Advances in Pure Mathematics, 2012, 2, 285290 http://dx.doi.org/10.4236/apm.2012.24037 Published Online July 2012 (http://www.SciRP.org/journal/apm) Totally Umbilical Screen Transversal Lightlike Submanifolds of SemiRiemannian Product Manifolds S. M. Khursheed Haider, Advin, Mamta Thakur Department of Mathematics, Jamia Millia Islamia, New Delhi, India Email: smkhaider@rediffmail.com, {advin.maseih, mthakur09}@gmail.com Received February 17, 2012; revised March 5, 2012; accepted March 12, 2012 ABSTRACT We study totally umbilical screen transversal lightlike submanifolds immersed in a semiRiemannian product manifold and obtain necessary and sufficient conditions for induced connection on a totally umbilical radical screen transver sal lightlike submanifold to be metric connection. We prove a theorem which classifies totally umbilical STantiin variant lightlike submanifold immersed in a semiRiemannian product manifold. Keywords: SemiRiemannian Product Manifolds; Lightlike Submanifolds; Totally Umbilical Radical STLightlike Submanifolds; Totally Umbilical STAntiInvariant Lightlike Submanifolds 1. Introduction It is well known that the geometry of lightlike submani folds of semiRiemannian manifolds is different from the geometry of submanifolds immersed in a Riemannian manifold since the normal vector bundle of lightlike submanifolds intersect with tangent bundle making it more interesting to study. The general theory of lightlike submanifolds of a semiRiemannian manifold has been developed by DuggalBejancu [1] and Kupeli [2]. Totally umbilical CRsubmanifolds of a Kaehler manifold with Riemannian metric were studied by Bejancu [3], Desh mukh and Husain [4] and many more whereas, totally umbilical lightlike submanifolds of semiRiemannian manifolds of constant curvature was investigated by DuggalJin [5] and totally umbilical CRlightlike sub manifolds of an indefinite Kaehler manifold were studied by DuggalBejancu [1] and Gogna et al. [6]. In [7], B. Sahin initiated the study of transversal lightlike sub manifolds of an indefinite Kaehler manifold and investi gated the existence of such lightlike submanifolds in an indefinite space form. These submanifolds in Sasakian setting were studied by Yildirim and Sahin [8]. As a gen eralization of real null curves of indefinite Kaehler manifolds, B. Sahin [9] introduced the notion of screen transversal lightlike submanifolds and obtained many interesting results. In this paper, we study totally umbili cal screen transversal lightlike submanifolds of semi Riemannian product manifolds. This paper is arranged as follows. In Sections 2 and 3, we give the basic concepts on lightlike submanifolds and semiRiemannian product manifolds needed for this pa per. In Section 4, we study the integrability of distribu tions involved in the definition of totally umbilical radi cal screen transversal lightlike submanifolds and obtain necessary and sufficient conditions for induced connec tion on totally umbilical radical screen transversal lightlike submanifolds to be metric connection. In Sec tion 5, we prove a theorem which shows that the induced connection on a totally umbilical STantiinvariant lightlike submanifold is a metric connection under some conditions. We also prove a theorem which classifies totally umbilical STantiinvariant lightlike submanifold immersed in a semiRiemannian product manifold. 2. Preliminaries We follow [1] for the notation and fundamental equation for lightlike submanifolds used in this paper. A sub manifold Mm immersed in a semiRiemannian manifold , mn g is called a lightlike submanifold if it is a lightlike manifold with respect to the metric g induced from and radical distribution RadTM is of rank r, where 1 ≤ r ≤ m. Let STM TMRadTMS TM be a screen distribution which is a semiRiemannian complementary distribution of RadTM in TM, i.e., STMConsider a screen transversal vector bundle , which is a semiRiemannian complementary vector bun C opyright © 2012 SciRes. APM
