On Submanifolds of an Almost r-paracontact Riemannian Manifold Endowed with a Quarter Symmetric Metric Connection

Transcription

1 Theoretcal Mathematcs & Applcatos vol. 4 o ISS: prt ole Scepress Ltd 04 O Submafolds of a Almost r-paracotact emaa Mafold Edowed wth a Quarter Symmetrc Metrc Coecto Mob Ahmad Abdullah. Al-Hossa ad Jaarda Prasad Ojha 3 Abstract We defe a quarter symmetrc metrc coecto a almost r paracotact emaa mafold ad we cosder submafolds of a almost r paracotact emaa mafold edowed wth a quarter symmetrc metrc coecto. We also obta Gauss ad Codazz equatos Wegarte equato ad curvature tesor for submafolds of a almost r paracotact emaa mafold edowed wth a quarter symmetrc metrc coecto. Mathematcs Subject Classfcato : 53C05; 53C07; 53C40; 53D35 Departmet of Mathematcs Faculty of Scece Jaza Uversty Jaza-079 KSA. Emal: mobahmad@redffmal.com Departmet of Mathematcs Faculty of Scece Jaza Uversty Jaza-079 KSA. Emal: aalhossa@jazau.edu.sa 3 Departmet of Mathematcs SMCEM Luckow-705 Ida. Emal: jaardaojha@gmal.com Artcle Ifo: eceved: May evsed: July Publshed ole : ovember 5 04.

2 O submafolds of a almost r-paracotact emaa mafold edowed Keywords: Almost r-paracotact emaa mafold; quarter symmetrc metrc coecto; submafolds; curvature tesor; Gauss ad Wegarte formulae Itroducto I []. S. Mshra studed almost complex ad almost cotact submafolds. I [] S. Al ad. vas cosdered submafolds of a emaa mafold wth quarter symmetrc coecto. Some propertes of submafolds of a emaa mafold wth quarter symmetrc sem-metrc coecto were studed [3] by L. S. Das. Moreover [4] I. Mha ad K. Matsumoto studed submafolds of a almost type. Let be a lear coecto a r paracotact emaa mafold of P Sasaka dmesoal dfferetable mafold M. The torso tesor T ad the curvature tesor of are gve respectvely by T [ ] ] [ The coecto s symmetrc f ts torso tesor T vashes otherwse t s o-symmetrc. The coecto s metrc coecto f there s a emaa metrc g M such that g 0 otherwse t s o-metrc. It s well kow that a lear coecto s symmetrc ad metrc f t s the Lev-Cvta coecto. I [5] S. Golab troduced the dea of a quarter-symmetrc lear coecto f ts torso tesor T s of the form T u u where u s a -form ad s a tesor feld of the type. I [6]. S. Mshra ad S.. Padey cosdered a quarter symmetrc metc coecto ad studed some of ts propertes. I [7] [8] [9] [0] ad [] some kds of quarter.

3 Mob Ahmad Abdullah. Al-Hossa ad Jaarda Prasad Ojha 3 symmetrc metrc coectos were studed. Let M be a dmesoal emaa mafold wth a postve defte metrc g. If there exst a tesor feld of type r vector felds ξ ξ ξ3... ξ r > r r -forms 3 r... such that. ξ δ β r { 3... r}. ξ.3 g ξ r. β.4 g g β where ad are vector felds o M the the structure ξ g r s sad to be a almost r paracotact emaa structure ad M s a almost From. through.4 we have.5 ξ 0 r.6 o 0 r r paracotact emaa mafold [7]..7 g g. def A almost r paracotact emaa mafold M wth structure ξ g r s sad to be s paracotact type f O almost ψ r r paracotact emaa mafold M wth a structure ξ g r s sad to be P Sasaka f t also satsfes ψ [ g [ g β β β β ] β β ] for all vector felds ad o M [].

