Non-existence of Screen Homothetic Lightlike Hypersurfaces of an Indefinite Kenmotsu Manifold

Applied Mathematical Sciences, Vol. 9, 2015, no. 18, 853-865 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2015.4121001 Non-existence of Screen Homothetic Lightlike Hypersurfaces of an Indefinite Kenmotsu Manifold D. H. Jin Department of Mathematics, Dongguk University Gyeongju 780-714, Republic of Korea Copyright c 2014 D. H. Jin. 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. Abstract We study screen homothetic lightlike hypersurfaces M of an indefinite Kenmotsu manifold M with flat transversal connection. We prove that there do not exist screen homothetic lightlike hypersurfaces of indefinite Kenmotsu manifolds with flat transversal connection subject to the conditions: (1) M is locally symmetric, or (2) M is semi-symmetric. Mathematics Subject Classification: 53C25, 53C40, 53C50 Keywords: screen homothetic, totally geodesic screen distribution, flat transversal connection, locally symmetric, semi-symmetric 1 Introduction The theory of lightlike submanifolds is an important topic of research in differential geometry due to its application in mathematical physics. The study of such notion was initiated by Duggal and Bejancu [3] and then studied by many authors (see two books [4, 5]). Now we have lightlike version of a large variety of Riemannian submanifolds. In this paper, we study lightlike hypersurfaces M of indefinite Kenmotsu manifolds M with flat transversal connection, whose shape operator is homothetic to the shape operator of its screen distribution by some constant ϕ. The motivation for this geometric restriction comes from the classical geometry of non-degenerate submanifolds for which there are only one type of

854 D. H. Jin shape operator with its one type of respective second fundamental form. The purpose of this paper is to prove that there do not exist screen homothetic lightlike hypersurfaces of indefinite Kenmotsu manifolds with flat transversal connection subject to the conditions: (1) M is locally symmetric, or (2) M is semi-symmetric. 2 Lightlike hypersurfaces An odd dimensional semi-riemannian manifold ( M, ḡ) is said to be an indefinite Kenmotsu manifold [10, 11] if there exists an almost contact metric structure (J, ζ, θ, ḡ), where J is a (1, 1)-type tensor field, ζ is a vector field and θ is a 1-form such that J 2 X = X + θ(x)ζ, Jζ = 0, θ J = 0, θ(ζ) = 1, (2.1) θ(x) = ḡ(ζ, X), ḡ(jx, JY ) = ḡ(x, Y ) θ(x)θ(y ), X ζ = X + θ(x)ζ, (2.2) ( X J)Y = ḡ(jx, Y )ζ + θ(y )JX, (2.3) for any vector fields X, Y on M, where is the Levi-Civita connection of M. A hypersurface (M, g) of M is called a lightlike hypersurface if the normal bundle T M of M is a subbundle of the tangent bundle T M of M and coincides with the radical distribution Rad(T M) = T M T M. Then there exists a non-degenerate complementary vector bundle S(T M) of Rad(T M) in T M, which is called a screen distribution on M, such that T M = Rad(T M) orth S(T M), (2.4) where orth denotes the orthogonal direct sum. We denote such a lightlike hypersurface by M = (M, g, S(T M)). Denote by F ( M) the algebra of smooth functions on M and by Γ(E) the F ( M) module of smooth sections of a vector bundle E over M. For any null section ξ of Rad(T M), there exists a unique null section N of a unique vector bundle tr(t M) [3] in the orthogonal complement S(T M) of S(T M) in M satisfying ḡ (ξ, N) = 1, ḡ(n, N) = ḡ(n, X) = 0, X Γ(S(T M)). In this case, the tangent bundle T M of M is decomposed as follow: T M = T M tr(t M) = {Rad(T M) tr(t M)} orth S(T M). (2.5) We call tr(t M) and N the transversal vector bundle and the null transversal vector field of M with respect to the screen distribution S(T M) respectively. From now and in the sequel, we denote by X, Y, Z, U, the vector fields on M unless otherwise specified. Let P be the projection morphism of T M