S. M. K. HAIDER ET AL. 286 i STM . dle of RadTM in . Since for any local basis TM of RadTM, there exists a local null frame i of sections with values in the orthogonal complement of N STM in such that ST M ,gN ij ij , it follows that there exists a lightlike transversal vector bundle ltr(TM) locally spanned by [[1]; pg144]. Let tr(TM) be complementary (but not orthogonal) vector bundle to TM in i N TM . Then tr TMltr TM , S TM M TMMSTM STM STM . S TMRadT ,M STM ltr M TM ST STM Following are four subcases of a lightlike submanifold ,,MgST . Case 1: rlightlike if r < min{m, n}. Case 2: Coisotropic if r = n < m; = 0. Case 3: Isotropic if r = m < n; = 0. Case 4: Totally lightlike if r = n = m; = 0 = . The Gauss and Weingarten formulae are XY TM,,XY XX YYh (2.1) and , X U , t XU UAX TM U tr TM (2.2) where and , XU YAX ,, t Y hX t X belong to and , respectively, and U TM tr TM are linear connection on M and on the vector bundle , respectively. Moreover, we have TM tr l YYhX h ,Y, s Y X XX (2.3) , ls DX X N XN NAX N (2.4) , sl DX X W XW WAX ,, W (2.5) YTMNltr TM and TM WS . Denote the projection of TM on by P. Then, by using (2.1), (2.3)(2.5) and the fact that STM is a metric connection, we obtain ,,, ,,, ,, , sl W sW. hXYW gYDXWgAXY gDXNWgNAX ,, , t XX h X PY (2.6) From the decomposition of the tangent bundle of a lightlike submanifold, we have XX PY PY AX , (2.7) for YTM and RadTM . In general, the induced connection on M is not a metric connection whereas is a metric connection on ,, ,MgSTMSTM be a lightlike submani Let folds of , g. For any vector field X tangent to M, we put XfX X (2.8) where fX and are the tangential and transversal parts of FX respectively. For VtrTM VBVCV π: (2.9) where BV and CV are the tangential and transversal parts of FV respectively. 3. SemiRiemannian Product Manifolds Let (M1, g1) and (M2, g2) be two m1 and m2dimensional semiRiemannian manifolds with constant indices q1 > 0 and q2 > 0 respectively. Let 12 1 MM 12 : , and 2 MM be the projections which are given by π, , yx yy for any and 12 , yMM . We denote the product manifold by 12 , MMg , where 12 ,π,π, XYgX YgX Y for any ,YTM 2 ππ 2 , where denotes the differen tial mapping. Then we have , ππ0 , π and where I is the iden tity map of 12 M . Thus , g is a (m1 + m2) dimensional semiRiemannian manifold with constant index (q1 + q2). The Riemannian product manifold , 12 MMg is characterized by M1 and M2 which are totally geodesic submanifolds of . πF Now, if we put then we can easily see that F2 = I and ,, FX YgXFY (3.1) for any , YTM , where F is called almost Rie mannian product structure on 12 M. If we denote the LeviCivita connection on by , then 0 XFY (3.2) for any , YTM , that is, F is parallel with respect to . 4. Totally Umbilical Radical STLightlike Submanifolds In this section, we study totally umbilical radical ST lightlike submanifolds of a semiRiemannian product manifold. We first recall the following definitions from [9]. Definition 4.1. A rlightlike submanifold M of a semi Riemannian product manifold M is said to be a screen transversal (ST) lightlike submanifold of if there Copyright © 2012 SciRes. APM
