CHAPTER-III HYPERBOLIC HSU-STRUCTURE METRIC MANIFOLD. Estelar
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1 CHAPE-III HPEBOLIC HSU-SUCUE MEIC MANIOLD I this chpte I hve obtied itebility coditios fo hypebolic Hsustuctue metic mifold. Pseudo Pojective d Pseudo H-Pojective cuvtue tesos hve bee defied i this mifold. uthemoe theoems of diffeet kids of ecuet ecuet symmetic biecuet biecuet symmetic - ecuet d -ecuet symmetic hypebolic Hsu-stuctue metic mifold ivolvi equivlet coditios with espect to vious cuvtue tesos hve lso bee discussed. Nijehuis d ssocited Nijehuis tesos hve bee defied d decompositios of Nijehuis teso hve bee doe. INEGABILI CONDIION heoem.. he ecessy d sufficiet coditio tht M be hypebolic Hsu-stuctue metic mifold is tht it cotis distibutio π of comple dimesio d distibutio ~ π cojute toπ such tht π d ~ π hve o diectio i commo d sp toethe lie mifold of dimesio pojectios of π d ~ π bei L d M ive by Estel L = M =. Poof. Let M be hypebolic Hsu-stuctue metic mifold coespodi to the eie vlues d. Let thee be liely idepedet eie vectos = 3... d liely idepedet comple cojute eie vectos S. he we hve 3
2 . = 0 = 0 fo ll. b S = 0 b = 0 fo ll. Now y c d S = 0 c d S = 0 hese equtios imply y y c d S = 0. y c = 0 d y S = 0 c = d = 0 fo d y. { S} is liely idepedet set. y Let us put.3 he we hve y L = M =. = L = L S S = M =. M S S y y y Estel hus we hve poved tht thee is distibutio π of comple dimesio d thee is comple cojute distibutio ~ π of dimesio which hs o commo diectio with π d sps with π lie mifold of dimesio pojectios of π d ~ π bei L d M. 4
3 Covesely we suppose tht thee is distibutio π of comple dimesio d distibutio ~ π comple cojute to π hvi o commo diectio with lie mifold of dimesio. Let d S be liely idepedet vectos i π d liely idepedet vectos i π espectively. Let { S} sp lie mifold of Estel dimesio. he { S} is liely idepedet set. Let us defie the ivese set { S} such tht.4 = S S which yields.4b = S S = δ y y y.4c S = S = 0. y y Let us put = = { S S} the.5 = { S S} But fom the equtios.4 d.5 we hve = { S S} =. his is we hve poved tht the mifold dmits lmost hypebolic Hsu-stuctue mifold. 5
4 . ECUENCE AND ECUENCE SMME O DIEEN KINDS Pseudo Pojective Cuvtue eso Defiitio. A cuvtue teso W defied by. W = K is sid to be Pseudo Pojective cuvtue teso. he mifold M is sid to be Pseudo Pojective ecuet if. W = A W whee A is ecuece pmete d Pseudo Pojective symmetic if.b W = 0. heoem.. If the hypebolic Hsu-stuctue metic mifold be ecuet the it is Pseudo Pojective ecuet fo the sme ecuece pmete. Poof. Diffeetiti the equtio. with espect to we hve.3 W = K Estel. Multiplyi the equtio. by A thouhout we et A.4 A W = A K. Now subtcti the equtio.4 fom the equtio.3 we et.5 W A W = K A K A 6
5 A. Now usi the fct tht if the mifold M be ecuet the fom the equtio.5 we et.6 W = A W. Sice the ecuet mifold is ci ecuet which shows tht the mifold is Pseudo Pojective ecuet. heoem.. If the hypebolic Hsu-stuctue metic mifold be ecuet symmetic the it is Pseudo Pojective ecuet symmetic fo the sme ecuece pmete. Poof. Sice the ecuet symmetic mifold is ci symmetic the fom the equtio.5 we hve.7 W = 0 which shows tht the mifold is Pseudo Pojective ecuet symmetic. Pseudo H-Pojective Cuvtue eso Defitio.. A cuvtue teso P defied by.8. P = K Estel is sid to be Pseudo H-Pojective cuvtue teso. he mifold M is sid to be Pseudo H-Pojective ecuet if.9 P = A P d Pseudo Pojective symmetic if.9b P = 0. heoem.3. If the hypebolic Hsu-stuctue metic mifold be ecuet the it is Pseudo H-Pojective ecuet fo the sme ecuece pmete. Poof. Diffeetiti the equtio.8 with espect to we hve 7
