Advances in Pure Mathematics, 2011, 1, 378383 doi:10.4236/apm.2011.16067 Published Online November 2011 (http://www.SciRP.org/journal/apm) Copyright © 2011 SciRes. APM Geodesic Lightlike Submanifolds of Indefinite Sasakian Manifolds* Junhong Dong1, Ximin Liu2 1Department of Mat hem at i cs, South China University of Technology, Guangzhou, China 2School of Mathematical Sciences, Dalian University of Technology, Dalian, Chi na Email: dongjunhong run@163.com, ximinliu@dlut.ed u.cn Received July 23, 201 1; revised August 15, 2011; accepted August 25, 2011 Abstract In this paper, we study geodesic contact CRlightlike submanifolds and geodesic screen CRlightlike (SCR) submanifolds of indefinite Sasakian manifolds. Some necessary and sufficient conditions for totally geodesic, mixed geodesic, geodesic and geodesic contact CRlightlike submanifolds and SCR submanifolds are obtained. D Keywords: CRLightlike Submanifolds, Sasakian Manifolds, Totally Geodesic Submanifolds 1. Introduction A submanifold of a se mi R ie ma n ni an ma ni f ol d is called lightlike submanifold if the induced metric on is degenerate. The general theory of a lightlike sub manifold has been developed by Kupeli [1] and Bejancu Duggal [2]. The geometry of CRlightlike submanifolds of inde finite Kaehler manifolds was studied by Guggal and Bejancu [2]. The geodesic CRlightlike submanifolds in indefinite Kaehler manifolds were studied by Sahin and Günes [3,4]. Lightlike submanifold of indefinite Sasakian mani folds can be defined according to the behavior of the almost contact structure, and contact CRlightlike sub manifolds and screen CRlightlike (SCR) submanifolds of indefinite Sasakian manifolds were studied by Duggal and Sahin in [5]. The study of the geometry of subma nifolds of indefinite Sasakian manifolds has been de veloped by [6] and others. In this paper, geodesic contact CRlightlike submani folds and geodesic screen CRlightlike (SCR) submani folds of indefinite Sasakian manifolds are considered. Some necessary and sufficient conditions for totally geodesic, mixed geodesic, Dgeodesic and D geo desic contact CRlightlike submanifolds and SCR sub manifolds are obtained. 2. Preliminaries A submanifold m immersed in a semiRiemannian manifold () mn , g is called a lightlike submanifold if it admits a degenerate metric induced from whose radical distribution is of rank where RadTM r 1rm , , where =RadTMTMTM = ,=0, xx xM TMuTMguvvTM . Let STM TM be a screen distribution which is semi Riemannian complementary distribution of in , i.e. RadTM TM =(RadTMS TM . As is a nondegenerate vector subbundle of STM  TM , we put .MS TM=S T M TM We consider a nondegenerate vector subbundle of STM RadTM, which is a complementary vector bundle of in TM . Since, for any local basis {} i of , there exists a local frame of sections with value in the orthogonal complement of RadTM {} i N STM such that =, ii gN ij and ,N =0 ij , there exists a lightlike, transversal vector bundle gN ltrTM locally spanned by . Let be the comple mentary (but not orthogonal) vector bundle to TM in {} i N trTM  TM . Then =,trTMltrTMSTM =.TMSTMRadTMltrTMS TM *This work is supporte d by NSFC (10931005).