Non-existence of screen homothetic lightlike hypersurfaces 855 on S(T M). Then the Gauss and Weingartan formulas of T M and S(T M) are given by X Y = X Y + B(X, Y )N, (2.6) X N = A N X + τ(x)n ; (2.7) X P Y = XP Y + C(X, P Y )ξ, (2.8) X ξ = A ξx τ(x)ξ, (2.9) where and are the liner connections on T M and S(T M) respectively, B and C are the local second fundamental forms on T M and S(T M) respectively, A N and A ξ are the shape operators on T M and S(T M) respectively and τ is a 1-form. As is torsion-free, is also torsion-free and B is symmetric on T M. From the fact B(X, Y ) = ḡ( X Y, ξ), we show that B is independent of the choice of a screen distribution S(T M) and satisfies B(X, ξ) = 0. (2.10) The above two local second fundamental forms B and C of M and S(T M) respectively are related to their shape operators by B(X, Y ) = g(a ξx, Y ), ḡ(a ξx, N) = 0, (2.11) C(X, P Y ) = g(a N X, P Y ), ḡ(a N X, N) = 0. (2.12) From (2.11), we show that A ξ is S(T M)-valued self-adjoint such that A ξξ = 0. Denote by R, R and R the curvature tensors of the connections, and respectively. Using (2.6) (2.9), we obtain the Gauss-Codazzi equations: R(X, Y )Z = R(X, Y )Z + B(X, Z)A N Y B(Y, Z)A N X (2.13) + {( X B)(Y, Z) ( Y B)(X, Z) + τ(x)b(y, Z) τ(y )B(X, Z)}N, R(X, Y )N = X (A N Y ) + Y (A N X) + A N [X, Y ] + τ(x)a N Y (2.14) τ(y )A N X + {B(Y, A N X) B(X, A N Y ) + 2dτ(X, Y )}N, R(X, Y )P Z = R (X, Y )P Z + C(X, P Z)A ξy C(Y, P Z)A ξx (2.15) + {( X C)(Y, P Z) ( Y C)(X, P Z) + τ(y )C(X, P Z) τ(x)c(y, P Z)}ξ, R(X, Y )ξ = X(A ξy ) + Y (A ξx) + A ξ[x, Y ] τ(x)a ξy (2.16) + τ(y )A ξx + {C(Y, A ξx) C(X, A ξy ) 2dτ(X, Y )}ξ. Let XN = π( X N), where π is the projection morphism of T M on tr(t M). Then is a linear connection on the transversal vector bundle

856 D. H. Jin tr(t M) of M. We say that is the transversal connection of M. We define the curvature tensor R on tr(t M) by R (X, Y )N = X Y N Y XN [X,Y ]N. The transversal connection of M is said to be flat [6] if R vanishes identically. This definition comes from the definition of flat normal connection [2] in the theory of classical geometry of non-degenerate submanifolds. We have the following result (see [8]). Theorem 2.1. Let M be a lightlike hypersurface of a semi-riemannian manifold M. Then the transversal connection of M is flat if and only if τ is closed, i.e., dτ = 0, on M. Note 1. dτ is independent of the choice of the null section ξ on Rad(T M). In fact, if we take ξ = γξ, then Ñ = γ 1 ξ. As τ(x) = ḡ( X N, ξ), we have τ(x) = τ(x) + X(In γ). If we take the exterior derivative d to this equation, then we get dτ = d τ [3]. Note 2. In case dτ = 0, by the cohomology theory there exist a smooth function l such that τ = dl. Thus τ(x) = X(l). If we take ξ = γξ, then τ(x) = τ(x) + X(In γ). Setting γ = exp(l) in this equation, we get τ(x) = 0. We call the pair {ξ, N} such that the corresponding 1-form τ vanishes the canonical null pair of M. Although S(T M) is not unique it is canonically isomorphic to the factor vector bundle S(T M) = T M/Rad(T M) due to Kupeli [12]. Thus all S(T M) are mutually isomorphic. In the sequel, we deal with only lightlike hypersurfaces M equipped with the canonical null pair {ξ, N}. 3 Locally symmetric lightlike hypersurfaces Definition 1. A lightlike hypersurface M is screen homothetic [4, 6] if the shape operators A N and A ξ of M and S(T M) respectively are related by A N = ϕ A ξ, or equivalently, C(X, P Y ) = ϕ B(X, Y ), (3.1) where ϕ is a constant on a coordinate neighborhood U in M. In particular, if ϕ = 0, i.e., C = 0 on U, then we say that S(T M) is totally geodesic [10] in M, and if ϕ 0 on U, then we say that M is proper screen homothetic [6, 7]. In the sequel, by saying that M is screen homothetic we shall mean not only M is proper screen homothetic but also S(T M) is totally geodesic in M. Theorem 3.1. Let M be a locally symmetric lightlike hypersurface of an indefinite Kenmotsu manifold M. Then S(T M) is not totally geodesic. Moreover, if the transversal connection is flat, then M is not proper screen homothetic.