4 4 O submafolds of a almost r-paracotact emaa mafold edowed The codtos are equvalet respectvely to for all r ad [ ξ ] [ g ] ξ β. I ths paper we study quarter symmetrc metrc coecto a almost r paracotact emaa mafold. We cosder hypesurfaces ad submafolds of almost r paracotact emaa mafold edowed wth a quarter symmetrc metrc coecto. We also obta Gauss ad Codazz equatos for hypersurfaces ad curvature tesor ad Wegarte equato for submafolds of almost r paracotact emaa mafold wth respect to quarter symmetrc metrc coecto. ξ β Prelmares Let M be a dmesoal dfferetable mafold of class C ad M be the hypersurface dfferetal the taget space of M by the mmerso τ : M M. The d τ of the mmerso τ s deoted by B. The vector feld Suppose that the evelopg mafold M correspods to a vector feld B that of M. M s a almost r paracotact emaa mafold wth metrc g. The the hypersurface M s also a almost r paracotact emaa mafold wth duced metrc g defed by g g B B where ad are the arbtrary vector felds ad s a tesor of type. If the emaa mafolds M ad a uque vector feld defed alog M such that g B 0 ad M are both oretable we ca choose

5 Mob Ahmad Abdullah. Al-Hossa ad Jaarda Prasad Ojha 5 g for arbtrary vector feld M. We call ths vector feld the ormal vector feld to the hypersurface M. We ow defe a quarter symmetrc metrc coecto by [7][8]. g ξ for arbtrary vector felds ad tagets to M where deotes the Lev-Cvta coecto wth respect to emaa metrc g s a tesor of type ad ξ s the vector feld defed by g ξ s a -form for a arbtrary vector feld of M. Also g g. ow suppose that ξ s a almost r g r paracotact emaa structure o M the every vector feld o M s decomposed as B l where l s a -form o M. For ay vector feld o M ad ormal we have b B b B B ad B where b s a -form o M. For each r we have [8]. B B b.3 ξ B a ξ where ξ s a vector feld ad a s defed as.4 a m ξ for each r o M.

6 6 O submafolds of a almost r-paracotact emaa mafold edowed ow we defe as.5 B r. Theorem. The coecto duced o the hypersurface of a emaa mafold wth a quarter symmetrc metrc coecto wth respect to the ut ormal s also quarter symmetrc metrc coecto. Proof: Let be the duced coecto from o the hypersurface wth respect to the ut ormal the we have.6 B B h B for arbtrary vector felds ad o M where h s a secod fudametal tesor of the hypersurface M. Let be coecto duced o the hypersurface from wth respect to the ut ormal the we have.7 B B m for arbtrary vector felds ad of B M m beg a tesor feld of type 0 o the hypersurface M. From equato. we have B B B B g B B Bξ B B Usg.5.6 ad.7 above equato we get B m B h B.8 b g Bξ a Comparg the tagetal ad ormal vector felds we get.9 g ξ ad a...0 m h b a g. Thus. [ ].

7 Mob Ahmad Abdullah. Al-Hossa ad Jaarda Prasad Ojha 7 Hece the coecto duced o coecto [5]. M s a quarter symmetrc metrc 3 Totally umblcal ad totally geodesc hypersurfaces ad We defe B ad B respectvely by B B B B B B B B B B where ad beg arbtrary vector felds o M. The.6 ad.7 take the form B h ad B m. These are Gauss equatos wth respect to duced coecto ad respectvely. fucto Let be -orthoormal vector felds the the s called the mea curvature of ad h M wth respect to emaa coecto m s called the mea curvature of M wth respect to the quarter symmetrc sem-metrc coecto. From ths we have followg deftos:

8 8 O submafolds of a almost r-paracotact emaa mafold edowed Defto 3. The hypersurface M s called totally geodesc hypersurface of M wth respect to the emaa coecto f h vashes. Defto 3. The hypersurface M s called totally umblcal wth respect to coecto f h s proportoal to the metrc tesor g. We call M totally geodesc ad totally umblcal wth respect to quarter symmetrc metrc coecto accordg as the fucto m vashes ad proportoal to the metrc g respectvely. ow we have followg theorems: Theorem 3. I order that the mea curvature of the hypersurface respect to cocdes wth that of vector feld ξ s taget to Proof: I vew of.0 we have Summg up for M ad M wth M wth respect to f ad oly f the M s varat. m h b a g 3... ad dvdg by we obta m h f ad oly f 0 ad b 0. Hece from.3 we have ξ B. ξ Thus the vector feld s taget to M ξ ad M s varat. Theorem 3. The hypersurface M wll be totally geodesc wth respect to emaa coecto f ad oly f t s totally geodesc wth respect to the quarter symmetrc metrc coecto ad b a g 0. Proof: The proof follows from.0 easly.

