Symmetries in Lightlike Hypersufaces of Indefinite Kenmotsu Manifolds

Transcription

1 International Journal of Contemporary Mathematical Sciences Vol. 12, 2017, no. 3, HIKARI Ltd, Symmetries in Lightlike Hypersufaces of Indefinite Kenmotsu Manifolds Pontsho Moile University of Botswana Department of Mathematics Private Bag 0022 Gaborone, Botswana Copyright c 2017 Pontsho Moile. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we deal with the symmetric property of lightlike hypersurfaces of indefinite Kenmotsu manifolds tangent to the vector field. We prove that there exist no weakly Ricci η-einstein (or screen locally conformal) lightlike hypersurfaces in indefinite Kenmotsu manifolds, tangent to the structure vector field, if α + β + γ is nowhere zero, where α, β and γ are 1-form defined on the submanifold.we also prove that the geometry of the special weakly lightlike hypersurfaces is closely related to that of geodesibility and umbilicality of its tangent space and screen distribution, respectively. Under some conditions, a special weakly Ricci symmetric screen locally (or globally) conformal (or η-einstein or Einstein) lightlike hypersurface of an indefinite Kenmotsu space form, tangent to the structure vector field, is locally symmetric, semi-symmetric and Ricci semi-symmetric. Mathematics Subject Classification: 53C25; 53C50 Keywords: Indefinite Kenmotsu maniflod ; η-einstein; Weakly Ricci symmetric lightlike hypersurfaces; Special weakly Ricci symmetric lightlike hypersurfaces

2 118 Pontsho Moile 1 Introduction The general theory of lightlike submanifolds was introduced and presented by Duggal and Bejancu. They, Duggal and Bejancu in [1] introduced a nondegenerate screen distribution to construct a non-intersecting lightlike transversal vector bundle of the tangent bundle. The induced objects on a lightlike submanifold depend on the choice of a screen distribution which, in general, is not unique. Several authors have studied lightlike hypersurfaces of indefinite almost contact manifolds (see [7], [8], [9], [10] and many more references therein). In this paper, we study symmetries in lightlike hypersurfaces of indefinite Kenmotsu manifolds M, tangent to the structure vector field, by introducing the condition of screen conformality on the Ricci tensor. We then pay, in Sections 3 and 4, a specific attention to Ricci symmetries (Weakly Ricci symmetry, Special weakly Ricci symmetry and Ricci semi-symmetry) and locally symmetry. A relationship between symmetries is established. 2 Preliminaries Let M be a (2n + 1)-dimensional manifold endowed with an almost contact structure (φ, ξ, η), i.e. φ is a tensor field of type (1, 1), ξ is a vector field, and η is a 1-form satisfying φ 2 = I + η ξ, η(ξ) = 1, η φ = 0 and φξ = 0. (1) Then (φ, ξ, η, g) is called an almost contact metric structure on M if (φ, ξ, η) is an almost contact structure on M and g is a semi-riemannian metric on M such that, for any vector field X, Y on M [5] If, moreover, η(x) = g(ξ, X), g(φ X, φ Y ) = g(x, Y ) η(x) η(y ). (2) ( X φ)y = g(φ X, Y )ξ η(y )φ X, (3) where is the Levi-Civita connection for the semi-riemannian metric g, we call M an indefinite Kenmotsu manifold. Here, without loss of generality, the vector field ξ is assumed to be spacelike, that is, g(ξ, ξ) = 1. In this case, the relation (3) implies X ξ = X η(x)ξ. (4) A plane section σ in T p M is called a φ-section if it is spanned by X and φ X, where X is a unit tangent vector field orthogonal to ξ. The sectional curvature of a φ-section σ is called a φ-sectional curvature. If a Kenmotsu manifold M

