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Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2011, Article ID 140259, 11 pages doi:10.1155/2011/140259 Research Article Characteristic Lightlike Submanifolds of an Indefinite S-Manifold Jae Won Lee Department of Mathematics, Sogang University, Sinsu-Dong, Mapo-Gu, Seoul 121-742, Republic of Korea Correspondence should be addressed to Jae Won Lee, [email protected] Received 12 April 2011; Revised 12 August 2011; Accepted 23 August 2011 Academic Editor: Christian Corda Copyright q 2011 Jae Won Lee. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We study characteristic r-lightlike submanifolds M tangent to the characteristic vector fields in an indefinite metric S-manifold, and we also discuss the existence of characteristic lightlike submanifolds of an indefinite S-space form under suitable hypotheses: 1 M is totally umbilical or 2 its screen distribution ST M is totally umbilical in M. 1. Introduction In the theory of submanifolds of semi-Riemannian manifolds, it is interesting to study the geometry of lightlike submanifolds due to the fact that the intersection of normal vector bundle and the tangent bundle is nontrivial, making it interesting and remarkably diﬀerent from the study of nondegenerate submanifolds. In particular, many authors study lightlike submanifolds on indefinite Sasakian manifolds e.g., 1–4. Similar to Riemannian geometry, it is natural that indefinite S-manifolds are generalizations of indefinite Sasakian manifolds. Brunetti and Pastore analyzed some properties of indefinite S-manifolds and gave some characterizations in terms of the Levi-Civita connection and of the characteristic vector fields 5. After then, they studied the geometry of lightlike hypersurfaces of indefinite S-manifold 6. As Jin’s generalizations of lightlike submanifolds of the Sasakian manifolds with the general codimension 3, 4, 7, Lee and Jin recently extended lightlike hypersurfaces on indefinite S-manifold to lightlike submanifolds with codimension 2 on an indefinite S-manifold, called characteristic half lightlike submanifolds 8. However, a general notion of characteristic lightlike submanifolds of an indefinite S-manifold have not been introduced as yet. The objective of this paper is to study characteristic r-lightlike submanifolds M of an indefinite S-manifold M subject to the conditions: 1 M is totally umbilcial, or 2 ST M 2 International Journal of Mathematics and Mathematical Sciences is totally umbilcal in M. In Section 2, we begin with some fundamental formulae in the theory of r-lightlike submanifolds. In Section 3, for an indefinite metric g.f.f-manifold we consider a lightlike submanifold M tangent to the characteristic vector fields, we recall some basic information about indefinite S-manifolds and deal with the existence of irrotational characteristic submanifolds of an indefinite S-space form. Afterwards, we study characteristic r-lightlike submanifolds of M in Sections 4 and 5. 2. Preliminaries Let M, g be an m-dimensional lightlike submanifold of an m n-dimensional semi-Riemannian manifold M, g. Then the radical distribution RadT M T M ∩ T M⊥ is a vector subbundle of the tangent bundle T M and the normal bundle T M⊥ , of rank r 1 ≤ r ≤ min{m, n}. In general, there exist two complementary nondegenerate distributions ST M and ST M⊥ of RadT M in T M and T M⊥ , respectively, called the screen and coscreen distributions on M, such that T M RadT M⊕orth ST M, T M⊥ RadT M⊕orth S T M⊥ , 2.1 where the symbol ⊕orth denotes the orthogonal direct sum. We denote such a lightlike submanifold by M, g, ST M, ST M⊥ . Denote by FM the algebra of smooth functions on M and by ΓE the FM module of smooth sections of a vector bundle E over M. We use the same notation for any other vector bundle. We use the following range of indices: i, j, k, . . . ∈ {1, . . . , r}, α, β, γ, . . . ∈ {r 1, . . . , n}. 