id	sid	tid	token	lemma	pos
ejpam-3435	1	1	european	european	PROPN
ejpam-3435	1	2	journal	journal	PROPN
ejpam-3435	1	3	of	of	ADP
ejpam-3435	1	4	pure	pure	ADJ
ejpam-3435	1	5	and	and	CCONJ
ejpam-3435	1	6	applied	apply	VERB
ejpam-3435	1	7	mathematics	mathematic	NOUN
ejpam-3435	1	8	vol	vol	NOUN
ejpam-3435	1	9	.	.	PROPN
ejpam-3435	2	1	12	12	NUM
ejpam-3435	2	2	,	,	PUNCT
ejpam-3435	2	3	no	no	INTJ
ejpam-3435	2	4	.	.	NOUN
ejpam-3435	2	5	4	4	NUM
ejpam-3435	2	6	,	,	PUNCT
ejpam-3435	2	7	2019	2019	NUM
ejpam-3435	2	8	,	,	PUNCT
ejpam-3435	2	9	1811	1811	NUM
ejpam-3435	2	10	-	-	SYM
ejpam-3435	2	11	1818	1818	NUM
ejpam-3435	2	12	issn	issn	PROPN
ejpam-3435	2	13	1307	1307	NUM
ejpam-3435	2	14	-	-	SYM
ejpam-3435	2	15	5543	5543	NUM
ejpam-3435	2	16	–	–	PUNCT
ejpam-3435	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-3435	2	18	published	publish	VERB
ejpam-3435	2	19	by	by	ADP
ejpam-3435	2	20	new	new	PROPN
ejpam-3435	2	21	york	york	PROPN
ejpam-3435	2	22	business	business	PROPN
ejpam-3435	2	23	global	global	ADJ
ejpam-3435	2	24	curvature	curvature	NOUN
ejpam-3435	2	25	inequalities	inequality	NOUN
ejpam-3435	2	26	for	for	ADP
ejpam-3435	2	27	submanifolds	submanifold	NOUN
ejpam-3435	2	28	of	of	ADP
ejpam-3435	2	29	s	s	NOUN
ejpam-3435	2	30	-	-	PUNCT
ejpam-3435	2	31	space	space	NOUN
ejpam-3435	2	32	form	form	NOUN
ejpam-3435	2	33	najma	najma	PROPN
ejpam-3435	2	34	abdul	abdul	PROPN
ejpam-3435	2	35	rehman	rehman	PROPN
ejpam-3435	2	36	department	department	PROPN
ejpam-3435	2	37	of	of	ADP
ejpam-3435	2	38	mathematics	mathematics	PROPN
ejpam-3435	2	39	,	,	PUNCT
ejpam-3435	2	40	comsats	comsats	PROPN
ejpam-3435	2	41	university	university	PROPN
ejpam-3435	2	42	islamabad	islamabad	PROPN
ejpam-3435	2	43	,	,	PUNCT
ejpam-3435	2	44	sahiwal	sahiwal	PROPN
ejpam-3435	2	45	campus	campus	PROPN
ejpam-3435	2	46	,	,	PUNCT
ejpam-3435	2	47	pakistan	pakistan	PROPN
ejpam-3435	2	48	abstract	abstract	NOUN
ejpam-3435	2	49	.	.	PUNCT
ejpam-3435	3	1	in	in	ADP
ejpam-3435	3	2	this	this	DET
ejpam-3435	3	3	paper	paper	NOUN
ejpam-3435	3	4	we	we	PRON
ejpam-3435	3	5	establish	establish	VERB
ejpam-3435	3	6	new	new	ADJ
ejpam-3435	3	7	results	result	NOUN
ejpam-3435	3	8	of	of	ADP
ejpam-3435	3	9	squared	square	VERB
ejpam-3435	3	10	mean	mean	ADJ
ejpam-3435	3	11	curvature	curvature	NOUN
ejpam-3435	3	12	and	and	CCONJ
ejpam-3435	3	13	ricci	ricci	NOUN
ejpam-3435	3	14	curvature	curvature	NOUN
ejpam-3435	3	15	for	for	ADP
ejpam-3435	3	16	the	the	DET
ejpam-3435	3	17	sub	sub	NOUN
ejpam-3435	3	18	manifolds	manifold	NOUN
ejpam-3435	3	19	of	of	ADP
ejpam-3435	3	20	s	s	NOUN
ejpam-3435	3	21	-	-	NOUN
ejpam-3435	3	22	space	space	NOUN
ejpam-3435	3	23	from	from	ADP
ejpam-3435	3	24	that	that	PRON
ejpam-3435	3	25	is	be	AUX
ejpam-3435	3	26	the	the	DET
ejpam-3435	3	27	generalization	generalization	NOUN
ejpam-3435	3	28	of	of	ADP
ejpam-3435	3	29	complex	complex	ADJ
ejpam-3435	3	30	and	and	CCONJ
ejpam-3435	3	31	contact	contact	NOUN
ejpam-3435	3	32	structures	structure	NOUN
ejpam-3435	3	33	.	.	PUNCT
ejpam-3435	4	1	obtained	obtain	VERB
ejpam-3435	4	2	results	result	NOUN
ejpam-3435	4	3	are	be	AUX
ejpam-3435	4	4	discussed	discuss	VERB
ejpam-3435	4	5	for	for	ADP
ejpam-3435	4	6	invariant	invariant	ADJ
ejpam-3435	4	7	,	,	PUNCT
ejpam-3435	4	8	anti	anti	ADJ
ejpam-3435	4	9	invariant	invariant	ADJ
ejpam-3435	4	10	and	and	CCONJ
ejpam-3435	4	11	cr	cr	PROPN
ejpam-3435	4	12	sub	sub	PROPN
ejpam-3435	4	13	manifolds	manifold	NOUN
ejpam-3435	4	14	of	of	ADP
ejpam-3435	4	15	s	s	NOUN
ejpam-3435	4	16	-	-	NOUN
ejpam-3435	4	17	space	space	NOUN
ejpam-3435	4	18	from	from	ADP
ejpam-3435	4	19	.	.	PUNCT
ejpam-3435	5	1	2010	2010	NUM
ejpam-3435	5	2	mathematics	mathematic	NOUN
ejpam-3435	5	3	subject	subject	NOUN
ejpam-3435	5	4	classifications	classification	NOUN
ejpam-3435	5	5	:	:	PUNCT
ejpam-3435	5	6	53c40	53c40	NUM
ejpam-3435	5	7	,	,	PUNCT
ejpam-3435	5	8	53c25	53c25	NUM
ejpam-3435	5	9	key	key	ADJ
ejpam-3435	5	10	words	word	NOUN
ejpam-3435	5	11	and	and	CCONJ
ejpam-3435	5	12	phrases	phrase	NOUN
ejpam-3435	5	13	:	:	PUNCT
ejpam-3435	5	14	curvature	curvature	NOUN
ejpam-3435	5	15	,	,	PUNCT
ejpam-3435	5	16	sub	sub	NOUN
ejpam-3435	5	17	manifolds	manifold	NOUN
ejpam-3435	5	18	,	,	PUNCT
ejpam-3435	5	19	s	s	NOUN
ejpam-3435	5	20	-	-	PUNCT
ejpam-3435	5	21	space	space	NOUN
ejpam-3435	5	22	form	form	NOUN
ejpam-3435	5	23	1	1	NUM
ejpam-3435	5	24	.	.	PUNCT
ejpam-3435	6	1	introduction	introduction	NOUN
ejpam-3435	6	2	one	one	NUM
ejpam-3435	6	3	of	of	ADP
ejpam-3435	6	4	the	the	DET
ejpam-3435	6	5	main	main	ADJ
ejpam-3435	6	6	and	and	CCONJ
ejpam-3435	6	7	useful	useful	ADJ
ejpam-3435	6	8	idea	idea	NOUN
ejpam-3435	6	9	in	in	ADP
ejpam-3435	6	10	submanifolds	submanifold	NOUN
ejpam-3435	6	11	conjectures	conjecture	VERB
ejpam-3435	6	12	is	be	AUX
ejpam-3435	6	13	to	to	PART
ejpam-3435	6	14	derive	derive	VERB
ejpam-3435	6	15	relationship	relationship	NOUN
ejpam-3435	6	16	among	among	ADP
ejpam-3435	6	17	squared	squared	ADJ
ejpam-3435	6	18	mean	mean	ADJ
ejpam-3435	6	19	curvature	curvature	NOUN
ejpam-3435	6	20	and	and	CCONJ
ejpam-3435	6	21	ricci	ricci	PROPN
ejpam-3435	6	22	curvature	curvature	NOUN
ejpam-3435	6	23	of	of	ADP
ejpam-3435	6	24	submanifolds	submanifold	NOUN
ejpam-3435	6	25	,	,	PUNCT
ejpam-3435	6	26	was	be	AUX
ejpam-3435	6	27	explained	explain	VERB
ejpam-3435	6	28	by	by	ADP
ejpam-3435	6	29	chen	chen	PROPN
ejpam-3435	7	1	[	[	X
ejpam-3435	7	2	6	6	NUM
ejpam-3435	7	3	]	]	PUNCT
ejpam-3435	7	4	,	,	PUNCT
ejpam-3435	7	5	[	[	X
ejpam-3435	7	6	7	7	NUM
ejpam-3435	7	7	]	]	PUNCT
ejpam-3435	7	8	.	.	PUNCT
ejpam-3435	8	1	after	after	SCONJ
ejpam-3435	8	2	this	this	DET
ejpam-3435	8	3	many	many	ADJ
ejpam-3435	8	4	authors	author	NOUN
ejpam-3435	8	5	derived	derive	VERB
ejpam-3435	8	6	chen	chen	PROPN
ejpam-3435	8	7	inequalities	inequality	NOUN
ejpam-3435	8	8	for	for	ADP
ejpam-3435	8	9	complex	complex	ADJ
ejpam-3435	8	10	and	and	CCONJ
ejpam-3435	8	11	contact	contact	NOUN
ejpam-3435	8	12	space	space	NOUN
ejpam-3435	8	13	forms	form	NOUN
ejpam-3435	8	14	[	[	X
ejpam-3435	8	15	1	1	NUM
ejpam-3435	8	16	]	]	PUNCT
ejpam-3435	8	17	,	,	PUNCT
ejpam-3435	8	18	[	[	X
ejpam-3435	8	19	11	11	NUM
ejpam-3435	8	20	]	]	PUNCT
ejpam-3435	8	21	,	,	PUNCT
ejpam-3435	8	22	[	[	X
ejpam-3435	8	23	12	12	NUM
ejpam-3435	8	24	]	]	PUNCT
ejpam-3435	8	25	and	and	CCONJ
ejpam-3435	8	26	on	on	ADP
ejpam-3435	8	27	hyper	hyper	ADJ
ejpam-3435	8	28	surfaces	surface	NOUN
ejpam-3435	8	29	of	of	ADP
ejpam-3435	8	30	a	a	DET
ejpam-3435	8	31	lorentzian	lorentzian	ADJ
ejpam-3435	8	32	manifold	manifold	NOUN
ejpam-3435	9	1	[	[	X
ejpam-3435	9	2	9	9	NUM
ejpam-3435	9	3	]	]	PUNCT
ejpam-3435	9	4	.	.	PUNCT
ejpam-3435	10	1	after	after	ADP
ejpam-3435	10	2	the	the	DET
ejpam-3435	10	3	generalization	generalization	NOUN
ejpam-3435	10	4	of	of	ADP
ejpam-3435	10	5	complex	complex	ADJ
ejpam-3435	10	6	and	and	CCONJ
ejpam-3435	10	7	contact	contact	NOUN
ejpam-3435	10	8	space	space	NOUN
ejpam-3435	10	9	forms	form	NOUN
ejpam-3435	10	10	into	into	ADP
ejpam-3435	10	11	s	s	NOUN
ejpam-3435	10	12	-	-	PUNCT
ejpam-3435	10	13	space	space	NOUN
ejpam-3435	10	14	form	form	NOUN
ejpam-3435	10	15	[	[	X
ejpam-3435	10	16	4	4	NUM
ejpam-3435	10	17	]	]	PUNCT
ejpam-3435	10	18	,	,	PUNCT
ejpam-3435	10	19	it	it	PRON
ejpam-3435	10	20	is	be	AUX
ejpam-3435	10	21	natural	natural	ADJ
ejpam-3435	10	22	to	to	PART
ejpam-3435	10	23	study	study	VERB
ejpam-3435	10	24	the	the	DET
ejpam-3435	10	25	inequalities	inequality	NOUN
ejpam-3435	10	26	of	of	ADP
ejpam-3435	10	27	ricci	ricci	PROPN
ejpam-3435	10	28	curvature	curvature	NOUN
ejpam-3435	10	29	for	for	ADP
ejpam-3435	10	30	submanifolds	submanifold	NOUN
ejpam-3435	10	31	of	of	ADP
ejpam-3435	10	32	s	s	NOUN
ejpam-3435	10	33	-	-	PUNCT
ejpam-3435	10	34	space	space	NOUN
ejpam-3435	10	35	forms	form	NOUN
ejpam-3435	10	36	.	.	PUNCT
ejpam-3435	11	1	geometry	geometry	NOUN
ejpam-3435	11	2	of	of	ADP
ejpam-3435	11	3	s	s	NOUN
ejpam-3435	11	4	-	-	PUNCT
ejpam-3435	11	5	space	space	NOUN
ejpam-3435	11	6	forms	form	NOUN
ejpam-3435	11	7	were	be	AUX
ejpam-3435	11	8	studied	study	VERB
ejpam-3435	11	9	by	by	ADP
ejpam-3435	11	10	many	many	ADJ
ejpam-3435	11	11	authors	author	NOUN
ejpam-3435	11	12	i.e.	i.e.	X
ejpam-3435	11	13	[	[	X
ejpam-3435	11	14	10	10	NUM
ejpam-3435	11	15	]	]	PUNCT
ejpam-3435	11	16	,	,	PUNCT
ejpam-3435	11	17	[	[	X
ejpam-3435	11	18	13	13	NUM
ejpam-3435	11	19	]	]	PUNCT
ejpam-3435	11	20	.	.	PUNCT
ejpam-3435	12	1	in	in	ADP
ejpam-3435	12	2	this	this	DET
ejpam-3435	12	3	paper	paper	NOUN
ejpam-3435	12	4	we	we	PRON
ejpam-3435	12	5	find	find	VERB
ejpam-3435	12	6	relations	relation	NOUN
ejpam-3435	12	7	between	between	ADP
ejpam-3435	12	8	squared	square	VERB
ejpam-3435	12	9	mean	mean	ADJ
ejpam-3435	12	10	curvature	curvature	NOUN
ejpam-3435	12	11	and	and	CCONJ
ejpam-3435	12	12	ricci	ricci	NOUN
ejpam-3435	12	13	curvature	curvature	NOUN
ejpam-3435	12	14	for	for	ADP
ejpam-3435	12	15	the	the	DET
ejpam-3435	12	16	sub	sub	NOUN
ejpam-3435	12	17	manifolds	manifold	NOUN
ejpam-3435	12	18	of	of	ADP
ejpam-3435	12	19	s	s	NOUN
ejpam-3435	12	20	-	-	PUNCT
ejpam-3435	12	21	space	space	NOUN
ejpam-3435	12	22	form	form	NOUN
ejpam-3435	12	23	and	and	CCONJ
ejpam-3435	12	24	discuss	discuss	VERB
ejpam-3435	12	25	this	this	DET
ejpam-3435	12	26	relation	relation	NOUN
ejpam-3435	12	27	for	for	ADP
ejpam-3435	12	28	invariant	invariant	ADJ
ejpam-3435	12	29	,	,	PUNCT
ejpam-3435	12	30	anti	anti	ADJ
ejpam-3435	12	31	invariant	invariant	ADJ
ejpam-3435	12	32	and	and	CCONJ
ejpam-3435	12	33	cr	cr	PROPN
ejpam-3435	12	34	sub	sub	PROPN
ejpam-3435	12	35	manifolds	manifold	NOUN
ejpam-3435	12	36	of	of	ADP
ejpam-3435	12	37	s	s	NOUN
ejpam-3435	12	38	-	-	PUNCT
ejpam-3435	12	39	space	space	NOUN
ejpam-3435	12	40	form	form	NOUN
ejpam-3435	12	41	.	.	PUNCT
ejpam-3435	13	1	after	after	ADP
ejpam-3435	13	2	introduction	introduction	NOUN
ejpam-3435	13	3	,	,	PUNCT
ejpam-3435	13	4	second	second	ADJ
ejpam-3435	13	5	section	section	NOUN
ejpam-3435	13	6	contains	contain	VERB
ejpam-3435	13	7	basics	basic	NOUN
ejpam-3435	13	8	of	of	ADP
ejpam-3435	13	9	s	s	NOUN
ejpam-3435	13	10	-	-	PUNCT
ejpam-3435	13	11	space	space	NOUN
ejpam-3435	13	12	forms	form	NOUN
ejpam-3435	13	13	and	and	CCONJ
ejpam-3435	13	14	submanifolds	submanifold	NOUN
ejpam-3435	13	15	.	.	PUNCT
ejpam-3435	14	1	third	third	ADJ
ejpam-3435	14	2	section	section	NOUN
ejpam-3435	14	3	contains	contain	VERB
ejpam-3435	14	4	main	main	ADJ
ejpam-3435	14	5	results	result	NOUN
ejpam-3435	14	6	.	.	PUNCT
ejpam-3435	15	1	2	2	X
ejpam-3435	15	2	.	.	X
ejpam-3435	15	3	preliminaries	preliminary	NOUN
ejpam-3435	15	4	this	this	DET
ejpam-3435	15	5	section	section	NOUN
ejpam-3435	15	6	presents	present	VERB
ejpam-3435	15	7	some	some	DET
ejpam-3435	15	8	well	well	ADV
ejpam-3435	15	9	known	know	VERB
ejpam-3435	15	10	facts	fact	NOUN
ejpam-3435	15	11	related	relate	VERB
ejpam-3435	15	12	to	to	ADP
ejpam-3435	15	13	s	s	NOUN
ejpam-3435	15	14	-	-	PUNCT
ejpam-3435	15	15	space	space	NOUN
ejpam-3435	15	16	form	form	NOUN
ejpam-3435	15	17	and	and	CCONJ
ejpam-3435	15	18	sub	sub	NOUN
ejpam-3435	15	19	manifolds	manifold	NOUN
ejpam-3435	15	20	.	.	PUNCT
ejpam-3435	16	1	yano[14	yano[14	PROPN
ejpam-3435	16	2	]	]	PUNCT
ejpam-3435	16	3	presented	present	VERB
ejpam-3435	16	4	that	that	SCONJ
ejpam-3435	16	5	almost	almost	ADV
ejpam-3435	16	6	complex	complex	ADJ
ejpam-3435	16	7	and	and	CCONJ
ejpam-3435	16	8	almost	almost	ADV
ejpam-3435	16	9	contact	contact	NOUN
ejpam-3435	16	10	structures	structure	NOUN
ejpam-3435	16	11	can	can	AUX
ejpam-3435	16	12	be	be	AUX
ejpam-3435	16	13	generalized	generalize	VERB
ejpam-3435	16	14	as	as	ADP
ejpam-3435	16	15	f	f	PROPN
ejpam-3435	16	16	structure	structure	NOUN
ejpam-3435	16	17	on	on	ADP
ejpam-3435	16	18	a	a	DET
ejpam-3435	16	19	smooth	smooth	ADJ
ejpam-3435	16	20	manifold	manifold	NOUN
ejpam-3435	16	21	of	of	ADP
ejpam-3435	16	22	dimension	dimension	NOUN
ejpam-3435	16	23	2	2	NUM
ejpam-3435	16	24	m	m	NOUN
ejpam-3435	16	25	+	+	NOUN
ejpam-3435	16	26	s	s	PART
ejpam-3435	16	27	.	.	PUNCT
ejpam-3435	17	1	the	the	DET
ejpam-3435	17	2	idea	idea	NOUN
ejpam-3435	17	3	for	for	ADP
ejpam-3435	17	4	the	the	DET
ejpam-3435	17	5	f	f	NOUN
ejpam-3435	17	6	-	-	PUNCT
ejpam-3435	17	7	structure	structure	NOUN
ejpam-3435	17	8	is	be	AUX
ejpam-3435	17	9	to	to	PART
ejpam-3435	17	10	consider	consider	VERB
ejpam-3435	17	11	a	a	DET
ejpam-3435	17	12	tensor	tensor	NOUN
ejpam-3435	17	13	field	field	NOUN
ejpam-3435	17	14	with	with	ADP
ejpam-3435	17	15	condition	condition	NOUN
ejpam-3435	17	16	f3	f3	NOUN
ejpam-3435	17	17	+	+	CCONJ
ejpam-3435	17	18	f	f	PROPN
ejpam-3435	17	19	=	=	SYM
ejpam-3435	17	20	0	0	PROPN
ejpam-3435	17	21	,	,	PUNCT
ejpam-3435	17	22	of	of	ADP
ejpam-3435	17	23	type	type	NOUN
ejpam-3435	17	24	(	(	PUNCT
ejpam-3435	17	25	1,1	1,1	NUM
ejpam-3435	17	26	)	)	PUNCT
ejpam-3435	17	27	and	and	CCONJ
ejpam-3435	17	28	rank	rank	NOUN
ejpam-3435	17	29	2	2	NUM
ejpam-3435	17	30	m	m	NOUN
ejpam-3435	17	31	.	.	PUNCT
ejpam-3435	18	1	doi	doi	NOUN
ejpam-3435	18	2	:	:	PUNCT
ejpam-3435	18	3	https://doi.org/10.29020/nybg.ejpam.v12i4.3435	https://doi.org/10.29020/nybg.ejpam.v12i4.3435	PRON
ejpam-3435	18	4	email	email	NOUN
ejpam-3435	18	5	address	address	NOUN
ejpam-3435	18	6	:	:	PUNCT
ejpam-3435	18	7	najma−ar@hotmail.com	najma−ar@hotmail.com	X
ejpam-3435	18	8	(	(	PUNCT
ejpam-3435	18	9	n.	n.	PROPN
ejpam-3435	18	10	a.	a.	PROPN
ejpam-3435	18	11	rehman	rehman	PROPN
ejpam-3435	18	12	)	)	PUNCT
ejpam-3435	18	13	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3435	18	14	1811	1811	NUM
ejpam-3435	19	1	c	c	X
ejpam-3435	19	2	©	©	PROPN
ejpam-3435	19	3	2019	2019	NUM
ejpam-3435	19	4	ejpam	ejpam	NOUN
ejpam-3435	19	5	all	all	DET
ejpam-3435	19	6	rights	right	NOUN
ejpam-3435	19	7	reserved	reserve	VERB
ejpam-3435	19	8	.	.	PUNCT
ejpam-3435	20	1	n.	n.	NOUN
ejpam-3435	20	2	a.	a.	PROPN
ejpam-3435	20	3	rehman	rehman	PROPN
ejpam-3435	20	4	/	/	SYM
ejpam-3435	20	5	eur	eur	PROPN
ejpam-3435	20	6	.	.	PUNCT
ejpam-3435	21	1	j.	j.	PROPN
ejpam-3435	21	2	pure	pure	PROPN
ejpam-3435	21	3	appl	appl	PROPN
ejpam-3435	21	4	.	.	PROPN
ejpam-3435	21	5	math	math	PROPN
ejpam-3435	21	6	,	,	PUNCT
ejpam-3435	21	7	12	12	NUM
ejpam-3435	21	8	(	(	PUNCT
ejpam-3435	21	9	4	4	NUM
ejpam-3435	21	10	)	)	PUNCT
ejpam-3435	21	11	(	(	PUNCT
ejpam-3435	21	12	2019	2019	NUM
ejpam-3435	21	13	)	)	PUNCT
ejpam-3435	21	14	,	,	PUNCT
ejpam-3435	21	15	1811	1811	NUM
ejpam-3435	21	16	-	-	SYM
ejpam-3435	21	17	1818	1818	NUM
ejpam-3435	21	18	1812	1812	NUM
ejpam-3435	21	19	consider	consider	VERB
ejpam-3435	21	20	manifold	manifold	ADJ
ejpam-3435	21	21	m2m+s	m2m+s	PROPN
ejpam-3435	21	22	along	along	ADP
ejpam-3435	21	23	an	an	DET
ejpam-3435	21	24	f	f	NOUN
ejpam-3435	21	25	-structure	-structure	NOUN
ejpam-3435	21	26	of	of	ADP
ejpam-3435	21	27	rank	rank	NOUN
ejpam-3435	21	28	2	2	NUM
ejpam-3435	21	29	m.	m.	NOUN
ejpam-3435	21	30	we	we	PRON
ejpam-3435	21	31	take	take	VERB
ejpam-3435	21	32	s	s	PRON
ejpam-3435	21	33	structural	structural	ADJ
ejpam-3435	21	34	vector	vector	NOUN
ejpam-3435	21	35	fields	field	NOUN
