Lightlike Hypersurfaces of an Indefinite Kaehler Manifold with an (`, m)-type Connection

Jin [1] defined an (`, m)-type connection on semi-Riemannian manifolds. Semi-symmetric nonmetric connection and non-metric φ-symmetric connection are two important examples of this connection such that (`, m) = (1, 0) and (`, m) = (0, 1), respectively. In semiRiemannian geometry, there are few literatures for the lightlike geometry, so we expose new theories for non-degenerate submanifolds in semi-Riemannian geometry. The goal of this paper is to study a characterization of a (Lie) recurrent lightlike hypersurface M of an indefinite Kaehler manifold with an (`, m)-type connection when the charateristic vector field is tangnet to M . In the special case that an indefinite Kaehler manifold of constant holomorphic sectional curvature is an indefinite complex space form, we investigate a lightlike hypersurface of an indefinite complex space form with an (`, m)-type connection when the charateristic vector field is tangnet to M . Moreover, we show that the total space, the complex space form, is characterized by the screen conformal lightlike hypersurface with an (`, m)-type connection. With a semi-symmetric non-metric connection, we show that an indefinite complex space form is flat.


Introduction
A linear connection∇ on a semi-Riemannian manifold (M ,ḡ) is called an ( , m)-type connection [1] if∇ and its torsion tensorT satisfy where and m are two smooth functions onM , J is a tensor field of type (1, 1) and θ is a 1-form associated with a smooth unit vector field ζ, which is called the characteristic vector field ofM , by θ(X) =ḡ(X, ζ). Throughout this paper, we denote byX,Ȳ andZ the smooth vector fields onM . Two special cases are important for both the mathematical study and the applications to physics: (1) In case ( , m) = (1, 0): This connection∇ becomes a semi-symmetric nonmetric connection. The notion of semi-symmetric non-metric connection was introduced by Ageshe-Chafle [2,3] and later, studied by several authors [4]. (2) In case ( , m) = (0, 1): ∇ becomes a non-metric φ-symmetric connection such that φ(X,Ȳ ) =ḡ(JX,Ȳ ). The notion of the non-metric φsymmetric connection was introduced by this author [5,6]. (1. 3) The objective of study in this paper is lightlike hypersurfaces of an indefinite Kaehler manifold M = (M ,ḡ, J) with an ( , m)-type connection subject to the conditions that (1) the tensor field J, defined by (1.1) and (1.2), is identical with the indefinite almost complex structure tensor J ofM and (2) the characteristic vector field ζ ofM is tangent to M . In this paper, we set ( , m) = (0, 0) and we shall assume that ζ is unit spacelike, without loss of generality.

Preliminaries
Let (M ,ḡ, J) be an indefinite Kaehler manifold equipped with an ( , m)-type connection∇ and a Levi-Civita connec-tion ∇, whereḡ is a semi-Riemannian metric and J is an indefinite almost complex structure such that By direct calculation from (1.3) and (2.1) 3 , we see that where ⊕ orth denotes the orthogonal direct sum. It is known [7] that, for any null section ξ of T M ⊥ on a coordinate neighborhood U ⊂ M , there exists a unique null section N of a unique vector bundle tr 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. Then the tangent bunde TM ofM is decomposed as follow: In case the vector field ζ is tangent to M . If ζ belongs to Rad(T M ), then ζ = aξ, 1 =ḡ(ζ, ζ) = a 2 g(ξ, ξ) = 0, a ∈ F (M ).
where ∇ and ∇ * are the induced 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 and τ is a 1-form.
In this case, the decomposition of T M is reduced to (2.8) Consider two null vector fields U and V and two 1-forms u and v such that Denote by S the projection morphism of T M on D. Any vector field X of M is expressed as X = SX +u(X)U . Applying J to this form, we have where F is a tensor field of type (1, 1) globally defined on M by F = J • S. Applying J to (2.10) and using (2.1) and (2.9), we have As u(U ) = 1 and F U = 0, the set (F, u, U ) defines an indefinite almost contact structure on M and F is called the structure tensor field of M .