3 Symmetries in lightlike hypersufaces has constant φ-sectional curvature c, then, by virtue of the Proposition 12 in [12], the curvature tensor R of M is given by, for any X, Y, Z Γ(T M), R(X, Y )Z = c 3 { } c + 1 { g(y, Z)X g(x, Z)Y + η(x)η(z)y 4 4 η(y )η(z)x + g(x, Z)η(Y )ξ g(y, Z)η(X)ξ +g(φ Y, Z)φ X g(φ X, Z)φ Y 2g(φ X, Y )φ Z }. (5) A Kenmotsu manifold M of constant φ-sectional curvature c will be called Kenmotsu space form and denoted by M(c). Let (M, g) be a (2n+1)-dimensional semi-riemannian manifold with index s, 0 < s < 2n + 1 and let (M, g) be a hypersurface of M, with g = g M. M is said to be a lightlike hypersurface of M if g is of constant rank 2n 1 and the orthogonal complement T M of tangent space T M, defined as T M = { Yp T p M : g p (X p, Y p ) = 0, X p T p M }, (6) p M is a distribution of rank 1 on M [1]: T M T M and then coincides with the radical distribution Rad T M = T M T M. A complementary bundle of T M in T M is a rank 2n 1 non-degenerate distribution over M. It is called a screen distribution and is often denoted by S(T M). Existence of S(T M) is secured provided M is paracompact. However, in general, S(T M) is not canonical (thus it is not unique) and the lightlike geometry depends on its choice. A lightlike hypersurface endowed with a specific screen distribution is denoted by the triple (M, g, S(T M)). As T M lies in the tangent bundle, the following result has an important role in studying the geometry of a lightlike hypersurface [1]. Theorem 2.1. (Duggal-Bejancu) Let (M, g, S(T M)) be a lightlike hypersurface of (M, g). Then, there exists a unique vector bundle N(T M) of rank 1 over M such that for any non-zero section E of T M on a coordinate neighborhood U M, there exists a unique section N of N(T M) on U satisfying g(n, E) = 1 and g(n, N) = g(n, W ) = 0, W Γ(S(T M) U ). (7) Throughout the project, all manifolds are supposed to be paracompact and smooth. Theorem 2.2. Let M be a lightlike hypersurface of an indefinite Kenmotsu space form M(c), with ξ T M. Then the Ricci tensor Ric of M is given by, for any X, Y Γ(T M), Ric(X, Y ) = ag(x, Y ) + B(X, Y )tra N B(A N X, Y ), (8)

4 120 Pontsho Moile where a = (2n 1) and trace tr is written with respect to g restricted to S(T M). Note that the Ricci tensor does not depend on the choice of the vector field E of the distribution T M. We also state the following theorem Theorem 2.3. [9] Let M be a lightlike hypersurface of an indefinite Kenmotsu space form M(c), with ξ T M. If the second fundamental form h of M is parallel, then M is totally geodesic. Lemma 2.4. Let M be a lightlike hypersurface of an indefinite Kenmotsu space form M(c), with ξ T M. Then, the covariant derivative of the Ricci tensor Ric of M is given by, for any X, Y, Z Γ(T M), ( X Ric)(Y, Z) = a{b(x, Y )θ(z) + B(X, Z)θ(Y )} + X B(Y, Z)trA N { X B(A N Y, Z) + B(( X A N )Y, Z)} + B(Y, Z)X tra N. (9) 3 Weakly and special wealky Ricci symmetric lightlike hypersurfaces Let M be a lightlike hypersurface of an indefinite Kenmotsu space form M(c) with ξ T M. Definition 3.1. A submanifold M of a semi-riemannian manifold M is said to be η-einstein if its Ricci tensor Ric satisfies Ric(X, Y ) = k 1 g(x, Y ) + k 2 η(x)η(y ), (10) where the non-zero functions k 1 and k 2 are not necessarily constant on M. If k 2 = 0, then M is said to be Einstein. By Theorem 2.2, the Ricci tensor of M is given by, for any X, Y Γ(T M), Ric(X, Y ) = ag(x, Y ) + B(X, Y )tra N B(A N X, Y ), (11) where a = (2n 1) and trace tr is written with respect to g restricted to S(T M). From this relation, we have Ric(X, Y ) Ric(Y, X) = B(A N X, Y ) B(A N Y, X). (12) This means that the Ricci tensor of a lightlike hypersurface M of an indefinite Kenmotsu space form M(c) is not symmetric in general. We recall the definition of screen conformal lightlike hypersurfaces of a semi-riemannian manifold M.

