On a Semi-symmetric Non-metric Connection Satisfying the Schur`s Theorem on a Riemannian Manifold

Transcription

1 O a Sem-symmetrc No-metrc oecto Satsfy te Scur`s Teorem o a emaa Mafod Ho Ta Yu Facuty of Matematcs, Km I Su versty, D.P..K Abstract: 99, Aace ad ae troduced te cocet of a sem-symmetrc o-metrc coecto[]. Te sem-symmetrc o-metrc coecto does ot satsfy te Scur`s teorem. Te urose of te reset aer s to study some roertes of a ew sem-symmetrc o-metrc coecto satsfy te Scur`s teorem a emaa mafod. Ad we cosdered ecessary ad suffcet codto tat a emaa mafod wt a sem-symmetrc o-metrc coecto be a emaa mafod wt costat curvature. Key words: sem-symmetrc o-metrc coecto, costat curvature. Itroducto I te reced aer a sem-symmetrc o-metrc coecto was studed te emaa mafod but te emaa mafod wt te sem-symmetrc o-metrc coecto Scur s teorem s ot roved. I [3] te statstca mafod wt costat curvature was studed ad [4] tat a statstca mafod wt costat curvature s a roectve fat mafod was roved. I [5] Scur s teorem was roved te ema mafod wt Lev-vta oecto. Ma resuts of te aer I ts aer we study te sem-symmetrc o-metrc coecto o a emaa mafod M, tat satsfes Z X, Y X Y, Z Y X, Z T X, Y Y X X Y Te reato betwee te sem-symmetrc o-metrc coecto ad te Lev-vta coecto of M, s ve by X Y X Y Y X X, Y P were s a -form ad P s a vector fed defed by X, P X. Te dua coecto mafod of te coecto satsfes o a emaa

2 Z X, Y X Y, Z Y X, Z 3 T X, Y Y X X Y Te reato betwee te coecto ve by X Y X Y Y X X, Y P If te oca exresso of,, s, ad Lev-vta coecto te oca exresso of,, 3 ad 4, resectvey s 4, { },, resectvey, te, T 5 } { 6, T 7 { } 8 Let M,, be a emaa mafod wt sem-symmetrc o-metrc coecto. Teorem. Suose M s a coected dmesoa emaa mafod wt a sem-symmetrc o-metrc coecto tat s everywere wader. If 3, te M s a costat curvature sace. Proof. By te secod Bac detfy of a curvature tesor o a emaa mafod wt a asymmetrc o-metrc coecto T m m T m m T m m we obta If a sectoa curvature at te ot s deedet of E a -dmesoa subsace of T M, te te curvature tesor s

3 . Hece [ [ From ts we obta 0 By cotract wt, we obta 0. otract dexes ad, we obta 0. osequety, from 3 we obta 0, tat s cost. oroary. Suose M s coected dmesoa emaa mafod wt a dua coecto tat s everywere wader. If 3, te M s a costat curvature sace. By us 6, 8 curvature tesor of ad are resectvey were K K K s te curvature tesor of. From 9 ad 0 we obta 9 0 Lemma. If te Wey coforma curvature of,, ad, resectvey, te ad are Proof. From 9 ad 0, we ave K 3

4 were, cotract dexes ad K By cotract wt 4 4 Substtut to 4 [ K ] 5 Aso substtut 5 to 3, we ave. Tus we ave: Teorem. If a emaa metrc admts a sem-symmetrc o-metrc coecto wose curvature tesor ad dua curvature tesor vases o a emaa mafod, te te emaa metrc s coforma fat. Teorem 3. I order tat a emaa metrc wt a costat curvature s admtted o M,,, t s ecessary ad suffcet tat a sem-symmetrc o-metrc coecto soud be a couate symmetrc ad a coforma fat coecto, ad tat ts cc curvature tesor satsfy te Este equato Proof. From,, we obta Tus s a couate symmetrc coecto. O te oter ad, from, cc curvature satsfes te Este equato. Ad from ad emma, 0. Tus s a coforma fat metrc. oversey, f s a couate symmetrc ad a coforma fat coecto, te from emma s te Esta equato, we fd. Tus te emaa metrc s of costat

5 curvature o M,,. Lemma. Te curvature tesor of a emaa mafod wt a sem-symmetrc o-metrc coecto satsfes te foow roertes: 0 3 If -form s cosed, te 0,, P 0 0,, P 0 were reresets crce ermutato ad P, P are te voume curvature tesor of,. Proof. From 9 ad 0, we obta By us tese, we rove Lemma. Now we cosder coecto trasformatos accord to coforma trasformato of te metrc a emaa mafod M, : e. orresod coecto trasformatos rereseted ocay resectvey; were ad _ are 6. x Teorem 4. If te tesor s ve by, te te varat of te coecto trasformato 6 s te Wey coforma curvature tesor of, tat s

6 7 Proof. From 6 te curvature tesors are By summ tese equatos we ave 8 were. otract dexes, 8. Aso cotract wt,. Substtut to, we et. Ad substtut to 8 ad utt we obta. Tus we ave: oroary. If te sem-symmetrc o-metrc coecto s a couate symmetrc coecto tat s, te te varat of te coecto trasformato 6 s te Wey coforma curvature tesor of te curvature tesor. oroary. If te sem-symmetrc o-metrc coecto s a

7 couate symmetrc coecto tat s, te te varat of te coecto trasformatos 6, 8 s te Wey coforma curvature tesor of te curvature tesor. efereces [] Aace Nmaa S, afe Maaa, A sem-symmetrc o-metrc coecto o a emaa mafod. Ida J. Pure. A. Mat. 399 No [] De.., Bswas. S., O a tye of sem-symmetrc o-metrc coecto o a emaa mafod. Istabu v. Fe Fa. Mat. Der. 55/56 996/ [3] Kurose T ad eto a, O te dvereces of -coformay fat statstca mafods. Toou Mat.J., 4b,47-433,994 [4] E. S. Steaova, Dua Symmetrc statstca mafods. Joura of Matematca Scece,vo,47, No, ,007 [5] S. S. er, W. H. e ad K. S. Lam, Lectures o Dffereta Geometry, Word Scetfc, 000, 60.