2.2 Let trT M and ltrT M be complementary but not orthogonal vector bundles to T M in T M|M and T M⊥ in ST M⊥ , respectively, and let {N1 , . . . , Nr } be a lightlike basis of ΓltrT M|U consisting of smooth sections of ST M⊥ ⊥|U , where U is a coordinate neighborhood of M, such that g Ni , ξj δij , g Ni , Nj 0, 2.3 where {ξ1 , . . . , ξr } is a lightlike basis of ΓRadT M. Then we have T M T M ⊕ trT M {RadT M ⊕ trT M}⊕orth ST M {RadT M ⊕ ltrT M}⊕orth ST M⊕orth S T M⊥ . 2.4 We say that a lightlike submanifolds M, g, ST M, ST M⊥ of M are characterized as follows: 1 r-lightlike if 1 ≤ r < min{m, n}; 2 coisotropic if 1 ≤ r n < m; 3 isotropic if 1 ≤ r m < n; 4 totally lightlike if 1 ≤ r m n. International Journal of Mathematics and Mathematical Sciences 3 The above three classes 2–4 are particular cases of the class 1 as follows: ST M⊥ {0}, ST M {0}, and ST M ST M⊥ {0}, respectively. The geometry of r-lightlike submanifolds is more general form than that of the other three type submanifolds. For this reason, in this paper we consider only r-lightlike submanifolds M ≡ M, g, ST M, ST M⊥ , with the following local quasiorthonormal field of frames on M: {ξ1 , . . . , ξr , N1 , . . . , Nr , Fr1 , . . . , Fm , Wr1 , . . . , Wn }, 2.5 where the sets {Fr1 , . . . , Fm } and {Wr1 , . . . , Wn } are orthonormal basis of ΓST M and ΓST M⊥ , respectively. Let ∇ be the Levi-Civita connection of M and P the projection morphism of ΓT M on ΓST M with respect to 2.1. For an r-lightlike submanifold, the local Gauss-Weingartan formulas are given by ∇X Y ∇X Y r n hi X, Y Ni hsα X, Y Wα , i1 ∇X Ni −ANi X r n τij XNj ρiα XWα , j1 ∇X Wα −AWα X 2.6 αr1 r n φαi XNi σαβ XWβ , i1 ∇X P Y ∇∗X P Y 2.7 αr1 2.8 βr1 r h∗i X, P Y ξi , 2.9 i1 ∇X ξi −A∗ξi X − r τji Xξj , 2.10 j1 for any X, Y ∈ ΓT M, where ∇ and ∇∗ are induced linear connections on T M and ST M, respectively, the bilinear forms hi and hsα on M are called the local lightlike and screen second fundamental forms on T M, respectively, h∗i are called the local radical second fundamental forms on ST M. ANi , A∗ξi , and AWα are linear operators on ΓT M and τij , ρiα , φαi , and σαβ are 1-forms on T M. Since ∇ is torsion-free, ∇ is also torsion-free and both hi and hsα are symmetric. From the fact hi X, Y g∇X Y, ξi , we know that hi are independent of the choice of a screen distribution. We say that hX, Y r hi X, Y Ni i1 n hsα X, Y Wα 2.11 αr1 is the second fundamental tensor of M. The induced connection ∇ on T M is not metric and satisfies r hi X, Y ηi Z hi X, Z ηi Y , ∇X g Y, Z i1 2.12 4 International Journal of Mathematics and Mathematical Sciences for all X, Y ∈ ΓT M, where ηi s are the 1-forms such that ηi X gX, Ni , ∀X ∈ ΓT M. 2.13 But the connection ∇∗ on ST M is metric. The above three local second fundamental forms are related to their shape operators by r hi X, Y g A∗ξi X, Y − hk X, ξi ηk Y , k1 hi X, P Y g A∗ξi X, P Y , α hsα X, Y gAWα X, Y − g A∗ξi X, Nj 0, r 2.14 2.15 φαi Xηi Y , 2.16 gAWα X, Ni α ρiα X, ηj ANi X ηi ANj X 0, 2.17 i1 α hsα X, P Y gAWα X, P Y , h∗i X, P Y gANi X, P Y , 2.18 where X, Y ∈ ΓT M and α is the sign of Wα but it is ±1 related to the causal character of Wα . From 2.18, we know that each ANi is shape operator related to the local second fundamental form h∗i on ST M. Replacing Y by ξj in 2.14, we have hi X, ξj hj X, ξi 0, 2.19 for all X ∈ ΓT M. It follows hi X, ξi 0, hi ξj , ξk 0. 