Non-existence of screen homothetic lightlike hypersurfaces 857 Proof. From the decomposition (2.5) of T M, ζ is decomposed as follow: ζ = W + fn, (3.2) where W is a non-vanishing smooth vector field on M and f = θ(ξ) is a smooth function. Substituting (3.2) into (2.2) and using (2.6) and (2.7), we have X W = X + θ(x)w + fa N X, (3.3) Xf + fτ(x) + B(X, W ) = fθ(x). (3.4) Substituting (3.4) into [X, Y ]f = X(Y f) Y (Xf), we have ( X B)(Y, W ) ( Y B)(X, W ) + τ(x)b(y, W ) τ(y )B(X, W )(3.5) + f{b(y, A N X) B(X, A N Y ) + 2dτ(X, Y )} = 2fdθ(X, Y ). Using (2.13), (2.14) and (3.2), the equation (3.5) reduce to ḡ( R(X, Y )ζ, ξ) = 2fdθ(X, Y ). (3.6) Substituting (3.3) into R(X, Y )W = X Y W Y X W [X, Y ] W and using (2.13), (2.14), (3.2) (3.6) and the fact that is torsion-free, we have R(X, Y )ζ = θ(x)y θ(y )X + 2dθ(X, Y )ζ. (3.7) Taking the scalar product with ζ to (3.7) and using the facts that g( R(X, Y )ζ, ζ) = 0 and θ(x) = ḡ(x, ζ), we have dθ = 0, i.e., the 1-form θ is closed on T M. Cǎlin [1] proved that if ζ is tangent to M, then it belongs to S(T M) which we assume in this case. Taking Y = ζ to (2.6) and using (2.2), we get X ζ = X + θ(x)ζ, B(X, ζ) = 0. Taking the scalar product with N to the first equation and using (2.8), we have C(X, ζ) = η(x), where η is a 1-form such that η(x) = ḡ(x, N). Assume that M is screen homothetic. Using the last two equations, we get η(x) = C(X, ζ) = ϕb(x, ζ) = 0. It is a contradiction as η(ξ) = 1. Thus ζ is not tangent to M and f 0. Let e = θ(n). Assume that e = 0. Applying X to ḡ(ζ, N) = 0 and using (2.2) and (2.7), we have ḡ(a N X, ζ) = η(x). Replacing X by ξ to this and using (3.1) and the fact that A ξξ = 0, we have 0 = ϕḡ(a ξξ, ζ) = η(ξ) = 1. It is a contradiction. Thus e is non-vanishing function.

858 D. H. Jin Case 1. Assume that S(T M) is totally geodesic in M, i.e., ϕ = 0. Substituting (3.2) into (3.7) and using (2.13), (2.14), (3.5) and the fact that dθ = 0, we have R(X, Y )W = θ(x)y θ(y )X. (3.8) Applying X to θ(y ) = g(y, ζ) and using (2.2) and (2.6), we have ( X θ)(y ) = eb(x, Y ) g(x, Y ) + θ(x)θ(y ). (3.9) Applying Z to (3.8) and using (3.3), (3.9) and the fact that Z R = 0, we have R(X, Y )Z = {g(x, Z) eb(x, Z)}Y {g(y, Z) eb(y, Z)}X. (3.10) Replacing Z by W to (3.10) and then, comparing this result with (3.8) and using the fact that θ(x) = g(x, W ) + fη(x), we have {fη(x) + eb(x, W )}Y = {fη(y ) + eb(y, W )}X. Replacing Y by ξ to this and using the fact that X = P X + η(x)ξ, we have fp X = eb(x, W )ξ, X Γ(T M). The left term of this equation belongs to S(T M) and the right term belongs to T M. Thus fp X = 0 and eb(x, W ) = 0 for all X Γ(T M). From the first equation of this results, we have f = 0. It is a contradiction as f 0. Thus S(T M) is not totally geodesic in M. Case 2. Assume that M is screen homothetic. Substituting (3.2) into (3.7) with dθ = 0 and then, taking the scalar product with N, we have g(r(x, Y )W, N) = θ(x)η(y ) θ(y )η(x). (3.11) As the transversal connection is flat, i.e., dτ = 0, we have τ = 0 by Note 2. Substituting W = P W +eξ to (3.11) and using (2.11), (2.15), (2.16) and (3.1), we have ( X C)(Y, P W ) ( Y C)(X, P W ) = θ(x)η(y ) θ(y )η(x). (3.12) Applying X to C(Y, P W ) = ϕb(y, W ) and using (2.8) (2.11), we have ( X C)(Y, P W ) = ϕ( X B)(Y, W ). Substituting this equation into (3.12) and using (3.5) with τ = 0, we have θ(x)η(y ) θ(y )η(x) = 0, X, Y Γ(T M).