S. M. K. HAIDER ET AL. 287 exists a screen transversal bundle such that STM .STM STM FRadTM ,, 0,gZWX 0 s H Definition 4.2. A STlightlike submanifold M of a semiRiemannian product manifold M is said to be a radical STlightlike submanifold if is invariant with respect to F. We also need the following definition of totally um bilical lightlike submanifolds of a semiRiemannian manifold. Definition 4.3. [5] A lightlike submanifold (M, g) of a semiRiemannian manifold , g is called totally umbilical in , if there is a smooth transversal vector field tr TM , of M, called the transversal cur vature vector of M, such that for all YTM , ,,XYHhXY g It is known that M is totally umbilical if and only if on each coordinate neighborhood U, there exists smooth vector fields l ltr TM and HSTM s such that ,, ss gXYH , , and , l hXY gXY DXW ,, 0 ll XYHh (4.1) for any Y TM and WS .TM In respect of the integrability of the distributions in volved in the definition of totally umbilical radical STlightlike submanifolds immersed in a semiRieman nian product manifold, we have: Theorem 4.4. Let M be a totally umbilical radical STlightlike submanifold of a SemiRiemannian product manifold. Then the screen distribution S(TM) is always integrable. Proof. From (2.3) and (3.2), a direct calculation shows that ,, , , ss , X YNhY FXFNghX FY (4.2) for , YS ltr TMTM and . Using (4.1) in (4.2), we get N 0N,,gXY , from which our assertion follows. Theorem 4.5. Let M be a totally umbilical radical STlightlike submanifold of a semiRiemannian product manifold. Then the distribution RadTM is always inte grable. Proof. For , W RadTM and STM , from (2.3) and (3.2) we get , , ,,. ,, s s ZWXg hZFXFW hW FX FZ (4.3) Taking account of (4.1) in (4.3), we obtain which proves our assertion. The necessary and sufficient conditions under which is given by the following result. Theorem 4.6. Let M be a totally umbilical radical STlightlike submanifold of a semiRiemannian product manifold ,0hXY if and only if . Then has no components in RadTM for any ,.XY STM Proof. Using (2.3) and (3.2), for any ,YSTM , we obtain ,, ,,. ls X ls X FYhXFYhX FY YFhXY FhXY (4.4) Taking inner product of (4.4) with FN for any NltrTM 2 and using the fact that I , we get ,, ,. sX hX FYFNgY N (4.5) From (2.7), (4.1) and (4.5), we have ,, ,,. s XFYgH FNghXYN (4.6) Thus, our assertion follows from (4.6). It is known that the induced connection on a lightlike submanifold immersed in a semiRiemannian manifold is not a metric connection. In view of this, it is interesting to see under what condition the induced connection on a totally umbilical radical STlightlike submanifold is a metric connection. The following theorem gives the geometric conditions for the induced connection to be a metric connection. Theorem 4.7. Let M be a totally umbilical radical STlightlike submanifold of a semiRiemannian product manifold . Then the induced connection on M is a metric connection if and only if for 0 F AX RadTM . TM , Proof. For TM RadTM , , from (3.2) we have . XX FF , ,,. sl FX XX ss AXF fFhX Bh XCh X (4.7) Using (2.3), (2.5), (2.8), (2.9) and (4.1) in (4.7), we obtain XF Taking tangential components of the above equation and then using (4.1), we arrive at AX , which proves our assertion. Corollary 4.8. Let M be a totally umbilical radical STlightlike submanifold of a semiRiemannian product manifold . Then the distribution RadTM is parallel if Copyright © 2012 SciRes. APM
