CR-submanifolds of a nearly trans-hyperbolic Sasakian manifold with a quarter symmetric non-metric connection

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1 IOSR Journal of Matematics IOSR-JM e-issn: p-issn: Volume 10 Issue 3 Ver I May-Jun wwwiosrjournalsor CR-submanifolds of a nearly trans-yperbolic Sasakian manifold wit a quarter symmetric non-metric connection Mobin mad * Janardan rasad Oja ** and Mod anis Siddiqui * *epartment of Matematics Jazan University KS ** epartment of Matematics SRMGC Lucknow-7105 India bstract: Tis paper deals wit te study of CR-submanifolds of a nearly trans-yperbolic Sasakian manifold wit a quarter symmetric non-metric connection We study parallel distribution relatin to vertical CRsubmanifolds of a nearly trans-yperbolic Sasakian manifold wit quarter symmetric non-metric connection Furter we obtain te parallel distributions on CR-submanifolds Keywords: CR-submanifolds nearly trans-yperbolic Sasakian manifold quarter symmetric non-metric connection parallel distribution I Introduction In 1978 urel Bejancu introduced te notion of CR-submanifold of Kaeler manifold 1 On te oter and CR-submanifold ave been studied by kobayasi 3 J Oubina introduced a new class of almost contact metric manifold known as trans-sasakian manifold 4 Gere studied on armonicity on nearly trans- Sasaki eometry of CR-submanifold of manifold 5 CR-submanifold of a trans-sasakian manifold ave been studied by Said 6 Later l-solamy studied te CR-submanifold of a nearly trans-sasakian manifold 7 In 1976 Upadyay and ube ave studied almost contact yperbolic structure 8 Batt and ube studied on CRsubmanifold of trans-yperbolic Sasakian manifold 9 Gill and ube ave also worked on CR-submanifold of trans-yperbolic Sasakian manifold 10 Kumar and ube studied CR-submanifold of a nearly trans-yperbolic Sasakian manifold 11 In tis paper we study CR-submanifold of a nearly trans-yperbolic Sasakian manifold endowed wit a quarter symmetric non-metric connection Let be a linear connection in an n dimensional differentiable manifold M Te torsion tensor T and curvature tensor R of are iven respectively by T R Te connection is symmetric if its torsion tensor T vanises oterwise it is non-symmetric Te connection is metric connection if tere is a Riemannian metric in M suc tat 0 oterwise it is non-metric It is well known tat a linear connection is symmetric and metric if and only if it is te Levi-Civita connection In 1 S Golab introduced te idea of a quarter symmetric connection linear connection is said to be a quarter symmetric connection if its torsion tensor T is of te form T Some properties of quarter symmetric non-metric connection was studied by several autors in Tis paper is oranized as follows: In section we ive a brief introduction of nearly trans-yperbolic Sasakian manifold In section 3 we ave proved some basic lemmas on nearly trans-yperbolic Sasakian manifold wit a quarter symmetric non-metric connection In section 4 we ave discussed parallel distributions II reliminaries Let M be an n -dimensional almost yperbolic contact metric manifold wit almost yperbolic were a tensor of type 11 a vector field called structure vector contact metric structure field and te dual 1-form of satisfyin te followin o 3 for any tanent to M 17 In tis case wwwiosrjournalsor 8 ae

2 CR-submanifolds of a nearly trans-yperbolic Sasakian manifold wit a quarter symmetric 4 n almost yperbolic contact metric structure and only if on M is called trans-yperbolic Sasakian 10 if 5 for all ave tanent to M and are functions on M On a trans-yperbolic Sasakian manifold M we 6 were is te Riemannian metric and is te Riemannian connection Let M be an m dimensional isometrically immersed submanifold of nearly trans-yperbolic Sasakian manifold M We denote by te Riemannian metric tensor field on M as well as M efinition 1 n m dimensional Riemannian submanifold M of a nearly trans-yperbolic Sasakian manifold M is called a CR-submanifold if is tanent to M and tere eists differentiable distribution : M T M suc tat is invariant under tat is for eac M ; i te distribution ii te complementary ortoonal distribution : T M of te distribution on M is anti-invariant under tat is M T M for all M were T M and T M are tanent space and normal space of M at M respectively If dim 0 resp dim 0 ten CR-submanifold is called an invariant resp anti-invariant Te distribution resp is called orizontal resp vertical distribution Te pair is called - for M orizontal resp -vertical if resp For any vector field tanent to M we write 8 were and belon to te distribution and respectively For any vector field N normal to M we put 9 N BN CN were BN resp CN denotes te tanential resp normal component of N Now we remark tat owin to te eistence of te 1-form we can define a quarter symmetric non-metric connection in almost contact metric manifold by 10 suc tat TM were for any vector field Usin 5 and 10 we et 11 Similarly we ave On addin above equations we obtain is te induced connection wit respect to on M is a 1-form and is a wwwiosrjournalsor 9 ae

