id	sid	tid	token	lemma	pos
ejpam-5492	1	1	european	european	PROPN
ejpam-5492	1	2	journal	journal	PROPN
ejpam-5492	1	3	of	of	ADP
ejpam-5492	1	4	pure	pure	ADJ
ejpam-5492	1	5	and	and	CCONJ
ejpam-5492	1	6	applied	apply	VERB
ejpam-5492	1	7	mathematics	mathematic	NOUN
ejpam-5492	1	8	vol	vol	NOUN
ejpam-5492	1	9	.	.	PROPN
ejpam-5492	2	1	17	17	NUM
ejpam-5492	2	2	,	,	PUNCT
ejpam-5492	2	3	no	no	INTJ
ejpam-5492	2	4	.	.	NOUN
ejpam-5492	2	5	4	4	NUM
ejpam-5492	2	6	,	,	PUNCT
ejpam-5492	2	7	2024	2024	NUM
ejpam-5492	2	8	,	,	PUNCT
ejpam-5492	2	9	2481	2481	NUM
ejpam-5492	2	10	-	-	SYM
ejpam-5492	2	11	2491	2491	NUM
ejpam-5492	2	12	issn	issn	VERB
ejpam-5492	2	13	1307	1307	NUM
ejpam-5492	2	14	-	-	SYM
ejpam-5492	2	15	5543	5543	NUM
ejpam-5492	2	16	–	–	PUNCT
ejpam-5492	2	17	ejpam.com	ejpam.com	X
ejpam-5492	2	18	published	publish	VERB
ejpam-5492	2	19	by	by	ADP
ejpam-5492	2	20	new	new	PROPN
ejpam-5492	2	21	york	york	PROPN
ejpam-5492	2	22	business	business	PROPN
ejpam-5492	2	23	global	global	PROPN
ejpam-5492	2	24	chen	chen	PROPN
ejpam-5492	2	25	’s	’s	PART
ejpam-5492	2	26	inequality	inequality	NOUN
ejpam-5492	2	27	for	for	ADP
ejpam-5492	2	28	cr	cr	NOUN
ejpam-5492	2	29	-	-	PUNCT
ejpam-5492	2	30	warped	warp	VERB
ejpam-5492	2	31	products	product	NOUN
ejpam-5492	2	32	in	in	ADP
ejpam-5492	2	33	locally	locally	ADV
ejpam-5492	2	34	metallic	metallic	ADJ
ejpam-5492	2	35	riemannian	riemannian	NOUN
ejpam-5492	2	36	manifolds	manifold	NOUN
ejpam-5492	2	37	lamia	lamia	PROPN
ejpam-5492	2	38	saeed	saeed	PROPN
ejpam-5492	2	39	alqahtani1,∗	alqahtani1,∗	PROPN
ejpam-5492	2	40	,	,	PUNCT
ejpam-5492	2	41	ebtehal	ebtehal	NOUN
ejpam-5492	2	42	m.	m.	PROPN
ejpam-5492	2	43	al	al	PROPN
ejpam-5492	2	44	-	-	PUNCT
ejpam-5492	2	45	husainy1,2	husainy1,2	PROPN
ejpam-5492	2	46	1	1	NUM
ejpam-5492	2	47	department	department	NOUN
ejpam-5492	2	48	of	of	ADP
ejpam-5492	2	49	mathematics	mathematic	NOUN
ejpam-5492	2	50	,	,	PUNCT
ejpam-5492	2	51	faculty	faculty	NOUN
ejpam-5492	2	52	of	of	ADP
ejpam-5492	2	53	science	science	NOUN
ejpam-5492	2	54	,	,	PUNCT
ejpam-5492	2	55	king	king	NOUN
ejpam-5492	2	56	abdulaziz	abdulaziz	PROPN
ejpam-5492	2	57	university	university	PROPN
ejpam-5492	2	58	,	,	PUNCT
ejpam-5492	2	59	21589	21589	NUM
ejpam-5492	2	60	jeddah	jeddah	PROPN
ejpam-5492	2	61	,	,	PUNCT
ejpam-5492	2	62	saudi	saudi	PROPN
ejpam-5492	2	63	arabia	arabia	PROPN
ejpam-5492	2	64	2	2	NUM
ejpam-5492	2	65	mathematics	mathematics	PROPN
ejpam-5492	2	66	department	department	NOUN
ejpam-5492	2	67	,	,	PUNCT
ejpam-5492	2	68	faculty	faculty	NOUN
ejpam-5492	2	69	of	of	ADP
ejpam-5492	2	70	science	science	NOUN
ejpam-5492	2	71	,	,	PUNCT
ejpam-5492	2	72	taibah	taibah	PROPN
ejpam-5492	2	73	university	university	PROPN
ejpam-5492	2	74	,	,	PUNCT
ejpam-5492	2	75	344	344	NUM
ejpam-5492	2	76	al	al	PROPN
ejpam-5492	2	77	-	-	PUNCT
ejpam-5492	2	78	madinah	madinah	PROPN
ejpam-5492	2	79	almunawara	almunawara	PROPN
ejpam-5492	2	80	,	,	PUNCT
ejpam-5492	2	81	saudi	saudi	PROPN
ejpam-5492	2	82	arabia	arabia	PROPN
ejpam-5492	2	83	abstract	abstract	NOUN
ejpam-5492	2	84	.	.	PUNCT
ejpam-5492	3	1	in	in	ADP
ejpam-5492	3	2	this	this	DET
ejpam-5492	3	3	paper	paper	NOUN
ejpam-5492	3	4	,	,	PUNCT
ejpam-5492	3	5	we	we	PRON
ejpam-5492	3	6	study	study	VERB
ejpam-5492	3	7	cr	cr	NOUN
ejpam-5492	3	8	-	-	PUNCT
ejpam-5492	3	9	warped	warp	VERB
ejpam-5492	3	10	product	product	NOUN
ejpam-5492	3	11	submanifolds	submanifold	NOUN
ejpam-5492	3	12	in	in	ADP
ejpam-5492	3	13	locally	locally	ADV
ejpam-5492	3	14	metallic	metallic	ADJ
ejpam-5492	3	15	riemannian	riemannian	ADJ
ejpam-5492	3	16	manifolds	manifold	NOUN
ejpam-5492	3	17	.	.	PUNCT
ejpam-5492	4	1	we	we	PRON
ejpam-5492	4	2	provide	provide	VERB
ejpam-5492	4	3	several	several	ADJ
ejpam-5492	4	4	non	non	ADJ
ejpam-5492	4	5	-	-	ADJ
ejpam-5492	4	6	trivial	trivial	ADJ
ejpam-5492	4	7	examples	example	NOUN
ejpam-5492	4	8	of	of	ADP
ejpam-5492	4	9	such	such	ADJ
ejpam-5492	4	10	submanifolds	submanifold	NOUN
ejpam-5492	4	11	.	.	PUNCT
ejpam-5492	5	1	we	we	PRON
ejpam-5492	5	2	establish	establish	VERB
ejpam-5492	5	3	a	a	DET
ejpam-5492	5	4	sharp	sharp	ADJ
ejpam-5492	5	5	inequality	inequality	NOUN
ejpam-5492	5	6	known	know	VERB
ejpam-5492	5	7	as	as	ADP
ejpam-5492	5	8	chen	chen	PROPN
ejpam-5492	5	9	’s	’s	PART
ejpam-5492	5	10	inequality	inequality	NOUN
ejpam-5492	5	11	for	for	ADP
ejpam-5492	5	12	the	the	DET
ejpam-5492	5	13	squared	square	VERB
ejpam-5492	5	14	norm	norm	NOUN
ejpam-5492	5	15	of	of	ADP
ejpam-5492	5	16	the	the	DET
ejpam-5492	5	17	second	second	ADJ
ejpam-5492	5	18	fundamental	fundamental	ADJ
ejpam-5492	5	19	form	form	NOUN
ejpam-5492	5	20	.	.	PUNCT
ejpam-5492	6	1	we	we	PRON
ejpam-5492	6	2	also	also	ADV
ejpam-5492	6	3	discuss	discuss	VERB
ejpam-5492	6	4	the	the	DET
ejpam-5492	6	5	equality	equality	NOUN
ejpam-5492	6	6	case	case	NOUN
ejpam-5492	6	7	of	of	ADP
ejpam-5492	6	8	chen	chen	PROPN
ejpam-5492	6	9	’s	’s	PART
ejpam-5492	6	10	inequality	inequality	NOUN
ejpam-5492	6	11	.	.	PUNCT
ejpam-5492	7	1	2020	2020	NUM
ejpam-5492	7	2	mathematics	mathematic	NOUN
ejpam-5492	7	3	subject	subject	NOUN
ejpam-5492	7	4	classifications	classification	NOUN
ejpam-5492	7	5	:	:	PUNCT
ejpam-5492	7	6	53b25	53b25	NUM
ejpam-5492	7	7	,	,	PUNCT
ejpam-5492	7	8	53c15	53c15	NUM
ejpam-5492	7	9	,	,	PUNCT
ejpam-5492	7	10	53c40	53c40	NUM
ejpam-5492	7	11	,	,	PUNCT
ejpam-5492	7	12	53c42	53c42	NUM
ejpam-5492	7	13	,	,	PUNCT
ejpam-5492	7	14	53d10	53d10	NUM
ejpam-5492	7	15	key	key	ADJ
ejpam-5492	7	16	words	word	NOUN
ejpam-5492	7	17	and	and	CCONJ
ejpam-5492	7	18	phrases	phrase	NOUN
ejpam-5492	7	19	:	:	PUNCT
ejpam-5492	7	20	warped	warped	ADJ
ejpam-5492	7	21	product	product	NOUN
ejpam-5492	7	22	,	,	PUNCT
ejpam-5492	7	23	cr	cr	PROPN
ejpam-5492	7	24	warped	warp	VERB
ejpam-5492	7	25	product	product	NOUN
ejpam-5492	7	26	,	,	PUNCT
ejpam-5492	7	27	metallic	metallic	ADJ
ejpam-5492	7	28	riemannian	riemannian	ADJ
ejpam-5492	7	29	structure	structure	NOUN
ejpam-5492	7	30	,	,	PUNCT
ejpam-5492	7	31	golden	golden	ADJ
ejpam-5492	7	32	structure	structure	NOUN
ejpam-5492	7	33	,	,	PUNCT
ejpam-5492	7	34	cr	cr	NOUN
ejpam-5492	7	35	submanifold	submanifold	VERB
ejpam-5492	7	36	1	1	NUM
ejpam-5492	7	37	.	.	PUNCT
ejpam-5492	8	1	introduction	introduction	NOUN
ejpam-5492	8	2	warped	warp	VERB
ejpam-5492	8	3	products	product	NOUN
ejpam-5492	8	4	are	be	AUX
ejpam-5492	8	5	considered	consider	VERB
ejpam-5492	8	6	a	a	DET
ejpam-5492	8	7	generalization	generalization	NOUN
ejpam-5492	8	8	of	of	ADP
ejpam-5492	8	9	cartesian	cartesian	ADJ
ejpam-5492	8	10	products	product	NOUN
ejpam-5492	8	11	.	.	PUNCT
ejpam-5492	9	1	the	the	DET
ejpam-5492	9	2	study	study	NOUN
ejpam-5492	9	3	of	of	ADP
ejpam-5492	9	4	warped	warped	ADJ
ejpam-5492	9	5	product	product	NOUN
ejpam-5492	9	6	manifolds	manifold	NOUN
ejpam-5492	9	7	was	be	AUX
ejpam-5492	9	8	developed	develop	VERB
ejpam-5492	9	9	by	by	ADP
ejpam-5492	9	10	bishop	bishop	PROPN
ejpam-5492	9	11	and	and	CCONJ
ejpam-5492	9	12	o’neill	o’neill	INTJ
ejpam-5492	9	13	in	in	ADP
ejpam-5492	9	14	[	[	X
ejpam-5492	9	15	1	1	NUM
ejpam-5492	9	16	]	]	PUNCT
ejpam-5492	9	17	,	,	PUNCT
ejpam-5492	9	18	who	who	PRON
ejpam-5492	9	19	obtained	obtain	VERB
ejpam-5492	9	20	fundamental	fundamental	ADJ
ejpam-5492	9	21	properties	property	NOUN
ejpam-5492	9	22	of	of	ADP
ejpam-5492	9	23	warped	warped	ADJ
ejpam-5492	9	24	product	product	NOUN
ejpam-5492	9	25	manifolds	manifold	NOUN
ejpam-5492	9	26	and	and	CCONJ
ejpam-5492	9	27	constructed	construct	VERB
ejpam-5492	9	28	a	a	DET
ejpam-5492	9	29	class	class	NOUN
ejpam-5492	9	30	of	of	ADP
ejpam-5492	9	31	complete	complete	ADJ
ejpam-5492	9	32	manifolds	manifold	NOUN
ejpam-5492	9	33	with	with	ADP
ejpam-5492	9	34	negative	negative	ADJ
ejpam-5492	9	35	curvature	curvature	NOUN
ejpam-5492	9	36	.	.	PUNCT
ejpam-5492	10	1	subsequently	subsequently	ADV
ejpam-5492	10	2	,	,	PUNCT
ejpam-5492	10	3	b.	b.	PROPN
ejpam-5492	10	4	y.	y.	PROPN
ejpam-5492	10	5	chen	chen	PROPN
ejpam-5492	10	6	studied	study	VERB
ejpam-5492	10	7	cr	cr	PROPN
ejpam-5492	10	8	-	-	PUNCT
ejpam-5492	10	9	submanifolds	submanifold	NOUN
ejpam-5492	10	10	of	of	ADP
ejpam-5492	10	11	kähler	kähler	PROPN
ejpam-5492	10	12	manifolds	manifold	NOUN
ejpam-5492	10	13	,	,	PUNCT
ejpam-5492	10	14	that	that	PRON
ejpam-5492	10	15	are	be	AUX
ejpam-5492	10	16	warped	warp	VERB
ejpam-5492	10	17	products	product	NOUN
ejpam-5492	10	18	of	of	ADP
ejpam-5492	10	19	complex	complex	ADJ
ejpam-5492	10	20	and	and	CCONJ
ejpam-5492	10	21	totally	totally	ADV
ejpam-5492	10	22	real	real	ADJ
ejpam-5492	10	23	submanifolds	submanifold	NOUN
ejpam-5492	10	24	,	,	PUNCT
ejpam-5492	10	25	and	and	CCONJ
ejpam-5492	10	26	published	publish	VERB
ejpam-5492	10	27	his	his	PRON
ejpam-5492	10	28	findings	finding	NOUN
ejpam-5492	10	29	in	in	ADP
ejpam-5492	10	30	a	a	DET
ejpam-5492	10	31	series	series	NOUN
ejpam-5492	10	32	of	of	ADP
ejpam-5492	10	33	papers	paper	NOUN
ejpam-5492	11	1	[	[	X
ejpam-5492	11	2	6–8	6–8	X
ejpam-5492	11	3	]	]	X
ejpam-5492	11	4	.	.	PUNCT
ejpam-5492	12	1	also	also	ADV
ejpam-5492	12	2	,	,	PUNCT
ejpam-5492	12	3	he	he	PRON
ejpam-5492	12	4	presented	present	VERB
ejpam-5492	12	5	a	a	DET
ejpam-5492	12	6	multitude	multitude	NOUN
ejpam-5492	12	7	of	of	ADP
ejpam-5492	12	8	properties	property	NOUN
ejpam-5492	12	9	for	for	ADP
ejpam-5492	12	10	warped	warped	ADJ
ejpam-5492	12	11	product	product	NOUN
ejpam-5492	12	12	manifolds	manifold	NOUN
ejpam-5492	12	13	and	and	CCONJ
ejpam-5492	12	14	submanifolds	submanifold	NOUN
ejpam-5492	12	15	in	in	ADP
ejpam-5492	12	16	[	[	X
ejpam-5492	12	17	9	9	NUM
ejpam-5492	12	18	]	]	PUNCT
ejpam-5492	12	19	and	and	CCONJ
ejpam-5492	12	20	discussed	discuss	VERB
ejpam-5492	12	21	the	the	DET
ejpam-5492	12	22	applications	application	NOUN
ejpam-5492	12	23	of	of	ADP
ejpam-5492	12	24	these	these	DET
ejpam-5492	12	25	properties	property	NOUN
ejpam-5492	12	26	to	to	PART
ejpam-5492	12	27	differential	differential	VERB
ejpam-5492	12	28	geometry	geometry	NOUN
ejpam-5492	12	29	and	and	CCONJ
ejpam-5492	12	30	geometric	geometric	ADJ
ejpam-5492	12	31	analysis	analysis	NOUN
ejpam-5492	12	32	.	.	PUNCT
ejpam-5492	13	1	hretcanu	hretcanu	NOUN
ejpam-5492	13	2	and	and	CCONJ
ejpam-5492	13	3	crasmareanu	crasmareanu	NOUN
ejpam-5492	13	4	introduced	introduce	VERB
ejpam-5492	13	5	the	the	DET
ejpam-5492	13	6	notion	notion	NOUN
ejpam-5492	13	7	of	of	ADP
ejpam-5492	13	8	a	a	DET
ejpam-5492	13	9	golden	golden	ADJ
ejpam-5492	13	10	structure	structure	NOUN
ejpam-5492	13	11	on	on	ADP
ejpam-5492	13	12	riemannian	riemannian	ADJ
ejpam-5492	13	13	manifolds	manifold	NOUN
ejpam-5492	13	14	in	in	ADP
ejpam-5492	13	15	[	[	X
ejpam-5492	13	16	15	15	NUM
ejpam-5492	13	17	]	]	PUNCT
ejpam-5492	13	18	.	.	PUNCT
ejpam-5492	14	1	they	they	PRON
ejpam-5492	14	2	showed	show	VERB
ejpam-5492	14	3	that	that	SCONJ
ejpam-5492	14	4	a	a	DET
ejpam-5492	14	5	golden	golden	ADJ
ejpam-5492	14	6	structure	structure	NOUN
ejpam-5492	14	7	is	be	AUX
ejpam-5492	14	8	a	a	DET
ejpam-5492	14	9	generalization	generalization	NOUN
ejpam-5492	14	10	of	of	ADP
ejpam-5492	14	11	an	an	DET
ejpam-5492	14	12	almost	almost	ADV
ejpam-5492	14	13	product	product	NOUN
ejpam-5492	14	14	structure	structure	NOUN
ejpam-5492	14	15	.	.	PUNCT
ejpam-5492	15	1	the	the	DET
ejpam-5492	15	2	properties	property	NOUN
ejpam-5492	15	3	of	of	ADP
ejpam-5492	15	4	submanifolds	submanifold	NOUN
ejpam-5492	15	5	in	in	ADP
ejpam-5492	15	6	golden	golden	ADJ
ejpam-5492	15	7	riemannian	riemannian	ADJ
ejpam-5492	15	8	manifolds	manifold	NOUN
ejpam-5492	15	9	were	be	AUX
ejpam-5492	15	10	then	then	ADV
ejpam-5492	15	11	studied	study	VERB
ejpam-5492	15	12	in	in	ADP
ejpam-5492	15	13	[	[	X
ejpam-5492	15	14	10	10	NUM
ejpam-5492	15	15	,	,	PUNCT
ejpam-5492	15	16	16	16	NUM
ejpam-5492	15	17	]	]	PUNCT
ejpam-5492	15	18	using	use	VERB
ejpam-5492	15	19	the	the	DET
ejpam-5492	15	20	correspondences	correspondence	NOUN
ejpam-5492	15	21	between	between	ADP
ejpam-5492	15	22	a	a	DET
ejpam-5492	15	23	golden	golden	ADJ
ejpam-5492	15	24	structure	structure	NOUN
ejpam-5492	15	25	and	and	CCONJ
ejpam-5492	15	26	an	an	DET
ejpam-5492	15	27	almost	almost	ADV
ejpam-5492	15	28	product	product	NOUN
ejpam-5492	15	29	structure	structure	NOUN
ejpam-5492	15	30	.	.	PUNCT
ejpam-5492	16	1	the	the	DET
ejpam-5492	16	2	metallic	metallic	ADJ
ejpam-5492	16	3	structure	structure	NOUN
ejpam-5492	16	4	,	,	PUNCT
ejpam-5492	16	5	defined	define	VERB
ejpam-5492	16	6	in	in	ADP
ejpam-5492	16	7	[	[	X
ejpam-5492	16	8	17	17	NUM
ejpam-5492	16	9	]	]	PUNCT
ejpam-5492	16	10	,	,	PUNCT
ejpam-5492	16	11	is	be	AUX
ejpam-5492	16	12	a	a	DET
ejpam-5492	16	13	further	further	ADJ
ejpam-5492	16	14	generalization	generalization	NOUN
ejpam-5492	16	15	of	of	ADP
ejpam-5492	16	16	the	the	DET
ejpam-5492	16	17	golden	golden	ADJ
ejpam-5492	16	18	structure	structure	NOUN
ejpam-5492	16	19	.	.	PUNCT
ejpam-5492	17	1	different	different	ADJ
ejpam-5492	17	2	types	type	NOUN
ejpam-5492	17	3	of	of	ADP
ejpam-5492	17	4	submanifolds	submanifold	NOUN
ejpam-5492	17	5	in	in	ADP
ejpam-5492	17	6	metallic	metallic	ADJ
ejpam-5492	17	7	riemannian	riemannian	ADJ
ejpam-5492	17	8	manifolds	manifold	NOUN
ejpam-5492	17	9	∗corresponding	∗corresponde	VERB
ejpam-5492	17	10	author	author	NOUN
ejpam-5492	17	11	.	.	PUNCT
ejpam-5492	18	1	doi	doi	NOUN
ejpam-5492	18	2	:	:	PUNCT
ejpam-5492	18	3	https://doi.org/10.29020/nybg.ejpam.v17i4.5492	https://doi.org/10.29020/nybg.ejpam.v17i4.5492	PROPN
ejpam-5492	18	4	email	email	NOUN
ejpam-5492	18	5	addresses	address	NOUN
ejpam-5492	18	6	:	:	PUNCT
ejpam-5492	18	7	lalqahtani@kau.edu.sa	lalqahtani@kau.edu.sa	PROPN
ejpam-5492	18	8	(	(	PUNCT
ejpam-5492	18	9	l.	l.	PROPN
ejpam-5492	18	10	alqahtani	alqahtani	PROPN
ejpam-5492	18	11	)	)	PUNCT
ejpam-5492	18	12	,	,	PUNCT
ejpam-5492	18	13	ealhusainy@stu.kau.edu.sa	ealhusainy@stu.kau.edu.sa	PROPN
ejpam-5492	18	14	(	(	PUNCT
ejpam-5492	18	15	e.	e.	PROPN
ejpam-5492	18	16	al	al	PROPN
ejpam-5492	18	17	-	-	PUNCT
ejpam-5492	18	18	husainy	husainy	PROPN
ejpam-5492	18	19	)	)	PUNCT
ejpam-5492	18	20	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-5492	19	1	2481	2481	NUM
ejpam-5492	19	2	copyright	copyright	NOUN
ejpam-5492	19	3	:	:	PUNCT
ejpam-5492	19	4	©	©	PROPN
ejpam-5492	19	5	2024	2024	NUM
ejpam-5492	19	6	the	the	DET
ejpam-5492	19	7	author(s	author(s	NOUN
ejpam-5492	19	8	)	)	PUNCT
ejpam-5492	19	9	.	.	PUNCT
ejpam-5492	20	1	(	(	PUNCT
ejpam-5492	20	2	cc	cc	NOUN
ejpam-5492	20	3	by	by	ADP
ejpam-5492	20	4	-	-	PUNCT
ejpam-5492	20	5	nc	nc	PROPN
ejpam-5492	20	6	4.0	4.0	NUM
ejpam-5492	20	7	)	)	PUNCT
ejpam-5492	20	8	l.	l.	PROPN
ejpam-5492	20	9	alqahtani	alqahtani	PROPN
ejpam-5492	20	10	,	,	PUNCT
ejpam-5492	20	11	e.	e.	PROPN
ejpam-5492	20	12	al	al	PROPN
ejpam-5492	20	13	-	-	PROPN
ejpam-5492	20	14	husainy	husainy	PROPN
ejpam-5492	20	15	/	/	SYM
ejpam-5492	20	16	eur	eur	PROPN
ejpam-5492	20	17	.	.	PUNCT
ejpam-5492	21	1	j.	j.	PROPN
ejpam-5492	21	2	pure	pure	PROPN
ejpam-5492	21	3	appl	appl	PROPN
ejpam-5492	21	4	.	.	PROPN
ejpam-5492	21	5	math	math	PROPN
ejpam-5492	21	6	,	,	PUNCT
ejpam-5492	21	7	17	17	NUM
ejpam-5492	21	8	(	(	PUNCT
ejpam-5492	21	9	4	4	NUM
ejpam-5492	21	10	)	)	PUNCT
ejpam-5492	21	11	(	(	PUNCT
ejpam-5492	21	12	2024	2024	NUM
ejpam-5492	21	13	)	)	PUNCT
ejpam-5492	21	14	,	,	PUNCT
ejpam-5492	21	15	2481	2481	NUM
ejpam-5492	21	16	-	-	SYM
ejpam-5492	21	17	2491	2491	NUM
ejpam-5492	21	18	2482	2482	NUM
ejpam-5492	21	19	were	be	AUX
ejpam-5492	21	20	studied	study	VERB
ejpam-5492	21	21	in	in	ADP
ejpam-5492	21	22	[	[	X
ejpam-5492	21	23	3	3	NUM
ejpam-5492	21	24	,	,	PUNCT
ejpam-5492	21	25	13	13	NUM
ejpam-5492	21	26	]	]	PUNCT
ejpam-5492	21	27	,	,	PUNCT
ejpam-5492	21	28	which	which	PRON
ejpam-5492	21	29	involved	involve	VERB
ejpam-5492	21	30	obtaining	obtain	VERB
ejpam-5492	21	31	different	different	ADJ
ejpam-5492	21	32	integrability	integrability	NOUN
ejpam-5492	21	33	conditions	condition	NOUN
ejpam-5492	21	34	for	for	ADP
ejpam-5492	21	35	the	the	DET
ejpam-5492	21	36	distributions	distribution	NOUN
ejpam-5492	21	37	related	relate	VERB
ejpam-5492	21	38	to	to	ADP
ejpam-5492	21	39	these	these	DET
ejpam-5492	21	40	submanifolds	submanifold	NOUN
ejpam-5492	21	41	.	.	PUNCT
ejpam-5492	22	1	the	the	DET
ejpam-5492	22	2	metallic	metallic	ADJ
ejpam-5492	22	3	warped	warped	ADJ
ejpam-5492	22	4	product	product	NOUN
ejpam-5492	22	5	riemannian	riemannian	NOUN
ejpam-5492	22	6	manifold	manifold	NOUN
ejpam-5492	22	7	was	be	AUX
ejpam-5492	22	8	studied	study	VERB
ejpam-5492	22	9	in	in	ADP
ejpam-5492	22	10	[	[	X
ejpam-5492	22	11	2	2	NUM
ejpam-5492	22	12	,	,	PUNCT
ejpam-5492	22	13	14	14	NUM
ejpam-5492	22	14	]	]	PUNCT
ejpam-5492	22	15	.	.	PUNCT
ejpam-5492	23	1	after	after	ADP
ejpam-5492	23	2	that	that	PRON
ejpam-5492	23	3	,	,	PUNCT
ejpam-5492	23	4	hretcanu	hretcanu	PROPN
ejpam-5492	23	5	and	and	CCONJ
ejpam-5492	23	6	blaga	blaga	PROPN
ejpam-5492	23	7	worked	work	VERB
ejpam-5492	23	8	on	on	ADP
ejpam-5492	23	9	the	the	DET
ejpam-5492	23	10	existence	existence	NOUN
ejpam-5492	23	11	problem	problem	NOUN
ejpam-5492	23	12	of	of	ADP
ejpam-5492	23	13	proper	proper	ADJ
ejpam-5492	23	14	warped	warped	ADJ
ejpam-5492	23	15	product	product	NOUN
ejpam-5492	23	16	bi	bi	ADJ
ejpam-5492	23	17	-	-	ADJ
ejpam-5492	23	18	slant	slant	ADJ
ejpam-5492	23	19	submanifolds	submanifold	NOUN
ejpam-5492	23	20	in	in	ADP
ejpam-5492	23	21	locally	locally	ADV
ejpam-5492	23	22	metallic	metallic	ADJ
ejpam-5492	23	23	riemannian	riemannian	NOUN
ejpam-5492	23	24	manifolds	manifold	NOUN
ejpam-5492	23	25	[	[	X
ejpam-5492	23	26	14	14	NUM
ejpam-5492	23	27	]	]	PUNCT
ejpam-5492	23	28	.	.	PUNCT
ejpam-5492	24	1	they	they	PRON
ejpam-5492	24	2	provided	provide	VERB
ejpam-5492	24	3	a	a	DET
ejpam-5492	24	4	brief	brief	ADJ
ejpam-5492	24	5	overview	overview	NOUN
ejpam-5492	24	6	of	of	ADP
ejpam-5492	24	7	metallic	metallic	ADJ
ejpam-5492	24	8	riemannian	riemannian	ADJ
ejpam-5492	24	9	manifolds	manifold	NOUN
ejpam-5492	24	10	and	and	CCONJ
ejpam-5492	24	11	their	their	PRON
ejpam-5492	24	12	submanifolds	submanifold	NOUN
ejpam-5492	24	13	,	,	PUNCT
ejpam-5492	24	14	and	and	CCONJ
ejpam-5492	24	15	then	then	ADV
ejpam-5492	24	16	discussed	discuss	VERB
ejpam-5492	24	17	slant	slant	NOUN
ejpam-5492	24	18	and	and	CCONJ
ejpam-5492	24	19	bi	bi	ADJ
ejpam-5492	24	20	-	-	ADJ
ejpam-5492	24	21	slant	slant	ADJ
ejpam-5492	24	22	submanifolds	submanifold	NOUN
ejpam-5492	24	23	(	(	PUNCT
ejpam-5492	24	24	including	include	VERB
ejpam-5492	24	25	semi	semi	ADJ
ejpam-5492	24	26	-	-	ADJ
ejpam-5492	24	27	slant	slant	ADJ
ejpam-5492	24	28	and	and	CCONJ
ejpam-5492	24	29	hemi	hemi	NOUN
ejpam-5492	24	30	-	-	PUNCT
ejpam-5492	24	31	slant	slant	ADJ
ejpam-5492	24	32	submanifolds	submanifold	NOUN
ejpam-5492	24	33	)	)	PUNCT
ejpam-5492	24	34	in	in	ADP
ejpam-5492	24	35	locally	locally	ADV
ejpam-5492	24	36	metallic	metallic	ADJ
ejpam-5492	24	37	riemannian	riemannian	ADJ
ejpam-5492	24	38	manifolds	manifold	NOUN
ejpam-5492	24	39	.	.	PUNCT
ejpam-5492	25	1	they	they	PRON
ejpam-5492	25	2	also	also	ADV
ejpam-5492	25	3	studied	study	VERB
ejpam-5492	25	4	the	the	DET
ejpam-5492	25	5	properties	property	NOUN
ejpam-5492	25	6	of	of	ADP
ejpam-5492	25	7	warped	warped	ADJ
ejpam-5492	25	8	product	product	NOUN
ejpam-5492	25	9	bi	bi	ADJ
ejpam-5492	25	10	-	-	ADJ
ejpam-5492	25	11	slant	slant	ADJ
ejpam-5492	25	12	submanifolds	submanifold	NOUN
ejpam-5492	25	13	in	in	ADP
ejpam-5492	25	14	metallic	metallic	ADJ
ejpam-5492	25	15	riemannian	riemannian	ADJ
ejpam-5492	25	16	manifolds	manifold	NOUN
ejpam-5492	25	17	and	and	CCONJ
ejpam-5492	25	18	investigated	investigate	VERB
ejpam-5492	25	19	the	the	DET
ejpam-5492	25	20	existence	existence	NOUN
ejpam-5492	25	21	of	of	ADP
ejpam-5492	25	22	various	various	ADJ
ejpam-5492	25	23	types	type	NOUN
ejpam-5492	25	24	of	of	ADP
ejpam-5492	25	25	warped	warped	ADJ
ejpam-5492	25	26	products	product	NOUN
ejpam-5492	25	27	,	,	PUNCT
ejpam-5492	25	28	including	include	VERB
ejpam-5492	25	29	warped	warped	ADJ
ejpam-5492	25	30	product	product	NOUN
ejpam-5492	25	31	cr	cr	PROPN
ejpam-5492	25	32	submanifolds	submanifold	NOUN
ejpam-5492	25	33	in	in	ADP
ejpam-5492	25	34	locally	locally	ADV
ejpam-5492	25	35	metallic	metallic	ADJ
ejpam-5492	25	36	riemannian	riemannian	NOUN
ejpam-5492	25	37	manifolds	manifold	NOUN
ejpam-5492	25	38	[	[	X
ejpam-5492	25	39	14	14	NUM
ejpam-5492	25	40	]	]	PUNCT
ejpam-5492	25	41	where	where	SCONJ
ejpam-5492	25	42	they	they	PRON
ejpam-5492	25	43	proved	prove	VERB
ejpam-5492	25	44	that	that	SCONJ
ejpam-5492	25	45	there	there	PRON
ejpam-5492	25	46	is	be	VERB
ejpam-5492	25	47	no	no	DET
ejpam-5492	25	48	proper	proper	ADJ
ejpam-5492	25	49	cr	cr	NOUN
ejpam-5492	25	50	warped	warped	ADJ
ejpam-5492	25	51	product	product	NOUN
ejpam-5492	25	52	of	of	ADP
ejpam-5492	25	53	the	the	DET
ejpam-5492	25	54	form	form	NOUN
ejpam-5492	25	55	mt	mt	PROPN
ejpam-5492	25	56	×f	×f	PROPN
ejpam-5492	25	57	m⊥	m⊥	PROPN
ejpam-5492	25	58	,	,	PUNCT
ejpam-5492	25	59	where	where	SCONJ
ejpam-5492	25	60	mt	mt	PROPN
ejpam-5492	25	61	and	and	CCONJ
ejpam-5492	25	62	m⊥	m⊥	NOUN
ejpam-5492	25	63	are	be	AUX
ejpam-5492	25	64	invariant	invariant	ADJ
ejpam-5492	25	65	and	and	CCONJ
ejpam-5492	25	66	anti	anti	ADJ
ejpam-5492	25	67	-	-	ADJ
ejpam-5492	25	68	invariant	invariant	ADJ
ejpam-5492	25	69	submanifolds	submanifold	NOUN
ejpam-5492	25	70	,	,	PUNCT
ejpam-5492	25	71	respectively	respectively	ADV
ejpam-5492	25	72	,	,	PUNCT
ejpam-5492	25	73	in	in	ADP
ejpam-5492	25	74	a	a	DET
ejpam-5492	25	75	locally	locally	ADV
ejpam-5492	25	76	metallic	metallic	ADJ
ejpam-5492	25	77	riemannian	riemannian	ADJ
ejpam-5492	25	78	manifold	manifold	NOUN