J. H. DONG ET AL.379 Now, let be the leviCivita connection on , we have ,= ,, ,, , X, X gYZgYZ gYZ XYZ TM (2.1) =,, , XX YYhXYXYTM , (2.2) =, , t XVX VAXVX TM VtrTM , (2.3) where and , XV YAX ,, t X hXY V belong to TM and , TMtr respectively. Using the projectors and :ltrTM STM :str TMltr TM , from [1], we have =,,,, ls XX YYh XYhXYXYTM , (2.4) =,,, ls XNX NAXNDXNN ltrTM (2.5) =,, sl XWX WAX WDXWW STM . ,, , X (2.6) Denote the projection of TM to by , we have the decomposition STM P = XX PYPYhX PY (2.7) = t X AX (2.8) for any ,, ,X YTMRadTMNltrTM . From the above equations we have ,,= ,, l hXY gAXY (2.9) ,,= , N, hXPYN gAXPY (2.10) ,,=0, =0. (2.11) l gh XA Definition 2.1 A (2n + 1)dimensional SemiRieman nian manifold , g is called a contact metric mani fold if there is a tensor field (1,1) , a vector field , called the characteristic vector field, and its dual 1form V such that ,= ,,,=gXY gXYXYgVV, (2.12) 2=,,= , XXVgXV X (2.13) ,=,, ,dXY gXYXYTM , (2.14) where =1 . From the above definiton, it follows that =0,=0,=1.V V (2.15) The (, ,, )Vg is called a contact metric structure of . If =0Nd V , we say that has a normal contact structure, where N is the Nijenhuis tensor field of . A normal contact metric manifold is called a Sasakian m a ni fo ld for which we have = XVX. (2.16) =, XYgXYV YX . (2.17) Let ,, ,MgSTMSTM be a lightlike submanifold of , g. For any vector field tangent to , we put =, PX QX (2.18) where and are the tangential and the transversal parts of PX QX , respectively. Let’s suppose is a spacelike vector field so that V =1 , it’s similar when is a timelike vector field. V 3. Geodesic Invariant L ightlike Submanifolds Definition 3.1 Let ,, ,MgSTM STM be a lightlike submanifold, tangent to the structure vector fi eld ,V VSTM, immersed in an indefinite Sasakian manifold , g, we say that is an invariant subma  nifolds of if the following conditions are satisfied =, =RadTMRadTMS TMSTM . (3.1) From (2.16), (2.17), (2.18) and (2.4) we hav e ,=,=0, = = ls XX hXVhXVVVPX, (3.2) ,= ,= ,,, l hXYhXYh XYXYTM . (3.3) From (3.1) and (2.12) we have =, =ltr TMltr TMSTMSTM . (3.4) Theorem 3.1 Let ,, ,MgSTM STM be an invariant lightlike submanifold of an indefinite Sasakian manifold , then is totally geodesic if and only if and l h h of are parallel. Proof. Suppose is parallel, for any l h ,, YZ TM , we have ,= ,, ,=0. lll XX lX hYVhYV hYV hY V X By (3.2), we h a v e ,= ,=0 ll X hYV hYV, Copyright © 2011 SciRes. APM
J. H. DONG ET AL. 380 so ,= lX hY Y00.. That is to say ,= l hYPX ,=0.hYPX In a similar way, we can get Thus, s is totally geodesic. Conversely, if since ,= ,=0 ls hXY hXY, ,= ,, ,=0, l XX X lX hYZhYZ hYZ hY Z ll ,= ,, ,=0, sss XX sX hYZhYZhYZ hY Z X so and l h h are parallel, which completes the proof. 