SEMI-INCLINE LIGHTLIKE SUBMANIFOLDS OF INDEFINITE COSYMPLECTIC MANIFOLDS

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper, we investigate the geometry of semi-incline lightlike submanifolds of indefinite Cosymplectic manifolds and acquires portrayal the hypothesis for such manifolds. Next, we give the integrability states of appropriations on semi-incline lightlike submanifolds of indefinite Cosymplectic manifolds. At last, we likewise acquire necessary and sufficient condition for foliation controlled by appropriations to be absolutely geodesic and give a few precedents.


INTRODUCTION
The hypothesis of submanifolds of semi-Riemannian manifolds is intriguing to think about the geometry of lightlike submanifolds due to the way that the crossing point of typical vector pack and the digression group is non-inconsequential. In this way, the examination turns out to be more fascinating and surprisingly unique in relation to the investigation of non-degenerate submanifolds. The geometry of lightlike submanifolds of inconclusive Kaehler manifolds was displayed in a book by Duggal furthermore, Bejancu [7]. B. Y. Chen has introduced the notion some results also fundamentals which are needed for the paper. In section three, we investigate the geometry of semi-incline lightlike submanifolds of indefinite Cosymplectic manifolds.
Section four is devoted to the study of foliations determined by distributions on semi-incline lightlike submanifolds of indefinite Cosymplectic manifolds. In section five, we provide some examples of semi-incline lightlike submanifolds of an indefinite Cosymplectic manifold.

PRELIMINARIES
An odd-dimensional semi-Riemannian manifoldM is said to be an indefinite almost contact metric manifold if there exist structure tensors (φ * ,V, η,ḡ) , where φ * is a (1,1) tensor field, V a vector field, η a 1-form andḡ is the semi-Riemannian metric onM satisfying for all X,Y ∈ Γ(TM), where ε = 1 or −1. It follows that for all X,Y ∈ TM, where TM denotes the Lie algebra of vector fields onM. An indefinite almost contact metric manifoldM is called an indefinite Cosymplectic structure iff [1], A semi-Riemannian manifold endowed with an indefinite Cosymplectic structure is called an indefinite Cosymplectic manifold.
Let (M,ḡ, φ * ,V, η, ) be an ε-almost contact metric manifold. If ε = 1, thenM is said to be a spacelike ε-almost contact metric manifold and if ε = −1, thenM is called a timelike ε-almost contact metric manifold. In this paper, we consider indefinite Cosymplectic manifolds with spacelike characteristic vector field V. Although S(T M) is not unique, it is canonically isomorphic to the factor vector bundle T M|Rad(T M) .We say that a submanifold (M, g, S(T M), S(T M ⊥ )) of M is The Gauss and Weingarten equations are where (∇ X Y, A V X) and (h(X,Y ),∇ t X V ) belong to Γ(T M) and Γ(tr(T M)) respectively.∇ and . L and S are the projection morphism of tr(T M) on ltr(T M) and S(T M ⊥ ) called the lightlike connection and screen transversal connection on M respectively. For any vector field X tangent to M, we put where PX and FX are tangential and transversal parts of φ * X respectively. Now, by using (2.12),(2.14)-(2.16) and metric connection∇, we obtain Denote the projection of T M on S(T M) byP. Then from the decomposition of the tangent bundle of a lightlike submanifold, we have for all X,Y ∈ Γ(T M).
Using above equation , we get Note that in general∇ is not a metric connection. Since∇ is metric connection, by using (2.14), we have

MANIFOLDS
This new class of lightlike submanifolds of an indefinite Cosymplectic manifold includes slant, contact Cauchy-Riemann lightlike submanifolds as its subcases which have been studied in [16], [5]. In this section, we introduce the definition of semi-incline lightlike submanifolds of indefinite Cosymplectic manifolds and some properties. At first, we state the following lemmas for later use:  The proofs of lemma 3.1 and lemma 3.2 follow as in lemma 3.1 and lemma 3.2 respectively of [5], so we omit here them.
As mentioned in the introduction, the purpose of this paper is to define semi-incline lightlike submanifolds of an indefinite Cosymplectic manifold. To define this notion, one needs to consider angle between two vector fields. As we can see from Section one, a lightlike submanifold has two distributions viz. radical and screen. The radical distribution is totally lightlike and therefore, it is not possible to define angle between two vector fields of radical distribution. On the other hand, the screen distribution is non-degenerate. Thus one way to define semi-incline lightlike submanifolds is to choose a Riemannian screen distribution on lightlike submanifolds, for which we use lemma 3.2.
Definition. [17] Let M be a q-lightlike submanifold of an indefinite Cosymplectic manifoldM of index 2q such that 2q < dim(M) with structure vector field tangent to M. Then we say that M is a semi-incline lightlike submanifold ofM if the following conditions are satisfied: (iv). the distribution D 2 is incline with angle θ ( == 0), i.e. for each p ∈ M and each nonzero vector X ∈ (D 2 ) p , the angle θ between φ * X and the vector subspace (D 2 ) p is a non-zero constant, which is independent of the choice of p ∈ M and X ∈ (D 2 ) p , From the above definition, we have the following decomposition In particular, we have for any X ∈ Γ(T M), we get By above, we have ). Where f P 5 X and FP 5 X are the tangential and transversal component of φ * P 5 X.
So M is a semi-incline lightlike submanifold of an indefinite Cosymplectic manifold. Proof. The proof of above Corollary follows as in proof of Corollary 3.2 of [17].

Proof. Let M be a semi-incline lightlike submanifold of an indefinite Cosymplectic manifold
M. Let X,Y ∈ Γ(RadT M). From equation (3.8), we have using (3.11), we obtain now, using (3.12), we get which gives in view of (3.13), we have also, using (3.14), we get which completes the prove of the Theorem.
Proof. Let M be a semi-incline lightlike submanifold of an indefinite Cosymplectic manifold using (3.9), we get which gives now, using (3.12), we obtain which gives in view of (3.13), we have also, using (3.14), we get which completes the prove of the Theorem.
By above, we can see this is a contradiction. Hence M does not have a metric connection.

FOLIATIONS ON SEMI-INCLINE LIGHTLIKE SUBMANIFOLDS OF INDEFINITE COSYMPLECTIC MANIFOLDS
In this section, we proceed to obtain necessary and sufficient conditions for foliations de- for all X,Y ∈ Γ(RadT M) and Z ∈ Γ(S(T M)).
Proof. Let M be a semi-incline lightlike submanifold of an indefinite Cosymplectic manifold M. To prove RadT M defines a totally geodesic foliation it is sufficient to show that∇ X Y ∈ Γ(RadT M), for all X,Y ∈ Γ(RadT M). Since∇is metric connection, using (2.2), (2.6), (2.14) and (3.4), for any X,Y ∈ Γ(RadT M) and ZΓ(S(T M)), we obtain which completes the prove of the Theorem.

EXAMPLES
q with its usual Cosymplectic structure given by where (x i , y i , z) are the cartesian coordinates on R 2m+1 q . Now, we construct some examples of semi-incline lightlike submanifold of an indefinite Cosymplectic manifold.