5 Symmetries in lightlike hypersufaces Definition 3.2. [3]A lightlike hypersurface (M, g, S(T M)) of a semi-riemannian manifold is screen locally conformal if the shape operator A N and A E of M and its screen distribution S(T M), respectively, are related by A N = ϕ A E, (13) where ϕ is a non-vanishing smooth function on U in M. In case U = M the screen conformality is said to be global. Theorem 3.3. Let (M, g, S(T M)) be a locally (or globally) screen conformal lightlike hypersurface of an indefinite Kenmotsu space form M(c), with ξ T M. Then the Ricci tensor of the induced connection is symmetric. A submanifold M is said to be weakly Ricci symmetric if there exist 1-forms α, β and γ such that the condition [4]: ( X Ric)(Y, Z) = α(x)ric(y, Z) + β(y )Ric(X, Z) + γ(z)ric(y, X), (14) holds for any vector fields X, Y, Z Γ(T M). We denote this kind of 2ndimensional submanifold by (W RS) 2n. Suppose that M is an η-einstein lightlike hypersurface, then the Ricci tensor Ric of M satisfies Ric(Y, Z) = k 1 g(y, Z) + k 2 η(y )η(z), (15) where non zero functions k 1 and k 2 are not necessarily constant on M and its covariant derivative is given by X Ric(Y, Z) = X k 1 g(y, Z) + k 1 {B(X, Y )θ(z) + B(X, Z)θ(Y )} + X k 2 η(y )η(z)} + k 2 {η(z)( X η)y + η(y )( X η)z}. (16) We recall, using (8), that Ric(ξ, ξ) = a = (2n 1). (17) Therefore, for an η-einstein lightlike hypersurface we get Ric(ξ, ξ) = k 1 + k 2, that is, k 1 + k 2 = (2n 1). This implies that the covariant derivatives of k 1 and k 2 are opposite, that is, for any X Γ(T M), If, moreover M is weakly Ricci symmetric, we have X k 1 = X k 2. (18) X Ric(Y, Z) = α(x){k 1 g(y, Z) + k 2 η(y )η(z)} + β(y ){k 1 g(x, Z) +k 2 η(x)η(z)} + γ(z){k 1 g(y, X) + k 2 η(y )η(x)}. (19)

6 122 Pontsho Moile Taking the pieces (16) and (19) together and using (18), we obtain, for X = ξ, ξ k 1 {g(y, Z) η(y )η(z)} = α(ξ){k 1 g(y, Z) + k 2 η(y )η(z)} +(k 1 + k 2 )β(y )η(z) + (k 1 + k 2 )γ(z)η(y ). (20) Taking Y = Z = ξ in (20) and since n > 1, we get α(ξ) + β(ξ) + γ(ξ) = 0. Putting Y = V and Z = U in (20), we have α(ξ) = ξ. ln k 1 and the other components of α are given by α(v ) = α(u) = α(f i ) = 0. Thus, for any X Γ(T M), α(x) = α(ξ)η(x). Taking Z = ξ in (20), we have β(y ) = {α(ξ) + γ(ξ)}η(y ). Using this again in (20), we have, for Y = ξ, γ(z) = γ(ξ)η(z). Therefore, we have Theorem 3.4. Let M be a weakly Ricci symmetric η-einstien lightlike hypersurface of an indefinite Kenmotsu space form (M 2n+1 (c)(n > 1) with ξ T M. Then, the 1-forms α, β and γ satisfy α + β + γ = 0. Corollary 3.5. There exist no weakly Ricci-symmetric η-einstien lightlike hypersurface of an indefinite Kenmotsu space form (M 2n+1 (c)(n > 1) with ξ T M if α + β + γ is not everywhere zero. The induced Ricci tensor Ric of submanifold M is said to be parallel if, for any X, Y, Z Γ(T M), ( X Ric)(Y, Z) = 0. (21) Theorem 3.6. Let M be a weakly Ricci symmetric η-einstien lightlike hypersurface of an indefinite Kenmotsu manifold M 2n+1 for n > 1 with ξ T M such that the Ricci tensor Ric of M is parallel. Then the 1-forms α, β and γ satisfy α = 0 and β + γ = 0. Proof. Let M be a weakly Ricci symmetric η-einstien lightlike hypersurface of an indefinite Kenmotsu manifold M 2n+1 for n > 1 with ξ T M. By Theorem 3.4, we have, for any X Γ(M), α(x) + β(x) + γ(x) = 0, where α(x) = α(ξ)η(x), β(x) = (α(ξ) + γ(ξ))η(x), γ(x) = γ(ξ)η(x) and α(ξ) + β(ξ) + γ(ξ) = 0. If the Ricci tensor Ric of M is parallel, using (19) we have 0 = α(x){k 1 g(y, Z) + k 2 η(y )η(z)} + β(y ){k 1 g(x, Z) + k 2 η(x)η(z)} + γ(z){k 1 g(y, X) + k 2 η(y )η(x)}, (22) which leads, by taking Y = V and Z = U, to k 1 α(ξ) = 0, i.e. α(ξ) = 0. Therefore α(x) = 0 and β(x) + γ(x) = 0. This completes the proof. It is easy to show that the above theorem still hods for k 2 = 0,ie for a weakly Ricci symmetric Einstein lightlike hypersurface.