ejpam-3435	21	36	ξ1	ξ1	NOUN
ejpam-3435	21	37	,	,	PUNCT
ejpam-3435	21	38	ξ2	ξ2	NOUN
ejpam-3435	21	39	,	,	PUNCT
ejpam-3435	21	40	.	.	PUNCT
ejpam-3435	21	41	.	.	PUNCT
ejpam-3435	22	1	.	.	PUNCT
ejpam-3435	23	1	,	,	PUNCT
ejpam-3435	23	2	ξs	ξs	VERB
ejpam-3435	23	3	on	on	ADP
ejpam-3435	23	4	m	m	PRON
ejpam-3435	23	5	such	such	ADJ
ejpam-3435	23	6	as	as	ADP
ejpam-3435	23	7	:	:	PUNCT
ejpam-3435	23	8	fξα	fξα	NOUN
ejpam-3435	23	9	=	=	SYM
ejpam-3435	23	10	0	0	PROPN
ejpam-3435	23	11	,	,	PUNCT
ejpam-3435	23	12	ηα	ηα	NOUN
ejpam-3435	23	13	◦	◦	NOUN
ejpam-3435	23	14	f	f	X
ejpam-3435	23	15	=	=	SYM
ejpam-3435	23	16	0	0	NUM
ejpam-3435	23	17	,	,	PUNCT
ejpam-3435	23	18	f2	f2	PROPN
ejpam-3435	23	19	=	=	PUNCT
ejpam-3435	23	20	−i	−i	PROPN
ejpam-3435	23	21	+	+	CCONJ
ejpam-3435	23	22	∑	∑	PUNCT
ejpam-3435	23	23	ξα	ξα	PROPN
ejpam-3435	23	24	⊗	⊗	PROPN
ejpam-3435	23	25	ηα	ηα	PROPN
ejpam-3435	23	26	,	,	PUNCT
ejpam-3435	23	27	(	(	PUNCT
ejpam-3435	23	28	1	1	X
ejpam-3435	23	29	)	)	PUNCT
ejpam-3435	23	30	where	where	SCONJ
ejpam-3435	23	31	ηα	ηα	NOUN
ejpam-3435	23	32	and	and	CCONJ
ejpam-3435	23	33	ξα	ξα	NOUN
ejpam-3435	23	34	are	be	AUX
ejpam-3435	23	35	the	the	DET
ejpam-3435	23	36	dual	dual	ADJ
ejpam-3435	23	37	forms	form	NOUN
ejpam-3435	23	38	to	to	ADP
ejpam-3435	23	39	each	each	DET
ejpam-3435	23	40	other	other	ADJ
ejpam-3435	23	41	,	,	PUNCT
ejpam-3435	23	42	therefore	therefore	ADV
ejpam-3435	23	43	complemented	complement	VERB
ejpam-3435	23	44	frames	frame	NOUN
ejpam-3435	23	45	exist	exist	VERB
ejpam-3435	23	46	on	on	ADP
ejpam-3435	23	47	f	f	PROPN
ejpam-3435	23	48	-structure	-structure	NOUN
ejpam-3435	23	49	.	.	PUNCT
ejpam-3435	24	1	for	for	ADP
ejpam-3435	24	2	fmanifold	fmanifold	ADJ
ejpam-3435	24	3	we	we	PRON
ejpam-3435	24	4	define	define	VERB
ejpam-3435	24	5	a	a	DET
ejpam-3435	24	6	riemannian	riemannian	ADJ
ejpam-3435	24	7	metric	metric	ADJ
ejpam-3435	24	8	g	g	NOUN
ejpam-3435	24	9	as	as	ADP
ejpam-3435	24	10	g(y	g(y	PROPN
ejpam-3435	24	11	,	,	PUNCT
ejpam-3435	24	12	x	x	NOUN
ejpam-3435	24	13	)	)	PUNCT
ejpam-3435	24	14	=	=	SYM
ejpam-3435	24	15	g(fy	g(fy	PROPN
ejpam-3435	24	16	,	,	PUNCT
ejpam-3435	24	17	fz	fz	NOUN
ejpam-3435	24	18	)	)	PUNCT
ejpam-3435	25	1	+	+	CCONJ
ejpam-3435	25	2	∑	∑	ADP
ejpam-3435	25	3	ηα(y	ηα(y	NOUN
ejpam-3435	25	4	)	)	PUNCT
ejpam-3435	25	5	ηα(z	ηα(z	NOUN
ejpam-3435	25	6	)	)	PUNCT
ejpam-3435	25	7	for	for	ADP
ejpam-3435	25	8	vector	vector	NOUN
ejpam-3435	25	9	fields	field	NOUN
ejpam-3435	25	10	y	y	PROPN
ejpam-3435	25	11	and	and	CCONJ
ejpam-3435	25	12	z	z	PROPN
ejpam-3435	25	13	on	on	ADP
ejpam-3435	25	14	m	m	PRON
ejpam-3435	25	15	[	[	X
ejpam-3435	25	16	4	4	NUM
ejpam-3435	25	17	]	]	PUNCT
ejpam-3435	25	18	.	.	PUNCT
ejpam-3435	26	1	an	an	DET
ejpam-3435	26	2	f	f	PROPN
ejpam-3435	26	3	-structure	-structure	PROPN
ejpam-3435	26	4	f	f	PROPN
ejpam-3435	26	5	is	be	AUX
ejpam-3435	26	6	normal	normal	ADJ
ejpam-3435	26	7	,	,	PUNCT
ejpam-3435	26	8	if	if	SCONJ
ejpam-3435	26	9	there	there	PRON
ejpam-3435	26	10	exist	exist	VERB
ejpam-3435	26	11	complemented	complemented	ADJ
ejpam-3435	26	12	frames	frame	NOUN
ejpam-3435	26	13	and	and	CCONJ
ejpam-3435	26	14	[	[	X
ejpam-3435	26	15	f	f	X
ejpam-3435	26	16	,	,	PUNCT
ejpam-3435	26	17	f	f	X
ejpam-3435	26	18	]	]	PUNCT
ejpam-3435	27	1	+	+	CCONJ
ejpam-3435	27	2	2	2	NUM
ejpam-3435	27	3	∑	∑	ADP
ejpam-3435	27	4	ξα	ξα	PROPN
ejpam-3435	27	5	⊗	⊗	PROPN
ejpam-3435	27	6	dηα	dηα	NOUN
ejpam-3435	27	7	=	=	SYM
ejpam-3435	27	8	0	0	PROPN
ejpam-3435	27	9	,	,	PUNCT
ejpam-3435	27	10	where	where	SCONJ
ejpam-3435	27	11	[	[	X
ejpam-3435	27	12	f	f	X
ejpam-3435	27	13	,	,	PUNCT
ejpam-3435	27	14	f	f	X
ejpam-3435	27	15	]	]	PUNCT
ejpam-3435	27	16	is	be	AUX
ejpam-3435	27	17	nijenhuis	nijenhuis	NOUN
ejpam-3435	27	18	torsion	torsion	NOUN
ejpam-3435	27	19	of	of	ADP
ejpam-3435	27	20	f	f	PROPN
ejpam-3435	27	21	.	.	PUNCT
ejpam-3435	28	1	let	let	VERB
ejpam-3435	28	2	fundamental	fundamental	ADJ
ejpam-3435	28	3	2	2	NUM
ejpam-3435	28	4	-	-	PUNCT
ejpam-3435	28	5	form	form	NOUN
ejpam-3435	28	6	b	b	NOUN
ejpam-3435	28	7	be	be	AUX
ejpam-3435	28	8	defined	define	VERB
ejpam-3435	28	9	as	as	ADP
ejpam-3435	28	10	b(y	b(y	PROPN
ejpam-3435	28	11	,	,	PUNCT
ejpam-3435	28	12	z	z	NOUN
ejpam-3435	28	13	)	)	PUNCT
ejpam-3435	28	14	=	=	SYM
ejpam-3435	28	15	g(y	g(y	PROPN
ejpam-3435	28	16	,	,	PUNCT
ejpam-3435	28	17	fz	fz	NOUN
ejpam-3435	28	18	)	)	PUNCT
ejpam-3435	28	19	,	,	PUNCT
ejpam-3435	28	20	y	y	PROPN
ejpam-3435	28	21	,	,	PUNCT
ejpam-3435	28	22	z	z	PROPN
ejpam-3435	28	23	∈	∈	PROPN
ejpam-3435	28	24	t	t	PROPN
ejpam-3435	28	25	(	(	PUNCT
ejpam-3435	28	26	m	m	NOUN
ejpam-3435	28	27	)	)	PUNCT
ejpam-3435	28	28	.	.	PUNCT
ejpam-3435	29	1	f	f	PROPN
ejpam-3435	29	2	-structure	-structure	NOUN
ejpam-3435	29	3	that	that	PRON
ejpam-3435	29	4	is	be	AUX
ejpam-3435	29	5	normal	normal	ADJ
ejpam-3435	29	6	and	and	CCONJ
ejpam-3435	29	7	dη1	dη1	X
ejpam-3435	29	8	=	=	PUNCT
ejpam-3435	29	9	·	·	PUNCT
ejpam-3435	29	10	·	·	PUNCT
ejpam-3435	29	11	·	·	PUNCT
ejpam-3435	30	1	=	=	SYM
ejpam-3435	30	2	dηs	dηs	VERB
ejpam-3435	30	3	=	=	SYM
ejpam-3435	30	4	b	b	PROPN
ejpam-3435	30	5	is	be	AUX
ejpam-3435	30	6	known	know	VERB
ejpam-3435	30	7	as	as	ADP
ejpam-3435	30	8	an	an	DET
ejpam-3435	30	9	s	s	NOUN
ejpam-3435	30	10	-	-	NOUN
ejpam-3435	30	11	structure	structure	NOUN
ejpam-3435	30	12	.	.	PUNCT
ejpam-3435	31	1	a	a	DET
ejpam-3435	31	2	smooth	smooth	ADJ
ejpam-3435	31	3	manifold	manifold	NOUN
ejpam-3435	31	4	along	along	ADP
ejpam-3435	31	5	with	with	ADP
ejpam-3435	31	6	an	an	DET
ejpam-3435	31	7	s	s	NOUN
ejpam-3435	31	8	-	-	PUNCT
ejpam-3435	31	9	structure	structure	NOUN
ejpam-3435	31	10	known	know	VERB
ejpam-3435	31	11	as	as	ADP
ejpam-3435	31	12	an	an	DET
ejpam-3435	31	13	s	s	NOUN
ejpam-3435	31	14	-	-	ADJ
ejpam-3435	31	15	manifold	manifold	ADJ
ejpam-3435	31	16	.	.	PUNCT
ejpam-3435	32	1	blair	blair	PROPN
ejpam-3435	32	2	described	describe	VERB
ejpam-3435	32	3	such	such	ADJ
ejpam-3435	32	4	types	type	NOUN
ejpam-3435	32	5	of	of	ADP
ejpam-3435	32	6	manifolds	manifold	NOUN
ejpam-3435	32	7	in	in	ADP
ejpam-3435	32	8	[	[	X
ejpam-3435	32	9	4	4	NUM
ejpam-3435	32	10	]	]	PUNCT
ejpam-3435	32	11	.	.	PUNCT
ejpam-3435	33	1	for	for	ADP
ejpam-3435	33	2	sasakian	sasakian	ADJ
ejpam-3435	33	3	manifolds	manifold	NOUN
ejpam-3435	33	4	we	we	PRON
ejpam-3435	33	5	take	take	VERB
ejpam-3435	33	6	s	s	NOUN
ejpam-3435	33	7	=	=	NOUN
ejpam-3435	33	8	1	1	X
ejpam-3435	33	9	.	.	PUNCT
ejpam-3435	34	1	for	for	ADP
ejpam-3435	34	2	s	s	PRON
ejpam-3435	34	3	≥	≥	NUM
ejpam-3435	34	4	2	2	NUM
ejpam-3435	34	5	we	we	PRON
ejpam-3435	34	6	may	may	AUX
ejpam-3435	34	7	have	have	VERB
ejpam-3435	34	8	some	some	DET
ejpam-3435	34	9	attractive	attractive	ADJ
ejpam-3435	34	10	applications	application	NOUN
ejpam-3435	34	11	discussed	discuss	VERB
ejpam-3435	34	12	in	in	ADP
ejpam-3435	34	13	[	[	X
ejpam-3435	34	14	4	4	NUM
ejpam-3435	34	15	]	]	PUNCT
ejpam-3435	34	16	.	.	PUNCT
ejpam-3435	35	1	if	if	SCONJ
ejpam-3435	35	2	m	m	NOUN
ejpam-3435	35	3	is	be	AUX
ejpam-3435	35	4	an	an	DET
ejpam-3435	35	5	s	s	NOUN
ejpam-3435	35	6	-	-	ADJ
ejpam-3435	35	7	manifold	manifold	ADJ
ejpam-3435	35	8	,	,	PUNCT
ejpam-3435	35	9	then	then	ADV
ejpam-3435	35	10	we	we	PRON
ejpam-3435	35	11	consider	consider	VERB
ejpam-3435	35	12	the	the	DET
ejpam-3435	35	13	formulas	formula	NOUN
ejpam-3435	35	14	[	[	X
ejpam-3435	35	15	4	4	NUM
ejpam-3435	35	16	]	]	X
ejpam-3435	35	17	:	:	PUNCT
ejpam-3435	35	18	∇̃y	∇̃y	NOUN
ejpam-3435	35	19	ξα	ξα	ADJ
ejpam-3435	35	20	=	=	NOUN
ejpam-3435	35	21	−fy	−fy	PROPN
ejpam-3435	35	22	,	,	PUNCT
ejpam-3435	35	23	y	y	PROPN
ejpam-3435	35	24	∈	∈	PROPN
ejpam-3435	35	25	t	t	PROPN
ejpam-3435	35	26	(	(	PUNCT
ejpam-3435	35	27	m	m	PROPN
ejpam-3435	35	28	)	)	PUNCT
ejpam-3435	35	29	,	,	PUNCT
ejpam-3435	35	30	α	α	NOUN
ejpam-3435	35	31	=	=	SYM
ejpam-3435	35	32	1	1	NUM
ejpam-3435	35	33	,	,	PUNCT
ejpam-3435	35	34	.	.	PUNCT
ejpam-3435	35	35	.	.	PUNCT
ejpam-3435	36	1	.	.	PUNCT
ejpam-3435	37	1	,	,	PUNCT
ejpam-3435	37	2	s	s	X
ejpam-3435	37	3	,	,	PUNCT
ejpam-3435	37	4	(	(	PUNCT
ejpam-3435	37	5	2	2	NUM
ejpam-3435	37	6	)	)	PUNCT
ejpam-3435	37	7	(	(	PUNCT
ejpam-3435	37	8	∇̃y	∇̃y	X
ejpam-3435	37	9	f)z	f)z	NOUN
ejpam-3435	38	1	=	=	PUNCT
ejpam-3435	38	2	∑	∑	PUNCT
ejpam-3435	38	3	α	α	PROPN
ejpam-3435	38	4	{	{	PUNCT
ejpam-3435	38	5	g(fy	g(fy	PROPN
ejpam-3435	38	6	,	,	PUNCT
ejpam-3435	38	7	fz)ξα	fz)ξα	NUM
ejpam-3435	38	8	+	+	NUM
ejpam-3435	38	9	ηα(z)f2y	ηα(z)f2y	NOUN
ejpam-3435	38	10	}	}	PUNCT
ejpam-3435	38	11	,	,	PUNCT
ejpam-3435	38	12	y	y	PROPN
ejpam-3435	38	13	,	,	PUNCT
ejpam-3435	38	14	z	z	PROPN
ejpam-3435	38	15	∈	∈	PROPN
ejpam-3435	38	16	t	t	PROPN
ejpam-3435	38	17	(	(	PUNCT
ejpam-3435	38	18	m	m	PROPN
ejpam-3435	38	19	)	)	PUNCT
ejpam-3435	38	20	,	,	PUNCT
ejpam-3435	38	21	(	(	PUNCT
ejpam-3435	38	22	3	3	X
ejpam-3435	38	23	)	)	PUNCT
ejpam-3435	38	24	where	where	SCONJ
ejpam-3435	38	25	∇̃	∇̃	PRON
ejpam-3435	38	26	is	be	AUX
ejpam-3435	38	27	the	the	DET
ejpam-3435	38	28	riemannian	riemannian	ADJ
ejpam-3435	38	29	connection	connection	NOUN
ejpam-3435	38	30	of	of	ADP
ejpam-3435	38	31	g.	g.	PROPN
ejpam-3435	38	32	the	the	DET
ejpam-3435	38	33	projection	projection	NOUN
ejpam-3435	38	34	tensor	tensor	NOUN
ejpam-3435	38	35	−f2	−f2	PROPN
ejpam-3435	38	36	determine	determine	VERB
ejpam-3435	38	37	the	the	DET
ejpam-3435	38	38	distribution	distribution	NOUN
ejpam-3435	38	39	l	l	NOUN
ejpam-3435	38	40	and	and	CCONJ
ejpam-3435	38	41	f2+i	f2+i	PROPN
ejpam-3435	38	42	determine	determine	VERB
ejpam-3435	38	43	the	the	DET
ejpam-3435	38	44	complementary	complementary	ADJ
ejpam-3435	38	45	distribution	distribution	NOUN
ejpam-3435	38	46	m	m	VERB
ejpam-3435	38	47	which	which	PRON
ejpam-3435	38	48	is	be	AUX
ejpam-3435	38	49	determined	determine	VERB
ejpam-3435	38	50	and	and	CCONJ
ejpam-3435	38	51	spanned	span	VERB
ejpam-3435	38	52	by	by	ADP
ejpam-3435	38	53	ξ1	ξ1	NOUN
ejpam-3435	38	54	,	,	PUNCT
ejpam-3435	38	55	.	.	PUNCT
ejpam-3435	38	56	.	.	PUNCT
ejpam-3435	38	57	.	.	PUNCT
ejpam-3435	39	1	,	,	PUNCT
ejpam-3435	39	2	ξs	ξs	VERB
ejpam-3435	39	3	.	.	PUNCT
ejpam-3435	40	1	it	it	PRON
ejpam-3435	40	2	can	can	AUX
ejpam-3435	40	3	be	be	AUX
ejpam-3435	40	4	observe	observe	VERB
ejpam-3435	40	5	that	that	SCONJ
ejpam-3435	40	6	if	if	SCONJ
ejpam-3435	40	7	y	y	PROPN
ejpam-3435	40	8	∈	∈	PROPN
ejpam-3435	40	9	l	l	NOUN
ejpam-3435	40	10	then	then	ADV
ejpam-3435	40	11	ηα(y	ηα(y	X
ejpam-3435	40	12	)	)	PUNCT
ejpam-3435	41	1	=	=	SYM
ejpam-3435	41	2	0	0	NUM
ejpam-3435	41	3	for	for	ADP
ejpam-3435	41	4	all	all	DET
ejpam-3435	41	5	α	α	NOUN
ejpam-3435	41	6	,	,	PUNCT
ejpam-3435	41	7	and	and	CCONJ
ejpam-3435	41	8	for	for	ADP
ejpam-3435	41	9	y	y	PROPN
ejpam-3435	41	10	∈	∈	PROPN
ejpam-3435	41	11	m	m	PROPN
ejpam-3435	41	12	,	,	PUNCT
ejpam-3435	41	13	we	we	PRON
ejpam-3435	41	14	have	have	VERB
ejpam-3435	41	15	fy	fy	PROPN
ejpam-3435	41	16	=	=	SYM
ejpam-3435	41	17	0	0	PROPN
ejpam-3435	41	18	.	.	PUNCT
ejpam-3435	42	1	a	a	DET
ejpam-3435	42	2	plane	plane	NOUN
ejpam-3435	42	3	section	section	NOUN
ejpam-3435	42	4	π	π	PROPN
ejpam-3435	42	5	on	on	ADP
ejpam-3435	42	6	m	m	PROPN
ejpam-3435	42	7	is	be	AUX
ejpam-3435	42	8	said	say	VERB
ejpam-3435	42	9	to	to	PART
ejpam-3435	42	10	be	be	AUX
ejpam-3435	42	11	f	f	PROPN
ejpam-3435	42	12	section	section	NOUN
ejpam-3435	42	13	if	if	SCONJ
ejpam-3435	42	14	it	it	PRON
ejpam-3435	42	15	is	be	AUX
ejpam-3435	42	16	established	establish	VERB
ejpam-3435	42	17	by	by	ADP
ejpam-3435	42	18	a	a	DET
ejpam-3435	42	19	vector	vector	NOUN
ejpam-3435	42	20	y	y	PROPN
ejpam-3435	42	21	∈	∈	PROPN
ejpam-3435	42	22	l(p	l(p	PROPN
ejpam-3435	42	23	)	)	PUNCT
ejpam-3435	42	24	,	,	PUNCT
ejpam-3435	42	25	p	p	PROPN
ejpam-3435	42	26	∈	∈	PROPN
ejpam-3435	42	27	m	m	PROPN
ejpam-3435	42	28	,	,	PUNCT
ejpam-3435	42	29	such	such	ADJ
ejpam-3435	42	30	that	that	SCONJ
ejpam-3435	42	31	{	{	PUNCT
ejpam-3435	42	32	y	y	PROPN
ejpam-3435	42	33	,	,	PUNCT
ejpam-3435	42	34	fy	fy	PROPN
ejpam-3435	42	35	}	}	PUNCT
ejpam-3435	42	36	span	span	VERB
ejpam-3435	42	37	the	the	DET
ejpam-3435	42	38	section	section	NOUN
ejpam-3435	42	39	.	.	PUNCT
ejpam-3435	43	1	we	we	PRON
ejpam-3435	43	2	take	take	VERB
ejpam-3435	43	3	the	the	DET
ejpam-3435	43	4	sectional	sectional	ADJ
ejpam-3435	43	5	curvature	curvature	NOUN
ejpam-3435	43	6	of	of	ADP
ejpam-3435	43	7	π	π	PROPN
ejpam-3435	43	8	as	as	ADP
ejpam-3435	43	9	the	the	DET
ejpam-3435	43	10	f	f	NOUN
ejpam-3435	43	11	-	-	PUNCT
ejpam-3435	43	12	sectional	sectional	ADJ
ejpam-3435	43	13	curvature	curvature	NOUN
ejpam-3435	43	14	.	.	PUNCT
ejpam-3435	44	1	if	if	SCONJ
ejpam-3435	44	2	m	m	NOUN
ejpam-3435	44	3	is	be	AUX
ejpam-3435	44	4	an	an	DET
ejpam-3435	44	5	s	s	NOUN
ejpam-3435	44	6	-	-	ADJ
ejpam-3435	44	7	manifold	manifold	ADJ
ejpam-3435	44	8	of	of	ADP
ejpam-3435	44	9	constant	constant	ADJ
ejpam-3435	44	10	f	f	PROPN
ejpam-3435	44	11	-sectional	-sectional	ADJ
ejpam-3435	44	12	curvature	curvature	NOUN
ejpam-3435	44	13	k	k	NOUN
ejpam-3435	44	14	,	,	PUNCT
ejpam-3435	44	15	then	then	ADV
ejpam-3435	44	16	its	its	PRON
ejpam-3435	44	17	curvature	curvature	NOUN
ejpam-3435	44	18	tensor	tensor	NOUN
ejpam-3435	44	19	is	be	AUX
ejpam-3435	44	20	as	as	ADP
ejpam-3435	44	21	:	:	PUNCT
ejpam-3435	44	22	r̃(y	r̃(y	PROPN
ejpam-3435	44	23	,	,	PUNCT
ejpam-3435	44	24	z)u	z)u	NOUN
ejpam-3435	44	25	=	=	SYM
ejpam-3435	44	26	∑	∑	PUNCT
ejpam-3435	44	27	α	α	X
ejpam-3435	44	28	,	,	PUNCT
ejpam-3435	44	29	β	β	X
ejpam-3435	44	30	{	{	PUNCT
ejpam-3435	44	31	ηα(y	ηα(y	X
ejpam-3435	44	32	)	)	PUNCT
ejpam-3435	44	33	ηβ(u)f2z	ηβ(u)f2z	NOUN
ejpam-3435	44	34	−	−	PROPN
ejpam-3435	44	35	ηα(z)ηβ(u)f2y	ηα(z)ηβ(u)f2y	NOUN
ejpam-3435	44	36	−	−	PROPN
ejpam-3435	44	37	g(fy	g(fy	PROPN
ejpam-3435	44	38	,	,	PUNCT
ejpam-3435	44	39	fu)ηα(z)ξβ	fu)ηα(z)ξβ	NOUN
ejpam-3435	44	40	+	+	NOUN
ejpam-3435	44	41	g(fz	g(fz	NOUN
ejpam-3435	44	42	,	,	PUNCT
ejpam-3435	44	43	fu)ηα(y	fu)ηα(y	PROPN
ejpam-3435	44	44	)	)	PUNCT
ejpam-3435	44	45	ξβ}+	ξβ}+	PROPN
ejpam-3435	44	46	k	k	PROPN
ejpam-3435	45	1	+	+	CCONJ
ejpam-3435	45	2	3s	3s	NUM
ejpam-3435	45	3	4	4	NUM
ejpam-3435	45	4	{	{	PUNCT
ejpam-3435	45	5	−g(fz	−g(fz	ADV
ejpam-3435	45	6	,	,	PUNCT
ejpam-3435	45	7	fu)f2y	fu)f2y	PROPN
ejpam-3435	45	8	+	+	SYM
ejpam-3435	45	9	g(fy	g(fy	PROPN
ejpam-3435	45	10	,	,	PUNCT
ejpam-3435	45	11	fu)f2z	fu)f2z	PROPN
ejpam-3435	45	12	}	}	PUNCT
ejpam-3435	45	13	+	+	NUM
ejpam-3435	46	1	k	k	PROPN
ejpam-3435	46	2	−	−	PROPN
ejpam-3435	46	3	s	s	PART
ejpam-3435	46	4	4	4	NUM
ejpam-3435	46	5	{	{	PUNCT
ejpam-3435	46	6	g(y	g(y	PROPN
ejpam-3435	46	7	,	,	PUNCT
ejpam-3435	46	8	fu)fz	fu)fz	PUNCT
ejpam-3435	46	9	−	−	X
ejpam-3435	46	10	g(z	g(z	PROPN
ejpam-3435	46	11	,	,	PUNCT
ejpam-3435	46	12	fu)fy	fu)fy	X
ejpam-3435	46	13	+	+	X
ejpam-3435	46	14	2g(y	2g(y	NUM
ejpam-3435	46	15	,	,	PUNCT
ejpam-3435	46	16	fz)fu	fz)fu	NUM
ejpam-3435	46	17	}	}	PUNCT
ejpam-3435	46	18	,	,	PUNCT
ejpam-3435	46	19	(	(	PUNCT
ejpam-3435	46	20	4	4	X
ejpam-3435	46	21	)	)	PUNCT
ejpam-3435	46	22	y	y	PROPN
ejpam-3435	46	23	,	,	PUNCT
ejpam-3435	46	24	z	z	PROPN
ejpam-3435	46	25	,	,	PUNCT