2.20 Also, replacing X by ξj in 2.14 and using 2.20, we have hi X, ξj g X, A∗ξi ξj , A∗ξi ξj A∗ξj ξi 0, A∗ξi ξi 0. 2.21 For an r-lightlike submanifold, replace Y by ξi in 2.16, we have hsα X, ξi −α φαi X, ∀X ∈ ΓT M. 2.22 Note 1. Using 2.14 and the fact that hi are symmetric, we have r hk X, ξi ηk Y − hk Y, ξi ηk X . g A∗ξi X, Y − g X, A∗ξi Y 2.23 k1 From this, 2.20 and 2.21, we show that A∗ξi are self-adjoint on ΓT M with respect to g if and only if hi X, ξj 0 for all X ∈ ΓT M, i and j if and only if A∗ξi ξj 0 for all i, j. We call International Journal of Mathematics and Mathematical Sciences 5 self-adjoint A∗ξi the lightlike shape operators of M. It follows from the above equivalence and 2.10 that the radical distribution RadT M of a lightlike submanifold M, with the lightlike shape operators A∗ξi , is always an integrable distribution. 3. Characteristic Lightlike Submanifolds A manifold M is called a globally framed f-manifold (or g.f.f-manifold) if it is endowed with a nonnull 1, 1-tensor field φ of constant rank, such that ker φ is parallelizable, that is, there 2 exist global vector fields ξα , α ∈ {1, . . . , k}, with their dual 1-forms ηα , satisfying φ −I ηα ⊗ ξα and ηα ξβ δβα . 2nr , φ, ξα , ηα , α ∈ {1, . . . , k}, is said to be an indefinite metric The g.f.f-manifold M g.f.f-manifold if g is a semi-Riemannian metric, with index ν, 0 < ν < 2n k, satisfying the following compatibility condtion r g φX, φY gX, Y − α ηα Xηα Y , 3.1 α1 for any X, Y ∈ ΓT M, being α ±1 according to whether ξ α is spacelike or timelike. Then, for any α ∈ {1, . . . , k}, one has ηα X α gX, ξ α . An indefinite metric g.f.f-manifold is called an indefinite S-manifold if it is normal and dηα Φ, for any α ∈ {1, . . . , k}, where ΦX, Y gX, φY for any X, Y ∈ ΓT M. The normality condition is expressed by the vanishing of the tensor field N Nφ 2dηα ⊗ξ α , Nφ being the Nijenhuis torsion of φ. Furthermore, as proved in 5, the Levi-Civita connection of an indefinite S-manifold satisfies: 2 ∇X φ Y g φX, φY ξ ηY φ X, 3.2 where ξ kα1 ξα and η kα1 α ηα . We recall that ∇X ξ α −α φX and ker φ is an integrable flat distribution since ∇ξ ξβ 0 more details in 5. α Following the notations in 9, we adopt the curvature tensor R, and thus we have RX, Y, Z ∇X ∇Y Z − ∇Y ∇X Z − ∇X,Y Z, and RX, Y, Z, W gRZ, W, Y , X, for any X, Y , Z, W ∈ ΓT M. An indefinite S-manifold M, φ, ξα , ηα is called an indefinite S-space form, denoted by Mc, if it has the constant φ-sectional curvature c 5. The curvature tensor R of this space form Mc is given by 4RX, Y, Z, W −c 3 g φY, φZ g φX, φW − g φX, φZ g φY, φW − c − {ΦW, XΦZ, Y − ΦZ, XΦW, Y 2ΦX, Y ΦW, Z} − 4 ηWηXg φZ, φY − ηWηY g φZ, φX ηY ηZg φW, φX − ηZηXg φW, φY , for any vector fields X, Y, Z, W ∈ ΓT M. 3.3 6 International Journal of Mathematics and Mathematical Sciences Note 2. Although ST M is not unique, it is canonically isomorphic to the factor vector bundle ST M∗ T M/RadT M considered by Kupeli 10. Thus all screen distributions ST M are mutually isomorphic. For this reason, we newly define generic lightlike submanifolds of M as follows. Definition 3.1. Let M be a