7 Symmetries in lightlike hypersufaces Definition 3.7. A non zero Ricci tensor of a lightlike hypersurface M is said to be cyclic parallel if, for any X, Y, Z Γ(T M), C( X Ric)(Y, Z) = 0, (23) where C( X Ric)(Y, Z) = ( X Ric)(Y, Z) + ( Y Ric)(Z, X) + ( Z Ric)(X, Y ). Let M be a weakly lightlike hypersurface of an indefinite Kenmotsu space form with ξ T M such its Ricci tensor Ric of M is symmetric. We then have, for X, Y, Z Γ(T M), C( X Ric)(Y, Z) = {α(x) + β(x) + γ(x)}ric(y, Z) + {α(y ) + β(y ) + γ(y )}Ric(X, Z) + {α(z) + β(z) + γ(z)}ric(y, X). (24) Taking Z = ξ in (24) and using the identity Ric(, ξ) = aη( ) and Ric(ξ, ) = aη( ) B(A N ξ, ), one obtains C( X Ric)(Y, ξ) = a{α(x) + β(x) + γ(x)}η(y ) For Y = ξ, this relation becomes + a{α(y ) + β(y ) + γ(y )}η(x) + {α(ξ) + β(ξ) + γ(ξ)}ric(y, X) (25) C( X Ric)(ξ, ξ) = a{α(x) + β(x) + γ(x)} + 2a{α(ξ) + β(ξ) + γ(ξ)}η(x) {α(ξ) + β(ξ) + γ(ξ)}b(a N ξ, X). (26) If the induced Ricci tensor Ric of the lightlike hypersurface M is cyclic parallel, then, using (26) and taking X = ξ, we have 3a{α(ξ) + β(ξ) + γ(ξ)} = 0. (27) If n > 1, then α(ξ)+β(ξ)+γ(ξ) = 0. Replacing this into (26), one obtains, for any X Γ(T M), α(x) + β(x) + γ(x) = 0. Therefore, we have the following result. Theorem 3.8. There exist no weakly Ricci symmetric screen locally conformal lightlike hypersuface M of an indefinite Kenmotsu space form M 2n+1 (c)(n > 1) with ξ T M and cyclic parallel Ricci tensor if α + β + γ is not zero everywhere. By Theorem 3.4 and using the relation (24), we have the following result.

8 124 Pontsho Moile Theorem 3.9. If M is weakly Ricci symmetric η-einstien (or Einstein) lightlike hypersurface of an indefinite kenmotsu manifold M 2n+1 (n > 1) with ξ T M, then M is cyclic parallel. Definition [13]A submanifold M is said to be special weakly Ricci symmetric if, for all X, Y, Z T M, X Ric(Y, Z) = 2α(X)Ric(Y, Z) + α(y )Ric(X, Z) + α(z)ric(y, X), (28) where α is a 1-form. Such a submanifold is denoted by (SW RS) 2n. Suppose that M is (SW RS) 2n. If the induced Ricci tensor Ric of M is cyclic parallel, then, using (24), we have 4α(X)Ric(Y, Z) + 4α(Y )Ric(Z, X) + 4α(Z)Ric(X, Y ) = 0 (29) Taking Z = ξ in (29), we get 4aα(X)η(Y ) + 4aα(Y )η(x) 4α(Y )B(A N ξ, X) + 4α(ξ)Ric(X, Y ) = 0, (30) which implies, by taking X = ξ, that 8aα(ξ)η(Y ) + 4aα(Y ) = 0, (31) and for Y = ξ, we have 12aα(ξ) = 0, that is, α(ξ) = 0. Again, taking X = E, Y = ξ and Z = ξ in (29), we obtain α(e) = 0. Similarly, we have α(v ) = α(u) = α(f i ) = 0. Therefore, We have α(x) = 0, X Γ(T M). (32) Theorem There exist no special weakly Ricci symmetric screen locally conformal lightlike hypersurfaces M of an indefinite Kenmotsu space form M 2n+1 (c)(n > 1) with ξ T M and Ricci tensor cyclic parallel if the 1-form α 0. Corollary If M is a special weakly Ricci symmetric screen locally conformal lightlike hypersurface of an indefinite Kenmotsu space form M 2n+1 (c)(n > 1) with ξ T M and the 1-form α 0, then M cannot be Einstein. If the Ricci tensor Ric of M is parallel, then M cannot be η-einstein Let M be a (SW RS) 2n. Then, replacing Z by ξ into the Definition 3.10 and since Ric(, ξ) = aη( ), we get ( X Ric)(Y, ξ) = 2α(X)Ric(Y, ξ) + α(y )Ric(X, ξ) + α(ξ)ric(y, X) = 2aα(X)η(Y ) + aα(y )η(x) + α(ξ)ric(y, X) (33)

