	,	,	PUNCT
cana-2204	168	8	𝑌	𝑌	PROPN
cana-2204	168	9	)	)	PUNCT
cana-2204	168	10	+	+	NUM
cana-2204	168	11	𝑞	𝑞	X
cana-2204	168	12	(	(	PUNCT
cana-2204	168	13	𝑋)𝜂(𝑌	𝑋)𝜂(𝑌	NOUN
cana-2204	168	14	)	)	PUNCT
cana-2204	168	15	(	(	PUNCT
cana-2204	168	16	4.6	4.6	NUM
cana-2204	168	17	)	)	PUNCT
cana-2204	168	18	where	where	SCONJ
cana-2204	168	19	,	,	PUNCT
cana-2204	168	20	𝑝	𝑝	NOUN
cana-2204	168	21	=	=	PUNCT
cana-2204	168	22	√	√	PROPN
cana-2204	168	23	{	{	PUNCT
cana-2204	168	24	𝑟	𝑟	NOUN
cana-2204	168	25	−	−	PROPN
cana-2204	168	26	2𝜖(𝑛	2𝜖(𝑛	NOUN
cana-2204	168	27	−	−	NOUN
cana-2204	168	28	1)(𝛼2	1)(𝛼2	NUM
cana-2204	168	29	+	+	NOUN
cana-2204	168	30	𝛽2	𝛽2	NOUN
cana-2204	168	31	)	)	PUNCT
cana-2204	168	32	(	(	PUNCT
cana-2204	168	33	𝑛	𝑛	PRON
cana-2204	168	34	−	−	NOUN
cana-2204	168	35	1)(𝑛	1)(𝑛	NUM
cana-2204	168	36	−	−	NOUN
cana-2204	168	37	2	2	NUM
cana-2204	168	38	)	)	PUNCT
cana-2204	168	39	}	}	PUNCT
cana-2204	168	40	and	and	CCONJ
cana-2204	168	41	𝑞	𝑞	X
cana-2204	168	42	=	=	PUNCT
cana-2204	168	43	{	{	PUNCT
cana-2204	168	44	𝑟	𝑟	NOUN
cana-2204	168	45	−	−	PROPN
cana-2204	168	46	𝜖𝑛(𝑛	𝜖𝑛(𝑛	NOUN
cana-2204	168	47	−	−	PROPN
cana-2204	168	48	1)(𝛼2	1)(𝛼2	NUM
cana-2204	168	49	+	+	SYM
cana-2204	168	50	𝛽2	𝛽2	NOUN
cana-2204	168	51	)	)	PUNCT
cana-2204	168	52	(	(	PUNCT
cana-2204	168	53	𝑛	𝑛	PRON
cana-2204	168	54	−	−	NOUN
cana-2204	168	55	1)(𝑛	1)(𝑛	NUM
cana-2204	168	56	−	−	NOUN
cana-2204	168	57	2	2	NUM
cana-2204	168	58	)	)	PUNCT
cana-2204	168	59	}	}	PUNCT
cana-2204	168	60	√	√	VERB
cana-2204	168	61	{	{	PUNCT
cana-2204	168	62	(	(	PUNCT
cana-2204	168	63	𝑛	𝑛	PRON
cana-2204	168	64	−	−	NOUN
cana-2204	168	65	1)(𝑛	1)(𝑛	NUM
cana-2204	168	66	−	−	NOUN
cana-2204	168	67	2	2	NUM
cana-2204	168	68	)	)	PUNCT
cana-2204	168	69	𝑟	𝑟	NOUN
cana-2204	168	70	−	−	PROPN
cana-2204	168	71	2𝜖(𝑛	2𝜖(𝑛	NOUN
cana-2204	168	72	−	−	NOUN
cana-2204	168	73	1)(𝛼2	1)(𝛼2	NUM
cana-2204	168	74	+	+	NOUN
cana-2204	168	75	𝛽2	𝛽2	NOUN
cana-2204	168	76	)	)	PUNCT
cana-2204	168	77	}	}	PUNCT
cana-2204	168	78	now	now	ADV
cana-2204	168	79	,	,	PUNCT
cana-2204	168	80	from	from	ADP
cana-2204	168	81	the	the	DET
cana-2204	168	82	equation	equation	NOUN
cana-2204	168	83	(	(	PUNCT
cana-2204	168	84	2.13	2.13	NUM
cana-2204	168	85	)	)	PUNCT
cana-2204	168	86	,	,	PUNCT
cana-2204	168	87	we	we	PRON
cana-2204	168	88	have	have	AUX
cana-2204	168	89	�	�	PROPN
cana-2204	168	90	̃	̃	NOUN
cana-2204	168	91	�	�	NOUN
cana-2204	168	92	(𝑋	(𝑋	NUM
cana-2204	168	93	,	,	PUNCT
cana-2204	168	94	𝑌	𝑌	PROPN
cana-2204	168	95	,	,	PUNCT
cana-2204	168	96	𝑍,𝑊	𝑍,𝑊	ADJ
cana-2204	168	97	)	)	PUNCT
cana-2204	168	98	=	=	SYM
cana-2204	168	99	𝐹(𝑌	𝐹(𝑌	NOUN
cana-2204	168	100	,	,	PUNCT
cana-2204	168	101	𝑍)𝐹(𝑋,𝑊	𝑍)𝐹(𝑋,𝑊	NUM
cana-2204	168	102	)	)	PUNCT
cana-2204	168	103	−	−	ADP
cana-2204	168	104	𝐹(𝑋	𝐹(𝑋	PROPN
cana-2204	168	105	,	,	PUNCT
cana-2204	168	106	𝑍)𝐹(𝑌,𝑊	𝑍)𝐹(𝑌,𝑊	PROPN
cana-2204	168	107	)	)	PUNCT
cana-2204	168	108	therefore	therefore	ADV
cana-2204	168	109	,	,	PUNCT
cana-2204	168	110	the	the	DET
cana-2204	168	111	manifold	manifold	NOUN
cana-2204	168	112	of	of	ADP
cana-2204	168	113	quasi	quasi	ADJ
cana-2204	168	114	-	-	NOUN
cana-2204	168	115	contact	contact	ADJ
cana-2204	168	116	curvature	curvature	NOUN
cana-2204	168	117	is	be	AUX
cana-2204	168	118	a	a	DET
cana-2204	168	119	𝜓(𝐹)𝑛.	𝜓(𝐹)𝑛.	NOUN
cana-2204	168	120	from	from	ADP
cana-2204	168	121	the	the	DET
cana-2204	168	122	above	above	ADJ
cana-2204	168	123	equation	equation	NOUN
cana-2204	168	124	&	&	CCONJ
cana-2204	168	125	theorem	theorem	VERB
cana-2204	168	126	4.2	4.2	NUM
cana-2204	168	127	,	,	PUNCT
cana-2204	168	128	we	we	PRON
cana-2204	168	129	have	have	VERB
cana-2204	168	130	the	the	DET
cana-2204	168	131	following	follow	VERB
cana-2204	168	132	result	result	VERB
cana-2204	168	133	theorem4.3.a	theorem4.3.a	PROPN
cana-2204	168	134	conformally	conformally	ADV
cana-2204	168	135	flat	flat	ADJ
cana-2204	168	136	(	(	PUNCT
cana-2204	168	137	𝜖	𝜖	NOUN
cana-2204	168	138	)	)	PUNCT
cana-2204	168	139	−	−	PROPN
cana-2204	168	140	lorentzian	lorentzian	ADJ
cana-2204	168	141	para	para	NOUN
cana-2204	168	142	-	-	PUNCT
cana-2204	168	143	sasakian	sasakian	NOUN
cana-2204	168	144	manifold	manifold	NOUN
cana-2204	168	145	with	with	ADP
cana-2204	168	146	generalized	generalized	ADJ
cana-2204	168	147	symmetric	symmetric	ADJ
cana-2204	168	148	metric	metric	ADJ
cana-2204	168	149	connection	connection	NOUN
cana-2204	168	150	is	be	AUX
cana-2204	168	151	a	a	DET
cana-2204	168	152	𝜓(𝐹)𝑛.	𝜓(𝐹)𝑛.	NOUN
cana-2204	168	153	5	5	NUM
cana-2204	168	154	.	.	PUNCT
cana-2204	169	1	weyl	weyl	VERB
cana-2204	169	2	-	-	PUNCT
cana-2204	169	3	semi	semi	ADV
cana-2204	169	4	-	-	ADJ
cana-2204	169	5	symmetric	symmetric	ADJ
cana-2204	169	6	(	(	PUNCT
cana-2204	169	7	𝝐	𝝐	NOUN
cana-2204	169	8	)	)	PUNCT
cana-2204	169	9	−	−	PROPN
cana-2204	169	10	lorentzian	lorentzian	ADJ
cana-2204	169	11	para	para	NOUN
cana-2204	169	12	-	-	PUNCT
cana-2204	169	13	sasakian	sasakian	NOUN
cana-2204	169	14	manifold	manifold	NOUN
cana-2204	169	15	with	with	ADP
cana-2204	169	16	parallelized	parallelize	VERB
cana-2204	169	17	generalized	generalize	VERB
cana-2204	169	18	symmetric	symmetric	ADJ
cana-2204	169	19	metric	metric	ADJ
cana-2204	169	20	connection	connection	NOUN
cana-2204	169	21	an	an	DET
cana-2204	169	22	(	(	PUNCT
cana-2204	169	23	𝜖	𝜖	NOUN
cana-2204	169	24	)	)	PUNCT
cana-2204	169	25	−	−	PROPN
cana-2204	169	26	lorentzian	lorentzian	ADJ
cana-2204	169	27	para	para	NOUN
cana-2204	169	28	-	-	PUNCT
cana-2204	169	29	sasakian	sasakian	PROPN
cana-2204	169	30	manifold	manifold	NOUN
cana-2204	169	31	is	be	AUX
cana-2204	169	32	said	say	VERB
cana-2204	169	33	to	to	PART
cana-2204	169	34	be	be	AUX
cana-2204	169	35	weyl	weyl	VERB
cana-2204	169	36	-	-	PUNCT
cana-2204	169	37	semi	semi	ADV
cana-2204	169	38	-	-	ADJ
cana-2204	169	39	symmetric	symmetric	ADJ
cana-2204	169	40	if	if	SCONJ
cana-2204	169	41	𝑅.	𝑅.	NOUN
cana-2204	169	42	𝐶	𝐶	NOUN
cana-2204	169	43	=	=	SYM
cana-2204	169	44	0	0	NUM
cana-2204	170	1	(	(	PUNCT
cana-2204	170	2	5.1	5.1	NUM
cana-2204	170	3	)	)	PUNCT
cana-2204	170	4	from	from	ADP
cana-2204	170	5	(	(	PUNCT
cana-2204	170	6	4.1	4.1	NUM
cana-2204	170	7	)	)	PUNCT
cana-2204	170	8	,	,	PUNCT
cana-2204	170	9	we	we	PRON
cana-2204	170	10	have	have	AUX
cana-2204	170	11	𝜂(𝐶(𝑋	𝜂(𝐶(𝑋	ADJ
cana-2204	170	12	,	,	PUNCT
cana-2204	170	13	𝑌)𝑍	𝑌)𝑍	PROPN
cana-2204	170	14	)	)	PUNCT
cana-2204	171	1	=	=	SYM
cana-2204	171	2	1	1	NUM
cana-2204	171	3	(	(	PUNCT
cana-2204	171	4	𝑛	𝑛	PRON
cana-2204	171	5	−	−	PROPN
cana-2204	171	6	2	2	NUM
cana-2204	171	7	)	)	PUNCT
cana-2204	172	1	[	[	X
cana-2204	172	2	{	{	PUNCT
cana-2204	172	3	𝑟	𝑟	X
cana-2204	172	4	(	(	PUNCT
cana-2204	172	5	𝑛	𝑛	PROPN
cana-2204	172	6	−	−	PROPN
cana-2204	172	7	1	1	NUM
cana-2204	172	8	)	)	PUNCT
cana-2204	172	9	–	–	PUNCT
cana-2204	172	10	𝜖(𝑛	𝜖(𝑛	NOUN
cana-2204	172	11	−	−	PROPN
cana-2204	172	12	1)(𝛼2	1)(𝛼2	NUM
cana-2204	172	13	+	+	NOUN
cana-2204	172	14	𝛽2	𝛽2	NOUN
cana-2204	172	15	)	)	PUNCT
cana-2204	172	16	}	}	PUNCT
cana-2204	172	17	{	{	PUNCT
cana-2204	172	18	𝑔(𝑌	𝑔(𝑌	PROPN
cana-2204	172	19	,	,	PUNCT
cana-2204	172	20	𝑍)𝜂(𝑋	𝑍)𝜂(𝑋	ADJ
cana-2204	172	21	)	)	PUNCT
cana-2204	172	22	−	−	PRON
cana-2204	172	23	𝑔(𝑋	𝑔(𝑋	PROPN
cana-2204	172	24	,	,	PUNCT
cana-2204	172	25	𝑍)𝜂(𝑌	𝑍)𝜂(𝑌	PROPN
cana-2204	172	26	)	)	PUNCT
cana-2204	172	27	−𝑆(𝑌	−𝑆(𝑌	PROPN
cana-2204	172	28	,	,	PUNCT
cana-2204	172	29	𝑍)𝜂(𝑋	𝑍)𝜂(𝑋	X
cana-2204	172	30	)	)	PUNCT
cana-2204	173	1	+	+	CCONJ
cana-2204	173	2	𝑆(𝑋	𝑆(𝑋	PROPN
cana-2204	173	3	,	,	PUNCT
cana-2204	173	4	𝑍)𝜂(𝑌	𝑍)𝜂(𝑌	NOUN