r-lightlike submanifold of M such that all the characteristic vector fields ξα are tangent to M. A screen distribution ST M is said to be characteristic if ker φ ⊂ ST M and φST M⊥ ⊂ ΓST M. Definition 3.2. A r-lightlike submanifold M of M is said to be characteristic if ker φ ⊂ T M and a characteristic screen distribution ST M is chosen. Proposition 3.3 see 6. Let M, g, ST M be a lightlike hypersurface of an indefinite S-manifold M, φ, ξα , ηα , g such that the characteristic vector fields are tangent to M. Then there exists a screen distribution such that ker φ ⊂ T M and φE ⊂ ΓST M, where E is a nonzero section of RadT M. Proposition 3.4 see 8. Let M, g, ST M be a 1-lightlike submanifold of codimension 2 of an indefinite S-manifold M, φ, ξα , ηα , g. Then M is a characteristic lightlike submanifold of M. Definition 3.5. A lightlike submanifold M is said to be irrotational 10 if ∇X ξi ∈ ΓT M for any X ∈ ΓT M and ξi ∈ ΓRadT M for all i. Note 3. For an r-lightlike M, the above definition is equivalent to hj X, ξi 0, hsα X, ξi φαi X 0, ∀X ∈ ΓT M. 3.4 The extrinsic geometry of lightlike hypersurfaces depends on a choice of screen distribution, or equivalently, normalization. Since the screen distribution is not uniquely determined, a well-defined concept of S-manifold is not possible for an arbitrary lightlike submanifold of a semi-Riemannian manifold, then one must look for a class of normalization for which the induced Riemannian curvature has the desired symmetries. Let M, g be a semiRiemannian manifold, p ∈ M. F⊗4 Tp∗ M is said to be an algebraic curvature tensor 11 on Tp M if it satisfies the following symmetries: FX, Y, Z, W −FY, X, Z, W FZ, W, X, Y , FX, Y, Z, W FY, Z, X, W FZ, X, Y, W 0. 3.5 Definition 3.6. A screen distribution ST M is said to be admissible if the associated induced Riemannian curvature is an algebraic curvature tensor. Theorem 3.7. Let M, g, ST M be an irrotational generic characteristic lightlike submanifold of an indefinite S-space form Mc, φ, ξα , ηα , g with an admissible screen distribution ST M. Then one has c . International Journal of Mathematics and Mathematical Sciences 7 Proof. Denote by R and R the curvature tensors of ∇ and ∇, respectively. Using the local Gauss-Weingarten formulas for M, we obtain RX, Y Z RX, Y Z r hi X, ZANi Y − hi Y, ZANi X i1 n {hsα X, ZAWα Y − hsα Y, ZAWα X} αr1 r ∇X hi Y, Z − ∇Y hi X, Z i1 r τji Xhj Y, Z − τji Y hj X, Z j1 n φαi Xhsα Y, Z − φαi Y hsα X, Z 3.6 Ni αr1 ⎧ n ⎨ ∇X hsα Y, Z − ∇Y hsα X, Z ⎩ αr1 r ρiα Xhi Y, Z − ρiα Y hsα X, Z i1 ⎫ n ⎬ σβα Xhsβ Y, Z − σβα Y hsβ X, Z Wα , ⎭ βr1 for all X, Y, Z ∈ ΓT M. Replace Z by ξk in 3.6 and use 2.10, 2.15, 2.17, and 3.4, we have RX, Y ξk RX, Y ξk r g A∗ξi Y, A∗ξk X − g A∗ξi X, A∗ξk Y Ni i1 n α g AWα Y, A∗ξk X − g AWα X, A∗ξk Y Wα . 3.7 αr1 Using 3.7, the fact RX, Y Z ∈ ΓT M for X, Y, Z ∈ ΓT M, and a screen distribution ST M is admissible, we get g RX, Y Z, ξk −g RX, Y ξk , Z −gRX, Y ξk , Z r g A∗ξi X, A∗ξk Y − g A∗ξi Y, A∗ξk X ηi Z i1 8 International Journal of Mathematics and Mathematical Sciences gRX, Y Z, ξk r g A∗ξi X, A∗ξk Y − g A∗ξi Y, A∗ξk X ηi Z i1 r g A∗ξi X, A∗ξk Y − g A∗ξi Y, A∗ξk X ηi Z, ∀X, Y, Z ∈ ΓT M. i1 3.8 On the other hand, since ηξα 0 and gφξα , φX 0 for any X ∈ ΓT M, Mc is an indefinite S-space form implies the Riemannian curvature R in 3.3 is given by 4RX, Y, Z, ξα c − {Φξα , XΦZ, Y − ΦZ, XΦξα , Y 2ΦX, Y Φξα , Z} −c − g φξα , X ΦZ, Y − ΦZ, Xg φξα , Y 2ΦX, Y g φξα , Z , 3.9 for any X, Y, Z, ∈ ΓT