Hemi-slant submanifolds of cosymplectic manifolds

In this paper we study the hemi-slant submanifolds of cosymplectic manifolds. Necessary and sufficient conditions for distributions to be integrable are worked out. Some important results are obtained in this direction.


Introduction
In 1990, B. Y. Chen introduced the notion of slant submanifold, which generalizes holomorphic and totally real submanifolds [1]. After that many research articles have been published by different geometers in this direction for different ambient spaces.
A. Lotta introduced the notion of slant immersions of a Riemannian manifolds into an almost contact metric manifolds [3]. After, these submanifolds were studied by J. L.Cabrerizo et. al in the setting of Sasakian manifolds [7]. In [8] Papaghiuc defines the semi-slant submanifolds as a generilization of slant submanifolds . Bislant submanifolds of an almost Hermitian manifold were introduced as natural generalization of semi-slant submanifolds by Carriazo [2]. One of the classes of bi-slant submanifolds is that of anti-slant submanifolds which are studied by A. Carriazo [2] but the name anti-slant seems to refer that it has no slant factor, so B. Sahin [4] give the name of hemi-slant submanifolds instead of anti-slant submanifolds. In [2] V. A. Khan and M. A. Khan studied the hemi-slant submanifolds of sasakian manifolds.
In this paper we study the hemi-slant submanifolds of cosymplectic manifolds. In section 2, we collect the basic formulae and definitions for a cosymplectic manifolds and their submanifolds for ready references. In section 3, we study the hemi-slant submanifolds of cosymplectic manifolds. We obtain the integrability conditions of the distributions which are involved in the definition.

Preliminaries
Let N be a (2m + 1)-dimensional almost contact metric manifold with structure (φ , ξ , η, g) where φ is a tensor field of type (1, 1), ξ a vector field, η is a one form and g is the Riemannian metric on N. Then they satisfy These conditions also imply that and for all vector fields X ,Y in T N. Where T N denotes the Lie algebra of vector fields on N. A normal almost contact metric manifold is called a cosymplectic manifold if where ∇ denotes the Levi-Civita connection of (N, g). Throughout, we denote by N a cosymplectic manifold, M a submanifold of N and ξ a structure vector field tangent to M. A and h denotes the shape operator and second fundamental form of immersion of M into N. If ∇ is the induced connection on M, the Gauss and Weingarten formulae of M into N are then given respectively by for all vector fields X ,Y on T M and V on T ⊥ M, where ∇ ⊥ denotes the connection on the normal bundle T ⊥ M of M. The shape operator and the second fundamental form are related by The mean curvature vector is defined by where n is the dimension of M and {e 1 , e 2 , ..., e n } is the local orthonormal frame of M.
For any X ∈ T M, we can write where T X and FX are the tangential and normal components of φ X respectively. Similarly for any V ∈ T ⊥ M, we have where tV and fV are the tangential and normal components of φV respectively. The covariant derivative of the tensor fields T , F, t and f are defined by the following and for all X , Y ∈ T M and V ∈ T ⊥ M.
A submanifold M of an almost contact metric manifold N is said to be totally umbilical if where H is the mean curvature vector. If h(X ,Y ) = 0 for any X ,Y ∈ T M, then M is said to be totally geodesic and if H = 0, then M is said to be a minimal submanifold.
A. Lotta has introduced the notion of slant immersion of a Riemannian manifold into an almost contact metric manifold [3] and slant submanifolds in Sasakian manifolds have been studied by J.L. Cabrerizo et al. [7].
For any x ∈ M and X ∈ T x M if the vectors X and ξ are linearly independent, the angle denoted by θ (X ) ∈ [0, π 2 ] between φ X and T x M is well defined. If θ (X ) does not depend on the choice of x ∈ M and X ∈ T x M, we say that M is slant in N . The constant angle θ is then called the slant angle of M in N . The anti-invariant submanifold of an almost contact metric manifold is a slant submanifold with slant angle θ = π 2 and an invariant submanifold is a slant submanifold with the slant angle θ = 0. If the slant angle θ of M is different from 0 and π 2 , then it is called a proper slant submanifold. If M is a slant submanifold of an almost contact manifold then the tangent bundle T M of M is decomposed as where ξ denotes the distribution spanned by the structure vector field ξ and D is a complementary distribution of ξ in T M, known as the slant distribution. For a proper slant submanifold M of an almost contact manifold N with a slant angle θ , Lotta [3] proved that Cabrerizo et al. [7] extended the above result into a characterization for a slant submanifold in a contact metric manifold. In fact, they obtained the following crucial theorems.
furthermore, in such case, if θ is the slant angle of M, then λ = cos 2 θ .
Theorem 2.2. [7] Let M be a slant submanifold of an almost contact metric manifold M with slant angle θ . Then for any X ,Y ∈ T M, we have