Non-existence of screen homothetic lightlike hypersurfaces 859 Replacing Y by ξ to this, we have g(x, W ) = 0. This implies W = eξ. Consequently the structure vector field ζ of M is decomposed as ζ = eξ + fn. (3.13) From the fact that ḡ(ζ, ζ) = 1 and (3.13), we get 2ef = 1. Applying X to (3.13) and using (2.2), (2.7), (2.9) and the fact that η(x) = 2eθ(X), we have P X eθ(x)ξ + fθ(x)n = (e + ϕf)a ξx + X[e]ξ + X[f]N. Taking the scalar product with ξ and N to this result by turns, we get X[f] = fθ(x), X[e] = eθ(x), (e + ϕf)a ξx = P X. (3.14) If e + ϕf = 0, then we obtain P X = 0 for all X Γ(T M). It is an impossible result if S(T M) {0}. Thus the function α = (e + ϕf) 1 is a non-vanishing one and we get A ξx = αp X, B(X, Y ) = αg(x, Y ). (3.15) Applying X to α = (e + ϕf) 1 and αe, and then, using (3.14) 1, 2, we have X[α] = α(2αe 1)θ(X), X[αe] = 2αe(αe 1)θ(X). (3.16) Applying X to θ(y ) = g(y, ζ) and using (2.2), (2.6) and (3.15), we have ( X θ)(y ) = (αe 1)g(X, Y ) + θ(x)θ(y ). (3.17) Substituting (3.2) into (3.7) with dθ = 0 and using (2.10), (2.13), (2.14), (3.5) and the facts that W = eξ, A N = ϕa ξ and α = (e + fϕ) 1, we have R(X, Y )ξ = α{θ(x)y θ(y )X}, X, Y Γ(T M). (3.18) From (2.9), (3.14) 3 and the facts that X = P X + η(x)ξ and 2ef = 1, we have X ξ = αx + 2αe θ(x)ξ. (3.19) Applying Z to (3.18) and using (3.16) (3.19), we have ( Z R)(X, Y )ξ = αr(x, Y )Z + α(αe 1){g(X, Z)Y g(y, Z)X}. (3.20) Assume that M be locally symmetric. The equation (3.20) is reduced to R(X, Y )Z = (αe 1){g(Y, Z)X g(x, Z)Y }. Taking Z = ξ to this and then, comparing this result with (3.18), we have α{θ(x)y θ(y )X} = 0. Replacing Y by ξ to this and using the facts that X = P X + η(x)ξ and θ(x) = fη(x), we have αp X = 0 for all X Γ(T M). Thus we have α = 0. It is a contradiction as α 0. Thus M is not proper screen homothetic.

860 D. H. Jin 4 Semi-symmetric lightlike hypersurfaces Definition 2. A lightlike hypersurface M is called semi-symmetric [5, 10] if R(X, Y )R = 0, X, Y Γ(T M). Theorem 4.1. Let M be a lightlike hypersurface of an indefinite Kenmotsu manifold M with flat transversal connection. If M is semi-symmetric and dim M > 2, then neither M is proper screen homothetic nor S(T M) is totally geodesic in M. Proof. It is well-known that, for any lightlike hypersurface of an indefinite almost contact metric manifold M, J(Rad(T M)) and J(tr(T M)) are vector subbundles of S(T M) of rank 1 [8, 9]. Assume that M is screen homothetic. Then we can use the equations (3.1) (3.7), (3.11) (3.20) in Section 3. Applying J to (3.13) and using Jζ = 0 and 2ef = 1, we get Jξ = 2f 2 JN. Thus J(Rad(T M)) J(tr(T M)) {0}. From this result we have J(Rad(T M)) = J(tr(T M)). Using this result, the tangent bundle T M of M is decomposed as follow: T M = Rad(T M) orth {J(Rad(T M)) orth H}, (4.1) where H is a non-degenerate and almost complex distribution on S(T M) with respect to J. Consider a timelike vector field V and its 1-form v defined by V = f 1 Jξ = e 1 JN, v(x) = g(x, V ), (4.2) for all X Γ(T M). Applying J to (4.2) 1 and using the fact that 2ef = 1, we get JV = eξ fn. (4.3) Denote by Q the projection morphism of T M on H with respect to the decomposition (4.1). Then any vector field X on M is expressed as follow: X = QX + v(x)v + η(x)ξ. Applying J to this and using (4.3) and the fact that θ(x) = fη(x), we have JX = F X θ(x)v + ev(x)ξ fv(x)n, (4.4) where F is a tensor field of type (1, 1) globally defined on M by F X = JQX, X Γ(T M).