4. Geodesic Contact CRLightlike Submanifolds Definition 4.1 Let be a lightlike submanifold, tangent to the structure vector field , immersed in an indefinite Sasakian manifold ,, ,MgSTM STM V , g. We say that is a contact CRlightlike sub manifold of if the following conditions are satisfied [(A)] is a distribution on RadTM such that (RadTadTM )={0}.MR 0 D [(B)] There exist vector bundles and over D such that 0 =,S TMRadTMDDV 00 12 =, =DDDLL, where 0 is nondegenerate and , 2 is a vector subbundle of ST . So we have the decomposition D 1=Lltr TM L M 0 =,=TMDDV DRadTMRadTMD . If we denote , then we have ˆ= DDV ˆˆ =,= TMDDD D ˆ . Definition 4.2 A contact CRlightlike submanifold of an indefinite Sasakian manifold is called geodesic contact CRlightlike submanifold if its second funda mental form satisfied ˆ D h ,=0hXY , for any ˆ , YD . Definition 4.3 A contact CRlightlike submanifold of an indefinite Sasakian manifold is called mixed geodesic contact CRlightlike submanifold if its second funda mental form satisfied for any h ,=0hXZ, ˆ D and D . Definition 4.4 A contact CRlightlike submanifold of an indefinite Sasakian manifold is called geodesic contact CRlightlike submanifold if its second funda mental form satisfied for any D h ,=0hZU , , UD . Theorem 4.1 Let be a contact CRlightlike submanifold of an indefinite Sasakian manifold . Then is totally geodesic if and only if ,=,, l w YAXgYDXW, XY has no compo nents in 1 L , spanVYTM or has no components in 1 L . Proof. We know that is totally geodesic if and only if =0,YhX , for any , Y TM. By the definition of the second fundamental form, ,=0YhX is equivalent to ,Y , =0W,,Y W =0,ghX STM ghX ,, for any . RadTM From (2.4) and (2.7) we have = , , , , X XX X X X ghXg Y gY Y gY gY ggXYV gY YgX ,, =, =, =, =, Y YX (4.1) and ,, =, =, , =, =, , =,, ,. sX X X sl WX l W ghXY WgYW XgYWgY W gY W YAXWDXW gYAXgYDXW (4.2) Thus, from (4.1) and (4.2), the proof is completed. Theorem 4.2 Let be a contact CRlightlike sub manifold of an indefinite Sasakian manifold . Then is mixed geodesic if and only if Y X has no components in 2 RadT LM . Proof. By the definition, is mixed geodesic if and only if ,,=0, ,, ˆ , . ghXYghXY W xDYD =0. Then we have ,,= , =, =, =, ,, =,(), =,(), =, X XX X X X Y Y ghXYg Y gY Y gY gY ggXYV gY YgX gAXYgX gAX YX Copyright © 2011 SciRes. APM
J. H. DONG ET AL.381 and ,, =, =, =, =,, , =, =,. X XX X X X Y ghXY WgYW gYW YW gYW YW ggXYVYXW gYW gAXW Thus, the proof of the theorem is complete. Theorem 4.3 Let be a contact CRlightlike sub manifold of an indefinite Sasakian manifold . Then is geodesic if and only if ˆ D 2XRadTM L XY , has no components in . ˆ ,,LXY D 2 Proof. is geodesic if and only if ˆ D ,,=0, ,,=0 ls ghXYghXYW , for any ˆ ,, YD RadTM and . WSTM Then we have ,,= , =, =, =, ,, =, =, =, X XX X X X X X ghXYg Y gY Y gY gY ggXYVYX gY gY gY and ,, =, =, =, =,, , =, =,. sX XX X X X X ghXY WgYW gYW YW gYW YW ggXYVYXW gYW gYW Thus the assertions of the theorem follows. Theorem 4.4 Let be a contact CRlightlike sub manifold of an indefinite Sasakian manifold . Then is geodesic if and only if DW X, X have no components in 2 , MXLRadT D Y . Proof. is geodesic if and, only if D ,,=0, ,,=0 ls ghXYghXY W , for any ,, YD RadTM and WSTM . So we have ,,=,= , =, XX ghXYg YgY gAXY and ,, =, =, =,. XX W hXY WgYWgYW gAXY Thus the assertions of the theorem follows. 