ejpam-3435	46	26	u	u	PROPN
ejpam-3435	46	27	∈	∈	PROPN
ejpam-3435	46	28	t	t	PROPN
ejpam-3435	46	29	(	(	PUNCT
ejpam-3435	46	30	m	m	PROPN
ejpam-3435	46	31	)	)	PUNCT
ejpam-3435	46	32	.	.	PUNCT
ejpam-3435	47	1	such	such	DET
ejpam-3435	47	2	a	a	DET
ejpam-3435	47	3	manifold	manifold	ADJ
ejpam-3435	47	4	m(k	m(k	PROPN
ejpam-3435	47	5	)	)	PUNCT
ejpam-3435	47	6	will	will	AUX
ejpam-3435	47	7	be	be	AUX
ejpam-3435	47	8	called	call	VERB
ejpam-3435	47	9	an	an	DET
ejpam-3435	47	10	s	s	NOUN
ejpam-3435	47	11	-	-	PUNCT
ejpam-3435	47	12	space	space	NOUN
ejpam-3435	47	13	form	form	NOUN
ejpam-3435	47	14	.	.	PUNCT
ejpam-3435	48	1	examples	example	NOUN
ejpam-3435	48	2	of	of	ADP
ejpam-3435	48	3	s	s	NOUN
ejpam-3435	48	4	-	-	PUNCT
ejpam-3435	48	5	space	space	NOUN
ejpam-3435	48	6	forms	form	NOUN
ejpam-3435	48	7	are	be	AUX
ejpam-3435	48	8	the	the	DET
ejpam-3435	48	9	euclidean	euclidean	ADJ
ejpam-3435	48	10	space	space	NOUN
ejpam-3435	48	11	e2n+s	e2n+s	NOUN
ejpam-3435	48	12	and	and	CCONJ
ejpam-3435	48	13	the	the	DET
ejpam-3435	48	14	hyperbolic	hyperbolic	ADJ
ejpam-3435	48	15	space	space	NOUN
ejpam-3435	48	16	h2n+s	h2n+s	PROPN
ejpam-3435	48	17	.	.	PUNCT
ejpam-3435	49	1	n.	n.	PROPN
ejpam-3435	49	2	a.	a.	PROPN
ejpam-3435	49	3	rehman	rehman	PROPN
ejpam-3435	49	4	/	/	SYM
ejpam-3435	49	5	eur	eur	PROPN
ejpam-3435	49	6	.	.	PUNCT
ejpam-3435	50	1	j.	j.	PROPN
ejpam-3435	50	2	pure	pure	PROPN
ejpam-3435	50	3	appl	appl	PROPN
ejpam-3435	50	4	.	.	PROPN
ejpam-3435	50	5	math	math	PROPN
ejpam-3435	50	6	,	,	PUNCT
ejpam-3435	50	7	12	12	NUM
ejpam-3435	50	8	(	(	PUNCT
ejpam-3435	50	9	4	4	NUM
ejpam-3435	50	10	)	)	PUNCT
ejpam-3435	50	11	(	(	PUNCT
ejpam-3435	50	12	2019	2019	NUM
ejpam-3435	50	13	)	)	PUNCT
ejpam-3435	50	14	,	,	PUNCT
ejpam-3435	50	15	1811	1811	NUM
ejpam-3435	50	16	-	-	SYM
ejpam-3435	50	17	1818	1818	NUM
ejpam-3435	50	18	1813	1813	NUM
ejpam-3435	50	19	consider	consider	VERB
ejpam-3435	50	20	immersed	immersed	ADJ
ejpam-3435	50	21	submanifold	submanifold	NOUN
ejpam-3435	50	22	mm	mm	PROPN
ejpam-3435	50	23	of	of	ADP
ejpam-3435	50	24	n2n+s	n2n+s	PROPN
ejpam-3435	50	25	.	.	PUNCT
ejpam-3435	51	1	then	then	ADV
ejpam-3435	51	2	m	m	PROPN
ejpam-3435	51	3	is	be	AUX
ejpam-3435	51	4	an	an	DET
ejpam-3435	51	5	invariant	invariant	ADJ
ejpam-3435	51	6	submnaifold	submnaifold	NOUN
ejpam-3435	51	7	if	if	SCONJ
ejpam-3435	51	8	ξα	ξα	NOUN
ejpam-3435	51	9	∈	∈	PROPN
ejpam-3435	51	10	tm	tm	NOUN
ejpam-3435	51	11	for	for	ADP
ejpam-3435	51	12	any	any	DET
ejpam-3435	51	13	α	α	NOUN
ejpam-3435	51	14	and	and	CCONJ
ejpam-3435	51	15	fy	fy	PROPN
ejpam-3435	51	16	∈	∈	PROPN
ejpam-3435	51	17	tm	tm	NOUN
ejpam-3435	51	18	for	for	ADP
ejpam-3435	51	19	any	any	DET
ejpam-3435	51	20	y	y	PROPN
ejpam-3435	51	21	∈	∈	PROPN
ejpam-3435	51	22	tm	tm	NOUN
ejpam-3435	51	23	.	.	PUNCT
ejpam-3435	52	1	it	it	PRON
ejpam-3435	52	2	is	be	AUX
ejpam-3435	52	3	said	say	VERB
ejpam-3435	52	4	to	to	PART
ejpam-3435	52	5	be	be	AUX
ejpam-3435	52	6	anti	anti	ADJ
ejpam-3435	52	7	-	-	ADJ
ejpam-3435	52	8	invariant	invariant	ADJ
ejpam-3435	52	9	submanifold	submanifold	NOUN
ejpam-3435	52	10	if	if	SCONJ
ejpam-3435	52	11	fy	fy	PROPN
ejpam-3435	52	12	∈	∈	PROPN
ejpam-3435	52	13	tm⊥	tm⊥	PROPN
ejpam-3435	52	14	for	for	ADP
ejpam-3435	52	15	any	any	DET
ejpam-3435	52	16	y	y	PROPN
ejpam-3435	52	17	∈	∈	PROPN
ejpam-3435	52	18	tm	tm	PROPN
ejpam-3435	52	19	.	.	PUNCT
ejpam-3435	53	1	for	for	ADP
ejpam-3435	53	2	a	a	DET
ejpam-3435	53	3	vector	vector	NOUN
ejpam-3435	53	4	field	field	NOUN
ejpam-3435	53	5	y	y	PROPN
ejpam-3435	53	6	∈	∈	PROPN
ejpam-3435	53	7	tm⊥	tm⊥	PROPN
ejpam-3435	53	8	,	,	PUNCT
ejpam-3435	53	9	it	it	PRON
ejpam-3435	53	10	can	can	AUX
ejpam-3435	53	11	be	be	AUX
ejpam-3435	53	12	written	write	VERB
ejpam-3435	53	13	as	as	ADP
ejpam-3435	53	14	fy	fy	PROPN
ejpam-3435	53	15	=	=	NOUN
ejpam-3435	53	16	ty	ty	INTJ
ejpam-3435	54	1	+	+	CCONJ
ejpam-3435	54	2	sy	sy	INTJ
ejpam-3435	54	3	,	,	PUNCT
ejpam-3435	54	4	where	where	SCONJ
ejpam-3435	54	5	ty	ty	PRON
ejpam-3435	54	6	represents	represent	VERB
ejpam-3435	54	7	tangent	tangent	ADJ
ejpam-3435	54	8	component	component	NOUN
ejpam-3435	54	9	of	of	ADP
ejpam-3435	54	10	fy	fy	PROPN
ejpam-3435	54	11	,	,	PUNCT
ejpam-3435	54	12	sy	sy	PROPN
ejpam-3435	54	13	shows	show	VERB
ejpam-3435	54	14	normal	normal	ADJ
ejpam-3435	54	15	component	component	NOUN
ejpam-3435	54	16	of	of	ADP
ejpam-3435	54	17	fy	fy	PROPN
ejpam-3435	54	18	.	.	PUNCT
ejpam-3435	55	1	if	if	SCONJ
ejpam-3435	55	2	s	s	PRON
ejpam-3435	55	3	does	do	AUX
ejpam-3435	55	4	not	not	PART
ejpam-3435	55	5	disappear	disappear	VERB
ejpam-3435	55	6	,	,	PUNCT
ejpam-3435	55	7	then	then	ADV
ejpam-3435	55	8	its	its	PRON
ejpam-3435	55	9	an	an	DET
ejpam-3435	55	10	f	f	NOUN
ejpam-3435	55	11	-	-	PUNCT
ejpam-3435	55	12	structure	structure	NOUN
ejpam-3435	55	13	[	[	X
ejpam-3435	55	14	5	5	NUM
ejpam-3435	55	15	]	]	PUNCT
ejpam-3435	55	16	.	.	PUNCT
ejpam-3435	56	1	for	for	ADP
ejpam-3435	56	2	a	a	DET
ejpam-3435	56	3	vector	vector	NOUN
ejpam-3435	56	4	field	field	NOUN
ejpam-3435	56	5	z	z	PROPN
ejpam-3435	56	6	∈	∈	PROPN
ejpam-3435	56	7	tm	tm	NOUN
ejpam-3435	56	8	,	,	PUNCT
ejpam-3435	56	9	it	it	PRON
ejpam-3435	56	10	can	can	AUX
ejpam-3435	56	11	be	be	AUX
ejpam-3435	56	12	written	write	VERB
ejpam-3435	56	13	as	as	ADP
ejpam-3435	56	14	fz	fz	NOUN
ejpam-3435	56	15	=	=	PROPN
ejpam-3435	56	16	pz	pz	PROPN
ejpam-3435	56	17	+	+	CCONJ
ejpam-3435	56	18	nz	nz	PROPN
ejpam-3435	56	19	,	,	PUNCT
ejpam-3435	56	20	where	where	SCONJ
ejpam-3435	56	21	pz	pz	NOUN
ejpam-3435	56	22	is	be	AUX
ejpam-3435	56	23	tangent	tangent	ADJ
ejpam-3435	56	24	component	component	NOUN
ejpam-3435	56	25	of	of	ADP
ejpam-3435	56	26	fz	fz	PROPN
ejpam-3435	56	27	,	,	PUNCT
ejpam-3435	56	28	nz	nz	PROPN
ejpam-3435	56	29	is	be	AUX
ejpam-3435	56	30	normal	normal	ADJ
ejpam-3435	56	31	component	component	NOUN
ejpam-3435	56	32	of	of	ADP
ejpam-3435	56	33	fz	fz	PROPN
ejpam-3435	56	34	.	.	PUNCT
ejpam-3435	57	1	let	let	VERB
ejpam-3435	57	2	dim(m	dim(m	NOUN
ejpam-3435	57	3	)	)	PUNCT
ejpam-3435	57	4	≥	≥	NOUN
ejpam-3435	57	5	s	s	PART
ejpam-3435	58	1	and	and	CCONJ
ejpam-3435	58	2	we	we	PRON
ejpam-3435	58	3	take	take	VERB
ejpam-3435	58	4	the	the	DET
ejpam-3435	58	5	structure	structure	NOUN
ejpam-3435	58	6	vector	vector	NOUN
ejpam-3435	58	7	fields	field	NOUN
ejpam-3435	58	8	ξ1	ξ1	NOUN
ejpam-3435	58	9	,	,	PUNCT
ejpam-3435	58	10	ξ2	ξ2	NOUN
ejpam-3435	58	11	,	,	PUNCT
ejpam-3435	58	12	.	.	PUNCT
ejpam-3435	58	13	.	.	PUNCT
ejpam-3435	59	1	.	.	PUNCT
ejpam-3435	60	1	,	,	PUNCT
ejpam-3435	60	2	ξs	ξs	VERB
ejpam-3435	60	3	as	as	SCONJ
ejpam-3435	60	4	tangents	tangent	NOUN
ejpam-3435	60	5	to	to	ADP
ejpam-3435	60	6	m.	m.	NOUN
ejpam-3435	60	7	m	m	VERB
ejpam-3435	60	8	is	be	AUX
ejpam-3435	60	9	known	know	VERB
ejpam-3435	60	10	as	as	ADP
ejpam-3435	60	11	cr	cr	NOUN
ejpam-3435	60	12	-	-	PUNCT
ejpam-3435	60	13	submaifold	submaifold	NOUN
ejpam-3435	60	14	of	of	ADP
ejpam-3435	60	15	n	n	PRON
ejpam-3435	60	16	if	if	SCONJ
ejpam-3435	60	17	we	we	PRON
ejpam-3435	60	18	have	have	VERB
ejpam-3435	60	19	two	two	NUM
ejpam-3435	60	20	differentiable	differentiable	ADJ
ejpam-3435	60	21	distributions	distribution	NOUN
ejpam-3435	60	22	d	d	NOUN
ejpam-3435	60	23	and	and	CCONJ
ejpam-3435	60	24	d⊥	d⊥	NOUN
ejpam-3435	60	25	on	on	ADP
ejpam-3435	60	26	m	m	PROPN
ejpam-3435	60	27	,	,	PUNCT
ejpam-3435	60	28	tm	tm	NOUN
ejpam-3435	60	29	=	=	PROPN
ejpam-3435	61	1	d	d	X
ejpam-3435	61	2	+	+	NOUN
ejpam-3435	61	3	d⊥	d⊥	NOUN
ejpam-3435	61	4	satisfying	satisfying	NOUN
ejpam-3435	61	5	•	•	NOUN
ejpam-3435	61	6	d	d	NOUN
ejpam-3435	61	7	and	and	CCONJ
ejpam-3435	61	8	d⊥	d⊥	NOUN
ejpam-3435	61	9	are	be	AUX
ejpam-3435	61	10	mutually	mutually	ADV
ejpam-3435	61	11	orthogonal	orthogonal	ADJ
ejpam-3435	61	12	to	to	ADP
ejpam-3435	61	13	each	each	DET
ejpam-3435	61	14	other	other	ADJ
ejpam-3435	61	15	.	.	PUNCT
ejpam-3435	62	1	•	•	INTJ
ejpam-3435	62	2	if	if	SCONJ
ejpam-3435	62	3	fdp	fdp	ADJ
ejpam-3435	62	4	=	=	SYM
ejpam-3435	62	5	dp	dp	PROPN
ejpam-3435	62	6	,	,	PUNCT
ejpam-3435	62	7	for	for	ADP
ejpam-3435	62	8	any	any	DET
ejpam-3435	62	9	p	p	NOUN
ejpam-3435	62	10	∈m	∈m	NOUN
ejpam-3435	62	11	then	then	ADV
ejpam-3435	62	12	the	the	DET
ejpam-3435	62	13	distribution	distribution	NOUN
ejpam-3435	62	14	d	d	NOUN
ejpam-3435	62	15	is	be	AUX
ejpam-3435	62	16	known	know	VERB
ejpam-3435	62	17	as	as	ADP
ejpam-3435	62	18	invariant	invariant	ADJ
ejpam-3435	62	19	under	under	ADP
ejpam-3435	62	20	f.	f.	PROPN
ejpam-3435	62	21	•	•	PROPN
ejpam-3435	62	22	if	if	SCONJ
ejpam-3435	62	23	fd⊥p	fd⊥p	NOUN
ejpam-3435	62	24	⊆	⊆	NUM
ejpam-3435	62	25	tpm⊥	tpm⊥	NOUN
ejpam-3435	62	26	,	,	PUNCT
ejpam-3435	62	27	for	for	ADP
ejpam-3435	62	28	any	any	DET
ejpam-3435	62	29	p	p	NOUN
ejpam-3435	62	30	∈m	∈m	NOUN
ejpam-3435	62	31	then	then	ADV
ejpam-3435	62	32	the	the	DET
ejpam-3435	62	33	distribution	distribution	NOUN
ejpam-3435	62	34	d⊥	d⊥	NOUN
ejpam-3435	62	35	is	be	AUX
ejpam-3435	62	36	known	know	VERB
ejpam-3435	62	37	as	as	ADP
ejpam-3435	62	38	anti	anti	X
ejpam-3435	62	39	invariant	invariant	PROPN
ejpam-3435	62	40	under	under	ADP
ejpam-3435	62	41	f.	f.	PROPN
ejpam-3435	62	42	the	the	DET
ejpam-3435	62	43	gauss	gauss	ADJ
ejpam-3435	62	44	equation	equation	NOUN
ejpam-3435	62	45	for	for	ADP
ejpam-3435	62	46	the	the	DET
ejpam-3435	62	47	submanifold	submanifold	NOUN
ejpam-3435	62	48	mm	mm	PROPN
ejpam-3435	62	49	into	into	ADP
ejpam-3435	62	50	a	a	DET
ejpam-3435	62	51	(	(	PUNCT
ejpam-3435	62	52	m+	m+	NUM
ejpam-3435	62	53	q)-dimensional	q)-dimensional	ADJ
ejpam-3435	62	54	riemannian	riemannian	NOUN
ejpam-3435	62	55	manifold	manifold	ADJ
ejpam-3435	62	56	nm+q	nm+q	PROPN
ejpam-3435	62	57	is	be	AUX
ejpam-3435	62	58	r̃′(y	r̃′(y	NOUN
ejpam-3435	62	59	,	,	PUNCT
ejpam-3435	62	60	z	z	PROPN
ejpam-3435	62	61	,	,	PUNCT
ejpam-3435	62	62	u	u	NOUN
ejpam-3435	62	63	,	,	PUNCT
ejpam-3435	62	64	w	w	NOUN
ejpam-3435	62	65	)	)	PUNCT
ejpam-3435	63	1	=	=	SYM
ejpam-3435	63	2	r′(y	r′(y	PROPN
ejpam-3435	63	3	,	,	PUNCT
ejpam-3435	63	4	z	z	PROPN
ejpam-3435	63	5	,	,	PUNCT
ejpam-3435	63	6	u	u	NOUN
ejpam-3435	63	7	,	,	PUNCT
ejpam-3435	63	8	w	w	NOUN
ejpam-3435	63	9	)	)	PUNCT
ejpam-3435	63	10	+	+	CCONJ
ejpam-3435	64	1	g(h′(y	g(h′(y	PROPN
ejpam-3435	64	2	,	,	PUNCT
ejpam-3435	64	3	u	u	NOUN
ejpam-3435	64	4	)	)	PUNCT
ejpam-3435	64	5	,	,	PUNCT
ejpam-3435	64	6	h′(z	h′(z	PROPN
ejpam-3435	64	7	,	,	PUNCT
ejpam-3435	64	8	w	w	NOUN
ejpam-3435	64	9	)	)	PUNCT
ejpam-3435	64	10	)	)	PUNCT
ejpam-3435	64	11	−	−	PROPN
ejpam-3435	65	1	g(h′(y	g(h′(y	PROPN
ejpam-3435	65	2	,	,	PUNCT
ejpam-3435	65	3	w	w	NOUN
ejpam-3435	65	4	)	)	PUNCT
ejpam-3435	65	5	,	,	PUNCT
ejpam-3435	65	6	h′(z	h′(z	PROPN
ejpam-3435	65	7	,	,	PUNCT
ejpam-3435	65	8	u	u	NOUN
ejpam-3435	65	9	)	)	PUNCT
ejpam-3435	65	10	)	)	PUNCT
ejpam-3435	65	11	.	.	PUNCT
ejpam-3435	66	1	consider	consider	VERB
ejpam-3435	66	2	k	k	NOUN
ejpam-3435	66	3	-	-	PUNCT
ejpam-3435	66	4	plan	plan	NOUN
ejpam-3435	66	5	section	section	NOUN
ejpam-3435	66	6	of	of	ADP
ejpam-3435	66	7	tpm	tpm	PROPN
ejpam-3435	66	8	denoted	denote	VERB
ejpam-3435	66	9	by	by	ADP
ejpam-3435	66	10	d	d	NOUN
ejpam-3435	66	11	and	and	CCONJ
ejpam-3435	66	12	let	let	VERB
ejpam-3435	66	13	x	x	PRON
ejpam-3435	66	14	be	be	AUX
ejpam-3435	66	15	a	a	DET
ejpam-3435	66	16	unit	unit	NOUN
ejpam-3435	66	17	vector	vector	NOUN
ejpam-3435	66	18	in	in	ADP
ejpam-3435	66	19	d.	d.	PROPN
ejpam-3435	66	20	we	we	PRON
ejpam-3435	66	21	can	can	AUX
ejpam-3435	66	22	choose	choose	VERB
ejpam-3435	66	23	orthonormal	orthonormal	ADJ
ejpam-3435	66	24	basis	basis	NOUN
ejpam-3435	66	25	e1	e1	NOUN
ejpam-3435	66	26	,	,	PUNCT
ejpam-3435	66	27	e2	e2	PROPN
ejpam-3435	66	28	,	,	PUNCT
ejpam-3435	66	29	.	.	PUNCT
ejpam-3435	66	30	.	.	PUNCT
ejpam-3435	67	1	.	.	PUNCT
ejpam-3435	68	1	,	,	PUNCT
ejpam-3435	68	2	ek	ek	NOUN
ejpam-3435	68	3	of	of	ADP
ejpam-3435	68	4	d	d	PROPN
ejpam-3435	68	5	and	and	CCONJ
ejpam-3435	68	6	we	we	PRON
ejpam-3435	68	7	consider	consider	VERB
ejpam-3435	68	8	e1	e1	NOUN
ejpam-3435	68	9	=	=	SYM
ejpam-3435	69	1	x	x	NOUN
ejpam-3435	69	2	,	,	PUNCT
ejpam-3435	69	3	then	then	ADV
ejpam-3435	69	4	ricci	ricci	PROPN
ejpam-3435	69	5	curvature	curvature	PROPN
ejpam-3435	69	6	ricd	ricd	PROPN
ejpam-3435	69	7	of	of	ADP
ejpam-3435	69	8	d	d	PROPN
ejpam-3435	69	9	at	at	ADP
ejpam-3435	69	10	x	x	X
ejpam-3435	69	11	can	can	AUX
ejpam-3435	69	12	be	be	AUX
ejpam-3435	69	13	define	define	VERB
ejpam-3435	69	14	as	as	ADP
ejpam-3435	69	15	ricd(x	ricd(x	VERB
ejpam-3435	69	16	)	)	PUNCT
ejpam-3435	70	1	=	=	SYM
ejpam-3435	70	2	k12	k12	NOUN
ejpam-3435	71	1	+	+	NOUN
ejpam-3435	71	2	k13	k13	ADJ
ejpam-3435	71	3	+	+	X
ejpam-3435	71	4	·	·	PUNCT
ejpam-3435	71	5	·	·	PUNCT
ejpam-3435	71	6	·	·	PUNCT
ejpam-3435	71	7	+	+	NUM
ejpam-3435	71	8	k1k	k1k	PROPN
ejpam-3435	71	9	,	,	PUNCT
ejpam-3435	71	10	where	where	SCONJ
ejpam-3435	71	11	kij	kij	PROPN
ejpam-3435	71	12	represents	represent	VERB
ejpam-3435	71	13	sectional	sectional	ADJ
ejpam-3435	71	14	curvature	curvature	NOUN
ejpam-3435	71	15	of	of	ADP
ejpam-3435	71	16	2	2	NUM
ejpam-3435	71	17	-	-	PUNCT
ejpam-3435	71	18	plane	plane	NOUN
ejpam-3435	71	19	section	section	NOUN
ejpam-3435	71	20	spanned	span	VERB
ejpam-3435	71	21	by	by	ADP
ejpam-3435	71	22	ei	ei	PROPN
ejpam-3435	71	23	,	,	PUNCT
ejpam-3435	71	24	ej	ej	PROPN
ejpam-3435	71	25	.	.	PUNCT
ejpam-3435	72	1	therefore	therefore	ADV
ejpam-3435	72	2	the	the	DET
ejpam-3435	72	3	scalar	scalar	ADJ
ejpam-3435	72	4	curvature	curvature	NOUN
ejpam-3435	72	5	τ	τ	PROPN
ejpam-3435	72	6	of	of	ADP
ejpam-3435	72	7	the	the	DET
ejpam-3435	72	8	k	k	ADJ
ejpam-3435	72	9	-	-	ADJ
ejpam-3435	72	10	plane	plane	NOUN
ejpam-3435	72	11	section	section	NOUN
ejpam-3435	72	12	d	d	NOUN
ejpam-3435	72	13	can	can	AUX
ejpam-3435	72	14	be	be	AUX
ejpam-3435	72	15	written	write	VERB
ejpam-3435	72	16	as	as	ADP
ejpam-3435	72	17	τ(d	τ(d	PROPN
ejpam-3435	72	18	)	)	PUNCT
ejpam-3435	72	19	=	=	PUNCT
ejpam-3435	73	1	∑	∑	PUNCT
ejpam-3435	73	2	1≤i	1≤i	PROPN
ejpam-3435	73	3	<	<	X
ejpam-3435	73	4	j≤k	j≤k	PROPN
ejpam-3435	73	5	kij	kij	PROPN
ejpam-3435	73	6	.	.	PUNCT
ejpam-3435	73	7	.	.	PUNCT
ejpam-3435	74	1	3	3	X
ejpam-3435	74	2	.	.	X
ejpam-3435	74	3	main	main	ADJ
ejpam-3435	74	4	results	result	NOUN
ejpam-3435	74	5	theorem	theorem	VERB
ejpam-3435	74	6	1	1	NUM
ejpam-3435	74	7	.	.	PUNCT
ejpam-3435	75	1	let	let	VERB
ejpam-3435	75	2	mm	mm	NOUN
ejpam-3435	75	3	,	,	PUNCT
ejpam-3435	75	4	be	be	AUX
ejpam-3435	75	5	a	a	DET
ejpam-3435	75	6	sub	sub	NOUN
ejpam-3435	75	7	manifold	manifold	NOUN
ejpam-3435	75	8	of	of	ADP
ejpam-3435	75	9	s	s	NOUN
ejpam-3435	75	10	-	-	PUNCT
ejpam-3435	75	11	space	space	NOUN
ejpam-3435	75	12	form	form	NOUN
ejpam-3435	75	13	n2n+s(k	n2n+s(k	NOUN
ejpam-3435	75	14	)	)	PUNCT
ejpam-3435	75	15	of	of	ADP
ejpam-3435	75	16	constant	constant	ADJ
ejpam-3435	75	17	f	f	PROPN
ejpam-3435	75	18	-sectional	-sectional	ADJ
ejpam-3435	75	19	curvature	curvature	NOUN
ejpam-3435	75	20	k	k	NOUN
ejpam-3435	75	21	,	,	PUNCT
ejpam-3435	75	22	with	with	ADP
ejpam-3435	75	23	structural	structural	ADJ
ejpam-3435	75	24	vector	vector	NOUN