6 .0 P = K. Multiplyi the equtio.8 by A thouhout we et A. A P = A K. Now subtcti the equtio. fom the equtio.0 we et. P A P = K A K A A A A A. Now usi the fct tht if the mifold M be ecuet the fom the equtio. we et Estel.3 P = A P. Sice the ecuet mifold is ci ecuet which shows tht the mifold is Pseudo H-Pojective ecuet. heoem.4. If the hypebolic Hsu-stuctue metic mifold be ecuet symmetic the it is Pseudo H-Pojective ecuet symmetic fo the sme ecuece pmete. Poof. Sice the ecuet symmetic mifold is ci symmetic the fom the equtio. we hve.4 P = 0 which shows tht the mifold is Pseudo H-Pojective ecuet symmetic. 8
7 heoem.5. I the hypebolic Hsu-stuctue metic mifold if y two of the followi coditios hold fo the sme ecuece pmete the the thid lso holds: i It is Pseudo H-Pojective ecuet ii It is Pseudo Pojective ecuet iii It is ci ecuet. Poof. om the equtios. d.8 we hve.5 P = W. Multiplyi the equtio.5 by A thouhout we et A.6 A P = A W A. Diffeetiti the equtio.5 with espect to we et.7 P = W Estel. Subtcti the equtio.6 fom the equtio.7 we et.8 P A P = W A W A } { { A } { A } { A } { A }. 9
8 Let the hypebolic Hsu-stuctue metic mifold M is Pseudo Pojective ecuet d ci ecuet fo the sme ecuece pmete. he fom the equtio.8 we et.9 P = A P which shows tht the mifold is Pseudo H-Pojective ecuet. heoem.6. I the hypebolic Hsu-stuctue metic mifold if y two of the followi coditios hold fo the sme ecuece pmete the the thid lso holds: i ii It is Pseudo H-Pojective ecuet symmetic It is Pseudo Pojective ecuet symmetic iii It is ci ecuet symmetic. Poof. If the hypebolic Hsu-stuctue metic mifold is Pseudo Pojective ecuet symmetic d ci ecuet symmetic the fom the equtio.8 we et.0 P = 0 which shows tht the mifold is Pseudo H-Pojective ecuet symmetic. heoem.7. I the hypebolic Hsu-stuctue metic mifold if y two of the followi coditios hold fo the sme ecuece pmete the the thid lso holds: i ii It is Pseudo Pojective -ecuet It is Pseudo Cocicul - ecuet iii It is ci - ecuet povided Estel {. } =0. Poof. om the equtios. d I.6 we hve. = W. Bi i equtio. we et.3 = W 30
9 . Multiplyi the equtio.3 by A the bi d usi the equtio I. i the esulti equtio we et A.4 A = A W A. Diffeetiti the equtio.3 with espect to d usi the equtio.3 i the esulti equtio we et.5 = W W. Bi i equtio.5 d usi the equtio I. the subtcti the equtio.4 fom the esulti equtio we et Estel.6 A = W W A W A A { A }. 3
10 Now usi the equtio. toethe with the fct tht the hypebolic Hsustuctue metic mifold is Pseudo Cocicul -ecuet d ci - ecuet i equtio.6 we et W W = A W which shows tht the mifold is Pseudo Pojective - ecuet. Similly it c be show tht if the mifold is eithe Pseudo Pojective - ecuet d ci -ecuet o Pseudo Pojective -ecuet d Pseudo Cocicul -ecuet the it is eithe Pseudo Cocicul -ecuet o ci -ecuet povided the equtio. is stisfied. heoem.8. I the hypebolic Hsu-stuctue metic mifold if y two of the followi coditios hold fo the sme ecuece pmete the the thid lso holds: i ii It is Pseudo Pojective - symmetic It is Pseudo Cocicul - symmetic iii It is ci - symmetic povided {.7 } =0. Poof. Let the hypebolic Hsu-stuctue metic mifold is Pseudo Cocicul - symmetic d ci - symmetic the fom the equtio.6 we et W W = 0 povided the equtio.7 is stisfied which shows tht the mifold is Pseudo Pojective - symmetic. Similly it c be show tht if the mifold is eithe Pseudo Pojective - symmetic d ci - symmetic o Pseudo Pojective - symmetic d Pseudo Cocicul - symmetic the it is eithe Pseudo Cocicul - symmetic o ci - symmetic fo the sme ecuece pmete povided the equtio.7 is stisfied. Estel heoem.9. I the hypebolic Hsu-stuctue metic mifold if y two of the followi coditios hold fo the sme ecuece pmete the the thid lso holds: i It is Pseudo Pojective -ecuet 3
11 ii It is Pseudo Cocicul - ecuet iii It is ci - ecuet povided.8 { } { } = 0. Poof. Bi d i equtio. we et.9 = W. Multiplyi the equtio.9 by A the bi d usi the equtio I. i the esulti equtio we et A.30 A = A W A. Diffeetiti the equtio.9 with espect to d usi the equtio.9 i the esulti equtio we et.3 = W W W Estel. Bi i equtio.3 d usi the equtio I. the subtcti the equtio.30 fom the esulti equtio we et 33