M. So, replacing X, Y , Z by P X, ξ, P Z in 3.9, we find 4RX, Y, Z, ξα −c − −g φξα , P X g P Z, φξα − 2g X, φξα g φξα , Z 3.10 3c − g φξα , P X g φξα , P Z . Then, using 3.3, 3.8, and 3.9, we get r 4 g A∗ξi X, A∗ξk Y − g A∗ξi Y, A∗ξk X ηi Z i1 −3c − g φξα , P X g φξα , P Z , 3.11 ∀X, Y, Z ∈ ΓT M. Choosing X Z φNα ∈ ΓST M, we obtain c . Corollary 3.8. There exist no irrotational characteristic r-lightlike submanifolds M, g, ST M of an indefinite S-space form Mc, φ, ξα , ηα , g with c / such that the screen distribution ST M is admissible. 4. Totally Umbilical Characteristic Lightlike Submanifolds Definition 4.1. An r-lightlike submanifold M of M is said to be totally umbilical 1 if there is a smooth vector field H ∈ ΓtrT M such that hX, Y HgX, P Y , for all X, Y ∈ ΓT M. In case H 0, we say that M is totally geodesic. 4.1 International Journal of Mathematics and Mathematical Sciences 9 It is easy to see that M is totally umbilical if and only if, on each coordinate neighborhood U, there exist smooth functions Ai and Bα such that hi X, Y Ai gX, Y , hsα X, Y Bα gX, Y , 4.2 for any X, Y ∈ ΓT M. From 4.2 we show that any totally umbilical r-lightlike submanifold of M is irrotational. Thus, by Theorem 3.7, we have the following. Theorem 4.2. Let M, g, ST M be a totally umbilical characteristic r-lightlike submanifold of an indefinite S-space form Mc, φ, ξα , ηα , g. Then one has c . Theorem 4.3. Let M, g, ST M be a totally umbilical characteristic r-lightlike submanifold of an indefinite S-manifold M, φ, ξα , ηα , g. Then M is totally geodesic. Proof. Apply ∇X to gφξi , Wα 0 with X ∈ ΓT M, for all i and α, and use 2.8, 2.10, 2.15, 2.17, 2.22, and 3.2, we have hi X, φWα α hsα X, φξi , ∀X ∈ ΓT M. 4.3 Assume that M is totally umbilical. Then we have Ai g X, φWα α Bα g X, φξi , ∀X ∈ ΓT M. 4.4 Replace X by φNi and X by φWα by turns, we get Ai 0 for all i and Bα 0 for all α. Thus we show that H ri1 Ai Ni nαr1 Bα Wα 0 and M is totally geodesic. Corollary 4.4 see 1. Let M, g, ST M be a totally umbilical characteristic r-lightlike submanifold of an indefinite S-manifold M, φ, ξα , ηα , g. Then there exists a unique torsion-free metric connection ∇ on M induced by the connection ∇ on M. Proof. From 4.2 and Theorem 4.3, we have hi X, Y 0 for all X, Y ∈ ΓT M and i. Thus, using 2.12, we obtain our assertion. 5. Totally Umbilical Screen Distributions Definition 5.1. A screen distribution ST M of M is said to be totally umbilical 1 in M if, for each locally second fundamental form h∗i , there exist smooth functions Ci on any coordinate neighborhood U in M such that h∗i X, P Y Ci gX, Y , ∀X, Y ∈ ΓT M. In case Ci 0 for all i, we say that ST M is totally geodesic in M. 5.1 10 International Journal of Mathematics and Mathematical Sciences Due to 2.18 and 5.1, we know that ST M is totally umbilical in M if and only if each shape operators ANi of ST M satisfies gANi X, P Y Ci gX, P Y , ∀X, Y ∈ ΓT M, 5.2 for some smooth functions Ci on U ⊆ M. In general, ST M is not necessarily integrable. The following result gives equivalent conditions for the integrability of a screen ST M. Theorem 5.2 see 1. Let M be an r-lightlike submanifold of a semi-Riemannian manifold M, g. Then the following assertions are equivalent: 1 ST M is integrable, 2 h∗i is symmetric on ΓST M, for each i, 3 ANi is self-adjoint on ΓST M with respect to g, for each i. We know that, from 5.2, each shape operator ANi is self-adjoint on ΓST M with respect to g, which further follows from