Non-existence of screen homothetic lightlike hypersurfaces 861 Applying X to fv = Jξ and v(y ) = g(y, V ) by turns and using (2.1), (2.3), (2.6), (2.9), (2.11), (3.14), (4.2) (4.4) and the fact that F ξ = 0, we get X V = e(ρ 1)v(X)ξ + ρf X, ρ = 2αe 1, (4.5) ( X v)y = ρg(f X, Y ) + (ρ + 1)v(X)θ(Y ). (4.6) As ρ = 2αe 1, we get αe = (ρ + 1)/2. Thus the equation (3.17) reduce to ( X θ)(y ) = 1 (ρ 1)g(X, Y ) + θ(x)θ(y ). (4.7) 2 Applying Y to (4.4) and using (2.3), (2.6) (2.9), (3.13) (3.15) and (4.2) (4.7), we get ( X F )Y = (ρ + 1)θ(Y )F X + ρv(y )P X + ρg(x, Y )V (4.8) + e(ρ 1)g(F X, Y )ξ, X, Y Γ(T M). Substituting (4.5) into the right term of R(X, Y )V = X Y V Y X V [X, Y ] V and using the fact that is torsion-free, we have R(X, Y )V = e{(xρ)v(y ) (Y ρ)v(x) (4.9) + (ρ 2 + ρ 2)[v(X)θ(Y ) v(y )θ(x)]}ξ + (Xρ)F Y (Y ρ)f X + ρ(ρ + 1){θ(Y )F X θ(x)f Y } + 1 2 (ρ2 + 1)[v(Y )P X v(x)p Y ]. Assume that M is semi-symmetric, i.e., R(U, Z)R = 0. In case M is proper screen homothetic. Applying U to (3.20) and using (3.2) and (3.20), we have ( U Z R)(X, Y )ξ = α( U R)(X, Y )Z + α( Z R)(X, Y )U αθ(u)r(x, Y )Z + αr(x, Y ) U Z + α(2αe 1)(αe 1)θ(U){g(X, Z)Y g(y, Z)X} + α(αe 1){2αeθ(Z)[g(U, X)Y g(u, Y )X] From the last equation and (3.20), we have + 2αeg(U, Z)[θ(X)Y θ(y )X] + g(x, U Z)Y g(y, U Z)X}. θ(z){r(x, Y )U + (αe 1)[g(X, U)Y g(y, U)X]} = θ(u){r(x, Y )Z + (αe 1)[g(X, Z)Y g(y, Z)X]}. Replacing U by ξ to this equation and using (3.18), R is given by R(X, Y )Z = (αe 1){g(Y, Z)X g(x, Z)Y } (4.10) +2αeθ(Z){θ(X)Y θ(y )X}.