ejpam-5492	25	79	.	.	PUNCT
ejpam-5492	26	1	a	a	DET
ejpam-5492	26	2	study	study	NOUN
ejpam-5492	26	3	related	relate	VERB
ejpam-5492	26	4	to	to	ADP
ejpam-5492	26	5	this	this	DET
ejpam-5492	26	6	field	field	NOUN
ejpam-5492	26	7	using	use	VERB
ejpam-5492	26	8	other	other	ADJ
ejpam-5492	26	9	mathematical	mathematical	ADJ
ejpam-5492	26	10	methods	method	NOUN
ejpam-5492	26	11	that	that	PRON
ejpam-5492	26	12	may	may	AUX
ejpam-5492	26	13	be	be	AUX
ejpam-5492	26	14	relevant	relevant	ADJ
ejpam-5492	26	15	from	from	ADP
ejpam-5492	26	16	another	another	DET
ejpam-5492	26	17	point	point	NOUN
ejpam-5492	26	18	of	of	ADP
ejpam-5492	26	19	view	view	NOUN
ejpam-5492	26	20	in	in	ADP
ejpam-5492	26	21	[	[	X
ejpam-5492	26	22	18	18	NUM
ejpam-5492	26	23	]	]	PUNCT
ejpam-5492	26	24	.	.	PUNCT
ejpam-5492	27	1	in	in	ADP
ejpam-5492	27	2	this	this	DET
ejpam-5492	27	3	paper	paper	NOUN
ejpam-5492	27	4	,	,	PUNCT
ejpam-5492	27	5	we	we	PRON
ejpam-5492	27	6	continue	continue	VERB
ejpam-5492	27	7	the	the	DET
ejpam-5492	27	8	research	research	NOUN
ejpam-5492	27	9	on	on	ADP
ejpam-5492	27	10	warped	warped	ADJ
ejpam-5492	27	11	product	product	NOUN
ejpam-5492	27	12	cr	cr	NOUN
ejpam-5492	27	13	-	-	PUNCT
ejpam-5492	27	14	submanifolds	submanifold	NOUN
ejpam-5492	27	15	of	of	ADP
ejpam-5492	27	16	the	the	DET
ejpam-5492	27	17	form	form	NOUN
ejpam-5492	27	18	m⊥	m⊥	NOUN
ejpam-5492	27	19	×f	×f	PROPN
ejpam-5492	27	20	mt	mt	PROPN
ejpam-5492	27	21	in	in	ADP
ejpam-5492	27	22	locally	locally	ADV
ejpam-5492	27	23	metallic	metallic	ADJ
ejpam-5492	27	24	riemannian	riemannian	ADJ
ejpam-5492	27	25	manifolds	manifold	NOUN
ejpam-5492	27	26	.	.	PUNCT
ejpam-5492	28	1	we	we	PRON
ejpam-5492	28	2	provide	provide	VERB
ejpam-5492	28	3	some	some	DET
ejpam-5492	28	4	examples	example	NOUN
ejpam-5492	28	5	of	of	ADP
ejpam-5492	28	6	warped	warped	ADJ
ejpam-5492	28	7	product	product	NOUN
ejpam-5492	28	8	cr	cr	NOUN
ejpam-5492	28	9	-	-	PUNCT
ejpam-5492	28	10	submanifolds	submanifold	NOUN
ejpam-5492	28	11	in	in	ADP
ejpam-5492	28	12	a	a	DET
ejpam-5492	28	13	metallic	metallic	ADJ
ejpam-5492	28	14	riemannian	riemannian	ADJ
ejpam-5492	28	15	manifold	manifold	NOUN
ejpam-5492	28	16	.	.	PUNCT
ejpam-5492	29	1	also	also	ADV
ejpam-5492	29	2	,	,	PUNCT
ejpam-5492	29	3	we	we	PRON
ejpam-5492	29	4	obtain	obtain	VERB
ejpam-5492	29	5	some	some	DET
ejpam-5492	29	6	useful	useful	ADJ
ejpam-5492	29	7	lemmas	lemma	NOUN
ejpam-5492	29	8	that	that	PRON
ejpam-5492	29	9	will	will	AUX
ejpam-5492	29	10	be	be	AUX
ejpam-5492	29	11	used	use	VERB
ejpam-5492	29	12	to	to	PART
ejpam-5492	29	13	prove	prove	VERB
ejpam-5492	29	14	our	our	PRON
ejpam-5492	29	15	main	main	ADJ
ejpam-5492	29	16	theorem	theorem	NOUN
ejpam-5492	29	17	.	.	PUNCT
ejpam-5492	30	1	we	we	PRON
ejpam-5492	30	2	derive	derive	VERB
ejpam-5492	30	3	a	a	DET
ejpam-5492	30	4	relation	relation	NOUN
ejpam-5492	30	5	for	for	ADP
ejpam-5492	30	6	the	the	DET
ejpam-5492	30	7	squared	square	VERB
ejpam-5492	30	8	norm	norm	NOUN
ejpam-5492	30	9	of	of	ADP
ejpam-5492	30	10	the	the	DET
ejpam-5492	30	11	second	second	ADJ
ejpam-5492	30	12	fundamental	fundamental	ADJ
ejpam-5492	30	13	form	form	NOUN
ejpam-5492	30	14	in	in	ADP
ejpam-5492	30	15	terms	term	NOUN
ejpam-5492	30	16	of	of	ADP
ejpam-5492	30	17	the	the	DET
ejpam-5492	30	18	components	component	NOUN
ejpam-5492	30	19	of	of	ADP
ejpam-5492	30	20	the	the	DET
ejpam-5492	30	21	gradient	gradient	NOUN
ejpam-5492	30	22	of	of	ADP
ejpam-5492	30	23	the	the	DET
ejpam-5492	30	24	warping	warp	VERB
ejpam-5492	30	25	function	function	NOUN
ejpam-5492	30	26	,	,	PUNCT
ejpam-5492	30	27	and	and	CCONJ
ejpam-5492	30	28	consider	consider	VERB
ejpam-5492	30	29	the	the	DET
ejpam-5492	30	30	equality	equality	NOUN
ejpam-5492	30	31	case	case	NOUN
ejpam-5492	30	32	.	.	PUNCT
ejpam-5492	31	1	2	2	X
ejpam-5492	31	2	.	.	X
ejpam-5492	31	3	preliminaries	preliminary	NOUN
ejpam-5492	31	4	let	let	VERB
ejpam-5492	31	5	m̃	m̃	PROPN
ejpam-5492	31	6	be	be	AUX
ejpam-5492	31	7	a	a	DET
ejpam-5492	31	8	smooth	smooth	ADJ
ejpam-5492	31	9	manifold	manifold	NOUN
ejpam-5492	31	10	of	of	ADP
ejpam-5492	31	11	dimension	dimension	NOUN
ejpam-5492	31	12	m.	m.	NOUN
ejpam-5492	31	13	the	the	DET
ejpam-5492	31	14	metallic	metallic	ADJ
ejpam-5492	31	15	structure	structure	NOUN
ejpam-5492	31	16	j	j	PROPN
ejpam-5492	31	17	is	be	AUX
ejpam-5492	31	18	a	a	DET
ejpam-5492	31	19	(	(	PUNCT
ejpam-5492	31	20	1,1	1,1	NUM
ejpam-5492	31	21	)	)	PUNCT
ejpam-5492	31	22	tensor	tensor	NOUN
ejpam-5492	31	23	field	field	NOUN
ejpam-5492	31	24	defined	define	VERB
ejpam-5492	31	25	by	by	ADP
ejpam-5492	31	26	the	the	DET
ejpam-5492	31	27	equation	equation	NOUN
ejpam-5492	31	28	j2	j2	PROPN
ejpam-5492	31	29	=	=	SYM
ejpam-5492	31	30	pj	pj	PROPN
ejpam-5492	31	31	+	+	CCONJ
ejpam-5492	31	32	qi	qi	PROPN
ejpam-5492	31	33	,	,	PUNCT
ejpam-5492	31	34	(	(	PUNCT
ejpam-5492	31	35	1	1	X
ejpam-5492	31	36	)	)	PUNCT
ejpam-5492	31	37	where	where	SCONJ
ejpam-5492	31	38	p	p	X
ejpam-5492	31	39	,	,	PUNCT
ejpam-5492	31	40	q∈	q∈	NOUN
ejpam-5492	31	41	n	n	NOUN
ejpam-5492	32	1	and	and	CCONJ
ejpam-5492	32	2	i	i	PRON
ejpam-5492	32	3	is	be	AUX
ejpam-5492	32	4	the	the	DET
ejpam-5492	32	5	identity	identity	NOUN
ejpam-5492	32	6	operator	operator	NOUN
ejpam-5492	32	7	on	on	ADP
ejpam-5492	32	8	the	the	DET
ejpam-5492	32	9	space	space	NOUN
ejpam-5492	32	10	of	of	ADP
ejpam-5492	32	11	all	all	DET
ejpam-5492	32	12	vector	vector	NOUN
ejpam-5492	32	13	fields	field	NOUN
ejpam-5492	32	14	on	on	ADP
ejpam-5492	32	15	m̃	m̃	PROPN
ejpam-5492	32	16	,	,	PUNCT
ejpam-5492	32	17	denoted	denote	VERB
ejpam-5492	32	18	by	by	ADP
ejpam-5492	32	19	γ(tm̃	γ(tm̃	PROPN
ejpam-5492	32	20	)	)	PUNCT
ejpam-5492	32	21	,	,	PUNCT
ejpam-5492	33	1	[	[	X
ejpam-5492	33	2	17	17	NUM
ejpam-5492	33	3	]	]	PUNCT
ejpam-5492	33	4	.	.	PUNCT
ejpam-5492	34	1	a	a	DET
ejpam-5492	34	2	metallic	metallic	ADJ
ejpam-5492	34	3	riemannian	riemannian	ADJ
ejpam-5492	34	4	manifold	manifold	NOUN
ejpam-5492	34	5	is	be	AUX
ejpam-5492	34	6	a	a	DET
ejpam-5492	34	7	riemannian	riemannian	ADJ
ejpam-5492	34	8	manifold	manifold	NOUN
ejpam-5492	34	9	(	(	PUNCT
ejpam-5492	34	10	m̃	m̃	PROPN
ejpam-5492	34	11	,	,	PUNCT
ejpam-5492	34	12	g̃	g̃	PROPN
ejpam-5492	34	13	)	)	PUNCT
ejpam-5492	34	14	where	where	SCONJ
ejpam-5492	34	15	the	the	DET
ejpam-5492	34	16	riemannian	riemannian	ADJ
ejpam-5492	34	17	metric	metric	ADJ
ejpam-5492	34	18	g̃	g̃	PROPN
ejpam-5492	34	19	is	be	AUX
ejpam-5492	34	20	j	j	NOUN
ejpam-5492	34	21	-	-	NOUN
ejpam-5492	34	22	compatible	compatible	ADJ
ejpam-5492	34	23	.	.	PUNCT
ejpam-5492	35	1	this	this	PRON
ejpam-5492	35	2	means	mean	VERB
ejpam-5492	35	3	that	that	SCONJ
ejpam-5492	35	4	g̃(jx	g̃(jx	PROPN
ejpam-5492	35	5	,	,	PUNCT
ejpam-5492	35	6	y	y	NOUN
ejpam-5492	35	7	)	)	PUNCT
ejpam-5492	35	8	=	=	PUNCT
ejpam-5492	36	1	g̃(x	g̃(x	PROPN
ejpam-5492	36	2	,	,	PUNCT
ejpam-5492	36	3	jy	jy	PROPN
ejpam-5492	36	4	)	)	PUNCT
ejpam-5492	36	5	(	(	PUNCT
ejpam-5492	36	6	2	2	X
ejpam-5492	36	7	)	)	PUNCT
ejpam-5492	36	8	holds	hold	VERB
ejpam-5492	36	9	for	for	ADP
ejpam-5492	36	10	all	all	DET
ejpam-5492	36	11	vector	vector	NOUN
ejpam-5492	36	12	fields	field	NOUN
ejpam-5492	36	13	x	x	PUNCT
ejpam-5492	36	14	and	and	CCONJ
ejpam-5492	36	15	y	y	PROPN
ejpam-5492	36	16	in	in	ADP
ejpam-5492	36	17	γ(tm̃)[17	γ(tm̃)[17	PROPN
ejpam-5492	36	18	]	]	PUNCT
ejpam-5492	36	19	.	.	PUNCT
ejpam-5492	37	1	specifically	specifically	ADV
ejpam-5492	37	2	,	,	PUNCT
ejpam-5492	37	3	a	a	DET
ejpam-5492	37	4	golden	golden	ADJ
ejpam-5492	37	5	structure	structure	NOUN
ejpam-5492	37	6	is	be	AUX
ejpam-5492	37	7	a	a	DET
ejpam-5492	37	8	special	special	ADJ
ejpam-5492	37	9	type	type	NOUN
ejpam-5492	37	10	of	of	ADP
ejpam-5492	37	11	a	a	DET
ejpam-5492	37	12	metallic	metallic	ADJ
ejpam-5492	37	13	structure	structure	NOUN
ejpam-5492	37	14	.	.	PUNCT
ejpam-5492	38	1	it	it	PRON
ejpam-5492	38	2	is	be	AUX
ejpam-5492	38	3	defined	define	VERB
ejpam-5492	38	4	by	by	ADP
ejpam-5492	38	5	the	the	DET
ejpam-5492	38	6	equation	equation	NOUN
ejpam-5492	39	1	j2	j2	NOUN
ejpam-5492	39	2	=	=	SYM
ejpam-5492	39	3	j	j	PROPN
ejpam-5492	40	1	+	+	CCONJ
ejpam-5492	40	2	i	i	PROPN
ejpam-5492	40	3	,	,	PUNCT
ejpam-5492	40	4	where	where	SCONJ
ejpam-5492	40	5	p	p	NOUN
ejpam-5492	40	6	=	=	X
ejpam-5492	40	7	q	q	NOUN
ejpam-5492	40	8	=	=	SYM
ejpam-5492	40	9	1	1	NUM
ejpam-5492	41	1	[	[	X
ejpam-5492	41	2	11	11	NUM
ejpam-5492	41	3	]	]	PUNCT
ejpam-5492	41	4	.	.	PUNCT
ejpam-5492	42	1	on	on	ADP
ejpam-5492	42	2	a	a	DET
ejpam-5492	42	3	metallic	metallic	ADJ
ejpam-5492	42	4	riemannian	riemannian	ADJ
ejpam-5492	42	5	manifold	manifold	ADJ
ejpam-5492	42	6	m̃	m̃	PROPN
ejpam-5492	42	7	,	,	PUNCT
ejpam-5492	42	8	the	the	DET
ejpam-5492	42	9	riemannian	riemannian	ADJ
ejpam-5492	42	10	metric	metric	PROPN
ejpam-5492	42	11	g̃	g̃	PROPN
ejpam-5492	42	12	satisfies	satisfy	VERB
ejpam-5492	42	13	the	the	DET
ejpam-5492	42	14	equation	equation	NOUN
ejpam-5492	42	15	g̃(jx	g̃(jx	PROPN
ejpam-5492	42	16	,	,	PUNCT
ejpam-5492	42	17	jy	jy	PROPN
ejpam-5492	42	18	)	)	PUNCT
ejpam-5492	43	1	=	=	SYM
ejpam-5492	43	2	pg̃(jx	pg̃(jx	PROPN
ejpam-5492	43	3	,	,	PUNCT
ejpam-5492	43	4	y	y	PROPN
ejpam-5492	43	5	)	)	PUNCT
ejpam-5492	44	1	+	+	CCONJ
ejpam-5492	44	2	qg̃(x	qg̃(x	X
ejpam-5492	44	3	,	,	PUNCT
ejpam-5492	44	4	y	y	PROPN
ejpam-5492	44	5	)	)	PUNCT
ejpam-5492	44	6	,	,	PUNCT
ejpam-5492	44	7	(	(	PUNCT
ejpam-5492	44	8	3	3	X
ejpam-5492	44	9	)	)	PUNCT
ejpam-5492	44	10	for	for	ADP
ejpam-5492	44	11	all	all	DET
ejpam-5492	44	12	x	x	NOUN
ejpam-5492	44	13	,	,	PUNCT
ejpam-5492	44	14	y	y	PROPN
ejpam-5492	44	15	∈	∈	PROPN
ejpam-5492	44	16	γ(tm̃	γ(tm̃	PROPN
ejpam-5492	44	17	)	)	PUNCT
ejpam-5492	44	18	.	.	PUNCT
ejpam-5492	45	1	this	this	DET
ejpam-5492	45	2	equation	equation	NOUN
ejpam-5492	45	3	can	can	AUX
ejpam-5492	45	4	be	be	AUX
ejpam-5492	45	5	derived	derive	VERB
ejpam-5492	45	6	from	from	ADP
ejpam-5492	45	7	equations	equation	NOUN
ejpam-5492	45	8	(	(	PUNCT
ejpam-5492	45	9	1	1	NUM
ejpam-5492	45	10	)	)	PUNCT
ejpam-5492	45	11	and	and	CCONJ
ejpam-5492	45	12	(	(	PUNCT
ejpam-5492	45	13	2	2	NUM
ejpam-5492	45	14	)	)	PUNCT
ejpam-5492	45	15	.	.	PUNCT
ejpam-5492	46	1	now	now	ADV
ejpam-5492	46	2	,	,	PUNCT
ejpam-5492	46	3	let	let	VERB
ejpam-5492	46	4	m	m	PRON
ejpam-5492	46	5	be	be	AUX
ejpam-5492	46	6	a	a	DET
ejpam-5492	46	7	submanifold	submanifold	NOUN
ejpam-5492	46	8	embedded	embed	VERB
ejpam-5492	46	9	in	in	ADP
ejpam-5492	46	10	a	a	DET
ejpam-5492	46	11	metallic	metallic	ADJ
ejpam-5492	46	12	riemannian	riemannian	NOUN
ejpam-5492	46	13	manifold	manifold	NOUN
ejpam-5492	46	14	(	(	PUNCT
ejpam-5492	46	15	m̃	m̃	PROPN
ejpam-5492	46	16	,	,	PUNCT
ejpam-5492	46	17	g̃	g̃	PROPN
ejpam-5492	46	18	,	,	PUNCT
ejpam-5492	46	19	j	j	PROPN
ejpam-5492	46	20	)	)	PUNCT
ejpam-5492	46	21	.	.	PUNCT
ejpam-5492	47	1	let	let	VERB
ejpam-5492	47	2	tx	tx	VERB
ejpam-5492	47	3	and	and	CCONJ
ejpam-5492	47	4	nx	nx	PROPN
ejpam-5492	47	5	be	be	AUX
ejpam-5492	47	6	the	the	DET
ejpam-5492	47	7	tangential	tangential	ADJ
ejpam-5492	47	8	and	and	CCONJ
ejpam-5492	47	9	normal	normal	ADJ
ejpam-5492	47	10	components	component	NOUN
ejpam-5492	47	11	of	of	ADP
ejpam-5492	47	12	jx	jx	PROPN
ejpam-5492	47	13	,	,	PUNCT
ejpam-5492	47	14	respectively	respectively	ADV
ejpam-5492	47	15	,	,	PUNCT
ejpam-5492	47	16	for	for	ADP
ejpam-5492	47	17	any	any	DET
ejpam-5492	47	18	l.	l.	PROPN
ejpam-5492	47	19	alqahtani	alqahtani	PROPN
ejpam-5492	47	20	,	,	PUNCT
ejpam-5492	47	21	e.	e.	PROPN
ejpam-5492	47	22	al	al	PROPN
ejpam-5492	47	23	-	-	PROPN
ejpam-5492	47	24	husainy	husainy	PROPN
ejpam-5492	47	25	/	/	SYM
ejpam-5492	47	26	eur	eur	PROPN
ejpam-5492	47	27	.	.	PUNCT
ejpam-5492	48	1	j.	j.	PROPN
ejpam-5492	48	2	pure	pure	PROPN
ejpam-5492	48	3	appl	appl	PROPN
ejpam-5492	48	4	.	.	PROPN
ejpam-5492	48	5	math	math	PROPN
ejpam-5492	48	6	,	,	PUNCT
ejpam-5492	48	7	17	17	NUM
ejpam-5492	48	8	(	(	PUNCT
ejpam-5492	48	9	4	4	NUM
ejpam-5492	48	10	)	)	PUNCT
ejpam-5492	48	11	(	(	PUNCT
ejpam-5492	48	12	2024	2024	NUM
ejpam-5492	48	13	)	)	PUNCT
ejpam-5492	48	14	,	,	PUNCT
ejpam-5492	48	15	2481	2481	NUM
ejpam-5492	48	16	-	-	SYM
ejpam-5492	48	17	2491	2491	NUM
ejpam-5492	48	18	2483	2483	NUM
ejpam-5492	48	19	x	x	SYM
ejpam-5492	48	20	∈	∈	PROPN
ejpam-5492	48	21	γ(tm	γ(tm	NOUN
ejpam-5492	48	22	)	)	PUNCT
ejpam-5492	48	23	.	.	PUNCT
ejpam-5492	49	1	similarly	similarly	ADV
ejpam-5492	49	2	,	,	PUNCT
ejpam-5492	49	3	let	let	VERB
ejpam-5492	49	4	tv	tv	NOUN
ejpam-5492	49	5	and	and	CCONJ
ejpam-5492	49	6	nv	nv	PROPN
ejpam-5492	49	7	be	be	AUX
ejpam-5492	49	8	the	the	DET
ejpam-5492	49	9	tangential	tangential	ADJ
ejpam-5492	49	10	and	and	CCONJ
ejpam-5492	49	11	normal	normal	ADJ
ejpam-5492	49	12	components	component	NOUN
ejpam-5492	49	13	of	of	ADP
ejpam-5492	49	14	jv	jv	NOUN
ejpam-5492	49	15	,	,	PUNCT
ejpam-5492	49	16	respectively	respectively	ADV
ejpam-5492	49	17	,	,	PUNCT
ejpam-5492	49	18	for	for	ADP
ejpam-5492	49	19	any	any	DET
ejpam-5492	49	20	v	v	NOUN
ejpam-5492	49	21	∈	∈	PROPN
ejpam-5492	49	22	γ(t⊥m	γ(t⊥m	NOUN
ejpam-5492	49	23	)	)	PUNCT
ejpam-5492	49	24	.	.	PUNCT
ejpam-5492	50	1	that	that	PRON
ejpam-5492	50	2	is	be	AUX
ejpam-5492	50	3	jx	jx	PROPN
ejpam-5492	50	4	=	=	PUNCT
ejpam-5492	50	5	tx	tx	PROPN
ejpam-5492	51	1	+	+	PROPN
ejpam-5492	51	2	nx	nx	X
ejpam-5492	51	3	,	,	PUNCT
ejpam-5492	51	4	(	(	PUNCT
ejpam-5492	51	5	4	4	X
ejpam-5492	51	6	)	)	PUNCT
ejpam-5492	51	7	jv	jv	NOUN
ejpam-5492	51	8	=	=	SYM
ejpam-5492	51	9	tv	tv	PROPN
ejpam-5492	51	10	+	+	X
ejpam-5492	51	11	nv	nv	PROPN
ejpam-5492	51	12	.	.	PUNCT
ejpam-5492	52	1	(	(	PUNCT
ejpam-5492	52	2	5	5	X
ejpam-5492	52	3	)	)	PUNCT
ejpam-5492	52	4	this	this	PRON
ejpam-5492	52	5	implies	imply	VERB
ejpam-5492	52	6	that	that	SCONJ
ejpam-5492	52	7	for	for	ADP
ejpam-5492	52	8	any	any	DET
ejpam-5492	52	9	x	x	NOUN
ejpam-5492	52	10	,	,	PUNCT
ejpam-5492	52	11	y	y	PROPN
ejpam-5492	52	12	in	in	ADP
ejpam-5492	52	13	γ(tm	γ(tm	NOUN
ejpam-5492	52	14	)	)	PUNCT
ejpam-5492	52	15	and	and	CCONJ
ejpam-5492	52	16	u	u	NOUN
ejpam-5492	52	17	,	,	PUNCT
ejpam-5492	52	18	v	v	NOUN
ejpam-5492	52	19	in	in	ADP
ejpam-5492	52	20	γ(t⊥m	γ(t⊥m	NOUN
ejpam-5492	52	21	)	)	PUNCT
ejpam-5492	52	22	,	,	PUNCT
ejpam-5492	52	23	we	we	PRON
ejpam-5492	52	24	have	have	VERB
ejpam-5492	52	25	[	[	X
ejpam-5492	52	26	3	3	NUM
ejpam-5492	52	27	]	]	X
ejpam-5492	52	28	g̃(tx	g̃(tx	NOUN
ejpam-5492	52	29	,	,	PUNCT
ejpam-5492	52	30	y	y	PROPN
ejpam-5492	52	31	)	)	PUNCT
ejpam-5492	53	1	=	=	PUNCT
ejpam-5492	53	2	g̃(x	g̃(x	NOUN
ejpam-5492	53	3	,	,	PUNCT
ejpam-5492	53	4	ty	ty	INTJ
ejpam-5492	53	5	)	)	PUNCT
ejpam-5492	53	6	,	,	PUNCT
ejpam-5492	53	7	(	(	PUNCT
ejpam-5492	53	8	6	6	X
ejpam-5492	53	9	)	)	PUNCT
ejpam-5492	53	10	g̃(nu	g̃(nu	NOUN
ejpam-5492	53	11	,	,	PUNCT
ejpam-5492	53	12	v	v	NOUN
ejpam-5492	53	13	)	)	PUNCT
ejpam-5492	53	14	=	=	SYM
ejpam-5492	53	15	g̃(u	g̃(u	PROPN
ejpam-5492	53	16	,	,	PUNCT
ejpam-5492	53	17	nv	nv	PROPN
ejpam-5492	53	18	)	)	PUNCT
ejpam-5492	53	19	,	,	PUNCT
ejpam-5492	53	20	(	(	PUNCT
ejpam-5492	53	21	7	7	X
ejpam-5492	53	22	)	)	PUNCT
ejpam-5492	53	23	g̃(nx	g̃(nx	NOUN
ejpam-5492	53	24	,	,	PUNCT
ejpam-5492	53	25	v	v	NOUN
ejpam-5492	53	26	)	)	PUNCT
ejpam-5492	53	27	=	=	PUNCT
ejpam-5492	53	28	g̃(x	g̃(x	NOUN
ejpam-5492	53	29	,	,	PUNCT
ejpam-5492	53	30	tv	tv	NOUN
ejpam-5492	53	31	)	)	PUNCT
ejpam-5492	53	32	.	.	PUNCT
ejpam-5492	54	1	(	(	PUNCT
ejpam-5492	54	2	8)	8)	NUM
ejpam-5492	54	3	clearly	clearly	ADV
ejpam-5492	54	4	,	,	PUNCT
ejpam-5492	54	5	the	the	DET
ejpam-5492	54	6	maps	map	NOUN
ejpam-5492	54	7	t	t	PROPN
ejpam-5492	54	8	and	and	CCONJ
ejpam-5492	54	9	n	n	PROPN
ejpam-5492	54	10	are	be	AUX
ejpam-5492	54	11	g̃	g̃	PROPN
ejpam-5492	54	12	-symmetric	-symmetric	NOUN
ejpam-5492	54	13	.	.	PUNCT
ejpam-5492	55	1	consequently	consequently	ADV
ejpam-5492	55	2	,	,	PUNCT
ejpam-5492	55	3	the	the	DET
ejpam-5492	55	4	following	follow	VERB
ejpam-5492	55	5	equations	equation	NOUN
ejpam-5492	55	6	hold	hold	VERB
ejpam-5492	55	7	for	for	ADP
ejpam-5492	55	8	any	any	DET
ejpam-5492	55	9	x	x	SYM
ejpam-5492	55	10	∈	∈	PROPN
ejpam-5492	55	11	γ(tm	γ(tm	NOUN
ejpam-5492	55	12	)	)	PUNCT
ejpam-5492	55	13	and	and	CCONJ
ejpam-5492	55	14	v	v	ADP
ejpam-5492	55	15	∈	∈	PROPN
ejpam-5492	55	16	γ(t⊥m	γ(t⊥m	NOUN
ejpam-5492	55	17	)	)	PUNCT
ejpam-5492	55	18	,	,	PUNCT
ejpam-5492	56	1	[	[	X
ejpam-5492	56	2	13	13	NUM
ejpam-5492	56	3	]	]	PUNCT
ejpam-5492	56	4	t	t	X
ejpam-5492	56	5	2x	2x	NUM
ejpam-5492	56	6	=	=	X
ejpam-5492	56	7	ptx	ptx	VERB
ejpam-5492	56	8	+	+	CCONJ
ejpam-5492	56	9	qx	qx	PROPN
ejpam-5492	56	10	−	−	PROPN
ejpam-5492	56	11	tnx	tnx	NOUN
ejpam-5492	56	12	,	,	PUNCT
ejpam-5492	56	13	pnx	pnx	NOUN
ejpam-5492	56	14	=	=	PUNCT
ejpam-5492	56	15	ntx	ntx	ADJ
ejpam-5492	56	16	+	+	CCONJ
ejpam-5492	56	17	nnx	nnx	PROPN
ejpam-5492	56	18	,	,	PUNCT
ejpam-5492	56	19	(	(	PUNCT
ejpam-5492	56	20	9	9	X
ejpam-5492	56	21	)	)	PUNCT
ejpam-5492	56	22	n2v	n2v	NOUN
ejpam-5492	56	23	=	=	SYM
ejpam-5492	56	24	pnv	pnv	NOUN
ejpam-5492	56	25	+	+	X
ejpam-5492	56	26	qv	qv	X
ejpam-5492	56	27	−ntv	−ntv	PROPN
ejpam-5492	56	28	,	,	PUNCT
ejpam-5492	56	29	ptv	ptv	X
ejpam-5492	56	30	=	=	NOUN
ejpam-5492	56	31	ttv	ttv	PROPN
ejpam-5492	56	32	+	+	CCONJ
ejpam-5492	56	33	tnv	tnv	NOUN
ejpam-5492	56	34	.	.	PUNCT
ejpam-5492	57	1	(	(	PUNCT
ejpam-5492	57	2	10	10	NUM
ejpam-5492	57	3	)	)	PUNCT
ejpam-5492	57	4	suppose	suppose	VERB
ejpam-5492	57	5	that	that	SCONJ
ejpam-5492	57	6	∇̃	∇̃	PRON
ejpam-5492	57	7	and	and	CCONJ
ejpam-5492	57	8	∇	∇	PROPN
ejpam-5492	57	9	be	be	VERB
ejpam-5492	57	10	the	the	DET
ejpam-5492	57	11	levi	levi	NOUN
ejpam-5492	57	12	-	-	PUNCT
ejpam-5492	57	13	civita	civita	NOUN
ejpam-5492	57	14	connections	connection	NOUN
ejpam-5492	57	15	on	on	ADP
ejpam-5492	57	16	riemannian	riemannian	ADJ
ejpam-5492	57	17	manifolds	manifold	NOUN
ejpam-5492	57	18	m̃	m̃	PROPN
ejpam-5492	57	19	)	)	PUNCT
ejpam-5492	57	20	and	and	CCONJ
ejpam-5492	57	21	m	m	PROPN
ejpam-5492	57	22	,	,	PUNCT
ejpam-5492	57	23	respectively	respectively	ADV
ejpam-5492	57	24	.	.	PUNCT
ejpam-5492	58	1	then	then	ADV
ejpam-5492	58	2	,	,	PUNCT
ejpam-5492	58	3	for	for	ADP
ejpam-5492	58	4	any	any	DET
ejpam-5492	58	5	x.y	x.y	PROPN
ejpam-5492	58	6	∈	∈	PROPN
ejpam-5492	58	7	γ(m	γ(m	PROPN
ejpam-5492	58	8	)	)	PUNCT
ejpam-5492	58	9	,	,	PUNCT
ejpam-5492	58	10	v	v	NOUN
ejpam-5492	58	11	∈	∈	PROPN
ejpam-5492	58	12	γ(t⊥m	γ(t⊥m	NOUN
ejpam-5492	58	13	)	)	PUNCT
ejpam-5492	58	14	,	,	PUNCT
ejpam-5492	58	15	the	the	DET
ejpam-5492	58	16	gauss	gauss	ADJ
ejpam-5492	58	17	and	and	CCONJ
ejpam-5492	58	18	weingarten	weingarten	ADJ
ejpam-5492	58	19	formulas	formula	NOUN
ejpam-5492	58	20	are	be	AUX
ejpam-5492	58	21	given	give	VERB
ejpam-5492	58	22	by	by	ADP
ejpam-5492	58	23	∇̃xy	∇̃xy	PRON
ejpam-5492	58	24	=	=	SYM
ejpam-5492	58	25	∇xy	∇xy	PROPN
ejpam-5492	58	26	+	+	NUM
ejpam-5492	58	27	h(x	h(x	PROPN
ejpam-5492	58	28	,	,	PUNCT
ejpam-5492	58	29	y	y	PROPN
ejpam-5492	58	30	)	)	PUNCT
ejpam-5492	58	31	,	,	PUNCT
ejpam-5492	58	32	(	(	PUNCT
ejpam-5492	58	33	11	11	NUM
ejpam-5492	58	34	)	)	PUNCT
ejpam-5492	58	35	∇̃xv	∇̃xv	PUNCT
ejpam-5492	59	1	=	=	PUNCT
ejpam-5492	59	2	−av	−av	NOUN
ejpam-5492	59	3	x	x	PUNCT
ejpam-5492	60	1	+	+	PUNCT
ejpam-5492	60	2	∇⊥	∇⊥	NOUN
ejpam-5492	60	3	xv	xv	PROPN
ejpam-5492	60	4	,	,	PUNCT
ejpam-5492	60	5	(	(	PUNCT
ejpam-5492	60	6	12	12	NUM
ejpam-5492	60	7	)	)	PUNCT
ejpam-5492	60	8	where	where	SCONJ
ejpam-5492	60	9	h	h	NOUN
ejpam-5492	60	10	and	and	CCONJ
ejpam-5492	60	11	av	av	PROPN
ejpam-5492	60	12	are	be	AUX
ejpam-5492	60	13	the	the	DET
ejpam-5492	60	14	second	second	ADJ
ejpam-5492	60	15	fundamental	fundamental	ADJ
ejpam-5492	60	16	form	form	NOUN
ejpam-5492	60	17	and	and	CCONJ
ejpam-5492	60	18	the	the	DET
ejpam-5492	60	19	shape	shape	NOUN
ejpam-5492	60	20	operator	operator	NOUN
ejpam-5492	60	21	on	on	ADP
ejpam-5492	60	22	m	m	PROPN
ejpam-5492	60	23	,	,	PUNCT
ejpam-5492	60	24	respectively	respectively	ADV
ejpam-5492	60	25	[	[	X
ejpam-5492	60	26	5	5	NUM
ejpam-5492	60	27	]	]	PUNCT
ejpam-5492	60	28	.	.	PUNCT
ejpam-5492	61	1	they	they	PRON
ejpam-5492	61	2	are	be	AUX
ejpam-5492	61	3	related	relate	VERB
ejpam-5492	61	4	by	by	ADP
ejpam-5492	61	5	g̃(h(x	g̃(h(x	PROPN
ejpam-5492	61	6	,	,	PUNCT
ejpam-5492	61	7	y	y	PROPN
ejpam-5492	61	8	)	)	PUNCT
ejpam-5492	61	9	,	,	PUNCT
ejpam-5492	61	10	v	v	NOUN
ejpam-5492	61	11	)	)	PUNCT
ejpam-5492	62	1	=	=	SYM
ejpam-5492	62	2	g̃(av	g̃(av	NOUN
ejpam-5492	62	3	x	x	X
ejpam-5492	62	4	,	,	PUNCT
ejpam-5492	62	5	y	y	PROPN
ejpam-5492	62	6	)	)	PUNCT
ejpam-5492	62	7	.	.	PUNCT
ejpam-5492	63	1	(	(	PUNCT
ejpam-5492	63	2	13	13	NUM
ejpam-5492	63	3	)	)	PUNCT
ejpam-5492	63	4	a	a	DET
ejpam-5492	63	5	locally	locally	ADV
ejpam-5492	63	6	metallic	metallic	ADJ
ejpam-5492	63	7	riemannian	riemannian	NOUN
ejpam-5492	63	8	manifold	manifold	NOUN
ejpam-5492	63	9	(	(	PUNCT
ejpam-5492	63	10	m̃	m̃	PROPN
ejpam-5492	63	11	,	,	PUNCT
ejpam-5492	63	12	g̃	g̃	PROPN
ejpam-5492	63	13	,	,	PUNCT
ejpam-5492	63	14	j	j	NOUN
ejpam-5492	63	15	)	)	PUNCT
ejpam-5492	63	16	is	be	AUX
ejpam-5492	63	17	a	a	DET
ejpam-5492	63	18	manifold	manifold	NOUN
ejpam-5492	63	19	that	that	PRON
ejpam-5492	63	20	has	have	VERB
ejpam-5492	63	21	a	a	DET
ejpam-5492	63	22	metallic	metallic	ADJ
ejpam-5492	63	23	riemannian	riemannian	ADJ
ejpam-5492	63	24	structure	structure	NOUN
ejpam-5492	63	25	such	such	ADJ
ejpam-5492	63	26	that	that	SCONJ
ejpam-5492	63	27	j	j	PROPN
ejpam-5492	63	28	is	be	AUX
ejpam-5492	63	29	parallel	parallel	ADJ
ejpam-5492	63	30	with	with	ADP
ejpam-5492	63	31	respect	respect	NOUN
ejpam-5492	63	32	to	to	ADP
ejpam-5492	63	33	the	the	DET
ejpam-5492	63	34	levi	levi	PROPN
ejpam-5492	63	35	-	-	PUNCT
ejpam-5492	63	36	civita	civita	PROPN
ejpam-5492	63	37	connection	connection	NOUN