that above theorem that any totally umbilical screen distribution ST M of M is integrable. Theorem 5.3. Let M, g, ST M be a characteristic r-lightlike submanifold of an indefinite S-manifold M, φ, ξα , ηα , g. If ST M is totally umbilical in M, then ST M is totally geodesic in M. Proof. Apply the operator ∇X to gφNk , Nj 0 for some k, j such that k / j, and use 2.7 and 2.18 h∗k X, φNj h∗j X, φNk , ∀X ∈ ΓT M. 5.3 Assume that ST M is totally umbilical in M. Then we have Ck g X, φNj Cj g X, φNk , ∀X ∈ ΓT M. 5.4 Replacing X by φξj in 5.4 and taking k, j 1, 2, 2, 3, . . . , r − 1, r and r, 1 by turns and use the above method, we have Ci 0 for all i ∈ {1, . . . , r}. Thus we have our assertion. Theorem 5.4. Let M, g, ST M be a characteristic r-lightlike submanifold of an indefinite S-manifold M, φ, ξα , ηα , g such that ST M is totally umbilical in M. Then M is not irrotational. Proof. Apply the operator ∇X to gφξi , Nj 0 for all i and j, and use 2.6, 2.10, 2.15, 2.18, and Theorem 5.3, we have hi X, φNj h∗j X, φξi , ∀X ∈ ΓT M. 5.5 International Journal of Mathematics and Mathematical Sciences 11 Since ST M is totally umbilical, by Theorem 5.3, we have that ST M is totally geodesic. Then, by 5.5, we have hi X, φNj 0, ∀X ∈ ΓT M. 5.6 Apply the operator ∇X to gξα , ξi 0 and use 2.6, 2.10, 2.15, we have hi X, ξα −α g X, φξi , ∀X ∈ ΓT M. 5.7 Replace X by φNi in this equation and 5.6, we have 0 hi φNi , ξα −g φNi , φξi −1. 5.8 It is a contradiction. Thus M is not irrotational. Since any totally umbilical r-lightlike submanifold of M is irrotational, by Theorem 5.4, we have the following result. Corollary 5.5. There exist no totally umbilical characteristic r-lightlike submanifolds M, g, ST M of an indefinite S-manifold M, φ, ξα , ηα , g equipped with a totally umbilical screen distribution ST M in M. References 1 K. L. Duggal and A. Bejancu, Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, Kluwer Academic, Dodrecht, The Netherlands, 1996. 2 K. L. Duggal and B. Sahin, “Lightlike submanifolds of indefinite Sasakian manifolds,” International Journal of Mathematics and Mathematical Sciences, vol. 2007, Article ID 57585, 21 pages, 2007. 3 D. H. Jin, “Geometry of lightlike hypersurfaces of an indefinite Sasakian manifold,” Indian Journal of Pure and Applied Mathematics, vol. 41, no. 4, pp. 569–581, 2010. 4 D. H. Jin, “Geometry of half lightlike submanifolds of an indefinite Sasakian manifold,” submitted to Bulletin of the Korean Mathematical Society. 5 L. Brunetti and A. M. Pastore, “Curvature of a class of indefinite globally framed f-manifolds,” Bulletin Mathématique de la Société des Sciences Mathématiques de Roumanie, vol. 5199, no. 3, pp. 183–204, 2008. 6 L. Brunetti and A. M. Pastore, “Lightlike hypersurfaces in indefinite S-manifolds,” Diﬀerential Geometry, vol. 12, pp. 18–40, 2010. 7 D. H. Jin, “Generic lightlike submanifolds of an indefinite Sasakian manifold”. 8 J. W. Lee and D. H. Jin, “Half lightlike submanifolds in indefinite S-manifolds,” to appear in Taiwanese Journal of Mathematics. 9 S. Kobayashi and K. Nomizu, Foundations of Diﬀerential Geometry, vol. 1-2, Interscience Publishers, New York, NY, USA, 1963. 10 D. N. Kupeli, Singular Semi-Riemannian Geometry, Kluwer Academic, Dodrecht, The Netherlands, 1996. 11 C. Atindogbe and K. L. Duggal, “Pseudo-Jacobi operators and Osserman lightlike hypersurfaces,” Kodai Mathematical Journal, vol. 32, no. 1, pp. 91–108, 2009. 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