12 .3 A = W W W A W A A { A }. Now usi the equtio.8 toethe with the fct tht the hypebolic Hsustuctue metic mifold is Pseudo Cocicul -ecuet d ci - ecuet i equtio.3 we et W W = A W W which shows tht the mifold is Pseudo Pojective - ecuet. Similly it c be show tht if the mifold is eithe Pseudo Pojective - ecuet d ci -ecuet o Pseudo Pojective -ecuet d Pseudo Cocicul -ecuet the it is eithe Pseudo Cocicul -ecuet o ci -ecuet povided the equtio.8 is stisfied. Estel heoem.0. I the hypebolic Hsu-stuctue metic mifold if y two of the followi coditios hold fo the sme ecuece pmete the the thid lso holds: i It is Pseudo Pojective - symmetic ii It is Pseudo Cocicul - symmetic iii It is ci - symmetic povided.33 { } { 34
13 } = 0. Poof. Let the hypebolic Hsu-stuctue metic mifold is Pseudo Cocicul - symmetic d ci - symmetic the fom the equtio.3 we et W W W = 0 povided the equtio.33 is stisfied which shows tht the mifold is Pseudo Pojective - symmetic. Similly it c be show tht if the mifold is eithe Pseudo Pojective - symmetic d ci - symmetic o Pseudo Pojective - symmetic d Pseudo Cocicul - symmetic the it is eithe Pseudo Cocicul - symmetic o ci - symmetic fo the sme ecuece pmete povided the equtio.33 is stisfied. heoem.. I the hypebolic Hsu-stuctue metic mifold if y two of the followi coditios hold fo the sme ecuece pmete the the thid lso holds: i ii It is Pseudo Pojective 3-ecuet It is Pseudo Cocicul 3- ecuet iii It is ci - ecuet povided.34 { } { Estel } = 0. Poof. Bi d i equtio. we et.35 = W. Multiplyi the equtio.35 by A d bi the usi the equtio I. i the esulti equtio we et A.36 A = A W 35
14 A. Diffeetiti the equtio.35 with espect to we et.37 = W W W W. Bi i equtio.37 d usi the I. the subtcti the equtio.36 fom the esulti equtio we et.38 A = W W W A W Estel A A { A } W 36
15 . Now usi the equtio.34 toethe with the fct tht the hypebolic Hsustuctue metic mifold is Pseudo Cocicul 3-ecuet d ci - ecuet i equtio.38 we et W W W W = A W which shows tht the mifold is Pseudo Pojective 3- ecuet. Similly it c be show tht if the mifold is eithe Pseudo Pojective 3- ecuet d ci -ecuet o Pseudo Pojective 3-ecuet d Pseudo Cocicul 3-ecuet the it is eithe Pseudo Cocicul 3-ecuet o ci -ecuet povided the equtio.34 is stisfied. heoem.. I the hypebolic Hsu-stuctue metic mifold if y two of the followi coditios hold fo the sme ecuece pmete the the thid lso holds: i ii It is Pseudo Pojective 3- symmetic It is Pseudo Cocicul 3- symmetic iii It is ci - symmetic povided Estel.39 { } { } = 0. Poof. Let the hypebolic Hsu-stuctue metic mifold is Pseudo Cocicul 3- symmetic d ci - symmetic the fom the equtio.38 we et W W W W = 0 37
16 povided the equtio.39 is stisfied which shows tht the mifold is Pseudo Pojective 3- ecuet symmetic. Similly it c be show tht if the mifold is eithe Pseudo Pojective 3- symmetic d ci - symmetic o Pseudo Pojective 3- symmetic d Pseudo Cocicul 3-symmetic the it is eithe Pseudo Cocicul 3- symmetic o ci - symmetic fo the sme ecuece pmete povided the equtio.39 is stisfied. 3. BIECUENCE AND BIECUENCE SMME O DIEEN KINDS he mifold M is sid to be Pseudo Pojective biecuet if 3. W = A W whee A is biecuece pmete d Pseudo Pojective biecuet symmetic if 3.b W = 0. heoem 3.. If the hypebolic Hsu-stuctue metic mifold be biecuet the it is Pseudo Pojective biecuet fo the sme biecuece pmete. Poof. We hve Estel 3. W = K Diffeetiti the equtio 3. with espect to d we hve 3.3 W = K. Multiplyi the equtio 3. by A thouhout we et A 3.4 A W = A K. 38