ejpam-5492	63	38	∇̃	∇̃	X
ejpam-5492	63	39	on	on	ADP
ejpam-5492	63	40	m̃	m̃	PROPN
ejpam-5492	63	41	,	,	PUNCT
ejpam-5492	63	42	that	that	PRON
ejpam-5492	63	43	is	be	AUX
ejpam-5492	63	44	∇̃j	∇̃j	NOUN
ejpam-5492	63	45	=	=	SYM
ejpam-5492	63	46	0	0	NUM
ejpam-5492	63	47	,	,	PUNCT
ejpam-5492	63	48	[	[	X
ejpam-5492	63	49	12	12	NUM
ejpam-5492	63	50	]	]	PUNCT
ejpam-5492	63	51	.	.	PUNCT
ejpam-5492	64	1	hence	hence	ADV
ejpam-5492	64	2	,	,	PUNCT
ejpam-5492	64	3	from	from	ADP
ejpam-5492	64	4	equation	equation	NOUN
ejpam-5492	64	5	(	(	PUNCT
ejpam-5492	64	6	1	1	NUM
ejpam-5492	64	7	)	)	PUNCT
ejpam-5492	64	8	,	,	PUNCT
ejpam-5492	64	9	one	one	PRON
ejpam-5492	64	10	can	can	AUX
ejpam-5492	64	11	see	see	VERB
ejpam-5492	64	12	that	that	SCONJ
ejpam-5492	64	13	the	the	DET
ejpam-5492	64	14	following	follow	VERB
ejpam-5492	64	15	equation	equation	NOUN
ejpam-5492	64	16	holds	hold	VERB
ejpam-5492	64	17	for	for	ADP
ejpam-5492	64	18	any	any	DET
ejpam-5492	64	19	x	x	NOUN
ejpam-5492	64	20	,	,	PUNCT
ejpam-5492	64	21	y	y	PROPN
ejpam-5492	64	22	,	,	PUNCT
ejpam-5492	64	23	z	z	PROPN
ejpam-5492	64	24	∈	∈	PROPN
ejpam-5492	64	25	γ(tm	γ(tm	NOUN
ejpam-5492	64	26	)	)	PUNCT
ejpam-5492	64	27	,	,	PUNCT
ejpam-5492	64	28	[	[	X
ejpam-5492	64	29	4	4	NUM
ejpam-5492	64	30	]	]	PUNCT
ejpam-5492	64	31	g̃((∇̃xj)y	g̃((∇̃xj)y	NOUN
ejpam-5492	64	32	,	,	PUNCT
ejpam-5492	64	33	z	z	NOUN
ejpam-5492	64	34	)	)	PUNCT
ejpam-5492	64	35	=	=	SYM
ejpam-5492	64	36	g̃(y	g̃(y	NOUN
ejpam-5492	64	37	,	,	PUNCT
ejpam-5492	64	38	(	(	PUNCT
ejpam-5492	64	39	∇̃xj)z	∇̃xj)z	NOUN
ejpam-5492	64	40	)	)	PUNCT
ejpam-5492	64	41	.	.	PUNCT
ejpam-5492	65	1	(	(	PUNCT
ejpam-5492	65	2	14	14	NUM
ejpam-5492	65	3	)	)	PUNCT
ejpam-5492	65	4	now	now	ADV
ejpam-5492	65	5	,	,	PUNCT
ejpam-5492	65	6	let	let	VERB
ejpam-5492	65	7	(	(	PUNCT
ejpam-5492	65	8	m1	m1	NOUN
ejpam-5492	65	9	,	,	PUNCT
ejpam-5492	65	10	g1	g1	PROPN
ejpam-5492	65	11	)	)	PUNCT
ejpam-5492	65	12	and	and	CCONJ
ejpam-5492	65	13	(	(	PUNCT
ejpam-5492	65	14	m2	m2	PROPN
ejpam-5492	65	15	,	,	PUNCT
ejpam-5492	65	16	g2	g2	PROPN
ejpam-5492	65	17	)	)	PUNCT
ejpam-5492	65	18	be	be	VERB
ejpam-5492	65	19	two	two	NUM
ejpam-5492	65	20	riemannian	riemannian	ADJ
ejpam-5492	65	21	manifolds	manifold	NOUN
ejpam-5492	65	22	,	,	PUNCT
ejpam-5492	65	23	then	then	ADV
ejpam-5492	65	24	the	the	DET
ejpam-5492	65	25	warped	warped	ADJ
ejpam-5492	65	26	product	product	NOUN
ejpam-5492	65	27	m1	m1	PROPN
ejpam-5492	65	28	×f	×f	PROPN
ejpam-5492	65	29	m2	m2	PROPN
ejpam-5492	65	30	is	be	AUX
ejpam-5492	65	31	a	a	DET
ejpam-5492	65	32	riemannian	riemannian	NOUN
ejpam-5492	65	33	manifold	manifold	NOUN
ejpam-5492	65	34	with	with	ADP
ejpam-5492	65	35	riemannian	riemannian	ADJ
ejpam-5492	65	36	metric	metric	ADJ
ejpam-5492	65	37	g	g	PROPN
ejpam-5492	65	38	=	=	PUNCT
ejpam-5492	65	39	g1	g1	PROPN
ejpam-5492	65	40	+	+	CCONJ
ejpam-5492	65	41	f2g2	f2g2	NOUN
ejpam-5492	65	42	,	,	PUNCT
ejpam-5492	65	43	where	where	SCONJ
ejpam-5492	65	44	f	f	PROPN
ejpam-5492	65	45	is	be	AUX
ejpam-5492	65	46	a	a	DET
ejpam-5492	65	47	positive	positive	ADJ
ejpam-5492	65	48	smooth	smooth	ADJ
ejpam-5492	65	49	function	function	NOUN
ejpam-5492	65	50	on	on	ADP
ejpam-5492	65	51	m1	m1	PROPN
ejpam-5492	65	52	,	,	PUNCT
ejpam-5492	65	53	called	call	VERB
ejpam-5492	65	54	the	the	DET
ejpam-5492	65	55	warping	warp	VERB
ejpam-5492	65	56	function	function	NOUN
ejpam-5492	65	57	[	[	X
ejpam-5492	65	58	1	1	NUM
ejpam-5492	65	59	]	]	PUNCT
ejpam-5492	65	60	.	.	PUNCT
ejpam-5492	66	1	note	note	VERB
ejpam-5492	66	2	that	that	SCONJ
ejpam-5492	66	3	if	if	SCONJ
ejpam-5492	66	4	m1	m1	PROPN
ejpam-5492	66	5	and	and	CCONJ
ejpam-5492	66	6	m2	m2	PROPN
ejpam-5492	66	7	have	have	AUX
ejpam-5492	66	8	dimension	dimension	NOUN
ejpam-5492	66	9	n1	n1	PROPN
ejpam-5492	66	10	and	and	CCONJ
ejpam-5492	66	11	n2	n2	ADJ
ejpam-5492	66	12	,	,	PUNCT
ejpam-5492	66	13	respectively	respectively	ADV
ejpam-5492	66	14	,	,	PUNCT
ejpam-5492	66	15	then	then	ADV
ejpam-5492	66	16	the	the	DET
ejpam-5492	66	17	dimension	dimension	NOUN
ejpam-5492	66	18	of	of	ADP
ejpam-5492	66	19	the	the	DET
ejpam-5492	66	20	warped	warped	ADJ
ejpam-5492	66	21	product	product	NOUN
ejpam-5492	66	22	m1	m1	PROPN
ejpam-5492	66	23	×f	×f	PROPN
ejpam-5492	66	24	m2	m2	PROPN
ejpam-5492	66	25	is	be	AUX
ejpam-5492	66	26	n	n	PRON
ejpam-5492	66	27	=	=	SYM
ejpam-5492	66	28	n1	n1	PROPN
ejpam-5492	66	29	+	+	CCONJ
ejpam-5492	66	30	n2	n2	ADJ
ejpam-5492	66	31	.	.	PUNCT
ejpam-5492	67	1	on	on	ADP
ejpam-5492	67	2	the	the	DET
ejpam-5492	67	3	warped	warped	ADJ
ejpam-5492	67	4	product	product	NOUN
ejpam-5492	67	5	m	m	NOUN
ejpam-5492	67	6	=	=	SYM
ejpam-5492	67	7	m1×f	m1×f	PROPN
ejpam-5492	67	8	m2	m2	PROPN
ejpam-5492	67	9	,	,	PUNCT
ejpam-5492	67	10	if	if	SCONJ
ejpam-5492	67	11	x	x	NOUN
ejpam-5492	67	12	,	,	PUNCT
ejpam-5492	67	13	y	y	PROPN
ejpam-5492	67	14	∈	∈	PROPN
ejpam-5492	67	15	γ(tm1	γ(tm1	PROPN
ejpam-5492	67	16	)	)	PUNCT
ejpam-5492	67	17	,	,	PUNCT
ejpam-5492	67	18	and	and	CCONJ
ejpam-5492	67	19	z	z	X
ejpam-5492	67	20	,	,	PUNCT
ejpam-5492	67	21	w	w	PROPN
ejpam-5492	67	22	∈	∈	PROPN
ejpam-5492	67	23	γ(tm2	γ(tm2	NOUN
ejpam-5492	67	24	)	)	PUNCT
ejpam-5492	67	25	,	,	PUNCT
ejpam-5492	67	26	then	then	ADV
ejpam-5492	67	27	[	[	X
ejpam-5492	67	28	9	9	NUM
ejpam-5492	67	29	]	]	PUNCT
ejpam-5492	67	30	,	,	PUNCT
ejpam-5492	67	31	∇xy	∇xy	PROPN
ejpam-5492	67	32	∈	∈	PROPN
ejpam-5492	67	33	γ(tm1	γ(tm1	PROPN
ejpam-5492	67	34	)	)	PUNCT
ejpam-5492	67	35	,	,	PUNCT
ejpam-5492	67	36	(	(	PUNCT
ejpam-5492	67	37	15	15	X
ejpam-5492	67	38	)	)	PUNCT
ejpam-5492	67	39	l.	l.	PROPN
ejpam-5492	67	40	alqahtani	alqahtani	PROPN
ejpam-5492	67	41	,	,	PUNCT
ejpam-5492	67	42	e.	e.	PROPN
ejpam-5492	67	43	al	al	PROPN
ejpam-5492	67	44	-	-	PROPN
ejpam-5492	67	45	husainy	husainy	PROPN
ejpam-5492	67	46	/	/	SYM
ejpam-5492	67	47	eur	eur	PROPN
ejpam-5492	67	48	.	.	PUNCT
ejpam-5492	68	1	j.	j.	PROPN
ejpam-5492	68	2	pure	pure	PROPN
ejpam-5492	68	3	appl	appl	PROPN
ejpam-5492	68	4	.	.	PROPN
ejpam-5492	68	5	math	math	PROPN
ejpam-5492	68	6	,	,	PUNCT
ejpam-5492	68	7	17	17	NUM
ejpam-5492	68	8	(	(	PUNCT
ejpam-5492	68	9	4	4	NUM
ejpam-5492	68	10	)	)	PUNCT
ejpam-5492	68	11	(	(	PUNCT
ejpam-5492	68	12	2024	2024	NUM
ejpam-5492	68	13	)	)	PUNCT
ejpam-5492	68	14	,	,	PUNCT
ejpam-5492	68	15	2481	2481	NUM
ejpam-5492	68	16	-	-	SYM
ejpam-5492	68	17	2491	2491	NUM
ejpam-5492	68	18	2484	2484	NUM
ejpam-5492	68	19	∇xz	∇xz	PROPN
ejpam-5492	68	20	=	=	SYM
ejpam-5492	68	21	∇zx	∇zx	PROPN
ejpam-5492	68	22	=	=	SYM
ejpam-5492	68	23	x(ln	x(ln	PROPN
ejpam-5492	68	24	f)z	f)z	NOUN
ejpam-5492	68	25	,	,	PUNCT
ejpam-5492	68	26	(	(	PUNCT
ejpam-5492	68	27	16	16	NUM
ejpam-5492	68	28	)	)	PUNCT
ejpam-5492	68	29	∇zw	∇zw	NOUN
ejpam-5492	68	30	=	=	PUNCT
ejpam-5492	69	1	∇́zw	∇́zw	ADP
ejpam-5492	69	2	−	−	X
ejpam-5492	70	1	g(z	g(z	PROPN
ejpam-5492	70	2	,	,	PUNCT
ejpam-5492	70	3	w	w	NOUN
ejpam-5492	70	4	)	)	PUNCT
ejpam-5492	70	5	∇⃗	∇⃗	X
ejpam-5492	70	6	ln	ln	PROPN
ejpam-5492	70	7	f.	f.	PROPN
ejpam-5492	70	8	(	(	PUNCT
ejpam-5492	70	9	17	17	NUM
ejpam-5492	70	10	)	)	PUNCT
ejpam-5492	70	11	where	where	SCONJ
ejpam-5492	70	12	∇	∇	PROPN
ejpam-5492	70	13	is	be	AUX
ejpam-5492	70	14	the	the	DET
ejpam-5492	70	15	levi	levi	PROPN
ejpam-5492	70	16	-	-	PUNCT
ejpam-5492	70	17	civita	civita	PROPN
ejpam-5492	70	18	connection	connection	NOUN
ejpam-5492	70	19	on	on	ADP
ejpam-5492	70	20	m	m	PROPN
ejpam-5492	70	21	and	and	CCONJ
ejpam-5492	70	22	∇⃗	∇⃗	PUNCT
ejpam-5492	71	1	ln	ln	X
ejpam-5492	71	2	f	f	PROPN
ejpam-5492	71	3	is	be	AUX
ejpam-5492	71	4	the	the	DET
ejpam-5492	71	5	gradient	gradient	NOUN
ejpam-5492	71	6	of	of	ADP
ejpam-5492	71	7	ln	ln	PROPN
ejpam-5492	71	8	f	f	PROPN
ejpam-5492	71	9	which	which	PRON
ejpam-5492	71	10	is	be	AUX
ejpam-5492	71	11	defined	define	VERB
ejpam-5492	71	12	for	for	ADP
ejpam-5492	71	13	any	any	DET
ejpam-5492	71	14	x	x	SYM
ejpam-5492	71	15	∈	∈	PROPN
ejpam-5492	71	16	γ(tm	γ(tm	NOUN
ejpam-5492	71	17	)	)	PUNCT
ejpam-5492	71	18	as	as	ADP
ejpam-5492	71	19	g(∇⃗f	g(∇⃗f	NOUN
ejpam-5492	71	20	,	,	PUNCT
ejpam-5492	71	21	x	x	NOUN
ejpam-5492	71	22	)	)	PUNCT
ejpam-5492	71	23	=	=	SYM
ejpam-5492	71	24	x(f	x(f	PROPN
ejpam-5492	71	25	)	)	PUNCT
ejpam-5492	71	26	.	.	PUNCT
ejpam-5492	72	1	(	(	PUNCT
ejpam-5492	72	2	18	18	NUM
ejpam-5492	72	3	)	)	PUNCT
ejpam-5492	72	4	this	this	PRON
ejpam-5492	72	5	implies	imply	VERB
ejpam-5492	72	6	that	that	SCONJ
ejpam-5492	72	7	∥	∥	PROPN
ejpam-5492	72	8	∇⃗f	∇⃗f	NUM
ejpam-5492	72	9	∥2=	∥2=	NOUN
ejpam-5492	72	10	n∑	n∑	X
ejpam-5492	72	11	i=1	i=1	PROPN
ejpam-5492	72	12	(	(	PUNCT
ejpam-5492	72	13	ei(f	ei(f	NOUN
ejpam-5492	72	14	)	)	PUNCT
ejpam-5492	72	15	)	)	PUNCT
ejpam-5492	72	16	2	2	NUM
ejpam-5492	72	17	(	(	PUNCT
ejpam-5492	72	18	19	19	NUM
ejpam-5492	72	19	)	)	PUNCT
ejpam-5492	72	20	where	where	SCONJ
ejpam-5492	72	21	{	{	PUNCT
ejpam-5492	72	22	e1	e1	NOUN
ejpam-5492	72	23	,	,	PUNCT
ejpam-5492	72	24	·	·	PUNCT
ejpam-5492	72	25	·	·	PUNCT
ejpam-5492	72	26	·	·	PUNCT
ejpam-5492	72	27	,	,	PUNCT
ejpam-5492	72	28	en	en	ADP
ejpam-5492	72	29	}	}	PUNCT
ejpam-5492	72	30	is	be	AUX
ejpam-5492	72	31	a	a	DET
ejpam-5492	72	32	local	local	ADJ
ejpam-5492	72	33	orthonormal	orthonormal	ADJ
ejpam-5492	72	34	frame	frame	NOUN
ejpam-5492	72	35	field	field	NOUN
ejpam-5492	72	36	on	on	ADP
ejpam-5492	72	37	m	m	PROPN
ejpam-5492	72	38	.	.	PUNCT
ejpam-5492	73	1	moreover	moreover	ADV
ejpam-5492	73	2	,	,	PUNCT
ejpam-5492	73	3	on	on	ADP
ejpam-5492	73	4	any	any	DET
ejpam-5492	73	5	warped	warped	ADJ
ejpam-5492	73	6	product	product	NOUN
ejpam-5492	73	7	m	m	NOUN
ejpam-5492	73	8	=	=	SYM
ejpam-5492	73	9	m1	m1	PROPN
ejpam-5492	73	10	×f	×f	PROPN
ejpam-5492	73	11	m2	m2	PROPN
ejpam-5492	73	12	,	,	PUNCT
ejpam-5492	73	13	m1	m1	PROPN
ejpam-5492	73	14	and	and	CCONJ
ejpam-5492	73	15	m2	m2	PROPN
ejpam-5492	73	16	are	be	AUX
ejpam-5492	73	17	totally	totally	ADV
ejpam-5492	73	18	geodesic	geodesic	ADJ
ejpam-5492	73	19	and	and	CCONJ
ejpam-5492	73	20	totally	totally	ADV
ejpam-5492	73	21	umbilical	umbilical	ADJ
ejpam-5492	73	22	submanifolds	submanifold	NOUN
ejpam-5492	73	23	of	of	ADP
ejpam-5492	73	24	m	m	PROPN
ejpam-5492	73	25	[	[	X
ejpam-5492	73	26	1	1	NUM
ejpam-5492	73	27	]	]	PUNCT
ejpam-5492	73	28	.	.	PUNCT
ejpam-5492	74	1	a	a	DET
ejpam-5492	74	2	submanifold	submanifold	NOUN
ejpam-5492	74	3	m	m	VERB
ejpam-5492	74	4	of	of	ADP
ejpam-5492	74	5	a	a	DET
ejpam-5492	74	6	metallic	metallic	ADJ
ejpam-5492	74	7	riemannian	riemannian	NOUN
ejpam-5492	74	8	manifold	manifold	ADJ
ejpam-5492	74	9	m̃	m̃	PROPN
ejpam-5492	74	10	is	be	AUX
ejpam-5492	74	11	called	call	VERB
ejpam-5492	74	12	invariant	invariant	ADJ
ejpam-5492	74	13	under	under	ADP
ejpam-5492	74	14	the	the	DET
ejpam-5492	74	15	metallic	metallic	ADJ
ejpam-5492	74	16	structure	structure	NOUN
ejpam-5492	74	17	j	j	PROPN
ejpam-5492	74	18	if	if	SCONJ
ejpam-5492	74	19	j(txm	j(txm	PROPN
ejpam-5492	74	20	)	)	PUNCT
ejpam-5492	75	1	⊂	⊂	PROPN
ejpam-5492	75	2	txm	txm	PROPN
ejpam-5492	75	3	,	,	PUNCT
ejpam-5492	75	4	for	for	ADP
ejpam-5492	75	5	any	any	DET
ejpam-5492	75	6	x	x	SYM
ejpam-5492	75	7	∈	∈	PROPN
ejpam-5492	75	8	m	m	NOUN
ejpam-5492	75	9	.	.	PUNCT
ejpam-5492	76	1	it	it	PRON
ejpam-5492	76	2	follows	follow	VERB
ejpam-5492	76	3	that	that	SCONJ
ejpam-5492	76	4	the	the	DET
ejpam-5492	76	5	orthogonal	orthogonal	ADJ
ejpam-5492	76	6	complement	complement	NOUN
ejpam-5492	76	7	of	of	ADP
ejpam-5492	76	8	the	the	DET
ejpam-5492	76	9	tangent	tangent	ADJ
ejpam-5492	76	10	space	space	NOUN
ejpam-5492	76	11	of	of	ADP
ejpam-5492	76	12	m	m	PROPN
ejpam-5492	76	13	is	be	AUX
ejpam-5492	76	14	also	also	ADV
ejpam-5492	76	15	invariant	invariant	ADJ
ejpam-5492	76	16	under	under	ADP
ejpam-5492	76	17	j	j	PROPN
ejpam-5492	76	18	,	,	PUNCT
ejpam-5492	76	19	j(txm	j(txm	PROPN
ejpam-5492	76	20	⊥	⊥	NUM
ejpam-5492	76	21	)	)	PUNCT
ejpam-5492	77	1	⊂	⊂	PROPN
ejpam-5492	77	2	txm	txm	PROPN
ejpam-5492	77	3	⊥	⊥	PROPN
ejpam-5492	77	4	,	,	PUNCT
ejpam-5492	77	5	for	for	ADP
ejpam-5492	77	6	any	any	DET
ejpam-5492	77	7	x	x	SYM
ejpam-5492	77	8	∈	∈	PROPN
ejpam-5492	77	9	m	m	NOUN
ejpam-5492	77	10	.	.	PUNCT
ejpam-5492	78	1	this	this	PRON
ejpam-5492	78	2	is	be	AUX
ejpam-5492	78	3	because	because	SCONJ
ejpam-5492	78	4	g(x	g(x	NOUN
ejpam-5492	78	5	,	,	PUNCT
ejpam-5492	78	6	ju	ju	NOUN
ejpam-5492	78	7	)	)	PUNCT
ejpam-5492	78	8	=	=	SYM
ejpam-5492	79	1	g(jx	g(jx	NOUN
ejpam-5492	79	2	,	,	PUNCT
ejpam-5492	79	3	u	u	NOUN
ejpam-5492	79	4	)	)	PUNCT
ejpam-5492	79	5	=	=	SYM
ejpam-5492	79	6	0	0	NUM
ejpam-5492	79	7	,	,	PUNCT
ejpam-5492	79	8	for	for	ADP
ejpam-5492	79	9	any	any	DET
ejpam-5492	79	10	u	u	PROPN
ejpam-5492	79	11	∈	∈	PROPN
ejpam-5492	79	12	γ(tm⊥	γ(tm⊥	PROPN
ejpam-5492	79	13	)	)	PUNCT
ejpam-5492	79	14	and	and	CCONJ
ejpam-5492	79	15	x	x	PUNCT
ejpam-5492	79	16	∈	∈	NOUN
ejpam-5492	79	17	γ(tm	γ(tm	NOUN
ejpam-5492	79	18	)	)	PUNCT
ejpam-5492	79	19	.	.	PUNCT
ejpam-5492	80	1	an	an	DET
ejpam-5492	80	2	anti	anti	ADJ
ejpam-5492	80	3	-	-	ADJ
ejpam-5492	80	4	invariant	invariant	ADJ
ejpam-5492	80	5	submanifold	submanifold	NOUN
ejpam-5492	80	6	m	m	NOUN
ejpam-5492	80	7	of	of	ADP
ejpam-5492	80	8	m̃	m̃	PROPN
ejpam-5492	80	9	is	be	AUX
ejpam-5492	80	10	a	a	DET
ejpam-5492	80	11	submanifold	submanifold	NOUN
ejpam-5492	80	12	such	such	ADJ
ejpam-5492	80	13	that	that	DET
ejpam-5492	80	14	j(txm	j(txm	NOUN
ejpam-5492	80	15	)	)	PUNCT
ejpam-5492	81	1	⊂	⊂	PROPN
ejpam-5492	81	2	txm	txm	PROPN
ejpam-5492	81	3	⊥	⊥	PROPN
ejpam-5492	81	4	,	,	PUNCT
ejpam-5492	81	5	for	for	ADP
ejpam-5492	81	6	any	any	DET
ejpam-5492	81	7	x	x	SYM
ejpam-5492	81	8	∈	∈	NOUN
ejpam-5492	81	9	m	m	VERB
ejpam-5492	81	10	[	[	X
ejpam-5492	81	11	4	4	NUM
ejpam-5492	81	12	]	]	PUNCT
ejpam-5492	81	13	.	.	PUNCT
ejpam-5492	82	1	the	the	DET
ejpam-5492	82	2	existence	existence	NOUN
ejpam-5492	82	3	of	of	ADP
ejpam-5492	82	4	warped	warped	ADJ
ejpam-5492	82	5	product	product	NOUN
ejpam-5492	82	6	cr	cr	PROPN
ejpam-5492	82	7	submanifolds	submanifold	NOUN
ejpam-5492	82	8	in	in	ADP
ejpam-5492	82	9	a	a	DET
ejpam-5492	82	10	locally	locally	ADV
ejpam-5492	82	11	metallic	metallic	ADJ
ejpam-5492	82	12	riemannian	riemannian	NOUN
ejpam-5492	82	13	manifold	manifold	ADJ
ejpam-5492	82	14	m̃	m̃	PROPN
ejpam-5492	82	15	has	have	AUX
ejpam-5492	82	16	been	be	AUX
ejpam-5492	82	17	studied	study	VERB
ejpam-5492	82	18	in[14	in[14	PROPN
ejpam-5492	82	19	]	]	PUNCT
ejpam-5492	82	20	,	,	PUNCT
ejpam-5492	82	21	and	and	CCONJ
ejpam-5492	82	22	proved	prove	VERB
ejpam-5492	82	23	the	the	DET
ejpam-5492	82	24	following	following	NOUN
ejpam-5492	82	25	:	:	PUNCT
ejpam-5492	82	26	theorem	theorem	NOUN
ejpam-5492	82	27	1	1	NUM
ejpam-5492	82	28	.	.	PUNCT
ejpam-5492	83	1	(	(	PUNCT
ejpam-5492	83	2	[	[	X
ejpam-5492	83	3	14	14	NUM
ejpam-5492	83	4	]	]	PUNCT
ejpam-5492	83	5	)	)	PUNCT
ejpam-5492	83	6	let	let	VERB
ejpam-5492	83	7	m	m	PROPN
ejpam-5492	83	8	=	=	PROPN
ejpam-5492	83	9	mt	mt	PROPN
ejpam-5492	83	10	×f	×f	PROPN
ejpam-5492	83	11	m⊥	m⊥	NOUN
ejpam-5492	83	12	be	be	VERB
ejpam-5492	83	13	a	a	DET
ejpam-5492	83	14	warped	warped	ADJ
ejpam-5492	83	15	product	product	NOUN
ejpam-5492	83	16	cr	cr	NOUN
ejpam-5492	83	17	-	-	PUNCT
ejpam-5492	83	18	submanifold	submanifold	NOUN
ejpam-5492	83	19	in	in	ADP
ejpam-5492	83	20	a	a	DET
ejpam-5492	83	21	locally	locally	ADV
ejpam-5492	83	22	metallic	metallic	ADJ
ejpam-5492	83	23	riemannian	riemannian	NOUN
ejpam-5492	83	24	manifold	manifold	NOUN
ejpam-5492	83	25	(	(	PUNCT
ejpam-5492	83	26	m̃	m̃	PROPN
ejpam-5492	83	27	,	,	PUNCT
ejpam-5492	83	28	g̃	g̃	PROPN
ejpam-5492	83	29	,	,	PUNCT
ejpam-5492	83	30	j	j	PROPN
ejpam-5492	83	31	)	)	PUNCT
ejpam-5492	83	32	,	,	PUNCT
ejpam-5492	83	33	where	where	SCONJ
ejpam-5492	83	34	mt	mt	PROPN
ejpam-5492	83	35	and	and	CCONJ
ejpam-5492	83	36	m⊥	m⊥	NOUN
ejpam-5492	83	37	are	be	AUX
ejpam-5492	83	38	invariant	invariant	ADJ
ejpam-5492	83	39	and	and	CCONJ
ejpam-5492	83	40	antiinvariant	antiinvariant	ADJ
ejpam-5492	83	41	submanifolds	submanifold	NOUN
ejpam-5492	83	42	of	of	ADP
ejpam-5492	83	43	m̃	m̃	PROPN
ejpam-5492	83	44	,	,	PUNCT
ejpam-5492	83	45	respectively	respectively	ADV
ejpam-5492	83	46	.	.	PUNCT
ejpam-5492	84	1	then	then	ADV
ejpam-5492	84	2	,	,	PUNCT
ejpam-5492	84	3	m	m	PROPN
ejpam-5492	84	4	=	=	SYM
ejpam-5492	84	5	mt	mt	PROPN
ejpam-5492	84	6	×f	×f	PROPN
ejpam-5492	84	7	m⊥	m⊥	NOUN
ejpam-5492	84	8	is	be	AUX
ejpam-5492	84	9	a	a	DET
ejpam-5492	84	10	non	non	ADJ
ejpam-5492	84	11	-	-	ADJ
ejpam-5492	84	12	proper	proper	ADJ
ejpam-5492	84	13	warped	warped	ADJ
ejpam-5492	84	14	product	product	NOUN
ejpam-5492	84	15	submanifold	submanifold	VERB
ejpam-5492	84	16	in	in	ADP
ejpam-5492	84	17	m̃	m̃	PROPN
ejpam-5492	84	18	,	,	PUNCT
ejpam-5492	84	19	that	that	ADV
ejpam-5492	84	20	is	is	ADV
ejpam-5492	84	21	,	,	PUNCT
ejpam-5492	84	22	the	the	DET
ejpam-5492	84	23	warping	warp	VERB
ejpam-5492	84	24	function	function	NOUN
ejpam-5492	84	25	f	f	PROPN
ejpam-5492	84	26	is	be	AUX
ejpam-5492	84	27	constant	constant	ADJ
ejpam-5492	84	28	on	on	ADP
ejpam-5492	84	29	mt	mt	PROPN
ejpam-5492	84	30	.	.	PUNCT
ejpam-5492	85	1	3	3	X
ejpam-5492	85	2	.	.	X
ejpam-5492	85	3	basic	basic	ADJ
ejpam-5492	85	4	lemmas	lemma	NOUN
ejpam-5492	85	5	and	and	CCONJ
ejpam-5492	85	6	examples	example	NOUN
ejpam-5492	85	7	in	in	ADP
ejpam-5492	85	8	this	this	DET
ejpam-5492	85	9	section	section	NOUN
ejpam-5492	85	10	,	,	PUNCT
ejpam-5492	85	11	we	we	PRON
ejpam-5492	85	12	consider	consider	VERB
ejpam-5492	85	13	warped	warped	ADJ
ejpam-5492	85	14	product	product	NOUN
ejpam-5492	85	15	cr	cr	NOUN
ejpam-5492	85	16	-	-	PUNCT
ejpam-5492	85	17	submanifolds	submanifold	NOUN
ejpam-5492	85	18	in	in	ADP
ejpam-5492	85	19	the	the	DET
ejpam-5492	85	20	form	form	NOUN
ejpam-5492	85	21	m	m	NOUN
ejpam-5492	85	22	=	=	PUNCT
ejpam-5492	85	23	m⊥×f	m⊥×f	PROPN
ejpam-5492	85	24	mt	mt	PROPN
ejpam-5492	85	25	such	such	ADJ
ejpam-5492	85	26	that	that	SCONJ
ejpam-5492	85	27	m⊥	m⊥	NOUN
ejpam-5492	85	28	is	be	AUX
ejpam-5492	85	29	an	an	DET
ejpam-5492	85	30	anti	anti	ADJ
ejpam-5492	85	31	-	-	ADJ
ejpam-5492	85	32	invariant	invariant	ADJ
ejpam-5492	85	33	submanifold	submanifold	NOUN
ejpam-5492	85	34	and	and	CCONJ
ejpam-5492	85	35	mt	mt	PROPN
ejpam-5492	85	36	is	be	AUX
ejpam-5492	85	37	an	an	DET
ejpam-5492	85	38	invariant	invariant	ADJ
ejpam-5492	85	39	submanifold	submanifold	NOUN
ejpam-5492	85	40	of	of	ADP
ejpam-5492	85	41	a	a	DET
ejpam-5492	85	42	locally	locally	ADV
ejpam-5492	85	43	metallic	metallic	ADJ
ejpam-5492	85	44	riemannian	riemannian	ADJ
ejpam-5492	85	45	manifold	manifold	ADJ
ejpam-5492	85	46	m̃	m̃	PROPN
ejpam-5492	85	47	.	.	PUNCT
ejpam-5492	86	1	lemma	lemma	PROPN
ejpam-5492	86	2	1	1	X
ejpam-5492	86	3	.	.	PUNCT
ejpam-5492	87	1	let	let	VERB
ejpam-5492	87	2	m	m	PROPN
ejpam-5492	87	3	=	=	VERB
ejpam-5492	87	4	m⊥×f	m⊥×f	PROPN
ejpam-5492	87	5	mt	mt	PROPN
ejpam-5492	87	6	be	be	AUX
ejpam-5492	87	7	a	a	DET
ejpam-5492	87	8	warped	warped	ADJ
ejpam-5492	87	9	product	product	NOUN
ejpam-5492	87	10	cr	cr	NOUN
ejpam-5492	87	11	-	-	PUNCT
ejpam-5492	87	12	submanifold	submanifold	NOUN
ejpam-5492	87	13	in	in	ADP
ejpam-5492	87	14	a	a	DET
ejpam-5492	87	15	locally	locally	ADV
ejpam-5492	87	16	metallic	metallic	ADJ
ejpam-5492	87	17	riemannian	riemannian	ADJ
ejpam-5492	87	18	manifold	manifold	ADJ
ejpam-5492	87	19	m̃	m̃	PROPN
ejpam-5492	87	20	,	,	PUNCT
ejpam-5492	87	21	then	then	ADV
ejpam-5492	87	22	for	for	ADP
ejpam-5492	87	23	any	any	DET
ejpam-5492	87	24	x	x	NOUN
ejpam-5492	87	25	,	,	PUNCT
ejpam-5492	87	26	y	y	PROPN
ejpam-5492	87	27	∈	∈	PROPN
ejpam-5492	87	28	γ(tmt	γ(tmt	PROPN
ejpam-5492	87	29	)	)	PUNCT
ejpam-5492	87	30	and	and	CCONJ
ejpam-5492	87	31	z	z	NOUN
ejpam-5492	87	32	,	,	PUNCT
ejpam-5492	87	33	w	w	PROPN
ejpam-5492	87	34	∈	∈	PROPN
ejpam-5492	87	35	γ(tm⊥	γ(tm⊥	PROPN
ejpam-5492	87	36	)	)	PUNCT
ejpam-5492	87	37	,	,	PUNCT
ejpam-5492	87	38	we	we	PRON
ejpam-5492	87	39	have	have	VERB
ejpam-5492	87	40	g(h	g(h	PROPN
ejpam-5492	87	41	(	(	PUNCT
ejpam-5492	87	42	x	x	X
ejpam-5492	87	43	,	,	PUNCT
ejpam-5492	87	44	y	y	PROPN
ejpam-5492	87	45	)	)	PUNCT
ejpam-5492	87	46	,	,	PUNCT
ejpam-5492	87	47	jz	jz	PROPN
ejpam-5492	87	48	)	)	PUNCT
ejpam-5492	88	1	=	=	SYM
ejpam-5492	88	2	−z	−z	NOUN
ejpam-5492	88	3	(	(	PUNCT
ejpam-5492	88	4	ln	ln	PROPN
ejpam-5492	88	5	f	f	NOUN
ejpam-5492	88	6	)	)	PUNCT
ejpam-5492	88	7	g(x	g(x	PROPN
ejpam-5492	88	8	,	,	PUNCT
ejpam-5492	88	9	jy	jy	PROPN
ejpam-5492	88	10	)	)	PUNCT
ejpam-5492	88	11	,	,	PUNCT
ejpam-5492	88	12	(	(	PUNCT
ejpam-5492	88	13	20	20	NUM
ejpam-5492	88	14	)	)	PUNCT
ejpam-5492	88	15	g(h	g(h	NUM
ejpam-5492	88	16	(	(	PUNCT
ejpam-5492	88	17	x	x	X
ejpam-5492	88	18	,	,	PUNCT
ejpam-5492	88	19	z	z	NOUN
ejpam-5492	88	20	)	)	PUNCT
ejpam-5492	88	21	,	,	PUNCT
ejpam-5492	88	22	jw	jw	PROPN
ejpam-5492	88	23	)	)	PUNCT
ejpam-5492	89	1	=	=	PUNCT
ejpam-5492	89	2	0	0	NUM
ejpam-5492	89	3	,	,	PUNCT
ejpam-5492	89	4	(	(	PUNCT
ejpam-5492	89	5	21	21	NUM
ejpam-5492	89	6	)	)	PUNCT
ejpam-5492	90	1	g(h	g(h	PROPN
ejpam-5492	90	2	(	(	PUNCT
ejpam-5492	90	3	jx	jx	PROPN
ejpam-5492	90	4	,	,	PUNCT
ejpam-5492	90	5	y	y	PROPN
ejpam-5492	90	6	)	)	PUNCT
ejpam-5492	90	7	,	,	PUNCT
ejpam-5492	90	8	jz	jz	PROPN
ejpam-5492	90	9	)	)	PUNCT
ejpam-5492	90	10	=	=	SYM
ejpam-5492	90	11	−pz	−pz	PROPN
ejpam-5492	90	12	(	(	PUNCT
ejpam-5492	90	13	ln	ln	NOUN
ejpam-5492	90	14	f	f	NOUN
ejpam-5492	90	15	)	)	PUNCT
ejpam-5492	90	16	g(jx	g(jx	NOUN
ejpam-5492	90	17	,	,	PUNCT
ejpam-5492	90	18	y	y	PROPN
ejpam-5492	90	19	)	)	PUNCT
ejpam-5492	91	1	+	+	CCONJ
ejpam-5492	91	2	qz	qz	NOUN
ejpam-5492	91	3	(	(	PUNCT
ejpam-5492	91	4	ln	ln	NOUN
ejpam-5492	91	5	f	f	NOUN
ejpam-5492	91	6	)	)	PUNCT
ejpam-5492	91	7	g(x	g(x	PROPN
ejpam-5492	91	8	,	,	PUNCT
ejpam-5492	91	9	y	y	PROPN