3 1 CR-submanifolds of a nearly trans-yperbolic Sasakian manifold wit a quarter symmetric Tis is te condition for an almost contact structure wit a quarter symmetric non-metric connection to be nearly trans-yperbolic Sasakian manifold From 10 and 6 we et 13 1 We denote by te metric tensor of M as well as tat induced on M Let be te quarter symmetric non-metric connection on M and be te induced connection on M wit respect to te unit normal N Teorem Te connection induced on te CR-submanifolds of a nearly trans-yperbolic Sasakian manifold wit a quarter symmetric non-metric connection is also a quarter symmetric non-metric connection roof Let be te induced connection wit respect to te unit normal N on a CR-submanifold of a nearly trans-yperbolic Sasakian manifold wit a quarter symmetric non-metric connection Ten 14 m were m is a tensor field of type 0 on CR-submanifold M If be te induced connection on CRsubmanifolds from Riemannian connection ten 15 were is a second fundamental tensor Now from 14 and 15 we ave m Equatin te tanential and normal components from bot te sides in te above equation we et m and Tus is also a quarter symmetric non-metric connection Now te Gauss formula for a CR-submanifold of a nearly trans-yperbolic Sasakian manifold wit a quarter symmetric non-metric connection is 16 and te Weinarten formula for M is iven by 17 N N for TM N T M denotes te operator of te normal connection Moreover we ave 18 N N were and are called te second fundamental tensors of M and III N Some Basic Lemmas Lemma 31 Let M be a CR-submanifold of a nearly trans-yperbolic Sasakian manifold M wit a quartersymmetric non-metric connection Ten 31 3 B wwwiosrjournalsor 10 ae

4 CR-submanifolds of a nearly trans-yperbolic Sasakian manifold wit a quarter symmetric wwwiosrjournalsor 11 ae 33 C for all TM roof By direct covariant differentiation we ave By virtue of and 17 we et Similarly ddin we obtain 34 C B C B for any TM Now equatin orizontal vertical and normal components in 34 we et te desired result Lemma 3 Let M be a CR-Submanifod of a nearly trans-yperbolic Sasakian manifold M wit a quarter symmetric non-metric connection Ten for any roof From Gauss formula 16 we ave 37 lso we ave 38 From 37 and 38 we et lso for nearly trans-yperbolic Sasakian manifold wit quarter symmetric non-metric connection we ave

5 CR-submanifolds of a nearly trans-yperbolic Sasakian manifold wit a quarter symmetric wwwiosrjournalsor 1 ae 310 ddin 39 and 310 we obtain Subtractin 39 from 310 we et Hence Lemma is proved Lemma 33 Let M be a CR-submanifold of a nearly trans-yperbolic Sasakian manifold M wit a quarter symmetric non-metric connection ten and for any roof From Weinarten formula 17 we ave 311 lso we ave 31 From 311 and 31 we et 313 On addin 313 and 314 we obtain Subtractin 313 and 314 we find Tis proves our assertions Lemma 34 Let M be a CR-submanifold of a nearly trans-yperbolic Sasakian manifold M wit a quarter symmetric non-metric connection ten

6 CR-submanifolds of a nearly trans-yperbolic Sasakian manifold wit a quarter symmetric wwwiosrjournalsor 13 ae for any and roof By usin Gauss and Weinarten equation for and respectively we et 315 lso we ave 316 From 315 and 316 we obtain 317 lso for nearly trans-yperbolic Sasakian manifold wit a quarter symmetric non-metric connection we ave 318 ddin 317 and 318 we find Subtractin 317 from 318 we et Hence Lemma is proved IV arallel istributions efinition 41 Te orizontal resp vertical distribution resp is said to be parallel wit respect to te quarter symmetric non-metric connection on M if resp W for any vector field resp W roposition 41 Let M be a -vertical CR-submanifold of a nearly trans-yperbolic Sasakian manifold M wit a quarter symmetric non-metric connection If te orizontal distribution is parallel ten 41 for all roof For orizontal distribution we ave 4 for any Usin te fact tat 0 for 3 ives 43 B for any lso since 44 C B Terefore 45 C for any From 33 we ave 46 C