17 Now subtcti the equtio 3.4 fom the equtio 3.3 we et 3.5 W A W = K A K A A. Now usi the fct tht if the mifold M be biecuet the fom the equtio 3.5 we et 3.6 W = A W. Sice the biecuet mifold is ci biecuet which shows tht the mifold is Pseudo Pojective biecuet. heoem 3.. If the hypebolic Hsu-stuctue metic mifold be biecuet symmetic the it is Pseudo Pojective biecuet symmetic fo the sme biecuece pmete. Poof. Sice the biecuet symmetic mifold is ci biecuet symmetic the fom the equtio 3.5 we hve 3.7 W = 0 which shows tht the mifold is Pseudo Pojective biecuet symmetic. Estel he mifold M is sid to be Pseudo H-Pojective biecuet if 3.8 P = A P d Pseudo H- Pojective biecuet symmetic if 3.8b P = 0. heoem 3.3. If the hypebolic Hsu-stuctue metic mifold be biecuet the it is Pseudo H-Pojective biecuet fo the sme biecuece pmete. Poof. We hve 3.9 P = K 39
18 . Diffeetiti the equtio 3.9 with espect to d we hve 3.0 P = K. Multiplyi the equtio 3.9 by A thouhout we et A 3. A P = A K. Now subtcti the equtio 3. fom the equtio 3.0 we et 3. P A P = K A K A A A A A. Now usi the fct tht if the mifold M be biecuet the fom the equtio 3. we et Estel 3.3 P = A P. Sice the biecuet mifold is ci biecuet which shows tht the mifold is Pseudo H-Pojective biecuet. heoem3.4. If the hypebolic Hsu-stuctue metic mifold be biecuet symmetic the it is Pseudo H-Pojective biecuet symmetic fo the sme biecuece pmete. 40
19 Poof. Sice the biecuet symmetic mifold is ci biecuet symmetic the fom the equtio 3. we hve 3.4 P = 0 which shows tht the mifold is Pseudo H-Pojective biecuet symmetic. heoem 3.5. I the hypebolic Hsu-stuctue metic mifold if y two of the followi coditios hold fo the sme biecuece pmete the the thid lso holds: i It is Pseudo H-Pojective biecuet ii It is Pseudo Pojective biecuet iii It is ci biecuet. Poof. om the equtios. d.8 we hve 3.5 P = W. Multiplyi the equtio 3.5 by A thouhout we et 3.6 A P = A W A Estel A. Diffeetiti the equtio 3.5 with espect to d we et 3.7 P = W. Subtcti the equtio 3.6 fom the equtio 3.7 we et 3.8 P A P = W 4
20 A W A } { { A } { A } { A } A }. { Let the hypebolic Hsu-stuctue metic mifold M is Pseudo Pojective biecuet d ci biecuet fo the sme biecuece pmete. he fom the equtio 3.8 we et 3.9 P = A P which shows tht the mifold is Pseudo H-Pojective biecuet. heoem 3.6. I the hypebolic Hsu-stuctue metic mifold if y two of the followi coditios hold fo the sme biecuece pmete the the thid lso holds: i It is Pseudo H-Pojective biecuet symmetic ii It is Pseudo Pojective biecuet symmetic iii It is ci biecuet symmetic. Poof. If the hypebolic Hsu-stuctue metic mifold is Pseudo Pojective biecuet symmetic d ci biecuet symmetic the fom the equtio 3.8 we et Estel 3.0 P = 0 which shows tht the mifold is Pseudo H-Pojective biecuet symmetic. heoem 3.7. I the hypebolic Hsu-stuctue metic mifold if y two of the followi coditios hold fo the sme biecuece pmete the the thid lso holds: i It is Pseudo Pojective -biecuet ii It is Pseudo Cocicul - biecuet 4
21 iii It is ci - biecuet povided 3. { } { } 0. = Poof. om the equtios. d I.6 we hve 3. = W Bi i equtio 3. we et. 3.3 = W. Multiplyi the equtio 3.3 by A the bi d usi the equtio I. i the esulti equtio we et 3.4 A = A W Estel A A. Diffeetiti the equtio 3.3 with espect to d usi the equtio 3.3 i the esulti equtio we et 3.5 = W W W W 43
22 . Bi i equtio 3.5 d usi the equtio I. the subtcti the equtio 3.4 fom the esulti equtio we et 3.6 A = W W W Estel W A W A A { A } 44
23 { } { } }. { Now usi the equtio 3. toethe with the fct tht the hypebolic Hsustuctue metic mifold is Pseudo Cocicul -biecuet d ci - biecuet i equtio 3.6 we et W W W W = A W which shows tht the mifold is Pseudo Pojective - biecuet. Similly it c be show tht if the mifold is eithe Pseudo Pojective - biecuet d ci -biecuet o Pseudo Pojective -biecuet d Pseudo Cocicul -biecuet the it is eithe Pseudo Cocicul -biecuet o ci -biecuet povided the equtio 3. is stisfied. heoem 3.8. I the hypebolic Hsu-stuctue metic mifold if y two of the followi coditios hold fo the sme biecuece pmete the the thid lso holds: i It is Pseudo Pojective -biecuet symmetic ii It is Pseudo Cocicul - biecuet symmetic iii It is ci - biecuet symmetic povided Estel 3.7 { } { } 0. = 45