9 Symmetries in lightlike hypersufaces Putting X = Y = ξ into this relation, one obtains ( ξ Ric)(ξ, ξ) = 4aα(ξ). Since ( ξ Ric)(ξ, ξ) = 0, we have α(ξ) = 0. Taking X = ξ in the covariant derivative given in Definition 3.10, we get ( ξ Ric)(Y, Z) = 2α(ξ)Ric(Y, Z) + α(y )Ric(ξ, Z) + α(z)ric(y, ξ) = 2α(ξ)Ric(Y, Z) + aα(y )η(z) α(y )B(A N ξ, Z) + aα(z)η(y ). (34) which implies, for Y = ξ and Z = U, that ( ξ Ric)(ξ, U) = 2α(ξ)Ric(ξ, U) + aα(ξ)η(u) + aα(u)η(ξ) = aα(u). On the other hand, ( ξ Ric)(ξ, U) = ξ Ric(U, ξ) Ric( ξ U, ξ) Ric(U, ξ ξ) = Ric( ξ U, ξ) = aη( ξ U) = 0. (35) From these last two relations and since n > 1, one obtains α(u) = 0. To find the general form of α(v ) where V = φe we replace X by ξ, Y by ξ and Z by V in the special weakly symmetric formula and we get ( ξ Ric)(ξ, V ) = 2α(ξ)Ric(ξ, V ) + aα(ξ)η(v ) + aα(v )η(ξ) = aα(v ). Since ( ξ Ric)(ξ, V ) = ξ Ric(V, ξ) Ric( ξ V, ξ) Ric(V, ξ ξ) = 0, therefore α(v ) = 0. For Z = ξ and Y = F i Γ(D 0 ), we have ( ξ Ric)(ξ, F i ) = 2α(ξ)Ric(F i, ξ) + α(f i )Ric(ξ, ξ) + α(ξ)ric(f i, ξ) = aα(f i ). (36) Since ( ξ Ric)(ξ, F i ) = ξ Ric(F i, ξ) Ric( ξ F i, ξ) Ric(F i, ξ ξ) = 0 and n > 1, we have α(f i ) = 0. We conclude that, for any X Γ(T M), α(x) = 0. We have Theorem The Ricci tensor of a special weakly Ricci symmetric screen locally (or globally) conformal (or η-einstein or Einstein) lightlike hypersurface of an indefinite Kenmotsu space form M 2n+1 (c)(n > 1) with ξ T M, is parallel. Suppose that S(T M) is parallel, then, the shape operator A N The relation (9) becomes vanishes. ( X Ric)(Y, Z) = a{b(x, Y )θ(z) + B(X, Z)θ(Y )}. (37) Taking Z = E in this relation, one obtains ( X Ric)(Y, E) = ab(x, Y ). (38) Therefore, we have the following result.