S. M. K. HAIDER ET AL. 288 and only if for any . 21F Proof. From (3.2), for any we obtain 0 12 , 12 , A RadTM RadTM 11 22 .FF Using (2.3), (2.5), (2.8), (2.9) and (4.1) in the above equation, we get 1 212 12 , ,, l s Fh 21 1 12 2 12 s s AFf Bh Ch F 21 12 Af (4.8) Considering the tangential components of (4.8) and using (4.1), we arrive at F , from which our assertion follows. Lemma 4.9. Let M be a totally umbilical STlightlike submanifold of a semiRiemannian product manifold . Then , s W XgHWX for any STM TM and WS . Proof. For , YTM , from (2.6) and (4.1), we have ,. s ,, W AXY gXYgH W (4.9) If RadTM , then from (4.9) we infer that Moreover, if 0. W AX STM , then due to nondegeneracy of , we have STM , s W XgHWX t , which proves the assertion. For the induced connection of a totally umbilical radical STlightlike submanifold in semiRiemannian product manifolds to be a metric connection on tr TM, we have: Theorem 4.10. Let M be a totally umbilical radical STlightlike submanifold of a semiRiemannian product manifold . Then is a metric connection on if and only if X has no component in t tr TMsN for any TM ,WXTM and TM TM .Nltr S NltrTM Proof. For and , using (2.2), (2.5), (2.9) and (3.2), we get 12 , , ts XFN ,, l X NWgAXFN BW CW CW DX FN ltr TM (4.10) where , 1 and RadTMCWBW 2 CW . Using (4.1) and (4.10), we obtain, Considering lemma 4.9, we get 12 , ts XFNX ,, ,. NWgAXCWgFNCW 2 ,,, . ts XX NW gFNCW (4.11) Thus our assertion follows from (4.11) and Theorem 2.3 page 159 of [1]. Theorem 4.11. Let M be a totally umbilical radical STlightlike submanifold of a semiRiemannian product manifold . Then 12 21FF AA ,.RadTM ,.RadTM for all 12 Proof. For any 12 , using product structure on , we get 11 22 ,FF from which we have 121 2121212 ,, , ls s F FFhFhAF (4.12) where we have used (2.3), (2.5) and (3.2). Interchanging 1 and 2 in (4.12) and then subtracting the resulting equation from (4.12), we obtain 12 212 21 11221 . ss FF FAA FF (4.13) Taking inner product of (4.13) with ,XSTM we get 12 12 21 21 ,, ,. FF FXgFXg AAX (4.14) Now, from (2.3) and (4.1), a direct calculation shows that 12 21 ,0, ,0.gFXgFX (4.15) Using (4.15) in (4.14), we get 12 21 ,0. FF gAA X (4.16) Thus our assertion follows from (4.16) together with nondegeneracy of STM . 5. Totally Umbilical STAntiInvariant Lightlike Submanifolds In this section, we study totally umbilical STantiin variant lightlike submanifolds immersed in a semiRie mannian product manifold. First we recall the following definition from [9]. Definition 5.1. [9] A STlightlike submanifold M of a semiRiemannian product Manifold is said to be a STantiinvariant lightlike submanifold of M if STM is screen transversal with respect to F, i.e., FSTM STM . Copyright © 2012 SciRes. APM
S. M. K. HAIDER ET AL. 289 The necessary and sufficient conditions for the in duced connection on a totally umbilical STanti invariant lightlike submanifold M to be a metric connec tion is given by the following result. Theorem 5.2. Let M be a totally umbilical STanti invariant lightlike submanifold of a semiRiemannian product manifold . Then the induced connection on M is a metric connection if and only if s X has no component in STM for all , TM ,, s Bh X . RadTM Proof. Using (2.3), (2.5), (2.8), (2.9), (3.2) and (4.1), we arrive at ,. sl FX X s AX FChX Ch X YSTM (5.1) Taking inner product of (5.1) with FY for and then using (4.1), we obtain s ,,, XX FFYg Y which proves the assertion. Theorem 5.3. Let M be a totally umbilical STanti invariant lightlike submanifold of a semiRiemannian product manifold . Then RadTM is parallel if and only if 12 s has no component in STM RadTM for all . 