24 Poof. Let the hypebolic Hsu-stuctue metic mifold is Pseudo Cocicul - biecuet symmetic d ci -biecuet symmetic the fom the equtio 3.6 we et W W W W = 0 povided the equtio 3.7 is stisfied which shows tht the mifold is Pseudo Pojective - biecuet symmetic. Similly it c be show tht if the mifold is eithe Pseudo Pojective - biecuet symmetic d ci -biecuet symmetic o Pseudo Pojective - biecuet symmetic d Pseudo Cocicul -biecuet symmetic the it is eithe Pseudo Cocicul -biecuet symmetic o ci -biecuet symmetic fo the sme biecuece pmete povided the equtio 3.7 is stisfied. heoem 3.9. I the hypebolic Hsu-stuctue metic mifold if y two of the followi coditios hold fo the sme biecuece pmete the the thid lso holds: i ii It is Pseudo Pojective -biecuet It is Pseudo Cocicul - biecuet iii It is ci - biecuet povided Estel 3.8 { } { 46
25 } 0. Poof. Bi d i equtio 3. we et = 3.9 = W. Multiplyi the equtio 3.9 by A the bi d usi the equtio I. i the esulti equtio we et 3.30 A = A W A A. Diffeetiti the equtio 3.9 with espect to d usi the equtio 3.9 i the esulti equtio we et Estel 3.3 = W W W W W W W W W 47
26 Estel. Bi i equtio 3.3 d usi the equtio I. the subtcti the equtio 3.30 fom the esulti equtio we et 3.3 A = W W 48
27 W W W W W W W A W A A { A } { } Estel { } { }. Now usi the equtio 3.8 toethe with the fct tht the hypebolic Hsustuctue metic mifold is Pseudo Cocicul -biecuet d ci - biecuet i equtio 3.3 we et 49
28 W W W W W W W W W = A W which shows tht the mifold is Pseudo Pojective - biecuet. Similly it c be show tht if the mifold is eithe Pseudo Pojective - biecuet d ci -biecuet o Pseudo Pojective -biecuet d Pseudo Cocicul -biecuet the it is eithe Pseudo Cocicul -biecuet o ci -biecuet povided the equtio 3.8 is stisfied. heoem 3.0. I the hypebolic Hsu-stuctue metic mifold if y two of the followi coditios hold fo the sme biecuece pmete the the thid lso holds: i ii It is Pseudo Pojective -biecuet symmetic It is Pseudo Cocicul - biecuet symmetic iii It is ci - biecuet symmetic povided 3.33 { Estel } { 50
29 } 0. = Poof. Let the hypebolic Hsu-stuctue metic mifold is Pseudo Cocicul - biecuet symmetic d ci -biecuet symmetic the fom the equtio 3.3 we et W W W W W W W W W = 0 povided the equtio 3.33 is stisfied which shows tht the mifold is Pseudo Pojective - biecuet symmetic. Similly it c be show tht if the mifold is eithe Pseudo Pojective - biecuet symmetic d ci -biecuet symmetic o Pseudo Pojective - biecuet symmetic d Pseudo Cocicul -biecuet symmetic the it is eithe Pseudo Cocicul -biecuet symmetic o ci -biecuet symmetic fo the sme biecuece pmete povided the equtio 3.33 is stisfied. heoem 3.. I the hypebolic Hsu-stuctue metic mifold if y two of the followi coditios hold fo the sme biecuece pmete the the thid lso holds: i ii Estel It is Pseudo Pojective 3-biecuet It is Pseudo Cocicul 3- biecuet iii It is ci - biecuet povided 3.34 { 5
30 } { } 0. = Estel Poof. Bi d i equtio 3. we et 3.35 = W. Multiplyi the equtio 3.35 by A the bi d usi the equtio I. i the esulti equtio we et 3.36 A = A W A 5
31 A. Diffeetiti the equtio 3.35 with espect to d we et 3.37 = W W W W W W W W W W W W W W W W Estel 53
32 Estel. Bi i equtio 3.37 d usi the equtio I. the subtcti the equtio 3.36 fom the esulti equtio we et 54
33 A W = W W W W W W W W W W W W W W W W A A Estel
34 A { A } { } { } { Estel }. Now usi the equtio 3.34 toethe with the fct tht the hypebolic Hsustuctue metic mifold is Pseudo Cocicul 3-biecuet d ci - biecuet i equtio 3.38 we et 56