10 126 Pontsho Moile Theorem Let (M, g, S(T M) be a special weakly Ricci symmetric screen locally (or globally) conformal (or η-einstein or Einstein) lightlike hypersurface of an indefinite Kenmotsu space form M 2n+1 (c)(n > 1) with ξ T M such that S(T M) is parallel. Then, M is totally geodesic. 4 Locally symmetric and Ricci semi symmetric lightlike hypersurfaces Let (M, g) be a lightlike hypersurface of an indefinite Kenmotsu space form (M(c), g) with ξ T M. Let us consider the pair {E, N} on U M (Theorem 2.1). Since the sectional curvature tensor c = 1, the curvature tensor R (5) of M becomes, for any X, Y, Z Γ(T M), which implies that R(X, Y )Z = g(x, Z)Y g(y, Z)X (39) g(r(x, Y )Z, E) = 0, (40) and g(r(x, Y )Z, N) = g(x, Z)θ(Y ) g(y, Z)θ(X). (41) Definition 4.1. The semi-riemmannian manifold (M, g) is locally symmetric, if its curvature tensor R satisfies the condition ( X R)(Y, Z)W = 0, (42) for any X, Y, Z and W on M, where is the Levi-Civita connection with respect to the metric g. Theorem 4.2. An indefinite Kenmotsu space form (M(c), g) is locally symmetric. Proof. The covariant derivative of R is given by, for any X, Y, Z Γ(T M), ( W R)(X, Y )Z = W R(X, Y )Z R( W X, Y )Z R(X, W Y )Z Using (39), one obtains R(X, Y ) W Z. (43) W R(X, Y )Z = W (g(x, Z)Y g(y, Z)X) = W g(x, Z)Y + g(x, Z) W Y W g(y, Z)X g(y, Z) W X R( W X, Y )Z = g( W X, Z)Y g(y, Z) W X, R(X, W Y )Z = g(x, Z) W Y g( W Y, Z)X, R(X, Y ) W Z = g(x, W Z)Y g(y, W Z)X.

11 Symmetries in lightlike hypersufaces Putting these pieces together into (43), we have ( W R)(X, Y )Z = 0. This completes the proof. The curvature tensor fieldr is given by, for any X, Y, Z Γ(T M), R(X, Y )Z = g(x, Z)Y g(y, Z)X + B(Y, Z)A N X B(X, Z)A N Y, (44) and ( X B)(Y, Z) ( Y B)(X, Z) = τ(y )B(X, Z) τ(x)b(y, Z). (45) From this definition, we have ( W R)(X, Y )Z = 0, the sequel, we need the following Lemma. X, Y, Z Γ(T M). In Lemma 4.3. Let (M, g, S(T M)) be a lightlike hypersurface of an indefinite Kenmotsu manifold (M, g) with ξ T M. Then, for any X, Y Γ(T M) and Z Γ(S(T M)), g(( X A E)Y, Z) = ( X B)(Y, Z) and g(( X A N )Y, Z) = ( X C)(Y, Z). Proof. For any X, Y Γ(T M) and Z Γ(S(T M)) we have g(( X A E)Y, Z) = g( X(A EY ), Z) g(a E( X Y ), Z) = X.g(A EY, Z) g(a EY, XZ) B( X Y, Z) = X.B(Y, Z) B(Y, X Z) B( X Y, Z) = ( X B)(Y, Z). The second relation is obtained by similar calculation. Definition 4.4. A lightlike hypersurface (M, g, S(T M)) of a semi-riemannian manifold (M, g) is said to be locally symmetric [11] if the curvature tensor R of M satisfies the following conditions g(( W R)(X, Y )Z, P T ) = 0 and g(( W R)(X, Y )Z, N) = 0, (46) for any X, Y, Z, W, T Γ(T M) and N Γ(N(T M)). Lemma 4.5. Let (M, g) be a lightlike hypersurface of an indefinite Kenmotsu space form (M(c), g) with ξ T M. Then, g(( W R)(X, Y )Z, T ) = B(Y, Z)( W C)(X, T ) B(X, Z)( W C)(Y, T ) + {B(W, X)g(Y, T ) B(W, Y )g(x, T )}θ(z) + ( W B)(Y, Z)C(X, T ) ( W B)(X, Z)C(Y, T ) + B(W, Z){θ(X)g(Y, T ) θ(y )g(x, T )}, (47)