( , m)-type connections
Using (1.1), (1.2), (2.4) and (2.10), we obtain where T is the torsion tensor with respect to ∇ and η is a 1-form such that η(X) =ḡ(X, N ). From the fact that B(X, Y ) =ḡ(∇ X Y, ξ), we know that B is independent of the choice of the screen distribution S(T M ) and satisfy The local second fundamental forms are related to their shape operators by As S(T M ) is non-degenerate, from (3.4) 2 and (3.5), we obtain Applying∇ X to (2.9) 1, 2, 4 and (2.10), we have 4 Some results Definition 1. The structure tensor field F of M is said to be recurrent [8] if there exists a 1-form on T M such that A lightlike hypersurface M of an indefinite Kaehler manifold M is said to be recurrent if it admits a recurrent structure tensor field F . Proof. From the above definition and (3.11), we get Taking the scalar product with N to this and using (3 Replacing Y by ξ to this and using the facts that F ξ = −V and Replacing Y by V to (4.1) and using the last equation, we obtain = 0. Replacing X by ξ and V to (4.1) with = 0 by turns, we obtain As ( , m) = (0, 0), from the last two equations we obtain ( 2 + m 2 )θ(X) = 0. Taking X = ζ to this, we get 2 + m 2 = 0. It follows that = 0 and m = 0. It is a contradiction to ( , m) = (0, 0). Thus we have our theorem.
Definition 2. The structure tensor field F of M is said to be Lie recurrent [8] if there exists a 1-form ϑ on M such that where L X denotes the Lie derivative on M with respect to X.  (1) the structure tensor field F is Lie parallel, (2) the 1-form τ satisfies τ = 0, and Proof. (1) Using the above definition, (2.10), (2.11), (3.2) and (3.11), we get Taking Y = ξ to (4.2) and using (3.4) 1 and the fact that Taking the scalar product with V to (4.3) and using g(F X, V ) = 0, we have Replacing Y by V to (4.2) and using the fact that F V = ξ, we have Applying F to this equation and using (2.11) and (4.4), we obtain (2) Taking X = U to ∇ V X + F ∇ ξ X = 0 and using (2.11) and (3.9), we get Taking the scalar product with N to this equation, we obtain g(A N V, U ) = 0. (4.5) Replacing X by V to (4.2) and using (2.11), (3.3), (3.5) and (3.10), we get Taking the scalar product with U to (4.6) and using (3.5) and (4.5), we have From this equation and (3.8), we see that Replacing X by U to (4.2) and using (2.11), (3.3) (3.8) and (3.9), we get Taking the scalar product with V to (4.9) and using (4.8), we get τ (F Y ) = 0.
(3) As τ = 0, from (4.7) we have B(X, U ) = mθ(U )u(X). Thus B(U, X) = mθ(X). Taking X = U to (3.5) and using (4.11), we get g(A * ξ U, X) = 0. Using this and the fact that S(T M ) is non-degenerate, we have A * ξ U = 0. Replacing X by ξ to (4.3) and using (2.7), (3.7) and the fact that τ = 0, we obtain A * ξ V = 0. On the other hand, replacing Y by U to (4.6) and using (4.10), we get A N V = A * ξ U . Thus we see that A N V = 0. where R is the curvature tensor of the Levi-Civita connection ∇ onM .

Indefinite complex space forms
Denote byR the curvature tensors of the ( , m)-type con-nection∇ onM . By directed calculations from (1.2) and (1.3), we see that  (5.4) respectively. Comparing the tangential and transversal components of the left and right terms of (5.2) and using (5.1), and (5.3), we obtain Definition 4. A screen distribution S(T M ) is called totally geodesic [7] in M if C = 0 on a cooerinate neighborhood U.
Theorem 5.1. Let M be a lightlike hypersurface of an indefinite complex space formM (c) with an ( , m)-type connection such that ζ is tangent to M . If one of the following four conditions is satisfied ; (1) M Lie recurrent, (2) U is parallel with respect to ∇, (3) V is parallal with respect to ∇, Proof.
(3) If V is parallel with respect to ∇, then, taking the scalar product with V and U to (3.10) such that ∇ X V = 0 by turns, we have respectively. From these two equations, we obtain θ(V ) = 0, τ (X) = −mθ(V )η(X).
(5.12) Applying∇ X to (5.12) 1 and using (2.4) and the fact that Using (5.12), the equation (3.10) is reduced to Taking the scalar product with N to this equation and using (3.5), (3.8), (5.12) 1 and the fact that ∇ X V = 0, we obtain Taking P Z = V to (5.7) and using (5.13) and the last two equations, we get Taking X = ξ and Y = U to this equation, we have c = 0.