862 D. H. Jin In case S(T M) is totally geodesic, ϕ = 0 and αe = 1. Thus R is given by R(X, Y )Z = 2θ(Z){θ(X)Y θ(y )X}. (4.11) Case 1. Let M be proper screen homothetic. Replacing Z by V to (4.10) and then, comparing the S(T M)-components of this result and (4.9), we get (αe 1){v(X)P Y v(y )P X} = (Xρ)F Y (Y ρ)f X + ρ(ρ + 1){θ(Y )F X θ(x)f Y } + 1 2 (ρ2 + 1){v(Y )P X v(x)p Y }, X, Y Γ(T M). Substituting ρ = 2αe 1 and X[ρ] = 4αe(αe 1)θ(X) to this, we have αe(2αe 1){v(X)P Y v(y )P X} + 2αe{θ(X)F Y θ(y )F X} = 0. (4.12) Replacing Y by ξ to this and using the facts that v(ξ) = 0 and P ξ = F ξ = 0, we have αf X = 0 for all X Γ(T M). Substituting (4.4) into the equation ḡ(jx, JY ) = g(x, Y ) θ(x)θ(y ) for any X, Y Γ(T M), we have g(f X, F Y ) = g(x, Y ) + v(x)v(y ). As α 0, we get α{g(x, Y ) + v(x)v(y )} = 0. This implies g(x, Y ) = 0 for any X, Y Γ(H). It is a contradiction as dim M > 2. Thus M is not proper screen homothetic. Case 2. Let S(T M) be totally geodesic in M. Then αe = 1 and (4.12) is reduced to v(x)p Y v(y )P X + 2{θ(X)F Y θ(y )F X} = 0. Replacing Y by V to this and using the facts that θ(v ) = 0 and F V = 0, we have P X = v(x)v, X Γ(T M). It is a contradiction as dim M > 2. Thus S(T M) is not totally geodesic. An odd dimensional semi-riemannian manifold ( M, g) is called an indefinite Sasakian manifold [5, 8] if there exists an almost contact metric structure (φ, U, u, g), where φ is a (1, 1)-type tensor field, U is a vector field and u is a 1-form such that φ 2 X = X + u(x)u, φ U = 0, u φ = 0, u(u) = 1, g(u, U) = ɛ, g(φx, φy ) = g(x, Y ) ɛu(x)u(y ), u(x) = ɛ g(x, U), du(x, Y ) = g(φx, Y ), ɛ = ±1, X U = ɛφx, ( X φ)y = g(x, Y )U ɛu(y )X, for any X, Y Γ(T M), where is the Levi-Civita connection on M.

Non-existence of screen homothetic lightlike hypersurfaces 863 The following result is a very marvellous one in the common sense. Theorem 4.2. Let M be a lightlike hypersurface of an indefinite Kenmotsu manifold M with flat transversal connection. If S(T M) is totally geodesic in M, then the leaf M of S(T M) is an indefinite Sasakian manifold with an almost contact metric structure (F, V, v, g) such that ɛ = g(v, V ) = 1. Proof. Assume that S(T M) is totally geodesic in M. Then we have αe = 1 and ρ = 1. Take X, Y Γ(S(T M)), then (4.5) and (4.8) reduce to X V = F X, ( X F )Y = g(x, Y )V + v(y )X, respectively. By direct calculations, we show that (F, V, v, g) is an almost contact metric structure on M. Thus M is an indefinite Sasakian manifold. 5 Lightlike hypersurfaces of a space form Definition 3. An indefinite Kenmotsu manifold M is called an indefinite Kenmotsu space form, denoted by M( c), if it has the constant J-sectional curvature c [11]. The curvature tensor R of this space form M( c) is given by 4 R(X, Y )Z = ( c 3){ḡ(Y, Z)X ḡ(x, Z)Y } + ( c + 1){θ(X)θ(Z)Y θ(y )θ(z)x + ḡ(x, Z)θ(Y )ζ ḡ(y, Z)θ(X)ζ + ḡ(jy, Z)JX + ḡ(jz, X)JY 2ḡ(JX, Y )JZ}, X, Y, Z Γ(T M). It is well known [11] that if an indefinite Kenmotsu manifold M is a space form, then it is Einstein and c = 1, i.e., R is given by R(X, Y )Z = ḡ(x, Z)Y ḡ(y, Z)X, X, Y, Z Γ(T M). (5.1) Theorem 5.1. Let M be a lightlike hypersurface of an indefinite Kenmotsu space form M(c). Then neither M is proper screen homothetic nor S(T M) is totally geodesic in M. Proof. Assume that M is screen homothetic. Taking the scalar product with ξ to (2.13) and (5.1) and then, comparing the resulting two equations, we have ( X B)(Y, Z) ( Y B)(X, Z) + τ(x)b(y, Z) τ(y )B(X, Z) = 0. (5.2) Comparing (3.5) with dθ = 0 and (5.2) with Z = W and using (3.1) and the fact that f 0, we have dτ = 0. By Theorem 2.1, we show that the transversal connection is flat. Thus we can use (3.1) (3.7) and (3.11) (3.20) in Section 3 and (4.1) (4.9) in Section 4.

