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 Advances in Pure Mathematics, 2012, 2, 285-290 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 Semi-Riemannian Product Manifolds S. M. Khursheed Haider, Advin, Mamta Thakur Department of Mathematics, Jamia Millia Islamia, New Delhi, India Email: [email protected], {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 semi-Riemannian 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 ST-anti-in- variant lightlike submanifold immersed in a semi-Riemannian product manifold. Keywords: Semi-Riemannian Product Manifolds; Lightlike Submanifolds; Totally Umbilical Radical ST-Lightlike Submanifolds; Totally Umbilical ST-Anti-Invariant Lightlike Submanifolds 1. Introduction It is well known that the geometry of lightlike submani- folds of semi-Riemannian 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 semi-Riemannian manifold has been developed by Duggal-Bejancu [1] and Kupeli [2]. Totally umbilical CR-submanifolds 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 semi-Riemannian manifolds of constant curvature was investigated by Duggal-Jin [5] and totally umbilical CR-lightlike sub- manifolds of an indefinite Kaehler manifold were studied by Duggal-Bejancu [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 semi-Riemannian 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 ST-anti-invariant lightlike submanifold is a metric connection under some conditions. We also prove a theorem which classifies totally umbilical ST-anti-invariant lightlike submanifold immersed in a semi-Riemannian 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 semi-Riemannian manifold , mn g M is called a lightlike submanifold if it is a lightlike manifold with respect to the metric g induced from g and radical distribution RadTM is of rank r, where 1 ≤ r ≤ m. Let STM TMRadTMS TM be a screen distribution which is a semi-Riemannian complementary distribution of RadTM in TM, i.e., STMConsider a screen transversal vector bundle , which is a semi-Riemannian 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]; pg-144]. Let tr(TM) be complementary (but not orthogonal) vector bundle to TM in i N M 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: r-lightlike if r < min{m, n}. Case 2: Co-isotropic 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 X 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) X 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. g 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 X 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 , M g. For any vector field X tangent to M, we put F XfX X (2.8) where fX and X are the tangential and transversal parts of FX respectively. For VtrTM F VBVCV π: (2.9) where BV and CV are the tangential and transversal parts of FV respectively. 3. Semi-Riemannian Product Manifolds Let (M1, g1) and (M2, g2) be two m1 and m2-dimensional semi-Riemannian manifolds with constant indices q1 > 0 and q2 > 0 respectively. Let 12 1 M MM 12 : , and 2 M MM be the projections which are given by π, , x yx x yy for any and 12 , x yMM . We denote the product manifold by 12 , M MMg , where 12 ,π,π, g XYgX YgX Y for any ,YTM 2 ππ 2 X , where denotes the differen- tial mapping. Then we have , ππ0 , π and I where I is the iden- tity map of 12 M M . Thus , M g is a (m1 + m2)- dimensional semi-Riemannian manifold with constant index (q1 + q2). The Riemannian product manifold , M 12 MMg is characterized by M1 and M2 which are totally geodesic submanifolds of M . πF Now, if we put then we can easily see that F2 = I and ,, g FX YgXFY (3.1) for any , X YTM , where F is called almost Rie- mannian product structure on 12 M M. If we denote the Levi-Civita connection on M by , then 0 XFY (3.2) for any , X YTM , that is, F is parallel with respect to . 4. Totally Umbilical Radical ST-Lightlike Submanifolds In this section, we study totally umbilical radical ST- lightlike submanifolds of a semi-Riemannian product manifold. We first recall the following definitions from [9]. Definition 4.1. A r-lightlike submanifold M of a semi- Riemannian product manifold M is said to be a screen transversal (ST) lightlike submanifold of M 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 ST-lightlike submanifold M of a semi-Riemannian product manifold M is said to be a radical ST-lightlike submanifold if is invariant with respect to F. We also need the following definition of totally um- bilical lightlike submanifolds of a semi-Riemannian manifold. Definition 4.3. [5] A lightlike submanifold (M, g) of a semi-Riemannian manifold , M g is called totally umbilical in M , if there is a smooth transversal vector field H tr TM , of M, called the transversal cur- vature vector of M, such that for all X YTM , ,,XYHhXY g It is known that M is totally umbilical if and only if on each co-ordinate neighborhood U, there exists smooth vector fields l H ltr TM and HSTM s such that ,, ss gXYH , , and , l hXY gXY DXW ,, 0 ll XYHh (4.1) for any X Y TM and WS .TM In respect of the integrability of the distributions in- volved in the definition of totally umbilical radical ST-lightlike submanifolds immersed in a semi-Rieman- nian product manifold, we have: Theorem 4.4. Let M be a totally umbilical radical ST-lightlike submanifold of a Semi-Riemannian product manifold. Then the screen distribution S(TM) is always integrable. Proof. From (2.3) and (3.2), a direct calculation shows that ,, , , ss , g X YNhY FXFNghX FY (4.2) for , X 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 ST-lightlike submanifold of a semi-Riemannian product manifold. Then the distribution RadTM is always inte- grable. Proof. For , Z W RadTM and X STM , from (2.3) and (3.2) we get , , ,,. ,, s s g ZWXg hZFXFW g 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 ST-lightlike submanifold of a semi-Riemannian product manifold ,0hXY if and only if s M . Then H has no components in F RadTM for any ,.XY STM X Proof. Using (2.3) and (3.2), for any ,YSTM , we obtain ,, ,,. ls X ls X FYhXFYhX FY F YFhXY FhXY (4.4) Taking inner product of (4.4) with FN for any NltrTM 2 and using the fact that F I , we get ,, ,. sX g hX FYFNgY N (4.5) From (2.7), (4.1) and (4.5), we have ,, ,,. s XFYgH FNghXYN g (4.6) Thus, our assertion follows from (4.6). It is known that the induced connection on a lightlike submanifold immersed in a semi-Riemannian 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 ST-lightlike 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 ST-lightlike submanifold of a semi-Riemannian product manifold M . Then the induced connection on M is a metric connection if and only if for 0 F AX RadTM . TM , X Proof. For X 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 f AX , which proves our assertion. Corollary 4.8. Let M be a totally umbilical radical ST-lightlike submanifold of a semi-Riemannian product manifold . Then the distribution RadTM is parallel if M 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 ST-lightlike submanifold of a semi-Riemannian product manifold M . Then , s W A XgHWX for any X STM TM and WS . Proof. For , X YTM , from (2.6) and (4.1), we have ,. s ,, W g AXY gXYgH W (4.9) If X RadTM , then from (4.9) we infer that Moreover, if 0. W AX X STM , then due to non-degeneracy of , we have STM , s W A XgHWX t , which proves the assertion. For the induced connection of a totally umbilical radical ST-lightlike submanifold in semi-Riemannian product manifolds to be a metric connection on tr TM, we have: Theorem 4.10. Let M be a totally umbilical radical ST-lightlike submanifold of a semi-Riemannian product manifold M . Then is a metric connection on if and only if X has no component in t tr TMsN for any X 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 g 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 ,, ,. g NWgAXCWgFNCW 2 ,,, . ts XX g 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 ST-lightlike submanifold of a semi-Riemannian product manifold M . Then 12 21FF AA ,.RadTM ,.RadTM for all 12 Proof. For any 12 , using product structure on M , 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 F FAA FF (4.13) Taking inner product of (4.13) with ,XSTM we get 12 12 21 21 ,, ,. FF g 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 non-degeneracy of STM . 5. Totally Umbilical ST-Anti-Invariant Lightlike Submanifolds In this section, we study totally umbilical ST-anti-in- variant lightlike submanifolds immersed in a semi-Rie- mannian product manifold. First we recall the following definition from [9]. Definition 5.1. [9] A ST-lightlike submanifold M of a semi-Riemannian product Manifold M is said to be a ST-anti-invariant 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 ST-anti- 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 ST-anti- invariant lightlike submanifold of a semi-Riemannian product manifold M . Then the induced connection on M is a metric connection if and only if s X F has no component in F STM for all , X 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 g FFYg Y which proves the assertion. Theorem 5.3. Let M be a totally umbilical ST-anti- invariant lightlike submanifold of a semi-Riemannian product manifold M . Then RadTM is parallel if and only if 12 s F has no component in F 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 ,,, g FFYg Y from which our assertion follows. Theorem 5.4. Let M be a totally umbilical ST-anti- invariant lightlike submanifold of a semi-Riemannian product manifold M . Then s H has no component in F ltr TM. Proof. For , X YSTM , using (2.3), (2.5) and (3.2) we get , ,, s X FY sl FX X l X A YFYD F 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 ST-anti- invariant lightlike submanifold of a semi-Riemannian product manifold ,0 s H F ,gXYg . M . Then if and only if 0 l H s X F X has no component in F STM for all X 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) X for any STM ,,. sls XX . From screen transversal parts of (5.4), we arrive at F XXCh XXChXX F 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- anti-invariant lightlike submanifold immersed in a semi- Riemannian product manifold. Theorem 5.6. Let M be a totally umbilical ST-anti- invariant lightlike submanifold of a semi-Riemannian product manifold M . Then either Hs has no components in dim 1STM . F STM or Proof. Taking inner product of the tangential compo- nents of (5.4) with Z STM and using (3.1) and (2.9), we get ,,, s FX g AXZghXXFZ (5.5) X for any STM . On the other hand, by virtue of (2.6) we have ,,, s FX g AXZ ghXZFX (5.6) Combining (5.5) and (5.6), we get ,, ,, ss g hXXFZghXZFX Using (4.1) in the above equation, we obtain ,, ,,. ss g XXgH FZgXZgH FX (5.7) Interchanging X and Z in (5.7) and rearranging the terms, we get , ,,. , ss gXZ g HFX gHFZ gZZ (5.8) From (5.7) and (5.8), we conclude that 2 , ,,. ,, ss gXZ g 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. 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