ejpam-5492	91	10	)	)	PUNCT
ejpam-5492	91	11	.	.	PUNCT
ejpam-5492	92	1	(	(	PUNCT
ejpam-5492	92	2	22	22	X
ejpam-5492	92	3	)	)	PUNCT
ejpam-5492	92	4	l.	l.	PROPN
ejpam-5492	92	5	alqahtani	alqahtani	PROPN
ejpam-5492	92	6	,	,	PUNCT
ejpam-5492	92	7	e.	e.	PROPN
ejpam-5492	92	8	al	al	PROPN
ejpam-5492	92	9	-	-	PROPN
ejpam-5492	92	10	husainy	husainy	PROPN
ejpam-5492	92	11	/	/	SYM
ejpam-5492	92	12	eur	eur	PROPN
ejpam-5492	92	13	.	.	PUNCT
ejpam-5492	93	1	j.	j.	PROPN
ejpam-5492	93	2	pure	pure	PROPN
ejpam-5492	93	3	appl	appl	PROPN
ejpam-5492	93	4	.	.	PROPN
ejpam-5492	93	5	math	math	PROPN
ejpam-5492	93	6	,	,	PUNCT
ejpam-5492	93	7	17	17	NUM
ejpam-5492	93	8	(	(	PUNCT
ejpam-5492	93	9	4	4	NUM
ejpam-5492	93	10	)	)	PUNCT
ejpam-5492	93	11	(	(	PUNCT
ejpam-5492	93	12	2024	2024	NUM
ejpam-5492	93	13	)	)	PUNCT
ejpam-5492	93	14	,	,	PUNCT
ejpam-5492	93	15	2481	2481	NUM
ejpam-5492	93	16	-	-	SYM
ejpam-5492	93	17	2491	2491	NUM
ejpam-5492	93	18	2485	2485	NUM
ejpam-5492	93	19	proof	proof	NOUN
ejpam-5492	93	20	.	.	PUNCT
ejpam-5492	94	1	since	since	SCONJ
ejpam-5492	94	2	m̃	m̃	PROPN
ejpam-5492	94	3	is	be	AUX
ejpam-5492	94	4	a	a	DET
ejpam-5492	94	5	locally	locally	ADV
ejpam-5492	94	6	metallic	metallic	ADJ
ejpam-5492	94	7	riemannian	riemannian	ADJ
ejpam-5492	94	8	manifold	manifold	NOUN
ejpam-5492	94	9	,	,	PUNCT
ejpam-5492	94	10	then	then	ADV
ejpam-5492	94	11	by	by	ADP
ejpam-5492	94	12	using	use	VERB
ejpam-5492	94	13	(	(	PUNCT
ejpam-5492	94	14	3	3	NUM
ejpam-5492	94	15	)	)	PUNCT
ejpam-5492	94	16	,	,	PUNCT
ejpam-5492	94	17	for	for	ADP
ejpam-5492	94	18	any	any	DET
ejpam-5492	94	19	z	z	PROPN
ejpam-5492	94	20	∈	∈	PROPN
ejpam-5492	94	21	γ(tm⊥	γ(tm⊥	PROPN
ejpam-5492	94	22	)	)	PUNCT
ejpam-5492	94	23	and	and	CCONJ
ejpam-5492	94	24	x	x	X
ejpam-5492	94	25	,	,	PUNCT
ejpam-5492	94	26	y	y	PROPN
ejpam-5492	94	27	∈	∈	PROPN
ejpam-5492	94	28	γ(tmt	γ(tmt	PROPN
ejpam-5492	94	29	)	)	PUNCT
ejpam-5492	94	30	,	,	PUNCT
ejpam-5492	94	31	we	we	PRON
ejpam-5492	94	32	have	have	VERB
ejpam-5492	94	33	g(h	g(h	PROPN
ejpam-5492	94	34	(	(	PUNCT
ejpam-5492	94	35	x	x	X
ejpam-5492	94	36	,	,	PUNCT
ejpam-5492	94	37	y	y	PROPN
ejpam-5492	94	38	)	)	PUNCT
ejpam-5492	94	39	,	,	PUNCT
ejpam-5492	94	40	jz	jz	PROPN
ejpam-5492	94	41	)	)	PUNCT
ejpam-5492	94	42	=	=	SYM
ejpam-5492	95	1	1	1	NUM
ejpam-5492	95	2	p	p	PROPN
ejpam-5492	95	3	g(∇̃xjy	g(∇̃xjy	PROPN
ejpam-5492	95	4	,	,	PUNCT
ejpam-5492	95	5	jz)−	jz)−	PROPN
ejpam-5492	95	6	q	q	PROPN
ejpam-5492	95	7	p	p	NOUN
ejpam-5492	95	8	g(∇̃xy	g(∇̃xy	NOUN
ejpam-5492	95	9	,	,	PUNCT
ejpam-5492	95	10	z	z	NOUN
ejpam-5492	95	11	)	)	PUNCT
ejpam-5492	95	12	(	(	PUNCT
ejpam-5492	95	13	23	23	NUM
ejpam-5492	95	14	)	)	PUNCT
ejpam-5492	95	15	also	also	ADV
ejpam-5492	95	16	,	,	PUNCT
ejpam-5492	95	17	by	by	ADP
ejpam-5492	95	18	gauss	gauss	ADJ
ejpam-5492	95	19	formula	formula	NOUN
ejpam-5492	95	20	(	(	PUNCT
ejpam-5492	95	21	11	11	NUM
ejpam-5492	95	22	)	)	PUNCT
ejpam-5492	95	23	and	and	CCONJ
ejpam-5492	95	24	(	(	PUNCT
ejpam-5492	95	25	16	16	NUM
ejpam-5492	95	26	)	)	PUNCT
ejpam-5492	95	27	,	,	PUNCT
ejpam-5492	95	28	we	we	PRON
ejpam-5492	95	29	have	have	VERB
ejpam-5492	95	30	g(h	g(h	PROPN
ejpam-5492	95	31	(	(	PUNCT
ejpam-5492	95	32	x	x	X
ejpam-5492	95	33	,	,	PUNCT
ejpam-5492	95	34	y	y	PROPN
ejpam-5492	95	35	)	)	PUNCT
ejpam-5492	95	36	,	,	PUNCT
ejpam-5492	95	37	jz	jz	PROPN
ejpam-5492	95	38	)	)	PUNCT
ejpam-5492	95	39	=	=	PUNCT
ejpam-5492	96	1	1	1	NUM
ejpam-5492	96	2	p	p	NOUN
ejpam-5492	96	3	g(h	g(h	PROPN
ejpam-5492	96	4	(	(	PUNCT
ejpam-5492	96	5	x	x	NOUN
ejpam-5492	96	6	,	,	PUNCT
ejpam-5492	96	7	jy	jy	PROPN
ejpam-5492	96	8	)	)	PUNCT
ejpam-5492	96	9	,	,	PUNCT
ejpam-5492	96	10	jz	jz	PROPN
ejpam-5492	96	11	)	)	PUNCT
ejpam-5492	96	12	+	+	CCONJ
ejpam-5492	96	13	q	q	PROPN
ejpam-5492	96	14	p	p	NOUN
ejpam-5492	96	15	z	z	NOUN
ejpam-5492	96	16	(	(	PUNCT
ejpam-5492	96	17	ln	ln	NOUN
ejpam-5492	96	18	f	f	NOUN
ejpam-5492	96	19	)	)	PUNCT
ejpam-5492	96	20	g(x	g(x	PROPN
ejpam-5492	96	21	,	,	PUNCT
ejpam-5492	96	22	y	y	PROPN
ejpam-5492	96	23	)	)	PUNCT
ejpam-5492	96	24	.	.	PUNCT
ejpam-5492	97	1	(	(	PUNCT
ejpam-5492	97	2	24	24	NUM
ejpam-5492	97	3	)	)	PUNCT
ejpam-5492	97	4	by	by	ADP
ejpam-5492	97	5	interchanging	interchange	VERB
ejpam-5492	97	6	y	y	PROPN
ejpam-5492	97	7	by	by	ADP
ejpam-5492	97	8	jy	jy	PROPN
ejpam-5492	97	9	in	in	ADP
ejpam-5492	97	10	(	(	PUNCT
ejpam-5492	97	11	24	24	NUM
ejpam-5492	97	12	)	)	PUNCT
ejpam-5492	97	13	,	,	PUNCT
ejpam-5492	97	14	we	we	PRON
ejpam-5492	97	15	have	have	VERB
ejpam-5492	97	16	g(h	g(h	PROPN
ejpam-5492	97	17	(	(	PUNCT
ejpam-5492	97	18	x	x	X
ejpam-5492	97	19	,	,	PUNCT
ejpam-5492	97	20	jy	jy	PROPN
ejpam-5492	97	21	)	)	PUNCT
ejpam-5492	97	22	,	,	PUNCT
ejpam-5492	97	23	jz	jz	PROPN
ejpam-5492	97	24	)	)	PUNCT
ejpam-5492	97	25	=	=	PUNCT
ejpam-5492	98	1	g(h	g(h	PROPN
ejpam-5492	98	2	(	(	PUNCT
ejpam-5492	98	3	x	x	NOUN
ejpam-5492	98	4	,	,	PUNCT
ejpam-5492	98	5	jy	jy	PROPN
ejpam-5492	98	6	)	)	PUNCT
ejpam-5492	98	7	,	,	PUNCT
ejpam-5492	98	8	jz	jz	PROPN
ejpam-5492	98	9	)	)	PUNCT
ejpam-5492	98	10	+	+	CCONJ
ejpam-5492	98	11	q	q	PROPN
ejpam-5492	98	12	p	p	X
ejpam-5492	98	13	g(h	g(h	PROPN
ejpam-5492	98	14	(	(	PUNCT
ejpam-5492	98	15	x	x	X
ejpam-5492	98	16	,	,	PUNCT
ejpam-5492	98	17	y	y	PROPN
ejpam-5492	98	18	)	)	PUNCT
ejpam-5492	98	19	,	,	PUNCT
ejpam-5492	98	20	jz)−	jz)−	PROPN
ejpam-5492	98	21	q	q	PROPN
ejpam-5492	98	22	p	p	PROPN
ejpam-5492	98	23	z	z	PROPN
ejpam-5492	98	24	(	(	PUNCT
ejpam-5492	98	25	ln	ln	NOUN
ejpam-5492	98	26	f	f	NOUN
ejpam-5492	98	27	)	)	PUNCT
ejpam-5492	98	28	g(x	g(x	PROPN
ejpam-5492	98	29	,	,	PUNCT
ejpam-5492	98	30	jy	jy	PROPN
ejpam-5492	98	31	)	)	PUNCT
ejpam-5492	98	32	.	.	PUNCT
ejpam-5492	99	1	(	(	PUNCT
ejpam-5492	99	2	25	25	NUM
ejpam-5492	99	3	)	)	PUNCT
ejpam-5492	99	4	then	then	ADV
ejpam-5492	99	5	,	,	PUNCT
ejpam-5492	99	6	we	we	PRON
ejpam-5492	99	7	get	get	VERB
ejpam-5492	99	8	g(h	g(h	PROPN
ejpam-5492	99	9	(	(	PUNCT
ejpam-5492	99	10	x	x	X
ejpam-5492	99	11	,	,	PUNCT
ejpam-5492	99	12	y	y	PROPN
ejpam-5492	99	13	)	)	PUNCT
ejpam-5492	99	14	,	,	PUNCT
ejpam-5492	99	15	jz	jz	PROPN
ejpam-5492	99	16	)	)	PUNCT
ejpam-5492	100	1	=	=	SYM
ejpam-5492	100	2	−z	−z	NOUN
ejpam-5492	100	3	(	(	PUNCT
ejpam-5492	100	4	ln	ln	PROPN
ejpam-5492	100	5	f	f	NOUN
ejpam-5492	100	6	)	)	PUNCT
ejpam-5492	100	7	g(x	g(x	PROPN
ejpam-5492	100	8	,	,	PUNCT
ejpam-5492	100	9	jy	jy	PROPN
ejpam-5492	100	10	)	)	PUNCT
ejpam-5492	100	11	,	,	PUNCT
ejpam-5492	100	12	(	(	PUNCT
ejpam-5492	100	13	26	26	NUM
ejpam-5492	100	14	)	)	PUNCT
ejpam-5492	100	15	which	which	PRON
ejpam-5492	100	16	proves	prove	VERB
ejpam-5492	100	17	(	(	PUNCT
ejpam-5492	100	18	20	20	NUM
ejpam-5492	100	19	)	)	PUNCT
ejpam-5492	100	20	.	.	PUNCT
ejpam-5492	101	1	now	now	ADV
ejpam-5492	101	2	,	,	PUNCT
ejpam-5492	101	3	by	by	ADP
ejpam-5492	101	4	using	use	VERB
ejpam-5492	101	5	the	the	DET
ejpam-5492	101	6	same	same	ADJ
ejpam-5492	101	7	strategies	strategy	NOUN
ejpam-5492	101	8	,	,	PUNCT
ejpam-5492	101	9	g(h	g(h	NUM
ejpam-5492	101	10	(	(	PUNCT
ejpam-5492	101	11	x	x	X
ejpam-5492	101	12	,	,	PUNCT
ejpam-5492	101	13	z	z	NOUN
ejpam-5492	101	14	)	)	PUNCT
ejpam-5492	101	15	,	,	PUNCT
ejpam-5492	101	16	jw	jw	PROPN
ejpam-5492	101	17	)	)	PUNCT
ejpam-5492	101	18	=	=	PUNCT
ejpam-5492	101	19	1	1	NUM
ejpam-5492	101	20	p	p	NOUN
ejpam-5492	101	21	g(∇̃zjx	g(∇̃zjx	PROPN
ejpam-5492	101	22	,	,	PUNCT
ejpam-5492	101	23	jw	jw	PROPN
ejpam-5492	101	24	)	)	PUNCT
ejpam-5492	101	25	−	−	PROPN
ejpam-5492	102	1	q	q	PROPN
ejpam-5492	102	2	p	p	NOUN
ejpam-5492	102	3	g(∇̃zx	g(∇̃zx	NOUN
ejpam-5492	102	4	,	,	PUNCT
ejpam-5492	102	5	w	w	NOUN
ejpam-5492	102	6	)	)	PUNCT
ejpam-5492	102	7	.	.	PUNCT
ejpam-5492	103	1	(	(	PUNCT
ejpam-5492	103	2	27	27	NUM
ejpam-5492	103	3	)	)	PUNCT
ejpam-5492	103	4	on	on	ADP
ejpam-5492	103	5	the	the	DET
ejpam-5492	103	6	other	other	ADJ
ejpam-5492	103	7	hand	hand	NOUN
ejpam-5492	103	8	,	,	PUNCT
ejpam-5492	103	9	we	we	PRON
ejpam-5492	103	10	derive	derive	VERB
ejpam-5492	103	11	g(h	g(h	PROPN
ejpam-5492	103	12	(	(	PUNCT
ejpam-5492	103	13	x	x	X
ejpam-5492	103	14	,	,	PUNCT
ejpam-5492	103	15	z	z	NOUN
ejpam-5492	103	16	)	)	PUNCT
ejpam-5492	103	17	,	,	PUNCT
ejpam-5492	103	18	jw	jw	PROPN
ejpam-5492	103	19	)	)	PUNCT
ejpam-5492	104	1	=	=	PUNCT
ejpam-5492	105	1	1	1	NUM
ejpam-5492	105	2	p	p	X
ejpam-5492	105	3	g(h	g(h	PROPN
ejpam-5492	105	4	(	(	PUNCT
ejpam-5492	105	5	z	z	PROPN
ejpam-5492	105	6	,	,	PUNCT
ejpam-5492	105	7	jx	jx	PROPN
ejpam-5492	105	8	)	)	PUNCT
ejpam-5492	105	9	,	,	PUNCT
ejpam-5492	105	10	jw	jw	PROPN
ejpam-5492	105	11	)	)	PUNCT
ejpam-5492	105	12	.	.	PUNCT
ejpam-5492	106	1	(	(	PUNCT
ejpam-5492	106	2	28	28	NUM
ejpam-5492	106	3	)	)	PUNCT
ejpam-5492	106	4	by	by	ADP
ejpam-5492	106	5	replacing	replace	VERB
ejpam-5492	106	6	x	x	PUNCT
ejpam-5492	106	7	with	with	ADP
ejpam-5492	106	8	jx	jx	PROPN
ejpam-5492	106	9	,	,	PUNCT
ejpam-5492	106	10	and	and	CCONJ
ejpam-5492	106	11	applying	apply	VERB
ejpam-5492	106	12	(	(	PUNCT
ejpam-5492	106	13	3	3	NUM
ejpam-5492	106	14	)	)	PUNCT
ejpam-5492	106	15	,	,	PUNCT
ejpam-5492	106	16	we	we	PRON
ejpam-5492	106	17	get	get	VERB
ejpam-5492	106	18	g(h	g(h	PROPN
ejpam-5492	106	19	(	(	PUNCT
ejpam-5492	106	20	jx	jx	PROPN
ejpam-5492	106	21	,	,	PUNCT
ejpam-5492	106	22	z	z	PROPN
ejpam-5492	106	23	)	)	PUNCT
ejpam-5492	106	24	,	,	PUNCT
ejpam-5492	106	25	jw	jw	PROPN
ejpam-5492	106	26	)	)	PUNCT
ejpam-5492	107	1	=	=	PUNCT
ejpam-5492	108	1	g(h	g(h	PROPN
ejpam-5492	108	2	(	(	PUNCT
ejpam-5492	108	3	z	z	NOUN
ejpam-5492	108	4	,	,	PUNCT
ejpam-5492	108	5	jx	jx	PROPN
ejpam-5492	108	6	)	)	PUNCT
ejpam-5492	108	7	,	,	PUNCT
ejpam-5492	108	8	jw	jw	PROPN
ejpam-5492	108	9	)	)	PUNCT
ejpam-5492	109	1	+	+	CCONJ
ejpam-5492	110	1	q	q	X
ejpam-5492	110	2	p	p	X
ejpam-5492	110	3	g(h	g(h	PROPN
ejpam-5492	110	4	(	(	PUNCT
ejpam-5492	110	5	z	z	NOUN
ejpam-5492	110	6	,	,	PUNCT
ejpam-5492	110	7	x	x	X
ejpam-5492	110	8	)	)	PUNCT
ejpam-5492	110	9	,	,	PUNCT
ejpam-5492	110	10	jw	jw	PROPN
ejpam-5492	110	11	)	)	PUNCT
ejpam-5492	110	12	.	.	PUNCT
ejpam-5492	111	1	(	(	PUNCT
ejpam-5492	111	2	29	29	NUM
ejpam-5492	111	3	)	)	PUNCT
ejpam-5492	111	4	this	this	PRON
ejpam-5492	111	5	implies	imply	VERB
ejpam-5492	111	6	that	that	SCONJ
ejpam-5492	112	1	g(h	g(h	PROPN
ejpam-5492	112	2	(	(	PUNCT
ejpam-5492	112	3	x	x	X
ejpam-5492	112	4	,	,	PUNCT
ejpam-5492	112	5	z	z	NOUN
ejpam-5492	112	6	)	)	PUNCT
ejpam-5492	112	7	,	,	PUNCT
ejpam-5492	112	8	jw	jw	PROPN
ejpam-5492	112	9	)	)	PUNCT
ejpam-5492	112	10	=	=	PUNCT
ejpam-5492	113	1	0	0	X
ejpam-5492	113	2	.	.	PUNCT
ejpam-5492	113	3	by	by	ADP
ejpam-5492	113	4	rewriting	rewrite	VERB
ejpam-5492	113	5	the	the	DET
ejpam-5492	113	6	equation	equation	NOUN
ejpam-5492	113	7	(	(	PUNCT
ejpam-5492	113	8	20	20	NUM
ejpam-5492	113	9	)	)	PUNCT
ejpam-5492	113	10	with	with	ADP
ejpam-5492	113	11	jx	jx	PROPN
ejpam-5492	113	12	instead	instead	ADV
ejpam-5492	113	13	of	of	ADP
ejpam-5492	113	14	x	x	PRON
ejpam-5492	113	15	,	,	PUNCT
ejpam-5492	113	16	and	and	CCONJ
ejpam-5492	113	17	by	by	ADP
ejpam-5492	113	18	using	use	VERB
ejpam-5492	113	19	(	(	PUNCT
ejpam-5492	113	20	3	3	NUM
ejpam-5492	113	21	)	)	PUNCT
ejpam-5492	113	22	,	,	PUNCT
ejpam-5492	113	23	we	we	PRON
ejpam-5492	113	24	can	can	AUX
ejpam-5492	113	25	see	see	VERB
ejpam-5492	113	26	that	that	PRON
ejpam-5492	113	27	(	(	PUNCT
ejpam-5492	113	28	22	22	NUM
ejpam-5492	113	29	)	)	PUNCT
ejpam-5492	113	30	holds	hold	VERB
ejpam-5492	113	31	.	.	PUNCT
ejpam-5492	114	1	this	this	PRON
ejpam-5492	114	2	completes	complete	VERB
ejpam-5492	114	3	the	the	DET
ejpam-5492	114	4	proof	proof	NOUN
ejpam-5492	114	5	of	of	ADP
ejpam-5492	114	6	the	the	DET
ejpam-5492	114	7	lemma	lemma	PROPN
ejpam-5492	114	8	.	.	PUNCT
ejpam-5492	115	1	corollary	corollary	ADJ
ejpam-5492	115	2	1	1	NUM
ejpam-5492	115	3	.	.	PUNCT
ejpam-5492	116	1	let	let	VERB
ejpam-5492	116	2	m	m	VERB
ejpam-5492	116	3	=	=	VERB
ejpam-5492	116	4	m⊥	m⊥	VERB
ejpam-5492	116	5	×f	×f	PROPN
ejpam-5492	116	6	mt	mt	PROPN
ejpam-5492	116	7	be	be	AUX
ejpam-5492	116	8	a	a	DET
ejpam-5492	116	9	warped	warped	ADJ
ejpam-5492	116	10	product	product	NOUN
ejpam-5492	116	11	cr	cr	NOUN
ejpam-5492	116	12	-	-	PUNCT
ejpam-5492	116	13	submanifold	submanifold	NOUN
ejpam-5492	116	14	m	m	VERB
ejpam-5492	116	15	in	in	ADP
ejpam-5492	116	16	a	a	DET
ejpam-5492	116	17	locally	locally	ADV
ejpam-5492	116	18	metallic	metallic	ADJ
ejpam-5492	116	19	riemannian	riemannian	ADJ
ejpam-5492	116	20	manifold	manifold	ADJ
ejpam-5492	116	21	m̃	m̃	PROPN
ejpam-5492	116	22	,	,	PUNCT
ejpam-5492	116	23	then	then	ADV
ejpam-5492	116	24	g(h	g(h	NUM
ejpam-5492	116	25	(	(	PUNCT
ejpam-5492	116	26	jx	jx	PROPN
ejpam-5492	116	27	,	,	PUNCT
ejpam-5492	116	28	jy	jy	PROPN
ejpam-5492	116	29	)	)	PUNCT
ejpam-5492	116	30	,	,	PUNCT
ejpam-5492	116	31	jz	jz	PROPN
ejpam-5492	116	32	)	)	PUNCT
ejpam-5492	116	33	=	=	SYM
ejpam-5492	117	1	qg(h	qg(h	X
ejpam-5492	117	2	(	(	PUNCT
ejpam-5492	117	3	x	x	X
ejpam-5492	117	4	,	,	PUNCT
ejpam-5492	117	5	y	y	PROPN
ejpam-5492	117	6	)	)	PUNCT
ejpam-5492	117	7	,	,	PUNCT
ejpam-5492	117	8	jz	jz	PROPN
ejpam-5492	117	9	)	)	PUNCT
ejpam-5492	117	10	+	+	CCONJ
ejpam-5492	117	11	pg(h	pg(h	NUM
ejpam-5492	117	12	(	(	PUNCT
ejpam-5492	117	13	jx	jx	PROPN
ejpam-5492	117	14	,	,	PUNCT
ejpam-5492	117	15	y	y	PROPN
ejpam-5492	117	16	)	)	PUNCT
ejpam-5492	117	17	,	,	PUNCT
ejpam-5492	117	18	jz	jz	PROPN
ejpam-5492	117	19	)	)	PUNCT
ejpam-5492	117	20	.	.	PUNCT
ejpam-5492	118	1	(	(	PUNCT
ejpam-5492	118	2	30	30	NUM
ejpam-5492	118	3	)	)	PUNCT
ejpam-5492	118	4	for	for	ADP
ejpam-5492	118	5	any	any	DET
ejpam-5492	118	6	x	x	NOUN
ejpam-5492	118	7	,	,	PUNCT
ejpam-5492	118	8	y	y	PROPN
ejpam-5492	118	9	∈	∈	PROPN
ejpam-5492	118	10	γ(tmt	γ(tmt	PROPN
ejpam-5492	118	11	)	)	PUNCT
ejpam-5492	118	12	and	and	CCONJ
ejpam-5492	118	13	z	z	NOUN
ejpam-5492	118	14	∈	∈	PROPN
ejpam-5492	118	15	γ(tm⊥	γ(tm⊥	PROPN
ejpam-5492	118	16	)	)	PUNCT
ejpam-5492	118	17	,	,	PUNCT
ejpam-5492	118	18	proof	proof	NOUN
ejpam-5492	118	19	.	.	PUNCT
ejpam-5492	119	1	after	after	ADP
ejpam-5492	119	2	interchanging	interchange	VERB
ejpam-5492	119	3	y	y	PROPN
ejpam-5492	119	4	by	by	ADP
ejpam-5492	119	5	jy	jy	PROPN
ejpam-5492	119	6	in	in	ADP
ejpam-5492	119	7	(	(	PUNCT
ejpam-5492	119	8	22	22	NUM
ejpam-5492	119	9	)	)	PUNCT
ejpam-5492	119	10	,	,	PUNCT
ejpam-5492	119	11	and	and	CCONJ
ejpam-5492	119	12	applying	apply	VERB
ejpam-5492	119	13	(	(	PUNCT
ejpam-5492	119	14	3	3	NUM
ejpam-5492	119	15	)	)	PUNCT
ejpam-5492	119	16	,	,	PUNCT
ejpam-5492	119	17	we	we	PRON
ejpam-5492	119	18	obtain	obtain	VERB
ejpam-5492	119	19	g(h	g(h	PROPN
ejpam-5492	119	20	(	(	PUNCT
ejpam-5492	119	21	jx	jx	PROPN
ejpam-5492	119	22	,	,	PUNCT
ejpam-5492	119	23	jy	jy	PROPN
ejpam-5492	119	24	)	)	PUNCT
ejpam-5492	119	25	,	,	PUNCT
ejpam-5492	119	26	jz	jz	PROPN
ejpam-5492	119	27	)	)	PUNCT
ejpam-5492	119	28	=	=	SYM
ejpam-5492	119	29	−(p2	−(p2	PART
ejpam-5492	119	30	+	+	NUM
ejpam-5492	119	31	q)z	q)z	X
ejpam-5492	119	32	(	(	PUNCT
ejpam-5492	119	33	ln	ln	ADJ
ejpam-5492	119	34	f	f	X
ejpam-5492	119	35	)	)	PUNCT
ejpam-5492	119	36	g(jx	g(jx	NOUN
ejpam-5492	119	37	,	,	PUNCT
ejpam-5492	119	38	y	y	NOUN
ejpam-5492	119	39	)	)	PUNCT
ejpam-5492	119	40	−	−	PROPN
ejpam-5492	120	1	pqz	pqz	PROPN
ejpam-5492	120	2	(	(	PUNCT
ejpam-5492	120	3	ln	ln	NOUN
ejpam-5492	120	4	f	f	NOUN
ejpam-5492	120	5	)	)	PUNCT
ejpam-5492	120	6	g(x	g(x	PROPN
ejpam-5492	120	7	,	,	PUNCT
ejpam-5492	120	8	y	y	PROPN
ejpam-5492	120	9	)	)	PUNCT
ejpam-5492	120	10	.	.	PUNCT
ejpam-5492	121	1	(	(	PUNCT
ejpam-5492	121	2	31	31	NUM
ejpam-5492	121	3	)	)	PUNCT
ejpam-5492	121	4	using	use	VERB
ejpam-5492	121	5	equation	equation	NOUN
ejpam-5492	121	6	(	(	PUNCT
ejpam-5492	121	7	31)and	31)and	NUM
ejpam-5492	121	8	(	(	PUNCT
ejpam-5492	121	9	20	20	NUM
ejpam-5492	121	10	)	)	PUNCT
ejpam-5492	121	11	along	along	ADP
ejpam-5492	121	12	with	with	ADP
ejpam-5492	121	13	(	(	PUNCT
ejpam-5492	121	14	22	22	NUM
ejpam-5492	121	15	)	)	PUNCT
ejpam-5492	121	16	,	,	PUNCT
ejpam-5492	121	17	we	we	PRON
ejpam-5492	121	18	get	get	VERB
ejpam-5492	121	19	(	(	PUNCT
ejpam-5492	121	20	30	30	NUM
ejpam-5492	121	21	)	)	PUNCT
ejpam-5492	121	22	.	.	PUNCT
ejpam-5492	122	1	here	here	ADV
ejpam-5492	122	2	,	,	PUNCT
ejpam-5492	122	3	we	we	PRON
ejpam-5492	122	4	provide	provide	VERB
ejpam-5492	122	5	some	some	DET
ejpam-5492	122	6	examples	example	NOUN
ejpam-5492	122	7	of	of	ADP
ejpam-5492	122	8	a	a	DET
ejpam-5492	122	9	cr	cr	ADV
ejpam-5492	122	10	-	-	PUNCT
ejpam-5492	122	11	warped	warp	VERB
ejpam-5492	122	12	product	product	NOUN
ejpam-5492	122	13	manifold	manifold	ADJ
ejpam-5492	122	14	m	m	NOUN
ejpam-5492	122	15	=	=	VERB
ejpam-5492	122	16	m⊥	m⊥	PROPN
ejpam-5492	122	17	×f	×f	PROPN
ejpam-5492	122	18	mt	mt	PROPN
ejpam-5492	122	19	in	in	ADP
ejpam-5492	122	20	a	a	DET
ejpam-5492	122	21	locally	locally	ADV
ejpam-5492	122	22	metallic	metallic	ADJ
ejpam-5492	122	23	riemannian	riemannian	NOUN
ejpam-5492	122	24	manifold	manifold	NOUN
ejpam-5492	122	25	(	(	PUNCT
ejpam-5492	122	26	m̃	m̃	PROPN
ejpam-5492	122	27	,	,	PUNCT
ejpam-5492	122	28	g̃	g̃	PROPN
ejpam-5492	122	29	,	,	PUNCT
ejpam-5492	122	30	j	j	PROPN
ejpam-5492	122	31	)	)	PUNCT
ejpam-5492	122	32	.	.	PUNCT
ejpam-5492	123	1	l.	l.	PROPN
ejpam-5492	123	2	alqahtani	alqahtani	PROPN
ejpam-5492	123	3	,	,	PUNCT
ejpam-5492	123	4	e.	e.	PROPN
ejpam-5492	123	5	al	al	PROPN
ejpam-5492	123	6	-	-	PROPN
ejpam-5492	123	7	husainy	husainy	PROPN
ejpam-5492	123	8	/	/	SYM
ejpam-5492	123	9	eur	eur	PROPN
ejpam-5492	123	10	.	.	PUNCT
ejpam-5492	124	1	j.	j.	PROPN
ejpam-5492	124	2	pure	pure	PROPN
ejpam-5492	124	3	appl	appl	PROPN
ejpam-5492	124	4	.	.	PROPN
ejpam-5492	124	5	math	math	PROPN
ejpam-5492	124	6	,	,	PUNCT
ejpam-5492	124	7	17	17	NUM
ejpam-5492	124	8	(	(	PUNCT
ejpam-5492	124	9	4	4	NUM
ejpam-5492	124	10	)	)	PUNCT
ejpam-5492	124	11	(	(	PUNCT
ejpam-5492	124	12	2024	2024	NUM
ejpam-5492	124	13	)	)	PUNCT
ejpam-5492	124	14	,	,	PUNCT
ejpam-5492	124	15	2481	2481	NUM
ejpam-5492	124	16	-	-	SYM
ejpam-5492	124	17	2491	2491	NUM
ejpam-5492	124	18	2486	2486	NUM
ejpam-5492	124	19	example	example	NOUN
ejpam-5492	124	20	1	1	NUM
ejpam-5492	124	21	.	.	PUNCT
ejpam-5492	125	1	let	let	VERB
ejpam-5492	125	2	m	m	PRON
ejpam-5492	125	3	be	be	AUX
ejpam-5492	125	4	a	a	DET
ejpam-5492	125	5	submanifold	submanifold	NOUN
ejpam-5492	125	6	of	of	ADP
ejpam-5492	125	7	m̃	m̃	PROPN
ejpam-5492	125	8	defined	define	VERB
ejpam-5492	125	9	by	by	ADP
ejpam-5492	125	10	the	the	DET
ejpam-5492	125	11	immersion	immersion	NOUN
ejpam-5492	125	12	i	i	PRON
ejpam-5492	125	13	as	as	SCONJ
ejpam-5492	125	14	follows	follow	VERB
ejpam-5492	125	15	:	:	PUNCT
ejpam-5492	125	16	i(f	i(f	NOUN
ejpam-5492	125	17	,	,	PUNCT
ejpam-5492	125	18	α	α	X
ejpam-5492	125	19	,	,	PUNCT
ejpam-5492	125	20	θ	θ	NOUN
ejpam-5492	125	21	)	)	PUNCT
ejpam-5492	126	1	=	=	SYM
ejpam-5492	127	1	(	(	PUNCT
ejpam-5492	127	2	f	f	PROPN
ejpam-5492	127	3	cos	cos	PROPN
ejpam-5492	127	4	θ	θ	PROPN
ejpam-5492	127	5	sinα	sinα	PROPN
ejpam-5492	127	6	,	,	PUNCT
ejpam-5492	127	7	f	f	PROPN
ejpam-5492	127	8	cos	cos	PROPN
ejpam-5492	127	9	θ	θ	PROPN
ejpam-5492	127	10	cosα	cosα	PROPN
ejpam-5492	127	11	,	,	PUNCT
ejpam-5492	127	12	f	f	PROPN
ejpam-5492	127	13	sin	sin	PROPN
ejpam-5492	127	14	θ	θ	PROPN
ejpam-5492	127	15	sinα	sinα	NOUN
ejpam-5492	127	16	,	,	PUNCT
ejpam-5492	127	17	f	f	PROPN
ejpam-5492	127	18	sin	sin	NOUN
ejpam-5492	127	19	θ	θ	PROPN
ejpam-5492	127	20	cosα	cosα	PROPN
ejpam-5492	127	21	,	,	PUNCT
ejpam-5492	127	22	f	f	PROPN
ejpam-5492	127	23	,	,	PUNCT
ejpam-5492	127	24	σ̃q	σ̃q	PROPN
ejpam-5492	127	25	−1	−1	NOUN
ejpam-5492	127	26	2	2	NUM
ejpam-5492	127	27	f	f	NOUN
ejpam-5492	127	28	,	,	PUNCT
ejpam-5492	127	29	σq	σq	VERB
ejpam-5492	127	30	−1	−1	NOUN
ejpam-5492	127	31	2	2	NUM
ejpam-5492	127	32	f	f	NOUN
ejpam-5492	127	33	)	)	PUNCT
ejpam-5492	127	34	,	,	PUNCT
ejpam-5492	127	35	where	where	SCONJ
ejpam-5492	127	36	f	f	PROPN
ejpam-5492	127	37	>	>	X
ejpam-5492	127	38	0	0	PROPN
ejpam-5492	127	39	,	,	PUNCT
ejpam-5492	127	40	α	α	X
ejpam-5492	127	41	,	,	PUNCT
ejpam-5492	127	42	θ	θ	PROPN
ejpam-5492	127	43	∈	∈	PROPN
ejpam-5492	127	44	(	(	PUNCT
ejpam-5492	127	45	0	0	NUM
ejpam-5492	127	46	,	,	PUNCT
ejpam-5492	127	47	π2	π2	NOUN
ejpam-5492	127	48	)	)	PUNCT
ejpam-5492	127	49	,	,	PUNCT
ejpam-5492	127	50	σ	σ	PROPN
ejpam-5492	127	51	is	be	AUX
ejpam-5492	127	52	a	a	DET
ejpam-5492	127	53	metallic	metallic	ADJ
ejpam-5492	127	54	number	number	NOUN
ejpam-5492	127	55	,	,	PUNCT
ejpam-5492	127	56	defined	define	VERB
ejpam-5492	127	57	by	by	ADP
ejpam-5492	127	58	σ	σ	PROPN
ejpam-5492	127	59	=	=	PROPN
ejpam-5492	127	60	p+	p+	NOUN
ejpam-5492	127	61	√	√	NOUN
ejpam-5492	127	62	p2	p2	VERB
ejpam-5492	127	63	+	+	NOUN
ejpam-5492	127	64	4q	4q	NOUN
ejpam-5492	127	65	2	2	NUM
ejpam-5492	127	66	,	,	PUNCT
ejpam-5492	127	67	and	and	CCONJ
ejpam-5492	128	1	σ̃	σ̃	PROPN
ejpam-5492	128	2	=	=	PUNCT
ejpam-5492	128	3	p−σ	p−σ	PROPN
ejpam-5492	128	4	,	,	PUNCT
ejpam-5492	128	5	for	for	ADP
ejpam-5492	128	6	some	some	DET
ejpam-5492	128	7	positive	positive	ADJ
ejpam-5492	128	8	integers	integer	NOUN
ejpam-5492	128	9	p	p	NOUN
ejpam-5492	128	10	and	and	CCONJ
ejpam-5492	128	11	q.	q.	NOUN
ejpam-5492	128	12	it	it	PRON
ejpam-5492	128	13	is	be	AUX
ejpam-5492	128	14	straightforward	straightforward	ADJ
ejpam-5492	128	15	to	to	PART
ejpam-5492	128	16	compute	compute	VERB
ejpam-5492	128	17	that	that	SCONJ
ejpam-5492	128	18	the	the	DET
ejpam-5492	128	19	tangent	tangent	NOUN
ejpam-5492	128	20	bundle	bundle	PROPN
ejpam-5492	128	21	ofm	ofm	PROPN
ejpam-5492	128	22	is	be	AUX
ejpam-5492	128	23	spanned	span	VERB
ejpam-5492	128	24	by	by	ADP
ejpam-5492	128	25	the	the	DET
ejpam-5492	128	26	vectors	vector	NOUN
ejpam-5492	128	27	{	{	PUNCT