5. Geodesic Contact SCRLightlike Submanifolds Definition 5.1 Let ,, ,MgSTM STM be a lightlike submanifold, tangent to the structure vector field , immersed in an indefinite Sasakian manifold V , g. We say that is a contact SCRlightlike sub manifold of if the following conditions are satisfied [(A)] There exist real nonnull distributions and D D , such that =, =0, STMD DVDSTM DD , where D is the orthogonal complementary to DV in MST . [(B)] TM=, D =, =RadTMRadTMltr TMltr D. Hence we have the decomposition =, =TMDD VDDRadTM . Let us denote ˆ=DDV . Definition 5.2 A contact SCRlightlike submanifold of an indefinite Sasakian man ifold is called mixed geodesic contact SCRlightlike submanifold if its second fun damental form satisfied for any h ,=0hXY , D and DY . Theorem 5.1 Let be a contact SCRlightlike submanifold of an indefinite Sasakian manifold . Then is totally g eodesic if and only if ,=,=0, , , , . W LgXYLgXYXYTM RadTMWS TM Proof. We know is totally geodesic if and only if ,,=0, ,,=0. ˆ , . ghXYghXY W XDYD Copyright © 2011 SciRes. APM
J. H. DONG ET AL. 382 From (2.1) and Lie derivative we obtain ,,= , =, , =,,, =,,,, =,,,,,, =,, =, ,,. X Y Y ghXYg Y XgYgY X gY X gYX gYXgXYgX Y YXgXYgXYgX Lg XYgX LgXYghXY Hence we have 2,,= ,. hXYLg XY In a similar way, we can get 2,,= , W, hXYWLgXY thus the proof is completed. Theorem 5.2 Let be a contact SCRlightlike sub manifold of an indefinite Sasakian manifold . Then is mixed geodesic if and only if ˆ , s XY YDAXD , for any ˆ, DY D . Proof. For any ˆ, , , XDYD RadTMWSTM denote by =, =, PX QXWBW CW where , , PXD QXDBWD and . CWSTMD If is mixed geodesic, then ,=Y=0hXY Y XX WSTM . From the definition, there exists such that =.WY Thus we have 0= = = =. XXXX t WX X tt WW XX WY WY AX WY PA XQA XBWCWY X =0 From the definition of the and C, we know that . So we have Q = t WX QA XCW ˆ , XW WDAX tD . From =WY and (2.13), we have =WY , thus the proof is completed. Theorem 5.3 Let be a contact SCRlightlike submanifold of an indefinite Sasakian manifold . Then D defines a totally geodesic foliatio n if and only if sZ,hX and , s hXN has no components in , DX ZD , D . Proof. From the definition, we have that D is a totally geodesic foliation if and only if XYD , for any , YD , which is equivalent to ,=,=0, ,. XX gYZgYN ZDNltrTM Then we have ,=,=, =, =, =, , =, =,, XX X XX X X X s YZ gYZgYZ gY ZYZ gYZ YZgXZVZX gYZ gYhXZ and ,= , =, =, =, =, =, =,,. X X XX X X X X s gYNgYN gYNYN gYN , YgXYV YXN gYN gYN gYhXN Thus the assertion is proved. 6. References [1] D. N. Kupeli, “Singular SemiRiemannian Geometry,” Kluwer, Dordrecht, 1996. [2] K. L. Duggal and A. Bejancu, “Lightlike Submanifolds of SemiRiemannian Manifolds and Applications,” Kluwer Academic, Dordrecht, 1996. [3] B. Sahin, “Transversal Lightlike Submanifolds of Indefi nite Kaehler Manifolds,” Analele Universitaii de Vest, Timisoara Seria Matematica—Informatica, Vol. 44, No. 1, 2006, pp. 119145. [4] B. Sahin and R. Günes, “Geodesic CRLightlike Sub manifolds,” Contributions to Algebra and Geometry, Vol. 42, No. 2, 2001, pp. 583594. Copyright © 2011 SciRes. APM
J. H. DONG ET AL. Copyright © 2011 SciRes. APM 383 [5] K. L. Duggal and B. Sahin, “Lightlike Submanifolds of Indefinite Sasakian Manifolds,” International Journal of Mathematics and Mathematical Sciences, Article ID 57585, 2007, 21 Pages. [6] K. L. Duggal and B. Sahin, “Generalized CauchyRieman Lightlike Submanifolds of Indefinite Sasakian Mani folds,” Acta Mathematica Hungarica, Vol. 122, No. 12, 2009, pp. 4558. doi:10.1007/s1047400872218