ejpam-3435	75	25	fields	field	NOUN
ejpam-3435	75	26	ξ1	ξ1	NOUN
ejpam-3435	75	27	,	,	PUNCT
ejpam-3435	75	28	ξ2	ξ2	NOUN
ejpam-3435	75	29	,	,	PUNCT
ejpam-3435	75	30	.	.	PUNCT
ejpam-3435	75	31	.	.	PUNCT
ejpam-3435	76	1	.	.	PUNCT
ejpam-3435	77	1	,	,	PUNCT
ejpam-3435	77	2	ξs	ξs	VERB
ejpam-3435	77	3	tangent	tangent	NOUN
ejpam-3435	77	4	to	to	ADP
ejpam-3435	77	5	mm	mm	PROPN
ejpam-3435	77	6	,	,	PUNCT
ejpam-3435	77	7	then	then	ADV
ejpam-3435	77	8	for	for	ADP
ejpam-3435	77	9	each	each	DET
ejpam-3435	77	10	unit	unit	NOUN
ejpam-3435	77	11	vector	vector	NOUN
ejpam-3435	77	12	x	x	PROPN
ejpam-3435	77	13	∈	∈	PROPN
ejpam-3435	77	14	tpm	tpm	PROPN
ejpam-3435	77	15	,	,	PUNCT
ejpam-3435	77	16	orthogonal	orthogonal	ADJ
ejpam-3435	77	17	to	to	ADP
ejpam-3435	77	18	ξ1	ξ1	NOUN
ejpam-3435	77	19	,	,	PUNCT
ejpam-3435	77	20	ξ2	ξ2	NOUN
ejpam-3435	77	21	,	,	PUNCT
ejpam-3435	77	22	.	.	PUNCT
ejpam-3435	77	23	.	.	PUNCT
ejpam-3435	78	1	.	.	PUNCT
ejpam-3435	79	1	,	,	PUNCT
ejpam-3435	79	2	ξs	ξs	VERB
ejpam-3435	79	3	ric(x	ric(x	NOUN
ejpam-3435	79	4	)	)	PUNCT
ejpam-3435	79	5	≤	≤	NOUN
ejpam-3435	79	6	1	1	NUM
ejpam-3435	79	7	4	4	NUM
ejpam-3435	79	8	[	[	PUNCT
ejpam-3435	79	9	m2	m2	PROPN
ejpam-3435	79	10	‖	‖	PROPN
ejpam-3435	79	11	h	h	NOUN
ejpam-3435	79	12	‖2	‖2	NOUN
ejpam-3435	80	1	+	+	PROPN
ejpam-3435	80	2	(	(	PUNCT
ejpam-3435	80	3	m−	m−	PROPN
ejpam-3435	80	4	1)(k	1)(k	NUM
ejpam-3435	80	5	+	+	CCONJ
ejpam-3435	80	6	3s	3s	NUM
ejpam-3435	80	7	)	)	PUNCT
ejpam-3435	81	1	+	+	CCONJ
ejpam-3435	81	2	3	3	NUM
ejpam-3435	81	3	2	2	NUM
ejpam-3435	81	4	‖	‖	PROPN
ejpam-3435	81	5	px	px	NOUN
ejpam-3435	81	6	‖2	‖2	NOUN
ejpam-3435	81	7	(	(	PUNCT
ejpam-3435	81	8	k	k	PROPN
ejpam-3435	81	9	−	−	PROPN
ejpam-3435	82	1	s)−	s)−	PROPN
ejpam-3435	82	2	1	1	NUM
ejpam-3435	82	3	2	2	NUM
ejpam-3435	82	4	(	(	PUNCT
ejpam-3435	82	5	3s−	3s−	NUM
ejpam-3435	82	6	1)(k	1)(k	NUM
ejpam-3435	82	7	+	+	SYM
ejpam-3435	82	8	3s−	3s−	NUM
ejpam-3435	82	9	4	4	NUM
ejpam-3435	82	10	)	)	PUNCT
ejpam-3435	82	11	]	]	PUNCT
ejpam-3435	82	12	(	(	PUNCT
ejpam-3435	82	13	5	5	X
ejpam-3435	82	14	)	)	PUNCT
ejpam-3435	82	15	n.	n.	NOUN
ejpam-3435	82	16	a.	a.	PROPN
ejpam-3435	82	17	rehman	rehman	PROPN
ejpam-3435	82	18	/	/	SYM
ejpam-3435	82	19	eur	eur	PROPN
ejpam-3435	82	20	.	.	PUNCT
ejpam-3435	83	1	j.	j.	PROPN
ejpam-3435	83	2	pure	pure	PROPN
ejpam-3435	83	3	appl	appl	PROPN
ejpam-3435	83	4	.	.	PROPN
ejpam-3435	83	5	math	math	PROPN
ejpam-3435	83	6	,	,	PUNCT
ejpam-3435	83	7	12	12	NUM
ejpam-3435	83	8	(	(	PUNCT
ejpam-3435	83	9	4	4	NUM
ejpam-3435	83	10	)	)	PUNCT
ejpam-3435	83	11	(	(	PUNCT
ejpam-3435	83	12	2019	2019	NUM
ejpam-3435	83	13	)	)	PUNCT
ejpam-3435	83	14	,	,	PUNCT
ejpam-3435	83	15	1811	1811	NUM
ejpam-3435	83	16	-	-	SYM
ejpam-3435	83	17	1818	1818	NUM
ejpam-3435	83	18	1814	1814	NUM
ejpam-3435	83	19	proof	proof	NOUN
ejpam-3435	83	20	.	.	PUNCT
ejpam-3435	84	1	let	let	VERB
ejpam-3435	84	2	mm	mm	PRON
ejpam-3435	84	3	be	be	AUX
ejpam-3435	84	4	a	a	DET
ejpam-3435	84	5	sub	sub	NOUN
ejpam-3435	84	6	manifold	manifold	NOUN
ejpam-3435	84	7	of	of	ADP
ejpam-3435	84	8	s	s	NOUN
ejpam-3435	84	9	-	-	PUNCT
ejpam-3435	84	10	space	space	NOUN
ejpam-3435	84	11	form	form	NOUN
ejpam-3435	84	12	n2n+s(k	n2n+s(k	NOUN
ejpam-3435	84	13	)	)	PUNCT
ejpam-3435	84	14	with	with	ADP
ejpam-3435	84	15	constant	constant	ADJ
ejpam-3435	84	16	f	f	PROPN
ejpam-3435	84	17	-sectional	-sectional	ADJ
ejpam-3435	84	18	curvature	curvature	NOUN
ejpam-3435	84	19	k.	k.	NOUN
ejpam-3435	84	20	for	for	ADP
ejpam-3435	84	21	a	a	DET
ejpam-3435	84	22	point	point	NOUN
ejpam-3435	84	23	p	p	X
ejpam-3435	84	24	∈	∈	PROPN
ejpam-3435	84	25	m	m	VERB
ejpam-3435	84	26	,	,	PUNCT
ejpam-3435	84	27	take	take	VERB
ejpam-3435	84	28	a	a	DET
ejpam-3435	84	29	unit	unit	NOUN
ejpam-3435	84	30	vector	vector	NOUN
ejpam-3435	84	31	x	x	PROPN
ejpam-3435	84	32	∈	∈	PROPN
ejpam-3435	84	33	tpm	tpm	PROPN
ejpam-3435	84	34	.	.	PUNCT
ejpam-3435	85	1	choose	choose	VERB
ejpam-3435	85	2	an	an	DET
ejpam-3435	85	3	orthonormal	orthonormal	ADJ
ejpam-3435	85	4	basis	basis	NOUN
ejpam-3435	85	5	e1	e1	NOUN
ejpam-3435	85	6	,	,	PUNCT
ejpam-3435	85	7	e2	e2	PROPN
ejpam-3435	85	8	,	,	PUNCT
ejpam-3435	85	9	.	.	PUNCT
ejpam-3435	85	10	.	.	PUNCT
ejpam-3435	86	1	.	.	PUNCT
ejpam-3435	87	1	,	,	PUNCT
ejpam-3435	87	2	em−s	em−s	PROPN
ejpam-3435	87	3	,	,	PUNCT
ejpam-3435	87	4	em−s+1	em−s+1	NOUN
ejpam-3435	87	5	,	,	PUNCT
ejpam-3435	87	6	.	.	PUNCT
ejpam-3435	87	7	.	.	PUNCT
ejpam-3435	88	1	.	.	PUNCT
ejpam-3435	89	1	,	,	PUNCT
ejpam-3435	89	2	em−s+s	em−s+s	PROPN
ejpam-3435	89	3	,	,	PUNCT
ejpam-3435	89	4	em−s+s+1	em−s+s+1	NOUN
ejpam-3435	89	5	,	,	PUNCT
ejpam-3435	89	6	.	.	PUNCT
ejpam-3435	89	7	.	.	PUNCT
ejpam-3435	90	1	.	.	PUNCT
ejpam-3435	91	1	,	,	PUNCT
ejpam-3435	91	2	e2n+s	e2n+s	NOUN
ejpam-3435	91	3	on	on	ADP
ejpam-3435	91	4	tpn	tpn	PROPN
ejpam-3435	91	5	.	.	PUNCT
ejpam-3435	92	1	then	then	ADV
ejpam-3435	92	2	it	it	PRON
ejpam-3435	92	3	is	be	AUX
ejpam-3435	92	4	clear	clear	ADJ
ejpam-3435	92	5	that	that	SCONJ
ejpam-3435	92	6	e1	e1	PROPN
ejpam-3435	92	7	,	,	PUNCT
ejpam-3435	92	8	e2	e2	PROPN
ejpam-3435	92	9	,	,	PUNCT
ejpam-3435	92	10	.	.	PUNCT
ejpam-3435	92	11	.	.	PUNCT
ejpam-3435	93	1	.	.	PUNCT
ejpam-3435	94	1	,	,	PUNCT
ejpam-3435	94	2	em−s	em−s	PROPN
ejpam-3435	94	3	,	,	PUNCT
ejpam-3435	94	4	em−s+1	em−s+1	NOUN
ejpam-3435	94	5	,	,	PUNCT
ejpam-3435	94	6	.	.	PUNCT
ejpam-3435	94	7	.	.	PUNCT
ejpam-3435	95	1	.	.	PUNCT
ejpam-3435	96	1	,	,	PUNCT
ejpam-3435	96	2	em−s+s	em−s+s	PROPN
ejpam-3435	96	3	are	be	AUX
ejpam-3435	96	4	unit	unit	NOUN
ejpam-3435	96	5	tangent	tangent	NOUN
ejpam-3435	96	6	vectors	vector	NOUN
ejpam-3435	96	7	to	to	ADP
ejpam-3435	96	8	m	m	PROPN
ejpam-3435	96	9	at	at	ADP
ejpam-3435	96	10	p.	p.	NOUN
ejpam-3435	96	11	we	we	PRON
ejpam-3435	96	12	take	take	VERB
ejpam-3435	96	13	x	x	SYM
ejpam-3435	96	14	=	=	SYM
ejpam-3435	96	15	e1	e1	PROPN
ejpam-3435	96	16	.	.	PUNCT
ejpam-3435	97	1	from	from	ADP
ejpam-3435	97	2	the	the	DET
ejpam-3435	97	3	curvature	curvature	NOUN
ejpam-3435	97	4	tensor	tensor	NOUN
ejpam-3435	97	5	of	of	ADP
ejpam-3435	97	6	s	s	NOUN
ejpam-3435	97	7	-	-	PUNCT
ejpam-3435	97	8	space	space	NOUN
ejpam-3435	97	9	form	form	NOUN
ejpam-3435	97	10	n2n+s(k	n2n+s(k	NOUN
ejpam-3435	97	11	)	)	PUNCT
ejpam-3435	97	12	,	,	PUNCT
ejpam-3435	97	13	after	after	SCONJ
ejpam-3435	97	14	summation	summation	NOUN
ejpam-3435	97	15	1	1	NUM
ejpam-3435	97	16	≤	≤	PUNCT
ejpam-3435	97	17	i	i	PRON
ejpam-3435	97	18	,	,	PUNCT
ejpam-3435	97	19	j	j	PROPN
ejpam-3435	97	20	≤	≤	PROPN
ejpam-3435	98	1	m	m	VERB
ejpam-3435	98	2	we	we	PRON
ejpam-3435	98	3	have	have	VERB
ejpam-3435	98	4	r̃′(ei	r̃′(ei	PROPN
ejpam-3435	98	5	,	,	PUNCT
ejpam-3435	98	6	ej	ej	INTJ
ejpam-3435	98	7	,	,	PUNCT
ejpam-3435	98	8	ej	ej	PROPN
ejpam-3435	98	9	,	,	PUNCT
ejpam-3435	98	10	ei	ei	PROPN
ejpam-3435	98	11	)	)	PUNCT
ejpam-3435	98	12	=	=	SYM
ejpam-3435	98	13	1	1	NUM
ejpam-3435	98	14	4	4	NUM
ejpam-3435	98	15	(	(	PUNCT
ejpam-3435	98	16	k	k	PROPN
ejpam-3435	98	17	+	+	NUM
ejpam-3435	98	18	3s)m(m−	3s)m(m−	NUM
ejpam-3435	98	19	1	1	NUM
ejpam-3435	98	20	)	)	PUNCT
ejpam-3435	98	21	+	+	CCONJ
ejpam-3435	99	1	k	k	X
ejpam-3435	99	2	+	+	X
ejpam-3435	99	3	3s−	3s−	NUM
ejpam-3435	99	4	4	4	NUM
ejpam-3435	99	5	4	4	NUM
ejpam-3435	99	6	(	(	PUNCT
ejpam-3435	99	7	2s−	2s−	NUM
ejpam-3435	99	8	2ms	2ms	NOUN
ejpam-3435	99	9	)	)	PUNCT
ejpam-3435	100	1	+	+	CCONJ
ejpam-3435	100	2	3	3	NUM
ejpam-3435	100	3	4	4	NUM
ejpam-3435	100	4	(	(	PUNCT
ejpam-3435	100	5	k	k	NOUN
ejpam-3435	100	6	−	−	PROPN
ejpam-3435	100	7	1	1	NUM
ejpam-3435	100	8	)	)	PUNCT
ejpam-3435	100	9	‖	‖	PROPN
ejpam-3435	100	10	p	p	NOUN
ejpam-3435	100	11	‖2	‖2	NOUN
ejpam-3435	100	12	.	.	PUNCT
ejpam-3435	101	1	using	use	VERB
ejpam-3435	101	2	gauss	gauss	ADJ
ejpam-3435	101	3	equation	equation	NOUN
ejpam-3435	101	4	1	1	NUM
ejpam-3435	101	5	4	4	NUM
ejpam-3435	101	6	(	(	PUNCT
ejpam-3435	101	7	k	k	PROPN
ejpam-3435	101	8	+	+	NUM
ejpam-3435	101	9	3s)m(m−	3s)m(m−	NUM
ejpam-3435	101	10	1	1	NUM
ejpam-3435	101	11	)	)	PUNCT
ejpam-3435	102	1	+	+	CCONJ
ejpam-3435	102	2	k	k	X
ejpam-3435	103	1	+	+	X
ejpam-3435	103	2	3s−	3s−	NUM
ejpam-3435	103	3	4	4	NUM
ejpam-3435	103	4	4	4	NUM
ejpam-3435	103	5	(	(	PUNCT
ejpam-3435	103	6	2s−	2s−	NUM
ejpam-3435	103	7	2ms	2ms	NOUN
ejpam-3435	103	8	)	)	PUNCT
ejpam-3435	104	1	+	+	CCONJ
ejpam-3435	104	2	3	3	NUM
ejpam-3435	104	3	4	4	NUM
ejpam-3435	104	4	(	(	PUNCT
ejpam-3435	104	5	k	k	NOUN
ejpam-3435	104	6	−	−	PROPN
ejpam-3435	104	7	1	1	NUM
ejpam-3435	104	8	)	)	PUNCT
ejpam-3435	104	9	‖	‖	PROPN
ejpam-3435	104	10	p	p	X
ejpam-3435	104	11	‖2=	‖2=	ADJ
ejpam-3435	104	12	2τ	2τ	NUM
ejpam-3435	105	1	−m2	−m2	NOUN
ejpam-3435	105	2	‖	‖	PROPN
ejpam-3435	105	3	h	h	NOUN
ejpam-3435	105	4	‖2	‖2	NOUN
ejpam-3435	106	1	+	+	CCONJ
ejpam-3435	106	2	‖	‖	ADJ
ejpam-3435	106	3	h	h	NOUN
ejpam-3435	106	4	‖2	‖2	NOUN
ejpam-3435	106	5	,	,	PUNCT
ejpam-3435	106	6	(	(	PUNCT
ejpam-3435	106	7	6	6	NUM
ejpam-3435	106	8	)	)	PUNCT
ejpam-3435	106	9	where	where	SCONJ
ejpam-3435	106	10	h	h	NOUN
ejpam-3435	106	11	represents	represent	VERB
ejpam-3435	106	12	mean	mean	ADJ
ejpam-3435	106	13	curvature	curvature	NOUN
ejpam-3435	106	14	,	,	PUNCT
ejpam-3435	106	15	h	h	NOUN
ejpam-3435	106	16	represents	represent	VERB
ejpam-3435	106	17	second	second	ADJ
ejpam-3435	106	18	fundamental	fundamental	ADJ
ejpam-3435	106	19	form	form	NOUN
ejpam-3435	106	20	and	and	CCONJ
ejpam-3435	106	21	τ	τ	PROPN
ejpam-3435	106	22	is	be	AUX
ejpam-3435	106	23	scalar	scalar	ADJ
ejpam-3435	106	24	curvature	curvature	NOUN
ejpam-3435	106	25	.	.	PUNCT
ejpam-3435	107	1	now	now	ADV
ejpam-3435	107	2	we	we	PRON
ejpam-3435	107	3	take	take	VERB
ejpam-3435	107	4	the	the	DET
ejpam-3435	107	5	notation	notation	NOUN
ejpam-3435	107	6	for	for	ADP
ejpam-3435	107	7	second	second	ADJ
ejpam-3435	107	8	fundamental	fundamental	ADJ
ejpam-3435	107	9	form	form	NOUN
ejpam-3435	107	10	h	h	NOUN
ejpam-3435	107	11	as	as	ADP
ejpam-3435	107	12	hrij	hrij	NOUN
ejpam-3435	107	13	=	=	PUNCT
ejpam-3435	108	1	g(h(ei	g(h(ei	PROPN
ejpam-3435	108	2	,	,	PUNCT
ejpam-3435	108	3	ej	ej	NOUN
ejpam-3435	108	4	)	)	PUNCT
ejpam-3435	108	5	,	,	PUNCT
ejpam-3435	108	6	er	er	INTJ
ejpam-3435	108	7	)	)	PUNCT
ejpam-3435	108	8	,	,	PUNCT
ejpam-3435	108	9	‖	‖	PROPN
ejpam-3435	109	1	h	h	PROPN
ejpam-3435	109	2	‖2=	‖2=	PROPN
ejpam-3435	109	3	∑m	∑m	PROPN
ejpam-3435	110	1	i	i	PROPN
ejpam-3435	110	2	,	,	PUNCT
ejpam-3435	110	3	j=1	j=1	PROPN
ejpam-3435	110	4	g(h(ei	g(h(ei	PROPN
ejpam-3435	110	5	,	,	PUNCT
ejpam-3435	110	6	ej	ej	NOUN
ejpam-3435	110	7	)	)	PUNCT
ejpam-3435	110	8	,	,	PUNCT
ejpam-3435	110	9	h(ei	h(ei	PROPN
ejpam-3435	110	10	,	,	PUNCT
ejpam-3435	110	11	ej	ej	NOUN
ejpam-3435	110	12	)	)	PUNCT
ejpam-3435	110	13	)	)	PUNCT
ejpam-3435	110	14	and	and	CCONJ
ejpam-3435	110	15	‖	‖	PROPN
ejpam-3435	110	16	p	p	PROPN
ejpam-3435	110	17	‖2=	‖2=	PROPN
ejpam-3435	110	18	∑m	∑m	PROPN
ejpam-3435	110	19	i	i	PROPN
ejpam-3435	110	20	,	,	PUNCT
ejpam-3435	110	21	j=1	j=1	PROPN
ejpam-3435	110	22	g	g	PROPN
ejpam-3435	110	23	2(pei	2(pei	NUM
ejpam-3435	110	24	,	,	PUNCT
ejpam-3435	110	25	ej	ej	PROPN
ejpam-3435	110	26	)	)	PUNCT
ejpam-3435	110	27	.	.	PUNCT
ejpam-3435	111	1	also	also	ADV
ejpam-3435	111	2	we	we	PRON
ejpam-3435	111	3	know	know	VERB
ejpam-3435	111	4	that	that	PRON
ejpam-3435	111	5	mean	mean	NOUN
ejpam-3435	111	6	curvature	curvature	NOUN
ejpam-3435	111	7	h(p	h(p	NOUN
ejpam-3435	111	8	)	)	PUNCT
ejpam-3435	111	9	=	=	SYM
ejpam-3435	112	1	1	1	NUM
ejpam-3435	112	2	m	m	NOUN
ejpam-3435	112	3	∑m	∑m	PROPN
ejpam-3435	112	4	i=1	i=1	PROPN
ejpam-3435	112	5	h(ei	h(ei	PROPN
ejpam-3435	112	6	,	,	PUNCT
ejpam-3435	112	7	ei	ei	NOUN
ejpam-3435	112	8	)	)	PUNCT
ejpam-3435	112	9	.	.	PUNCT
ejpam-3435	113	1	then	then	ADV
ejpam-3435	113	2	from	from	ADP
ejpam-3435	113	3	(	(	PUNCT
ejpam-3435	113	4	6	6	NUM
ejpam-3435	113	5	)	)	PUNCT
ejpam-3435	113	6	,	,	PUNCT
ejpam-3435	113	7	we	we	PRON
ejpam-3435	113	8	have	have	VERB
ejpam-3435	113	9	n2	n2	ADJ
ejpam-3435	113	10	‖	‖	PROPN
ejpam-3435	113	11	h	h	NOUN
ejpam-3435	113	12	‖2	‖2	NOUN
ejpam-3435	113	13	=	=	PUNCT
ejpam-3435	113	14	2τ	2τ	NUM
ejpam-3435	114	1	+	+	CCONJ
ejpam-3435	114	2	1	1	NUM
ejpam-3435	114	3	2	2	NUM
ejpam-3435	114	4	2n+s∑	2n+s∑	NUM
ejpam-3435	114	5	r	r	NOUN
ejpam-3435	114	6	=	=	NOUN
ejpam-3435	114	7	m+1	m+1	X
ejpam-3435	114	8	[	[	PUNCT
ejpam-3435	114	9	(	(	PUNCT
ejpam-3435	114	10	hr11	hr11	PROPN
ejpam-3435	114	11	+	+	CCONJ
ejpam-3435	114	12	·	·	PUNCT
ejpam-3435	114	13	·	·	PUNCT
ejpam-3435	114	14	·	·	PUNCT
ejpam-3435	115	1	+	+	CCONJ
ejpam-3435	115	2	hrmm)2	hrmm)2	X
ejpam-3435	115	3	+	+	CCONJ
ejpam-3435	115	4	(	(	PUNCT
ejpam-3435	115	5	hr11	hr11	PROPN
ejpam-3435	115	6	−	−	PROPN
ejpam-3435	116	1	hr22	hr22	PROPN
ejpam-3435	116	2	−	−	PROPN
ejpam-3435	116	3	·	·	PUNCT
ejpam-3435	116	4	·	·	PUNCT
ejpam-3435	116	5	·	·	PUNCT
ejpam-3435	117	1	−	−	PUNCT
ejpam-3435	118	1	hrmm)2	hrmm)2	X
ejpam-3435	118	2	]	]	X
ejpam-3435	119	1	+	+	CCONJ
ejpam-3435	119	2	2	2	NUM
ejpam-3435	119	3	2n+s∑	2n+s∑	NUM
ejpam-3435	119	4	r	r	NOUN
ejpam-3435	119	5	=	=	NOUN
ejpam-3435	119	6	m+1	m+1	NUM
ejpam-3435	119	7	∑	∑	PUNCT
ejpam-3435	119	8	i	i	PRON
ejpam-3435	119	9	<	<	X
ejpam-3435	119	10	j	j	PROPN
ejpam-3435	119	11	(	(	PUNCT
ejpam-3435	119	12	hrij	hrij	PROPN
ejpam-3435	119	13	)	)	PUNCT
ejpam-3435	119	14	2	2	NUM
ejpam-3435	119	15	−	−	PROPN
ejpam-3435	119	16	2	2	NUM
ejpam-3435	119	17	2n+s∑	2n+s∑	NUM
ejpam-3435	119	18	r	r	NOUN
ejpam-3435	119	19	=	=	NOUN
ejpam-3435	119	20	m+1	m+1	NUM
ejpam-3435	119	21	∑	∑	PUNCT
ejpam-3435	119	22	2≤i	2≤i	PROPN
ejpam-3435	119	23	<	<	X
ejpam-3435	119	24	j≤n	j≤n	NOUN
ejpam-3435	119	25	hriih	hriih	ADJ
ejpam-3435	119	26	r	r	NOUN
ejpam-3435	119	27	jj	jj	PROPN
ejpam-3435	119	28	−	−	PROPN
ejpam-3435	119	29	1	1	NUM
ejpam-3435	119	30	4	4	NUM
ejpam-3435	119	31	(	(	PUNCT
ejpam-3435	119	32	k	k	PROPN
ejpam-3435	119	33	+	+	NUM
ejpam-3435	119	34	3s)m(m−	3s)m(m−	PROPN
ejpam-3435	119	35	1)−	1)−	PROPN
ejpam-3435	119	36	k	k	PROPN
ejpam-3435	120	1	+	+	X
ejpam-3435	120	2	3s−	3s−	NUM
ejpam-3435	120	3	4	4	NUM
ejpam-3435	120	4	4	4	NUM
ejpam-3435	120	5	(	(	PUNCT
ejpam-3435	120	6	2s−	2s−	NUM
ejpam-3435	120	7	2ms)−	2ms)−	NUM
ejpam-3435	120	8	3	3	NUM
ejpam-3435	120	9	4	4	NUM
ejpam-3435	120	10	(	(	PUNCT
ejpam-3435	120	11	k	k	NOUN
ejpam-3435	120	12	−	−	PROPN
ejpam-3435	120	13	s	s	PART
ejpam-3435	120	14	)	)	PUNCT
ejpam-3435	120	15	‖	‖	PROPN
ejpam-3435	120	16	p	p	NOUN
ejpam-3435	120	17	‖2	‖2	NOUN
ejpam-3435	120	18	(	(	PUNCT
ejpam-3435	120	19	7	7	NUM
ejpam-3435	120	20	)	)	PUNCT
ejpam-3435	120	21	since	since	SCONJ
ejpam-3435	120	22	sectional	sectional	ADJ
ejpam-3435	120	23	curvature	curvature	NOUN
ejpam-3435	120	24	for	for	ADP
ejpam-3435	120	25	sub	sub	NOUN
ejpam-3435	120	26	manifold	manifold	ADJ
ejpam-3435	120	27	m	m	VERB
ejpam-3435	120	28	can	can	AUX
ejpam-3435	120	29	be	be	AUX
ejpam-3435	120	30	represented	represent	VERB
ejpam-3435	120	31	as∑	as∑	PROPN
ejpam-3435	120	32	2≤i	2≤i	NOUN
ejpam-3435	120	33	,	,	PUNCT
ejpam-3435	120	34	j≤n	j≤n	NOUN
ejpam-3435	120	35	kij	kij	NOUN
ejpam-3435	120	36	=	=	SYM
ejpam-3435	120	37	1	1	NUM