cana-2204	173	5	)	)	PUNCT
cana-2204	173	6	}	}	PUNCT
cana-2204	173	7	]	]	PUNCT
cana-2204	173	8	(	(	PUNCT
cana-2204	173	9	5.2	5.2	X
cana-2204	173	10	)	)	PUNCT
cana-2204	173	11	putting	put	VERB
cana-2204	173	12	𝑍	𝑍	NOUN
cana-2204	173	13	=	=	PUNCT
cana-2204	173	14	𝜉	𝜉	NOUN
cana-2204	173	15	in	in	ADP
cana-2204	173	16	above	above	ADP
cana-2204	173	17	equation	equation	NOUN
cana-2204	173	18	,	,	PUNCT
cana-2204	173	19	we	we	PRON
cana-2204	173	20	get	get	VERB
cana-2204	173	21	𝜂(𝐶(𝑋	𝜂(𝐶(𝑋	ADJ
cana-2204	173	22	,	,	PUNCT
cana-2204	173	23	𝑌)𝜉	𝑌)𝜉	PRON
cana-2204	173	24	)	)	PUNCT
cana-2204	174	1	=	=	SYM
cana-2204	174	2	0	0	NUM
cana-2204	174	3	(	(	PUNCT
cana-2204	174	4	5.3	5.3	NUM
cana-2204	174	5	)	)	PUNCT
cana-2204	174	6	again	again	ADV
cana-2204	174	7	putting	put	VERB
cana-2204	174	8	𝑋	𝑋	NOUN
cana-2204	174	9	=	=	PUNCT
cana-2204	174	10	𝜉	𝜉	NOUN
cana-2204	174	11	in	in	ADP
cana-2204	174	12	equation	equation	NOUN
cana-2204	174	13	(	(	PUNCT
cana-2204	174	14	5.2	5.2	NUM
cana-2204	174	15	)	)	PUNCT
cana-2204	174	16	,	,	PUNCT
cana-2204	174	17	we	we	PRON
cana-2204	174	18	get	get	VERB
cana-2204	174	19	𝜂(𝐶(𝜉	𝜂(𝐶(𝜉	NOUN
cana-2204	174	20	,	,	PUNCT
cana-2204	174	21	𝑌)𝑍	𝑌)𝑍	NUM
cana-2204	174	22	)	)	PUNCT
cana-2204	174	23	=	=	SYM
cana-2204	174	24	1	1	NUM
cana-2204	174	25	(	(	PUNCT
cana-2204	174	26	𝑛	𝑛	PRON
cana-2204	174	27	−	−	PROPN
cana-2204	174	28	2	2	NUM
cana-2204	174	29	)	)	PUNCT
cana-2204	175	1	[	[	X
cana-2204	175	2	{	{	PUNCT
cana-2204	175	3	𝑟	𝑟	X
cana-2204	175	4	(	(	PUNCT
cana-2204	175	5	𝑛	𝑛	PROPN
cana-2204	175	6	−	−	PROPN
cana-2204	175	7	1	1	NUM
cana-2204	175	8	)	)	PUNCT
cana-2204	175	9	−	−	PROPN
cana-2204	175	10	𝜖(𝑛	𝜖(𝑛	NOUN
cana-2204	175	11	−	−	PROPN
cana-2204	175	12	1)(𝛼2	1)(𝛼2	NUM
cana-2204	175	13	+	+	NOUN
cana-2204	175	14	𝛽2	𝛽2	NOUN
cana-2204	175	15	)	)	PUNCT
cana-2204	175	16	}	}	PUNCT
cana-2204	176	1	+	+	CCONJ
cana-2204	176	2	𝑆(𝑌	𝑆(𝑌	NOUN
cana-2204	176	3	,	,	PUNCT
cana-2204	176	4	𝑍	𝑍	PROPN
cana-2204	176	5	)	)	PUNCT
cana-2204	176	6	+	+	PROPN
cana-2204	176	7	(	(	PUNCT
cana-2204	176	8	𝑛	𝑛	DET
cana-2204	176	9	−	−	NOUN
cana-2204	176	10	1)(𝛼2	1)(𝛼2	NUM
cana-2204	176	11	+	+	NUM
cana-2204	176	12	𝛽2)𝜂(𝑋)𝜂(𝑌	𝛽2)𝜂(𝑋)𝜂(𝑌	PROPN
cana-2204	176	13	)	)	PUNCT
cana-2204	176	14	−	−	PRON
cana-2204	176	15	𝑔(𝑋	𝑔(𝑋	PROPN
cana-2204	176	16	,	,	PUNCT
cana-2204	176	17	𝑍	𝑍	PROPN
cana-2204	176	18	)	)	PUNCT
cana-2204	176	19	−	−	PROPN
cana-2204	176	20	𝜖𝜂(𝑋)𝜂(𝑌	𝜖𝜂(𝑋)𝜂(𝑌	NOUN
cana-2204	176	21	)	)	PUNCT
cana-2204	176	22	}	}	PUNCT
cana-2204	176	23	]	]	PUNCT
cana-2204	176	24	(	(	PUNCT
cana-2204	176	25	5.4	5.4	NUM
cana-2204	176	26	)	)	PUNCT
cana-2204	176	27	communications	communication	NOUN
cana-2204	176	28	on	on	ADP
cana-2204	176	29	applied	apply	VERB
cana-2204	176	30	nonlinear	nonlinear	ADJ
cana-2204	176	31	analysis	analysis	NOUN
cana-2204	176	32	issn	issn	NOUN
cana-2204	176	33	:	:	PUNCT
cana-2204	176	34	1074	1074	NUM
cana-2204	176	35	-	-	PUNCT
cana-2204	176	36	133x	133x	NUM
cana-2204	176	37	vol	vol	NOUN
cana-2204	176	38	32	32	NUM
cana-2204	176	39	no	no	NOUN
cana-2204	176	40	.	.	PUNCT
cana-2204	177	1	1s	1s	NUM
cana-2204	177	2	(	(	PUNCT
cana-2204	177	3	2025	2025	NUM
cana-2204	177	4	)	)	PUNCT
cana-2204	177	5	409	409	NUM
cana-2204	177	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2204	177	7	if	if	SCONJ
cana-2204	177	8	the	the	DET
cana-2204	177	9	manifold	manifold	NOUN
cana-2204	177	10	is	be	AUX
cana-2204	177	11	weyl	weyl	VERB
cana-2204	177	12	-	-	PUNCT
cana-2204	177	13	semi	semi	ADV
cana-2204	177	14	-	-	ADJ
cana-2204	177	15	symmetric	symmetric	ADJ
cana-2204	177	16	then	then	ADV
cana-2204	177	17	,	,	PUNCT
cana-2204	177	18	we	we	PRON
cana-2204	177	19	have	have	VERB
cana-2204	177	20	𝑔[𝑅(𝜉	𝑔[𝑅(𝜉	NOUN
cana-2204	177	21	,	,	PUNCT
cana-2204	177	22	𝑌)𝐶(𝑈	𝑌)𝐶(𝑈	ADJ
cana-2204	177	23	,	,	PUNCT
cana-2204	177	24	𝑉)𝑊	𝑉)𝑊	NOUN
cana-2204	177	25	,	,	PUNCT
cana-2204	177	26	𝜉	𝜉	X
cana-2204	177	27	]	]	X
cana-2204	177	28	−	−	PROPN
cana-2204	177	29	𝑔[𝐶(𝑅(𝜉	𝑔[𝐶(𝑅(𝜉	PROPN
cana-2204	177	30	,	,	PUNCT
cana-2204	177	31	𝑌)𝑈	𝑌)𝑈	PROPN
cana-2204	177	32	,	,	PUNCT
cana-2204	177	33	𝑉)𝑊	𝑉)𝑊	PROPN
cana-2204	177	34	,	,	PUNCT
cana-2204	177	35	𝜉	𝜉	X
cana-2204	177	36	]	]	X
cana-2204	177	37	−	−	PRON
cana-2204	177	38	𝑔[𝐶(𝑈	𝑔[𝐶(𝑈	PROPN
cana-2204	177	39	,	,	PUNCT
cana-2204	177	40	𝑅(𝜉	𝑅(𝜉	SYM
cana-2204	177	41	,	,	PUNCT
cana-2204	177	42	𝑌)𝑊	𝑌)𝑊	ADJ
cana-2204	177	43	,	,	PUNCT
cana-2204	177	44	𝜉	𝜉	X
cana-2204	177	45	]	]	X
cana-2204	177	46	−	−	PRON
cana-2204	177	47	𝑔[𝐶(𝑈	𝑔[𝐶(𝑈	ADJ
cana-2204	177	48	,	,	PUNCT
cana-2204	177	49	𝑉)𝑅(𝜉	𝑉)𝑅(𝜉	PROPN
cana-2204	177	50	,	,	PUNCT
cana-2204	177	51	𝑌)𝑊	𝑌)𝑊	NOUN
cana-2204	177	52	,	,	PUNCT
cana-2204	177	53	𝜉	𝜉	X
cana-2204	177	54	]	]	X
cana-2204	177	55	=	=	SYM
cana-2204	177	56	0	0	NUM
cana-2204	177	57	(	(	PUNCT
cana-2204	177	58	5.5	5.5	NUM
cana-2204	177	59	)	)	PUNCT
cana-2204	177	60	from	from	ADP
cana-2204	177	61	equation	equation	NOUN
cana-2204	177	62	(	(	PUNCT
cana-2204	177	63	3.16	3.16	NUM
cana-2204	177	64	)	)	PUNCT
cana-2204	177	65	,	,	PUNCT
cana-2204	177	66	we	we	PRON
cana-2204	177	67	obtained	obtain	VERB
cana-2204	177	68	𝑔(𝑅(𝜉	𝑔(𝑅(𝜉	PROPN
cana-2204	177	69	,	,	PUNCT
cana-2204	177	70	𝑋)𝑌	𝑋)𝑌	NOUN
cana-2204	177	71	,	,	PUNCT
cana-2204	177	72	𝜉	𝜉	NOUN
cana-2204	177	73	)	)	PUNCT
cana-2204	177	74	=	=	SYM
cana-2204	177	75	−(𝛼2	−(𝛼2	X
cana-2204	177	76	+	+	CCONJ
cana-2204	177	77	𝛽2){𝑔(𝑋	𝛽2){𝑔(𝑋	NUM
cana-2204	177	78	,	,	PUNCT
cana-2204	177	79	𝑌	𝑌	PROPN
cana-2204	177	80	)	)	PUNCT
cana-2204	177	81	+	+	NUM
cana-2204	177	82	𝜖𝜂(𝑋)𝜂(𝑌	𝜖𝜂(𝑋)𝜂(𝑌	NOUN
cana-2204	177	83	)	)	PUNCT
cana-2204	177	84	}	}	PUNCT
cana-2204	178	1	+	+	CCONJ
cana-2204	178	2	𝜖(𝜉𝛼)𝜂(𝑋	𝜖(𝜉𝛼)𝜂(𝑋	NOUN
cana-2204	178	3	)	)	PUNCT
cana-2204	178	4	+	+	CCONJ
cana-2204	178	5	(	(	PUNCT
cana-2204	178	6	𝛼𝜉)𝑔(𝑋	𝛼𝜉)𝑔(𝑋	PROPN
cana-2204	178	7	,	,	PUNCT
cana-2204	178	8	𝑌)𝜂(𝑌	𝑌)𝜂(𝑌	PROPN
cana-2204	178	9	)	)	PUNCT
cana-2204	179	1	+	+	NOUN
cana-2204	179	2	𝜖(𝑋𝛼	𝜖(𝑋𝛼	NOUN
cana-2204	179	3	)	)	PUNCT
cana-2204	179	4	−	−	PROPN
cana-2204	179	5	𝜖(𝑋𝛼)𝜂(𝑌)𝜂(𝑌	𝜖(𝑋𝛼)𝜂(𝑌)𝜂(𝑌	PROPN
cana-2204	179	6	)	)	PUNCT
cana-2204	179	7	(	(	PUNCT
cana-2204	179	8	5.6	5.6	NUM
cana-2204	179	9	)	)	PUNCT
cana-2204	179	10	using	use	VERB
cana-2204	179	11	equation	equation	NOUN
cana-2204	179	12	(	(	PUNCT
cana-2204	179	13	5.5	5.5	NUM
cana-2204	179	14	)	)	PUNCT
cana-2204	179	15	and	and	CCONJ
cana-2204	179	16	(	(	PUNCT
cana-2204	179	17	5.6	5.6	NUM
cana-2204	179	18	)	)	PUNCT
cana-2204	179	19	,	,	PUNCT
cana-2204	179	20	we	we	PRON
cana-2204	179	21	obtained	obtain	VERB
cana-2204	179	22	(	(	PUNCT
cana-2204	179	23	𝛼2	𝛼2	PROPN
cana-2204	179	24	+	+	CCONJ
cana-2204	179	25	𝛽2){𝐶′(𝑈	𝛽2){𝐶′(𝑈	PROPN
cana-2204	179	26	,	,	PUNCT
cana-2204	179	27	𝑉,𝑊	𝑉,𝑊	NOUN
cana-2204	179	28	,	,	PUNCT
cana-2204	179	29	𝑌	𝑌	PROPN
cana-2204	179	30	)	)	PUNCT
cana-2204	179	31	+	+	CCONJ
cana-2204	180	1	𝜖𝜂(𝑌)𝜂(𝐶(𝑈	𝜖𝜂(𝑌)𝜂(𝐶(𝑈	PROPN
cana-2204	180	2	,	,	PUNCT
cana-2204	180	3	𝑉)𝑊	𝑉)𝑊	NOUN
cana-2204	180	4	)	)	PUNCT
cana-2204	180	5	}	}	PUNCT
cana-2204	180	6	−	−	ADP
cana-2204	180	7	𝜖(𝜉𝛼)𝜂(𝑌	𝜖(𝜉𝛼)𝜂(𝑌	PROPN
cana-2204	180	8	)	)	PUNCT
cana-2204	180	9	−	−	PROPN
cana-2204	181	1	(	(	PUNCT
cana-2204	181	2	𝛼𝜉)𝐶	𝛼𝜉)𝐶	PROPN
cana-2204	181	3	′(𝑈	′(𝑈	NOUN
cana-2204	181	4	,	,	PUNCT
cana-2204	181	5	𝑉,𝑊,𝑈)𝜂(𝐶(𝑈	𝑉,𝑊,𝑈)𝜂(𝐶(𝑈	PROPN
cana-2204	181	6	,	,	PUNCT
cana-2204	181	7	𝑉)𝑊	𝑉)𝑊	NOUN
cana-2204	181	8	)	)	PUNCT
cana-2204	181	9	−	−	ADP
cana-2204	181	10	𝜖(𝑌𝛼	𝜖(𝑌𝛼	NOUN
cana-2204	181	11	)	)	PUNCT
cana-2204	182	1	+	+	CCONJ
cana-2204	182	2	𝜖(𝑌𝛼)𝜂(𝐶(𝑈	𝜖(𝑌𝛼)𝜂(𝐶(𝑈	ADJ
cana-2204	182	3	,	,	PUNCT