12 128 Pontsho Moile and g(( W R)(X, Y )Z, N) = B(Y, Z)C(X, A N W ) B(X, Z)C(Y, A N W ) + {B(W, X)θ(Y ) B(W, Y )θ(x)}θ(z), (48) for any X, Y, Z, W Γ(T M), T Γ(S(T M)) and N Γ(N(T M)). Theorem 4.6. Let (M, g) be a lightlike hypersurface of an indefinite Kenmotsu space form (M(c), g) with ξ T M. Then, M is locally symmetric if and only if it is totally geodesic. Proof. Let (M, g) be a lightlike hypersurface of an indefinite Kenmotsu space form (M(c), g) with ξ T M. Suppose that M is locally symmetric. Then, for any W, Y, Z Γ(T M), ( W R)(X, Y )Z = 0. Taking Y = E and Z = ξ in (47), one obtains 0 = g(( W R)(X, E)ξ, N) = B(W, X), (49) which implies that M is totally geodesic. The converse is obvious. In virtue of Theorem 2.3, we have the following result Theorem 4.7. Let (M, g) be a lightlike hypersurface of an indefinite Kenmotsu space form (M(c), g) with ξ T M.Then, M is locally symmetric if and only if it is parallel. It is known that lightlike submanifolds whose screen distribution is integrable have interesting properties. For any X, Y Γ(T M), u([x, Y ]) = B(X, φy ) B(φX, Y ). (50) It is easy to check that the distribution D ξ is integrable if and only if B(X, φy ) = B(φX, Y ), X, Y Γ(T M). Lemma 4.8. Let (M, g, S(T M)) be a locally symmetric lightlike hypersurface of an indefinite Kenmotsu space form M(c) with ξ T M. Then, for any X, Y, Z, T Γ(T M), we have, R(E, Y, Z, T ) = 0, R(X, E, Z, T ) = 0, R(X, Y, E, T ) = 0. (51) Proof. By Theorem 4.6, we have B = 0 and the proof is completed by using relations (44).

13 Symmetries in lightlike hypersufaces Let (M, g, S(T M)) be a screen integrable lightlike hypersurface of an indefinite Kenmotsu space form M(c) with ξ T M. Using Gauss and Weingarten equations, we have, R(X, Y )Z = R (X, Y )Z + C(X, Z)A EY C(Y, Z)A EX + {( X C)(Y, Z) ( Y C)(X, Z) + τ(y )C(X, Z) τ(x)c(y, Z)}E, X, Y, Z Γ(T M ), (52) where ( X C)(Y, Z) = X.C(Y, Z) C( X Y, Z) C(Y, XZ). By covariant derivative, we have any W, Y, T Γ(T M ) and by virtue of Lemma 4.3, that g(( W A )Y, T ) = ( W B)(Y, T ). If M is locally symmetric, then, using Theorem 4.6, B = 0. By Lemma 4.8, g(( W R )(X, Y )Z, T ) = 0, that is M is locally symmetric in M. Therefore, Theorem 4.9. Let (M, g, S(T M)) be a screen integrable lightlike hypersurface of an indefinite Kenmotsu space form M(c) with ξ T M. If M is locally symmetric, then any leaf M of S(T M) immersed in M as a non-degenerate submanifold is locally symmetric. Note that the locally symmetry has an integrability condition, namely, the semi-symmetry. Now, we deal with semi-symmetric lightlike hypersurfaces in indefinite Kenmotsu spaces form, tangent to the structure vector field ξ. Definition A lightlike hypersurface M of a semi-riemannian manifold M is said to be semi-symmetric if the following condition is satisfied ([2]) (R(W 1, W 2 ) R)(X, Y, Z, T ) = 0, W 1, W 2, X, Y, Z, T Γ(T M), (53) where R is the induced Riemann curvature on M. This is equivalent to R(R(W 1, W 2 )X, Y, Z, T )... R(X, Y, Z, R(W 1, W 2 )T ) = 0. In general the condition (53) is not equivalent to (R(W 1, W 2 ) R)(X, Y )Z = 0 like in the non-degenerate case. Indeed, by direct calculation we have, for any W 1, W 2, X, Y, Z, T Γ(T M), (R(W 1, W 2 ) R)(X, Y, Z, T ) = g((r(w 1, W 2 ) R)(X, Y )Z, T ) + (R(W 1, W 2 ) g)(r(x, Y )Z, T ). (54) Next, we investigate the effect of semi-symmetry condition on geometry of lightlike hypersurfaces in an indefinite Kenmotsu space form. We state the following theorem.