35 W W W W W W W W W W W W W W W W = A W which shows tht the mifold is Pseudo Pojective 3- biecuet. Similly it c be show tht if the mifold is eithe Pseudo Pojective 3-biecuet d ci -biecuet o Pseudo Pojective 3-biecuet d Pseudo Cocicul 3-biecuet the it is eithe Pseudo Cocicul 3- biecuet o ci -biecuet povided the equtio 3.34 is stisfied. heoem 3.. I the hypebolic Hsu-stuctue metic mifold if y two of the followi coditios hold fo the sme biecuece pmete the the thid lso holds: i ii It is Pseudo Pojective 3-biecuet symmetic It is Pseudo Cocicul 3- biecuet symmetic iii It is ci - biecuet symmetic povided Estel 3.39 { 57
36 } { } 0. = Poof. Let the hypebolic Hsu-stuctue metic mifold is Pseudo Cocicul 3- biecuet symmetic d ci -biecuet symmetic the fom the equtio 3.38 we et W W W W Estel W W W W W W W W W W W W = 0 58
37 povided the equtio 3.39 is stisfied which shows tht the mifold is Pseudo Pojective 3-biecuet symmetic. Similly it c be show tht if the mifold is eithe Pseudo Pojective 3-biecuet symmetic d ci -biecuet symmetic o Pseudo Pojective 3-biecuet symmetic d Pseudo Cocicul 3-biecuet symmetic the it is eithe Pseudo Cocicul 3-biecuet symmetic o ci -ecuet symmetic fo the sme biecuece pmete povided the equtio 3.39 is stisfied. 4. -ECUENCE AND -ECUENCE SMME O DIEEN KINDS he mifold M is sid to be Pseudo Pojective -ecuet if W... = A. W whee A. is -ecuece pmete d Pseudo Pojective -ecuet symmetic if 4.b... W = Estel heoem 4.. If the hypebolic Hsu-stuctue metic mifold be -ecuet the it is Pseudo Pojective -ecuet fo the sme -ecuece pmete. Poof. We hve 4. W = K Diffeetiti the equtio 4. with espect to we hve W... =... K
38 Multiplyi the equtio 4. by A... thouhout we et 4.4 A... W A... K = A... Now subtcti the equtio 4.4 fom the equtio 4.3 we et 4.5. W... A... W =... K... A... K A... A Now usi the fct tht if the mifold M be -ecuet the fom the equtio 4.5 we et 4.6. W... = A.. W. Sice the -ecuet mifold is ci -ecuet which shows tht the mifold is Pseudo Pojective -ecuet. heoem 4.. If the hypebolic Hsu-stuctue metic mifold be -ecuet symmetic the it is Pseudo Pojective -ecuet symmetic fo the sme - ecuece pmete. Estel Poof. Sice the -ecuet symmetic mifold is ci -ecuet symmetic the fom the equtio 4.5 we hve 4.7. W... 0 = which shows tht the mifold is Pseudo Pojective -ecuet symmetic. he mifold M is sid to be Pseudo H-Pojective -ecuet if P... = A... P 60
39 d Pseudo H- Pojective -ecuet symmetic if 4.8b... P = heoem 4.3. If the hypebolic Hsu-stuctue metic mifold be -ecuet the it is Pseudo H-Pojective -ecuet fo the sme -ecuece pmete. Poof. We hve 4.9 P = K. Diffeetiti the equtio 4.9 with espect to. we hve P... =... K Multiplyi the equtio 4.9 by A.. thouhout we et 4. A.. P A.. K = Estel A... Now subtcti the equtio 4. fom the equtio 4.0 we et P... A. P =... K. { A... A }... K A... A... 6
40 A A.... Now usi the fct tht if the mifold M be -ecuet the fom the equtio 4. we et P... A. P. = Sice the -ecuet mifold is ci -ecuet which shows tht the mifold is Pseudo H-Pojective -ecuet. heoem 4.4. If the hypebolic Hsu-stuctue metic mifold be -ecuet symmetic the it is Pseudo H-Pojective -ecuet symmetic fo the sme - ecuece pmete. Poof. Sice the -ecuet symmetic mifold is ci -ecuet symmetic the fom the equtio 4. we hve P... = 0 which shows tht the mifold is Pseudo H-Pojective -ecuet symmetic. heoem 4.5. I the hypebolic Hsu-stuctue metic mifold if y two of the followi coditios hold fo the sme -ecuece pmete the the thid lso holds: i ii It is Pseudo H-Pojective -ecuet It is Pseudo Pojective -ecuet iii It is ci -ecuet. Poof. om the equtios. d.8 we hve 4.5 P = W Estel. Multiplyi the equtio 4.5 by A.. thouhout we et 4.6 A.. P A... W = 6
41 A. A... Diffeetiti the equtio 4.5 with espect to.. we et P... =... W Subtcti the equtio 4.6 fom the equtio 4.7 we et P... A... P = W... A... W { Estel A } A.. } { { A... } { A... } A. }.... {... Let the hypebolic Hsu-stuctue metic mifold M is Pseudo Pojective -ecuet d ci -ecuet fo the sme -ecuece pmete. he fom the equtio 4.8 we et 63