ejpam-5492	128	28	z1	z1	PROPN
ejpam-5492	128	29	,	,	PUNCT
ejpam-5492	128	30	z2	z2	PROPN
ejpam-5492	128	31	,	,	PUNCT
ejpam-5492	128	32	z3	z3	PROPN
ejpam-5492	128	33	}	}	PUNCT
ejpam-5492	128	34	,	,	PUNCT
ejpam-5492	128	35	where	where	SCONJ
ejpam-5492	128	36	z1	z1	PROPN
ejpam-5492	128	37	=	=	SYM
ejpam-5492	128	38	cos	cos	PROPN
ejpam-5492	128	39	θ	θ	PROPN
ejpam-5492	128	40	sinα	sinα	PROPN
ejpam-5492	128	41	∂	∂	NOUN
ejpam-5492	128	42	∂x1	∂x1	NOUN
ejpam-5492	128	43	+	+	PUNCT
ejpam-5492	128	44	cos	cos	PROPN
ejpam-5492	128	45	θ	θ	PROPN
ejpam-5492	128	46	cosα	cosα	PROPN
ejpam-5492	128	47	∂	∂	NOUN
ejpam-5492	128	48	∂x2	∂x2	NOUN
ejpam-5492	128	49	+	+	CCONJ
ejpam-5492	128	50	sin	sin	NOUN
ejpam-5492	128	51	θ	θ	PROPN
ejpam-5492	128	52	sinα	sinα	NOUN
ejpam-5492	128	53	∂	∂	NOUN
ejpam-5492	128	54	∂x3	∂x3	ADV
ejpam-5492	128	55	+	+	CCONJ
ejpam-5492	128	56	sin	sin	NOUN
ejpam-5492	128	57	θ	θ	PROPN
ejpam-5492	128	58	cosα	cosα	PROPN
ejpam-5492	128	59	∂	∂	NOUN
ejpam-5492	128	60	∂x4	∂x4	VERB
ejpam-5492	128	61	+	+	NOUN
ejpam-5492	128	62	∂	∂	NUM
ejpam-5492	128	63	∂x5	∂x5	NOUN
ejpam-5492	129	1	+	+	CCONJ
ejpam-5492	129	2	σ̃	σ̃	PROPN
ejpam-5492	129	3	√	√	PROPN
ejpam-5492	129	4	q	q	NOUN
ejpam-5492	129	5	∂	∂	NOUN
ejpam-5492	129	6	∂x6	∂x6	NOUN
ejpam-5492	130	1	+	+	CCONJ
ejpam-5492	131	1	σ	σ	NUM
ejpam-5492	131	2	√	√	PROPN
ejpam-5492	131	3	q	q	NOUN
ejpam-5492	131	4	∂	∂	NUM
ejpam-5492	131	5	∂x7	∂x7	NOUN
ejpam-5492	131	6	,	,	PUNCT
ejpam-5492	131	7	z2	z2	PROPN
ejpam-5492	131	8	=	=	SYM
ejpam-5492	131	9	f	f	PROPN
ejpam-5492	131	10	cos	cos	PROPN
ejpam-5492	131	11	θ	θ	PROPN
ejpam-5492	131	12	cosα	cosα	NOUN
ejpam-5492	131	13	∂	∂	NOUN
ejpam-5492	132	1	∂x1	∂x1	NOUN
ejpam-5492	132	2	−	−	PUNCT
ejpam-5492	132	3	f	f	PROPN
ejpam-5492	132	4	cos	cos	PROPN
ejpam-5492	132	5	θ	θ	PROPN
ejpam-5492	132	6	sinα	sinα	PROPN
ejpam-5492	132	7	∂	∂	NOUN
ejpam-5492	132	8	∂x2	∂x2	NOUN
ejpam-5492	132	9	+	+	CCONJ
ejpam-5492	132	10	f	f	PROPN
ejpam-5492	132	11	sin	sin	NOUN
ejpam-5492	132	12	θ	θ	PROPN
ejpam-5492	132	13	cosα	cosα	NOUN
ejpam-5492	132	14	∂	∂	NUM
ejpam-5492	132	15	∂x3	∂x3	ADV
ejpam-5492	132	16	−	−	PROPN
ejpam-5492	132	17	f	f	NOUN
ejpam-5492	132	18	sin	sin	PROPN
ejpam-5492	132	19	θ	θ	PROPN
ejpam-5492	132	20	sinα	sinα	PROPN
ejpam-5492	132	21	∂	∂	NOUN
ejpam-5492	132	22	∂x4	∂x4	VERB
ejpam-5492	132	23	,	,	PUNCT
ejpam-5492	132	24	z3	z3	PROPN
ejpam-5492	132	25	=	=	SYM
ejpam-5492	132	26	−f	−f	ADJ
ejpam-5492	132	27	sin	sin	NOUN
ejpam-5492	132	28	θ	θ	PROPN
ejpam-5492	132	29	sinα	sinα	PROPN
ejpam-5492	132	30	∂	∂	NOUN
ejpam-5492	133	1	∂x1	∂x1	NOUN
ejpam-5492	133	2	−	−	PUNCT
ejpam-5492	133	3	f	f	X
ejpam-5492	133	4	sin	sin	VERB
ejpam-5492	133	5	θ	θ	PROPN
ejpam-5492	133	6	cosα	cosα	PROPN
ejpam-5492	133	7	∂	∂	NOUN
ejpam-5492	133	8	∂x2	∂x2	PROPN
ejpam-5492	133	9	+	+	NUM
ejpam-5492	133	10	f	f	PROPN
ejpam-5492	133	11	cos	cos	PROPN
ejpam-5492	133	12	θ	θ	PROPN
ejpam-5492	133	13	sinα	sinα	PROPN
ejpam-5492	133	14	∂	∂	NOUN
ejpam-5492	133	15	∂∂x3	∂∂x3	PROPN
ejpam-5492	134	1	+	+	CCONJ
ejpam-5492	134	2	f	f	PROPN
ejpam-5492	134	3	cos	cos	PROPN
ejpam-5492	134	4	θ	θ	PROPN
ejpam-5492	134	5	cosα	cosα	PROPN
ejpam-5492	134	6	∂	∂	NOUN
ejpam-5492	134	7	∂x4	∂x4	VERB
ejpam-5492	134	8	.	.	PUNCT
ejpam-5492	135	1	by	by	ADP
ejpam-5492	135	2	using	use	VERB
ejpam-5492	135	3	the	the	DET
ejpam-5492	135	4	metallic	metallic	ADJ
ejpam-5492	135	5	structure	structure	NOUN
ejpam-5492	135	6	j	j	PROPN
ejpam-5492	135	7	of	of	ADP
ejpam-5492	135	8	m̃	m̃	PROPN
ejpam-5492	135	9	which	which	PRON
ejpam-5492	135	10	is	be	AUX
ejpam-5492	135	11	j(x1	j(x1	ADJ
ejpam-5492	135	12	,	,	PUNCT
ejpam-5492	135	13	x2	x2	PROPN
ejpam-5492	135	14	,	,	PUNCT
ejpam-5492	135	15	x3	x3	PROPN
ejpam-5492	135	16	,	,	PUNCT
ejpam-5492	135	17	x4	x4	PROPN
ejpam-5492	135	18	,	,	PUNCT
ejpam-5492	135	19	x5	x5	PROPN
ejpam-5492	135	20	,	,	PUNCT
ejpam-5492	135	21	x6	x6	NOUN
ejpam-5492	135	22	,	,	PUNCT
ejpam-5492	135	23	x7	x7	NOUN
ejpam-5492	135	24	)	)	PUNCT
ejpam-5492	135	25	=	=	SYM
ejpam-5492	135	26	(	(	PUNCT
ejpam-5492	135	27	σx1	σx1	ADV
ejpam-5492	135	28	,	,	PUNCT
ejpam-5492	135	29	σx2	σx2	PROPN
ejpam-5492	135	30	,	,	PUNCT
ejpam-5492	135	31	σx3	σx3	PROPN
ejpam-5492	135	32	,	,	PUNCT
ejpam-5492	135	33	σx4	σx4	NOUN
ejpam-5492	135	34	,	,	PUNCT
ejpam-5492	135	35	σ̄x5	σ̄x5	NOUN
ejpam-5492	135	36	,	,	PUNCT
ejpam-5492	135	37	σx6	σx6	PROPN
ejpam-5492	135	38	,	,	PUNCT
ejpam-5492	135	39	σ̄x7	σ̄x7	PROPN
ejpam-5492	135	40	)	)	PUNCT
ejpam-5492	135	41	,	,	PUNCT
ejpam-5492	135	42	then	then	ADV
ejpam-5492	135	43	,	,	PUNCT
ejpam-5492	135	44	we	we	PRON
ejpam-5492	135	45	find	find	VERB
ejpam-5492	135	46	that	that	SCONJ
ejpam-5492	135	47	jz1	jz1	PROPN
ejpam-5492	135	48	=	=	SYM
ejpam-5492	135	49	σ	σ	PROPN
ejpam-5492	135	50	cos	cos	PROPN
ejpam-5492	135	51	θ	θ	PROPN
ejpam-5492	135	52	sinα	sinα	PROPN
ejpam-5492	135	53	∂	∂	NOUN
ejpam-5492	135	54	∂x1	∂x1	NOUN
ejpam-5492	135	55	+	+	CCONJ
ejpam-5492	135	56	σ	σ	PROPN
ejpam-5492	135	57	cos	cos	PROPN
ejpam-5492	135	58	θ	θ	PROPN
ejpam-5492	135	59	cosα	cosα	PROPN
ejpam-5492	135	60	∂	∂	NOUN
ejpam-5492	135	61	∂x2	∂x2	PROPN
ejpam-5492	135	62	+	+	CCONJ
ejpam-5492	135	63	σ	σ	PROPN
ejpam-5492	135	64	sin	sin	NOUN
ejpam-5492	135	65	θ	θ	PROPN
ejpam-5492	135	66	sinα	sinα	NOUN
ejpam-5492	135	67	∂	∂	NOUN
ejpam-5492	135	68	∂x3	∂x3	ADV
ejpam-5492	135	69	+	+	CCONJ
ejpam-5492	135	70	σ	σ	NUM
ejpam-5492	135	71	sin	sin	NOUN
ejpam-5492	135	72	θ	θ	PROPN
ejpam-5492	135	73	cosα	cosα	PROPN
ejpam-5492	135	74	∂	∂	NOUN
ejpam-5492	135	75	∂x4	∂x4	VERB
ejpam-5492	135	76	+	+	CCONJ
ejpam-5492	136	1	σ̃	σ̃	NOUN
ejpam-5492	136	2	∂	∂	NOUN
ejpam-5492	136	3	∂x5	∂x5	NOUN
ejpam-5492	136	4	+	+	CCONJ
ejpam-5492	136	5	σσ̃	σσ̃	NUM
ejpam-5492	136	6	√	√	NUM
ejpam-5492	136	7	q	q	NOUN
ejpam-5492	136	8	∂	∂	NOUN
ejpam-5492	136	9	∂x6	∂x6	NOUN
ejpam-5492	137	1	+	+	CCONJ
ejpam-5492	137	2	σ̃σ	σ̃σ	ADP
ejpam-5492	137	3	√	√	VERB
ejpam-5492	137	4	q	q	NOUN
ejpam-5492	137	5	∂	∂	NOUN
ejpam-5492	137	6	∂x7	∂x7	PROPN
ejpam-5492	137	7	,	,	PUNCT
ejpam-5492	137	8	jz2	jz2	X
ejpam-5492	137	9	=	=	PUNCT
ejpam-5492	137	10	fσ	fσ	PROPN
ejpam-5492	137	11	cos	co	NOUN
ejpam-5492	137	12	θ	θ	PROPN
ejpam-5492	137	13	cosα	cosα	NOUN
ejpam-5492	137	14	∂	∂	NOUN
ejpam-5492	137	15	∂x1	∂x1	NOUN
ejpam-5492	137	16	−	−	PUNCT
ejpam-5492	137	17	fσ	fσ	NOUN
ejpam-5492	137	18	cos	co	NOUN
ejpam-5492	137	19	θ	θ	PROPN
ejpam-5492	137	20	sinα	sinα	PROPN
ejpam-5492	137	21	∂	∂	NOUN
ejpam-5492	137	22	∂x2	∂x2	NOUN
ejpam-5492	137	23	+	+	CCONJ
ejpam-5492	137	24	fσ	fσ	ADP
ejpam-5492	137	25	sin	sin	NOUN
ejpam-5492	137	26	θ	θ	PROPN
ejpam-5492	137	27	cosα	cosα	NOUN
ejpam-5492	137	28	∂	∂	NUM
ejpam-5492	137	29	∂x3	∂x3	ADV
ejpam-5492	137	30	−	−	PROPN
ejpam-5492	138	1	fσ	fσ	NOUN
ejpam-5492	138	2	sin	sin	VERB
ejpam-5492	138	3	θ	θ	PROPN
ejpam-5492	138	4	sinα	sinα	PROPN
ejpam-5492	138	5	∂	∂	NOUN
ejpam-5492	138	6	∂x4	∂x4	VERB
ejpam-5492	138	7	,	,	PUNCT
ejpam-5492	138	8	jz3	jz3	NOUN
ejpam-5492	138	9	=	=	SYM
ejpam-5492	139	1	−fσ	−fσ	ADP
ejpam-5492	139	2	sin	sin	NOUN
ejpam-5492	139	3	θ	θ	PROPN
ejpam-5492	139	4	sinα	sinα	PROPN
ejpam-5492	139	5	∂	∂	NOUN
ejpam-5492	139	6	∂x1	∂x1	NOUN
ejpam-5492	139	7	−	−	PROPN
ejpam-5492	140	1	fσ	fσ	PRON
ejpam-5492	140	2	sin	sin	VERB
ejpam-5492	140	3	θ	θ	PROPN
ejpam-5492	140	4	cosα	cosα	PROPN
ejpam-5492	140	5	∂	∂	NOUN
ejpam-5492	140	6	∂x2	∂x2	NOUN
ejpam-5492	140	7	+	+	CCONJ
ejpam-5492	140	8	fσ	fσ	NOUN
ejpam-5492	140	9	cos	cos	ADP
ejpam-5492	140	10	θ	θ	PROPN
ejpam-5492	140	11	sinα	sinα	NOUN
ejpam-5492	140	12	∂	∂	NOUN
ejpam-5492	140	13	∂x3	∂x3	ADV
ejpam-5492	140	14	+	+	CCONJ
ejpam-5492	140	15	fσ	fσ	X
ejpam-5492	140	16	cos	cos	ADP
ejpam-5492	140	17	θ	θ	PROPN
ejpam-5492	140	18	cosα	cosα	PROPN
ejpam-5492	140	19	∂	∂	NOUN
ejpam-5492	140	20	∂x4	∂x4	VERB
ejpam-5492	140	21	,	,	PUNCT
ejpam-5492	140	22	now	now	ADV
ejpam-5492	140	23	,	,	PUNCT
ejpam-5492	140	24	we	we	PRON
ejpam-5492	140	25	define	define	VERB
ejpam-5492	140	26	two	two	NUM
ejpam-5492	140	27	vector	vector	NOUN
ejpam-5492	140	28	spaces	space	NOUN
ejpam-5492	140	29	dt	dt	NOUN
ejpam-5492	140	30	and	and	CCONJ
ejpam-5492	140	31	d⊥	d⊥	PROPN
ejpam-5492	140	32	,	,	PUNCT
ejpam-5492	140	33	where	where	SCONJ
ejpam-5492	140	34	dt	dt	NOUN
ejpam-5492	140	35	=	=	SYM
ejpam-5492	140	36	span{z2	span{z2	ADJ
ejpam-5492	140	37	,	,	PUNCT
ejpam-5492	140	38	z3	z3	PROPN
ejpam-5492	140	39	}	}	PUNCT
ejpam-5492	140	40	is	be	AUX
ejpam-5492	140	41	the	the	DET
ejpam-5492	140	42	invariant	invariant	ADJ
ejpam-5492	140	43	distribution	distribution	NOUN
ejpam-5492	140	44	and	and	CCONJ
ejpam-5492	140	45	d⊥	d⊥	NOUN
ejpam-5492	140	46	=	=	SYM
ejpam-5492	140	47	span{z1	span{z1	NOUN
ejpam-5492	140	48	}	}	PUNCT
ejpam-5492	140	49	is	be	AUX
ejpam-5492	140	50	the	the	DET
ejpam-5492	140	51	anti	anti	ADJ
ejpam-5492	140	52	-	-	ADJ
ejpam-5492	140	53	invariant	invariant	ADJ
ejpam-5492	140	54	distribution	distribution	NOUN
ejpam-5492	140	55	,	,	PUNCT
ejpam-5492	140	56	which	which	PRON
ejpam-5492	140	57	are	be	AUX
ejpam-5492	140	58	preserved	preserve	VERB
ejpam-5492	140	59	by	by	ADP
ejpam-5492	140	60	the	the	DET
ejpam-5492	140	61	action	action	NOUN
ejpam-5492	140	62	of	of	ADP
ejpam-5492	140	63	j	j	PROPN
ejpam-5492	140	64	.	.	PUNCT
ejpam-5492	141	1	hence	hence	ADV
ejpam-5492	141	2	,	,	PUNCT
ejpam-5492	141	3	the	the	DET
ejpam-5492	141	4	riemannian	riemannian	ADJ
ejpam-5492	141	5	metric	metric	NOUN
ejpam-5492	141	6	of	of	ADP
ejpam-5492	141	7	the	the	DET
ejpam-5492	141	8	warped	warped	ADJ
ejpam-5492	141	9	product	product	NOUN
ejpam-5492	141	10	crsubmanifold	crsubmanifold	ADJ
ejpam-5492	141	11	m	m	VERB
ejpam-5492	141	12	is	be	AUX
ejpam-5492	141	13	given	give	VERB
ejpam-5492	141	14	by	by	ADP
ejpam-5492	141	15	the	the	DET
ejpam-5492	141	16	following	following	NOUN
ejpam-5492	141	17	:	:	PUNCT
ejpam-5492	141	18	g	g	PROPN
ejpam-5492	141	19	=	=	SYM
ejpam-5492	141	20	(	(	PUNCT
ejpam-5492	141	21	2	2	NUM
ejpam-5492	141	22	+	+	NUM
ejpam-5492	141	23	σ̃2q−1	σ̃2q−1	NOUN
ejpam-5492	142	1	+	+	CCONJ
ejpam-5492	142	2	σ2q−1)d2f	σ2q−1)d2f	NOUN
ejpam-5492	142	3	+	+	NUM
ejpam-5492	142	4	f2(d2θ	f2(d2θ	PROPN
ejpam-5492	142	5	+	+	CCONJ
ejpam-5492	142	6	d2α	d2α	NOUN
ejpam-5492	142	7	)	)	PUNCT
ejpam-5492	142	8	.	.	PUNCT
ejpam-5492	143	1	then	then	ADV
ejpam-5492	143	2	,	,	PUNCT
ejpam-5492	143	3	m	m	VERB
ejpam-5492	143	4	=	=	VERB
ejpam-5492	143	5	m⊥	m⊥	NOUN
ejpam-5492	143	6	×	×	NOUN
ejpam-5492	143	7	fmt	fmt	NOUN
ejpam-5492	143	8	is	be	AUX
ejpam-5492	143	9	a	a	DET
ejpam-5492	143	10	warped	warped	ADJ
ejpam-5492	143	11	product	product	NOUN
ejpam-5492	143	12	cr	cr	NOUN
ejpam-5492	143	13	-	-	PUNCT
ejpam-5492	143	14	submanifold	submanifold	NOUN
ejpam-5492	143	15	in	in	ADP
ejpam-5492	143	16	the	the	DET
ejpam-5492	143	17	metallic	metallic	ADJ
ejpam-5492	143	18	riemannian	riemannian	NOUN
ejpam-5492	143	19	manifold	manifold	ADJ
ejpam-5492	143	20	m̃	m̃	PROPN
ejpam-5492	143	21	.	.	PUNCT
ejpam-5492	144	1	l.	l.	PROPN
ejpam-5492	144	2	alqahtani	alqahtani	PROPN
ejpam-5492	144	3	,	,	PUNCT
ejpam-5492	144	4	e.	e.	PROPN
ejpam-5492	144	5	al	al	PROPN
ejpam-5492	144	6	-	-	PROPN
ejpam-5492	144	7	husainy	husainy	PROPN
ejpam-5492	144	8	/	/	SYM
ejpam-5492	144	9	eur	eur	PROPN
ejpam-5492	144	10	.	.	PUNCT
ejpam-5492	145	1	j.	j.	PROPN
ejpam-5492	145	2	pure	pure	PROPN
ejpam-5492	145	3	appl	appl	PROPN
ejpam-5492	145	4	.	.	PROPN
ejpam-5492	145	5	math	math	PROPN
ejpam-5492	145	6	,	,	PUNCT
ejpam-5492	145	7	17	17	NUM
ejpam-5492	145	8	(	(	PUNCT
ejpam-5492	145	9	4	4	NUM
ejpam-5492	145	10	)	)	PUNCT
ejpam-5492	145	11	(	(	PUNCT
ejpam-5492	145	12	2024	2024	NUM
ejpam-5492	145	13	)	)	PUNCT
ejpam-5492	145	14	,	,	PUNCT
ejpam-5492	145	15	2481	2481	NUM
ejpam-5492	145	16	-	-	SYM
ejpam-5492	145	17	2491	2491	NUM
ejpam-5492	145	18	2487	2487	NUM
ejpam-5492	145	19	example	example	NOUN
ejpam-5492	146	1	2	2	NUM
ejpam-5492	146	2	.	.	X
ejpam-5492	147	1	let	let	VERB
ejpam-5492	147	2	m	m	PRON
ejpam-5492	147	3	be	be	AUX
ejpam-5492	147	4	a	a	DET
ejpam-5492	147	5	submanifold	submanifold	NOUN
ejpam-5492	147	6	of	of	ADP
ejpam-5492	147	7	m̃	m̃	PROPN
ejpam-5492	147	8	defined	define	VERB
ejpam-5492	147	9	by	by	ADP
ejpam-5492	147	10	the	the	DET
ejpam-5492	147	11	immersion	immersion	NOUN
ejpam-5492	147	12	i	i	PRON
ejpam-5492	147	13	as	as	SCONJ
ejpam-5492	147	14	follows	follow	VERB
ejpam-5492	147	15	:	:	PUNCT
ejpam-5492	147	16	i(u	i(u	PROPN
ejpam-5492	147	17	,	,	PUNCT
ejpam-5492	147	18	v	v	PROPN
ejpam-5492	147	19	,	,	PUNCT
ejpam-5492	147	20	θ	θ	PROPN
ejpam-5492	147	21	,	,	PUNCT
ejpam-5492	147	22	ϕ	ϕ	NOUN
ejpam-5492	147	23	)	)	PUNCT
ejpam-5492	147	24	=(	=(	NOUN
ejpam-5492	147	25	u	u	PROPN
ejpam-5492	147	26	cos	cos	PROPN
ejpam-5492	147	27	θ	θ	PROPN
ejpam-5492	147	28	,	,	PUNCT
ejpam-5492	147	29	u	u	PRON
ejpam-5492	147	30	sin	sin	NOUN
ejpam-5492	147	31	θ	θ	PROPN
ejpam-5492	147	32	,	,	PUNCT
ejpam-5492	147	33	v	v	NOUN
ejpam-5492	147	34	cos	cos	PROPN
ejpam-5492	147	35	θ	θ	PROPN
ejpam-5492	147	36	,	,	PUNCT
ejpam-5492	147	37	v	v	ADP
ejpam-5492	147	38	sin	sin	NOUN
ejpam-5492	147	39	θ	θ	NOUN
ejpam-5492	147	40	,	,	PUNCT
ejpam-5492	147	41	u	u	NOUN
ejpam-5492	147	42	cosϕ	cosϕ	NOUN
ejpam-5492	147	43	,	,	PUNCT
ejpam-5492	147	44	u	u	NOUN
ejpam-5492	147	45	sinϕ	sinϕ	NOUN
ejpam-5492	147	46	,	,	PUNCT
ejpam-5492	147	47	v	v	ADP
ejpam-5492	147	48	cosϕ	cosϕ	NOUN
ejpam-5492	147	49	,	,	PUNCT
ejpam-5492	147	50	v	v	NOUN
ejpam-5492	147	51	sinϕ	sinϕ	NOUN
ejpam-5492	147	52	,	,	PUNCT
ejpam-5492	147	53	1	1	NUM
ejpam-5492	147	54	√	√	PROPN
ejpam-5492	147	55	q	q	NOUN
ejpam-5492	148	1	σu	σu	INTJ
ejpam-5492	148	2	,	,	PUNCT
ejpam-5492	148	3	1	1	NUM
ejpam-5492	148	4	√	√	PROPN
ejpam-5492	148	5	q	q	PROPN
ejpam-5492	148	6	σ̃v	σ̃v	PROPN
ejpam-5492	148	7	,	,	PUNCT
ejpam-5492	148	8	1	1	NUM
ejpam-5492	148	9	√	√	PROPN
ejpam-5492	148	10	q	q	NOUN
ejpam-5492	148	11	σ̃u	σ̃u	PROPN
ejpam-5492	148	12	,	,	PUNCT
ejpam-5492	148	13	1	1	NUM
ejpam-5492	148	14	√	√	PROPN
ejpam-5492	148	15	q	q	NOUN
ejpam-5492	148	16	σv	σv	PROPN
ejpam-5492	148	17	)	)	PUNCT
ejpam-5492	148	18	,	,	PUNCT
ejpam-5492	148	19	where	where	SCONJ
ejpam-5492	148	20	u	u	NOUN
ejpam-5492	148	21	,	,	PUNCT
ejpam-5492	148	22	v	v	X
ejpam-5492	148	23	>	>	X
ejpam-5492	148	24	0	0	NUM
ejpam-5492	148	25	,	,	PUNCT
ejpam-5492	148	26	θ	θ	PROPN
ejpam-5492	148	27	,	,	PUNCT
ejpam-5492	148	28	ϕ	ϕ	PROPN
ejpam-5492	148	29	∈	∈	PROPN
ejpam-5492	148	30	(	(	PUNCT
ejpam-5492	148	31	0	0	NUM
ejpam-5492	148	32	,	,	PUNCT
ejpam-5492	148	33	π2	π2	NOUN
ejpam-5492	148	34	)	)	PUNCT
ejpam-5492	148	35	,	,	PUNCT
ejpam-5492	148	36	σ	σ	PROPN
ejpam-5492	148	37	is	be	AUX
ejpam-5492	148	38	a	a	DET
ejpam-5492	148	39	metallic	metallic	ADJ
ejpam-5492	148	40	number	number	NOUN
ejpam-5492	148	41	,	,	PUNCT
ejpam-5492	148	42	defined	define	VERB
ejpam-5492	148	43	by	by	ADP
ejpam-5492	148	44	σ	σ	PROPN
ejpam-5492	148	45	=	=	PROPN
ejpam-5492	148	46	p+	p+	NOUN
ejpam-5492	148	47	√	√	NOUN
ejpam-5492	148	48	p2	p2	VERB
ejpam-5492	148	49	+	+	NOUN
ejpam-5492	148	50	4q	4q	NOUN
ejpam-5492	148	51	2	2	NUM
ejpam-5492	148	52	,	,	PUNCT
ejpam-5492	148	53	and	and	CCONJ
ejpam-5492	148	54	σ̃	σ̃	PROPN
ejpam-5492	148	55	=	=	NOUN
ejpam-5492	148	56	p−	p−	NOUN
ejpam-5492	148	57	σ	σ	NOUN
ejpam-5492	148	58	,	,	PUNCT
ejpam-5492	148	59	for	for	ADP
ejpam-5492	148	60	some	some	DET
ejpam-5492	148	61	positive	positive	ADJ
ejpam-5492	148	62	integers	integer	NOUN
ejpam-5492	148	63	p	p	NOUN
ejpam-5492	148	64	and	and	CCONJ
ejpam-5492	148	65	q.	q.	NOUN
ejpam-5492	148	66	by	by	ADP
ejpam-5492	148	67	some	some	DET
ejpam-5492	148	68	computation	computation	NOUN
ejpam-5492	148	69	,	,	PUNCT
ejpam-5492	148	70	it	it	PRON
ejpam-5492	148	71	is	be	AUX
ejpam-5492	148	72	easy	easy	ADJ
ejpam-5492	148	73	to	to	PART
ejpam-5492	148	74	find	find	VERB
ejpam-5492	148	75	that	that	SCONJ
ejpam-5492	148	76	the	the	DET
ejpam-5492	148	77	tangent	tangent	ADJ
ejpam-5492	148	78	bundle	bundle	NOUN
ejpam-5492	148	79	of	of	ADP
ejpam-5492	148	80	m	m	PROPN
ejpam-5492	148	81	is	be	AUX
ejpam-5492	148	82	spanned	span	VERB
ejpam-5492	148	83	by	by	ADP
ejpam-5492	148	84	{	{	PUNCT
ejpam-5492	148	85	z1	z1	PROPN
ejpam-5492	148	86	,	,	PUNCT
ejpam-5492	148	87	z2	z2	PROPN
ejpam-5492	148	88	,	,	PUNCT
ejpam-5492	148	89	z3	z3	PROPN
ejpam-5492	148	90	,	,	PUNCT
ejpam-5492	148	91	z4	z4	PROPN
ejpam-5492	148	92	}	}	PUNCT
ejpam-5492	148	93	,	,	PUNCT
ejpam-5492	148	94	where	where	SCONJ
ejpam-5492	148	95	z1	z1	PROPN
ejpam-5492	148	96	=	=	SYM
ejpam-5492	148	97	cos	cos	PROPN
ejpam-5492	148	98	θ	θ	PROPN
ejpam-5492	148	99	∂	∂	X
ejpam-5492	148	100	∂x1	∂x1	NOUN
ejpam-5492	148	101	+	+	CCONJ
ejpam-5492	148	102	sin	sin	PROPN
ejpam-5492	148	103	θ	θ	PROPN
ejpam-5492	148	104	∂	∂	NUM
ejpam-5492	148	105	∂x2	∂x2	NOUN
ejpam-5492	148	106	+	+	CCONJ
ejpam-5492	148	107	cosϕ	cosϕ	NOUN
ejpam-5492	148	108	∂	∂	NUM
ejpam-5492	148	109	∂x5	∂x5	PROPN
ejpam-5492	148	110	+	+	CCONJ
ejpam-5492	148	111	sinϕ	sinϕ	PROPN
ejpam-5492	148	112	∂	∂	NOUN
ejpam-5492	148	113	∂x6	∂x6	NOUN
ejpam-5492	149	1	+	+	CCONJ
ejpam-5492	149	2	σ	σ	NUM
ejpam-5492	149	3	√	√	PROPN
ejpam-5492	149	4	q	q	NOUN
ejpam-5492	149	5	∂	∂	NOUN
ejpam-5492	149	6	∂x9	∂x9	PROPN
ejpam-5492	149	7	+	+	CCONJ
ejpam-5492	149	8	σ̃	σ̃	PROPN
ejpam-5492	149	9	√	√	PROPN
ejpam-5492	149	10	q	q	NOUN
ejpam-5492	149	11	∂	∂	NOUN
ejpam-5492	149	12	∂x11	∂x11	PROPN
ejpam-5492	149	13	,	,	PUNCT
ejpam-5492	149	14	z2	z2	PROPN
ejpam-5492	149	15	=	=	SYM
ejpam-5492	149	16	cos	cos	PROPN
ejpam-5492	149	17	θ	θ	PROPN
ejpam-5492	149	18	∂	∂	NUM
ejpam-5492	149	19	∂x3	∂x3	ADV
ejpam-5492	149	20	+	+	CCONJ
ejpam-5492	149	21	sin	sin	NOUN
ejpam-5492	149	22	θ	θ	PROPN
ejpam-5492	149	23	∂	∂	X
ejpam-5492	149	24	∂x4	∂x4	VERB
ejpam-5492	149	25	+	+	CCONJ
ejpam-5492	149	26	cosϕ	cosϕ	NUM
ejpam-5492	149	27	∂	∂	NOUN
ejpam-5492	149	28	∂x7	∂x7	NOUN
ejpam-5492	149	29	+	+	CCONJ
ejpam-5492	150	1	sinϕ	sinϕ	PROPN
ejpam-5492	150	2	∂	∂	NOUN
ejpam-5492	150	3	∂x8	∂x8	X
ejpam-5492	151	1	+	+	CCONJ
ejpam-5492	151	2	σ̃	σ̃	NOUN
ejpam-5492	151	3	√	√	ADJ
ejpam-5492	151	4	q	q	NOUN
ejpam-5492	151	5	∂	∂	NUM
ejpam-5492	151	6	∂x10	∂x10	PROPN
ejpam-5492	151	7	+	+	CCONJ
ejpam-5492	151	8	σ	σ	PROPN
ejpam-5492	151	9	√	√	NUM
ejpam-5492	151	10	q	q	NOUN
ejpam-5492	151	11	∂	∂	NUM
ejpam-5492	151	12	∂x12	∂x12	NOUN
ejpam-5492	151	13	,	,	PUNCT
ejpam-5492	151	14	z3	z3	PROPN
ejpam-5492	151	15	=	=	SYM
ejpam-5492	151	16	−u	−u	PROPN
ejpam-5492	151	17	sin	sin	NOUN
ejpam-5492	151	18	θ	θ	PROPN
ejpam-5492	151	19	∂	∂	NOUN
ejpam-5492	152	1	∂x1	∂x1	NOUN
ejpam-5492	152	2	+	+	CCONJ
ejpam-5492	152	3	u	u	PROPN
ejpam-5492	152	4	cos	cos	PROPN
ejpam-5492	152	5	θ	θ	PROPN
ejpam-5492	152	6	∂	∂	NUM
ejpam-5492	152	7	∂x2	∂x2	NOUN
ejpam-5492	152	8	−	−	NOUN
ejpam-5492	152	9	v	v	NOUN
ejpam-5492	152	10	sin	sin	NOUN
ejpam-5492	152	11	θ	θ	NOUN
ejpam-5492	152	12	∂	∂	NUM
ejpam-5492	153	1	∂x3	∂x3	ADV
ejpam-5492	153	2	+	+	CCONJ
ejpam-5492	153	3	v	v	ADJ
ejpam-5492	153	4	cos	cos	ADP
ejpam-5492	153	5	θ	θ	PROPN
ejpam-5492	153	6	∂	∂	NUM
ejpam-5492	153	7	∂x4	∂x4	VERB
ejpam-5492	153	8	,	,	PUNCT
ejpam-5492	153	9	z4	z4	PROPN
ejpam-5492	153	10	=	=	SYM
ejpam-5492	153	11	−u	−u	PROPN
ejpam-5492	153	12	sinϕ	sinϕ	PROPN
ejpam-5492	153	13	∂	∂	PROPN
ejpam-5492	154	1	∂x5	∂x5	PROPN
ejpam-5492	154	2	+	+	CCONJ
ejpam-5492	154	3	u	u	NOUN
ejpam-5492	154	4	cosϕ	cosϕ	NOUN
ejpam-5492	154	5	∂	∂	NUM
ejpam-5492	154	6	∂x6	∂x6	NUM
ejpam-5492	154	7	−	−	PROPN
ejpam-5492	154	8	v	v	PROPN
ejpam-5492	154	9	sinϕ	sinϕ	PROPN
ejpam-5492	154	10	∂	∂	X
ejpam-5492	154	11	∂∂x7	∂∂x7	PROPN
ejpam-5492	154	12	+	+	CCONJ
ejpam-5492	154	13	v	v	NUM
ejpam-5492	154	14	cosϕ	cosϕ	NOUN
ejpam-5492	154	15	∂	∂	NUM
ejpam-5492	154	16	∂x8	∂x8	NUM
ejpam-5492	154	17	,	,	PUNCT
ejpam-5492	154	18	by	by	ADP
ejpam-5492	154	19	using	use	VERB
ejpam-5492	154	20	the	the	DET
ejpam-5492	154	21	metallic	metallic	ADJ
ejpam-5492	154	22	structure	structure	NOUN
ejpam-5492	154	23	j	j	PROPN
ejpam-5492	154	24	of	of	ADP
ejpam-5492	154	25	m̃	m̃	PROPN
ejpam-5492	154	26	which	which	PRON
ejpam-5492	154	27	defines	define	VERB
ejpam-5492	154	28	as	as	ADP
ejpam-5492	154	29	j(x1	j(x1	PROPN
ejpam-5492	154	30	,	,	PUNCT
ejpam-5492	154	31	x2	x2	PROPN
ejpam-5492	154	32	,	,	PUNCT
ejpam-5492	154	33	x3	x3	PROPN
ejpam-5492	154	34	,	,	PUNCT
ejpam-5492	154	35	x4	x4	PROPN
ejpam-5492	154	36	,	,	PUNCT
ejpam-5492	154	37	x5	x5	PROPN
ejpam-5492	154	38	,	,	PUNCT
ejpam-5492	154	39	x6	x6	PROPN
ejpam-5492	154	40	,	,	PUNCT
ejpam-5492	154	41	x7	x7	NOUN
ejpam-5492	154	42	,	,	PUNCT
ejpam-5492	154	43	x8,x9	x8,x9	PROPN
ejpam-5492	154	44	,	,	PUNCT
ejpam-5492	154	45	x10	x10	PROPN
ejpam-5492	154	46	,	,	PUNCT
ejpam-5492	154	47	x11	x11	PROPN
ejpam-5492	154	48	,	,	PUNCT
ejpam-5492	154	49	x12	x12	NUM
ejpam-5492	154	50	)	)	PUNCT
ejpam-5492	154	51	=	=	PRON
ejpam-5492	154	52	(	(	PUNCT
ejpam-5492	154	53	σx1	σx1	ADV
ejpam-5492	154	54	,	,	PUNCT
ejpam-5492	154	55	σx2	σx2	PROPN
ejpam-5492	154	56	,	,	PUNCT
ejpam-5492	154	57	σx3	σx3	PROPN
ejpam-5492	154	58	,	,	PUNCT
ejpam-5492	154	59	σx4	σx4	NOUN
ejpam-5492	154	60	,	,	PUNCT
ejpam-5492	154	61	σ̄x5	σ̄x5	NOUN
ejpam-5492	154	62	,	,	PUNCT
ejpam-5492	154	63	σ̄x6	σ̄x6	NOUN
ejpam-5492	154	64	,	,	PUNCT
ejpam-5492	154	65	σ̄x7	σ̄x7	PROPN
ejpam-5492	154	66	,	,	PUNCT
ejpam-5492	154	67	σ̄x8	σ̄x8	NOUN
ejpam-5492	154	68	,	,	PUNCT
ejpam-5492	154	69	σ̄x9	σ̄x9	NOUN
ejpam-5492	154	70	,	,	PUNCT
ejpam-5492	154	71	σx10	σx10	PROPN
ejpam-5492	154	72	,	,	PUNCT
ejpam-5492	154	73	σx11	σx11	PROPN
ejpam-5492	154	74	,	,	PUNCT
ejpam-5492	154	75	σ̄x12	σ̄x12	NOUN
ejpam-5492	154	76	)	)	PUNCT
ejpam-5492	154	77	.	.	PUNCT
ejpam-5492	155	1	following	follow	VERB
ejpam-5492	155	2	this	this	PRON
ejpam-5492	155	3	,	,	PUNCT
ejpam-5492	155	4	we	we	PRON
ejpam-5492	155	5	obtain	obtain	VERB