ejpam-3435	120	38	4	4	NUM
ejpam-3435	120	39	(	(	PUNCT
ejpam-3435	120	40	k	k	PROPN
ejpam-3435	120	41	+	+	PROPN
ejpam-3435	120	42	3s)(m−	3s)(m−	NUM
ejpam-3435	120	43	2)(m−	2)(m−	NUM
ejpam-3435	121	1	1)−	1)−	NUM
ejpam-3435	121	2	k	k	PROPN
ejpam-3435	122	1	+	+	X
ejpam-3435	122	2	3s−	3s−	NUM
ejpam-3435	122	3	4	4	NUM
ejpam-3435	122	4	4	4	NUM
ejpam-3435	122	5	(	(	PUNCT
ejpam-3435	122	6	2(m−	2(m−	PROPN
ejpam-3435	122	7	2)s−	2)s−	NUM
ejpam-3435	122	8	(	(	PUNCT
ejpam-3435	122	9	s−	s−	PROPN
ejpam-3435	122	10	1	1	NUM
ejpam-3435	122	11	)	)	PUNCT
ejpam-3435	122	12	)	)	PUNCT
ejpam-3435	123	1	+	+	CCONJ
ejpam-3435	123	2	1	1	NUM
ejpam-3435	123	3	4	4	NUM
ejpam-3435	123	4	(	(	PUNCT
ejpam-3435	123	5	k	k	NOUN
ejpam-3435	123	6	−	−	PROPN
ejpam-3435	123	7	s	s	PART
ejpam-3435	123	8	)	)	PUNCT
ejpam-3435	123	9	(	(	PUNCT
ejpam-3435	123	10	3	3	NUM
ejpam-3435	123	11	‖	‖	PROPN
ejpam-3435	123	12	p	p	PROPN
ejpam-3435	123	13	‖2	‖2	NOUN
ejpam-3435	123	14	−3	−3	PROPN
ejpam-3435	123	15	‖	‖	PROPN
ejpam-3435	123	16	pe1	pe1	PROPN
ejpam-3435	123	17	‖2	‖2	NOUN
ejpam-3435	123	18	)	)	PUNCT
ejpam-3435	123	19	.	.	PUNCT
ejpam-3435	124	1	(	(	PUNCT
ejpam-3435	124	2	8)	8)	NUM
ejpam-3435	124	3	by	by	ADP
ejpam-3435	124	4	gauss	gauss	ADJ
ejpam-3435	124	5	equation	equation	NOUN
ejpam-3435	124	6	,	,	PUNCT
ejpam-3435	124	7	on	on	ADP
ejpam-3435	124	8	s	s	NOUN
ejpam-3435	124	9	-	-	PUNCT
ejpam-3435	124	10	space	space	NOUN
ejpam-3435	124	11	form	form	NOUN
ejpam-3435	124	12	we	we	PRON
ejpam-3435	124	13	can	can	AUX
ejpam-3435	124	14	write∑	write∑	VERB
ejpam-3435	124	15	2≤i	2≤i	NOUN
ejpam-3435	124	16	,	,	PUNCT
ejpam-3435	124	17	j≤n	j≤n	NOUN
ejpam-3435	124	18	k̃ij	k̃ij	PROPN
ejpam-3435	124	19	=	=	NOUN
ejpam-3435	124	20	1	1	NUM
ejpam-3435	124	21	4	4	NUM
ejpam-3435	124	22	(	(	PUNCT
ejpam-3435	124	23	k	k	PROPN
ejpam-3435	124	24	+	+	PROPN
ejpam-3435	124	25	3s)(m−	3s)(m−	NUM
ejpam-3435	124	26	2)(m−	2)(m−	NUM
ejpam-3435	125	1	1)−	1)−	NUM
ejpam-3435	125	2	k	k	PROPN
ejpam-3435	126	1	+	+	X
ejpam-3435	126	2	3s−	3s−	NUM
ejpam-3435	126	3	4	4	NUM
ejpam-3435	126	4	4	4	NUM
ejpam-3435	126	5	(	(	PUNCT
ejpam-3435	126	6	2(m−	2(m−	PROPN
ejpam-3435	126	7	2)s−	2)s−	NUM
ejpam-3435	126	8	(	(	PUNCT
ejpam-3435	126	9	s−	s−	PROPN
ejpam-3435	126	10	1	1	NUM
ejpam-3435	126	11	)	)	PUNCT
ejpam-3435	126	12	)	)	PUNCT
ejpam-3435	127	1	+	+	CCONJ
ejpam-3435	127	2	1	1	NUM
ejpam-3435	127	3	4	4	NUM
ejpam-3435	127	4	(	(	PUNCT
ejpam-3435	127	5	k	k	NOUN
ejpam-3435	127	6	−	−	PROPN
ejpam-3435	127	7	s	s	PART
ejpam-3435	127	8	)	)	PUNCT
ejpam-3435	127	9	(	(	PUNCT
ejpam-3435	127	10	3	3	NUM
ejpam-3435	127	11	‖	‖	PROPN
ejpam-3435	127	12	p	p	PROPN
ejpam-3435	127	13	‖2	‖2	NOUN
ejpam-3435	127	14	−3	−3	PROPN
ejpam-3435	127	15	‖	‖	PROPN
ejpam-3435	127	16	pe1	pe1	PROPN
ejpam-3435	127	17	‖2	‖2	NOUN
ejpam-3435	127	18	)	)	PUNCT
ejpam-3435	128	1	+	+	CCONJ
ejpam-3435	128	2	2m+s∑	2m+s∑	NUM
ejpam-3435	128	3	r	r	NOUN
ejpam-3435	128	4	=	=	NOUN
ejpam-3435	128	5	m+1	m+1	NUM
ejpam-3435	128	6	∑	∑	PUNCT
ejpam-3435	128	7	2≤i	2≤i	NOUN
ejpam-3435	128	8	<	<	X
ejpam-3435	128	9	j≤n	j≤n	NOUN
ejpam-3435	128	10	(	(	PUNCT
ejpam-3435	128	11	hriih	hriih	NOUN
ejpam-3435	128	12	r	r	NOUN
ejpam-3435	128	13	jj	jj	PROPN
ejpam-3435	128	14	−	−	PROPN
ejpam-3435	128	15	(	(	PUNCT
ejpam-3435	128	16	hrij	hrij	PROPN
ejpam-3435	128	17	)	)	PUNCT
ejpam-3435	128	18	2	2	NUM
ejpam-3435	128	19	)	)	PUNCT
ejpam-3435	128	20	(	(	PUNCT
ejpam-3435	128	21	9	9	X
ejpam-3435	128	22	)	)	PUNCT
ejpam-3435	128	23	substituting	substitute	VERB
ejpam-3435	128	24	(	(	PUNCT
ejpam-3435	128	25	9	9	NUM
ejpam-3435	128	26	)	)	PUNCT
ejpam-3435	128	27	in	in	ADP
ejpam-3435	128	28	(	(	PUNCT
ejpam-3435	128	29	7	7	NUM
ejpam-3435	128	30	)	)	PUNCT
ejpam-3435	128	31	,	,	PUNCT
ejpam-3435	128	32	we	we	PRON
ejpam-3435	128	33	get	get	VERB
ejpam-3435	128	34	n.	n.	NOUN
ejpam-3435	128	35	a.	a.	PROPN
ejpam-3435	128	36	rehman	rehman	PROPN
ejpam-3435	128	37	/	/	SYM
ejpam-3435	128	38	eur	eur	PROPN
ejpam-3435	128	39	.	.	PUNCT
ejpam-3435	129	1	j.	j.	PROPN
ejpam-3435	129	2	pure	pure	PROPN
ejpam-3435	129	3	appl	appl	PROPN
ejpam-3435	129	4	.	.	PROPN
ejpam-3435	129	5	math	math	PROPN
ejpam-3435	129	6	,	,	PUNCT
ejpam-3435	129	7	12	12	NUM
ejpam-3435	129	8	(	(	PUNCT
ejpam-3435	129	9	4	4	NUM
ejpam-3435	129	10	)	)	PUNCT
ejpam-3435	129	11	(	(	PUNCT
ejpam-3435	129	12	2019	2019	NUM
ejpam-3435	129	13	)	)	PUNCT
ejpam-3435	129	14	,	,	PUNCT
ejpam-3435	129	15	1811	1811	NUM
ejpam-3435	129	16	-	-	SYM
ejpam-3435	129	17	1818	1818	NUM
ejpam-3435	129	18	1815	1815	NUM
ejpam-3435	129	19	1	1	NUM
ejpam-3435	129	20	2	2	NUM
ejpam-3435	129	21	n2	n2	NOUN
ejpam-3435	129	22	‖	‖	PROPN
ejpam-3435	129	23	h	h	NOUN
ejpam-3435	129	24	‖2	‖2	NOUN
ejpam-3435	129	25	=	=	SYM
ejpam-3435	129	26	2ric(x	2ric(x	NUM
ejpam-3435	129	27	)	)	PUNCT
ejpam-3435	130	1	+	+	CCONJ
ejpam-3435	130	2	1	1	NUM
ejpam-3435	130	3	2	2	NUM
ejpam-3435	130	4	2n+s∑	2n+s∑	NUM
ejpam-3435	130	5	r	r	NOUN
ejpam-3435	130	6	=	=	NOUN
ejpam-3435	130	7	m+1	m+1	X
ejpam-3435	130	8	(	(	PUNCT
ejpam-3435	130	9	hr11	hr11	PROPN
ejpam-3435	130	10	−	−	PROPN
ejpam-3435	131	1	hr22	hr22	PROPN
ejpam-3435	131	2	−	−	PROPN
ejpam-3435	131	3	·	·	PUNCT
ejpam-3435	131	4	·	·	PUNCT
ejpam-3435	131	5	·	·	PUNCT
ejpam-3435	132	1	−	−	PUNCT
ejpam-3435	133	1	hrmm)2	hrmm)2	NOUN
ejpam-3435	134	1	+	+	CCONJ
ejpam-3435	134	2	2n+s∑	2n+s∑	NUM
ejpam-3435	134	3	r	r	NOUN
ejpam-3435	134	4	=	=	SYM
ejpam-3435	134	5	m+1	m+1	NOUN
ejpam-3435	134	6	m∑	m∑	ADP
ejpam-3435	134	7	j=1	j=1	NOUN
ejpam-3435	134	8	(	(	PUNCT
ejpam-3435	134	9	hr1j	hr1j	PROPN
ejpam-3435	134	10	)	)	PUNCT
ejpam-3435	134	11	2	2	NUM
ejpam-3435	134	12	−	−	PROPN
ejpam-3435	134	13	2(m−	2(m−	NUM
ejpam-3435	134	14	1	1	NUM
ejpam-3435	134	15	)	)	PUNCT
ejpam-3435	134	16	k	k	NOUN
ejpam-3435	135	1	+	+	PUNCT
ejpam-3435	135	2	3s	3s	NUM
ejpam-3435	135	3	4	4	NUM
ejpam-3435	135	4	−	−	NOUN
ejpam-3435	135	5	3	3	NUM
ejpam-3435	135	6	‖	‖	PROPN
ejpam-3435	135	7	px	px	PROPN
ejpam-3435	135	8	‖2	‖2	NOUN
ejpam-3435	135	9	k	k	PROPN
ejpam-3435	136	1	−	−	PROPN
ejpam-3435	136	2	s	s	PART
ejpam-3435	136	3	4	4	NUM
ejpam-3435	136	4	+	+	CCONJ
ejpam-3435	136	5	(	(	PUNCT
ejpam-3435	136	6	3s−	3s−	NUM
ejpam-3435	136	7	1	1	NUM
ejpam-3435	136	8	)	)	PUNCT
ejpam-3435	136	9	k	k	NOUN
ejpam-3435	137	1	+	+	PUNCT
ejpam-3435	137	2	3s−	3s−	NUM
ejpam-3435	137	3	4	4	NUM
ejpam-3435	137	4	4	4	NUM
ejpam-3435	137	5	,	,	PUNCT
ejpam-3435	137	6	(	(	PUNCT
ejpam-3435	137	7	10	10	NUM
ejpam-3435	137	8	)	)	PUNCT
ejpam-3435	137	9	therefore	therefore	ADV
ejpam-3435	137	10	1	1	NUM
ejpam-3435	137	11	2	2	NUM
ejpam-3435	137	12	n2	n2	NOUN
ejpam-3435	137	13	‖	‖	PROPN
ejpam-3435	137	14	h	h	PROPN
ejpam-3435	137	15	‖2≥	‖2≥	PUNCT
ejpam-3435	137	16	2ric(x)−	2ric(x)−	PROPN
ejpam-3435	137	17	2(m−	2(m−	NUM
ejpam-3435	137	18	1	1	NUM
ejpam-3435	137	19	)	)	PUNCT
ejpam-3435	137	20	k	k	NOUN
ejpam-3435	138	1	+	+	PUNCT
ejpam-3435	138	2	3s	3s	NUM
ejpam-3435	138	3	4	4	NUM
ejpam-3435	138	4	−	−	NOUN
ejpam-3435	138	5	3	3	NUM
ejpam-3435	138	6	‖	‖	PROPN
ejpam-3435	138	7	px	px	PROPN
ejpam-3435	138	8	‖2	‖2	NOUN
ejpam-3435	138	9	k	k	PROPN
ejpam-3435	139	1	−	−	PROPN
ejpam-3435	139	2	s	s	PART
ejpam-3435	139	3	4	4	NUM
ejpam-3435	139	4	+	+	CCONJ
ejpam-3435	139	5	(	(	PUNCT
ejpam-3435	139	6	3s−	3s−	NUM
ejpam-3435	139	7	1	1	NUM
ejpam-3435	139	8	)	)	PUNCT
ejpam-3435	139	9	k	k	NOUN
ejpam-3435	140	1	+	+	PUNCT
ejpam-3435	140	2	3s−	3s−	NUM
ejpam-3435	140	3	4	4	NUM
ejpam-3435	140	4	4	4	NUM
ejpam-3435	140	5	,	,	PUNCT
ejpam-3435	140	6	(	(	PUNCT
ejpam-3435	140	7	11	11	NUM
ejpam-3435	140	8	)	)	PUNCT
ejpam-3435	140	9	that	that	PRON
ejpam-3435	140	10	gives	give	VERB
ejpam-3435	140	11	required	require	VERB
ejpam-3435	140	12	result	result	NOUN
ejpam-3435	140	13	.	.	PUNCT
ejpam-3435	141	1	corollary	corollary	ADJ
ejpam-3435	141	2	1	1	NUM
ejpam-3435	141	3	.	.	PUNCT
ejpam-3435	142	1	in	in	ADP
ejpam-3435	142	2	theorem	theorem	NOUN
ejpam-3435	142	3	1	1	NUM
ejpam-3435	142	4	,	,	PUNCT
ejpam-3435	142	5	equality	equality	NOUN
ejpam-3435	142	6	holds	hold	VERB
ejpam-3435	142	7	at	at	ADP
ejpam-3435	142	8	point	point	NOUN
ejpam-3435	142	9	p	p	NOUN
ejpam-3435	142	10	∈m	∈m	NOUN
ejpam-3435	142	11	,	,	PUNCT
ejpam-3435	142	12	if	if	SCONJ
ejpam-3435	142	13	(	(	PUNCT
ejpam-3435	142	14	i	i	NOUN
ejpam-3435	142	15	)	)	PUNCT
ejpam-3435	142	16	h(p	h(p	PROPN
ejpam-3435	142	17	)	)	PUNCT
ejpam-3435	143	1	=	=	SYM
ejpam-3435	143	2	0	0	PUNCT
ejpam-3435	144	1	and	and	CCONJ
ejpam-3435	144	2	x	x	PRON
ejpam-3435	144	3	is	be	AUX
ejpam-3435	144	4	normal	normal	ADJ
ejpam-3435	144	5	to	to	ADP
ejpam-3435	144	6	tpm	tpm	PROPN
ejpam-3435	144	7	(	(	PUNCT
ejpam-3435	144	8	ii	ii	NOUN
ejpam-3435	144	9	)	)	PUNCT
ejpam-3435	144	10	p	p	NOUN
ejpam-3435	144	11	is	be	AUX
ejpam-3435	144	12	totally	totally	ADV
ejpam-3435	144	13	geodesic	geodesic	ADJ
ejpam-3435	144	14	point	point	NOUN
ejpam-3435	144	15	proof	proof	NOUN
ejpam-3435	144	16	.	.	PUNCT
ejpam-3435	145	1	(	(	PUNCT
ejpam-3435	145	2	1	1	X
ejpam-3435	145	3	)	)	PUNCT
ejpam-3435	145	4	when	when	SCONJ
ejpam-3435	145	5	h(p	h(p	NOUN
ejpam-3435	145	6	)	)	PUNCT
ejpam-3435	145	7	=	=	SYM
ejpam-3435	145	8	0	0	NUM
ejpam-3435	145	9	,	,	PUNCT
ejpam-3435	145	10	we	we	PRON
ejpam-3435	145	11	get	get	VERB
ejpam-3435	145	12	hr11	hr11	PROPN
ejpam-3435	145	13	=	=	PUNCT
ejpam-3435	146	1	hr22	hr22	PROPN
ejpam-3435	146	2	=	=	SYM
ejpam-3435	146	3	·	·	PUNCT
ejpam-3435	146	4	·	·	PUNCT
ejpam-3435	146	5	·	·	PUNCT
ejpam-3435	147	1	=	=	SYM
ejpam-3435	147	2	hrmm	hrmm	NOUN
ejpam-3435	147	3	=	=	NOUN
ejpam-3435	147	4	0	0	PUNCT
ejpam-3435	148	1	and	and	CCONJ
ejpam-3435	148	2	when	when	SCONJ
ejpam-3435	148	3	x	x	PRON
ejpam-3435	148	4	is	be	AUX
ejpam-3435	148	5	normal	normal	ADJ
ejpam-3435	148	6	to	to	ADP
ejpam-3435	148	7	tpm	tpm	PROPN
ejpam-3435	148	8	,	,	PUNCT
ejpam-3435	148	9	we	we	PRON
ejpam-3435	148	10	have	have	VERB
ejpam-3435	149	1	hr12	hr12	PROPN
ejpam-3435	149	2	=	=	SYM
ejpam-3435	149	3	hr13	hr13	PROPN
ejpam-3435	149	4	=	=	PUNCT
ejpam-3435	149	5	·	·	PUNCT
ejpam-3435	149	6	·	·	PUNCT
ejpam-3435	149	7	·	·	PUNCT
ejpam-3435	150	1	=	=	PUNCT
ejpam-3435	150	2	hr1	hr1	PROPN
ejpam-3435	150	3	m	m	PROPN
ejpam-3435	150	4	=	=	SYM
ejpam-3435	150	5	0	0	NUM
ejpam-3435	150	6	,	,	PUNCT
ejpam-3435	150	7	r	r	NOUN
ejpam-3435	150	8	∈	∈	PROPN
ejpam-3435	150	9	{	{	PUNCT
ejpam-3435	150	10	m	m	PROPN
ejpam-3435	150	11	+	+	ADJ
ejpam-3435	150	12	1	1	NUM
ejpam-3435	150	13	,	,	PUNCT
ejpam-3435	150	14	.	.	PUNCT
ejpam-3435	150	15	.	.	PUNCT
ejpam-3435	150	16	.	.	PUNCT
ejpam-3435	151	1	,	,	PUNCT
ejpam-3435	151	2	2n	2n	NUM
ejpam-3435	151	3	+	+	CCONJ
ejpam-3435	151	4	s	s	X
ejpam-3435	151	5	}	}	PUNCT
ejpam-3435	151	6	and	and	CCONJ
ejpam-3435	151	7	from	from	ADP
ejpam-3435	151	8	(	(	PUNCT
ejpam-3435	151	9	10	10	NUM
ejpam-3435	151	10	)	)	PUNCT
ejpam-3435	151	11	equality	equality	NOUN
ejpam-3435	151	12	holds	hold	VERB
ejpam-3435	151	13	.	.	PUNCT
ejpam-3435	152	1	(	(	PUNCT
ejpam-3435	152	2	2)when	2)when	NUM
ejpam-3435	152	3	p	p	NOUN
ejpam-3435	152	4	is	be	AUX
ejpam-3435	152	5	totally	totally	ADV
ejpam-3435	152	6	geodesic	geodesic	ADJ
ejpam-3435	152	7	point	point	NOUN
ejpam-3435	152	8	,	,	PUNCT
ejpam-3435	152	9	hrij	hrij	PROPN
ejpam-3435	152	10	=	=	PUNCT
ejpam-3435	152	11	0	0	NUM
ejpam-3435	152	12	for	for	ADP
ejpam-3435	152	13	all	all	DET
ejpam-3435	152	14	i	i	PROPN
ejpam-3435	152	15	,	,	PUNCT
ejpam-3435	152	16	j	j	PROPN
ejpam-3435	152	17	,	,	PUNCT
ejpam-3435	152	18	r	r	NOUN
ejpam-3435	152	19	∈	∈	PROPN
ejpam-3435	152	20	{	{	PUNCT
ejpam-3435	152	21	m+	m+	NOUN
ejpam-3435	152	22	1	1	NUM
ejpam-3435	152	23	,	,	PUNCT
ejpam-3435	152	24	.	.	PUNCT
ejpam-3435	152	25	.	.	PUNCT
ejpam-3435	152	26	.	.	PUNCT
ejpam-3435	153	1	,	,	PUNCT
ejpam-3435	153	2	2n	2n	NUM
ejpam-3435	153	3	}	}	PUNCT
ejpam-3435	153	4	and	and	CCONJ
ejpam-3435	153	5	from	from	ADP
ejpam-3435	153	6	(	(	PUNCT
ejpam-3435	153	7	10	10	NUM
ejpam-3435	153	8	)	)	PUNCT
ejpam-3435	153	9	equality	equality	NOUN
ejpam-3435	153	10	holds	hold	VERB
ejpam-3435	153	11	.	.	PUNCT
ejpam-3435	154	1	corollary	corollary	ADJ
ejpam-3435	154	2	2	2	NUM
ejpam-3435	154	3	.	.	PUNCT
ejpam-3435	155	1	let	let	VERB
ejpam-3435	155	2	m	m	PRON
ejpam-3435	155	3	be	be	AUX
ejpam-3435	155	4	an	an	DET
ejpam-3435	155	5	m	m	ADJ
ejpam-3435	155	6	-	-	ADJ
ejpam-3435	155	7	dimensional	dimensional	ADJ
ejpam-3435	155	8	invariant	invariant	ADJ
ejpam-3435	155	9	sub	sub	NOUN
ejpam-3435	155	10	manifold	manifold	ADJ
ejpam-3435	155	11	tangent	tangent	NOUN
ejpam-3435	155	12	to	to	PART
ejpam-3435	155	13	structure	structure	VERB
ejpam-3435	155	14	vector	vector	NOUN
ejpam-3435	155	15	fields	field	NOUN
ejpam-3435	155	16	ξ1	ξ1	NOUN
ejpam-3435	155	17	,	,	PUNCT
ejpam-3435	155	18	ξ2	ξ2	NOUN
ejpam-3435	155	19	,	,	PUNCT
ejpam-3435	155	20	.	.	PUNCT
ejpam-3435	155	21	.	.	PUNCT
ejpam-3435	156	1	.	.	PUNCT
ejpam-3435	157	1	,	,	PUNCT
ejpam-3435	157	2	.	.	PUNCT
ejpam-3435	157	3	.	.	PUNCT
ejpam-3435	158	1	.	.	PUNCT
ejpam-3435	159	1	,	,	PUNCT
ejpam-3435	159	2	ξs	ξs	VERB
ejpam-3435	159	3	in	in	ADP
ejpam-3435	159	4	s	s	NOUN
ejpam-3435	159	5	-	-	PUNCT
ejpam-3435	159	6	space	space	NOUN
ejpam-3435	159	7	form	form	NOUN
ejpam-3435	159	8	n2n+s(k	n2n+s(k	NOUN
ejpam-3435	159	9	)	)	PUNCT
ejpam-3435	159	10	,	,	PUNCT
ejpam-3435	159	11	then	then	ADV
ejpam-3435	159	12	for	for	SCONJ
ejpam-3435	159	13	each	each	DET
ejpam-3435	159	14	unit	unit	NOUN
ejpam-3435	159	15	vector	vector	NOUN
ejpam-3435	159	16	x	x	PROPN
ejpam-3435	159	17	∈	∈	PROPN
ejpam-3435	159	18	tpm	tpm	PROPN
ejpam-3435	159	19	orthogonal	orthogonal	NOUN
ejpam-3435	159	20	to	to	ADP
ejpam-3435	159	21	ξ1	ξ1	NOUN
ejpam-3435	159	22	,	,	PUNCT
ejpam-3435	159	23	ξ2	ξ2	NOUN
ejpam-3435	159	24	,	,	PUNCT
ejpam-3435	159	25	.	.	PUNCT
ejpam-3435	159	26	.	.	PUNCT
ejpam-3435	159	27	.	.	PUNCT
ejpam-3435	160	1	,	,	PUNCT
ejpam-3435	160	2	.	.	PUNCT
ejpam-3435	160	3	.	.	PUNCT
ejpam-3435	161	1	.	.	PUNCT
ejpam-3435	162	1	,	,	PUNCT
ejpam-3435	162	2	ξs	ξs	PROPN
ejpam-3435	162	3	,	,	PUNCT
ejpam-3435	162	4	we	we	PRON
ejpam-3435	162	5	have	have	VERB
ejpam-3435	162	6	ric(x	ric(x	NOUN
ejpam-3435	162	7	)	)	PUNCT
ejpam-3435	162	8	≤	≤	NOUN
ejpam-3435	162	9	1	1	NUM
ejpam-3435	162	10	4	4	NUM
ejpam-3435	162	11	[	[	PUNCT
ejpam-3435	162	12	(	(	PUNCT
ejpam-3435	162	13	m−	m−	PROPN
ejpam-3435	162	14	1)(k	1)(k	NUM
ejpam-3435	162	15	+	+	CCONJ
ejpam-3435	162	16	3s	3s	NUM
ejpam-3435	162	17	)	)	PUNCT
ejpam-3435	163	1	+	+	CCONJ
ejpam-3435	163	2	3	3	NUM
ejpam-3435	163	3	2	2	NUM
ejpam-3435	163	4	(	(	PUNCT
ejpam-3435	163	5	k	k	NOUN
ejpam-3435	163	6	−	−	PROPN
ejpam-3435	163	7	s)−	s)−	PROPN
ejpam-3435	163	8	1	1	NUM
ejpam-3435	163	9	2	2	NUM
ejpam-3435	163	10	(	(	PUNCT
ejpam-3435	163	11	3s−	3s−	NUM
ejpam-3435	163	12	1)(k	1)(k	NUM
ejpam-3435	163	13	+	+	SYM
ejpam-3435	163	14	3s−	3s−	NUM
ejpam-3435	163	15	4	4	NUM
ejpam-3435	163	16	)	)	PUNCT
ejpam-3435	163	17	]	]	PUNCT
ejpam-3435	163	18	(	(	PUNCT
ejpam-3435	163	19	12	12	NUM
ejpam-3435	163	20	)	)	PUNCT
ejpam-3435	163	21	proof	proof	NOUN
ejpam-3435	163	22	.	.	PUNCT
ejpam-3435	164	1	we	we	PRON
ejpam-3435	164	2	know	know	VERB