9 Mob Ahmad Abdullah. Al-Hossa ad Jaarda Prasad Ojha 9 4 Gauss Wegarte ad Codazz equatos I ths secto we shall obta Wegarte equato wth respect to the quarter symmetrc metrc coecto. For the emaa coecto these equatos are gve by [] 4. BH B for ay vector feld M where h s a tesor feld of type of defed by 4. g H h. From equato.. ad.4 we have 4.3 a B Bb. B B Usg 4. we have 4.4 BM where B M H a b ξ ξ for ay vector feld M. Equato 4.4 s Wegarte equato. We shall fd equato of Gauss ad Codazz wth respect to the quarter symmetrc metrc coecto. The curvature tesor wth respect to quarter symmetrc metrc coecto of M s 4.5. Puttg B B ad B we have [ ] M B B B B B B B B B [ B B ] B. By vrtue of ad. we get B B B B{ m M m M} 4.6 { m m m }

10 0 O submafolds of a almost r-paracotact emaa mafold edowed where [ ] s the curvature tesor of the quarter symmetrc metrc coecto. Substtutg U g U ad U g U 4.6 we ca easly obta B B B BU U m m U 4.7 m m U ad B B B m m 4.8 m. Equatos 4.7 ad 4.8 are the equato of Gauss ad Codazz wth respect to the quarter symmetrc metrc coecto. 5 Submafolds of codmeso Let M be a dmesoal dfferetable mafold of dfferetablty class C ad M be dmesoal mafold mmersed dfferetablty M by mmerso : M M. τ We deote the d τ of the mmerso τ by B so that the vector feld the taget space of Suppose that M correspods to a vector feld B that of M. M s a almost paracotact emaa mafold wth metrc tesor g. The the submafold emaa mafold wth metrc tesor g such that M s also a almost paracotact

11 Mob Ahmad Abdullah. Al-Hossa ad Jaarda Prasad Ojha g B B g for ay arbtrary vector felds If the mafolds ad M ad g B for arbtrary vector feld M []. M are both oretable such that g B g 0 g g M [3]. We suppose that the evelopg mafold symmetrc metrc coecto gve by [7]. g ξ for arbtrary vector felds wth respect to the emaa metrc g Let us ow put 5. B B a b 5. ξ B a b ξ M admts a quarter M deotes the Lev Cvta coecto where a ad b are -forms o s a -form. M ξ s a vector feld the taget space o M ad a b are fuctos o M defed by 5.3 a b. Theorem 5. The coecto duced o the submafold M of co-dmeso two of the emaa mafold M wth quarter symmetrc metrc coecto s also quarter symmetrc metrc coecto. Proof: Let be the coecto duced o the submafolds M from the coecto o the evelopg mafold wth respect to ut ormals ad the we have []

12 O submafolds of a almost r-paracotact emaa mafold edowed 5.4 k h B B B for arbtrary vector felds of M where h ad k are secod fudametal tesors of. M Smlarly f be coecto duced o M from the quarter symmetrc metrc coecto o M we have 5.5 m B B B m ad beg tesor felds of type 0 of the submafold M. I vew of equato. we have. ξ B B g B B B B B B I vew of ad 5.5 we fd 5.6 b a B g b a B k h B m B ξ where B ad. g B B g Comparg tagetal ad ormal vector felds to M we get 5.7 ξ g 5.8 a g a a h m b. g b b k Thus 5.9. ] [ Hece the coecto duced o M s quarter symmetrc metrc coecto. 6 Totally umblcal ad totally geodesc submafolds Let be --orthoormal vector felds o the