42 4.9. P. A. P = which shows tht the mifold is Pseudo H-Pojective -ecuet. heoem 4.6. I the hypebolic Hsu-stuctue metic mifold if y two of the followi coditios hold fo the sme -ecuece pmete the the thid lso holds: i It is Pseudo H-Pojective -ecuet symmetic ii It is Pseudo Pojective -ecuet symmetic iii It is ci -ecuet symmetic. Poof. If the hypebolic Hsu-stuctue metic mifold is Pseudo Pojective - ecuet symmetic d ci -ecuet symmetic the fom the equtio 4.8 we et 4.0. P. = 0 which shows tht the mifold is Pseudo H-Pojective -ecuet symmetic. heoem 4.7. I the hypebolic Hsu-stuctue metic mifold if y two of the followi coditios hold fo the sme -ecuece pmete the the thid lso holds: i It is Pseudo Pojective --ecuet ii It is Pseudo Cocicul - -ecuet iii It is ci - -ecuet povided 4. {.... Estel
43 } { Estel
44 } = 0. Poof. om the equtios. d I.6 we hve 4. = W. Bi i equtio 4. we et 4.3 = W. Multiplyi the equtio 4.3 by A. the bi d usi the Estel equtio I. i the esulti equtio we et 4.4 A. A... W = A. A.... Diffeetiti the equtio 4.3 with espect to... usi the equtio 4.3 the bi d usi the equtio I. the subtcti the equtio 4.4 fom the esulti equtio we et
45 W =..... W W W W W W W W W Estel
46 W.... W A A Estel
47 69 } { A..... { } { Estel
48 }. Now usi the equtio 4. toethe with the fct tht the hypebolic Hsustuctue metic mifold is Pseudo Cocicul --ecuet d ci - - ecuet i equtio 4.6 we et... W W.... W W W 3.. Estel 3 70
49 ... W W W W W.... W W.... = A... W which shows tht the mifold is Pseudo Pojective - -ecuet. Similly it c be show tht if the mifold is eithe Pseudo Pojective - -ecuet d ci --ecuet o Pseudo Pojective --ecuet d Pseudo Cocicul --ecuet the it is eithe Pseudo Cocicul --ecuet o ci --ecuet povided the equtio 4. is stisfied. heoem 4.8. I the hypebolic Hsu-stuctue metic mifold if y two of the Estel followi coditios hold fo the sme -ecuece pmete the the thid lso holds: i It is Pseudo Pojective --ecuet symmetic ii It is Pseudo Cocicul - -ecuet symmetic iii It is ci - -ecuet symmetic povided the equtio 5. is stisfied. Poof. Let the hypebolic Hsu-stuctue metic mifold is Pseudo Cocicul -ecuet symmetic d ci --ecuet symmetic the fom the equtio 4.6 we et.. W W... 7
50 ... W W W W W W W W.... W W = 0 povided the equtio 4. is stisfied which shows tht the mifold is Pseudo Pojective - -ecuet symmetic. Similly it c be show tht if the mifold is eithe Pseudo Pojective - -ecuet symmetic d ci --ecuet symmetic o Pseudo Pojective - -ecuet symmetic d Pseudo Cocicul --ecuet symmetic the it is eithe Pseudo Cocicul --ecuet symmetic o ci --ecuet symmetic fo the sme -ecuece pmete povided the equtio 4. is stisfied. Estel 5. Nijehuis eso he Nijehuis teso with espect to i hypebolic Hsu-stuctue metic mifold is defied by 5. N =. 7
51 73 heoem 5.. I the hypebolic Hsu-stuctue metic mifold we hve 5. N N = = i.e. Nijehuis teso is skew-symmetic i d. 5.3 } { N N = = i.e. N is pue i d. 5.4 N N = = 5.5 N N N = = = 5.6 } { N N N = = = 5.7 } { N N = =. Poof. Itechi the vectos d i equtio 5. d usi the fct tht = we obti the equtio 5.. Bi the vectos d i equtio 5. usi I. d 5. we et the equtio 5.3. Bi the equtio 5. thouhout d usi the equtio I. we obti 5.4. Bi d septely i equtio 5. usi I. d 5.4 the equti the esulti equtios we hve the equtio 5.5. Poof of the equtios 5.6 d 5.7 follows the sme ptte. heoem 5.. Let us put 5.8 P = he 5.9 P P = = 5.9b P P = = 5.9c P P = = 5.9d P P = =. Cosequetly 5.0 N P P = 5.0b N P P = 5.0c N P P =. Estel