cana-2204	182	4	𝑉)𝑊)𝜂(𝐶(𝑈	𝑉)𝑊)𝜂(𝐶(𝑈	ADJ
cana-2204	182	5	,	,	PUNCT
cana-2204	182	6	𝑉)𝑊	𝑉)𝑊	NOUN
cana-2204	182	7	)	)	PUNCT
cana-2204	183	1	+	+	NUM
cana-2204	183	2	𝑔[𝐶((𝛼2	𝑔[𝐶((𝛼2	NOUN
cana-2204	183	3	+	+	SYM
cana-2204	183	4	𝛽2)(𝜖𝑔(𝑌	𝛽2)(𝜖𝑔(𝑌	NOUN
cana-2204	183	5	,	,	PUNCT
cana-2204	183	6	𝑈)𝜉	𝑈)𝜉	ADJ
cana-2204	183	7	−	−	ADP
cana-2204	183	8	𝜂(𝑈)𝑌	𝜂(𝑈)𝑌	NOUN
cana-2204	183	9	)	)	PUNCT
cana-2204	184	1	+	+	CCONJ
cana-2204	184	2	(	(	PUNCT
cana-2204	184	3	𝜉𝛽	𝜉𝛽	ADJ
cana-2204	184	4	−	−	PROPN
cana-2204	184	5	2𝛼𝛽𝜂(𝑈))𝜙𝑌	2𝛼𝛽𝜂(𝑈))𝜙𝑌	PROPN
cana-2204	184	6	+	+	CCONJ
cana-2204	184	7	(	(	PUNCT
cana-2204	184	8	𝜉𝛼)𝑌	𝜉𝛼)𝑌	X
cana-2204	184	9	+	+	SYM
cana-2204	184	10	𝜖𝛼𝜉𝑔(𝑌	𝜖𝛼𝜉𝑔(𝑌	PROPN
cana-2204	184	11	,	,	PUNCT
cana-2204	184	12	𝑈)𝑈	𝑈)𝑈	VERB
cana-2204	184	13	−	−	PROPN
cana-2204	184	14	(	(	PUNCT
cana-2204	184	15	𝑌𝛼)𝜉	𝑌𝛼)𝜉	PROPN
cana-2204	184	16	−	−	PROPN
cana-2204	184	17	(	(	PUNCT
cana-2204	184	18	𝑌𝛼)𝜂(𝑈)𝑈	𝑌𝛼)𝜂(𝑈)𝑈	PROPN
cana-2204	184	19	,	,	PUNCT
cana-2204	184	20	𝑉)𝑊	𝑉)𝑊	PROPN
cana-2204	184	21	,	,	PUNCT
cana-2204	184	22	𝜉	𝜉	AUX
cana-2204	184	23	]	]	X
cana-2204	184	24	+	+	X
cana-2204	184	25	𝑔[𝐶(𝑈	𝑔[𝐶(𝑈	ADJ
cana-2204	184	26	,	,	PUNCT
cana-2204	184	27	(	(	PUNCT
cana-2204	184	28	𝛼2	𝛼2	PROPN
cana-2204	184	29	+	+	NUM
cana-2204	184	30	𝛽2)(𝜖𝑔(𝑌	𝛽2)(𝜖𝑔(𝑌	NOUN
cana-2204	184	31	,	,	PUNCT
cana-2204	184	32	𝑉)𝜉	𝑉)𝜉	VERB
cana-2204	184	33	−	−	PROPN
cana-2204	184	34	𝜂(𝑉)𝑌	𝜂(𝑉)𝑌	PROPN
cana-2204	184	35	)	)	PUNCT
cana-2204	185	1	+	+	CCONJ
cana-2204	185	2	(	(	PUNCT
cana-2204	185	3	𝜉𝛽	𝜉𝛽	ADJ
cana-2204	185	4	−	−	PROPN
cana-2204	185	5	2𝛼𝛽𝜂(𝑉))𝜙𝑌	2𝛼𝛽𝜂(𝑉))𝜙𝑌	NUM
cana-2204	185	6	+	+	CCONJ
cana-2204	185	7	(	(	PUNCT
cana-2204	185	8	𝜉𝛼)𝑌	𝜉𝛼)𝑌	X
cana-2204	185	9	+	+	SYM
cana-2204	185	10	𝜖𝛼𝜉𝑔(𝑌	𝜖𝛼𝜉𝑔(𝑌	PROPN
cana-2204	185	11	,	,	PUNCT
cana-2204	185	12	𝑉)𝑉	𝑉)𝑉	NOUN
cana-2204	185	13	−	−	PROPN
cana-2204	185	14	(	(	PUNCT
cana-2204	185	15	𝑌𝛼)𝜉	𝑌𝛼)𝜉	PROPN
cana-2204	185	16	−	−	PROPN
cana-2204	185	17	(	(	PUNCT
cana-2204	185	18	𝑌𝛼)𝜂(𝑉)𝑉,𝑊	𝑌𝛼)𝜂(𝑉)𝑉,𝑊	PROPN
cana-2204	185	19	)	)	PUNCT
cana-2204	185	20	,	,	PUNCT
cana-2204	185	21	𝜉	𝜉	AUX
cana-2204	185	22	]	]	X
cana-2204	185	23	+	+	CCONJ
cana-2204	185	24	𝑔[𝐶(𝑈	𝑔[𝐶(𝑈	ADJ
cana-2204	185	25	,	,	PUNCT
cana-2204	185	26	𝑉)((𝛼2	𝑉)((𝛼2	NOUN
cana-2204	185	27	+	+	CCONJ
cana-2204	185	28	𝛽2)(𝜖𝑔(𝑌,𝑊)𝜉	𝛽2)(𝜖𝑔(𝑌,𝑊)𝜉	PROPN
cana-2204	185	29	−	−	PROPN
cana-2204	185	30	𝜂(𝑊)𝑌	𝜂(𝑊)𝑌	NOUN
cana-2204	185	31	)	)	PUNCT
cana-2204	186	1	+	+	CCONJ
cana-2204	186	2	(	(	PUNCT
cana-2204	186	3	𝜉𝛽	𝜉𝛽	ADJ
cana-2204	186	4	−	−	PROPN
cana-2204	186	5	2𝛼𝛽𝜂(𝑊))𝜙𝑌	2𝛼𝛽𝜂(𝑊))𝜙𝑌	PROPN
cana-2204	186	6	+	+	CCONJ
cana-2204	186	7	(	(	PUNCT
cana-2204	186	8	𝜉𝛼)𝑌	𝜉𝛼)𝑌	NOUN
cana-2204	186	9	+	+	CCONJ
cana-2204	186	10	𝜖𝛼𝜉𝑔(𝑌,𝑊)𝑊	𝜖𝛼𝜉𝑔(𝑌,𝑊)𝑊	ADJ
cana-2204	186	11	−(𝑌𝛼)𝜉	−(𝑌𝛼)𝜉	NOUN
cana-2204	186	12	−	−	PROPN
cana-2204	186	13	(	(	PUNCT
cana-2204	186	14	𝑌𝛼)𝜂(𝑊)𝑊	𝑌𝛼)𝜂(𝑊)𝑊	PROPN
cana-2204	186	15	)	)	PUNCT
cana-2204	186	16	,	,	PUNCT
cana-2204	186	17	𝜉	𝜉	X
cana-2204	186	18	]	]	X
cana-2204	186	19	=	=	SYM
cana-2204	186	20	0	0	NUM
cana-2204	186	21	(	(	PUNCT
cana-2204	186	22	5.7	5.7	NUM
cana-2204	186	23	)	)	PUNCT
cana-2204	186	24	where	where	SCONJ
cana-2204	186	25	𝐶	𝐶	PROPN
cana-2204	186	26	′(𝑈	′(𝑈	NOUN
cana-2204	186	27	,	,	PUNCT
cana-2204	186	28	𝑉,𝑊	𝑉,𝑊	NOUN
cana-2204	186	29	,	,	PUNCT
cana-2204	186	30	𝑌	𝑌	PROPN
cana-2204	186	31	)	)	PUNCT
cana-2204	186	32	=	=	SYM
cana-2204	186	33	𝑔(𝐶(𝑈	𝑔(𝐶(𝑈	PROPN
cana-2204	186	34	,	,	PUNCT
cana-2204	186	35	𝑉)𝑊	𝑉)𝑊	NOUN
cana-2204	186	36	,	,	PUNCT
cana-2204	186	37	𝑌	𝑌	PROPN
cana-2204	186	38	)	)	PUNCT
cana-2204	186	39	putting	put	VERB
cana-2204	186	40	𝑌	𝑌	PROPN
cana-2204	186	41	=	=	SYM
cana-2204	186	42	𝑈	𝑈	PROPN
cana-2204	186	43	in	in	ADP
cana-2204	186	44	(	(	PUNCT
cana-2204	186	45	5.7	5.7	NUM
cana-2204	186	46	)	)	PUNCT
cana-2204	186	47	,	,	PUNCT
cana-2204	186	48	we	we	PRON
cana-2204	186	49	have	have	VERB
cana-2204	186	50	(	(	PUNCT
cana-2204	186	51	𝛼2	𝛼2	VERB
cana-2204	186	52	+	+	CCONJ
cana-2204	186	53	𝛽2){𝐶′(𝑈	𝛽2){𝐶′(𝑈	PROPN
cana-2204	186	54	,	,	PUNCT
cana-2204	186	55	𝑉,𝑊,𝑈	𝑉,𝑊,𝑈	NOUN
cana-2204	186	56	)	)	PUNCT
cana-2204	187	1	+	+	CCONJ
cana-2204	188	1	𝜖𝜂(𝑈)𝜂(𝐶(𝑈	𝜖𝜂(𝑈)𝜂(𝐶(𝑈	PROPN
cana-2204	188	2	,	,	PUNCT
cana-2204	188	3	𝑉)𝑊	𝑉)𝑊	NOUN
cana-2204	188	4	)	)	PUNCT
cana-2204	188	5	}	}	PUNCT
cana-2204	188	6	−	−	ADP
cana-2204	188	7	𝜖(𝜉𝛼)𝜂(𝑈	𝜖(𝜉𝛼)𝜂(𝑈	NUM
cana-2204	188	8	)	)	PUNCT
cana-2204	188	9	−(𝛼𝜉)𝐶	−(𝛼𝜉)𝐶	PROPN
cana-2204	188	10	′(𝑈	′(𝑈	NOUN
cana-2204	188	11	,	,	PUNCT
cana-2204	188	12	𝑉,𝑊,𝑈)𝜂(𝐶(𝑈	𝑉,𝑊,𝑈)𝜂(𝐶(𝑈	PROPN
cana-2204	188	13	,	,	PUNCT
cana-2204	188	14	𝑉)𝑊	𝑉)𝑊	NOUN
cana-2204	188	15	)	)	PUNCT
cana-2204	188	16	−	−	PROPN
cana-2204	189	1	𝜖(𝑈𝛼	𝜖(𝑈𝛼	X
cana-2204	189	2	)	)	PUNCT
cana-2204	189	3	+	+	CCONJ
cana-2204	189	4	𝜖(𝑈𝛼)𝜂(𝐶(𝑈	𝜖(𝑈𝛼)𝜂(𝐶(𝑈	VERB
cana-2204	189	5	,	,	PUNCT
cana-2204	189	6	𝑉)𝑊)𝜂(𝐶(𝑈	𝑉)𝑊)𝜂(𝐶(𝑈	ADJ
cana-2204	189	7	,	,	PUNCT
cana-2204	189	8	𝑉)𝑊	𝑉)𝑊	NOUN
cana-2204	189	9	)	)	PUNCT
cana-2204	189	10	+	+	NUM
cana-2204	189	11	𝑔[𝐶((𝛼2	𝑔[𝐶((𝛼2	NOUN
cana-2204	189	12	+	+	CCONJ
cana-2204	189	13	𝛽2)(𝜖𝑔(𝑈	𝛽2)(𝜖𝑔(𝑈	VERB
cana-2204	189	14	,	,	PUNCT
cana-2204	189	15	𝑈)𝜉	𝑈)𝜉	ADJ
cana-2204	189	16	−	−	NOUN
cana-2204	189	17	𝜂(𝑈)𝑈	𝜂(𝑈)𝑈	NOUN
cana-2204	189	18	)	)	PUNCT
cana-2204	189	19	+	+	CCONJ
cana-2204	189	20	(	(	PUNCT
cana-2204	189	21	𝜉𝛽	𝜉𝛽	ADJ
cana-2204	189	22	−	−	PROPN
cana-2204	189	23	2𝛼𝛽𝜂(𝑈))𝜙𝑈	2𝛼𝛽𝜂(𝑈))𝜙𝑈	NUM
cana-2204	189	24	+	+	CCONJ
cana-2204	189	25	(	(	PUNCT
cana-2204	189	26	𝜉𝛼)𝑈	𝜉𝛼)𝑈	PROPN
cana-2204	189	27	+	+	NOUN
cana-2204	189	28	𝜖𝛼𝜉𝑔(𝑈	𝜖𝛼𝜉𝑔(𝑈	VERB
cana-2204	189	29	,	,	PUNCT
cana-2204	189	30	𝑈)𝑈	𝑈)𝑈	VERB
cana-2204	189	31	−	−	PROPN
cana-2204	189	32	(	(	PUNCT
cana-2204	189	33	𝑈𝛼)𝜉	𝑈𝛼)𝜉	NOUN
cana-2204	189	34	−	−	X
cana-2204	189	35	(	(	PUNCT
cana-2204	189	36	𝑈𝛼)𝜂(𝑈)𝑈	𝑈𝛼)𝜂(𝑈)𝑈	PROPN
cana-2204	189	37	,	,	PUNCT
cana-2204	189	38	𝑉)𝑊	𝑉)𝑊	NOUN
cana-2204	189	39	,	,	PUNCT
cana-2204	189	40	𝜉	𝜉	AUX
cana-2204	189	41	]	]	X
cana-2204	189	42	+	+	NOUN
cana-2204	189	43	𝑔[𝐶(𝑈	𝑔[𝐶(𝑈	ADJ
cana-2204	189	44	,	,	PUNCT
cana-2204	189	45	(	(	PUNCT
cana-2204	189	46	𝛼2	𝛼2	PROPN
cana-2204	189	47	+	+	CCONJ
cana-2204	189	48	𝛽2)(𝜖𝑔(𝑈	𝛽2)(𝜖𝑔(𝑈	VERB
cana-2204	189	49	,	,	PUNCT
cana-2204	189	50	𝑉)𝜉	𝑉)𝜉	VERB
cana-2204	190	1	−	−	NOUN
cana-2204	190	2	𝜂(𝑉)𝑈	𝜂(𝑉)𝑈	NOUN
cana-2204	190	3	)	)	PUNCT
cana-2204	191	1	+	+	PROPN
cana-2204	191	2	(	(	PUNCT
cana-2204	191	3	𝜉𝛽	𝜉𝛽	ADJ
cana-2204	191	4	−	−	PROPN
cana-2204	191	5	2𝛼𝛽𝜂(𝑉))𝜙𝑈	2𝛼𝛽𝜂(𝑉))𝜙𝑈	NUM
cana-2204	191	6	+	+	CCONJ
cana-2204	191	7	(	(	PUNCT
cana-2204	191	8	𝜉𝛼)𝑈	𝜉𝛼)𝑈	PROPN
cana-2204	191	9	+	+	CCONJ
cana-2204	191	10	𝜖𝛼𝜉𝑔(𝑈	𝜖𝛼𝜉𝑔(𝑈	ADJ
cana-2204	191	11	,	,	PUNCT
cana-2204	191	12	𝑉)𝑉	𝑉)𝑉	NOUN
cana-2204	191	13	−	−	PROPN
cana-2204	191	14	(	(	PUNCT
cana-2204	191	15	𝑈𝛼)𝜉	𝑈𝛼)𝜉	NOUN
cana-2204	191	16	−	−	PROPN
cana-2204	191	17	(	(	PUNCT
cana-2204	191	18	𝑈𝛼)𝜂(𝑉)𝑉,𝑊	𝑈𝛼)𝜂(𝑉)𝑉,𝑊	PROPN
cana-2204	191	19	)	)	PUNCT
cana-2204	191	20	,	,	PUNCT
cana-2204	191	21	𝜉	𝜉	X
cana-2204	191	22	]	]	X
cana-2204	192	1	+	+	ADJ
cana-2204	192	2	𝑔	𝑔	PRON
cana-2204	192	3	[	[	VERB
cana-2204	192	4	𝐶(𝑈	𝐶(𝑈	VERB
cana-2204	192	5	,	,	PUNCT
cana-2204	192	6	𝑉	𝑉	PROPN
cana-2204	192	7	)	)	PUNCT
cana-2204	192	8	(	(	PUNCT
cana-2204	192	9	(	(	PUNCT
cana-2204	192	10	𝛼2	𝛼2	VERB
cana-2204	192	11	+	+	CCONJ
cana-2204	192	12	𝛽2)(𝜖𝑔(𝑈,𝑊)𝜉	𝛽2)(𝜖𝑔(𝑈,𝑊)𝜉	NOUN
cana-2204	192	13	−	−	NOUN
cana-2204	192	14	𝜂(𝑊)𝑈	𝜂(𝑊)𝑈	NOUN
cana-2204	192	15	)	)	PUNCT
cana-2204	192	16	+	+	CCONJ
cana-2204	192	17	(	(	PUNCT
cana-2204	192	18	𝜉𝛽	𝜉𝛽	ADJ
cana-2204	192	19	−	−	PROPN
cana-2204	192	20	2𝛼𝛽𝜂(𝑊))𝜙𝑈	2𝛼𝛽𝜂(𝑊))𝜙𝑈	NUM
cana-2204	193	1	+	+	CCONJ
cana-2204	193	2	(	(	PUNCT
cana-2204	193	3	𝜉𝛼)𝑈	𝜉𝛼)𝑈	NOUN
cana-2204	193	4	+	+	CCONJ