ejpam-3435	164	3	that	that	SCONJ
ejpam-3435	164	4	every	every	DET
ejpam-3435	164	5	invariant	invariant	ADJ
ejpam-3435	164	6	sub	sub	NOUN
ejpam-3435	164	7	manifold	manifold	NOUN
ejpam-3435	164	8	of	of	ADP
ejpam-3435	164	9	s	s	NOUN
ejpam-3435	164	10	-	-	PUNCT
ejpam-3435	164	11	space	space	NOUN
ejpam-3435	164	12	form	form	NOUN
ejpam-3435	164	13	is	be	AUX
ejpam-3435	164	14	minimal	minimal	ADJ
ejpam-3435	164	15	[	[	X
ejpam-3435	164	16	14	14	NUM
ejpam-3435	164	17	]	]	PUNCT
ejpam-3435	164	18	i.e.	i.e.	X
ejpam-3435	164	19	mean	mean	NOUN
ejpam-3435	164	20	curvature	curvature	NOUN
ejpam-3435	164	21	vanishes	vanish	VERB
ejpam-3435	164	22	and	and	CCONJ
ejpam-3435	164	23	since	since	SCONJ
ejpam-3435	164	24	x	x	PRON
ejpam-3435	164	25	in	in	ADP
ejpam-3435	164	26	unit	unit	NOUN
ejpam-3435	164	27	tangent	tangent	PROPN
ejpam-3435	164	28	vector	vector	NOUN
ejpam-3435	164	29	to	to	ADP
ejpam-3435	164	30	m	m	PROPN
ejpam-3435	164	31	,	,	PUNCT
ejpam-3435	164	32	‖	‖	PROPN
ejpam-3435	164	33	fx	fx	PROPN
ejpam-3435	165	1	‖=‖	‖=‖	INTJ
ejpam-3435	165	2	px	px	INTJ
ejpam-3435	166	1	‖=‖	‖=‖	NOUN
ejpam-3435	166	2	x	x	SYM
ejpam-3435	166	3	‖=	‖=	PROPN
ejpam-3435	166	4	1	1	NUM
ejpam-3435	166	5	,	,	PUNCT
ejpam-3435	166	6	therefore	therefore	ADV
ejpam-3435	166	7	from	from	ADP
ejpam-3435	166	8	(	(	PUNCT
ejpam-3435	166	9	5	5	NUM
ejpam-3435	166	10	)	)	PUNCT
ejpam-3435	166	11	,	,	PUNCT
ejpam-3435	166	12	we	we	PRON
ejpam-3435	166	13	get	get	VERB
ejpam-3435	166	14	result	result	NOUN
ejpam-3435	166	15	.	.	PUNCT
ejpam-3435	167	1	corollary	corollary	ADJ
ejpam-3435	167	2	3	3	X
ejpam-3435	167	3	.	.	PUNCT
ejpam-3435	168	1	let	let	VERB
ejpam-3435	168	2	m	m	PRON
ejpam-3435	168	3	be	be	AUX
ejpam-3435	168	4	an	an	DET
ejpam-3435	168	5	m	m	ADV
ejpam-3435	168	6	-	-	ADJ
ejpam-3435	168	7	dimensional	dimensional	ADJ
ejpam-3435	168	8	anti	anti	ADJ
ejpam-3435	168	9	-	-	ADJ
ejpam-3435	168	10	invariant	invariant	ADJ
ejpam-3435	168	11	sub	sub	NOUN
ejpam-3435	168	12	manifold	manifold	ADJ
ejpam-3435	168	13	tangent	tangent	NOUN
ejpam-3435	168	14	to	to	PART
ejpam-3435	168	15	structure	structure	VERB
ejpam-3435	168	16	vector	vector	NOUN
ejpam-3435	168	17	fields	field	NOUN
ejpam-3435	168	18	ξ1	ξ1	NOUN
ejpam-3435	168	19	,	,	PUNCT
ejpam-3435	168	20	ξ2	ξ2	NOUN
ejpam-3435	168	21	,	,	PUNCT
ejpam-3435	168	22	.	.	PUNCT
ejpam-3435	168	23	.	.	PUNCT
ejpam-3435	169	1	.	.	PUNCT
ejpam-3435	170	1	,	,	PUNCT
ejpam-3435	170	2	.	.	PUNCT
ejpam-3435	170	3	.	.	PUNCT
ejpam-3435	171	1	.	.	PUNCT
ejpam-3435	172	1	,	,	PUNCT
ejpam-3435	172	2	ξs	ξs	VERB
ejpam-3435	172	3	in	in	ADP
ejpam-3435	172	4	s	s	NOUN
ejpam-3435	172	5	-	-	PUNCT
ejpam-3435	172	6	space	space	NOUN
ejpam-3435	172	7	form	form	NOUN
ejpam-3435	172	8	n2n+s(k	n2n+s(k	NOUN
ejpam-3435	172	9	)	)	PUNCT
ejpam-3435	172	10	,	,	PUNCT
ejpam-3435	172	11	then	then	ADV
ejpam-3435	172	12	for	for	SCONJ
ejpam-3435	172	13	each	each	DET
ejpam-3435	172	14	unit	unit	NOUN
ejpam-3435	172	15	vector	vector	NOUN
ejpam-3435	172	16	x	x	PROPN
ejpam-3435	172	17	∈	∈	PROPN
ejpam-3435	172	18	tpm	tpm	PROPN
ejpam-3435	172	19	orthogonal	orthogonal	NOUN
ejpam-3435	172	20	to	to	ADP
ejpam-3435	172	21	ξ1	ξ1	NOUN
ejpam-3435	172	22	,	,	PUNCT
ejpam-3435	172	23	ξ2	ξ2	NOUN
ejpam-3435	172	24	,	,	PUNCT
ejpam-3435	172	25	.	.	PUNCT
ejpam-3435	172	26	.	.	PUNCT
ejpam-3435	172	27	.	.	PUNCT
ejpam-3435	173	1	,	,	PUNCT
ejpam-3435	173	2	.	.	PUNCT
ejpam-3435	173	3	.	.	PUNCT
ejpam-3435	174	1	.	.	PUNCT
ejpam-3435	175	1	,	,	PUNCT
ejpam-3435	175	2	ξs	ξs	PROPN
ejpam-3435	175	3	,	,	PUNCT
ejpam-3435	175	4	we	we	PRON
ejpam-3435	175	5	have	have	VERB
ejpam-3435	175	6	ric(x	ric(x	NOUN
ejpam-3435	175	7	)	)	PUNCT
ejpam-3435	175	8	≤	≤	NOUN
ejpam-3435	175	9	1	1	NUM
ejpam-3435	175	10	4	4	NUM
ejpam-3435	175	11	[	[	PUNCT
ejpam-3435	175	12	m2	m2	PROPN
ejpam-3435	175	13	‖	‖	PROPN
ejpam-3435	175	14	h	h	NOUN
ejpam-3435	175	15	‖2	‖2	NOUN
ejpam-3435	176	1	+	+	PROPN
ejpam-3435	176	2	(	(	PUNCT
ejpam-3435	176	3	m−	m−	PROPN
ejpam-3435	176	4	1)(k	1)(k	NUM
ejpam-3435	176	5	+	+	CCONJ
ejpam-3435	176	6	3s)−	3s)−	NUM
ejpam-3435	176	7	1	1	NUM
ejpam-3435	176	8	2	2	NUM
ejpam-3435	176	9	(	(	PUNCT
ejpam-3435	176	10	3s−	3s−	NUM
ejpam-3435	176	11	1)(k	1)(k	NUM
ejpam-3435	176	12	+	+	SYM
ejpam-3435	176	13	3s−	3s−	NUM
ejpam-3435	176	14	4	4	NUM
ejpam-3435	176	15	)	)	PUNCT
ejpam-3435	176	16	]	]	PUNCT
ejpam-3435	176	17	.	.	PUNCT
ejpam-3435	177	1	(	(	PUNCT
ejpam-3435	177	2	13	13	NUM
ejpam-3435	177	3	)	)	PUNCT
ejpam-3435	177	4	proof	proof	NOUN
ejpam-3435	177	5	.	.	PUNCT
ejpam-3435	178	1	here	here	ADV
ejpam-3435	178	2	for	for	ADP
ejpam-3435	178	3	anti	anti	ADJ
ejpam-3435	178	4	invariant	invariant	ADJ
ejpam-3435	178	5	sub	sub	NOUN
ejpam-3435	178	6	manifolds	manifold	NOUN
ejpam-3435	178	7	,	,	PUNCT
ejpam-3435	178	8	‖	‖	PROPN
ejpam-3435	178	9	px	px	PROPN
ejpam-3435	178	10	‖=	‖=	PROPN
ejpam-3435	178	11	0	0	PUNCT
ejpam-3435	179	1	and	and	CCONJ
ejpam-3435	179	2	we	we	PRON
ejpam-3435	179	3	have	have	AUX
ejpam-3435	179	4	required	require	VERB
ejpam-3435	179	5	result	result	NOUN
ejpam-3435	179	6	.	.	PUNCT
ejpam-3435	180	1	n.	n.	NOUN
ejpam-3435	180	2	a.	a.	PROPN
ejpam-3435	180	3	rehman	rehman	PROPN
ejpam-3435	180	4	/	/	SYM
ejpam-3435	180	5	eur	eur	PROPN
ejpam-3435	180	6	.	.	PUNCT
ejpam-3435	181	1	j.	j.	PROPN
ejpam-3435	181	2	pure	pure	PROPN
ejpam-3435	181	3	appl	appl	PROPN
ejpam-3435	181	4	.	.	PROPN
ejpam-3435	181	5	math	math	PROPN
ejpam-3435	181	6	,	,	PUNCT
ejpam-3435	181	7	12	12	NUM
ejpam-3435	181	8	(	(	PUNCT
ejpam-3435	181	9	4	4	NUM
ejpam-3435	181	10	)	)	PUNCT
ejpam-3435	181	11	(	(	PUNCT
ejpam-3435	181	12	2019	2019	NUM
ejpam-3435	181	13	)	)	PUNCT
ejpam-3435	181	14	,	,	PUNCT
ejpam-3435	181	15	1811	1811	NUM
ejpam-3435	181	16	-	-	SYM
ejpam-3435	181	17	1818	1818	NUM
ejpam-3435	181	18	1816	1816	NUM
ejpam-3435	181	19	corollary	corollary	NOUN
ejpam-3435	181	20	4	4	NUM
ejpam-3435	181	21	.	.	PUNCT
ejpam-3435	182	1	let	let	VERB
ejpam-3435	182	2	m	m	PRON
ejpam-3435	182	3	be	be	AUX
ejpam-3435	182	4	an	an	DET
ejpam-3435	182	5	m	m	ADV
ejpam-3435	182	6	-	-	ADJ
ejpam-3435	182	7	dimensional	dimensional	ADJ
ejpam-3435	182	8	cr	cr	NOUN
ejpam-3435	182	9	-	-	PUNCT
ejpam-3435	182	10	submanifold	submanifold	NOUN
ejpam-3435	182	11	of	of	ADP
ejpam-3435	182	12	s	s	NOUN
ejpam-3435	182	13	-	-	PUNCT
ejpam-3435	182	14	space	space	NOUN
ejpam-3435	182	15	form	form	NOUN
ejpam-3435	182	16	n2n+s(k	n2n+s(k	NOUN
ejpam-3435	182	17	)	)	PUNCT
ejpam-3435	182	18	then	then	ADV
ejpam-3435	182	19	(	(	PUNCT
ejpam-3435	182	20	i	i	NOUN
ejpam-3435	182	21	)	)	PUNCT
ejpam-3435	182	22	for	for	ADP
ejpam-3435	182	23	each	each	DET
ejpam-3435	182	24	unit	unit	NOUN
ejpam-3435	182	25	vector	vector	NOUN
ejpam-3435	182	26	x	x	SYM
ejpam-3435	182	27	∈	∈	PROPN
ejpam-3435	182	28	dp	dp	NOUN
ejpam-3435	182	29	,	,	PUNCT
ejpam-3435	182	30	ric(x	ric(x	NOUN
ejpam-3435	182	31	)	)	PUNCT
ejpam-3435	182	32	≤	≤	NOUN
ejpam-3435	182	33	1	1	NUM
ejpam-3435	182	34	4	4	NUM
ejpam-3435	182	35	[	[	PUNCT
ejpam-3435	182	36	m2	m2	PROPN
ejpam-3435	182	37	‖	‖	PROPN
ejpam-3435	182	38	h	h	NOUN
ejpam-3435	182	39	‖2	‖2	NOUN
ejpam-3435	183	1	+	+	PROPN
ejpam-3435	183	2	(	(	PUNCT
ejpam-3435	183	3	m−	m−	PROPN
ejpam-3435	183	4	1)(k	1)(k	NUM
ejpam-3435	183	5	+	+	CCONJ
ejpam-3435	183	6	3s	3s	NUM
ejpam-3435	183	7	)	)	PUNCT
ejpam-3435	184	1	+	+	CCONJ
ejpam-3435	184	2	3	3	NUM
ejpam-3435	184	3	2	2	NUM
ejpam-3435	184	4	(	(	PUNCT
ejpam-3435	184	5	k	k	NOUN
ejpam-3435	184	6	−	−	PROPN
ejpam-3435	184	7	s)−	s)−	PROPN
ejpam-3435	184	8	1	1	NUM
ejpam-3435	184	9	2	2	NUM
ejpam-3435	184	10	(	(	PUNCT
ejpam-3435	184	11	3s−	3s−	NUM
ejpam-3435	184	12	1)(k	1)(k	NUM
ejpam-3435	184	13	+	+	SYM
ejpam-3435	184	14	3s−	3s−	NUM
ejpam-3435	184	15	4	4	NUM
ejpam-3435	184	16	)	)	PUNCT
ejpam-3435	184	17	]	]	PUNCT
ejpam-3435	184	18	.	.	PUNCT
ejpam-3435	185	1	(	(	PUNCT
ejpam-3435	185	2	14	14	NUM
ejpam-3435	185	3	)	)	PUNCT
ejpam-3435	185	4	(	(	PUNCT
ejpam-3435	185	5	i	i	NOUN
ejpam-3435	185	6	)	)	PUNCT
ejpam-3435	185	7	for	for	ADP
ejpam-3435	185	8	each	each	DET
ejpam-3435	185	9	unit	unit	NOUN
ejpam-3435	185	10	vector	vector	NOUN
ejpam-3435	185	11	x	x	PROPN
ejpam-3435	185	12	∈	∈	PROPN
ejpam-3435	185	13	d⊥p	d⊥p	NOUN
ejpam-3435	185	14	,	,	PUNCT
ejpam-3435	185	15	ric(x	ric(x	NOUN
ejpam-3435	185	16	)	)	PUNCT
ejpam-3435	185	17	≤	≤	NOUN
ejpam-3435	185	18	1	1	NUM
ejpam-3435	185	19	4	4	NUM
ejpam-3435	185	20	[	[	PUNCT
ejpam-3435	185	21	m2	m2	PROPN
ejpam-3435	185	22	‖	‖	PROPN
ejpam-3435	185	23	h	h	NOUN
ejpam-3435	185	24	‖2	‖2	NOUN
ejpam-3435	186	1	+	+	PROPN
ejpam-3435	186	2	(	(	PUNCT
ejpam-3435	186	3	m−	m−	PROPN
ejpam-3435	186	4	1)(k	1)(k	NUM
ejpam-3435	186	5	+	+	CCONJ
ejpam-3435	186	6	3s)−	3s)−	NUM
ejpam-3435	186	7	1	1	NUM
ejpam-3435	186	8	2	2	NUM
ejpam-3435	186	9	(	(	PUNCT
ejpam-3435	186	10	3s−	3s−	NUM
ejpam-3435	186	11	1)(k	1)(k	NUM
ejpam-3435	186	12	+	+	SYM
ejpam-3435	186	13	3s−	3s−	NUM
ejpam-3435	186	14	4	4	NUM
ejpam-3435	186	15	)	)	PUNCT
ejpam-3435	186	16	]	]	PUNCT
ejpam-3435	186	17	.	.	PUNCT
ejpam-3435	187	1	(	(	PUNCT
ejpam-3435	187	2	15	15	X
ejpam-3435	187	3	)	)	PUNCT
ejpam-3435	187	4	let	let	VERB
ejpam-3435	187	5	mm	mm	NOUN
ejpam-3435	187	6	,	,	PUNCT
ejpam-3435	187	7	be	be	AUX
ejpam-3435	187	8	a	a	DET
ejpam-3435	187	9	submanifold	submanifold	NOUN
ejpam-3435	187	10	of	of	ADP
ejpam-3435	187	11	n2n+s(k	n2n+s(k	NOUN
ejpam-3435	187	12	)	)	PUNCT
ejpam-3435	187	13	s	s	NOUN
ejpam-3435	187	14	-	-	PUNCT
ejpam-3435	187	15	space	space	NOUN
ejpam-3435	187	16	form	form	NOUN
ejpam-3435	187	17	of	of	ADP
ejpam-3435	187	18	constant	constant	ADJ
ejpam-3435	187	19	f	f	PROPN
ejpam-3435	187	20	-sectional	-sectional	ADJ
ejpam-3435	187	21	curvature	curvature	NOUN
ejpam-3435	187	22	k	k	NOUN
ejpam-3435	187	23	,	,	PUNCT
ejpam-3435	187	24	using	use	VERB
ejpam-3435	187	25	the	the	DET
ejpam-3435	187	26	results	result	NOUN
ejpam-3435	187	27	of	of	ADP
ejpam-3435	187	28	theorem	theorem	NOUN
ejpam-3435	187	29	1	1	NUM
ejpam-3435	187	30	,	,	PUNCT
ejpam-3435	187	31	we	we	PRON
ejpam-3435	187	32	can	can	AUX
ejpam-3435	187	33	find	find	VERB
ejpam-3435	187	34	a	a	DET
ejpam-3435	187	35	relation	relation	NOUN
ejpam-3435	187	36	between	between	ADP
ejpam-3435	187	37	the	the	DET
ejpam-3435	187	38	k	k	PROPN
ejpam-3435	187	39	-	-	PUNCT
ejpam-3435	187	40	ricci	ricci	PROPN
ejpam-3435	187	41	curvature	curvature	NOUN
ejpam-3435	187	42	of	of	ADP
ejpam-3435	187	43	mm	mm	PROPN
ejpam-3435	187	44	and	and	CCONJ
ejpam-3435	187	45	the	the	DET
ejpam-3435	187	46	squared	square	VERB
ejpam-3435	187	47	mean	mean	ADJ
ejpam-3435	187	48	curvature	curvature	NOUN
ejpam-3435	187	49	‖	‖	PROPN
ejpam-3435	187	50	h	h	NOUN
ejpam-3435	187	51	‖2	‖2	NOUN
ejpam-3435	187	52	.	.	PUNCT
ejpam-3435	188	1	before	before	ADP
ejpam-3435	188	2	this	this	DET
ejpam-3435	188	3	result	result	NOUN
ejpam-3435	188	4	first	first	ADV
ejpam-3435	188	5	we	we	PRON
ejpam-3435	188	6	define	define	VERB
ejpam-3435	188	7	shape	shape	NOUN
ejpam-3435	188	8	operator	operator	NOUN
ejpam-3435	188	9	.	.	PUNCT
ejpam-3435	189	1	let	let	VERB
ejpam-3435	189	2	p	p	PRON
ejpam-3435	189	3	∈	∈	PROPN
ejpam-3435	189	4	mm	mm	PROPN
ejpam-3435	189	5	and	and	CCONJ
ejpam-3435	189	6	{	{	PUNCT
ejpam-3435	189	7	e1	e1	PROPN
ejpam-3435	189	8	,	,	PUNCT
ejpam-3435	189	9	e2	e2	PROPN
ejpam-3435	189	10	,	,	PUNCT
ejpam-3435	189	11	.	.	PUNCT
ejpam-3435	189	12	.	.	PUNCT
ejpam-3435	190	1	.	.	PUNCT
ejpam-3435	191	1	,	,	PUNCT
ejpam-3435	191	2	em	em	PRON
ejpam-3435	191	3	}	}	PUNCT
ejpam-3435	191	4	an	an	DET
ejpam-3435	191	5	orthonormal	orthonormal	ADJ
ejpam-3435	191	6	basis	basis	NOUN
ejpam-3435	191	7	of	of	ADP
ejpam-3435	191	8	tpm	tpm	PROPN
ejpam-3435	191	9	m.	m.	NOUN
ejpam-3435	191	10	we	we	PRON
ejpam-3435	191	11	choose	choose	VERB
ejpam-3435	191	12	the	the	DET
ejpam-3435	191	13	ortho	ortho	PROPN
ejpam-3435	191	14	normal	normal	ADJ
ejpam-3435	191	15	basis	basis	NOUN
ejpam-3435	191	16	{	{	PUNCT
ejpam-3435	191	17	e1	e1	PROPN
ejpam-3435	191	18	,	,	PUNCT
ejpam-3435	191	19	e2	e2	PROPN
ejpam-3435	191	20	,	,	PUNCT
ejpam-3435	191	21	.	.	PUNCT
ejpam-3435	191	22	.	.	PUNCT
ejpam-3435	192	1	.	.	PUNCT
ejpam-3435	193	1	,	,	PUNCT
ejpam-3435	193	2	em	em	PRON
ejpam-3435	193	3	,	,	PUNCT
ejpam-3435	193	4	em+1	em+1	PROPN
ejpam-3435	193	5	,	,	PUNCT
ejpam-3435	193	6	.	.	PUNCT
ejpam-3435	193	7	.	.	PUNCT
ejpam-3435	194	1	.	.	PUNCT
ejpam-3435	195	1	,	,	PUNCT
ejpam-3435	195	2	e2n+s	e2n+s	NOUN
ejpam-3435	195	3	}	}	PUNCT
ejpam-3435	195	4	on	on	ADP
ejpam-3435	195	5	tpn	tpn	PROPN
ejpam-3435	195	6	2n+s	2n+s	NUM
ejpam-3435	195	7	.	.	PUNCT
ejpam-3435	196	1	then	then	ADV
ejpam-3435	196	2	shape	shape	NOUN
ejpam-3435	196	3	operator	operator	NOUN
ejpam-3435	196	4	takes	take	VERB
ejpam-3435	196	5	the	the	DET
ejpam-3435	196	6	form	form	NOUN
ejpam-3435	196	7	(	(	PUNCT
ejpam-3435	196	8	see	see	VERB
ejpam-3435	196	9	[	[	X
ejpam-3435	196	10	12	12	NUM
ejpam-3435	196	11	]	]	PUNCT
ejpam-3435	196	12	)	)	PUNCT
ejpam-3435	196	13	aem+1	aem+1	PROPN
ejpam-3435	196	14	=	=	PUNCT
ejpam-3435	196	15			NOUN
ejpam-3435	196	16	a1	a1	NOUN
ejpam-3435	196	17	0	0	NUM
ejpam-3435	196	18	.	.	PUNCT
ejpam-3435	196	19	.	.	PUNCT
ejpam-3435	197	1	.	.	PUNCT
ejpam-3435	198	1	0	0	NUM
ejpam-3435	198	2	0	0	NUM
ejpam-3435	198	3	a2	a2	PROPN
ejpam-3435	198	4	.	.	PUNCT
ejpam-3435	198	5	.	.	PUNCT
ejpam-3435	198	6	.	.	PUNCT
ejpam-3435	199	1	0	0	PUNCT
ejpam-3435	199	2	.	.	PUNCT
ejpam-3435	199	3	.	.	PUNCT
ejpam-3435	199	4	.	.	PUNCT
ejpam-3435	199	5	.	.	PUNCT
ejpam-3435	199	6	.	.	PUNCT
ejpam-3435	199	7	.	.	PUNCT
ejpam-3435	199	8	.	.	PUNCT
ejpam-3435	199	9	.	.	PUNCT
ejpam-3435	199	10	.	.	PUNCT
ejpam-3435	199	11	.	.	PUNCT
ejpam-3435	199	12	.	.	PUNCT
ejpam-3435	200	1	.	.	PUNCT
ejpam-3435	201	1	0	0	NUM
ejpam-3435	202	1	0	0	NUM
ejpam-3435	202	2	0	0	NUM
ejpam-3435	202	3	am	be	AUX
ejpam-3435	202	4			SCONJ
ejpam-3435	202	5	aer	aer	PROPN
ejpam-3435	202	6	=	=	PRON
ejpam-3435	202	7	(	(	PUNCT
ejpam-3435	202	8	hrij	hrij	PROPN
ejpam-3435	202	9	)	)	PUNCT
ejpam-3435	202	10	,	,	PUNCT
ejpam-3435	202	11	i	i	PRON
ejpam-3435	202	12	,	,	PUNCT
ejpam-3435	202	13	j	j	PROPN
ejpam-3435	202	14	=	=	SYM
ejpam-3435	202	15	1	1	NUM
ejpam-3435	202	16	,	,	PUNCT
ejpam-3435	202	17	.	.	PUNCT
ejpam-3435	202	18	.	.	PUNCT
ejpam-3435	202	19	.	.	PUNCT
ejpam-3435	203	1	,	,	PUNCT
ejpam-3435	203	2	m	m	PROPN
ejpam-3435	203	3	;	;	PUNCT
ejpam-3435	203	4	r	r	NOUN
ejpam-3435	203	5	=	=	PUNCT
ejpam-3435	203	6	m+	m+	NUM
ejpam-3435	203	7	2	2	NUM
ejpam-3435	203	8	,	,	PUNCT
ejpam-3435	203	9	.	.	PUNCT
ejpam-3435	203	10	.	.	PUNCT
ejpam-3435	203	11	.	.	PUNCT
ejpam-3435	204	1	,	,	PUNCT
ejpam-3435	204	2	2n+	2n+	NUM
ejpam-3435	204	3	s	s	NOUN
ejpam-3435	204	4	,	,	PUNCT
ejpam-3435	204	5	traceaer	traceaer	NOUN
ejpam-3435	204	6	=	=	SYM
ejpam-3435	204	7	0	0	X
ejpam-3435	204	8	.	.	PUNCT
ejpam-3435	205	1	we	we	PRON
ejpam-3435	205	2	can	can	AUX
ejpam-3435	205	3	take	take	VERB
ejpam-3435	205	4	m2	m2	PROPN
ejpam-3435	205	5	‖	‖	PROPN
ejpam-3435	205	6	h	h	PROPN
ejpam-3435	205	7	‖2=	‖2=	PROPN
ejpam-3435	205	8	(	(	PUNCT
ejpam-3435	205	9	m∑	m∑	INTJ
ejpam-3435	205	10	i=1	i=1	PROPN