13 Mob Ahmad Abdullah. Al-Hossa ad Jaarda Prasad Ojha 3 submafold M. The the fucto { h k } s mea curvature of M wth respect to the emaa coecto ad s the mea curvature of ow we have the followg deftos: { m M wth respect to [3]. Defto 6. If h ad k vash separately the submafold totally geodesc wth respect to the emaa coecto. Defto 6. The submafold the f h ad k are proportoal to the metrc g. We call } M s called M s called totally umblcal wth respect to M totally geodesc ad totally umblcal wth respect to the quarter symmetrc metrc coecto accordg as the fuctos m ad vash separately ad are proportoal to metrc tesor g respectvely. Theorem 6. The mea curvature of coecto cocdes wth that of symmetrc metrc coecto f ad oly f { a Proof: I vew of 5.8 we have m a b g b M wth respect to the emaa M wth respect to the quarter a h b a g k b } 0. k Summg up for... ad dvdg by we get { m } { h. k }

14 4 O submafolds of a almost r-paracotact emaa mafold edowed f ad oly f { a b a b g } 0 whch proves our asserto. Theorem 6. The submafold M s totally geodesc wth respect to the emaa coecto f ad oly f t s totally geodesc wth respect to the quarter symmetrc metrc coecto provded that ad a a g 0 b b g 0. Proof: The proof follows easly from equatos 5.8 a ad b. 7 Curvature tesor ad Wegarte equato by [] For the emaa coecto the Wegarte equatos are gve 7. a B BH ad b B BK where H ad K are tesor felds of type such that g H h ad g K k. Also makg use of. ad 7. a we get B B H a a ξ ab b a 7.3 BM L where B.

15 Mob Ahmad Abdullah. Al-Hossa ad Jaarda Prasad Ojha 5 ad M H a a ξ L a b b a. Smlarly from. ad 7. b we obta 7.4 BM L where B M K b b ξ. Equatos 7.3 ad 7.4 are Wegarte equatos wth respect to the quarter symmetrc metrc coecto. 8 emaa curvature tesor for quarter symmetrc metrc coecto Let be the emaa curvature tesor of the evelopg mafold the M wth respect to the quarter symmetrc metrc coecto. [ ] eplacg by B by B ad by B we get B B B Usg 7.3 we obta B B B B { B B { B { B [ ] B B B m m m[ ] Aga usg ad 5.9 we fd B B B [ B B ] B. } } [ ] }.

16 6 O submafolds of a almost r-paracotact emaa mafold edowed } { } { } { } { } { } { } { m L m L m m m M M M m M m B B B B B where beg the emaa curvature tesor of the submafold wth respect to the duced coecto o. M efereces [] Mshra. S. Almost complex ad almost cotact submafold Tesor [] Al S. ad vas. O submafolds of a mafold wth quarter symmetrc coecto v. Mat. U. Parma [3] Das L. S. vas. Al S. ad Ahmad M. Study of submafolds mmersed a mafold wth quarter symmetrc sem-metrc coecto Tesor. S [4] Mha I. ad Matsumoto K. Submafolds of a almost r-paracotact mafold of P Sasaka type Tesor. S [5] Golab S. O sem-symmetrc ad quarter-symmetrc lear coectos Tesor [6] Mshra. S ad Padey S.. O quarter symmetrc metrc F-coectos Tesor. S [7] Ahmad M. Ju J. B. ad Haseeb A. Hypersurfaces of almost r-paracotact emaa mafold edowed wth quarter symmetrc metrc coecto Bull. Korea Math. Soc

17 Mob Ahmad Abdullah. Al-Hossa ad Jaarda Prasad Ojha 7 [8] Ahmad M. Ozgur C. ad Haseeb A. Hypersurfaces of almost r-paracotact emaa mafold edowed wth a quarter symmetrc o-metrc coecto KUGPOOK Math J [9] vas. ad Verma Geeta O quarter-symmetrc o-metrc coecto a emaa mafold J. ajastha Acad. Phys. Sc [0] Pujar S. S. A ote o Loretza para Sasaka mafold wth a LP-cotact o-metrc quarter symmetrc -coecto Joural of the Ida Math. Soc [] Trpath M. M. A ew coecto a emaa mafold It. Elec. J. Geo [] Che B.. Geometry of submafolds Marcel Dekker ew ork 973.