cana-2204	193	5	𝜖𝛼𝜉𝑔(𝑈,𝑊)𝑊	𝜖𝛼𝜉𝑔(𝑈,𝑊)𝑊	PROPN
cana-2204	193	6	−	−	PROPN
cana-2204	193	7	(	(	PUNCT
cana-2204	193	8	𝑈𝛼)𝜉	𝑈𝛼)𝜉	NOUN
cana-2204	193	9	−	−	PROPN
cana-2204	193	10	(	(	PUNCT
cana-2204	193	11	𝑈𝛼)𝜂(𝑊)𝑊	𝑈𝛼)𝜂(𝑊)𝑊	PROPN
cana-2204	193	12	)	)	PUNCT
cana-2204	193	13	,	,	PUNCT
cana-2204	193	14	𝜉	𝜉	X
cana-2204	193	15	]	]	X
cana-2204	193	16	=	=	SYM
cana-2204	193	17	0	0	NUM
cana-2204	193	18	(	(	PUNCT
cana-2204	193	19	5.8	5.8	NUM
cana-2204	193	20	)	)	PUNCT
cana-2204	193	21	taking	take	VERB
cana-2204	193	22	an	an	DET
cana-2204	193	23	orthogonal	orthogonal	ADJ
cana-2204	193	24	frame	frame	NOUN
cana-2204	193	25	in	in	ADP
cana-2204	193	26	the	the	DET
cana-2204	193	27	equation	equation	NOUN
cana-2204	193	28	(	(	PUNCT
cana-2204	193	29	5.8	5.8	NUM
cana-2204	193	30	)	)	PUNCT
cana-2204	193	31	over	over	ADP
cana-2204	193	32	𝑈	𝑈	PROPN
cana-2204	193	33	,	,	PUNCT
cana-2204	193	34	we	we	PRON
cana-2204	193	35	obtained	obtain	VERB
cana-2204	193	36	∑	∑	PROPN
cana-2204	193	37	𝐶	𝐶	PROPN
cana-2204	193	38	′(𝑒𝑖	′(𝑒𝑖	NOUN
cana-2204	193	39	,	,	PUNCT
cana-2204	193	40	𝑉,𝑊	𝑉,𝑊	NOUN
cana-2204	193	41	,	,	PUNCT
cana-2204	193	42	𝑒𝑖	𝑒𝑖	NOUN
cana-2204	193	43	)	)	PUNCT
cana-2204	193	44	𝑛	𝑛	PRON
cana-2204	193	45	𝑖=1	𝑖=1	PUNCT
cana-2204	194	1	=	=	SYM
cana-2204	194	2	0	0	NUM
cana-2204	194	3	(	(	PUNCT
cana-2204	194	4	5.9	5.9	NUM
cana-2204	194	5	)	)	PUNCT
cana-2204	194	6	and	and	CCONJ
cana-2204	194	7	using	use	VERB
cana-2204	194	8	equation	equation	NOUN
cana-2204	194	9	(	(	PUNCT
cana-2204	194	10	5.3	5.3	NUM
cana-2204	194	11	)	)	PUNCT
cana-2204	194	12	in	in	ADP
cana-2204	194	13	(	(	PUNCT
cana-2204	194	14	5.8	5.8	NUM
cana-2204	194	15	)	)	PUNCT
cana-2204	194	16	,	,	PUNCT
cana-2204	194	17	we	we	PRON
cana-2204	194	18	have	have	VERB
cana-2204	194	19	𝜂(𝐶(𝜉	𝜂(𝐶(𝜉	X
cana-2204	194	20	,	,	PUNCT
cana-2204	194	21	𝑉)𝑊	𝑉)𝑊	NOUN
cana-2204	194	22	)	)	PUNCT
cana-2204	194	23	=	=	SYM
cana-2204	194	24	0	0	PUNCT
cana-2204	194	25	(	(	PUNCT
cana-2204	194	26	5.10	5.10	NUM
cana-2204	194	27	)	)	PUNCT
cana-2204	194	28	using	use	VERB
cana-2204	194	29	equation	equation	NOUN
cana-2204	194	30	(	(	PUNCT
cana-2204	194	31	5.3	5.3	NUM
cana-2204	194	32	)	)	PUNCT
cana-2204	194	33	and	and	CCONJ
cana-2204	194	34	(	(	PUNCT
cana-2204	194	35	5.8	5.8	X
cana-2204	194	36	)	)	PUNCT
cana-2204	194	37	we	we	PRON
cana-2204	194	38	have	have	VERB
cana-2204	194	39	𝐶	𝐶	PROPN
cana-2204	194	40	′(𝑈	′(𝑈	NOUN
cana-2204	194	41	,	,	PUNCT
cana-2204	194	42	𝑉,𝑊	𝑉,𝑊	NOUN
cana-2204	194	43	,	,	PUNCT
cana-2204	194	44	𝑌	𝑌	PROPN
cana-2204	194	45	)	)	PUNCT
cana-2204	195	1	+	+	CCONJ
cana-2204	195	2	𝜂(𝑌)𝜂(𝐶(𝑈	𝜂(𝑌)𝜂(𝐶(𝑈	ADJ
cana-2204	195	3	,	,	PUNCT
cana-2204	195	4	𝑉)𝑊	𝑉)𝑊	NOUN
cana-2204	195	5	)	)	PUNCT
cana-2204	196	1	+	+	CCONJ
cana-2204	197	1	𝜖𝜂(𝑈)𝜂(𝐶(𝑌	𝜖𝜂(𝑈)𝜂(𝐶(𝑌	PROPN
cana-2204	197	2	,	,	PUNCT
cana-2204	197	3	𝑉)𝑊	𝑉)𝑊	NOUN
cana-2204	197	4	)	)	PUNCT
cana-2204	198	1	+	+	CCONJ
cana-2204	198	2	𝜖𝜂(𝑉)𝜂(𝐶(𝑈	𝜖𝜂(𝑉)𝜂(𝐶(𝑈	ADJ
cana-2204	198	3	,	,	PUNCT
cana-2204	198	4	𝑌)𝑊	𝑌)𝑊	ADJ
cana-2204	198	5	)	)	PUNCT
cana-2204	198	6	communications	communication	NOUN
cana-2204	198	7	on	on	ADP
cana-2204	198	8	applied	apply	VERB
cana-2204	198	9	nonlinear	nonlinear	ADJ
cana-2204	198	10	analysis	analysis	NOUN
cana-2204	198	11	issn	issn	NOUN
cana-2204	198	12	:	:	PUNCT
cana-2204	198	13	1074	1074	NUM
cana-2204	198	14	-	-	PUNCT
cana-2204	198	15	133x	133x	NUM
cana-2204	198	16	vol	vol	NOUN
cana-2204	198	17	32	32	NUM
cana-2204	198	18	no	no	NOUN
cana-2204	198	19	.	.	PUNCT
cana-2204	199	1	1s	1s	NUM
cana-2204	199	2	(	(	PUNCT
cana-2204	199	3	2025	2025	NUM
cana-2204	199	4	)	)	PUNCT
cana-2204	199	5	410	410	NUM
cana-2204	199	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2204	200	1	+	+	PUNCT
cana-2204	200	2	𝜖𝜂(𝑊)𝜂(𝐶(𝑈	𝜖𝜂(𝑊)𝜂(𝐶(𝑈	ADJ
cana-2204	200	3	,	,	PUNCT
cana-2204	200	4	𝑉)𝑌	𝑉)𝑌	NOUN
cana-2204	200	5	)	)	PUNCT
cana-2204	201	1	=	=	SYM
cana-2204	201	2	0	0	NUM
cana-2204	201	3	(	(	PUNCT
cana-2204	201	4	5.11	5.11	NUM
cana-2204	201	5	)	)	PUNCT
cana-2204	201	6	using	use	VERB
cana-2204	201	7	equation	equation	NOUN
cana-2204	201	8	(	(	PUNCT
cana-2204	201	9	5.2	5.2	NUM
cana-2204	201	10	)	)	PUNCT
cana-2204	201	11	and	and	CCONJ
cana-2204	201	12	(	(	PUNCT
cana-2204	201	13	5.11	5.11	NUM
cana-2204	201	14	)	)	PUNCT
cana-2204	201	15	,	,	PUNCT
cana-2204	201	16	we	we	PRON
cana-2204	201	17	have	have	VERB
cana-2204	201	18	𝐶′(𝑈	𝐶′(𝑈	PROPN
cana-2204	201	19	,	,	PUNCT
cana-2204	201	20	𝑉,𝑊	𝑉,𝑊	NOUN
cana-2204	201	21	,	,	PUNCT
cana-2204	201	22	𝑌	𝑌	PROPN
cana-2204	201	23	)	)	PUNCT
cana-2204	201	24	+	+	X
cana-2204	201	25	𝜂(𝑌	𝜂(𝑌	NOUN
cana-2204	201	26	)	)	PUNCT
cana-2204	201	27	1	1	NUM
cana-2204	201	28	(	(	PUNCT
cana-2204	201	29	𝑛	𝑛	PRON
cana-2204	201	30	−	−	PROPN
cana-2204	201	31	2	2	NUM
cana-2204	201	32	)	)	PUNCT
cana-2204	202	1	[	[	X
cana-2204	202	2	{	{	PUNCT
cana-2204	202	3	𝑟	𝑟	X
cana-2204	202	4	(	(	PUNCT
cana-2204	202	5	𝑛	𝑛	PROPN
cana-2204	202	6	−	−	PROPN
cana-2204	202	7	1	1	NUM
cana-2204	202	8	)	)	PUNCT
cana-2204	202	9	–	–	PUNCT
cana-2204	202	10	𝜖(𝑛	𝜖(𝑛	NOUN
cana-2204	202	11	−	−	PROPN
cana-2204	202	12	1)(𝛼2	1)(𝛼2	NUM
cana-2204	202	13	+	+	NOUN
cana-2204	202	14	𝛽2	𝛽2	NOUN
cana-2204	202	15	)	)	PUNCT
cana-2204	202	16	}	}	PUNCT
cana-2204	202	17	{	{	PUNCT
cana-2204	202	18	𝑔(𝑉,𝑊)𝜂(𝑈	𝑔(𝑉,𝑊)𝜂(𝑈	PROPN
cana-2204	202	19	)	)	PUNCT
cana-2204	202	20	−	−	PROPN
cana-2204	202	21	𝑔(𝑈,𝑊)𝜂(𝑉	𝑔(𝑈,𝑊)𝜂(𝑉	NOUN
cana-2204	202	22	)	)	PUNCT
cana-2204	202	23	−	−	NOUN
cana-2204	202	24	𝑆(𝑉,𝑊)𝜂(𝑈	𝑆(𝑉,𝑊)𝜂(𝑈	NOUN
cana-2204	202	25	)	)	PUNCT
cana-2204	203	1	+	+	CCONJ
cana-2204	203	2	𝑆(𝑈,𝑊)𝜂(𝑉	𝑆(𝑈,𝑊)𝜂(𝑉	NOUN
cana-2204	203	3	)	)	PUNCT
cana-2204	203	4	}	}	PUNCT
cana-2204	203	5	]	]	PUNCT
cana-2204	204	1	+	+	CCONJ
cana-2204	204	2	𝜖𝜂(𝑈	𝜖𝜂(𝑈	NUM
cana-2204	204	3	)	)	PUNCT
cana-2204	204	4	1	1	NUM
cana-2204	204	5	(	(	PUNCT
cana-2204	204	6	𝑛	𝑛	PRON
cana-2204	204	7	−	−	PROPN
cana-2204	204	8	2	2	NUM
cana-2204	204	9	)	)	PUNCT
cana-2204	204	10	[	[	X
cana-2204	204	11	{	{	PUNCT
cana-2204	204	12	𝑟	𝑟	X
cana-2204	204	13	(	(	PUNCT
cana-2204	204	14	𝑛	𝑛	PROPN
cana-2204	204	15	−	−	PROPN
cana-2204	204	16	1	1	NUM
cana-2204	204	17	)	)	PUNCT
cana-2204	204	18	–	–	PUNCT
cana-2204	204	19	𝜖(𝑛	𝜖(𝑛	NOUN
cana-2204	204	20	−	−	PROPN
cana-2204	204	21	1)(𝛼2	1)(𝛼2	NUM
cana-2204	204	22	+	+	NOUN
cana-2204	204	23	𝛽2	𝛽2	NOUN
cana-2204	204	24	)	)	PUNCT
cana-2204	204	25	}	}	PUNCT
cana-2204	204	26	{	{	PUNCT
cana-2204	204	27	𝑔(𝑉,𝑊)𝜂(𝑌	𝑔(𝑉,𝑊)𝜂(𝑌	PROPN
cana-2204	204	28	)	)	PUNCT
cana-2204	204	29	−	−	PROPN
cana-2204	204	30	𝑔(𝑌,𝑊)𝜂(𝑉	𝑔(𝑌,𝑊)𝜂(𝑉	PROPN
cana-2204	204	31	)	)	PUNCT
cana-2204	204	32	−	−	PROPN
cana-2204	204	33	𝑆(𝑉,𝑊)𝜂(𝑌	𝑆(𝑉,𝑊)𝜂(𝑌	NOUN
cana-2204	204	34	)	)	PUNCT
cana-2204	205	1	+	+	CCONJ
cana-2204	205	2	𝑆(𝑌,𝑊)𝜂(𝑉	𝑆(𝑌,𝑊)𝜂(𝑉	NOUN
cana-2204	205	3	)	)	PUNCT
cana-2204	205	4	}	}	PUNCT
cana-2204	205	5	]	]	PUNCT
cana-2204	206	1	+	+	CCONJ
cana-2204	206	2	𝜖𝜂(𝑉	𝜖𝜂(𝑉	NOUN
cana-2204	206	3	)	)	PUNCT
cana-2204	206	4	1	1	NUM
cana-2204	206	5	(	(	PUNCT
cana-2204	206	6	𝑛	𝑛	PRON
cana-2204	206	7	−	−	PROPN
cana-2204	206	8	2	2	NUM
cana-2204	206	9	)	)	PUNCT
cana-2204	206	10	[	[	X
cana-2204	206	11	{	{	PUNCT
cana-2204	206	12	𝑟	𝑟	X
cana-2204	206	13	(	(	PUNCT
cana-2204	206	14	𝑛	𝑛	PROPN
cana-2204	206	15	−	−	PROPN
cana-2204	206	16	1	1	NUM
cana-2204	206	17	)	)	PUNCT
cana-2204	206	18	–	–	PUNCT
cana-2204	206	19	𝜖(𝑛	𝜖(𝑛	NOUN
cana-2204	206	20	−	−	PROPN
cana-2204	206	21	1)(𝛼2	1)(𝛼2	NUM
cana-2204	206	22	+	+	NOUN
cana-2204	206	23	𝛽2	𝛽2	NOUN
cana-2204	206	24	)	)	PUNCT
cana-2204	206	25	}	}	PUNCT
cana-2204	206	26	{	{	PUNCT
cana-2204	206	27	𝑔(𝑉,𝑊)𝜂(𝑈	𝑔(𝑉,𝑊)𝜂(𝑈	PROPN
cana-2204	206	28	)	)	PUNCT