ejpam-3435	205	11	ai	ai	VERB
ejpam-3435	205	12	)	)	PUNCT
ejpam-3435	205	13	2	2	NUM
ejpam-3435	205	14	=	=	SYM
ejpam-3435	206	1	m∑	m∑	INTJ
ejpam-3435	206	2	i=1	i=1	X
ejpam-3435	207	1	a2i	a2i	NOUN
ejpam-3435	208	1	+	+	CCONJ
ejpam-3435	208	2	2	2	NUM
ejpam-3435	208	3	∑	∑	PUNCT
ejpam-3435	208	4	i	i	PROPN
ejpam-3435	208	5	<	<	X
ejpam-3435	208	6	j	j	PROPN
ejpam-3435	208	7	aiaj	aiaj	PROPN
ejpam-3435	208	8	≤	≤	NUM
ejpam-3435	208	9	m	m	VERB
ejpam-3435	208	10	m∑	m∑	NOUN
ejpam-3435	208	11	i=1	i=1	X
ejpam-3435	209	1	a2i	a2i	NOUN
ejpam-3435	209	2	.	.	PUNCT
ejpam-3435	210	1	since	since	SCONJ
ejpam-3435	210	2	we	we	PRON
ejpam-3435	210	3	know	know	VERB
ejpam-3435	210	4	that	that	SCONJ
ejpam-3435	210	5	0	0	NUM
ejpam-3435	210	6	≤	≤	NOUN
ejpam-3435	210	7	∑	∑	PUNCT
ejpam-3435	211	1	i	i	PRON
ejpam-3435	211	2	<	<	X
ejpam-3435	211	3	j(ai	j(ai	PROPN
ejpam-3435	211	4	−	−	NOUN
ejpam-3435	211	5	aj)2	aj)2	PROPN
ejpam-3435	211	6	=	=	PUNCT
ejpam-3435	211	7	(	(	PUNCT
ejpam-3435	211	8	n−	n−	NOUN
ejpam-3435	211	9	1	1	NUM
ejpam-3435	211	10	)	)	PUNCT
ejpam-3435	211	11	∑m	∑m	PROPN
ejpam-3435	211	12	i=1	i=1	PROPN
ejpam-3435	212	1	a	a	DET
ejpam-3435	212	2	2	2	NUM
ejpam-3435	212	3	i	i	PRON
ejpam-3435	212	4	−	−	PROPN
ejpam-3435	212	5	2	2	NUM
ejpam-3435	212	6	∑	∑	PROPN
ejpam-3435	212	7	i	i	PROPN
ejpam-3435	212	8	<	<	X
ejpam-3435	212	9	j	j	PROPN
ejpam-3435	212	10	aiaj	aiaj	PROPN
ejpam-3435	212	11	,	,	PUNCT
ejpam-3435	212	12	therefore	therefore	ADV
ejpam-3435	212	13	m∑	m∑	INTJ
ejpam-3435	212	14	i=1	i=1	PROPN
ejpam-3435	212	15	a2i	a2i	VERB
ejpam-3435	212	16	≥	≥	NUM
ejpam-3435	212	17	m	m	PROPN
ejpam-3435	212	18	‖	‖	PROPN
ejpam-3435	212	19	h	h	NOUN
ejpam-3435	212	20	‖2	‖2	NOUN
ejpam-3435	212	21	.	.	PUNCT
ejpam-3435	213	1	(	(	PUNCT
ejpam-3435	213	2	16	16	NUM
ejpam-3435	213	3	)	)	PUNCT
ejpam-3435	213	4	proposition	proposition	NOUN
ejpam-3435	213	5	3.1	3.1	NUM
ejpam-3435	213	6	.	.	PUNCT
ejpam-3435	214	1	let	let	VERB
ejpam-3435	214	2	mm	mm	NOUN
ejpam-3435	214	3	,	,	PUNCT
ejpam-3435	214	4	be	be	AUX
ejpam-3435	214	5	a	a	DET
ejpam-3435	214	6	submanifold	submanifold	NOUN
ejpam-3435	214	7	of	of	ADP
ejpam-3435	214	8	s	s	NOUN
ejpam-3435	214	9	-	-	PUNCT
ejpam-3435	214	10	space	space	NOUN
ejpam-3435	214	11	form	form	NOUN
ejpam-3435	214	12	n2n+s(k	n2n+s(k	NOUN
ejpam-3435	214	13	)	)	PUNCT
ejpam-3435	214	14	of	of	ADP
ejpam-3435	214	15	constant	constant	ADJ
ejpam-3435	214	16	f	f	PROPN
ejpam-3435	214	17	sectional	sectional	ADJ
ejpam-3435	214	18	curvature	curvature	NOUN
ejpam-3435	214	19	k	k	NOUN
ejpam-3435	214	20	,	,	PUNCT
ejpam-3435	214	21	tangent	tangent	NOUN
ejpam-3435	214	22	to	to	ADP
ejpam-3435	214	23	ξ1	ξ1	NOUN
ejpam-3435	214	24	,	,	PUNCT
ejpam-3435	214	25	ξ2	ξ2	NOUN
ejpam-3435	214	26	,	,	PUNCT
ejpam-3435	214	27	.	.	PUNCT
ejpam-3435	214	28	.	.	PUNCT
ejpam-3435	215	1	.	.	PUNCT
ejpam-3435	216	1	,	,	PUNCT
ejpam-3435	216	2	ξs	ξs	VERB
ejpam-3435	216	3	then	then	ADV
ejpam-3435	216	4	we	we	PRON
ejpam-3435	216	5	have	have	VERB
ejpam-3435	216	6	m2	m2	PROPN
ejpam-3435	216	7	‖	‖	PROPN
ejpam-3435	216	8	h	h	PROPN
ejpam-3435	216	9	‖2	‖2	NOUN
ejpam-3435	216	10	≥	≥	NOUN
ejpam-3435	216	11	2τ	2τ	NUM
ejpam-3435	216	12	m(m−	m(m−	PROPN
ejpam-3435	216	13	1	1	NUM
ejpam-3435	216	14	)	)	PUNCT
ejpam-3435	216	15	−	−	NOUN
ejpam-3435	216	16	1	1	NUM
ejpam-3435	216	17	4	4	NUM
ejpam-3435	216	18	(	(	PUNCT
ejpam-3435	216	19	k	k	PROPN
ejpam-3435	217	1	+	+	CCONJ
ejpam-3435	217	2	3s)−	3s)−	NUM
ejpam-3435	217	3	k	k	NOUN
ejpam-3435	217	4	+	+	X
ejpam-3435	217	5	3s−	3s−	NUM
ejpam-3435	217	6	4	4	NUM
ejpam-3435	217	7	4	4	NUM
ejpam-3435	217	8	(	(	PUNCT
ejpam-3435	217	9	2s−	2s−	NUM
ejpam-3435	217	10	2ms	2ms	NOUN
ejpam-3435	217	11	)	)	PUNCT
ejpam-3435	217	12	m(m−	m(m−	PROPN
ejpam-3435	217	13	1	1	NUM
ejpam-3435	217	14	)	)	PUNCT
ejpam-3435	217	15	−	−	NOUN
ejpam-3435	217	16	3	3	NUM
ejpam-3435	217	17	4	4	NUM
ejpam-3435	217	18	(	(	PUNCT
ejpam-3435	217	19	k	k	NOUN
ejpam-3435	217	20	−	−	PROPN
ejpam-3435	217	21	1	1	NUM
ejpam-3435	217	22	)	)	PUNCT
ejpam-3435	217	23	‖	‖	PROPN
ejpam-3435	217	24	px	px	NOUN
ejpam-3435	217	25	‖2	‖2	NOUN
ejpam-3435	217	26	m(m−	m(m−	PROPN
ejpam-3435	217	27	1	1	X
ejpam-3435	217	28	)	)	PUNCT
ejpam-3435	217	29	references	reference	NOUN
ejpam-3435	217	30	1817	1817	NUM
ejpam-3435	217	31	proof	proof	NOUN
ejpam-3435	217	32	.	.	PUNCT
ejpam-3435	218	1	we	we	PRON
ejpam-3435	218	2	choose	choose	VERB
ejpam-3435	218	3	orthonormal	orthonormal	ADJ
ejpam-3435	218	4	basis	basis	NOUN
ejpam-3435	218	5	e1	e1	NOUN
ejpam-3435	218	6	,	,	PUNCT
ejpam-3435	218	7	e2	e2	PROPN
ejpam-3435	218	8	,	,	PUNCT
ejpam-3435	218	9	.	.	PUNCT
ejpam-3435	218	10	.	.	PUNCT
ejpam-3435	219	1	.	.	PUNCT
ejpam-3435	220	1	,	,	PUNCT
ejpam-3435	220	2	em−s	em−s	PROPN
ejpam-3435	220	3	,	,	PUNCT
ejpam-3435	220	4	em−s+1	em−s+1	NOUN
ejpam-3435	220	5	,	,	PUNCT
ejpam-3435	220	6	.	.	PUNCT
ejpam-3435	220	7	.	.	PUNCT
ejpam-3435	221	1	.	.	PUNCT
ejpam-3435	222	1	,	,	PUNCT
ejpam-3435	222	2	em−s+s	em−s+s	PROPN
ejpam-3435	222	3	,	,	PUNCT
ejpam-3435	222	4	em−s+s+1	em−s+s+1	NOUN
ejpam-3435	222	5	,	,	PUNCT
ejpam-3435	222	6	.	.	PUNCT
ejpam-3435	222	7	.	.	PUNCT
ejpam-3435	223	1	.	.	PUNCT
ejpam-3435	224	1	,	,	PUNCT
ejpam-3435	224	2	e2n+s	e2n+s	NOUN
ejpam-3435	224	3	on	on	ADP
ejpam-3435	224	4	tpn	tpn	VERB
ejpam-3435	224	5	with	with	ADP
ejpam-3435	224	6	em+1	em+1	PROPN
ejpam-3435	224	7	parallel	parallel	NOUN
ejpam-3435	224	8	to	to	PART
ejpam-3435	224	9	mean	mean	VERB
ejpam-3435	224	10	curvature	curvature	NOUN
ejpam-3435	224	11	vector	vector	NOUN
ejpam-3435	224	12	h(p	h(p	NOUN
ejpam-3435	224	13	)	)	PUNCT
ejpam-3435	224	14	.	.	PUNCT
ejpam-3435	225	1	from	from	ADP
ejpam-3435	225	2	(	(	PUNCT
ejpam-3435	225	3	6	6	NUM
ejpam-3435	225	4	)	)	PUNCT
ejpam-3435	225	5	m2	m2	PROPN
ejpam-3435	225	6	‖	‖	PROPN
ejpam-3435	225	7	h	h	PROPN
ejpam-3435	225	8	‖2	‖2	NOUN
ejpam-3435	225	9	=	=	SYM
ejpam-3435	225	10	2τ+	2τ+	NUM
ejpam-3435	225	11	‖	‖	PROPN
ejpam-3435	225	12	h	h	PROPN
ejpam-3435	225	13	‖2	‖2	NOUN
ejpam-3435	225	14	−1	−1	NOUN
ejpam-3435	225	15	4	4	NUM
ejpam-3435	225	16	(	(	PUNCT
ejpam-3435	225	17	k	k	PROPN
ejpam-3435	225	18	+	+	NUM
ejpam-3435	225	19	3s)m(m−	3s)m(m−	PROPN
ejpam-3435	225	20	1)−	1)−	PROPN
ejpam-3435	225	21	k	k	PROPN
ejpam-3435	226	1	+	+	X
ejpam-3435	226	2	3s−	3s−	NUM
ejpam-3435	226	3	4	4	NUM
ejpam-3435	226	4	4	4	NUM
ejpam-3435	226	5	(	(	PUNCT
ejpam-3435	226	6	2s−	2s−	NUM
ejpam-3435	226	7	2ms)−	2ms)−	NUM
ejpam-3435	226	8	3	3	NUM
ejpam-3435	226	9	4	4	NUM
ejpam-3435	226	10	(	(	PUNCT
ejpam-3435	226	11	k	k	NOUN
ejpam-3435	226	12	−	−	PROPN
ejpam-3435	226	13	1	1	NUM
ejpam-3435	226	14	)	)	PUNCT
ejpam-3435	226	15	‖	‖	PROPN
ejpam-3435	226	16	p	p	NOUN
ejpam-3435	226	17	‖2	‖2	NOUN
ejpam-3435	226	18	=	=	NOUN
ejpam-3435	226	19	2τ	2τ	NOUN
ejpam-3435	226	20	+	+	CCONJ
ejpam-3435	226	21	m∑	m∑	INTJ
ejpam-3435	226	22	i=1	i=1	X
ejpam-3435	226	23	a2i	a2i	NOUN
ejpam-3435	227	1	+	+	CCONJ
ejpam-3435	228	1	2n∑	2n∑	NUM
ejpam-3435	228	2	r	r	NOUN
ejpam-3435	228	3	=	=	NOUN
ejpam-3435	228	4	m+2	m+2	NOUN
ejpam-3435	228	5	m∑	m∑	NOUN
ejpam-3435	229	1	i	i	PRON
ejpam-3435	229	2	,	,	PUNCT
ejpam-3435	229	3	j=1	j=1	PROPN
ejpam-3435	229	4	(	(	PUNCT
ejpam-3435	229	5	hrij	hrij	PROPN
ejpam-3435	229	6	)	)	PUNCT
ejpam-3435	229	7	2	2	NUM
ejpam-3435	229	8	−	−	NOUN
ejpam-3435	229	9	1	1	NUM
ejpam-3435	229	10	4	4	NUM
ejpam-3435	229	11	(	(	PUNCT
ejpam-3435	229	12	k	k	PROPN
ejpam-3435	229	13	+	+	NUM
ejpam-3435	229	14	3s)m(m−	3s)m(m−	NUM
ejpam-3435	229	15	1	1	NUM
ejpam-3435	229	16	)	)	PUNCT
ejpam-3435	229	17	−	−	PROPN
ejpam-3435	230	1	k	k	NOUN
ejpam-3435	231	1	+	+	CCONJ
ejpam-3435	231	2	3s−	3s−	NUM
ejpam-3435	231	3	4	4	NUM
ejpam-3435	231	4	4	4	NUM
ejpam-3435	231	5	(	(	PUNCT
ejpam-3435	231	6	2s−	2s−	NUM
ejpam-3435	231	7	2ms)−	2ms)−	NUM
ejpam-3435	231	8	3	3	NUM
ejpam-3435	231	9	4	4	NUM
ejpam-3435	231	10	(	(	PUNCT
ejpam-3435	231	11	k	k	NOUN
ejpam-3435	231	12	−	−	PROPN
ejpam-3435	231	13	1	1	NUM
ejpam-3435	231	14	)	)	PUNCT
ejpam-3435	231	15	‖	‖	PROPN
ejpam-3435	231	16	px	px	NOUN
ejpam-3435	231	17	‖2	‖2	NOUN
ejpam-3435	231	18	by	by	ADP
ejpam-3435	231	19	(	(	PUNCT
ejpam-3435	231	20	16	16	NUM
ejpam-3435	231	21	)	)	PUNCT
ejpam-3435	231	22	m2	m2	PROPN
ejpam-3435	231	23	‖	‖	PROPN
ejpam-3435	231	24	h	h	PROPN
ejpam-3435	231	25	‖2	‖2	NOUN
ejpam-3435	231	26	≥	≥	NOUN
ejpam-3435	231	27	2τ	2τ	NUM
ejpam-3435	232	1	+	+	ADV
ejpam-3435	232	2	m	m	PROPN
ejpam-3435	232	3	‖	‖	ADJ
ejpam-3435	232	4	h	h	NOUN
ejpam-3435	232	5	‖2	‖2	NOUN
ejpam-3435	232	6	−1	−1	NOUN
ejpam-3435	232	7	4	4	NUM
ejpam-3435	232	8	(	(	PUNCT
ejpam-3435	232	9	k	k	PROPN
ejpam-3435	232	10	+	+	NUM
ejpam-3435	232	11	3s)m(m−	3s)m(m−	NUM
ejpam-3435	232	12	1	1	NUM
ejpam-3435	232	13	)	)	PUNCT
ejpam-3435	232	14	−	−	PROPN
ejpam-3435	233	1	k	k	NOUN
ejpam-3435	234	1	+	+	CCONJ
ejpam-3435	234	2	3s−	3s−	NUM
ejpam-3435	234	3	4	4	NUM
ejpam-3435	234	4	4	4	NUM
ejpam-3435	234	5	(	(	PUNCT
ejpam-3435	234	6	2s−	2s−	NUM
ejpam-3435	234	7	2ms)−	2ms)−	NUM
ejpam-3435	234	8	3	3	NUM
ejpam-3435	234	9	4	4	NUM
ejpam-3435	234	10	(	(	PUNCT
ejpam-3435	234	11	k	k	NOUN
ejpam-3435	234	12	−	−	PROPN
ejpam-3435	234	13	1	1	NUM
ejpam-3435	234	14	)	)	PUNCT
ejpam-3435	234	15	‖	‖	PROPN
ejpam-3435	234	16	px	px	NOUN
ejpam-3435	234	17	‖2	‖2	NOUN
ejpam-3435	234	18	which	which	PRON
ejpam-3435	234	19	proves	prove	VERB
ejpam-3435	234	20	the	the	DET
ejpam-3435	234	21	required	require	VERB
ejpam-3435	234	22	result	result	NOUN
ejpam-3435	234	23	.	.	PUNCT
ejpam-3435	235	1	remark	remark	NOUN
ejpam-3435	235	2	1	1	NUM
ejpam-3435	235	3	.	.	NOUN
ejpam-3435	235	4	•	•	NOUN
ejpam-3435	235	5	for	for	ADP
ejpam-3435	235	6	s	s	NOUN
ejpam-3435	235	7	=	=	SYM
ejpam-3435	235	8	0	0	NUM
ejpam-3435	235	9	,	,	PUNCT
ejpam-3435	235	10	we	we	PRON
ejpam-3435	235	11	have	have	VERB
ejpam-3435	235	12	the	the	DET
ejpam-3435	235	13	results	result	NOUN
ejpam-3435	235	14	of	of	ADP
ejpam-3435	235	15	theorem	theorem	ADJ
ejpam-3435	235	16	1	1	NUM
ejpam-3435	235	17	,	,	PUNCT
ejpam-3435	235	18	corollary	corollary	ADJ
ejpam-3435	235	19	3	3	NUM
ejpam-3435	235	20	,	,	PUNCT
ejpam-3435	235	21	corollary	corollary	ADJ
ejpam-3435	235	22	2	2	NUM
ejpam-3435	235	23	,	,	PUNCT
ejpam-3435	235	24	proposition	proposition	NOUN
ejpam-3435	235	25	3.1	3.1	NUM
ejpam-3435	235	26	for	for	ADP
ejpam-3435	235	27	khaler	khaler	PROPN
ejpam-3435	235	28	manifold	manifold	NOUN
ejpam-3435	235	29	.	.	PUNCT
ejpam-3435	236	1	•	•	X
ejpam-3435	236	2	for	for	ADP
ejpam-3435	236	3	s	s	NOUN
ejpam-3435	236	4	=	=	SYM
ejpam-3435	236	5	1	1	NUM
ejpam-3435	236	6	,	,	PUNCT
ejpam-3435	236	7	we	we	PRON
ejpam-3435	236	8	have	have	VERB
ejpam-3435	236	9	the	the	DET
ejpam-3435	236	10	results	result	NOUN
ejpam-3435	236	11	of	of	ADP
ejpam-3435	236	12	theorem	theorem	ADJ
ejpam-3435	236	13	1	1	NUM
ejpam-3435	236	14	,	,	PUNCT
ejpam-3435	236	15	corollary	corollary	ADJ
ejpam-3435	236	16	3	3	NUM
ejpam-3435	236	17	,	,	PUNCT
ejpam-3435	236	18	corollary	corollary	ADJ
ejpam-3435	236	19	2	2	NUM
ejpam-3435	236	20	,	,	PUNCT
ejpam-3435	236	21	proposition	proposition	NOUN
ejpam-3435	236	22	3.1	3.1	NUM
ejpam-3435	236	23	for	for	ADP
ejpam-3435	236	24	sasakian	sasakian	ADJ
ejpam-3435	236	25	manifolds	manifold	NOUN
ejpam-3435	236	26	.	.	PUNCT
ejpam-3435	237	1	acknowledgements	acknowledgement	NOUN
ejpam-3435	237	2	this	this	DET
ejpam-3435	237	3	work	work	NOUN
ejpam-3435	237	4	is	be	AUX
ejpam-3435	237	5	partially	partially	ADV
ejpam-3435	237	6	supported	support	VERB
ejpam-3435	237	7	by	by	ADP
ejpam-3435	237	8	higher	high	ADJ
ejpam-3435	237	9	education	education	PROPN
ejpam-3435	237	10	commission	commission	PROPN
ejpam-3435	237	11	pakistan	pakistan	PROPN
ejpam-3435	237	12	.	.	PUNCT
ejpam-3435	238	1	references	reference	NOUN
ejpam-3435	238	2	[	[	X
ejpam-3435	238	3	1	1	NUM
ejpam-3435	238	4	]	]	PUNCT
ejpam-3435	238	5	p.	p.	NOUN
ejpam-3435	238	6	alegre	alegre	PROPN
ejpam-3435	238	7	,	,	PUNCT
ejpam-3435	238	8	a.	a.	NOUN
ejpam-3435	238	9	carriazo	carriazo	NOUN
ejpam-3435	238	10	,	,	PUNCT
ejpam-3435	238	11	y.	y.	PROPN
ejpam-3435	238	12	h.	h.	PROPN
ejpam-3435	238	13	kim	kim	PROPN
ejpam-3435	238	14	,	,	PUNCT
ejpam-3435	238	15	d.	d.	PROPN
ejpam-3435	238	16	w.	w.	PROPN
ejpam-3435	238	17	yoon	yoon	PROPN
ejpam-3435	238	18	,	,	PUNCT
ejpam-3435	238	19	b.y	b.y	PROPN
ejpam-3435	238	20	.	.	PROPN
ejpam-3435	238	21	chens	chens	PROPN
ejpam-3435	238	22	inequality	inequality	NOUN
ejpam-3435	238	23	for	for	ADP
ejpam-3435	238	24	submanifolds	submanifold	NOUN
ejpam-3435	238	25	of	of	ADP
ejpam-3435	238	26	generalized	generalized	ADJ
ejpam-3435	238	27	space	space	NOUN
ejpam-3435	238	28	forms	form	NOUN
ejpam-3435	238	29	,	,	PUNCT
ejpam-3435	238	30	indian	indian	ADJ
ejpam-3435	238	31	j.	j.	PROPN
ejpam-3435	238	32	pure	pure	PROPN
ejpam-3435	238	33	appl	appl	PROPN
ejpam-3435	238	34	.	.	PUNCT
ejpam-3435	238	35	math	math	PROPN
ejpam-3435	238	36	.	.	PUNCT
ejpam-3435	239	1	,	,	PUNCT
ejpam-3435	239	2	38	38	NUM
ejpam-3435	239	3	(	(	PUNCT
ejpam-3435	239	4	2007	2007	NUM
ejpam-3435	239	5	)	)	PUNCT
ejpam-3435	239	6	,	,	PUNCT
ejpam-3435	239	7	no	no	INTJ
ejpam-3435	239	8	.	.	NOUN
ejpam-3435	239	9	3	3	NUM
ejpam-3435	239	10	,	,	PUNCT
ejpam-3435	239	11	185	185	NUM
ejpam-3435	239	12	-	-	SYM
ejpam-3435	239	13	201	201	NUM
ejpam-3435	239	14	.	.	PUNCT
ejpam-3435	240	1	[	[	X
ejpam-3435	240	2	2	2	X
ejpam-3435	240	3	]	]	PUNCT
ejpam-3435	240	4	h.	h.	PROPN
ejpam-3435	240	5	aytimur	aytimur	PROPN
ejpam-3435	240	6	and	and	CCONJ
ejpam-3435	240	7	c.	c.	PROPN
ejpam-3435	240	8	ozgur	ozgur	PROPN
ejpam-3435	240	9	,	,	PUNCT
ejpam-3435	240	10	inequalities	inequality	NOUN
ejpam-3435	240	11	for	for	ADP
ejpam-3435	240	12	submanifolds	submanifold	NOUN
ejpam-3435	240	13	in	in	ADP
ejpam-3435	240	14	statistical	statistical	ADJ
ejpam-3435	240	15	manifolds	manifold	NOUN
ejpam-3435	240	16	of	of	ADP
ejpam-3435	240	17	quasi	quasi	ADJ
ejpam-3435	240	18	-	-	ADJ
ejpam-3435	240	19	constant	constant	ADJ
ejpam-3435	240	20	curvature	curvature	NOUN
ejpam-3435	240	21	,	,	PUNCT
ejpam-3435	240	22	ann	ann	PROPN
ejpam-3435	240	23	.	.	PROPN
ejpam-3435	240	24	polon	polon	PROPN
ejpam-3435	240	25	.	.	PUNCT
ejpam-3435	241	1	math	math	NOUN
ejpam-3435	241	2	.	.	PUNCT
ejpam-3435	242	1	121	121	NUM
ejpam-3435	242	2	(	(	PUNCT
ejpam-3435	242	3	2018	2018	NUM
ejpam-3435	242	4	)	)	PUNCT
ejpam-3435	242	5	,	,	PUNCT
ejpam-3435	242	6	no	no	INTJ
ejpam-3435	242	7	.	.	NOUN
ejpam-3435	242	8	3	3	NUM
ejpam-3435	242	9	,	,	PUNCT
ejpam-3435	242	10	197	197	NUM
ejpam-3435	242	11	-	-	SYM
ejpam-3435	242	12	215	215	NUM
ejpam-3435	242	13	.	.	PUNCT
ejpam-3435	243	1	[	[	X
ejpam-3435	243	2	3	3	X
ejpam-3435	243	3	]	]	X
ejpam-3435	243	4	h.	h.	PROPN
ejpam-3435	243	5	aytimur	aytimur	PROPN
ejpam-3435	243	6	and	and	CCONJ
ejpam-3435	243	7	c.	c.	PROPN
ejpam-3435	243	8	ozgur	ozgur	PROPN
ejpam-3435	243	9	,	,	PUNCT
ejpam-3435	243	10	inequalities	inequality	NOUN
ejpam-3435	243	11	for	for	ADP
ejpam-3435	243	12	submanifolds	submanifold	NOUN
ejpam-3435	243	13	of	of	ADP
ejpam-3435	243	14	sasaki	sasaki	NOUN
ejpam-3435	243	15	-	-	PUNCT
ejpam-3435	243	16	like	like	ADJ