14 130 Pontsho Moile Theorem [10] Let M be a lightlike hypersurface of an indefinite Kenmotsu space form M(c), with ξ T M. Then M is semi-symmetric if and only if it is totally geodesic. In virtue of Theorem 4.6 and Theorem 4.11, we have the following result. Theorem [10] Let M be a lightlike hypersurface of an indefinite Kenmotsu space form M(c) with ξ T M. Then M is locally symmetric if and only if M is semi-symmetric. A lightlike submanifold M of a semi-riemannian manifold M is said to be Ricci semi-symmetric if the following condition is satisfied ([6]) (R(W 1, W 2 ) Ric)(X, Y ) = 0, W 1, W 2, X, Y Γ(T M), (55) where R and Ric are induced Riemannian curvature and Ricci tensor on M, respectively. The latter condition is eqivalent to Ric((R(W 1, W 2 )X, Y ) Ric(X, (R(W 1, W 2 )Y ) = 0. In the following theorems we quote the results found by Massamba in [10] which show the effect of Ricci semi-symmetric condition on the geometry of lightlike hypersurfaces of an indefinite Kenmotsu space form. Theorem [10] Let M be a Ricci semi-symmetric lightlike hypersurface of an indefinite Kenmotsu space form M(c) with ξ T M. Then either M is totally geodesic or Ric(E, A N E) = 0. Using the results above, we have the following. Theorem Let (M, g, S(T M) be a special weakly Ricci symmetric screen locally (or globally) conformal (or η-einstein or Einstein) lightlike hypersurface of an indefinite Kenmotsu space form M 2n+1 (c)(n > 1) with ξ T M such that S(T M) is parallel. Then, the following assertions hold. (i) M is locally symmetric, (ii) M is semi-symmetric. Moreover, if Ric(E, A N E) = 0, then M is Ricci semi-symmetric. It is well known that the screen distribution of a manifold M is not unique but the local second fundamental form B of M is independent of the choice of the screen distribution therefore all of the results which depends only on B are independent of the choice of the screen distribution and are stable with respect to the vector bundles (S(T M), S(T M )) and N(T M). It was shown in [1, page 87] that a change from a screen distribution S(T M) to another S(T M),for C

15 Symmetries in lightlike hypersufaces and C the local second fundamental forms of S(T M) and S(T M), respectively the above results are independent of the screen distribution if and only if ω( X P Y + B(X, Y )W ) = 0, where ω the dual of 1-form of W = 2n 1 i=1 c i W i, called a characteristic vector field of the screen change, with respect to the induced metric g of M, that is, ω(x) = g(x, W ), X Γ(T M). And P a projection of T M on S(T M) with respect to the orthogonal decomposition of T M. Here c i are smooth functions on U. Acknowledgements. The author expresses his thanks to Prof. Fortuné Massamba (University of KwaZulu-Natal, South Africa) for very valuable suggestions that really improved this paper. References [1] A. Bejancu and K. L. Duggal, Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, Springer Netherlands Publishers, [2] B. Sahin, Lightlike hypersurfaces of semi-euclidean spaces satisfying curvature conditions of semi-symmetric type, Turkish J. Math., 31 (2007), [3] C. Atindogbe and K. L. Duggal, Conformal screen on lightlike hypersurfaces, Int. J. Pure and Applied Math., 11 (2004), no. 4, [4] C. Özgür, On weakly symmetric Kenmotsu manifolds, Differential Geometry- Dynamical Systems, 8 (2006), [5] D. E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds, Vol. 203, Birkhauser Boston, Inc., Boston, MA, [6] F. Defever, Ricci semi-symmetric hypersurfaces, Balkan J. of Geometry and Its Appl., 5 (2000), no. 1, [7] F. Massamba, Symmetries of null geometry in indefinite Kenmotsu manifolds, Mediterr. J. Math., 10 (2013), no. 2, [8] F. Massamba, On weakly Ricci symmetric lightlike hypersurfaces of indefinite Sasakian manifolds, SUT J. Math., 44 (2008), no. 2, [9] F. Massamba, Relative nullity foliations and lightlike hypersurfaces in indefinite Kenmotsu manifolds, Turk. J. Math., 35 (2011),

16 132 Pontsho Moile [10] F. Massamba, Screen almost conformal lightlike geometryin indefinite Kenmotsu space forms, Int. Electron. J. Geom., 5 (2012), no. 2, [11] R. Günes, B. Sahin, E. Kilic, On lightlike hypersurfaces of a semi- Riemannian space form, Turkish J. Math., 27 (2003), [12] K. Kenmotsu, A class of almost contact Riemannian manifolds, Tohoku Math. J., 24 (1972), [13] N. Aktan, A. Görgülü and E. Özüsa glam, On special weakly Riccisymmetric Kenmotsu manifolds, Sarajevo Journal of Mathematics, 3 (2007), Received: January 26, 2016; Published: May 5, 2017