cana-2204	206	29	−	−	PROPN
cana-2204	206	30	𝑔(𝑈,𝑊)𝜂(𝑉	𝑔(𝑈,𝑊)𝜂(𝑉	NOUN
cana-2204	206	31	)	)	PUNCT
cana-2204	206	32	−	−	NOUN
cana-2204	206	33	𝑆(𝑉,𝑊)𝜂(𝑈	𝑆(𝑉,𝑊)𝜂(𝑈	NOUN
cana-2204	206	34	)	)	PUNCT
cana-2204	207	1	+	+	CCONJ
cana-2204	207	2	𝑆(𝑈,𝑊)𝜂(𝑉	𝑆(𝑈,𝑊)𝜂(𝑉	NOUN
cana-2204	207	3	)	)	PUNCT
cana-2204	207	4	}	}	PUNCT
cana-2204	207	5	]	]	PUNCT
cana-2204	208	1	+	+	CCONJ
cana-2204	208	2	𝜖𝜂(𝑊	𝜖𝜂(𝑊	NOUN
cana-2204	208	3	)	)	PUNCT
cana-2204	208	4	1	1	NUM
cana-2204	208	5	(	(	PUNCT
cana-2204	208	6	𝑛	𝑛	PRON
cana-2204	208	7	−	−	PROPN
cana-2204	208	8	2	2	NUM
cana-2204	208	9	)	)	PUNCT
cana-2204	208	10	[	[	X
cana-2204	208	11	{	{	PUNCT
cana-2204	208	12	𝑟	𝑟	X
cana-2204	208	13	(	(	PUNCT
cana-2204	208	14	𝑛	𝑛	PROPN
cana-2204	208	15	−	−	PROPN
cana-2204	208	16	1	1	NUM
cana-2204	208	17	)	)	PUNCT
cana-2204	208	18	–	–	PUNCT
cana-2204	208	19	𝜖(𝑛	𝜖(𝑛	NOUN
cana-2204	208	20	−	−	PROPN
cana-2204	208	21	1)(𝛼2	1)(𝛼2	NUM
cana-2204	208	22	+	+	NOUN
cana-2204	208	23	𝛽2	𝛽2	NOUN
cana-2204	208	24	)	)	PUNCT
cana-2204	208	25	}	}	PUNCT
cana-2204	208	26	{	{	PUNCT
cana-2204	208	27	𝑔(𝑉	𝑔(𝑉	PROPN
cana-2204	208	28	,	,	PUNCT
cana-2204	208	29	𝑌)𝜂(𝑈	𝑌)𝜂(𝑈	NUM
cana-2204	208	30	)	)	PUNCT
cana-2204	208	31	−	−	PROPN
cana-2204	208	32	𝑔(𝑈	𝑔(𝑈	PROPN
cana-2204	208	33	,	,	PUNCT
cana-2204	208	34	𝑌)𝜂(𝑉	𝑌)𝜂(𝑉	NUM
cana-2204	208	35	)	)	PUNCT
cana-2204	208	36	−	−	PROPN
cana-2204	208	37	𝑆(𝑉	𝑆(𝑉	PROPN
cana-2204	208	38	,	,	PUNCT
cana-2204	208	39	𝑌)𝜂(𝑈	𝑌)𝜂(𝑈	NUM
cana-2204	208	40	)	)	PUNCT
cana-2204	209	1	+	+	CCONJ
cana-2204	209	2	𝑆(𝑈	𝑆(𝑈	NUM
cana-2204	209	3	,	,	PUNCT
cana-2204	209	4	𝑌)𝜂(𝑉	𝑌)𝜂(𝑉	NOUN
cana-2204	209	5	)	)	PUNCT
cana-2204	209	6	}	}	PUNCT
cana-2204	209	7	]	]	PUNCT
cana-2204	210	1	=	=	SYM
cana-2204	210	2	0	0	PUNCT
cana-2204	210	3	(	(	PUNCT
cana-2204	210	4	5.12	5.12	NUM
cana-2204	210	5	)	)	PUNCT
cana-2204	210	6	from	from	ADP
cana-2204	210	7	equation	equation	NOUN
cana-2204	210	8	(	(	PUNCT
cana-2204	210	9	5.4	5.4	NUM
cana-2204	210	10	)	)	PUNCT
cana-2204	210	11	and	and	CCONJ
cana-2204	210	12	(	(	PUNCT
cana-2204	210	13	5.10	5.10	NUM
cana-2204	210	14	)	)	PUNCT
cana-2204	210	15	,	,	PUNCT
cana-2204	210	16	we	we	PRON
cana-2204	210	17	have	have	VERB
cana-2204	210	18	𝑆(𝑌	𝑆(𝑌	NOUN
cana-2204	210	19	,	,	PUNCT
cana-2204	210	20	𝑍	𝑍	PROPN
cana-2204	210	21	)	)	PUNCT
cana-2204	210	22	=	=	SYM
cana-2204	210	23	{	{	PUNCT
cana-2204	210	24	𝑟	𝑟	X
cana-2204	210	25	(	(	PUNCT
cana-2204	210	26	𝑛−1	𝑛−1	NUM
cana-2204	210	27	)	)	PUNCT
cana-2204	210	28	−	−	NOUN
cana-2204	210	29	𝜖(𝛼2	𝜖(𝛼2	NOUN
cana-2204	210	30	+	+	SYM
cana-2204	210	31	𝛽2	𝛽2	NOUN
cana-2204	210	32	)	)	PUNCT
cana-2204	210	33	}	}	PUNCT
cana-2204	210	34	𝑔(𝑌	𝑔(𝑌	NUM
cana-2204	210	35	,	,	PUNCT
cana-2204	210	36	𝑍	𝑍	PROPN
cana-2204	210	37	)	)	PUNCT
cana-2204	210	38	+	+	CCONJ
cana-2204	210	39	{	{	PUNCT
cana-2204	210	40	𝑟	𝑟	X
cana-2204	210	41	(	(	PUNCT
cana-2204	210	42	𝑛−1	𝑛−1	NUM
cana-2204	210	43	)	)	PUNCT
cana-2204	210	44	−	−	NOUN
cana-2204	210	45	𝜖𝑛(𝛼2	𝜖𝑛(𝛼2	ADP
cana-2204	210	46	+	+	CCONJ
cana-2204	210	47	𝛽2	𝛽2	NOUN
cana-2204	210	48	)	)	PUNCT
cana-2204	210	49	}	}	PUNCT
cana-2204	210	50	𝜂(𝑌)𝜂(𝑍	𝜂(𝑌)𝜂(𝑍	NOUN
cana-2204	210	51	)	)	PUNCT
cana-2204	210	52	(	(	PUNCT
cana-2204	210	53	5.13	5.13	NUM
cana-2204	210	54	)	)	PUNCT
cana-2204	210	55	using	use	VERB
cana-2204	210	56	equation	equation	NOUN
cana-2204	210	57	(	(	PUNCT
cana-2204	210	58	5.12	5.12	NUM
cana-2204	210	59	)	)	PUNCT
cana-2204	210	60	and	and	CCONJ
cana-2204	210	61	(	(	PUNCT
cana-2204	210	62	5.11	5.11	NUM
cana-2204	210	63	)	)	PUNCT
cana-2204	210	64	,	,	PUNCT
cana-2204	210	65	we	we	PRON
cana-2204	210	66	have	have	VERB
cana-2204	210	67	𝐶	𝐶	PROPN
cana-2204	210	68	′(𝑈	′(𝑈	NOUN
cana-2204	210	69	,	,	PUNCT
cana-2204	210	70	𝑉,𝑊	𝑉,𝑊	NOUN
cana-2204	210	71	,	,	PUNCT
cana-2204	210	72	𝑌	𝑌	PROPN
cana-2204	210	73	)	)	PUNCT
cana-2204	210	74	=	=	SYM
cana-2204	210	75	0	0	NUM
cana-2204	210	76	(	(	PUNCT
cana-2204	210	77	5.14	5.14	NUM
cana-2204	210	78	)	)	PUNCT
cana-2204	210	79	from	from	ADP
cana-2204	210	80	the	the	DET
cana-2204	210	81	above	above	ADJ
cana-2204	210	82	equation	equation	NOUN
cana-2204	210	83	we	we	PRON
cana-2204	210	84	can	can	AUX
cana-2204	210	85	see	see	VERB
cana-2204	210	86	that	that	DET
cana-2204	210	87	𝑅.	𝑅.	NOUN
cana-2204	210	88	𝐶	𝐶	NOUN
cana-2204	210	89	=	=	SYM
cana-2204	210	90	0	0	PROPN
cana-2204	210	91	,	,	PUNCT
cana-2204	210	92	this	this	PRON
cana-2204	210	93	implies	imply	VERB
cana-2204	210	94	that	that	SCONJ
cana-2204	210	95	𝐶	𝐶	PROPN
cana-2204	210	96	=	=	NOUN
cana-2204	210	97	0	0	PROPN
cana-2204	210	98	.	.	PUNCT
cana-2204	211	1	hence	hence	ADV
cana-2204	211	2	this	this	DET
cana-2204	211	3	condition	condition	NOUN
cana-2204	211	4	with	with	ADP
cana-2204	211	5	the	the	DET
cana-2204	211	6	help	help	NOUN
cana-2204	211	7	of	of	ADP
cana-2204	211	8	theorem	theorem	ADJ
cana-2204	211	9	4.2	4.2	NUM
cana-2204	211	10	,	,	PUNCT
cana-2204	211	11	gives	give	VERB
cana-2204	211	12	the	the	DET
cana-2204	211	13	following	follow	VERB
cana-2204	211	14	results	result	NOUN
cana-2204	211	15	:	:	PUNCT
cana-2204	211	16	theorem	theorem	VERB
cana-2204	211	17	5.1.an𝑛	5.1.an𝑛	NUM
cana-2204	211	18	−dimensional	−dimensional	ADJ
cana-2204	211	19	weyl	weyl	VERB
cana-2204	211	20	-	-	PUNCT
cana-2204	211	21	semi	semi	ADJ
cana-2204	211	22	-	-	ADJ
cana-2204	211	23	symmetric	symmetric	ADJ
cana-2204	211	24	(	(	PUNCT
cana-2204	211	25	𝜖	𝜖	NOUN
cana-2204	211	26	)	)	PUNCT
cana-2204	211	27	−	−	PROPN
cana-2204	211	28	lorentzian	lorentzian	ADJ
cana-2204	211	29	para	para	NOUN
cana-2204	211	30	-	-	PUNCT
cana-2204	211	31	sasakian	sasakian	NOUN
cana-2204	211	32	manifolds	manifold	NOUN
cana-2204	211	33	with	with	ADP
cana-2204	211	34	parallelized	parallelize	VERB
cana-2204	211	35	generalized	generalize	VERB
cana-2204	211	36	symmetric	symmetric	ADJ
cana-2204	211	37	metric	metric	ADJ
cana-2204	211	38	connection	connection	NOUN
cana-2204	211	39	is	be	AUX
cana-2204	211	40	of	of	ADP
cana-2204	211	41	quasi	quasi	ADJ
cana-2204	211	42	-	-	ADJ
cana-2204	211	43	constant	constant	ADJ
cana-2204	211	44	curvature	curvature	NOUN
cana-2204	211	45	.	.	PUNCT
cana-2204	212	1	theorem	theorem	VERB
cana-2204	212	2	4.3	4.3	NUM
cana-2204	212	3	and	and	CCONJ
cana-2204	212	4	equation	equation	NOUN
cana-2204	212	5	(	(	PUNCT
cana-2204	212	6	5.14	5.14	NUM
cana-2204	212	7	)	)	PUNCT
cana-2204	212	8	leads	lead	VERB
cana-2204	212	9	to	to	ADP
cana-2204	212	10	the	the	DET
cana-2204	212	11	following	following	ADJ
cana-2204	212	12	result	result	NOUN
cana-2204	212	13	:	:	PUNCT
cana-2204	212	14	corollary	corollary	ADJ
cana-2204	212	15	5.2.an𝑛	5.2.an𝑛	NUM
cana-2204	212	16	−dimensional	−dimensional	ADP
cana-2204	212	17	weyl	weyl	VERB
cana-2204	212	18	-	-	PUNCT
cana-2204	212	19	semi	semi	ADJ
cana-2204	212	20	-	-	ADJ
cana-2204	212	21	symmetric	symmetric	ADJ
cana-2204	212	22	(	(	PUNCT
cana-2204	212	23	𝜖	𝜖	NOUN
cana-2204	212	24	)	)	PUNCT
cana-2204	212	25	−	−	PROPN
cana-2204	212	26	lorentzian	lorentzian	ADJ
cana-2204	212	27	para	para	NOUN
cana-2204	212	28	-	-	PUNCT
cana-2204	212	29	sasakian	sasakian	NOUN
cana-2204	212	30	manifolds	manifold	NOUN
cana-2204	212	31	with	with	ADP
cana-2204	212	32	parallelized	parallelize	VERB
cana-2204	212	33	generalized	generalize	VERB
cana-2204	212	34	symmetric	symmetric	ADJ
cana-2204	212	35	metric	metric	ADJ
cana-2204	212	36	connection	connection	NOUN
cana-2204	212	37	is	be	AUX
cana-2204	212	38	a	a	DET
cana-2204	212	39	𝜓(𝐹)𝑛.	𝜓(𝐹)𝑛.	NOUN
cana-2204	212	40	application	application	NOUN
cana-2204	212	41	.	.	PUNCT