864 D. H. Jin Case 1. If M is proper screen homothetic, then, replacing Z by V to (2.13) and using (5.1) and (5.2), for any X, Y Γ(T M), we have R(X, Y )V = (1 α 2 ϕ){v(y )P X v(x)p Y } + {v(y )η(x) v(x)η(y )}ξ. Comparing the Rad(T M)-components of this equation and (4.9), we obtain (Xρ)v(Y ) (Y ρ)v(x) + (ρ 2 + ρ 2){v(X)θ(Y ) v(y )θ(x)} = 2{θ(X)v(Y ) θ(y )v(x)}, X, Y Γ(T M). Substituting ρ = 2αe 1 and X[ρ] = 4αe(αe 1)θ(X) to this, we have αe{θ(x)v(y ) θ(y )v(x)} = 0, X, Y Γ(T M). Replacing Y by ξ to this equation and using v(ξ) = 0, θ(ξ) = f and 2ef = 1, we have αv(x) = 0 for all X Γ(T M). Taking X = V, we get α = 0. It is a contradiction as α 0. Thus M is not proper screen homothetic. Case 2. If S(T M) is totally geodesic in M, then we show that αe = ρ = 1. As the transversal connection is flat, (4.9) is reduced to R(X, Y )V = v(y )P X v(x)p Y + 2{θ(Y )F X θ(x)f Y }. Replacing Z by V to (2.13) and using (5.1) and (5.2), we have R(X, Y )V = v(y )P X v(x)p Y + {v(y )η(x) v(x)η(y )}ξ, for all X, Y Γ(T M). Comparing the last two equations, we obtain 2{θ(Y )F X θ(x)f Y } = {v(y )η(x) v(x)η(y )}ξ, for all X, Y Γ(T M). Taking the scalar product with N to this, we have v(y )η(x) v(x)η(y ) = 0, X, Y Γ(T M). Taking Y = ξ to this equation, we get v(x) = 0 for any X Γ(T M). It is a contradiction as v(v ) = 1. Thus S(T M) is not totally geodesic in M. References [1] C. Cǎlin, Contributions to geometry of CR-submanifold, Thesis, University of Iasi (Romania), 1998. [2] B. Y. Chen, Geometry of Submanifolds, Marcel Dekker, New York, 1973.

Non-existence of screen homothetic lightlike hypersurfaces 865 [3] K. L. Duggal and A. Bejancu, Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, Kluwer Acad. Publishers, Dordrecht, 1996. http://dx.doi.org/10.1007/978-94-017-2089-2 [4] K. L. Duggal and D. H. Jin, Null Curves and Hypersurfaces of Semi-Riemannian Manifolds, World Scientific, 2007. http://dx.doi.org/10.1142/6449 [5] K.L. Duggal and B. Sahin, Differential geometry of lightlike submanifolds, Frontiers in Mathematics, Birkhäuser, 2010. http://dx.doi.org/10.1007/978-3-0346-0251-8 [6] D. H. Jin, Screen conformal Einstein lightlike hypersurfaces of a Lorentzian space form, Commun. Korean Math. Soc., 25(2), 2010, 225-234. http://dx.doi.org/10.4134/ckms.2010.25.2.225 [7] D. H. Jin, Screen conformal lightlike real hypersurfaces of an indefinite complex space form, Bull. Korean Math. Soc., 47(2), 2010, 341-353. http://dx.doi.org/10.4134/bkms.2010.47.2.341 [8] D. H. Jin, Geometry of lightlike hypersurfaces of an indefinite Sasakian manifold, Indian J. of Pure and Applied Math., 41(4), 2010, 569-581. http://dx.doi.org/10.1007/s13226-010-0032-y [9] D. H. Jin, Geometry of lightlike hypersurfaces of an indefinite cosymplectic manifold, Commun. Korean Math. Soc. 27(2), 2012, 569-581. [10] D. H. Jin, The curvatures of lightlike hypersurfaces of an indefinite Kenmotsu manifold, Balkan J. of Geo. and Appl., 17(2), 2012, 49-57. [11] K. Kenmotsu, A class of almost contact Riemannian manifolds, Tôhoku Math. J., 21, 1972, pp.93-103. http://dx.doi.org/10.2748/tmj/1178241594 [12] D. N. Kupeli, Singular Semi-Riemannian Geometry, Mathematics and Its Applications, vol. 366, Kluwer Acad. Publishers, Dordrecht, 1996. http://dx.doi.org/10.1007/978-94-015-8761-7 Received: December 12, 2014; Published: January 29, 2015