Hemi-slant submanifolds of cosymplectic manifolds
In the present section, we introduce the hemi-slant submanifolds and obtain the necessary and sufficient conditions for the distributions of hemi-slant submanifolds of cosymplectic manifolds to be integrable.
it is clear from above that CR-submanifolds and slant submanifolds are hemi-slant submanifolds with slant angle θ = π 2 and D θ = 0, respectively.
In the rest of this paper, we use M a hemi-slant submanifold of almost contact metric manifold N.
On the other hand, if we denote the dimensions of the distributions D ⊥ and D θ by m 1 and m 2 respectively, then we have the following cases: (1) If m 2 = 0, then M is anti-invariant submanifold, Suppose M to be a hemi-slant submanifold of an almost contact metric manifold N, then for any X ∈ T M, we put where P 1 and P 2 are projection maps on the distribution D ⊥ and D θ . Now operating φ on both sides of (16), we arrive at Operating φ on both sides, we get It is easy to see on comparing that If we denote the orthogonal complement of φ T M in T ⊥ M by µ, then the normal bundle T ⊥ M can be decomposed as As N(D ⊥ ) and N(D θ ) are orthogonal distributions on F . g(Z,W) = 0 for each Z ∈ D ⊥ and W ∈ D θ . Thus, by (1), (3) and (9), we have which shows that the distributions F(D ⊥ ) and F(D θ ) are mutually perpendicular. In fact, the decomposition (17) is an orthogonal direct decomposition.

Lemma 3.1. Let M be a hemi-slant submanifolds of a cosymplectic manifold N. Then we have
for all X ,Y ∈ T M.

Lemma 3.2. Let M be a hemi-slant submanifolds of a cosymplectic manifold N. Then we have
for all X ∈ T M and V ∈ T ⊥ M.

Lemma 3.3. Let M be a hemi-slant submanifolds of a cosymplectic manifold N, then
Proof. We know that for ξ ∈ T M, we have From (4), it follows that Thus result follows directly from the above equation.

Theorem 3.4. Let M be a hemi-slant submanifold of a cosymplectic manifold N, Then
Proof. Using (7), we have Whereby using (4), we have By use of h(X ,Y ) = h(Y, X ), we arrive at Hence the result.

Theorem 3.5. Let M be a submanifold of a cosymplectic manifold N. Then the distribution D ⊥ is integrable if and only if
for any Z,W in D ⊥ .

Theorem 3.7. Let M be a hemi-slant submanifold of a cosymplectic manifold N. Then the anti-invariant distribution D ⊥ is integrable if and only if
for any X ,Y ∈ D ⊥ .
Proof. For X ,Y ∈ D ⊥ , we have whereby we have Comparing the tangential components we have, from which we conclude that for any X ,Y ∈ D θ .
Proof. For Z ∈ D ⊥ and X ,Y ∈ D θ , we have Using (1), (2) and (4), we get whereby use of (5), (6), we obtain As φ X ∈ φ (D ⊥ ) and F(D θ ) and F(D ⊥ are orthogonal to each other in T ⊥ M, thus we conclude the result. Proof. Since (∇ Z φ )W = 0. From(4), we have Using (5), (6) and (9), we obtain For X ∈ D θ , we have Therefore, we have The leaves of D ⊥ are totally geodesic in M, if for Z,W ∈ D ⊥ , ∇ Z W ∈ D ⊥ . Therefore from (27), we get the result.
for X ,Y ∈ D θ and Z ∈ D ⊥ .
The above equation has a solutio if either dim(D ⊥ ) = 1 or H ∈ µ or D ⊥ = 0, this completes the proof.