cana-2204	213	1	(	(	PUNCT
cana-2204	213	2	𝜖	𝜖	NOUN
cana-2204	213	3	)	)	PUNCT
cana-2204	213	4	−	−	PROPN
cana-2204	213	5	lorentzian	lorentzian	ADJ
cana-2204	213	6	para	para	NOUN
cana-2204	213	7	-	-	PUNCT
cana-2204	213	8	sasakian	sasakian	NOUN
cana-2204	213	9	manifolds	manifold	NOUN
cana-2204	213	10	with	with	ADP
cana-2204	213	11	parallelized	parallelize	VERB
cana-2204	213	12	generalized	generalize	VERB
cana-2204	213	13	symmetric	symmetric	ADJ
cana-2204	213	14	metric	metric	ADJ
cana-2204	213	15	connection	connection	NOUN
cana-2204	213	16	are	be	AUX
cana-2204	213	17	used	use	VERB
cana-2204	213	18	in	in	ADP
cana-2204	213	19	the	the	DET
cana-2204	213	20	newtons	newton	NOUN
cana-2204	213	21	law	law	NOUN
cana-2204	213	22	of	of	ADP
cana-2204	213	23	gravitational	gravitational	ADJ
cana-2204	213	24	field	field	NOUN
cana-2204	213	25	and	and	CCONJ
cana-2204	213	26	theory	theory	NOUN
cana-2204	213	27	of	of	ADP
cana-2204	213	28	relativity	relativity	NOUN
cana-2204	213	29	.	.	PUNCT
cana-2204	214	1	acknowledgement	acknowledgement	NOUN
cana-2204	214	2	.	.	PUNCT
cana-2204	215	1	the	the	DET
cana-2204	215	2	authors	author	NOUN
cana-2204	215	3	are	be	AUX
cana-2204	215	4	thankful	thankful	ADJ
cana-2204	215	5	to	to	ADP
cana-2204	215	6	integral	integral	ADJ
cana-2204	215	7	university	university	NOUN
cana-2204	215	8	,	,	PUNCT
cana-2204	215	9	lucknow	lucknow	NOUN
cana-2204	215	10	for	for	ADP
cana-2204	215	11	giving	give	VERB
cana-2204	215	12	manuscript	manuscript	NOUN
cana-2204	215	13	communication	communication	NOUN
cana-2204	215	14	number	number	NOUN
cana-2204	215	15	iu	iu	ADP
cana-2204	215	16	/	/	SYM
cana-2204	215	17	r&d/2024	r&d/2024	NOUN
cana-2204	215	18	-	-	PUNCT
cana-2204	215	19	mcn0002708	mcn0002708	NOUN
cana-2204	215	20	.	.	PUNCT
cana-2204	216	1	references	reference	NOUN
cana-2204	216	2	[	[	X
cana-2204	216	3	1	1	NUM
cana-2204	216	4	]	]	PUNCT
cana-2204	216	5	a.	a.	NOUN
cana-2204	216	6	bejancu	bejancu	PROPN
cana-2204	216	7	and	and	CCONJ
cana-2204	216	8	k.	k.	PROPN
cana-2204	216	9	l.	l.	PROPN
cana-2204	217	1	duggal	duggal	PROPN
cana-2204	217	2	,	,	PUNCT
cana-2204	217	3	real	real	ADJ
cana-2204	217	4	hypersurfaces	hypersurface	NOUN
cana-2204	217	5	of	of	ADP
cana-2204	217	6	indefinite	indefinite	ADJ
cana-2204	217	7	kaehler	kaehler	NOUN
cana-2204	217	8	manifolds	manifold	NOUN
cana-2204	217	9	,	,	PUNCT
cana-2204	217	10	int	int	NOUN
cana-2204	217	11	.	.	PUNCT
cana-2204	218	1	j.	j.	PROPN
cana-2204	218	2	math	math	PROPN
cana-2204	218	3	.	.	PUNCT
cana-2204	219	1	math	math	NOUN
cana-2204	219	2	.	.	PUNCT
cana-2204	220	1	sci	sci	PROPN
cana-2204	220	2	.	.	PUNCT
cana-2204	221	1	16no	16no	ADJ
cana-2204	221	2	.	.	PUNCT
cana-2204	222	1	3	3	NUM
cana-2204	222	2	,	,	PUNCT
cana-2204	222	3	(	(	PUNCT
cana-2204	222	4	1993	1993	NUM
cana-2204	222	5	)	)	PUNCT
cana-2204	222	6	,	,	PUNCT
cana-2204	222	7	545	545	NUM
cana-2204	222	8	-	-	SYM
cana-2204	222	9	556	556	NUM
cana-2204	222	10	.	.	PUNCT
cana-2204	223	1	[	[	X
cana-2204	223	2	2	2	NUM
cana-2204	223	3	]	]	X
cana-2204	223	4	b.y	b.y	PROPN
cana-2204	223	5	.	.	PROPN
cana-2204	223	6	chen	chen	PROPN
cana-2204	223	7	and	and	CCONJ
cana-2204	223	8	k.yano	k.yano	PROPN
cana-2204	223	9	,	,	PUNCT
cana-2204	223	10	hypersurfaces	hypersurface	NOUN
cana-2204	223	11	of	of	ADP
cana-2204	223	12	a	a	DET
cana-2204	223	13	conformally	conformally	ADV
cana-2204	223	14	flat	flat	ADJ
cana-2204	223	15	space	space	NOUN
cana-2204	223	16	.	.	PUNCT
cana-2204	224	1	tensor	tensor	NOUN
cana-2204	224	2	(	(	PUNCT
cana-2204	224	3	n.s	n.s	PROPN
cana-2204	224	4	)	)	PUNCT
cana-2204	224	5	26(1972	26(1972	PROPN
cana-2204	224	6	)	)	PUNCT
cana-2204	224	7	,	,	PUNCT
cana-2204	224	8	318	318	NUM
cana-2204	224	9	-	-	SYM
cana-2204	224	10	322	322	NUM
cana-2204	224	11	.	.	PUNCT
cana-2204	224	12	communications	communication	NOUN
cana-2204	224	13	on	on	ADP
cana-2204	224	14	applied	apply	VERB
cana-2204	224	15	nonlinear	nonlinear	ADJ
cana-2204	224	16	analysis	analysis	NOUN
cana-2204	224	17	issn	issn	NOUN
cana-2204	224	18	:	:	PUNCT
cana-2204	224	19	1074	1074	NUM
cana-2204	224	20	-	-	PUNCT
cana-2204	224	21	133x	133x	NUM
cana-2204	224	22	vol	vol	NOUN
cana-2204	224	23	32	32	NUM
cana-2204	224	24	no	no	NOUN
cana-2204	224	25	.	.	PUNCT
cana-2204	225	1	1s	1s	NUM
cana-2204	225	2	(	(	PUNCT
cana-2204	225	3	2025	2025	NUM
cana-2204	225	4	)	)	PUNCT
cana-2204	225	5	411	411	NUM
cana-2204	225	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2204	226	1	[	[	X
cana-2204	226	2	3	3	X
cana-2204	226	3	]	]	X
cana-2204	226	4	s.s	s.s	PROPN
cana-2204	226	5	.	.	PROPN
cana-2204	226	6	chern	chern	PROPN
cana-2204	226	7	,	,	PUNCT
cana-2204	226	8	on	on	ADP
cana-2204	226	9	the	the	DET
cana-2204	226	10	curvature	curvature	NOUN
cana-2204	226	11	and	and	CCONJ
cana-2204	226	12	characteristics	characteristic	NOUN
cana-2204	226	13	classes	class	NOUN
cana-2204	226	14	of	of	ADP
cana-2204	226	15	a	a	DET
cana-2204	226	16	riemannian	riemannian	ADJ
cana-2204	226	17	manifold	manifold	NOUN
cana-2204	226	18	,	,	PUNCT
cana-2204	226	19	abh	abh	PROPN
cana-2204	226	20	.	.	PUNCT
cana-2204	226	21	math	math	PROPN
cana-2204	226	22	.	.	PUNCT
cana-2204	227	1	semin	semin	PROPN
cana-2204	227	2	.	.	PUNCT
cana-2204	228	1	univ	univ	PROPN
cana-2204	228	2	.	.	PROPN
cana-2204	229	1	hambg	hambg	VERB
cana-2204	229	2	20	20	NUM
cana-2204	229	3	(	(	PUNCT
cana-2204	229	4	1956	1956	NUM
cana-2204	229	5	)	)	PUNCT
cana-2204	229	6	,	,	PUNCT
cana-2204	229	7	117	117	NUM
cana-2204	229	8	-	-	SYM
cana-2204	229	9	126	126	NUM
cana-2204	229	10	.	.	PUNCT
cana-2204	230	1	[	[	X
cana-2204	230	2	4	4	X
cana-2204	230	3	]	]	PUNCT
cana-2204	230	4	x.	x.	NOUN
cana-2204	230	5	xufeng	xufeng	PROPN
cana-2204	230	6	and	and	CCONJ
cana-2204	230	7	c.	c.	PROPN
cana-2204	230	8	xiaol	xiaol	PROPN
cana-2204	230	9	,	,	PUNCT
cana-2204	230	10	two	two	NUM
cana-2204	230	11	theorems	theorem	NOUN
cana-2204	230	12	on	on	ADP
cana-2204	230	13	(	(	PUNCT
cana-2204	230	14	ϵ	ϵ	NOUN
cana-2204	230	15	)	)	PUNCT
cana-2204	230	16	−sasakian	−sasakian	PROPN
cana-2204	230	17	manifolds	manifold	NOUN
cana-2204	230	18	,	,	PUNCT
cana-2204	230	19	int	int	NOUN
cana-2204	230	20	.	.	PUNCT
cana-2204	231	1	j.	j.	PROPN
cana-2204	231	2	math	math	PROPN
cana-2204	231	3	.	.	PUNCT
cana-2204	232	1	math	math	NOUN
cana-2204	232	2	.	.	PUNCT
cana-2204	233	1	sci	sci	PROPN
cana-2204	233	2	.	.	PROPN
cana-2204	233	3	21	21	NUM
cana-2204	233	4	(	(	PUNCT
cana-2204	233	5	1998	1998	NUM
cana-2204	233	6	)	)	PUNCT
cana-2204	233	7	,	,	PUNCT
cana-2204	233	8	no	no	INTJ
cana-2204	233	9	.	.	NOUN
cana-2204	233	10	2	2	NUM
cana-2204	233	11	,	,	PUNCT
cana-2204	233	12	249254	249254	NUM
cana-2204	233	13	.	.	PUNCT
cana-2204	234	1	[	[	X
cana-2204	234	2	5	5	X
cana-2204	234	3	]	]	PUNCT
cana-2204	234	4	k.	k.	PROPN
cana-2204	234	5	l.	l.	PROPN
cana-2204	234	6	duggal	duggal	PROPN
cana-2204	234	7	,	,	PUNCT
cana-2204	234	8	space	space	NOUN
cana-2204	234	9	time	time	NOUN
cana-2204	234	10	manifold	manifold	ADJ
cana-2204	234	11	and	and	CCONJ
cana-2204	234	12	contact	contact	NOUN
cana-2204	234	13	structures	structure	NOUN
cana-2204	234	14	,	,	PUNCT
cana-2204	234	15	int	int	NOUN
cana-2204	234	16	.	.	PUNCT
cana-2204	235	1	j.	j.	PROPN
cana-2204	235	2	math	math	PROPN
cana-2204	235	3	.	.	PUNCT
cana-2204	236	1	mat	mat	PROPN
cana-2204	236	2	.	.	PUNCT
cana-2204	237	1	sci	sci	PROPN
cana-2204	237	2	.	.	PROPN
cana-2204	237	3	13	13	NUM
cana-2204	237	4	(	(	PUNCT
cana-2204	237	5	1990	1990	NUM
cana-2204	237	6	)	)	PUNCT
cana-2204	237	7	,	,	PUNCT