7 CR-submanifolds of a nearly trans-yperbolic Sasakian manifold wit a quarter symmetric wwwiosrjournalsor 14 ae for any uttin in 46 we et 47 or 48 Similarly puttin in 46 we find 49 Hence from 48 and 49 we ave 410 Operatin on bot sides of 410 and usin 0 we et 411 for all Now for te distribution we prove te followin proposition roposition 4 Let M be a -vertical CR-submanifold of a nearly trans-yperbolic Sasakian manifold M wit a quarter symmetric non-metric connection If te distribution is parallel wit respect to te connection on M ten 41 for any roof Usin Gauss and Weinarten formula we obtain 413 for any Takin inner product wit in 313 we et 414 If te distribution is parallel ten and for any So from 414 we et or 0 wic is equivalent to 416 for any Tis completes te proof efinition 43 CR-submanifold wit a quarter-symmetric non-metric connection is said to be mied totally eodesic if 0 for all and Te followin Lemma is an easy consequence of 18 Lemma 44 Let M be a CR-submanifold of a nearly trans-yperbolic Sasakian manifold M wit a quartersymmetric non-metric connection Ten M is mied totally eodesic if and only if N for all efinition 45 normal vector field 0 N is called -parallel normal section if 0 N for all Now we ave te followin proposition roposition 46 Let M be a mied totally eodesic -vertical CR-submanifold of a nearly trans-yperbolic Sasakian manifold M wit a quarter symmetric non-metric connection Ten te normal section N is -parallel if and only if N for all roof Let N Ten from 3 we ave

8 CR-submanifolds of a nearly trans-yperbolic Sasakian manifold wit a quarter symmetric In particular we ave By usin it in 33 we et 418 Tus if te normal section 0 definition and 418 we et 419 N 0 wic is equivalent to N for all Te converse part easily follows from 418 Tis completes te proof of te proposition for any 0 or N N N wit quarter symmetric non-metric connection is -parallel ten by References 1 Bejancu CR-submsnifolds of a Kaeler manifold I roc mer Mat Soc Bejancu Geometry of CR-submanifolds Reidel ublisin Company Holland M Kobayasi CR-submanifolds of Sasakian manifold Tensor NS no J Oubina New class of almost contact metric structures ubl Mat ebrecen C Gere Harmonicity on nearly trans-sasaki manifolds emonstratio Mat no M H Said CR-submanifolds of a trans-sasakian manifold Indian J ure ppl Mat F R l-solamy CR-submanifolds of a nearly trans-sasakian manifold IJMMS M Upadyay and K K ube lmost contact yperbolic f -structure cta Mat cad Scient Hun Tomus L Batt and K K ube On CR-submanifolds of a trans-yperbolic Sasakian manifold cta Ciencia Indica H S Gill and K K ube On CR-submanifolds of trans-yperbolic Sasakian manifolds emonstratio Mat S Kumar and K K ube CR-submanifolds of a nearly trans-yperbolic Sasakian manifold emonstratio Matematica Vol LI No S Golab On semi symmetric and uarter symmetric linear connections Tensor M mad M Siddiqui and J Oja Semi-invariant submanifolds of Kenmotsu manifolds immersed in a eneralized almost r-contact structure admittin a quarter symmetric non-metric connection J Mat comput Sci 01 No M mad CR-submanifolds of L-Sasakian manifolds endowed wit a quarter symmetric metric connection Bull Korean Mat Soc No M mad J B Jun and Haseeb Submanifolds of an almost r-paracontact Reimannian manifold endowed wit a quarter symmetric non-metric connection J Cunceon Mat Soc Vol 4 No M mad C Ozur and Haseeb Hypersurfaces of an almost r-paracontact Reimannian manifold endowed wit a quarter symmetric non-metric connection Kyunpook Mat j E Blair Contact manifolds in Riemannian Geometry Lecture Notes in Matematics vol 509 Spriner-Verla Berlin 1976 wwwiosrjournalsor 15 ae