52 Poof. Bi the equtio 5.8 thouhout o diffeet vectos i it the usi the equtio I. we et the equtios 5.9 to 5.9d. Now subtcti the equtios 5.9 fom 5.9c usi the equtio 5.3 we hve the equtio 5.0. Poof of the equtios 5.0b d 5.0c follows similly. heoem 5.3. Let us put 5. Q =. he 5. Q = Q 5.b Q = Q 5.c Q = Q 5.d Q = Q. Cosequetly 5.3 Q Q = N = N 5.3b Q Q = N 5.3c Q Q = N 5.3d Q Q = N = N. Poof. Bi the equtio 5. thouhout o diffeet vectos i it usi the equtios I. we et the equtios 5. to 5.d. Now subtcti the equtio 5.b fom 5.d we et Estel 5.4 Q Q =. Usi the equtio 5.3 i 5.4 d usi the fct tht Nijehuis teso is pue i d we et the equtio 5.3. Similly we c obti the othe equtios. Coolly 5.. I the hypebolic Hsu-stuctue metic mifold we hve 5.5 P = Q 5.5b P = Q 5.5c P = Q 74
53 5.5d P = Q 5.5e P = Q. Poof. Bi d i equtios 5.8 d 5. espectively the compi the esulti equtios we et 5.5. Similly we c obti the othe equtios. Coolly 5.. I the hypebolic Hsu-stuctue metic mifold we hve 5.6 P Q = N 5.6b P Q = N = N 5.6c P Q = N = N 5.6d P Q = N = N. Poof. Addi the equtios 5.8 d 5. the compi the esulti equtio with 5. we et 5.6. Bi i 5.8 i 5. the ddi the esulti equtios d compi this equtio with 5.4 we obti the equtio 5.6b. Poof of the equtios 5.6c d 5.6d follows similly. heoem 5.4. Let us put 5.7 =. he 5.8 = = { } i.e. is skew-symmetic i d. Estel 5.8b = = { } 5.8c = = 5.8d = 5.8e = = { } 5.8f = = { }. Cosequetly 5.9 = N 5.9b = N 5.9c = N 75
54 5.9d = N 5.9e = N. Poof. Itechi the vectos d i equtio 5.7 d usi the fct tht = we et the equtio 5.8. Bi the vectos d i equtio 5.7 the usi the equtios I. d 5.7 we et the equtio 5.8b. Bi d septely i equtio 5.7 d usi the equtio I. the equti the esulti equtios we obti the equtio 5.8c. Bi the equtio 5.7 thouhout d usi the equtio I. the we et the equtio 5.8d. Poof of the equtios 5.8e d 5.8f follows the sme ptte. Now subtcti the equtio 5.8d fom the equtio 5.8c d usi the equtio 5.4 we et the equtio 5.9. Similly we c pove the equtios 5.9b.5.9e. heoem 5.5. Let us put 5.0 =. he 5. = = i.e. is skew-symmetic i d. 5.b = = { } 5.c = = 5.d = = 5.e = = { } 5.f =. Cosequetly Estel 5. = N 5.b = N 5.c = N = N. Poof. Itechi d i equtio 5.0 the compi the esulti equtio with 5.0 we et 5.. Bi d i 5.0 d compi the 76
55 esulti equtio with the equtio obtied by multiplyi the equtio 5.0 by we et 5.b. Similly we c obti 5.c. 5.f. Multiplyi the equtio 5.c by d ddi the esulti equtio with the equtio 5.e the compi this equtio with 5.7 we et 5.. Similly we c pove the equtios 5.b d 5.c. Coolly 5.3. I the hypebolic Hsu-stuctue metic mifold we hve 5.3 = 5.3b = 5.3c = 5.3d =. Cosequetly 5.4 = N 5.4b = N 5.4c = N 5.4d = N 5.4e = N. Poof. Compi the equtios 5.8c d 5.f we et 5.3. Multiplyi the equtio 5.c by d compi the esulti equtio with 5.8f we obti 5.3b. Similly we c obti the equtios 5.3c d 5.3d. Subtcti the equtio 5.8c fom 5.c the compi the esulti equtio with 5.5 we et 5.4. Poof of the emii equtios follows similly. Estel 6. Associted Nijehuis teso I the hypebolic Hsu-stuctue metic mifold the ssocited Nijehuis teso is defied s 6. N = N. 77
56 78 heoem 6.. I the hypebolic Hsu-stuctue metic mifold we hve 6. N N = i.e. N is skew-symmetic i d. 6.b N N N = = 6.c N N = i.e. N is pue i d. Poof. om the equtios 5. d 6. we et 6. which shows tht N is skew-symmetic i d. Usi the equtio 6. i 5.5 toethe with the fct tht = we hve the equtio6.b. om the equtios 5.3 d 6. we obti 6.c which shows tht N is pue i d. Coolly 6.. Let us defie 6.3 P P def =. he 6.4 P P = 6.4b P P = 6.4c P P = 6.4d P P = 6.4e P P = 6.4f N P P = 6.4 N P P =. Poof. Usi the equtio 6.3 i equtios 5.9 d 5.0 toethe with the fct tht = we et the equtios 6.4. Coolly 6.. Let us defie 6.5 Q Q def =. he 6.6 Q Q = 6.6b Q Q = Estel
57 79 6.6c Q Q = 6.6d Q Q =. Poof. Usi the equtio 6.5 i equtios 5. toethe with fct tht = we obti the equtios 6.6. Coolly 6.3. Let us defie 6.7 def =. he 6.8 = 6.8b = 6.8c =. 6.8d = 6.8e =. Poof. Usi the equtio 6.7 i 5.8 toethe with the fct tht = we et the equtios e. Coolly 6.4. Let us put 6.9 def =. he 6.0 = 6.0b = 6.0c = 6.0d =. Poof. Usi the equtio 6.9 i equtios 5. toethe with the fct tht = we et the equtios d. Estel
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