ejpam-3435	243	17	statistical	statistical	ADJ
ejpam-3435	243	18	manifolds	manifold	NOUN
ejpam-3435	243	19	,	,	PUNCT
ejpam-3435	243	20	turkish	turkish	ADJ
ejpam-3435	243	21	j.	j.	PROPN
ejpam-3435	243	22	math	math	PROPN
ejpam-3435	243	23	.	.	PUNCT
ejpam-3435	244	1	42	42	NUM
ejpam-3435	244	2	(	(	PUNCT
ejpam-3435	244	3	2018	2018	NUM
ejpam-3435	244	4	)	)	PUNCT
ejpam-3435	244	5	,	,	PUNCT
ejpam-3435	244	6	no	no	INTJ
ejpam-3435	244	7	.	.	NOUN
ejpam-3435	244	8	6	6	NUM
ejpam-3435	244	9	,	,	PUNCT
ejpam-3435	244	10	3149	3149	NUM
ejpam-3435	244	11	-	-	SYM
ejpam-3435	244	12	3163	3163	NUM
ejpam-3435	244	13	.	.	PUNCT
ejpam-3435	245	1	[	[	X
ejpam-3435	245	2	4	4	X
ejpam-3435	245	3	]	]	X
ejpam-3435	245	4	d.e	d.e	PROPN
ejpam-3435	245	5	.	.	PROPN
ejpam-3435	245	6	blair	blair	PROPN
ejpam-3435	245	7	,	,	PUNCT
ejpam-3435	245	8	geometry	geometry	NOUN
ejpam-3435	245	9	of	of	ADP
ejpam-3435	245	10	manifolds	manifold	NOUN
ejpam-3435	245	11	with	with	ADP
ejpam-3435	245	12	structural	structural	ADJ
ejpam-3435	245	13	group	group	NOUN
ejpam-3435	245	14	u(n)×o(s	u(n)×o(s	PROPN
ejpam-3435	245	15	)	)	PUNCT
ejpam-3435	245	16	,	,	PUNCT
ejpam-3435	245	17	j.	j.	PROPN
ejpam-3435	245	18	differential	differential	PROPN
ejpam-3435	245	19	geom	geom	PROPN
ejpam-3435	245	20	.	.	PUNCT
ejpam-3435	245	21	4	4	NUM
ejpam-3435	245	22	(	(	PUNCT
ejpam-3435	245	23	1970	1970	NUM
ejpam-3435	245	24	)	)	PUNCT
ejpam-3435	245	25	.	.	PUNCT
ejpam-3435	246	1	155	155	NUM
ejpam-3435	246	2	-	-	SYM
ejpam-3435	246	3	167	167	NUM
ejpam-3435	246	4	.	.	PUNCT
ejpam-3435	246	5	references	reference	NOUN
ejpam-3435	246	6	1818	1818	NUM
ejpam-3435	246	7	[	[	X
ejpam-3435	246	8	5	5	X
ejpam-3435	246	9	]	]	X
ejpam-3435	246	10	jose	jose	PROPN
ejpam-3435	246	11	l.	l.	PROPN
ejpam-3435	246	12	cabrerizo	cabrerizo	PROPN
ejpam-3435	246	13	,	,	PUNCT
ejpam-3435	246	14	luis	luis	PROPN
ejpam-3435	246	15	m.	m.	PROPN
ejpam-3435	246	16	fernandez	fernandez	PROPN
ejpam-3435	246	17	,	,	PUNCT
ejpam-3435	246	18	manuel	manuel	PROPN
ejpam-3435	246	19	fernandez	fernandez	PROPN
ejpam-3435	246	20	(	(	PUNCT
ejpam-3435	246	21	sevilla	sevilla	PROPN
ejpam-3435	246	22	)	)	PUNCT
ejpam-3435	246	23	,	,	PUNCT
ejpam-3435	246	24	on	on	ADP
ejpam-3435	246	25	normal	normal	ADJ
ejpam-3435	246	26	crsubmanifolds	crsubmanifold	NOUN
ejpam-3435	246	27	of	of	ADP
ejpam-3435	246	28	s	s	NOUN
ejpam-3435	246	29	-	-	PUNCT
ejpam-3435	246	30	manifolds	manifold	NOUN
ejpam-3435	246	31	,	,	PUNCT
ejpam-3435	246	32	colloquim	colloquim	ADJ
ejpam-3435	246	33	mathematicum	mathematicum	NOUN
ejpam-3435	246	34	,	,	PUNCT
ejpam-3435	246	35	vol	vol	NOUN
ejpam-3435	246	36	.	.	PUNCT
ejpam-3435	246	37	lxiv	lxiv	PROPN
ejpam-3435	246	38	,	,	PUNCT
ejpam-3435	246	39	(	(	PUNCT
ejpam-3435	246	40	1993	1993	NUM
ejpam-3435	246	41	)	)	PUNCT
ejpam-3435	246	42	,	,	PUNCT
ejpam-3435	246	43	fasc	fasc	PROPN
ejpam-3435	246	44	.	.	PROPN
ejpam-3435	246	45	2	2	NUM
ejpam-3435	246	46	.	.	PUNCT
ejpam-3435	247	1	[	[	X
ejpam-3435	247	2	6	6	NUM
ejpam-3435	247	3	]	]	X
ejpam-3435	247	4	b.y	b.y	PROPN
ejpam-3435	247	5	.	.	PROPN
ejpam-3435	247	6	chen	chen	PROPN
ejpam-3435	247	7	,	,	PUNCT
ejpam-3435	247	8	strings	string	NOUN
ejpam-3435	247	9	of	of	ADP
ejpam-3435	247	10	riemannian	riemannian	ADJ
ejpam-3435	247	11	invariants	invariant	NOUN
ejpam-3435	247	12	,	,	PUNCT
ejpam-3435	247	13	inequalities	inequality	NOUN
ejpam-3435	247	14	,	,	PUNCT
ejpam-3435	247	15	ideal	ideal	ADJ
ejpam-3435	247	16	immersions	immersion	NOUN
ejpam-3435	247	17	and	and	CCONJ
ejpam-3435	247	18	their	their	PRON
ejpam-3435	247	19	applications	application	NOUN
ejpam-3435	247	20	,	,	PUNCT
ejpam-3435	247	21	the	the	DET
ejpam-3435	247	22	third	third	ADJ
ejpam-3435	247	23	pacific	pacific	PROPN
ejpam-3435	247	24	rim	rim	PROPN
ejpam-3435	247	25	geometry	geometry	NOUN
ejpam-3435	247	26	conference	conference	PROPN
ejpam-3435	247	27	(	(	PUNCT
ejpam-3435	247	28	seoul	seoul	PROPN
ejpam-3435	247	29	,	,	PUNCT
ejpam-3435	247	30	1996	1996	NUM
ejpam-3435	247	31	)	)	PUNCT
ejpam-3435	247	32	,	,	PUNCT
ejpam-3435	247	33	760	760	NUM
ejpam-3435	247	34	,	,	PUNCT
ejpam-3435	247	35	monogr	monogr	PROPN
ejpam-3435	247	36	.	.	PUNCT
ejpam-3435	248	1	geom	geom	PROPN
ejpam-3435	248	2	.	.	PUNCT
ejpam-3435	249	1	topology	topology	PROPN
ejpam-3435	249	2	,	,	PUNCT
ejpam-3435	249	3	25	25	NUM
ejpam-3435	249	4	,	,	PUNCT
ejpam-3435	249	5	int	int	NOUN
ejpam-3435	249	6	.	.	PUNCT
ejpam-3435	250	1	press	press	PROPN
ejpam-3435	250	2	,	,	PUNCT
ejpam-3435	250	3	cambridge	cambridge	PROPN
ejpam-3435	250	4	,	,	PUNCT
ejpam-3435	250	5	ma	ma	PROPN
ejpam-3435	250	6	,	,	PUNCT
ejpam-3435	250	7	1998	1998	NUM
ejpam-3435	250	8	.	.	PUNCT
ejpam-3435	251	1	[	[	X
ejpam-3435	251	2	7	7	X
ejpam-3435	251	3	]	]	PUNCT
ejpam-3435	251	4	b.	b.	PROPN
ejpam-3435	251	5	y.	y.	PROPN
ejpam-3435	251	6	chen	chen	PROPN
ejpam-3435	251	7	,	,	PUNCT
ejpam-3435	251	8	some	some	DET
ejpam-3435	251	9	pinching	pinching	NOUN
ejpam-3435	251	10	and	and	CCONJ
ejpam-3435	251	11	classification	classification	NOUN
ejpam-3435	251	12	theorems	theorem	NOUN
ejpam-3435	251	13	for	for	ADP
ejpam-3435	251	14	minimal	minimal	ADJ
ejpam-3435	251	15	submanifolds	submanifold	NOUN
ejpam-3435	251	16	,	,	PUNCT
ejpam-3435	251	17	arch	arch	NOUN
ejpam-3435	251	18	.	.	PUNCT
ejpam-3435	252	1	math	math	NOUN
ejpam-3435	252	2	.	.	PUNCT
ejpam-3435	253	1	(	(	PUNCT
ejpam-3435	253	2	basel	basel	PROPN
ejpam-3435	253	3	)	)	PUNCT
ejpam-3435	253	4	,	,	PUNCT
ejpam-3435	253	5	60	60	NUM
ejpam-3435	253	6	(	(	PUNCT
ejpam-3435	253	7	1993	1993	NUM
ejpam-3435	253	8	)	)	PUNCT
ejpam-3435	253	9	,	,	PUNCT
ejpam-3435	254	1	no	no	INTJ
ejpam-3435	254	2	.	.	NOUN
ejpam-3435	254	3	6	6	NUM
ejpam-3435	254	4	,	,	PUNCT
ejpam-3435	254	5	568	568	NUM
ejpam-3435	254	6	-	-	SYM
ejpam-3435	254	7	578	578	NUM
ejpam-3435	254	8	.	.	PUNCT
ejpam-3435	255	1	[	[	X
ejpam-3435	255	2	8	8	NUM
ejpam-3435	255	3	]	]	PUNCT
ejpam-3435	255	4	a.	a.	NOUN
ejpam-3435	255	5	friedmann	friedmann	PROPN
ejpam-3435	255	6	and	and	CCONJ
ejpam-3435	255	7	j.	j.	PROPN
ejpam-3435	255	8	a.	a.	PROPN
ejpam-3435	255	9	schouten	schouten	PROPN
ejpam-3435	255	10	,	,	PUNCT
ejpam-3435	255	11	uber	uber	ADJ
ejpam-3435	255	12	die	die	NOUN
ejpam-3435	255	13	geometrie	geometrie	NOUN
ejpam-3435	255	14	der	der	NOUN
ejpam-3435	255	15	halbsymmetrischen	halbsymmetrischen	PROPN
ejpam-3435	255	16	ubertragungen	ubertragungen	PROPN
ejpam-3435	255	17	,	,	PUNCT
ejpam-3435	255	18	math	math	NOUN
ejpam-3435	255	19	.	.	PUNCT
ejpam-3435	256	1	z.	z.	PROPN
ejpam-3435	256	2	,	,	PUNCT
ejpam-3435	256	3	21	21	NUM
ejpam-3435	256	4	(	(	PUNCT
ejpam-3435	256	5	1924	1924	NUM
ejpam-3435	256	6	)	)	PUNCT
ejpam-3435	256	7	,	,	PUNCT
ejpam-3435	256	8	no	no	INTJ
ejpam-3435	256	9	.	.	NOUN
ejpam-3435	256	10	1	1	NUM
ejpam-3435	256	11	,	,	PUNCT
ejpam-3435	256	12	211	211	NUM
ejpam-3435	256	13	-	-	SYM
ejpam-3435	256	14	223	223	NUM
ejpam-3435	256	15	.	.	PUNCT
ejpam-3435	257	1	[	[	X
ejpam-3435	257	2	9	9	NUM
ejpam-3435	257	3	]	]	PUNCT
ejpam-3435	257	4	m.	m.	NOUN
ejpam-3435	257	5	glbahar	glbahar	PROPN
ejpam-3435	257	6	,	,	PUNCT
ejpam-3435	257	7	erol	erol	NOUN
ejpam-3435	257	8	kilic	kilic	NOUN
ejpam-3435	257	9	and	and	CCONJ
ejpam-3435	257	10	sadik	sadik	PROPN
ejpam-3435	257	11	kele	kele	PROPN
ejpam-3435	257	12	,	,	PUNCT
ejpam-3435	257	13	chen	chen	PROPN
ejpam-3435	257	14	-	-	PUNCT
ejpam-3435	257	15	like	like	ADJ
ejpam-3435	257	16	inequalities	inequality	NOUN
ejpam-3435	257	17	on	on	ADP
ejpam-3435	257	18	lightlike	lightlike	ADJ
ejpam-3435	257	19	hypersurfaces	hypersurface	NOUN
ejpam-3435	257	20	of	of	ADP
ejpam-3435	257	21	a	a	DET
ejpam-3435	257	22	lorentzian	lorentzian	ADJ
ejpam-3435	257	23	manifold	manifold	NOUN
ejpam-3435	257	24	,	,	PUNCT
ejpam-3435	257	25	journal	journal	NOUN
ejpam-3435	257	26	of	of	ADP
ejpam-3435	257	27	inequalities	inequality	NOUN
ejpam-3435	257	28	and	and	CCONJ
ejpam-3435	257	29	applications	application	NOUN
ejpam-3435	257	30	2013	2013	NUM
ejpam-3435	257	31	,	,	PUNCT
ejpam-3435	257	32	2013:266	2013:266	AUX
ejpam-3435	257	33	.	.	PUNCT
ejpam-3435	258	1	[	[	X
ejpam-3435	258	2	10	10	NUM
ejpam-3435	258	3	]	]	X
ejpam-3435	258	4	s.	s.	PROPN
ejpam-3435	258	5	ianus	ianus	PROPN
ejpam-3435	258	6	,	,	PUNCT
ejpam-3435	258	7	g.	g.	PROPN
ejpam-3435	258	8	e.	e.	PROPN
ejpam-3435	258	9	vlcu	vlcu	PROPN
ejpam-3435	258	10	,	,	PUNCT
ejpam-3435	258	11	and	and	CCONJ
ejpam-3435	258	12	r.	r.	PROPN
ejpam-3435	258	13	c.	c.	PROPN
ejpam-3435	258	14	voicu	voicu	PROPN
ejpam-3435	258	15	,	,	PUNCT
ejpam-3435	258	16	harmonic	harmonic	ADJ
ejpam-3435	258	17	maps	map	NOUN
ejpam-3435	258	18	and	and	CCONJ
ejpam-3435	258	19	riemannian	riemannian	ADJ
ejpam-3435	258	20	submersions	submersion	NOUN
ejpam-3435	258	21	between	between	ADP
ejpam-3435	258	22	manifolds	manifold	NOUN
ejpam-3435	258	23	endowed	endow	VERB
ejpam-3435	258	24	with	with	ADP
ejpam-3435	258	25	special	special	ADJ
ejpam-3435	258	26	structures	structure	NOUN
ejpam-3435	258	27	,	,	PUNCT
ejpam-3435	258	28	algebra	algebra	NOUN
ejpam-3435	258	29	,	,	PUNCT
ejpam-3435	258	30	geometry	geometry	NOUN
ejpam-3435	258	31	and	and	CCONJ
ejpam-3435	258	32	mathematical	mathematical	ADJ
ejpam-3435	258	33	physics	physics	NOUN
ejpam-3435	258	34	,	,	PUNCT
ejpam-3435	258	35	vol	vol	NOUN
ejpam-3435	258	36	.	.	PROPN
ejpam-3435	258	37	93	93	NUM
ejpam-3435	258	38	,	,	PUNCT
ejpam-3435	258	39	pp	pp	ADJ
ejpam-3435	258	40	.	.	PUNCT
ejpam-3435	258	41	277288	277288	NUM
ejpam-3435	258	42	,	,	PUNCT
ejpam-3435	258	43	banach	banach	NOUN
ejpam-3435	258	44	center	center	NOUN
ejpam-3435	258	45	publications	publication	NOUN
ejpam-3435	258	46	,	,	PUNCT
ejpam-3435	258	47	2011	2011	NUM
ejpam-3435	258	48	.	.	PUNCT
ejpam-3435	259	1	[	[	X
ejpam-3435	259	2	11	11	NUM
ejpam-3435	259	3	]	]	PUNCT
ejpam-3435	259	4	t.	t.	PROPN
ejpam-3435	259	5	imai	imai	PROPN
ejpam-3435	259	6	,	,	PUNCT
ejpam-3435	259	7	notes	note	NOUN
ejpam-3435	259	8	on	on	ADP
ejpam-3435	259	9	semi	semi	ADJ
ejpam-3435	259	10	-	-	ADJ
ejpam-3435	259	11	symmetric	symmetric	ADJ
ejpam-3435	259	12	metric	metric	ADJ
ejpam-3435	259	13	connections	connection	NOUN
ejpam-3435	259	14	,	,	PUNCT
ejpam-3435	259	15	vol	vol	NOUN
ejpam-3435	259	16	.	.	PUNCT
ejpam-3435	259	17	i.	i.	PROPN
ejpam-3435	259	18	tensor	tensor	PROPN
ejpam-3435	259	19	(	(	PUNCT
ejpam-3435	259	20	n.s	n.s	PROPN
ejpam-3435	259	21	.	.	PROPN
ejpam-3435	259	22	)	)	PUNCT
ejpam-3435	259	23	,	,	PUNCT
ejpam-3435	259	24	24	24	NUM
ejpam-3435	259	25	(	(	PUNCT
ejpam-3435	259	26	1972	1972	NUM
ejpam-3435	259	27	)	)	PUNCT
ejpam-3435	259	28	,	,	PUNCT
ejpam-3435	259	29	293	293	NUM
ejpam-3435	259	30	-	-	SYM
ejpam-3435	259	31	296	296	NUM
ejpam-3435	259	32	.	.	PUNCT
ejpam-3435	260	1	[	[	X
ejpam-3435	260	2	12	12	NUM
ejpam-3435	260	3	]	]	PUNCT
ejpam-3435	260	4	a.	a.	NOUN
ejpam-3435	260	5	mihai	mihai	PROPN
ejpam-3435	260	6	and	and	CCONJ
ejpam-3435	260	7	cihan	cihan	VERB
ejpam-3435	260	8	ozgur	ozgur	PROPN
ejpam-3435	260	9	,	,	PUNCT
ejpam-3435	260	10	chen	chen	PROPN
ejpam-3435	260	11	inequalities	inequality	NOUN
ejpam-3435	260	12	for	for	ADP
ejpam-3435	260	13	submanifolds	submanifold	NOUN
ejpam-3435	260	14	of	of	ADP
ejpam-3435	260	15	real	real	ADJ
ejpam-3435	260	16	space	space	NOUN
ejpam-3435	260	17	forms	form	NOUN
ejpam-3435	260	18	with	with	ADP
ejpam-3435	260	19	a	a	DET
ejpam-3435	260	20	semi	semi	ADJ
ejpam-3435	260	21	-	-	ADJ
ejpam-3435	260	22	symmetric	symmetric	ADJ
ejpam-3435	260	23	metric	metric	ADJ
ejpam-3435	260	24	connection	connection	NOUN
ejpam-3435	260	25	,	,	PUNCT
ejpam-3435	260	26	taiwanese	taiwanese	PROPN
ejpam-3435	260	27	j.	j.	PROPN
ejpam-3435	260	28	math	math	PROPN
ejpam-3435	260	29	.	.	PUNCT
ejpam-3435	260	30	,	,	PUNCT
ejpam-3435	260	31	to	to	PART
ejpam-3435	260	32	appear	appear	VERB
ejpam-3435	260	33	.	.	PUNCT
ejpam-3435	261	1	[	[	X
ejpam-3435	261	2	13	13	NUM
ejpam-3435	261	3	]	]	PUNCT
ejpam-3435	261	4	najma	najma	PROPN
ejpam-3435	261	5	abdul	abdul	PROPN
ejpam-3435	261	6	rehman	rehman	PROPN
ejpam-3435	261	7	,	,	PUNCT
ejpam-3435	261	8	mehwish	mehwish	PROPN
ejpam-3435	261	9	bari	bari	NOUN
ejpam-3435	261	10	,	,	PUNCT
ejpam-3435	261	11	biharmonic	biharmonic	NOUN
ejpam-3435	261	12	maps	map	NOUN
ejpam-3435	261	13	into	into	ADP
ejpam-3435	261	14	s	s	NOUN
ejpam-3435	261	15	-	-	PUNCT
ejpam-3435	261	16	space	space	NOUN
ejpam-3435	261	17	forms	form	NOUN
ejpam-3435	261	18	,	,	PUNCT
ejpam-3435	261	19	european	european	ADJ
ejpam-3435	261	20	journal	journal	NOUN
ejpam-3435	261	21	of	of	ADP
ejpam-3435	261	22	pure	pure	ADJ
ejpam-3435	261	23	and	and	CCONJ
ejpam-3435	261	24	applied	applied	ADJ
ejpam-3435	261	25	mathematics	mathematic	NOUN
ejpam-3435	261	26	,	,	PUNCT
ejpam-3435	261	27	vol	vol	NOUN
ejpam-3435	261	28	.	.	PROPN
ejpam-3435	261	29	11	11	NUM
ejpam-3435	261	30	,	,	PUNCT
ejpam-3435	261	31	no	no	INTJ
ejpam-3435	261	32	.	.	NOUN
ejpam-3435	261	33	1	1	NUM
ejpam-3435	261	34	,	,	PUNCT
ejpam-3435	261	35	2018	2018	NUM
ejpam-3435	261	36	,	,	PUNCT
ejpam-3435	261	37	150	150	NUM
ejpam-3435	261	38	-	-	SYM
ejpam-3435	261	39	159	159	NUM
ejpam-3435	261	40	.	.	PUNCT
ejpam-3435	262	1	[	[	X
ejpam-3435	262	2	14	14	NUM
ejpam-3435	262	3	]	]	PUNCT
ejpam-3435	262	4	k.	k.	PROPN
ejpam-3435	262	5	yano	yano	PROPN
ejpam-3435	262	6	and	and	CCONJ
ejpam-3435	262	7	m.	m.	PROPN
ejpam-3435	262	8	kon	kon	PROPN
ejpam-3435	262	9	,	,	PUNCT
ejpam-3435	262	10	structures	structure	NOUN
ejpam-3435	262	11	on	on	ADP
ejpam-3435	262	12	manifolds	manifold	NOUN
ejpam-3435	262	13	,	,	PUNCT
ejpam-3435	262	14	vol	vol	NOUN
ejpam-3435	262	15	.	.	PROPN
ejpam-3435	262	16	3	3	NUM
ejpam-3435	262	17	,	,	PUNCT
ejpam-3435	262	18	series	series	NOUN
ejpam-3435	262	19	in	in	ADP
ejpam-3435	262	20	pure	pure	ADJ
ejpam-3435	262	21	math	math	NOUN
ejpam-3435	262	22	.	.	PUNCT
ejpam-3435	262	23	,	,	PUNCT
ejpam-3435	262	24	world	world	PROPN
ejpam-3435	262	25	scientific	scientific	PROPN
ejpam-3435	262	26	,	,	PUNCT
ejpam-3435	262	27	singapore	singapore	PROPN
ejpam-3435	262	28	,	,	PUNCT
ejpam-3435	262	29	1984	1984	NUM
ejpam-3435	262	30	.	.	PUNCT
ejpam-3435	263	1	[	[	X
ejpam-3435	263	2	15	15	NUM
ejpam-3435	263	3	]	]	PUNCT
ejpam-3435	263	4	k.	k.	PROPN
ejpam-3435	263	5	yano	yano	PROPN
ejpam-3435	263	6	and	and	CCONJ
ejpam-3435	263	7	m.	m.	PROPN
ejpam-3435	263	8	kon	kon	PROPN
ejpam-3435	263	9	,	,	PUNCT
ejpam-3435	263	10	on	on	ADP
ejpam-3435	263	11	semi	semi	ADJ
ejpam-3435	263	12	-	-	ADJ
ejpam-3435	263	13	symmetric	symmetric	ADJ
ejpam-3435	263	14	metric	metric	ADJ
ejpam-3435	263	15	connection	connection	NOUN
ejpam-3435	263	16	,	,	PUNCT
ejpam-3435	263	17	rev	rev	PROPN
ejpam-3435	263	18	.	.	PROPN
ejpam-3435	264	1	roun	roun	PROPN
ejpam-3435	264	2	.	.	PUNCT
ejpam-3435	264	3	math	math	PROPN
ejpam-3435	264	4	.	.	PUNCT
ejpam-3435	265	1	pures	pure	NOUN
ejpam-3435	265	2	appl	appl	PROPN
ejpam-3435	265	3	.	.	PROPN
ejpam-3435	266	1	15	15	NUM
ejpam-3435	266	2	(	(	PUNCT
ejpam-3435	266	3	1970	1970	NUM
ejpam-3435	266	4	)	)	PUNCT
ejpam-3435	266	5	,	,	PUNCT
ejpam-3435	266	6	1579	1579	NUM
ejpam-3435	266	7	-	-	SYM
ejpam-3435	266	8	1586	1586	NUM
ejpam-3435	266	9	.	.	PUNCT