ejpam-5492	155	6	that	that	DET
ejpam-5492	155	7	jz1	jz1	PROPN
ejpam-5492	155	8	=	=	SYM
ejpam-5492	155	9	σ	σ	PROPN
ejpam-5492	155	10	cos	cos	PROPN
ejpam-5492	155	11	θ	θ	PROPN
ejpam-5492	155	12	∂	∂	X
ejpam-5492	155	13	∂x1	∂x1	NOUN
ejpam-5492	155	14	+	+	CCONJ
ejpam-5492	155	15	σ	σ	PROPN
ejpam-5492	155	16	sin	sin	NOUN
ejpam-5492	155	17	θ	θ	PROPN
ejpam-5492	155	18	∂	∂	NUM
ejpam-5492	155	19	∂x2	∂x2	NOUN
ejpam-5492	155	20	+	+	CCONJ
ejpam-5492	156	1	σ̃	σ̃	PROPN
ejpam-5492	156	2	cosϕ	cosϕ	NOUN
ejpam-5492	156	3	∂	∂	NUM
ejpam-5492	156	4	∂x5	∂x5	NOUN
ejpam-5492	157	1	+	+	CCONJ
ejpam-5492	157	2	σ̃	σ̃	PROPN
ejpam-5492	157	3	sinϕ	sinϕ	NOUN
ejpam-5492	157	4	∂	∂	NOUN
ejpam-5492	157	5	∂x6	∂x6	NOUN
ejpam-5492	158	1	+	+	CCONJ
ejpam-5492	158	2	σ̃σ	σ̃σ	ADP
ejpam-5492	158	3	√	√	VERB
ejpam-5492	158	4	q	q	NOUN
ejpam-5492	158	5	∂	∂	NOUN
ejpam-5492	158	6	∂x9	∂x9	PROPN
ejpam-5492	158	7	+	+	CCONJ
ejpam-5492	158	8	σ̃σ	σ̃σ	ADP
ejpam-5492	158	9	√	√	VERB
ejpam-5492	158	10	q	q	SYM
ejpam-5492	158	11	∂	∂	NOUN
ejpam-5492	158	12	∂x11	∂x11	PROPN
ejpam-5492	158	13	,	,	PUNCT
ejpam-5492	158	14	jz2	jz2	PROPN
ejpam-5492	158	15	=	=	PROPN
ejpam-5492	158	16	σ	σ	PROPN
ejpam-5492	158	17	cos	cos	PROPN
ejpam-5492	158	18	θ	θ	PROPN
ejpam-5492	158	19	∂	∂	NUM
ejpam-5492	158	20	∂x3	∂x3	ADV
ejpam-5492	158	21	+	+	CCONJ
ejpam-5492	158	22	σ	σ	NUM
ejpam-5492	158	23	sin	sin	NOUN
ejpam-5492	158	24	θ	θ	PROPN
ejpam-5492	158	25	∂	∂	X
ejpam-5492	158	26	∂x4	∂x4	VERB
ejpam-5492	158	27	+	+	CCONJ
ejpam-5492	159	1	σ̃	σ̃	PROPN
ejpam-5492	159	2	cosϕ	cosϕ	NOUN
ejpam-5492	159	3	∂	∂	NOUN
ejpam-5492	159	4	∂x7	∂x7	NOUN
ejpam-5492	159	5	+	+	CCONJ
ejpam-5492	159	6	σ̃	σ̃	PROPN
ejpam-5492	159	7	sinϕ	sinϕ	NOUN
ejpam-5492	159	8	∂	∂	NOUN
ejpam-5492	159	9	∂x8	∂x8	X
ejpam-5492	159	10	+	+	CCONJ
ejpam-5492	159	11	σ̃σ	σ̃σ	ADP
ejpam-5492	159	12	√	√	VERB
ejpam-5492	159	13	q	q	NOUN
ejpam-5492	159	14	∂	∂	NUM
ejpam-5492	159	15	∂x10	∂x10	NOUN
ejpam-5492	160	1	+	+	CCONJ
ejpam-5492	160	2	σ̃σ	σ̃σ	ADP
ejpam-5492	160	3	√	√	VERB
ejpam-5492	160	4	q	q	NOUN
ejpam-5492	160	5	∂	∂	NUM
ejpam-5492	160	6	∂x12	∂x12	NOUN
ejpam-5492	160	7	,	,	PUNCT
ejpam-5492	160	8	jz3	jz3	NOUN
ejpam-5492	160	9	=	=	SYM
ejpam-5492	160	10	−uσ	−uσ	ADJ
ejpam-5492	160	11	sin	sin	NOUN
ejpam-5492	160	12	θ	θ	NOUN
ejpam-5492	160	13	∂	∂	NOUN
ejpam-5492	161	1	∂x1	∂x1	NOUN
ejpam-5492	161	2	+	+	CCONJ
ejpam-5492	161	3	uσ	uσ	PROPN
ejpam-5492	161	4	cos	cos	PROPN
ejpam-5492	161	5	θ	θ	PROPN
ejpam-5492	161	6	∂	∂	NUM
ejpam-5492	161	7	∂x2	∂x2	NOUN
ejpam-5492	161	8	−	−	NOUN
ejpam-5492	161	9	vσ	vσ	NOUN
ejpam-5492	161	10	sin	sin	PROPN
ejpam-5492	161	11	θ	θ	PROPN
ejpam-5492	161	12	∂	∂	NUM
ejpam-5492	161	13	∂∂x3	∂∂x3	PROPN
ejpam-5492	162	1	+	+	CCONJ
ejpam-5492	162	2	vσ	vσ	PROPN
ejpam-5492	162	3	cos	cos	PROPN
ejpam-5492	162	4	θ	θ	PROPN
ejpam-5492	162	5	∂	∂	X
ejpam-5492	162	6	∂x4	∂x4	VERB
ejpam-5492	162	7	,	,	PUNCT
ejpam-5492	162	8	jz4	jz4	PROPN
ejpam-5492	162	9	=	=	PUNCT
ejpam-5492	162	10	−uσ̃	−uσ̃	PROPN
ejpam-5492	162	11	sinϕ	sinϕ	PROPN
ejpam-5492	162	12	∂	∂	PROPN
ejpam-5492	162	13	∂x5	∂x5	PROPN
ejpam-5492	163	1	+	+	CCONJ
ejpam-5492	163	2	uσ̃	uσ̃	NUM
ejpam-5492	163	3	cosϕ	cosϕ	NOUN
ejpam-5492	163	4	∂	∂	NUM
ejpam-5492	163	5	∂x6	∂x6	NOUN
ejpam-5492	163	6	−	−	PROPN
ejpam-5492	164	1	vσ̃	vσ̃	PROPN
ejpam-5492	164	2	sinϕ	sinϕ	PROPN
ejpam-5492	164	3	∂	∂	PROPN
ejpam-5492	165	1	∂∂x7	∂∂x7	PROPN
ejpam-5492	165	2	+	+	CCONJ
ejpam-5492	165	3	vσ̃	vσ̃	PROPN
ejpam-5492	165	4	cosϕ	cosϕ	NOUN
ejpam-5492	165	5	∂	∂	NUM
ejpam-5492	165	6	∂x8	∂x8	NOUN
ejpam-5492	165	7	,	,	PUNCT
ejpam-5492	165	8	now	now	ADV
ejpam-5492	165	9	,	,	PUNCT
ejpam-5492	165	10	we	we	PRON
ejpam-5492	165	11	define	define	VERB
ejpam-5492	165	12	two	two	NUM
ejpam-5492	165	13	vector	vector	NOUN
ejpam-5492	165	14	spaces	space	NOUN
ejpam-5492	165	15	dt	dt	NOUN
ejpam-5492	165	16	and	and	CCONJ
ejpam-5492	165	17	d⊥	d⊥	NOUN
ejpam-5492	165	18	such	such	ADJ
ejpam-5492	165	19	that	that	SCONJ
ejpam-5492	165	20	the	the	DET
ejpam-5492	165	21	invariant	invariant	ADJ
ejpam-5492	165	22	distribution	distribution	NOUN
ejpam-5492	165	23	dt	dt	NOUN
ejpam-5492	165	24	is	be	AUX
ejpam-5492	165	25	spanned	span	VERB
ejpam-5492	165	26	by	by	ADP
ejpam-5492	165	27	{	{	PUNCT
ejpam-5492	165	28	z3	z3	PROPN
ejpam-5492	165	29	,	,	PUNCT
ejpam-5492	165	30	z4	z4	PROPN
ejpam-5492	165	31	}	}	PUNCT
ejpam-5492	165	32	and	and	CCONJ
ejpam-5492	165	33	the	the	DET
ejpam-5492	165	34	anti	anti	ADJ
ejpam-5492	165	35	-	-	ADJ
ejpam-5492	165	36	invariant	invariant	ADJ
ejpam-5492	165	37	distribution	distribution	NOUN
ejpam-5492	165	38	d⊥	d⊥	NOUN
ejpam-5492	165	39	is	be	AUX
ejpam-5492	165	40	spanned	span	VERB
ejpam-5492	165	41	by	by	ADP
ejpam-5492	165	42	{	{	PUNCT
ejpam-5492	165	43	z1	z1	PROPN
ejpam-5492	165	44	,	,	PUNCT
ejpam-5492	165	45	z2	z2	PROPN
ejpam-5492	165	46	}	}	PUNCT
ejpam-5492	165	47	.	.	PUNCT
ejpam-5492	166	1	these	these	DET
ejpam-5492	166	2	distributions	distribution	NOUN
ejpam-5492	166	3	are	be	AUX
ejpam-5492	166	4	preserved	preserve	VERB
ejpam-5492	166	5	by	by	ADP
ejpam-5492	166	6	the	the	DET
ejpam-5492	166	7	action	action	NOUN
ejpam-5492	166	8	of	of	ADP
ejpam-5492	166	9	j	j	PROPN
ejpam-5492	166	10	.	.	PUNCT
ejpam-5492	167	1	the	the	DET
ejpam-5492	167	2	riemannian	riemannian	ADJ
ejpam-5492	167	3	metric	metric	NOUN
ejpam-5492	167	4	of	of	ADP
ejpam-5492	167	5	the	the	DET
ejpam-5492	167	6	warped	warped	ADJ
ejpam-5492	167	7	product	product	NOUN
ejpam-5492	167	8	cr	cr	NOUN
ejpam-5492	167	9	-	-	PUNCT
ejpam-5492	167	10	submanifold	submanifold	NOUN
ejpam-5492	167	11	m	m	VERB
ejpam-5492	167	12	is	be	AUX
ejpam-5492	167	13	given	give	VERB
ejpam-5492	167	14	by	by	ADP
ejpam-5492	167	15	the	the	DET
ejpam-5492	167	16	following	follow	VERB
ejpam-5492	167	17	l.	l.	PROPN
ejpam-5492	167	18	alqahtani	alqahtani	PROPN
ejpam-5492	167	19	,	,	PUNCT
ejpam-5492	167	20	e.	e.	PROPN
ejpam-5492	167	21	al	al	PROPN
ejpam-5492	167	22	-	-	PROPN
ejpam-5492	167	23	husainy	husainy	PROPN
ejpam-5492	167	24	/	/	SYM
ejpam-5492	167	25	eur	eur	PROPN
ejpam-5492	167	26	.	.	PUNCT
ejpam-5492	168	1	j.	j.	PROPN
ejpam-5492	168	2	pure	pure	PROPN
ejpam-5492	168	3	appl	appl	PROPN
ejpam-5492	168	4	.	.	PROPN
ejpam-5492	168	5	math	math	PROPN
ejpam-5492	168	6	,	,	PUNCT
ejpam-5492	168	7	17	17	NUM
ejpam-5492	168	8	(	(	PUNCT
ejpam-5492	168	9	4	4	NUM
ejpam-5492	168	10	)	)	PUNCT
ejpam-5492	168	11	(	(	PUNCT
ejpam-5492	168	12	2024	2024	NUM
ejpam-5492	168	13	)	)	PUNCT
ejpam-5492	168	14	,	,	PUNCT
ejpam-5492	168	15	2481	2481	NUM
ejpam-5492	168	16	-	-	SYM
ejpam-5492	168	17	2491	2491	NUM
ejpam-5492	168	18	2488	2488	NUM
ejpam-5492	168	19	g	g	NOUN
ejpam-5492	168	20	=	=	PUNCT
ejpam-5492	168	21	(	(	PUNCT
ejpam-5492	168	22	2	2	NUM
ejpam-5492	168	23	+	+	SYM
ejpam-5492	168	24	1	1	NUM
ejpam-5492	168	25	q	q	NOUN
ejpam-5492	168	26	σ̃2	σ̃2	PROPN
ejpam-5492	168	27	+	+	CCONJ
ejpam-5492	168	28	1	1	NUM
ejpam-5492	168	29	q	q	NOUN
ejpam-5492	168	30	σ2)(d2u+	σ2)(d2u+	NOUN
ejpam-5492	168	31	d2v	d2v	PROPN
ejpam-5492	168	32	)	)	PUNCT
ejpam-5492	169	1	+	+	CCONJ
ejpam-5492	169	2	(	(	PUNCT
ejpam-5492	169	3	u2	u2	NOUN
ejpam-5492	169	4	+	+	CCONJ
ejpam-5492	169	5	v2)(d2θ	v2)(d2θ	PRON
ejpam-5492	169	6	+	+	CCONJ
ejpam-5492	169	7	d2ϕ	d2ϕ	PROPN
ejpam-5492	169	8	)	)	PUNCT
ejpam-5492	169	9	.	.	PUNCT
ejpam-5492	170	1	in	in	ADP
ejpam-5492	170	2	other	other	ADJ
ejpam-5492	170	3	words	word	NOUN
ejpam-5492	170	4	,	,	PUNCT
ejpam-5492	170	5	m	m	VERB
ejpam-5492	170	6	=	=	VERB
ejpam-5492	170	7	m⊥	m⊥	PROPN
ejpam-5492	170	8	×f	×f	PROPN
ejpam-5492	170	9	mt	mt	PROPN
ejpam-5492	170	10	is	be	AUX
ejpam-5492	170	11	a	a	DET
ejpam-5492	170	12	warped	warped	ADJ
ejpam-5492	170	13	product	product	NOUN
ejpam-5492	170	14	cr	cr	NOUN
ejpam-5492	170	15	-	-	PUNCT
ejpam-5492	170	16	submanifold	submanifold	NOUN
ejpam-5492	170	17	in	in	ADP
ejpam-5492	170	18	the	the	DET
ejpam-5492	170	19	metallic	metallic	ADJ
ejpam-5492	170	20	riemannian	riemannian	NOUN
ejpam-5492	170	21	manifold	manifold	ADJ
ejpam-5492	170	22	m̃	m̃	PROPN
ejpam-5492	170	23	.	.	PUNCT
ejpam-5492	171	1	4	4	X
ejpam-5492	171	2	.	.	X
ejpam-5492	171	3	main	main	ADJ
ejpam-5492	171	4	theorem	theorem	NOUN
ejpam-5492	171	5	in	in	ADP
ejpam-5492	171	6	this	this	DET
ejpam-5492	171	7	section	section	NOUN
ejpam-5492	171	8	,	,	PUNCT
ejpam-5492	171	9	we	we	PRON
ejpam-5492	171	10	prove	prove	VERB
ejpam-5492	171	11	the	the	DET
ejpam-5492	171	12	main	main	ADJ
ejpam-5492	171	13	result	result	NOUN
ejpam-5492	171	14	,	,	PUNCT
ejpam-5492	171	15	which	which	PRON
ejpam-5492	171	16	is	be	AUX
ejpam-5492	171	17	based	base	VERB
ejpam-5492	171	18	on	on	ADP
ejpam-5492	171	19	lemma	lemma	PROPN
ejpam-5492	171	20	1	1	NUM
ejpam-5492	171	21	.	.	PUNCT
ejpam-5492	172	1	now	now	ADV
ejpam-5492	172	2	,	,	PUNCT
ejpam-5492	172	3	we	we	PRON
ejpam-5492	172	4	can	can	AUX
ejpam-5492	172	5	define	define	VERB
ejpam-5492	172	6	a	a	DET
ejpam-5492	172	7	canonical	canonical	ADJ
ejpam-5492	172	8	frame	frame	NOUN
ejpam-5492	172	9	field	field	NOUN
ejpam-5492	172	10	for	for	ADP
ejpam-5492	172	11	an	an	DET
ejpam-5492	172	12	n	n	ADV
ejpam-5492	172	13	-	-	PUNCT
ejpam-5492	172	14	dimensional	dimensional	ADJ
ejpam-5492	172	15	warped	warped	ADJ
ejpam-5492	172	16	product	product	NOUN
ejpam-5492	172	17	cr	cr	NOUN
ejpam-5492	172	18	-	-	PUNCT
ejpam-5492	172	19	submanifold	submanifold	NOUN
ejpam-5492	172	20	m	m	NOUN
ejpam-5492	172	21	=	=	ADJ
ejpam-5492	172	22	m⊥	m⊥	NOUN
ejpam-5492	172	23	×	×	NOUN
ejpam-5492	172	24	fmt	fmt	NOUN
ejpam-5492	172	25	of	of	ADP
ejpam-5492	172	26	an	an	DET
ejpam-5492	172	27	m	m	ADV
ejpam-5492	172	28	-	-	ADJ
ejpam-5492	172	29	dimensional	dimensional	ADJ
ejpam-5492	172	30	locally	locally	ADV
ejpam-5492	172	31	metallic	metallic	ADJ
ejpam-5492	172	32	riemannian	riemannian	ADJ
ejpam-5492	172	33	manifold	manifold	ADJ
ejpam-5492	172	34	m̃	m̃	PROPN
ejpam-5492	172	35	.	.	PUNCT
ejpam-5492	173	1	let	let	VERB
ejpam-5492	173	2	dim(mt	dim(mt	PRON
ejpam-5492	173	3	)	)	PUNCT
ejpam-5492	174	1	=	=	SYM
ejpam-5492	174	2	n1	n1	NOUN
ejpam-5492	174	3	and	and	CCONJ
ejpam-5492	174	4	dim(m⊥	dim(m⊥	PROPN
ejpam-5492	174	5	)	)	PUNCT
ejpam-5492	174	6	=	=	SYM
ejpam-5492	174	7	n2	n2	NOUN
ejpam-5492	174	8	,	,	PUNCT
ejpam-5492	174	9	and	and	CCONJ
ejpam-5492	174	10	so	so	ADV
ejpam-5492	174	11	n	n	NOUN
ejpam-5492	174	12	=	=	SYM
ejpam-5492	174	13	n1	n1	PROPN
ejpam-5492	174	14	+	+	CCONJ
ejpam-5492	174	15	n2	n2	ADJ
ejpam-5492	174	16	.	.	PUNCT
ejpam-5492	175	1	also	also	ADV
ejpam-5492	175	2	,	,	PUNCT
ejpam-5492	175	3	let	let	VERB
ejpam-5492	175	4	d	d	NOUN
ejpam-5492	175	5	and	and	CCONJ
ejpam-5492	175	6	d⊥	d⊥	NOUN
ejpam-5492	175	7	be	be	VERB
ejpam-5492	175	8	the	the	DET
ejpam-5492	175	9	tangent	tangent	NOUN
ejpam-5492	175	10	bundles	bundle	NOUN
ejpam-5492	175	11	of	of	ADP
ejpam-5492	175	12	mt	mt	PROPN
ejpam-5492	175	13	and	and	CCONJ
ejpam-5492	175	14	m⊥	m⊥	NOUN
ejpam-5492	175	15	,	,	PUNCT
ejpam-5492	175	16	respectively	respectively	ADV
ejpam-5492	175	17	.	.	PUNCT
ejpam-5492	176	1	the	the	DET
ejpam-5492	176	2	canonical	canonical	ADJ
ejpam-5492	176	3	frame	frame	NOUN
ejpam-5492	176	4	field	field	NOUN
ejpam-5492	176	5	of	of	ADP
ejpam-5492	176	6	d	d	NOUN
ejpam-5492	176	7	and	and	CCONJ
ejpam-5492	176	8	d⊥	d⊥	NOUN
ejpam-5492	176	9	are	be	AUX
ejpam-5492	176	10	given	give	VERB
ejpam-5492	176	11	by	by	ADP
ejpam-5492	176	12	the	the	DET
ejpam-5492	176	13	orthonormal	orthonormal	ADJ
ejpam-5492	176	14	vectors	vector	NOUN
ejpam-5492	176	15	{	{	PUNCT
ejpam-5492	176	16	e1	e1	PROPN
ejpam-5492	176	17	,	,	PUNCT
ejpam-5492	176	18	e2	e2	PROPN
ejpam-5492	176	19	,	,	PUNCT
ejpam-5492	176	20	·	·	PUNCT
ejpam-5492	176	21	·	·	PUNCT
ejpam-5492	176	22	·	·	PUNCT
ejpam-5492	176	23	,	,	PUNCT
ejpam-5492	176	24	et	et	NOUN
ejpam-5492	176	25	,	,	PUNCT
ejpam-5492	176	26	et+1	et+1	PROPN
ejpam-5492	176	27	=	=	SYM
ejpam-5492	176	28	je1	je1	PROPN
ejpam-5492	176	29	σ	σ	PROPN
ejpam-5492	176	30	,	,	PUNCT
ejpam-5492	176	31	et+2	et+2	PROPN
ejpam-5492	176	32	=	=	SYM
ejpam-5492	176	33	je2	je2	PROPN
ejpam-5492	176	34	σ	σ	PROPN
ejpam-5492	176	35	,	,	PUNCT
ejpam-5492	176	36	·	·	PUNCT
ejpam-5492	176	37	·	·	PUNCT
ejpam-5492	176	38	·	·	PUNCT
ejpam-5492	176	39	,	,	PUNCT
ejpam-5492	176	40	e2	e2	PROPN
ejpam-5492	176	41	t	t	PROPN
ejpam-5492	176	42	=	=	SYM
ejpam-5492	176	43	en1	en1	PROPN
ejpam-5492	176	44	=	=	PUNCT
ejpam-5492	176	45	jet	jet	NOUN
ejpam-5492	176	46	σ	σ	PROPN
ejpam-5492	176	47	}	}	PUNCT
ejpam-5492	176	48	,	,	PUNCT
ejpam-5492	176	49	(	(	PUNCT
ejpam-5492	176	50	32	32	NUM
ejpam-5492	176	51	)	)	PUNCT
ejpam-5492	176	52	{	{	PUNCT
ejpam-5492	176	53	en1	en1	VERB
ejpam-5492	176	54	+	+	NOUN
ejpam-5492	176	55	1	1	NUM
ejpam-5492	176	56	,	,	PUNCT
ejpam-5492	176	57	en1	en1	ADJ
ejpam-5492	176	58	+	+	NOUN
ejpam-5492	176	59	2	2	NUM
ejpam-5492	176	60	,	,	PUNCT
ejpam-5492	176	61	·	·	PUNCT
ejpam-5492	176	62	·	·	PUNCT
ejpam-5492	176	63	·	·	PUNCT
ejpam-5492	176	64	,	,	PUNCT
ejpam-5492	176	65	en	en	PROPN
ejpam-5492	176	66	=	=	PROPN
ejpam-5492	176	67	n1+n2	n1+n2	PROPN
ejpam-5492	176	68	}	}	PUNCT
ejpam-5492	176	69	,	,	PUNCT
ejpam-5492	176	70	(	(	PUNCT
ejpam-5492	176	71	33	33	NUM
ejpam-5492	176	72	)	)	PUNCT
ejpam-5492	176	73	respectively	respectively	ADV
ejpam-5492	176	74	,	,	PUNCT
ejpam-5492	176	75	where	where	SCONJ
ejpam-5492	176	76	σ	σ	PROPN
ejpam-5492	176	77	is	be	AUX
ejpam-5492	176	78	a	a	DET
ejpam-5492	176	79	metallic	metallic	ADJ
ejpam-5492	176	80	number	number	NOUN
ejpam-5492	176	81	.	.	PUNCT
ejpam-5492	177	1	on	on	ADP
ejpam-5492	177	2	the	the	DET
ejpam-5492	177	3	other	other	ADJ
ejpam-5492	177	4	hand	hand	NOUN
ejpam-5492	177	5	,	,	PUNCT
ejpam-5492	177	6	the	the	DET
ejpam-5492	177	7	orthonormal	orthonormal	ADJ
ejpam-5492	177	8	frame	frame	NOUN
ejpam-5492	177	9	fields	field	NOUN
ejpam-5492	177	10	of	of	ADP
ejpam-5492	177	11	the	the	DET
ejpam-5492	177	12	normal	normal	ADJ
ejpam-5492	177	13	subbundles	subbundle	NOUN
ejpam-5492	177	14	of	of	ADP
ejpam-5492	177	15	jd⊥	jd⊥	PROPN
ejpam-5492	177	16	and	and	CCONJ
ejpam-5492	177	17	µ	µ	PROPN
ejpam-5492	177	18	are	be	AUX
ejpam-5492	177	19	respectively	respectively	ADV
ejpam-5492	177	20	{	{	PUNCT
ejpam-5492	177	21	e∗1	e∗1	ADJ
ejpam-5492	177	22	=	=	PUNCT
ejpam-5492	177	23	jen1	jen1	PROPN
ejpam-5492	177	24	+	+	PROPN
ejpam-5492	177	25	1	1	NUM
ejpam-5492	177	26	,	,	PUNCT
ejpam-5492	177	27	e	e	NOUN
ejpam-5492	177	28	∗	∗	NOUN
ejpam-5492	177	29	2	2	NUM
ejpam-5492	177	30	=	=	SYM
ejpam-5492	177	31	jen1	jen1	PROPN
ejpam-5492	177	32	+	+	PROPN
ejpam-5492	177	33	2	2	NUM
ejpam-5492	177	34	,	,	PUNCT
ejpam-5492	177	35	·	·	PUNCT
ejpam-5492	177	36	·	·	PUNCT
ejpam-5492	177	37	·	·	PUNCT
ejpam-5492	177	38	,	,	PUNCT
ejpam-5492	177	39	e∗n2	e∗n2	PROPN
ejpam-5492	177	40	=	=	SYM
ejpam-5492	177	41	jen	jen	PROPN
ejpam-5492	177	42	=	=	PROPN
ejpam-5492	177	43	n1+n2	n1+n2	PROPN
ejpam-5492	177	44	}	}	PUNCT
ejpam-5492	177	45	(	(	PUNCT
ejpam-5492	177	46	34	34	NUM
ejpam-5492	177	47	)	)	PUNCT
ejpam-5492	177	48	{	{	PUNCT
ejpam-5492	177	49	e∗n2	e∗n2	PROPN
ejpam-5492	177	50	+	+	PROPN
ejpam-5492	177	51	1	1	NUM
ejpam-5492	177	52	=	=	SYM
ejpam-5492	177	53	en+n2	en+n2	NOUN
ejpam-5492	177	54	+	+	NOUN
ejpam-5492	177	55	1	1	NUM
ejpam-5492	177	56	,	,	PUNCT
ejpam-5492	177	57	e	e	NOUN
ejpam-5492	177	58	∗	∗	NOUN
ejpam-5492	177	59	n2	n2	X
ejpam-5492	177	60	+	+	PROPN
ejpam-5492	177	61	2	2	NUM
ejpam-5492	177	62	=	=	SYM
ejpam-5492	177	63	en+n2	en+n2	NOUN
ejpam-5492	177	64	+	+	NOUN
ejpam-5492	177	65	2	2	NUM
ejpam-5492	177	66	,	,	PUNCT
ejpam-5492	177	67	...	...	PUNCT
ejpam-5492	177	68	,	,	PUNCT
ejpam-5492	177	69	e	e	NOUN
ejpam-5492	177	70	∗	∗	NOUN
ejpam-5492	177	71	m−n	m−n	NOUN
ejpam-5492	177	72	=	=	PUNCT
ejpam-5492	177	73	em	em	PROPN
ejpam-5492	177	74	}	}	PUNCT
ejpam-5492	177	75	(	(	PUNCT
ejpam-5492	177	76	35	35	NUM
ejpam-5492	177	77	)	)	PUNCT
ejpam-5492	177	78	theorem	theorem	NOUN
ejpam-5492	177	79	2	2	NUM
ejpam-5492	177	80	.	.	PUNCT
ejpam-5492	178	1	let	let	VERB
ejpam-5492	178	2	m	m	PRON
ejpam-5492	178	3	be	be	AUX
ejpam-5492	178	4	a	a	DET
ejpam-5492	178	5	warped	warped	ADJ
ejpam-5492	178	6	product	product	NOUN
ejpam-5492	178	7	cr	cr	NOUN
ejpam-5492	178	8	-	-	PUNCT
ejpam-5492	178	9	submanifold	submanifold	NOUN
ejpam-5492	178	10	of	of	ADP
ejpam-5492	178	11	a	a	DET
ejpam-5492	178	12	locally	locally	ADV
ejpam-5492	178	13	metallic	metallic	ADJ
ejpam-5492	178	14	riemannian	riemannian	NOUN
ejpam-5492	178	15	manifold	manifold	NOUN
ejpam-5492	178	16	(	(	PUNCT
ejpam-5492	178	17	m̃	m̃	PROPN
ejpam-5492	178	18	,	,	PUNCT
ejpam-5492	178	19	g̃	g̃	PROPN
ejpam-5492	178	20	,	,	PUNCT
ejpam-5492	178	21	j	j	PROPN
ejpam-5492	178	22	)	)	PUNCT
ejpam-5492	178	23	,	,	PUNCT
ejpam-5492	178	24	where	where	SCONJ
ejpam-5492	178	25	mt	mt	PROPN
ejpam-5492	178	26	and	and	CCONJ
ejpam-5492	178	27	m⊥	m⊥	NOUN
ejpam-5492	178	28	are	be	AUX
ejpam-5492	178	29	invariant	invariant	ADJ
ejpam-5492	178	30	and	and	CCONJ
ejpam-5492	178	31	anti	anti	ADJ
ejpam-5492	178	32	-	-	ADJ
ejpam-5492	178	33	invariant	invariant	ADJ
ejpam-5492	178	34	submanifolds	submanifold	NOUN
ejpam-5492	178	35	of	of	ADP
ejpam-5492	178	36	m̃	m̃	PROPN
ejpam-5492	178	37	,	,	PUNCT
ejpam-5492	178	38	respectively	respectively	ADV
ejpam-5492	178	39	.	.	PUNCT
ejpam-5492	179	1	in	in	ADP
ejpam-5492	179	2	accordance	accordance	NOUN
ejpam-5492	179	3	with	with	ADP
ejpam-5492	179	4	this	this	PRON
ejpam-5492	179	5	,	,	PUNCT
ejpam-5492	179	6	we	we	PRON
ejpam-5492	179	7	have	have	VERB
ejpam-5492	179	8	(	(	PUNCT
ejpam-5492	179	9	i	i	NOUN
ejpam-5492	179	10	)	)	PUNCT
ejpam-5492	179	11	the	the	DET
ejpam-5492	179	12	squared	square	VERB
ejpam-5492	179	13	norm	norm	NOUN
ejpam-5492	179	14	of	of	ADP
ejpam-5492	179	15	the	the	DET
ejpam-5492	179	16	second	second	ADJ
ejpam-5492	179	17	fundamental	fundamental	ADJ
ejpam-5492	179	18	form	form	NOUN
ejpam-5492	179	19	of	of	ADP
ejpam-5492	179	20	m	m	PROPN
ejpam-5492	179	21	satisfies	satisfie	NOUN
ejpam-5492	179	22	the	the	DET
ejpam-5492	179	23	following	follow	VERB
ejpam-5492	179	24	inequality	inequality	NOUN
ejpam-5492	179	25	∥h∥2	∥h∥2	ADV
ejpam-5492	179	26	≥	≥	NOUN
ejpam-5492	179	27	(	(	PUNCT
ejpam-5492	179	28	p2	p2	PROPN
ejpam-5492	179	29	2	2	NUM
ejpam-5492	179	30	+	+	NUM
ejpam-5492	179	31	q2	q2	NOUN
ejpam-5492	179	32	σ2	σ2	PROPN
ejpam-5492	179	33	)	)	PUNCT
ejpam-5492	179	34	n1∥∇⃗⊥	n1∥∇⃗⊥	PROPN
ejpam-5492	179	35	ln	ln	ADJ
ejpam-5492	179	36	f∥2	f∥2	NOUN
ejpam-5492	179	37	,	,	PUNCT
ejpam-5492	179	38	(	(	PUNCT
ejpam-5492	179	39	36	36	NUM
ejpam-5492	179	40	)	)	PUNCT
ejpam-5492	179	41	where	where	SCONJ
ejpam-5492	179	42	dim(mt	dim(mt	PUNCT
ejpam-5492	179	43	)	)	PUNCT
ejpam-5492	180	1	=	=	SYM
ejpam-5492	180	2	n1	n1	NOUN
ejpam-5492	180	3	,	,	PUNCT
ejpam-5492	180	4	dim(m⊥	dim(m⊥	PROPN
ejpam-5492	180	5	)	)	PUNCT
ejpam-5492	180	6	=	=	SYM
ejpam-5492	180	7	n2	n2	NOUN
ejpam-5492	180	8	,	,	PUNCT
ejpam-5492	180	9	and	and	CCONJ
ejpam-5492	180	10	∇⃗⊥	∇⃗⊥	PROPN
ejpam-5492	181	1	ln	ln	PROPN
ejpam-5492	181	2	f	f	PROPN
ejpam-5492	181	3	is	be	AUX
ejpam-5492	181	4	gradient	gradient	ADJ
ejpam-5492	181	5	of	of	ADP
ejpam-5492	181	6	ln	ln	ADJ
ejpam-5492	181	7	f	f	PROPN
ejpam-5492	181	8	in	in	ADP
ejpam-5492	181	9	the	the	DET
ejpam-5492	181	10	normal	normal	ADJ
ejpam-5492	181	11	direction	direction	NOUN
ejpam-5492	181	12	to	to	ADP
ejpam-5492	181	13	m	m	PROPN
ejpam-5492	181	14	.	.	PUNCT
ejpam-5492	182	1	(	(	PUNCT
ejpam-5492	182	2	ii	ii	NOUN
ejpam-5492	182	3	)	)	PUNCT
ejpam-5492	182	4	if	if	SCONJ
ejpam-5492	182	5	equality	equality	NOUN
ejpam-5492	182	6	sign	sign	NOUN
ejpam-5492	182	7	,	,	PUNCT
ejpam-5492	182	8	in	in	ADP
ejpam-5492	182	9	the	the	DET
ejpam-5492	182	10	above	above	ADJ
ejpam-5492	182	11	inequality	inequality	NOUN
ejpam-5492	182	12	,	,	PUNCT
ejpam-5492	182	13	holds	hold	VERB
ejpam-5492	182	14	identically	identically	ADV
ejpam-5492	182	15	,	,	PUNCT
ejpam-5492	182	16	then	then	ADV
ejpam-5492	182	17	m⊥	m⊥	PROPN
ejpam-5492	182	18	and	and	CCONJ
ejpam-5492	182	19	mt	mt	PROPN
ejpam-5492	182	20	are	be	AUX
ejpam-5492	182	21	totally	totally	ADV
ejpam-5492	182	22	geodesic	geodesic	ADJ
ejpam-5492	182	23	and	and	CCONJ
ejpam-5492	182	24	totally	totally	ADV
ejpam-5492	182	25	umbilical	umbilical	ADJ
ejpam-5492	182	26	submanifolds	submanifold	NOUN
ejpam-5492	182	27	of	of	ADP
ejpam-5492	182	28	m̃	m̃	PROPN
ejpam-5492	182	29	,	,	PUNCT
ejpam-5492	182	30	respectively	respectively	ADV
ejpam-5492	182	31	.	.	PUNCT
ejpam-5492	183	1	proof	proof	NOUN
ejpam-5492	183	2	.	.	PUNCT
ejpam-5492	184	1	as	as	SCONJ
ejpam-5492	184	2	can	can	AUX
ejpam-5492	184	3	be	be	AUX
ejpam-5492	184	4	inferred	infer	VERB
ejpam-5492	184	5	from	from	ADP
ejpam-5492	184	6	the	the	DET
ejpam-5492	184	7	definition	definition	NOUN
ejpam-5492	184	8	of	of	ADP
ejpam-5492	184	9	h	h	NOUN
ejpam-5492	184	10	,	,	PUNCT
ejpam-5492	184	11	we	we	PRON
ejpam-5492	184	12	have	have	VERB
ejpam-5492	184	13	∥h∥2	∥h∥2	ADJ
ejpam-5492	184	14	=	=	SYM
ejpam-5492	184	15	n∑	n∑	NOUN
ejpam-5492	185	1	i	i	PROPN
ejpam-5492	185	2	,	,	PUNCT
ejpam-5492	185	3	j=1	j=1	PROPN
ejpam-5492	185	4	g(h(ei	g(h(ei	PROPN
ejpam-5492	185	5	,	,	PUNCT
ejpam-5492	185	6	ej	ej	NOUN
ejpam-5492	185	7	)	)	PUNCT
ejpam-5492	185	8	,	,	PUNCT
ejpam-5492	185	9	h(ei	h(ei	PROPN
ejpam-5492	185	10	,	,	PUNCT
ejpam-5492	185	11	ej	ej	NOUN
ejpam-5492	185	12	)	)	PUNCT
ejpam-5492	185	13	)	)	PUNCT
ejpam-5492	186	1	=	=	PUNCT
ejpam-5492	186	2	m−n∑	m−n∑	PART
ejpam-5492	186	3	r=1	r=1	PROPN
ejpam-5492	186	4	n∑	n∑	PROPN
ejpam-5492	186	5	i	i	PROPN
ejpam-5492	186	6	,	,	PUNCT
ejpam-5492	186	7	j=1	j=1	PROPN
ejpam-5492	186	8	g(h(ei	g(h(ei	PROPN
ejpam-5492	186	9	,	,	PUNCT
ejpam-5492	186	10	ej	ej	NOUN
ejpam-5492	186	11	)	)	PUNCT
ejpam-5492	186	12	,	,	PUNCT
ejpam-5492	186	13	e	e	NOUN