cana-2204	237	8	no	no	INTJ
cana-2204	237	9	.	.	NOUN
cana-2204	237	10	3	3	NUM
cana-2204	237	11	,	,	PUNCT
cana-2204	237	12	545	545	NUM
cana-2204	237	13	-	-	SYM
cana-2204	237	14	553	553	NUM
cana-2204	237	15	.	.	PUNCT
cana-2204	238	1	[	[	X
cana-2204	238	2	6	6	NUM
cana-2204	238	3	]	]	PUNCT
cana-2204	238	4	a.	a.	NOUN
cana-2204	238	5	hasseb	hasseb	PROPN
cana-2204	238	6	,	,	PUNCT
cana-2204	238	7	a.	a.	PROPN
cana-2204	238	8	prakash	prakash	PROPN
cana-2204	238	9	and	and	CCONJ
cana-2204	238	10	m.d	m.d	PROPN
cana-2204	238	11	.	.	PROPN
cana-2204	238	12	siddiqui	siddiqui	PROPN
cana-2204	238	13	,	,	PUNCT
cana-2204	238	14	a	a	DET
cana-2204	238	15	quarter	quarter	NOUN
cana-2204	238	16	symmetric	symmetric	ADJ
cana-2204	238	17	metric	metric	ADJ
cana-2204	238	18	connection	connection	NOUN
cana-2204	238	19	in	in	ADP
cana-2204	238	20	an	an	DET
cana-2204	238	21	(	(	PUNCT
cana-2204	238	22	ϵ	ϵ	NOUN
cana-2204	238	23	)	)	PUNCT
cana-2204	238	24	−	−	PROPN
cana-2204	238	25	lorentzian	lorentzian	ADJ
cana-2204	238	26	parasasakian	parasasakian	PROPN
cana-2204	238	27	manifold	manifold	NOUN
cana-2204	238	28	,	,	PUNCT
cana-2204	238	29	acta	acta	PROPN
cana-2204	238	30	math	math	PROPN
cana-2204	238	31	.	.	PUNCT
cana-2204	239	1	univ	univ	PROPN
cana-2204	239	2	.	.	PUNCT
cana-2204	239	3	comenianae	comenianae	PROPN
cana-2204	239	4	,	,	PUNCT
cana-2204	239	5	88(1	88(1	NOUN
cana-2204	239	6	)	)	PUNCT
cana-2204	239	7	,	,	PUNCT
cana-2204	239	8	(	(	PUNCT
cana-2204	239	9	2017	2017	NUM
cana-2204	239	10	)	)	PUNCT
cana-2204	239	11	,	,	PUNCT
cana-2204	239	12	143	143	NUM
cana-2204	239	13	-	-	SYM
cana-2204	239	14	152	152	NUM
cana-2204	239	15	.	.	PUNCT
cana-2204	240	1	[	[	X
cana-2204	240	2	7	7	X
cana-2204	240	3	]	]	PUNCT
cana-2204	240	4	k.	k.	PROPN
cana-2204	240	5	matsumoto	matsumoto	PROPN
cana-2204	240	6	,	,	PUNCT
cana-2204	240	7	on	on	ADP
cana-2204	240	8	lorentzian	lorentzian	ADJ
cana-2204	240	9	para	para	NOUN
cana-2204	240	10	-	-	PUNCT
cana-2204	240	11	contact	contact	NOUN
cana-2204	240	12	manifolds	manifold	NOUN
cana-2204	240	13	,	,	PUNCT
cana-2204	240	14	bull	bull	NOUN
cana-2204	240	15	.	.	PUNCT
cana-2204	241	1	yamagata	yamagata	PROPN
cana-2204	241	2	univ	univ	PROPN
cana-2204	241	3	.	.	PROPN
cana-2204	241	4	,	,	PUNCT
cana-2204	241	5	12(2	12(2	NUM
cana-2204	241	6	)	)	PUNCT
cana-2204	241	7	,	,	PUNCT
cana-2204	241	8	(	(	PUNCT
cana-2204	241	9	1989	1989	NUM
cana-2204	241	10	)	)	PUNCT
cana-2204	241	11	,	,	PUNCT
cana-2204	241	12	151	151	NUM
cana-2204	241	13	-	-	SYM
cana-2204	241	14	156	156	NUM
cana-2204	241	15	.	.	PUNCT
cana-2204	242	1	[	[	X
cana-2204	242	2	8	8	NUM
cana-2204	242	3	]	]	X
cana-2204	242	4	u.	u.	PROPN
cana-2204	242	5	c.	c.	PROPN
cana-2204	242	6	de	de	PROPN
cana-2204	242	7	and	and	CCONJ
cana-2204	242	8	a.	a.	NOUN
cana-2204	242	9	sarkar	sarkar	PROPN
cana-2204	242	10	,	,	PUNCT
cana-2204	242	11	on	on	ADP
cana-2204	242	12	(	(	PUNCT
cana-2204	242	13	ϵ	ϵ	NOUN
cana-2204	242	14	)	)	PUNCT
cana-2204	242	15	−kenmotsu	−kenmotsu	PROPN
cana-2204	242	16	manifolds	manifold	NOUN
cana-2204	242	17	,	,	PUNCT
cana-2204	242	18	hadronic	hadronic	ADJ
cana-2204	242	19	j.	j.	PROPN
cana-2204	242	20	32	32	NUM
cana-2204	242	21	(	(	PUNCT
cana-2204	242	22	2009	2009	NUM
cana-2204	242	23	)	)	PUNCT
cana-2204	242	24	,	,	PUNCT
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cana-2204	242	27	2	2	NUM
cana-2204	242	28	,	,	PUNCT
cana-2204	242	29	231	231	NUM
cana-2204	242	30	-	-	SYM
cana-2204	242	31	242	242	NUM
cana-2204	242	32	.	.	PUNCT
cana-2204	243	1	[	[	X
cana-2204	243	2	9	9	NUM
cana-2204	243	3	]	]	SYM
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cana-2204	243	8	rosca	rosca	PROPN
cana-2204	243	9	,	,	PUNCT
cana-2204	243	10	on	on	ADP
cana-2204	243	11	lorentzian	lorentzian	ADJ
cana-2204	243	12	p	p	PROPN
cana-2204	243	13	-	-	PUNCT
cana-2204	243	14	sasakian	sasakian	ADJ
cana-2204	243	15	manifolds	manifold	NOUN
cana-2204	243	16	,	,	PUNCT
cana-2204	243	17	classical	classical	ADJ
cana-2204	243	18	analysis	analysis	NOUN
cana-2204	243	19	,	,	PUNCT
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cana-2204	243	23	.	.	PUNCT
cana-2204	243	24	,	,	PUNCT
cana-2204	243	25	singapore	singapore	PROPN
cana-2204	243	26	,	,	PUNCT
cana-2204	243	27	(	(	PUNCT
cana-2204	243	28	1992	1992	NUM
cana-2204	243	29	)	)	PUNCT
cana-2204	243	30	,	,	PUNCT
cana-2204	243	31	155	155	NUM
cana-2204	243	32	-	-	SYM
cana-2204	243	33	169	169	NUM
cana-2204	243	34	.	.	PUNCT
cana-2204	244	1	[	[	X
cana-2204	244	2	10	10	NUM
cana-2204	244	3	]	]	X
cana-2204	244	4	i.	i.	PROPN
cana-2204	244	5	mihai	mihai	PROPN
cana-2204	244	6	and	and	CCONJ
cana-2204	244	7	a.	a.	NOUN
cana-2204	244	8	a.	a.	NOUN
cana-2204	244	9	shaikh	shaikh	PROPN
cana-2204	244	10	and	and	CCONJ
cana-2204	244	11	u.	u.	PROPN
cana-2204	244	12	c.	c.	PROPN
cana-2204	244	13	de	de	PROPN
cana-2204	244	14	,	,	PUNCT
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cana-2204	244	16	lorentzian	lorentzian	ADJ
cana-2204	244	17	para	para	PROPN
cana-2204	244	18	-	-	PUNCT
cana-2204	244	19	sasakian	sasakian	ADJ
cana-2204	244	20	manifolds	manifold	NOUN
cana-2204	244	21	,	,	PUNCT
cana-2204	244	22	rendicontisem	rendicontisem	PROPN
cana-2204	244	23	.	.	PROPN
cana-2204	244	24	math	math	PROPN
cana-2204	244	25	.	.	PUNCT
cana-2204	245	1	messina	messina	PROPN
cana-2204	245	2	,	,	PUNCT
cana-2204	245	3	series	series	PROPN
cana-2204	245	4	ii	ii	PROPN
cana-2204	245	5	1999	1999	NUM
cana-2204	245	6	.	.	PUNCT
cana-2204	246	1	[	[	X
cana-2204	246	2	11	11	NUM
cana-2204	246	3	]	]	PUNCT
cana-2204	246	4	k.	k.	PROPN
cana-2204	246	5	matsumoto	matsumoto	PROPN
cana-2204	246	6	and	and	CCONJ
cana-2204	246	7	i.	i.	PROPN
cana-2204	246	8	mihai	mihai	PROPN
cana-2204	246	9	,	,	PUNCT
cana-2204	246	10	on	on	ADP
cana-2204	246	11	a	a	DET
cana-2204	246	12	certain	certain	ADJ
cana-2204	246	13	transformation	transformation	NOUN
cana-2204	246	14	in	in	ADP
cana-2204	246	15	a	a	DET
cana-2204	246	16	lorentzian	lorentzian	ADJ
cana-2204	246	17	para	para	NOUN
cana-2204	246	18	-	-	PUNCT
cana-2204	246	19	sasakian	sasakian	NOUN
cana-2204	246	20	manifold	manifold	NOUN
cana-2204	246	21	,	,	PUNCT
cana-2204	246	22	tensor	tensor	NOUN
cana-2204	246	23	(	(	PUNCT
cana-2204	246	24	n.s	n.s	PROPN
cana-2204	246	25	.	.	PROPN
cana-2204	246	26	)	)	PUNCT
cana-2204	247	1	47	47	NUM
cana-2204	247	2	(	(	PUNCT
cana-2204	247	3	1988	1988	NUM
cana-2204	247	4	)	)	PUNCT
cana-2204	247	5	.	.	PUNCT
cana-2204	248	1	no	no	INTJ
cana-2204	248	2	.	.	NOUN
cana-2204	248	3	2	2	NUM
cana-2204	248	4	,	,	PUNCT
cana-2204	248	5	189	189	NUM
cana-2204	248	6	-	-	SYM
cana-2204	248	7	197	197	NUM
cana-2204	248	8	.	.	PUNCT
cana-2204	249	1	[	[	X
cana-2204	249	2	12	12	NUM
cana-2204	249	3	]	]	PUNCT
cana-2204	249	4	a.	a.	PROPN
cana-2204	249	5	k.	k.	PROPN
cana-2204	249	6	hakami	hakami	PROPN
cana-2204	249	7	,	,	PUNCT
cana-2204	249	8	m.	m.	PROPN
cana-2204	249	9	d.	d.	PROPN
cana-2204	249	10	siddiqi	siddiqi	PROPN
cana-2204	249	11	,	,	PUNCT
cana-2204	249	12	o.	o.	PROPN
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cana-2204	249	15	t.	t.	PROPN
cana-2204	249	16	khan	khan	PROPN
cana-2204	249	17	,	,	PUNCT
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cana-2204	249	21	(	(	PUNCT
cana-2204	249	22	α	α	NOUN
cana-2204	249	23	,	,	PUNCT
cana-2204	249	24	β	β	NOUN
cana-2204	249	25	)	)	PUNCT
cana-2204	249	26	−	−	NOUN
cana-2204	249	27	type	type	NOUN
cana-2204	249	28	almost	almost	ADV
cana-2204	249	29	contact	contact	NOUN
cana-2204	249	30	manifolds	manifold	NOUN