12 Proof. From (2.3), (2.5), (2.8), (2.9), (3.2) and (4.1), we have , 2 12 ,, s Bh 21 1 12 21 12 , sl F s AF Ch Ch RadTM 1 12 2 for any 12 . Using (4.1) in the above equation, we get , 21 s F AF YSTM . (5.2) Taking inner product of (5.2) with FY for , we obtain s 11 22 ,,, FFYg Y from which our assertion follows. Theorem 5.4. Let M be a totally umbilical STanti invariant lightlike submanifold of a semiRiemannian product manifold . Then has no component in ltr TM. Proof. For , YSTM , using (2.3), (2.5) and (3.2) we get , ,, s X FY sl FX X l X A YFYD YFhXY FhXY RadTM (5.3) Taking inner product of (5.3) with and then using (4.1) we obtain from which we have our assertion. Theorem 5.5. Let M be a totally umbilical STanti invariant lightlike submanifold of a semiRiemannian product manifold ,0 s H F ,gXYg . . Then if and only if 0 l H s X X has no component in STM for all STM , ,, sl FX XX ss AX FXXChXX Bh XXCh XX . Proof. Using (2.3), (2.5), (2.8), (2.9), (3.2) and (4.1) we get (5.4) for any STM ,,. sls XX . From screen transversal parts of (5.4), we arrive at XXCh XXChXX Taking inner product of the above equation with for RadTM and using (2.8), (4.1) we get ,,,, sl X gFXFgXXgH which proves our assertion. The following theorem classifies totally umbilical ST antiinvariant lightlike submanifold immersed in a semi Riemannian product manifold. Theorem 5.6. Let M be a totally umbilical STanti invariant lightlike submanifold of a semiRiemannian product manifold . Then either Hs has no components in dim 1STM . STM or Proof. Taking inner product of the tangential compo nents of (5.4) with STM and using (3.1) and (2.9), we get ,,, s FX AXZghXXFZ (5.5) for any STM . On the other hand, by virtue of (2.6) we have ,,, s FX AXZ ghXZFX (5.6) Combining (5.5) and (5.6), we get ,, ,, ss hXXFZghXZFX Using (4.1) in the above equation, we obtain ,, ,,. ss XXgH FZgXZgH FX (5.7) Interchanging X and Z in (5.7) and rearranging the terms, we get , ,,. , ss gXZ HFX gHFZ gZZ (5.8) From (5.7) and (5.8), we conclude that 2 , ,,. ,, ss gXZ HFX gHFX gXXgZZ (5.9) Thus our assertion follows from (5.9). Copyright © 2012 SciRes. APM
S. M. K. HAIDER ET AL. Copyright © 2012 SciRes. APM 290 REFERENCES [1] K. L. Duggal and A. Bejancu, “Lightlike Submanifolds of SemiRiemannian Manifolds and Applications,” Klu werAcademic Publishers, Dordrecht, 1996. [2] D. N. Kupeli, “Singular SemiRiemannian Geometry,” Kluwer Academic Publishers, Dordrecht, 1996. [3] A. Bejancu, “Umbilical CRSubmanifolds of a Kaehler Manifold,” Rendiconti di Matematica, Vol. 13, 1980, pp. 431466. [4] S. Deshmukh and S. I. Husain, “Totally Umbilical CR Submanifolds of a Kaehler Manifold,” Kodai Mathe matical Journal, Vol. 9, No. 3, 1986, pp. 425429. doi:10.2996/kmj/1138037271 [5] K. L. Duggal and D. H. Jin, “Totally Umbilical Lightlike Submanifolds,” Kodai Mathematical Journal, Vol. 26, No. 1, 2003, pp. 4968. doi:10.2996/kmj/1050496648 [6] M. Gogna, R. Kumar and R. K. Nagaich, “Totally Um bilical CRlightlike Submanifolds of Indefinite Kaehler Manifolds,” Bulletin of Mathematical Analysis and Ap plications, Vol. 2, No. 4, 2010, pp. 5461. [7] B. Sahin, “Transversal Lightlike Submanifolds of Indefi nite Kaehler Manifolds,” Analele Universitaii de Vest, Timisoara Seria MatematicaInformatica, Vol. XLIV, No. 1, 2006, pp. 119145. [8] C. Yildirim and B. Sahin, “Transversal Lightlike Sub manifolds of Indefinite Sasakian Manifolds,” Turkish Journal of Mathematics, Vol. 33, 2009, pp. 123. [9] B. Sahin, “Screen Transversal Lightlike Submanifolds of Indefinite Kaehler Manifolds,” Chaos, Solitons and Frac tals, Vol. 38, No. 5, 2008, pp. 14391448. doi:10.1016/j.chaos.2008.04.008