ejpam-5492	186	14	∗	∗	NOUN
ejpam-5492	186	15	r	r	NOUN
ejpam-5492	186	16	)	)	PUNCT
ejpam-5492	186	17	2	2	NUM
ejpam-5492	186	18	.	.	PUNCT
ejpam-5492	186	19	(	(	PUNCT
ejpam-5492	186	20	37	37	NUM
ejpam-5492	186	21	)	)	PUNCT
ejpam-5492	186	22	l.	l.	PROPN
ejpam-5492	186	23	alqahtani	alqahtani	PROPN
ejpam-5492	186	24	,	,	PUNCT
ejpam-5492	186	25	e.	e.	PROPN
ejpam-5492	186	26	al	al	PROPN
ejpam-5492	186	27	-	-	PROPN
ejpam-5492	186	28	husainy	husainy	PROPN
ejpam-5492	186	29	/	/	SYM
ejpam-5492	186	30	eur	eur	PROPN
ejpam-5492	186	31	.	.	PUNCT
ejpam-5492	187	1	j.	j.	PROPN
ejpam-5492	187	2	pure	pure	PROPN
ejpam-5492	187	3	appl	appl	PROPN
ejpam-5492	187	4	.	.	PROPN
ejpam-5492	187	5	math	math	PROPN
ejpam-5492	187	6	,	,	PUNCT
ejpam-5492	187	7	17	17	NUM
ejpam-5492	187	8	(	(	PUNCT
ejpam-5492	187	9	4	4	NUM
ejpam-5492	187	10	)	)	PUNCT
ejpam-5492	187	11	(	(	PUNCT
ejpam-5492	187	12	2024	2024	NUM
ejpam-5492	187	13	)	)	PUNCT
ejpam-5492	187	14	,	,	PUNCT
ejpam-5492	187	15	2481	2481	NUM
ejpam-5492	187	16	-	-	SYM
ejpam-5492	187	17	2491	2491	NUM
ejpam-5492	187	18	2489	2489	NUM
ejpam-5492	187	19	in	in	ADP
ejpam-5492	187	20	view	view	NOUN
ejpam-5492	187	21	of	of	ADP
ejpam-5492	187	22	the	the	DET
ejpam-5492	187	23	above	above	ADJ
ejpam-5492	187	24	equation	equation	NOUN
ejpam-5492	187	25	and	and	CCONJ
ejpam-5492	187	26	the	the	DET
ejpam-5492	187	27	definitions	definition	NOUN
ejpam-5492	187	28	of	of	ADP
ejpam-5492	187	29	the	the	DET
ejpam-5492	187	30	frame	frame	NOUN
ejpam-5492	187	31	fields	field	NOUN
ejpam-5492	187	32	of	of	ADP
ejpam-5492	187	33	d	d	PROPN
ejpam-5492	187	34	,	,	PUNCT
ejpam-5492	187	35	d⊥	d⊥	PROPN
ejpam-5492	187	36	,	,	PUNCT
ejpam-5492	187	37	jd⊥	jd⊥	PROPN
ejpam-5492	187	38	and	and	CCONJ
ejpam-5492	187	39	µ	µ	NOUN
ejpam-5492	187	40	,	,	PUNCT
ejpam-5492	187	41	we	we	PRON
ejpam-5492	187	42	can	can	AUX
ejpam-5492	187	43	derive	derive	VERB
ejpam-5492	187	44	the	the	DET
ejpam-5492	187	45	following	following	NOUN
ejpam-5492	187	46	:	:	PUNCT
ejpam-5492	187	47	∥h∥2	∥h∥2	ADJ
ejpam-5492	187	48	=	=	SYM
ejpam-5492	187	49	n2∑	n2∑	PROPN
ejpam-5492	187	50	r=1	r=1	NOUN
ejpam-5492	187	51	n1∑	n1∑	PROPN
ejpam-5492	187	52	i	i	PRON
ejpam-5492	187	53	,	,	PUNCT
ejpam-5492	187	54	j=1	j=1	PROPN
ejpam-5492	187	55	g(h(ei	g(h(ei	PROPN
ejpam-5492	187	56	,	,	PUNCT
ejpam-5492	187	57	ej	ej	NOUN
ejpam-5492	187	58	)	)	PUNCT
ejpam-5492	187	59	,	,	PUNCT
ejpam-5492	187	60	e	e	NOUN
ejpam-5492	187	61	∗	∗	NOUN
ejpam-5492	187	62	r	r	NOUN
ejpam-5492	187	63	)	)	PUNCT
ejpam-5492	187	64	2	2	NUM
ejpam-5492	188	1	+	+	CCONJ
ejpam-5492	188	2	m−n∑	m−n∑	X
ejpam-5492	188	3	r	r	NOUN
ejpam-5492	188	4	=	=	NOUN
ejpam-5492	188	5	n2	n2	ADJ
ejpam-5492	188	6	+	+	PROPN
ejpam-5492	188	7	1	1	PROPN
ejpam-5492	188	8	n1∑	n1∑	PROPN
ejpam-5492	188	9	i	i	PRON
ejpam-5492	188	10	,	,	PUNCT
ejpam-5492	188	11	j=1	j=1	PROPN
ejpam-5492	188	12	g(h(ei	g(h(ei	PROPN
ejpam-5492	188	13	,	,	PUNCT
ejpam-5492	188	14	ej	ej	NOUN
ejpam-5492	188	15	)	)	PUNCT
ejpam-5492	188	16	,	,	PUNCT
ejpam-5492	188	17	e	e	NOUN
ejpam-5492	188	18	∗	∗	NOUN
ejpam-5492	188	19	r	r	NOUN
ejpam-5492	188	20	)	)	PUNCT
ejpam-5492	188	21	2	2	NUM
ejpam-5492	188	22	+	+	SYM
ejpam-5492	188	23	2	2	NUM
ejpam-5492	188	24	n2∑	n2∑	PROPN
ejpam-5492	188	25	r=1	r=1	NOUN
ejpam-5492	188	26	n1∑	n1∑	PROPN
ejpam-5492	188	27	i=1	i=1	PROPN
ejpam-5492	188	28	n∑	n∑	PROPN
ejpam-5492	188	29	j	j	PROPN
ejpam-5492	189	1	=	=	NOUN
ejpam-5492	189	2	n1	n1	PROPN
ejpam-5492	189	3	+	+	NOUN
ejpam-5492	189	4	1	1	NUM
ejpam-5492	189	5	g(h(ei	g(h(ei	ADJ
ejpam-5492	189	6	,	,	PUNCT
ejpam-5492	189	7	ej	ej	NOUN
ejpam-5492	189	8	)	)	PUNCT
ejpam-5492	189	9	,	,	PUNCT
ejpam-5492	189	10	e	e	NOUN
ejpam-5492	189	11	∗	∗	NOUN
ejpam-5492	189	12	r	r	NOUN
ejpam-5492	189	13	)	)	PUNCT
ejpam-5492	189	14	2	2	NUM
ejpam-5492	189	15	+	+	CCONJ
ejpam-5492	189	16	2	2	NUM
ejpam-5492	189	17	m−n∑	m−n∑	NOUN
ejpam-5492	189	18	r	r	NOUN
ejpam-5492	189	19	=	=	NOUN
ejpam-5492	189	20	n2	n2	ADJ
ejpam-5492	189	21	+	+	PROPN
ejpam-5492	189	22	1	1	NUM
ejpam-5492	189	23	n1∑	n1∑	PROPN
ejpam-5492	189	24	i=1	i=1	PROPN
ejpam-5492	189	25	n∑	n∑	PROPN
ejpam-5492	189	26	j	j	PROPN
ejpam-5492	190	1	=	=	NOUN
ejpam-5492	190	2	n1	n1	PROPN
ejpam-5492	190	3	+	+	NOUN
ejpam-5492	190	4	1	1	NUM
ejpam-5492	190	5	g(h(ei	g(h(ei	ADJ
ejpam-5492	190	6	,	,	PUNCT
ejpam-5492	190	7	ej	ej	NOUN
ejpam-5492	190	8	)	)	PUNCT
ejpam-5492	190	9	,	,	PUNCT
ejpam-5492	190	10	e	e	NOUN
ejpam-5492	190	11	∗	∗	NOUN
ejpam-5492	190	12	r	r	NOUN
ejpam-5492	190	13	)	)	PUNCT
ejpam-5492	190	14	2	2	NUM
ejpam-5492	190	15	+	+	NUM
ejpam-5492	190	16	n2∑	n2∑	PROPN
ejpam-5492	190	17	r=1	r=1	NOUN
ejpam-5492	190	18	n∑	n∑	PROPN
ejpam-5492	190	19	i	i	PROPN
ejpam-5492	190	20	,	,	PUNCT
ejpam-5492	190	21	j	j	PROPN
ejpam-5492	190	22	=	=	PROPN
ejpam-5492	190	23	n1	n1	PROPN
ejpam-5492	190	24	+	+	NOUN
ejpam-5492	190	25	1	1	NUM
ejpam-5492	190	26	g(h(ei	g(h(ei	ADJ
ejpam-5492	190	27	,	,	PUNCT
ejpam-5492	190	28	ej	ej	NOUN
ejpam-5492	190	29	)	)	PUNCT
ejpam-5492	190	30	,	,	PUNCT
ejpam-5492	190	31	e	e	NOUN
ejpam-5492	190	32	∗	∗	NOUN
ejpam-5492	190	33	r	r	NOUN
ejpam-5492	190	34	)	)	PUNCT
ejpam-5492	190	35	2	2	NUM
ejpam-5492	191	1	+	+	CCONJ
ejpam-5492	191	2	m−n∑	m−n∑	X
ejpam-5492	191	3	r	r	NOUN
ejpam-5492	191	4	=	=	NOUN
ejpam-5492	191	5	n2	n2	ADJ
ejpam-5492	191	6	+	+	PROPN
ejpam-5492	191	7	1	1	NUM
ejpam-5492	191	8	n∑	n∑	NOUN
ejpam-5492	191	9	i	i	PROPN
ejpam-5492	191	10	,	,	PUNCT
ejpam-5492	191	11	j	j	PROPN
ejpam-5492	191	12	=	=	PROPN
ejpam-5492	191	13	n1	n1	PROPN
ejpam-5492	191	14	+	+	NOUN
ejpam-5492	191	15	1	1	NUM
ejpam-5492	191	16	g(h(ei	g(h(ei	ADJ
ejpam-5492	191	17	,	,	PUNCT
ejpam-5492	191	18	ej	ej	NOUN
ejpam-5492	191	19	)	)	PUNCT
ejpam-5492	191	20	,	,	PUNCT
ejpam-5492	191	21	e	e	NOUN
ejpam-5492	191	22	∗	∗	NOUN
ejpam-5492	191	23	r	r	NOUN
ejpam-5492	191	24	)	)	PUNCT
ejpam-5492	191	25	2	2	NUM
ejpam-5492	191	26	.	.	PUNCT
ejpam-5492	192	1	(	(	PUNCT
ejpam-5492	192	2	38	38	NUM
ejpam-5492	192	3	)	)	PUNCT
ejpam-5492	192	4	leaving	leave	VERB
ejpam-5492	192	5	the	the	DET
ejpam-5492	192	6	µ-components	µ-component	NOUN
ejpam-5492	192	7	in	in	ADP
ejpam-5492	192	8	(	(	PUNCT
ejpam-5492	192	9	38	38	NUM
ejpam-5492	192	10	)	)	PUNCT
ejpam-5492	192	11	,	,	PUNCT
ejpam-5492	192	12	and	and	CCONJ
ejpam-5492	192	13	using	use	VERB
ejpam-5492	192	14	(	(	PUNCT
ejpam-5492	192	15	21	21	NUM
ejpam-5492	192	16	)	)	PUNCT
ejpam-5492	192	17	,	,	PUNCT
ejpam-5492	192	18	we	we	PRON
ejpam-5492	192	19	obtain	obtain	VERB
ejpam-5492	192	20	that	that	DET
ejpam-5492	192	21	∥h∥2	∥h∥2	ADJ
ejpam-5492	192	22	≥	≥	NOUN
ejpam-5492	192	23	n2∑	n2∑	PROPN
ejpam-5492	192	24	r=1	r=1	NOUN
ejpam-5492	192	25	n1∑	n1∑	PROPN
ejpam-5492	192	26	i	i	PRON
ejpam-5492	192	27	,	,	PUNCT
ejpam-5492	192	28	j=1	j=1	PROPN
ejpam-5492	192	29	g(h(ei	g(h(ei	PROPN
ejpam-5492	192	30	,	,	PUNCT
ejpam-5492	192	31	ej	ej	NOUN
ejpam-5492	192	32	)	)	PUNCT
ejpam-5492	192	33	,	,	PUNCT
ejpam-5492	192	34	e	e	NOUN
ejpam-5492	192	35	∗	∗	NOUN
ejpam-5492	192	36	r	r	NOUN
ejpam-5492	192	37	)	)	PUNCT
ejpam-5492	192	38	2	2	NUM
ejpam-5492	192	39	.	.	PUNCT
ejpam-5492	192	40	with	with	ADP
ejpam-5492	192	41	the	the	DET
ejpam-5492	192	42	help	help	NOUN
ejpam-5492	192	43	of	of	ADP
ejpam-5492	192	44	equations	equation	NOUN
ejpam-5492	192	45	(	(	PUNCT
ejpam-5492	192	46	32	32	NUM
ejpam-5492	192	47	)	)	PUNCT
ejpam-5492	192	48	then	then	ADV
ejpam-5492	192	49	(	(	PUNCT
ejpam-5492	192	50	34	34	NUM
ejpam-5492	192	51	)	)	PUNCT
ejpam-5492	192	52	,	,	PUNCT
ejpam-5492	192	53	we	we	PRON
ejpam-5492	192	54	can	can	AUX
ejpam-5492	192	55	find	find	VERB
ejpam-5492	192	56	∥h∥2	∥h∥2	ADJ
ejpam-5492	192	57	≥	≥	NOUN
ejpam-5492	192	58	n∑	n∑	NOUN
ejpam-5492	192	59	r	r	NOUN
ejpam-5492	192	60	=	=	NOUN
ejpam-5492	192	61	n1	n1	NOUN
ejpam-5492	192	62	+	+	NOUN
ejpam-5492	192	63	1	1	NUM
ejpam-5492	192	64	t∑	t∑	ADJ
ejpam-5492	192	65	i	i	PRON
ejpam-5492	192	66	,	,	PUNCT
ejpam-5492	192	67	j=1	j=1	PROPN
ejpam-5492	192	68	g(h(ei	g(h(ei	PROPN
ejpam-5492	192	69	,	,	PUNCT
ejpam-5492	192	70	ej	ej	NOUN
ejpam-5492	192	71	)	)	PUNCT
ejpam-5492	192	72	,	,	PUNCT
ejpam-5492	192	73	jer	jer	PROPN
ejpam-5492	192	74	)	)	PUNCT
ejpam-5492	192	75	2	2	NUM
ejpam-5492	193	1	+	+	SYM
ejpam-5492	193	2	2	2	NUM
ejpam-5492	193	3	n∑	n∑	NOUN
ejpam-5492	193	4	r	r	NOUN
ejpam-5492	193	5	=	=	NOUN
ejpam-5492	193	6	n1	n1	NOUN
ejpam-5492	193	7	+	+	NOUN
ejpam-5492	193	8	1	1	NUM
ejpam-5492	193	9	t∑	t∑	ADJ
ejpam-5492	193	10	i=1	i=1	PROPN
ejpam-5492	194	1	n1∑	n1∑	PROPN
ejpam-5492	194	2	j	j	PROPN
ejpam-5492	195	1	=	=	PROPN
ejpam-5492	195	2	t+1	t+1	PROPN
ejpam-5492	195	3	g(h(ei	g(h(ei	PROPN
ejpam-5492	195	4	,	,	PUNCT
ejpam-5492	195	5	ej	ej	NOUN
ejpam-5492	195	6	)	)	PUNCT
ejpam-5492	195	7	,	,	PUNCT
ejpam-5492	195	8	jer	jer	PROPN
ejpam-5492	195	9	)	)	PUNCT
ejpam-5492	195	10	2	2	NUM
ejpam-5492	195	11	+	+	NUM
ejpam-5492	195	12	n∑	n∑	ADJ
ejpam-5492	195	13	r	r	NOUN
ejpam-5492	195	14	=	=	NOUN
ejpam-5492	195	15	n1	n1	ADJ
ejpam-5492	195	16	+	+	NOUN
ejpam-5492	195	17	1	1	NUM
ejpam-5492	195	18	n1∑	n1∑	PROPN
ejpam-5492	195	19	i	i	PRON
ejpam-5492	195	20	,	,	PUNCT
ejpam-5492	195	21	j	j	PROPN
ejpam-5492	195	22	=	=	PROPN
ejpam-5492	195	23	t+1	t+1	PROPN
ejpam-5492	195	24	g(h(ei	g(h(ei	PROPN
ejpam-5492	195	25	,	,	PUNCT
ejpam-5492	195	26	ej	ej	NOUN
ejpam-5492	195	27	)	)	PUNCT
ejpam-5492	195	28	,	,	PUNCT
ejpam-5492	195	29	jer	jer	PROPN
ejpam-5492	195	30	)	)	PUNCT
ejpam-5492	195	31	2	2	NUM
ejpam-5492	195	32	.	.	PUNCT
ejpam-5492	195	33	(	(	PUNCT
ejpam-5492	195	34	39	39	NUM
ejpam-5492	195	35	)	)	PUNCT
ejpam-5492	195	36	using	use	VERB
ejpam-5492	195	37	(	(	PUNCT
ejpam-5492	195	38	20	20	NUM
ejpam-5492	195	39	)	)	PUNCT
ejpam-5492	195	40	with	with	ADP
ejpam-5492	195	41	the	the	DET
ejpam-5492	195	42	help	help	NOUN
ejpam-5492	195	43	the	the	DET
ejpam-5492	195	44	canonical	canonical	ADJ
ejpam-5492	195	45	frame	frame	NOUN
ejpam-5492	195	46	field	field	NOUN
ejpam-5492	195	47	of	of	ADP
ejpam-5492	195	48	d	d	PROPN
ejpam-5492	195	49	for	for	ADP
ejpam-5492	195	50	all	all	DET
ejpam-5492	195	51	r	r	NOUN
ejpam-5492	195	52	=	=	SYM
ejpam-5492	195	53	1	1	NUM
ejpam-5492	195	54	,	,	PUNCT
ejpam-5492	195	55	·	·	PUNCT
ejpam-5492	195	56	·	·	PUNCT
ejpam-5492	195	57	·	·	PUNCT
ejpam-5492	195	58	,	,	PUNCT
ejpam-5492	195	59	n1	n1	PROPN
ejpam-5492	195	60	,	,	PUNCT
ejpam-5492	195	61	we	we	PRON
ejpam-5492	195	62	can	can	AUX
ejpam-5492	195	63	simplify	simplify	VERB
ejpam-5492	195	64	the	the	DET
ejpam-5492	195	65	last	last	ADJ
ejpam-5492	195	66	inequality	inequality	NOUN
ejpam-5492	195	67	as	as	SCONJ
ejpam-5492	195	68	follows	follow	VERB
ejpam-5492	195	69	∥h∥2	∥h∥2	ADJ
ejpam-5492	195	70	≥	≥	NOUN
ejpam-5492	195	71	n∑	n∑	NOUN
ejpam-5492	195	72	r	r	NOUN
ejpam-5492	195	73	=	=	SYM
ejpam-5492	195	74	n1	n1	ADJ
ejpam-5492	195	75	+	+	NOUN
ejpam-5492	195	76	1	1	NUM
ejpam-5492	195	77	n1∑	n1∑	PROPN
ejpam-5492	195	78	j	j	PROPN
ejpam-5492	195	79	=	=	PRON
ejpam-5492	195	80	t+1	t+1	PROPN
ejpam-5492	195	81	t∑	t∑	PROPN
ejpam-5492	195	82	i=1	i=1	PROPN
ejpam-5492	195	83	(	(	PUNCT
ejpam-5492	195	84	σ(er	σ(er	X
ejpam-5492	195	85	ln	ln	X
ejpam-5492	195	86	f)g(ei	f)g(ei	PROPN
ejpam-5492	195	87	,	,	PUNCT
ejpam-5492	195	88	ej	ej	NOUN
ejpam-5492	195	89	)	)	PUNCT
ejpam-5492	195	90	)	)	PUNCT
ejpam-5492	195	91	2	2	NUM
ejpam-5492	196	1	+	+	SYM
ejpam-5492	196	2	2	2	NUM
ejpam-5492	196	3	n∑	n∑	NOUN
ejpam-5492	196	4	r	r	NOUN
ejpam-5492	196	5	=	=	NOUN
ejpam-5492	196	6	n1	n1	NOUN
ejpam-5492	196	7	+	+	NOUN
ejpam-5492	196	8	1	1	NUM
ejpam-5492	196	9	t∑	t∑	ADJ
ejpam-5492	196	10	i	i	PRON
ejpam-5492	196	11	,	,	PUNCT
ejpam-5492	196	12	j=1	j=1	PROPN
ejpam-5492	196	13	(	(	PUNCT
ejpam-5492	196	14	(	(	PUNCT
ejpam-5492	196	15	er	er	INTJ
ejpam-5492	196	16	ln	ln	ADJ
ejpam-5492	196	17	f)g(ei	f)g(ei	NOUN
ejpam-5492	196	18	,	,	PUNCT
ejpam-5492	196	19	j2ej	j2ej	X
ejpam-5492	196	20	σ	σ	NOUN
ejpam-5492	196	21	)	)	PUNCT
ejpam-5492	196	22	)	)	PUNCT
ejpam-5492	196	23	2	2	NUM
ejpam-5492	197	1	+	+	NUM
ejpam-5492	197	2	n∑	n∑	ADJ
ejpam-5492	197	3	r	r	NOUN
ejpam-5492	197	4	=	=	NOUN
ejpam-5492	197	5	n1	n1	NOUN
ejpam-5492	197	6	+	+	NOUN
ejpam-5492	197	7	1	1	NUM
ejpam-5492	197	8	t∑	t∑	ADJ
ejpam-5492	197	9	i	i	PRON
ejpam-5492	197	10	,	,	PUNCT
ejpam-5492	197	11	j=1	j=1	PROPN
ejpam-5492	197	12	(	(	PUNCT
ejpam-5492	197	13	(	(	PUNCT
ejpam-5492	197	14	er	er	INTJ
ejpam-5492	197	15	ln	ln	INTJ
ejpam-5492	197	16	f)g	f)g	X
ejpam-5492	197	17	(	(	PUNCT
ejpam-5492	197	18	jei	jei	PROPN
ejpam-5492	197	19	σ	σ	PROPN
ejpam-5492	197	20	,	,	PUNCT
ejpam-5492	197	21	j2ej	j2ej	PROPN
ejpam-5492	197	22	σ	σ	NOUN
ejpam-5492	197	23	)	)	PUNCT
ejpam-5492	197	24	)	)	PUNCT
ejpam-5492	197	25	2	2	X
ejpam-5492	197	26	.	.	X
ejpam-5492	197	27	from	from	ADP
ejpam-5492	197	28	(	(	PUNCT
ejpam-5492	197	29	1	1	NUM
ejpam-5492	197	30	)	)	PUNCT
ejpam-5492	197	31	,	,	PUNCT
ejpam-5492	197	32	we	we	PRON
ejpam-5492	197	33	obtain	obtain	VERB
ejpam-5492	197	34	the	the	DET
ejpam-5492	197	35	following	following	NOUN
ejpam-5492	197	36	result	result	VERB
ejpam-5492	197	37	∥h∥2	∥h∥2	ADV
ejpam-5492	197	38	≥	≥	X
ejpam-5492	197	39	(	(	PUNCT
ejpam-5492	197	40	p2	p2	PROPN
ejpam-5492	197	41	2	2	NUM
ejpam-5492	197	42	+	+	NUM
ejpam-5492	197	43	q2	q2	NOUN
ejpam-5492	197	44	σ2	σ2	PROPN
ejpam-5492	197	45	)	)	PUNCT
ejpam-5492	197	46	2t∥∇⃗⊥	2t∥∇⃗⊥	PROPN
ejpam-5492	197	47	ln	ln	ADJ
ejpam-5492	197	48	f∥2	f∥2	NOUN
ejpam-5492	198	1	=	=	SYM
ejpam-5492	199	1	(	(	PUNCT
ejpam-5492	199	2	p2	p2	PROPN
ejpam-5492	199	3	2	2	NUM
ejpam-5492	199	4	+	+	NUM
ejpam-5492	199	5	q2	q2	NOUN
ejpam-5492	199	6	σ2	σ2	PROPN
ejpam-5492	199	7	)	)	PUNCT
ejpam-5492	199	8	n1∥∇⃗⊥	n1∥∇⃗⊥	PROPN
ejpam-5492	199	9	ln	ln	ADJ
ejpam-5492	199	10	f∥2	f∥2	NOUN
ejpam-5492	199	11	.	.	PUNCT
ejpam-5492	200	1	focusing	focus	VERB
ejpam-5492	200	2	on	on	ADP
ejpam-5492	200	3	the	the	DET
ejpam-5492	200	4	fifth	fifth	ADJ
ejpam-5492	200	5	and	and	CCONJ
ejpam-5492	200	6	sixth	sixth	ADJ
ejpam-5492	200	7	terms	term	NOUN
ejpam-5492	200	8	of	of	ADP
ejpam-5492	200	9	(	(	PUNCT
ejpam-5492	200	10	38	38	NUM
ejpam-5492	200	11	)	)	PUNCT
ejpam-5492	200	12	,	,	PUNCT
ejpam-5492	200	13	we	we	PRON
ejpam-5492	200	14	get	get	VERB
ejpam-5492	200	15	h(d⊥	h(d⊥	ADJ
ejpam-5492	200	16	,	,	PUNCT
ejpam-5492	200	17	d⊥	d⊥	NOUN
ejpam-5492	200	18	)	)	PUNCT
ejpam-5492	200	19	=	=	SYM
ejpam-5492	201	1	0	0	X
ejpam-5492	201	2	.	.	PUNCT
ejpam-5492	202	1	(	(	PUNCT
ejpam-5492	202	2	40	40	NUM
ejpam-5492	202	3	)	)	PUNCT
ejpam-5492	202	4	similarly	similarly	ADV
ejpam-5492	202	5	,	,	PUNCT
ejpam-5492	202	6	we	we	PRON
ejpam-5492	202	7	obtain	obtain	VERB
ejpam-5492	202	8	the	the	DET
ejpam-5492	202	9	following	following	NOUN
ejpam-5492	202	10	from	from	ADP
ejpam-5492	202	11	leaving	leave	VERB
ejpam-5492	202	12	the	the	DET
ejpam-5492	202	13	second	second	ADJ
ejpam-5492	202	14	term	term	NOUN
ejpam-5492	202	15	of	of	ADP
ejpam-5492	202	16	(	(	PUNCT
ejpam-5492	202	17	34	34	NUM
ejpam-5492	202	18	)	)	PUNCT
ejpam-5492	202	19	h(d	h(d	PROPN
ejpam-5492	202	20	,	,	PUNCT
ejpam-5492	202	21	d	d	NOUN
ejpam-5492	202	22	)	)	PUNCT
ejpam-5492	202	23	⊆	⊆	NUM
ejpam-5492	202	24	jd⊥.	jd⊥.	NOUN
ejpam-5492	202	25	(	(	PUNCT
ejpam-5492	202	26	41	41	NUM
ejpam-5492	202	27	)	)	PUNCT
ejpam-5492	202	28	references	reference	NOUN
ejpam-5492	202	29	2490	2490	NUM
ejpam-5492	202	30	while	while	SCONJ
ejpam-5492	202	31	the	the	DET
ejpam-5492	202	32	other	other	ADJ
ejpam-5492	202	33	terms	term	NOUN
ejpam-5492	202	34	in	in	ADP
ejpam-5492	202	35	(	(	PUNCT
ejpam-5492	202	36	34	34	NUM
ejpam-5492	202	37	)	)	PUNCT
ejpam-5492	202	38	vanish	vanish	VERB
ejpam-5492	202	39	,	,	PUNCT
ejpam-5492	202	40	the	the	DET
ejpam-5492	202	41	fourth	fourth	ADJ
ejpam-5492	202	42	term	term	NOUN
ejpam-5492	202	43	gives	give	VERB
ejpam-5492	202	44	the	the	DET
ejpam-5492	202	45	following	following	NOUN
ejpam-5492	202	46	:	:	PUNCT
ejpam-5492	202	47	h(d	h(d	NOUN
ejpam-5492	202	48	,	,	PUNCT
ejpam-5492	202	49	d⊥	d⊥	NOUN
ejpam-5492	202	50	)	)	PUNCT
ejpam-5492	202	51	⊆	⊆	NUM
ejpam-5492	202	52	jd⊥.	jd⊥.	NOUN
ejpam-5492	202	53	(	(	PUNCT
ejpam-5492	202	54	42	42	NUM
ejpam-5492	202	55	)	)	PUNCT
ejpam-5492	202	56	then	then	ADV
ejpam-5492	202	57	we	we	PRON
ejpam-5492	202	58	can	can	AUX
ejpam-5492	202	59	find	find	VERB
ejpam-5492	202	60	that	that	SCONJ
ejpam-5492	202	61	m⊥	m⊥	NOUN
ejpam-5492	202	62	is	be	AUX
ejpam-5492	202	63	totally	totally	ADV
ejpam-5492	202	64	geodesic	geodesic	ADJ
ejpam-5492	202	65	in	in	ADP
ejpam-5492	202	66	m̃	m̃	PROPN
ejpam-5492	202	67	due	due	ADP
ejpam-5492	202	68	to	to	ADP
ejpam-5492	202	69	its	its	PRON
ejpam-5492	202	70	totally	totally	ADV
ejpam-5492	202	71	geodesic	geodesic	NOUN
ejpam-5492	202	72	in	in	ADP
ejpam-5492	202	73	m	m	PROPN
ejpam-5492	202	74	and	and	CCONJ
ejpam-5492	202	75	(	(	PUNCT
ejpam-5492	202	76	40	40	NUM
ejpam-5492	202	77	)	)	PUNCT
ejpam-5492	202	78	.	.	PUNCT
ejpam-5492	203	1	similarly	similarly	ADV
ejpam-5492	203	2	,	,	PUNCT
ejpam-5492	203	3	equations	equation	NOUN
ejpam-5492	203	4	(	(	PUNCT
ejpam-5492	203	5	41	41	NUM
ejpam-5492	203	6	)	)	PUNCT
ejpam-5492	203	7	and	and	CCONJ
ejpam-5492	203	8	(	(	PUNCT
ejpam-5492	203	9	42	42	NUM
ejpam-5492	203	10	)	)	PUNCT
ejpam-5492	203	11	imply	imply	VERB
ejpam-5492	203	12	that	that	SCONJ
ejpam-5492	203	13	mt	mt	PROPN
ejpam-5492	203	14	is	be	AUX
ejpam-5492	203	15	totally	totally	ADV
ejpam-5492	203	16	umbilical	umbilical	ADJ
ejpam-5492	203	17	in	in	ADP
ejpam-5492	203	18	m̃	m̃	PROPN
ejpam-5492	203	19	due	due	ADP
ejpam-5492	203	20	to	to	ADP
ejpam-5492	203	21	mt	mt	PROPN
ejpam-5492	203	22	being	be	AUX
ejpam-5492	203	23	totally	totally	ADV
ejpam-5492	203	24	umbilical	umbilical	ADJ
ejpam-5492	203	25	in	in	ADP
ejpam-5492	203	26	m	m	PROPN
ejpam-5492	203	27	,	,	PUNCT
ejpam-5492	203	28	which	which	PRON
ejpam-5492	203	29	ends	end	VERB
ejpam-5492	203	30	the	the	DET
ejpam-5492	203	31	proof	proof	NOUN
ejpam-5492	203	32	.	.	PUNCT
ejpam-5492	204	1	clearly	clearly	ADV
ejpam-5492	204	2	,	,	PUNCT
ejpam-5492	204	3	theorem	theorem	ADJ
ejpam-5492	204	4	2	2	NUM
ejpam-5492	204	5	is	be	AUX
ejpam-5492	204	6	true	true	ADJ
ejpam-5492	204	7	for	for	SCONJ
ejpam-5492	204	8	the	the	DET
ejpam-5492	204	9	golden	golden	ADJ
ejpam-5492	204	10	riemannian	riemannian	NOUN
ejpam-5492	204	11	manifolds	manifold	NOUN
ejpam-5492	204	12	i.e.	i.e.	X
ejpam-5492	204	13	p	p	NOUN
ejpam-5492	204	14	=	=	NOUN
ejpam-5492	204	15	q	q	NOUN
ejpam-5492	204	16	=	=	NOUN
ejpam-5492	204	17	1	1	NUM
ejpam-5492	204	18	,	,	PUNCT
ejpam-5492	204	19	and	and	CCONJ
ejpam-5492	204	20	locally	locally	ADV
ejpam-5492	204	21	product	product	NOUN
ejpam-5492	204	22	riemannian	riemannian	NOUN
ejpam-5492	204	23	manifolds	manifold	VERB
ejpam-5492	204	24	p	p	NOUN
ejpam-5492	204	25	=	=	NOUN
ejpam-5492	204	26	0	0	NUM
ejpam-5492	204	27	,	,	PUNCT
ejpam-5492	204	28	q	q	NOUN
ejpam-5492	205	1	=	=	NOUN
ejpam-5492	205	2	1	1	NUM
ejpam-5492	205	3	.	.	NOUN
ejpam-5492	205	4	5	5	NUM
ejpam-5492	205	5	.	.	X
ejpam-5492	205	6	conclusion	conclusion	NOUN
ejpam-5492	205	7	the	the	DET
ejpam-5492	205	8	exploration	exploration	NOUN
ejpam-5492	205	9	of	of	ADP
ejpam-5492	205	10	warped	warped	ADJ
ejpam-5492	205	11	product	product	NOUN
ejpam-5492	205	12	submanifolds	submanifold	NOUN
ejpam-5492	205	13	in	in	ADP
ejpam-5492	205	14	locally	locally	ADV
ejpam-5492	205	15	metallic	metallic	ADJ
ejpam-5492	205	16	riemannian	riemannian	ADJ
ejpam-5492	205	17	manifolds	manifold	NOUN
ejpam-5492	205	18	opens	open	VERB
ejpam-5492	205	19	several	several	ADJ
ejpam-5492	205	20	intriguing	intriguing	ADJ
ejpam-5492	205	21	avenues	avenue	NOUN
ejpam-5492	205	22	for	for	ADP
ejpam-5492	205	23	future	future	ADJ
ejpam-5492	205	24	research	research	NOUN
ejpam-5492	205	25	.	.	PUNCT
ejpam-5492	206	1	in	in	ADP
ejpam-5492	206	2	particular	particular	ADJ
ejpam-5492	206	3	,	,	PUNCT
ejpam-5492	206	4	focusing	focus	VERB
ejpam-5492	206	5	on	on	ADP
ejpam-5492	206	6	warped	warped	ADJ
ejpam-5492	206	7	products	product	NOUN
ejpam-5492	206	8	formed	form	VERB
ejpam-5492	206	9	by	by	ADP
ejpam-5492	206	10	the	the	DET
ejpam-5492	206	11	product	product	NOUN
ejpam-5492	206	12	of	of	ADP
ejpam-5492	206	13	a	a	DET
ejpam-5492	206	14	proper	proper	ADJ
ejpam-5492	206	15	slant	slant	ADJ
ejpam-5492	206	16	submanifold	submanifold	NOUN
ejpam-5492	206	17	with	with	ADP
ejpam-5492	206	18	an	an	DET
ejpam-5492	206	19	invariant	invariant	ADJ
ejpam-5492	206	20	submanifold	submanifold	NOUN
ejpam-5492	206	21	,	,	PUNCT
ejpam-5492	206	22	called	call	VERB
ejpam-5492	206	23	warped	warped	ADJ
ejpam-5492	206	24	product	product	NOUN
ejpam-5492	206	25	semi	semi	ADJ
ejpam-5492	206	26	-	-	ADJ
ejpam-5492	206	27	slant	slant	ADJ
ejpam-5492	206	28	,	,	PUNCT
ejpam-5492	206	29	or	or	CCONJ
ejpam-5492	206	30	with	with	ADP
ejpam-5492	206	31	an	an	DET
ejpam-5492	206	32	anti	anti	ADJ
ejpam-5492	206	33	-	-	ADJ
ejpam-5492	206	34	invariant	invariant	ADJ
ejpam-5492	206	35	submanifold	submanifold	NOUN
ejpam-5492	206	36	,	,	PUNCT
ejpam-5492	206	37	called	call	VERB
ejpam-5492	206	38	warped	warped	ADJ
ejpam-5492	206	39	product	product	NOUN
ejpam-5492	206	40	hemi	hemi	NOUN
ejpam-5492	206	41	-	-	PUNCT
ejpam-5492	206	42	slant	slant	NOUN
ejpam-5492	206	43	,	,	PUNCT
ejpam-5492	206	44	opens	open	VERB
ejpam-5492	206	45	significant	significant	ADJ
ejpam-5492	206	46	aspects	aspect	NOUN
ejpam-5492	206	47	for	for	ADP
ejpam-5492	206	48	upcoming	upcoming	ADJ
ejpam-5492	206	49	studies	study	NOUN
ejpam-5492	206	50	.	.	PUNCT
ejpam-5492	207	1	further	further	ADJ
ejpam-5492	207	2	investigation	investigation	NOUN
ejpam-5492	207	3	into	into	ADP
ejpam-5492	207	4	the	the	DET
ejpam-5492	207	5	geometric	geometric	ADJ
ejpam-5492	207	6	properties	property	NOUN