cana-2204	249	31	with	with	ADP
cana-2204	249	32	quasi	quasi	ADJ
cana-2204	249	33	-	-	ADJ
cana-2204	249	34	hemislant	hemislant	ADJ
cana-2204	249	35	factors	factor	NOUN
cana-2204	249	36	,	,	PUNCT
cana-2204	249	37	symmetry	symmetry	NOUN
cana-2204	249	38	,	,	PUNCT
cana-2204	249	39	15	15	NUM
cana-2204	249	40	(	(	PUNCT
cana-2204	249	41	60	60	NUM
cana-2204	249	42	)	)	PUNCT
cana-2204	249	43	(	(	PUNCT
cana-2204	249	44	2023	2023	NUM
cana-2204	249	45	)	)	PUNCT
cana-2204	249	46	,	,	PUNCT
cana-2204	249	47	1270	1270	NUM
cana-2204	249	48	;	;	PUNCT
cana-2204	249	49	doi.org/10.3390/sym15061270	doi.org/10.3390/sym15061270	ADJ
cana-2204	249	50	.	.	PUNCT
cana-2204	250	1	[	[	X
cana-2204	250	2	13	13	NUM
cana-2204	250	3	]	]	X
cana-2204	250	4	o.	o.	NOUN
cana-2204	250	5	bahadir	bahadir	NOUN
cana-2204	250	6	,	,	PUNCT
cana-2204	250	7	m.	m.	NOUN
cana-2204	250	8	a.	a.	PROPN
cana-2204	250	9	choudhary	choudhary	PROPN
cana-2204	250	10	and	and	CCONJ
cana-2204	250	11	s.	s.	PROPN
cana-2204	250	12	pandey	pandey	PROPN
cana-2204	250	13	,	,	PUNCT
cana-2204	250	14	lp	lp	ADJ
cana-2204	250	15	-	-	ADJ
cana-2204	250	16	sasakian	sasakian	ADJ
cana-2204	250	17	manifolds	manifold	NOUN
cana-2204	250	18	with	with	ADP
cana-2204	250	19	generalized	generalized	ADJ
cana-2204	250	20	symmetric	symmetric	ADJ
cana-2204	250	21	metric	metric	ADJ
cana-2204	250	22	connection	connection	NOUN
cana-2204	250	23	,	,	PUNCT
cana-2204	250	24	novi	novi	PROPN
cana-2204	250	25	sad	sad	PROPN
cana-2204	250	26	j.	j.	PROPN
cana-2204	250	27	math	math	PROPN
cana-2204	250	28	.	.	PUNCT
cana-2204	251	1	vol	vol	NOUN
cana-2204	251	2	.	.	PROPN
cana-2204	252	1	51	51	NUM
cana-2204	252	2	,	,	PUNCT
cana-2204	252	3	no	no	INTJ
cana-2204	252	4	.	.	NOUN
cana-2204	252	5	2	2	NUM
cana-2204	252	6	,	,	PUNCT
cana-2204	252	7	(	(	PUNCT
cana-2204	252	8	2021	2021	NUM
cana-2204	252	9	)	)	PUNCT
cana-2204	252	10	,	,	PUNCT
cana-2204	252	11	75	75	NUM
cana-2204	252	12	-	-	SYM
cana-2204	252	13	87	87	NUM
cana-2204	252	14	.	.	PUNCT
cana-2204	253	1	[	[	X
cana-2204	253	2	14	14	NUM
cana-2204	253	3	]	]	X
cana-2204	253	4	t.	t.	PROPN
cana-2204	253	5	khan	khan	PROPN
cana-2204	253	6	,	,	PUNCT
cana-2204	253	7	s.	s.	PROPN
cana-2204	253	8	a.	a.	PROPN
cana-2204	253	9	khan	khan	PROPN
cana-2204	253	10	and	and	CCONJ
cana-2204	253	11	m.	m.	PROPN
cana-2204	253	12	ahmad	ahmad	PROPN
cana-2204	253	13	,	,	PUNCT
cana-2204	253	14	on	on	ADP
cana-2204	253	15	semi	semi	ADJ
cana-2204	253	16	-	-	ADJ
cana-2204	253	17	invariant	invariant	ADJ
cana-2204	253	18	submanifolds	submanifold	NOUN
cana-2204	253	19	of	of	ADP
cana-2204	253	20	nearly	nearly	ADV
cana-2204	253	21	hyperbolic	hyperbolic	ADJ
cana-2204	253	22	kenmotsu	kenmotsu	NOUN
cana-2204	253	23	manifolds	manifold	NOUN
cana-2204	253	24	with	with	ADP
cana-2204	253	25	semi	semi	ADJ
cana-2204	253	26	-	-	ADJ
cana-2204	253	27	symmetric	symmetric	ADJ
cana-2204	253	28	metric	metric	ADJ
cana-2204	253	29	connection	connection	NOUN
cana-2204	253	30	,	,	PUNCT
cana-2204	253	31	int	int	NOUN
cana-2204	253	32	.	.	PUNCT
cana-2204	254	1	journal	journal	PROPN
cana-2204	254	2	of	of	ADP
cana-2204	254	3	engineering	engineering	NOUN
cana-2204	254	4	research	research	NOUN
cana-2204	254	5	and	and	CCONJ
cana-2204	254	6	application	application	NOUN
cana-2204	254	7	,	,	PUNCT
cana-2204	254	8	vol	vol	NOUN
cana-2204	254	9	.	.	PROPN
cana-2204	254	10	4	4	NUM
cana-2204	254	11	,	,	PUNCT
cana-2204	254	12	issue	issue	NOUN
cana-2204	254	13	9	9	NUM
cana-2204	254	14	(	(	PUNCT
cana-2204	254	15	version	version	NOUN
cana-2204	254	16	3	3	NUM
cana-2204	254	17	)	)	PUNCT
cana-2204	254	18	(	(	PUNCT
cana-2204	254	19	2014	2014	NUM
cana-2204	254	20	)	)	PUNCT
cana-2204	254	21	,	,	PUNCT
cana-2204	254	22	61	61	NUM
cana-2204	254	23	-	-	SYM
cana-2204	254	24	69	69	NUM
cana-2204	254	25	.	.	PUNCT
cana-2204	255	1	[	[	X
cana-2204	255	2	15	15	NUM
cana-2204	255	3	]	]	X
cana-2204	255	4	n.	n.	PROPN
cana-2204	255	5	s.	s.	PROPN
cana-2204	255	6	agashe	agashe	PROPN
cana-2204	255	7	and	and	CCONJ
cana-2204	255	8	m.	m.	PROPN
cana-2204	255	9	r.	r.	PROPN
cana-2204	255	10	chafle	chafle	PROPN
cana-2204	255	11	,	,	PUNCT
cana-2204	255	12	a	a	DET
cana-2204	255	13	semi	semi	ADJ
cana-2204	255	14	-	-	ADJ
cana-2204	255	15	symmetric	symmetric	ADJ
cana-2204	255	16	non	non	ADJ
cana-2204	255	17	-	-	ADJ
cana-2204	255	18	metric	metric	ADJ
cana-2204	255	19	connection	connection	NOUN
cana-2204	255	20	in	in	ADP
cana-2204	255	21	a	a	DET
cana-2204	255	22	riemannian	riemannian	ADJ
cana-2204	255	23	manifold	manifold	NOUN
cana-2204	255	24	,	,	PUNCT
cana-2204	255	25	indian	indian	ADJ
cana-2204	255	26	j.	j.	PROPN
cana-2204	255	27	pure	pure	PROPN
cana-2204	255	28	appl	appl	PROPN
cana-2204	255	29	.	.	PUNCT
cana-2204	255	30	math	math	PROPN
cana-2204	255	31	.	.	PUNCT
cana-2204	255	32	,	,	PUNCT
cana-2204	255	33	vol	vol	NOUN
cana-2204	255	34	.	.	PROPN
cana-2204	255	35	23	23	NUM
cana-2204	255	36	(	(	PUNCT
cana-2204	255	37	1992	1992	NUM
cana-2204	255	38	)	)	PUNCT
cana-2204	255	39	,	,	PUNCT
cana-2204	255	40	399	399	NUM
cana-2204	255	41	-	-	SYM
cana-2204	255	42	409	409	NUM
cana-2204	255	43	.	.	PUNCT
cana-2204	256	1	[	[	X
cana-2204	256	2	16	16	NUM
cana-2204	256	3	]	]	PUNCT
cana-2204	256	4	m.	m.	NOUN
cana-2204	256	5	ahmad	ahmad	PROPN
cana-2204	256	6	,	,	PUNCT
cana-2204	256	7	s.a.khan	s.a.khan	PROPN
cana-2204	256	8	and	and	CCONJ
cana-2204	256	9	t.	t.	PROPN
cana-2204	256	10	khan	khan	PROPN
cana-2204	256	11	,	,	PUNCT
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cana-2204	256	13	non	non	ADJ
cana-2204	256	14	-	-	ADJ
cana-2204	256	15	invariant	invariant	ADJ
cana-2204	256	16	hypersurfaces	hypersurface	NOUN
cana-2204	256	17	of	of	ADP
cana-2204	256	18	a	a	DET
cana-2204	256	19	nearly	nearly	ADV
cana-2204	256	20	hyperbolic	hyperbolic	ADJ
cana-2204	256	21	sasakian	sasakian	ADJ
cana-2204	256	22	manifold	manifold	NOUN
cana-2204	256	23	,	,	PUNCT
cana-2204	256	24	international	international	ADJ
cana-2204	256	25	journal	journal	NOUN
cana-2204	256	26	of	of	ADP
cana-2204	256	27	mathematics	mathematic	NOUN
cana-2204	256	28	,	,	PUNCT
cana-2204	256	29	vol.28	vol.28	PROPN
cana-2204	256	30	no.8(2014	no.8(2014	NUM
cana-2204	256	31	)	)	PUNCT
cana-2204	256	32	,	,	PUNCT
cana-2204	256	33	1750064	1750064	NUM
cana-2204	256	34	-	-	SYM
cana-2204	256	35	1	1	NUM
cana-2204	256	36	-	-	SYM
cana-2204	256	37	8	8	NUM
cana-2204	256	38	.	.	PUNCT
cana-2204	257	1	[	[	X
cana-2204	257	2	17	17	NUM
cana-2204	257	3	]	]	PUNCT
cana-2204	257	4	t.	t.	PROPN
cana-2204	257	5	khan	khan	PROPN
cana-2204	257	6	,	,	PUNCT
cana-2204	257	7	s.	s.	PROPN
cana-2204	257	8	rizvi	rizvi	PROPN
cana-2204	257	9	and	and	CCONJ
cana-2204	257	10	o.	o.	NOUN
cana-2204	257	11	bahadir	bahadir	NOUN
cana-2204	257	12	,	,	PUNCT
cana-2204	257	13	quasi	quasi	PROPN
cana-2204	257	14	hemi	hemi	PROPN
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cana-2204	257	16	submanifolds	submanifold	NOUN
cana-2204	257	17	of	of	ADP
cana-2204	257	18	lorentzian	lorentzian	ADJ
cana-2204	257	19	con	con	ADJ
cana-2204	257	20	-	-	PUNCT
cana-2204	257	21	circular	circular	ADJ
cana-2204	257	22	structures	structure	NOUN
cana-2204	257	23	,	,	PUNCT
cana-2204	257	24	proof	proof	NOUN
cana-2204	257	25	,	,	PUNCT
cana-2204	257	26	vol.4(2024	vol.4(2024	NOUN
cana-2204	257	27	)	)	PUNCT
cana-2204	257	28	,	,	PUNCT
cana-2204	257	29	1	1	NUM
cana-2204	257	30	-	-	SYM
cana-2204	257	31	10;doi:10.38394/232020.2024.4.1	10;doi:10.38394/232020.2024.4.1	NUM
cana-2204	257	32	.	.	PUNCT