ejpam-5492	207	7	and	and	CCONJ
ejpam-5492	207	8	characteristics	characteristic	NOUN
ejpam-5492	207	9	of	of	ADP
ejpam-5492	207	10	these	these	DET
ejpam-5492	207	11	submanifolds	submanifold	NOUN
ejpam-5492	207	12	can	can	AUX
ejpam-5492	207	13	deepen	deepen	VERB
ejpam-5492	207	14	our	our	PRON
ejpam-5492	207	15	understanding	understanding	NOUN
ejpam-5492	207	16	of	of	ADP
ejpam-5492	207	17	their	their	PRON
ejpam-5492	207	18	structure	structure	NOUN
ejpam-5492	207	19	and	and	CCONJ
ejpam-5492	207	20	the	the	DET
ejpam-5492	207	21	implications	implication	NOUN
ejpam-5492	207	22	of	of	ADP
ejpam-5492	207	23	the	the	DET
ejpam-5492	207	24	local	local	ADJ
ejpam-5492	207	25	metallicity	metallicity	NOUN
ejpam-5492	207	26	of	of	ADP
ejpam-5492	207	27	the	the	DET
ejpam-5492	207	28	ambient	ambient	NOUN
ejpam-5492	207	29	manifold	manifold	NOUN
ejpam-5492	207	30	.	.	PUNCT
ejpam-5492	208	1	additionally	additionally	ADV
ejpam-5492	208	2	,	,	PUNCT
ejpam-5492	208	3	establishing	establish	VERB
ejpam-5492	208	4	a	a	DET
ejpam-5492	208	5	comprehensive	comprehensive	ADJ
ejpam-5492	208	6	classification	classification	NOUN
ejpam-5492	208	7	of	of	ADP
ejpam-5492	208	8	warped	warped	ADJ
ejpam-5492	208	9	product	product	NOUN
ejpam-5492	208	10	submanifolds	submanifold	NOUN
ejpam-5492	208	11	,	,	PUNCT
ejpam-5492	208	12	alongside	alongside	ADP
ejpam-5492	208	13	concrete	concrete	ADJ
ejpam-5492	208	14	examples	example	NOUN
ejpam-5492	208	15	,	,	PUNCT
ejpam-5492	208	16	will	will	AUX
ejpam-5492	208	17	enhance	enhance	VERB
ejpam-5492	208	18	the	the	DET
ejpam-5492	208	19	literature	literature	NOUN
ejpam-5492	208	20	and	and	CCONJ
ejpam-5492	208	21	provide	provide	VERB
ejpam-5492	208	22	benchmarks	benchmark	NOUN
ejpam-5492	208	23	for	for	ADP
ejpam-5492	208	24	further	further	ADJ
ejpam-5492	208	25	studies	study	NOUN
ejpam-5492	208	26	.	.	PUNCT
ejpam-5492	209	1	obtaining	obtain	VERB
ejpam-5492	209	2	chen	chen	PROPN
ejpam-5492	209	3	’s	’s	PART
ejpam-5492	209	4	inequality	inequality	NOUN
ejpam-5492	209	5	for	for	ADP
ejpam-5492	209	6	semi	semi	ADJ
ejpam-5492	209	7	-	-	ADJ
ejpam-5492	209	8	slant	slant	ADJ
ejpam-5492	209	9	and	and	CCONJ
ejpam-5492	209	10	hemi	hemi	NOUN
ejpam-5492	209	11	-	-	PUNCT
ejpam-5492	209	12	slant	slant	ADJ
ejpam-5492	209	13	warped	warped	ADJ
ejpam-5492	209	14	product	product	NOUN
ejpam-5492	209	15	submanifolds	submanifold	NOUN
ejpam-5492	209	16	in	in	ADP
ejpam-5492	209	17	locally	locally	ADV
ejpam-5492	209	18	metallic	metallic	ADJ
ejpam-5492	209	19	riemannian	riemannian	ADJ
ejpam-5492	209	20	manifolds	manifold	NOUN
ejpam-5492	209	21	reveals	reveal	VERB
ejpam-5492	209	22	significant	significant	ADJ
ejpam-5492	209	23	insights	insight	NOUN
ejpam-5492	209	24	about	about	ADP
ejpam-5492	209	25	the	the	DET
ejpam-5492	209	26	relationship	relationship	NOUN
ejpam-5492	209	27	between	between	ADP
ejpam-5492	209	28	the	the	DET
ejpam-5492	209	29	second	second	ADJ
ejpam-5492	209	30	fundamental	fundamental	ADJ
ejpam-5492	209	31	form	form	NOUN
ejpam-5492	209	32	and	and	CCONJ
ejpam-5492	209	33	the	the	DET
ejpam-5492	209	34	warping	warp	VERB
ejpam-5492	209	35	function	function	NOUN
ejpam-5492	209	36	of	of	ADP
ejpam-5492	209	37	such	such	ADJ
ejpam-5492	209	38	submanifolds	submanifold	NOUN
ejpam-5492	209	39	.	.	PUNCT
ejpam-5492	210	1	this	this	DET
ejpam-5492	210	2	study	study	NOUN
ejpam-5492	210	3	also	also	ADV
ejpam-5492	210	4	raises	raise	VERB
ejpam-5492	210	5	the	the	DET
ejpam-5492	210	6	question	question	NOUN
ejpam-5492	210	7	of	of	ADP
ejpam-5492	210	8	whether	whether	SCONJ
ejpam-5492	210	9	warped	warped	ADJ
ejpam-5492	210	10	product	product	NOUN
ejpam-5492	210	11	pointwise	pointwise	VERB
ejpam-5492	210	12	semi	semi	ADJ
ejpam-5492	210	13	-	-	ADJ
ejpam-5492	210	14	slant	slant	ADJ
ejpam-5492	210	15	and	and	CCONJ
ejpam-5492	210	16	hemi	hemi	NOUN
ejpam-5492	210	17	-	-	PUNCT
ejpam-5492	210	18	slant	slant	NOUN
ejpam-5492	210	19	submanifolds	submanifold	NOUN
ejpam-5492	210	20	can	can	AUX
ejpam-5492	210	21	be	be	AUX
ejpam-5492	210	22	discussed	discuss	VERB
ejpam-5492	210	23	in	in	ADP
ejpam-5492	210	24	the	the	DET
ejpam-5492	210	25	context	context	NOUN
ejpam-5492	210	26	of	of	ADP
ejpam-5492	210	27	metallic	metallic	ADJ
ejpam-5492	210	28	riemannian	riemannian	ADJ
ejpam-5492	210	29	manifolds	manifold	NOUN
ejpam-5492	210	30	.	.	PUNCT
ejpam-5492	211	1	references	reference	NOUN
ejpam-5492	211	2	[	[	X
ejpam-5492	211	3	1	1	NUM
ejpam-5492	211	4	]	]	X
ejpam-5492	211	5	r.l	r.l	PROPN
ejpam-5492	211	6	.	.	PROPN
ejpam-5492	211	7	bishop	bishop	PROPN
ejpam-5492	211	8	and	and	CCONJ
ejpam-5492	211	9	b.	b.	PROPN
ejpam-5492	211	10	o’neill	o’neill	PROPN
ejpam-5492	211	11	.	.	PUNCT
ejpam-5492	212	1	manifolds	manifold	NOUN
ejpam-5492	212	2	of	of	ADP
ejpam-5492	212	3	negative	negative	ADJ
ejpam-5492	212	4	curvature	curvature	NOUN
ejpam-5492	212	5	.	.	PUNCT
ejpam-5492	213	1	trans	trans	AUX
ejpam-5492	213	2	.	.	PUNCT
ejpam-5492	213	3	am	be	AUX
ejpam-5492	213	4	.	.	PUNCT
ejpam-5492	214	1	math	math	NOUN
ejpam-5492	214	2	.	.	PUNCT
ejpam-5492	215	1	soc	soc	PROPN
ejpam-5492	215	2	.	.	PUNCT
ejpam-5492	215	3	,	,	PUNCT
ejpam-5492	215	4	145:1–49	145:1–49	NUM
ejpam-5492	215	5	,	,	PUNCT
ejpam-5492	215	6	1969	1969	NUM
ejpam-5492	215	7	.	.	PUNCT
ejpam-5492	216	1	[	[	X
ejpam-5492	216	2	2	2	NUM
ejpam-5492	216	3	]	]	PUNCT
ejpam-5492	216	4	a.m.	a.m.	NOUN
ejpam-5492	216	5	blaga	blaga	PROPN
ejpam-5492	216	6	and	and	CCONJ
ejpam-5492	216	7	c.e	c.e	PROPN
ejpam-5492	216	8	.	.	PROPN
ejpam-5492	216	9	hretcanu	hretcanu	PROPN
ejpam-5492	216	10	.	.	PUNCT
ejpam-5492	217	1	golden	golden	ADJ
ejpam-5492	217	2	warped	warped	ADJ
ejpam-5492	217	3	product	product	NOUN
ejpam-5492	217	4	riemannian	riemannian	NOUN
ejpam-5492	217	5	manifolds	manifold	NOUN
ejpam-5492	217	6	.	.	PUNCT
ejpam-5492	218	1	lib	lib	PROPN
ejpam-5492	218	2	.	.	PUNCT
ejpam-5492	219	1	math	math	PROPN
ejpam-5492	219	2	.	.	PUNCT
ejpam-5492	219	3	,	,	PUNCT
ejpam-5492	219	4	37:39–50	37:39–50	NUM
ejpam-5492	219	5	,	,	PUNCT
ejpam-5492	219	6	2018	2018	NUM
ejpam-5492	219	7	.	.	PUNCT
ejpam-5492	220	1	[	[	X
ejpam-5492	220	2	3	3	NUM
ejpam-5492	220	3	]	]	PUNCT
ejpam-5492	220	4	a.m.	a.m.	NOUN
ejpam-5492	220	5	blaga	blaga	PROPN
ejpam-5492	220	6	and	and	CCONJ
ejpam-5492	220	7	c.e	c.e	PROPN
ejpam-5492	220	8	.	.	PROPN
ejpam-5492	220	9	hretcanu	hretcanu	PROPN
ejpam-5492	220	10	.	.	PUNCT
ejpam-5492	221	1	invariant	invariant	ADJ
ejpam-5492	221	2	,	,	PUNCT
ejpam-5492	221	3	anti	anti	ADJ
ejpam-5492	221	4	-	-	ADJ
ejpam-5492	221	5	invariant	invariant	ADJ
ejpam-5492	221	6	and	and	CCONJ
ejpam-5492	221	7	slant	slant	ADJ
ejpam-5492	221	8	submanifolds	submanifold	NOUN
ejpam-5492	221	9	of	of	ADP
ejpam-5492	221	10	a	a	DET
ejpam-5492	221	11	metallic	metallic	ADJ
ejpam-5492	221	12	riemannian	riemannian	ADJ
ejpam-5492	221	13	manifold	manifold	NOUN
ejpam-5492	221	14	.	.	PUNCT
ejpam-5492	222	1	novi	novi	PROPN
ejpam-5492	222	2	.	.	PUNCT
ejpam-5492	223	1	sad	sad	PROPN
ejpam-5492	223	2	.	.	PUNCT
ejpam-5492	224	1	j.	j.	PROPN
ejpam-5492	224	2	math	math	PROPN
ejpam-5492	224	3	.	.	PUNCT
ejpam-5492	224	4	,	,	PUNCT
ejpam-5492	224	5	48:55–80	48:55–80	PROPN
ejpam-5492	224	6	,	,	PUNCT
ejpam-5492	224	7	2018	2018	NUM
ejpam-5492	224	8	.	.	PUNCT
ejpam-5492	225	1	[	[	X
ejpam-5492	225	2	4	4	NUM
ejpam-5492	225	3	]	]	PUNCT
ejpam-5492	225	4	a.m.	a.m.	NOUN
ejpam-5492	225	5	blaga	blaga	PROPN
ejpam-5492	225	6	and	and	CCONJ
ejpam-5492	225	7	c.e	c.e	PROPN
ejpam-5492	225	8	.	.	PROPN
ejpam-5492	225	9	hretcanu	hretcanu	PROPN
ejpam-5492	225	10	.	.	PUNCT
ejpam-5492	226	1	metallic	metallic	ADJ
ejpam-5492	226	2	conjugate	conjugate	ADJ
ejpam-5492	226	3	connections	connection	NOUN
ejpam-5492	226	4	.	.	PUNCT
ejpam-5492	227	1	rev	rev	PROPN
ejpam-5492	227	2	.	.	PROPN
ejpam-5492	227	3	un	un	PROPN
ejpam-5492	227	4	.	.	PROPN
ejpam-5492	227	5	mat	mat	PROPN
ejpam-5492	227	6	.	.	PROPN
ejpam-5492	227	7	argentina	argentina	PROPN
ejpam-5492	227	8	,	,	PUNCT
ejpam-5492	227	9	59:179–192	59:179–192	PROPN
ejpam-5492	227	10	,	,	PUNCT
ejpam-5492	227	11	2018	2018	NUM
ejpam-5492	227	12	.	.	PUNCT
ejpam-5492	228	1	[	[	X
ejpam-5492	228	2	5	5	NUM
ejpam-5492	228	3	]	]	PUNCT
ejpam-5492	228	4	b.-y	b.-y	NOUN
ejpam-5492	228	5	.	.	PUNCT
ejpam-5492	229	1	chen	chen	PROPN
ejpam-5492	229	2	.	.	PUNCT
ejpam-5492	229	3	geometry	geometry	NOUN
ejpam-5492	229	4	of	of	ADP
ejpam-5492	229	5	submanifolds	submanifolds	PROPN
ejpam-5492	229	6	.	.	PUNCT
ejpam-5492	230	1	marcel	marcel	PROPN
ejpam-5492	230	2	dekker	dekker	PROPN
ejpam-5492	230	3	inc	inc	PROPN
ejpam-5492	230	4	.	.	PROPN
ejpam-5492	230	5	,	,	PUNCT
ejpam-5492	230	6	new	new	PROPN
ejpam-5492	230	7	york	york	PROPN
ejpam-5492	230	8	,	,	PUNCT
ejpam-5492	230	9	1973	1973	NUM
ejpam-5492	230	10	.	.	PUNCT
ejpam-5492	231	1	references	reference	NOUN
ejpam-5492	231	2	2491	2491	NUM
ejpam-5492	231	3	[	[	X
ejpam-5492	231	4	6	6	NUM
ejpam-5492	231	5	]	]	ADJ
ejpam-5492	231	6	b.-y	b.-y	NOUN
ejpam-5492	231	7	.	.	PUNCT
ejpam-5492	232	1	chen	chen	PROPN
ejpam-5492	232	2	.	.	PUNCT
ejpam-5492	232	3	geometry	geometry	NOUN
ejpam-5492	232	4	of	of	ADP
ejpam-5492	232	5	warped	warped	ADJ
ejpam-5492	232	6	product	product	NOUN
ejpam-5492	232	7	cr	cr	NOUN
ejpam-5492	232	8	-	-	PUNCT
ejpam-5492	232	9	submanifolds	submanifold	NOUN
ejpam-5492	232	10	in	in	ADP
ejpam-5492	232	11	kaehler	kaehler	PROPN
ejpam-5492	232	12	manifolds	manifold	NOUN
ejpam-5492	232	13	.	.	PUNCT
ejpam-5492	233	1	monatsh	monatsh	PROPN
ejpam-5492	233	2	.	.	PUNCT
ejpam-5492	233	3	math	math	NOUN
ejpam-5492	233	4	.	.	PUNCT
ejpam-5492	233	5	,	,	PUNCT
ejpam-5492	233	6	133:177–195	133:177–195	NUM
ejpam-5492	233	7	,	,	PUNCT
ejpam-5492	233	8	2001	2001	NUM
ejpam-5492	233	9	.	.	PUNCT
ejpam-5492	234	1	[	[	X
ejpam-5492	234	2	7	7	NUM
ejpam-5492	234	3	]	]	PUNCT
ejpam-5492	234	4	b.-y	b.-y	NOUN
ejpam-5492	234	5	chen	chen	PROPN
ejpam-5492	234	6	.	.	PUNCT
ejpam-5492	235	1	geometry	geometry	NOUN
ejpam-5492	235	2	of	of	ADP
ejpam-5492	235	3	warped	warped	ADJ
ejpam-5492	235	4	product	product	NOUN
ejpam-5492	235	5	cr	cr	NOUN
ejpam-5492	235	6	-	-	PUNCT
ejpam-5492	235	7	submanifolds	submanifold	NOUN
ejpam-5492	235	8	in	in	ADP
ejpam-5492	235	9	kaehler	kaehler	PROPN
ejpam-5492	235	10	manifolds	manifolds	PROPN
ejpam-5492	235	11	,	,	PUNCT
ejpam-5492	235	12	ii	ii	PROPN
ejpam-5492	235	13	.	.	PROPN
ejpam-5492	235	14	monatsh	monatsh	PROPN
ejpam-5492	235	15	.	.	PUNCT
ejpam-5492	236	1	math	math	NOUN
ejpam-5492	236	2	.	.	PUNCT
ejpam-5492	236	3	,	,	PUNCT
ejpam-5492	236	4	134:103–119	134:103–119	NUM
ejpam-5492	236	5	,	,	PUNCT
ejpam-5492	236	6	2001	2001	NUM
ejpam-5492	236	7	.	.	PUNCT
ejpam-5492	237	1	[	[	X
ejpam-5492	237	2	8	8	NUM
ejpam-5492	237	3	]	]	PUNCT
ejpam-5492	237	4	b.-y	b.-y	NOUN
ejpam-5492	237	5	.	.	PUNCT
ejpam-5492	238	1	chen	chen	PROPN
ejpam-5492	238	2	.	.	PUNCT
ejpam-5492	238	3	geometry	geometry	NOUN
ejpam-5492	238	4	of	of	ADP
ejpam-5492	238	5	warped	warped	ADJ
ejpam-5492	238	6	products	product	NOUN
ejpam-5492	238	7	as	as	ADP
ejpam-5492	238	8	riemannian	riemannian	ADJ
ejpam-5492	238	9	submanifolds	submanifold	NOUN
ejpam-5492	238	10	and	and	CCONJ
ejpam-5492	238	11	related	related	ADJ
ejpam-5492	238	12	problems	problem	NOUN
ejpam-5492	238	13	.	.	PUNCT
ejpam-5492	239	1	soochow	soochow	VERB
ejpam-5492	239	2	j.math	j.math	PROPN
ejpam-5492	239	3	.	.	PROPN
ejpam-5492	239	4	,	,	PUNCT
ejpam-5492	239	5	28:125–156	28:125–156	PROPN
ejpam-5492	239	6	,	,	PUNCT
ejpam-5492	239	7	2002	2002	NUM
ejpam-5492	239	8	.	.	PUNCT
ejpam-5492	240	1	[	[	X
ejpam-5492	240	2	9	9	NUM
ejpam-5492	240	3	]	]	PUNCT
ejpam-5492	240	4	b.-y	b.-y	NOUN
ejpam-5492	240	5	.	.	PUNCT
ejpam-5492	241	1	chen	chen	PROPN
ejpam-5492	241	2	.	.	PUNCT
ejpam-5492	241	3	differential	differential	PROPN
ejpam-5492	241	4	geometry	geometry	NOUN
ejpam-5492	241	5	of	of	ADP
ejpam-5492	241	6	warped	warped	ADJ
ejpam-5492	241	7	product	product	NOUN
ejpam-5492	241	8	manifolds	manifold	NOUN
ejpam-5492	241	9	and	and	CCONJ
ejpam-5492	241	10	submanifolds	submanifold	NOUN
ejpam-5492	241	11	.	.	PUNCT
ejpam-5492	242	1	world	world	PROPN
ejpam-5492	242	2	scientific	scientific	PROPN
ejpam-5492	242	3	,	,	PUNCT
ejpam-5492	242	4	singapore	singapore	PROPN
ejpam-5492	242	5	,	,	PUNCT
ejpam-5492	242	6	2017	2017	NUM
ejpam-5492	242	7	.	.	PUNCT
ejpam-5492	243	1	[	[	X
ejpam-5492	243	2	10	10	NUM
ejpam-5492	243	3	]	]	PUNCT
ejpam-5492	243	4	m.	m.	NOUN
ejpam-5492	243	5	crasmareanu	crasmareanu	NOUN
ejpam-5492	243	6	,	,	PUNCT
ejpam-5492	243	7	c.e	c.e	PROPN
ejpam-5492	243	8	.	.	PROPN
ejpam-5492	243	9	hretcanu	hretcanu	PROPN
ejpam-5492	243	10	,	,	PUNCT
ejpam-5492	243	11	and	and	CCONJ
ejpam-5492	243	12	m.i	m.i	PROPN
ejpam-5492	243	13	.	.	PROPN
ejpam-5492	243	14	munteanu	munteanu	PROPN
ejpam-5492	243	15	.	.	PUNCT
ejpam-5492	244	1	golden	golden	ADJ
ejpam-5492	244	2	-	-	PUNCT
ejpam-5492	244	3	and	and	CCONJ
ejpam-5492	244	4	product	product	NOUN
ejpam-5492	244	5	-	-	PUNCT
ejpam-5492	244	6	shaped	shape	VERB
ejpam-5492	244	7	hypersurfaces	hypersurface	NOUN
ejpam-5492	244	8	in	in	ADP
ejpam-5492	244	9	real	real	ADJ
ejpam-5492	244	10	space	space	NOUN
ejpam-5492	244	11	forms	form	NOUN
ejpam-5492	244	12	.	.	PUNCT
ejpam-5492	245	1	int	int	NOUN
ejpam-5492	245	2	.	.	PUNCT
ejpam-5492	246	1	j.	j.	PROPN
ejpam-5492	246	2	geom	geom	PROPN
ejpam-5492	246	3	.	.	PUNCT
ejpam-5492	247	1	methods	methods	PROPN
ejpam-5492	247	2	mod	mod	PROPN
ejpam-5492	247	3	.	.	PUNCT
ejpam-5492	248	1	phys	phy	NOUN
ejpam-5492	248	2	.	.	PUNCT
ejpam-5492	248	3	,	,	PUNCT
ejpam-5492	248	4	10:1320006	10:1320006	NUM
ejpam-5492	248	5	,	,	PUNCT
ejpam-5492	248	6	2013	2013	NUM
ejpam-5492	248	7	.	.	PUNCT
ejpam-5492	249	1	[	[	X
ejpam-5492	249	2	11	11	NUM
ejpam-5492	249	3	]	]	X
ejpam-5492	249	4	s.i	s.i	PROPN
ejpam-5492	249	5	.	.	PROPN
ejpam-5492	249	6	goldberg	goldberg	PROPN
ejpam-5492	249	7	and	and	CCONJ
ejpam-5492	249	8	k.	k.	PROPN
ejpam-5492	249	9	yano	yano	PROPN
ejpam-5492	249	10	.	.	PUNCT
ejpam-5492	250	1	polynomial	polynomial	ADJ
ejpam-5492	250	2	structures	structure	NOUN
ejpam-5492	250	3	on	on	ADP
ejpam-5492	250	4	manifolds	manifold	NOUN
ejpam-5492	250	5	.	.	PUNCT
ejpam-5492	251	1	kodai	kodai	PROPN
ejpam-5492	251	2	math	math	PROPN
ejpam-5492	251	3	.	.	PUNCT
ejpam-5492	252	1	sem	sem	PROPN
ejpam-5492	252	2	.	.	PUNCT
ejpam-5492	253	1	rep	rep	PROPN
ejpam-5492	253	2	.	.	PROPN
ejpam-5492	253	3	,	,	PUNCT
ejpam-5492	253	4	22:199–218	22:199–218	PROPN
ejpam-5492	253	5	,	,	PUNCT
ejpam-5492	253	6	1970	1970	NUM
ejpam-5492	253	7	.	.	PUNCT
ejpam-5492	254	1	[	[	X
ejpam-5492	254	2	12	12	NUM
ejpam-5492	254	3	]	]	X
ejpam-5492	254	4	c.e	c.e	PROPN
ejpam-5492	254	5	.	.	PROPN
ejpam-5492	254	6	hretcanu	hretcanu	PROPN
ejpam-5492	254	7	and	and	CCONJ
ejpam-5492	254	8	a.m.	a.m.	PROPN
ejpam-5492	254	9	blaga	blaga	PROPN
ejpam-5492	254	10	.	.	PUNCT
ejpam-5492	255	1	submanifolds	submanifold	NOUN
ejpam-5492	255	2	in	in	ADP
ejpam-5492	255	3	metallic	metallic	ADJ
ejpam-5492	255	4	riemannian	riemannian	ADJ
ejpam-5492	255	5	manifolds	manifold	NOUN
ejpam-5492	255	6	.	.	PUNCT
ejpam-5492	255	7	differ	differ	VERB
ejpam-5492	255	8	.	.	PUNCT
ejpam-5492	256	1	geom	geom	PROPN
ejpam-5492	256	2	.	.	PUNCT
ejpam-5492	257	1	dyn	dyn	PROPN
ejpam-5492	257	2	.	.	PUNCT
ejpam-5492	258	1	syst	syst	PROPN
ejpam-5492	258	2	.	.	PROPN
ejpam-5492	258	3	,	,	PUNCT
ejpam-5492	258	4	20:83–97	20:83–97	PROPN
ejpam-5492	258	5	,	,	PUNCT
ejpam-5492	258	6	2018	2018	NUM
ejpam-5492	258	7	.	.	PUNCT
ejpam-5492	259	1	[	[	X
ejpam-5492	259	2	13	13	NUM
ejpam-5492	259	3	]	]	X
ejpam-5492	259	4	c.e	c.e	PROPN
ejpam-5492	259	5	.	.	PROPN
ejpam-5492	259	6	hretcanu	hretcanu	PROPN
ejpam-5492	259	7	and	and	CCONJ
ejpam-5492	259	8	a.m.	a.m.	PROPN
ejpam-5492	259	9	blaga	blaga	PROPN
ejpam-5492	259	10	.	.	PUNCT
ejpam-5492	260	1	hemi	hemi	NOUN
ejpam-5492	260	2	-	-	PUNCT
ejpam-5492	260	3	slant	slant	ADJ
ejpam-5492	260	4	submanifolds	submanifold	NOUN
ejpam-5492	260	5	in	in	ADP
ejpam-5492	260	6	metallic	metallic	ADJ
ejpam-5492	260	7	riemannian	riemannian	ADJ
ejpam-5492	260	8	manifolds	manifold	NOUN
ejpam-5492	260	9	.	.	PUNCT
ejpam-5492	261	1	carpathian	carpathian	PROPN
ejpam-5492	261	2	j.	j.	PROPN
ejpam-5492	261	3	math	math	PROPN
ejpam-5492	261	4	.	.	PUNCT
ejpam-5492	261	5	,	,	PUNCT
ejpam-5492	261	6	35:59–68	35:59–68	PROPN
ejpam-5492	261	7	,	,	PUNCT
ejpam-5492	261	8	2019	2019	NUM
ejpam-5492	261	9	.	.	PUNCT
ejpam-5492	262	1	[	[	X
ejpam-5492	262	2	14	14	NUM
ejpam-5492	262	3	]	]	X
ejpam-5492	262	4	c.e	c.e	PROPN
ejpam-5492	262	5	.	.	PROPN
ejpam-5492	262	6	hretcanu	hretcanu	PROPN
ejpam-5492	262	7	and	and	CCONJ
ejpam-5492	262	8	a.m.	a.m.	PROPN
ejpam-5492	262	9	blaga	blaga	PROPN
ejpam-5492	262	10	.	.	PUNCT
ejpam-5492	263	1	warped	warp	VERB
ejpam-5492	263	2	product	product	NOUN
ejpam-5492	263	3	submanifolds	submanifold	NOUN
ejpam-5492	263	4	in	in	ADP
ejpam-5492	263	5	metallic	metallic	ADJ
ejpam-5492	263	6	riemannian	riemannian	ADJ
ejpam-5492	263	7	manifolds	manifold	NOUN
ejpam-5492	263	8	.	.	PUNCT
ejpam-5492	264	1	tamkang	tamkang	PROPN
ejpam-5492	264	2	j.	j.	PROPN
ejpam-5492	264	3	math	math	PROPN
ejpam-5492	264	4	.	.	PUNCT
ejpam-5492	264	5	,	,	PUNCT
ejpam-5492	264	6	51:161–186	51:161–186	NUM
ejpam-5492	264	7	,	,	PUNCT
ejpam-5492	264	8	2020	2020	NUM
ejpam-5492	264	9	.	.	PUNCT
ejpam-5492	265	1	[	[	X
ejpam-5492	265	2	15	15	NUM
ejpam-5492	265	3	]	]	X
ejpam-5492	265	4	c.e	c.e	PROPN
ejpam-5492	265	5	.	.	PROPN
ejpam-5492	265	6	hretcanu	hretcanu	PROPN
ejpam-5492	265	7	and	and	CCONJ
ejpam-5492	265	8	m.	m.	NOUN
ejpam-5492	265	9	crasmareanu	crasmareanu	NOUN
ejpam-5492	265	10	.	.	PUNCT
ejpam-5492	266	1	on	on	ADP
ejpam-5492	266	2	some	some	DET
ejpam-5492	266	3	invariant	invariant	ADJ
ejpam-5492	266	4	submanifolds	submanifold	NOUN
ejpam-5492	266	5	in	in	ADP
ejpam-5492	266	6	a	a	DET
ejpam-5492	266	7	riemannian	riemannian	NOUN
ejpam-5492	266	8	manifold	manifold	NOUN
ejpam-5492	266	9	with	with	ADP
ejpam-5492	266	10	golden	golden	ADJ
ejpam-5492	266	11	structure	structure	NOUN
ejpam-5492	266	12	.	.	PUNCT
ejpam-5492	267	1	an	an	DET
ejpam-5492	267	2	.	.	NOUN
ejpam-5492	267	3	stiint	stiint	PROPN
ejpam-5492	267	4	.	.	PUNCT
ejpam-5492	268	1	univ	univ	PROPN
ejpam-5492	268	2	.	.	PUNCT
ejpam-5492	269	1	al	al	PROPN
ejpam-5492	269	2	.	.	PROPN
ejpam-5492	269	3	i.	i.	PROPN
ejpam-5492	269	4	cuza	cuza	PROPN
ejpam-5492	269	5	iasi	iasi	PROPN
ejpam-5492	269	6	.	.	PUNCT
ejpam-5492	270	1	mat	mat	PROPN
ejpam-5492	270	2	.	.	PROPN
ejpam-5492	270	3	,	,	PUNCT
ejpam-5492	270	4	53:199–211	53:199–211	NUM
ejpam-5492	270	5	,	,	PUNCT
ejpam-5492	270	6	2007	2007	NUM
ejpam-5492	270	7	.	.	PUNCT
ejpam-5492	271	1	[	[	X
ejpam-5492	271	2	16	16	NUM
ejpam-5492	271	3	]	]	X
ejpam-5492	271	4	c.e	c.e	PROPN
ejpam-5492	271	5	.	.	PROPN
ejpam-5492	271	6	hretcanu	hretcanu	PROPN
ejpam-5492	271	7	and	and	CCONJ
ejpam-5492	271	8	m.c	m.c	PROPN
ejpam-5492	271	9	crasmareanu	crasmareanu	NOUN
ejpam-5492	271	10	.	.	PUNCT
ejpam-5492	272	1	applications	application	NOUN
ejpam-5492	272	2	of	of	ADP
ejpam-5492	272	3	the	the	DET
ejpam-5492	272	4	golden	golden	ADJ
ejpam-5492	272	5	ratio	ratio	NOUN
ejpam-5492	272	6	on	on	ADP
ejpam-5492	272	7	riemannian	riemannian	ADJ
ejpam-5492	272	8	manifolds	manifold	NOUN
ejpam-5492	272	9	.	.	PUNCT
ejpam-5492	273	1	turk	turk	PROPN
ejpam-5492	273	2	.	.	PUNCT
ejpam-5492	274	1	j.	j.	PROPN
ejpam-5492	274	2	math	math	PROPN
ejpam-5492	274	3	.	.	PUNCT
ejpam-5492	274	4	,	,	PUNCT
ejpam-5492	274	5	33:179–191	33:179–191	PROPN
ejpam-5492	274	6	,	,	PUNCT
ejpam-5492	274	7	2009	2009	NUM
ejpam-5492	274	8	.	.	PUNCT
ejpam-5492	275	1	[	[	X
ejpam-5492	275	2	17	17	NUM
ejpam-5492	275	3	]	]	X
ejpam-5492	275	4	c.e	c.e	PROPN
ejpam-5492	275	5	.	.	PROPN
ejpam-5492	275	6	hretcanu	hretcanu	PROPN
ejpam-5492	275	7	and	and	CCONJ
ejpam-5492	275	8	m.c	m.c	PROPN
ejpam-5492	275	9	.	.	PROPN
ejpam-5492	275	10	crasmareanu	crasmareanu	PROPN
ejpam-5492	275	11	.	.	PUNCT
ejpam-5492	276	1	metallic	metallic	ADJ
ejpam-5492	276	2	structures	structure	NOUN
ejpam-5492	276	3	on	on	ADP
ejpam-5492	276	4	riemannian	riemannian	ADJ
ejpam-5492	276	5	manifolds	manifold	NOUN
ejpam-5492	276	6	.	.	PUNCT
ejpam-5492	277	1	rev	rev	PROPN
ejpam-5492	277	2	.	.	PROPN
ejpam-5492	277	3	un	un	PROPN
ejpam-5492	277	4	.	.	PROPN
ejpam-5492	277	5	mat	mat	PROPN
ejpam-5492	277	6	.	.	PROPN
ejpam-5492	277	7	argentina	argentina	PROPN
ejpam-5492	277	8	,	,	PUNCT
ejpam-5492	277	9	54:15–27	54:15–27	NUM
ejpam-5492	277	10	,	,	PUNCT
ejpam-5492	277	11	2013	2013	NUM
ejpam-5492	277	12	.	.	PUNCT
ejpam-5492	278	1	[	[	X
ejpam-5492	278	2	18	18	NUM
ejpam-5492	278	3	]	]	X
ejpam-5492	278	4	w.w	w.w	PROPN
ejpam-5492	278	5	.	.	PROPN
ejpam-5492	278	6	mohammed	mohammed	PROPN
ejpam-5492	278	7	and	and	CCONJ
ejpam-5492	278	8	c.	c.	PROPN
ejpam-5492	278	9	cesarano	cesarano	PROPN
ejpam-5492	278	10	.	.	PUNCT
ejpam-5492	279	1	the	the	DET
ejpam-5492	279	2	soliton	soliton	NOUN
ejpam-5492	279	3	solutions	solution	NOUN
ejpam-5492	279	4	for	for	ADP
ejpam-5492	279	5	the	the	DET
ejpam-5492	279	6	(	(	PUNCT
ejpam-5492	279	7	4	4	NUM
ejpam-5492	279	8	+	+	NUM
ejpam-5492	279	9	1)-dimensional	1)-dimensional	NUM
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ejpam-5492	279	11	fokas	fokas	ADJ
ejpam-5492	279	12	equation	equation	NOUN
ejpam-5492	279	13	.	.	PUNCT
ejpam-5492	280	1	math	math	NOUN
ejpam-5492	280	2	.	.	PUNCT
ejpam-5492	281	1	methods	method	NOUN
ejpam-5492	281	2	appl	appl	PROPN
ejpam-5492	281	3	.	.	PUNCT
ejpam-5492	282	1	sci	sci	PROPN
ejpam-5492	282	2	.	.	PROPN
ejpam-5492	282	3	,	,	PUNCT
ejpam-5492	282	4	46:7589–7597	46:7589–7597	PROPN
ejpam-5492	282	5	,	,	PUNCT
ejpam-5492	282	6	2023	2023	NUM
ejpam-5492	282	7	.	.	PUNCT