QUASI HEMI-SLANT SUBMANIFOLDS OF SASAKIAN MANIFOLDS

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper, we define and study quasi hemi-slant submanifolds as a generalization of slant submanifolds, semi-slant submanifolds and hemi-slant submanifolds of contact metric manifolds [ In particular, for a Sasakian manifold]. Further, we obtain necessary and sufficient conditions for integrability of distributions which are involved in the definition of quasi hemi-slant submanifolds of Sasakian manifolds. After it, we investigate the necessary and sufficient condition for quasi hemi-slant submanifolds of Sasakian manifolds to be totally geodesic. Finally, we obtain the necessary and sufficient condition for a quasi hemi-slant submanifold to be local product Riemannian manifold and also give an example of such submanifolds.


INTRODUCTION
Presently, the theory of submanifolds has gained prominence in computer design, image processing, economic modeling as well as in mathematical physics and in mechanics. These extensive applications of this topic makes it an active and interesting field of research for geometers. The notion of geometry of submanifolds begin with the idea of the extrinsic geometry of surface and it is developed for ambient space in the course of time. [15], [21], [23].
Motivated from above studies, we study quasi hemi-slant submanifolds of Sasakian manifolds as a generalization of semi-slant submanifolds and hemi-slant submanifolds. The present paper is organized as follows : In section 2, we mention basic definitions and some properties of Sasakian manifolds. In section 3, we define quasi-hemi-slant submanifolds and some basic properties of the submanifolds. Section 4 deals with necessary and sufficient conditions for integrability of distributions which are involved in the definition. We also find necessary and sufficient condition for the submanifolds to be totally geodesic. In the last section, we provide an example of such submanifolds.
Since any almost contact manifold ( M, φ , ξ , η) admits a Riemannian metric g such that for any vector fields X,Y ∈ Γ(T M), where Γ(T M) represents the Lie algebra of vector fields on M. The manifold M together with the structure (φ , ξ , η, g) is called an almost contact metric manifold.
The immediate consequence of (2.2), we have for all vector fields X,Y ∈ Γ(T M).
An almost contact structure (φ , ξ , η) is said to be normal [3] if the almost complex structure J on the product manifold M × R is given by where J 2 = −I and f is the differentiable function on M ×R. J has no torsion i.e., J is integrable.
The condition for normality in terms of φ , ξ and η is the Nijenhuis tensor of φ . A Sasakian manifold [6] is normal contact metric manifold and every Sasakian manifold is K−contact manifold. It is easy to show that an almost contact metric manifold is a Sasakian manifold if and only if for all vector fields X,Y ∈ Γ(T M).
Let M be a Riemannian manifold isometrically immersed in M and the induced Riemannian metric on M is denoted by the same symbol g throughout this paper. Let A and h denote the shape operator and second fundamental form, respectively, of immersion of M into M. The Gauss and Weingarten formulas of M into M are given by [8] (2.6) for any vector fields X,Y ∈ Γ(T M) and V ∈ Γ(T ⊥ M), where ∇ is the induced connection on M and ∇ ⊥ represents the connection on the normal bundle T ⊥ M of M and A V is the shape operator of M with respect to normal vector V ∈ Γ(T ⊥ M). Moreover, A V and the second fundamental for any vector fields X,Y ∈ Γ(T M) and V ∈ Γ(T ⊥ M).
The mean curvature vector is defined by where n denotes the dimension of submanifold M and {e 1 , e 2 , ...., e n } is the orthonormal basis of tangent space of M.
For any X ∈ Γ(T M), we can write where T X and NX are the tangential and normal components of φ X on M respectively. Similarly for any V ∈ T ⊥ M, we have where tV and nV are the tangential and normal components of φV on M respectively.
A submanifold M of Sasakian manifold M is said to be totally umbilical if where H is the mean curvature vector. If h(X,Y ) = 0 for all X,Y ∈ Γ(T M), then M is said to be totally geodesic [9] and if H = 0, then M is said to be a minimal submanifold.
The covariant derivative of tangential and normal components of (2.10) and (2.11) are given as for any X,Y ∈ Γ(T M) and V ∈ Γ(T ⊥ M).
We note that on a slant submanifold M if θ = 0, then it is an invariant submanifold and if θ = π 2 , then it is an anti-invariant submanifold. This means slant submanifold is a generalization of invariant and anti-invariant submanifolds.
where D is invariant and D θ is slant with slant angle θ . In this case, the angle θ is called semi-slant angle.
Definition 6. A submanifold M of an almost contact metric manifold M is said to be hemi-slant [20], if there exist two orthogonal complementary distributions D θ and D ⊥ on M such that where D θ is slant with slant angle θ and D ⊥ is anti-invariant. In this case, the angle θ is called hemi-slant angle.

QUASI HEMI-SLANT SUBMANIFOLDS OF SASAKIAN MANIFOLDS
In the present section of the paper, we introduce the definition of quasi hemi-slant submanifolds of Sasakian manifolds and obtain some related results for later use.
(iii) The distribution D θ is slant with constant angle θ . The angle θ is called slant angle.
In this case, we call θ the quasi hemi-slant angle of M. Suppose the dimension of distributions D, D θ and D ⊥ are n 1 , n 2 and n 3 respectively. Then we easily see the following particular cases: We say that the quasi hemi- This means quasi hemi-slant submanifold is a generalization of invariant, anti-invariant, semiinvariant, slant, hemi-slant, semi-slant submanifolds and also they are the examples of quasi hemi-slant submanifolds.
Let M be a quasi hemi-slant submanifold of a Sasakian manifold M. We denote the projections of X ∈ Γ(T M) on the distributions D, D θ and D ⊥ by P, Q and R respectively. Then we can write for any X ∈ Γ(T M) where T X and NX are tangential and normal components of φ X on M.
Using Since φ D = D and φ D ⊥ ⊆ T ⊥ M, we have NPX = 0 and T RX = 0. Therefore, we get Then for any X ∈ Γ(T M), it is easy to see that T X = T PX + T QX and NX = NQX + NRX.
Thus from (3.5) , we have the following decomposition where '⊕' denotes orthogonal direct sum. Since where µ is the orthogonal complement of ND θ ⊕ ND ⊥ in Γ(T ⊥ M) and it is invariant with respect to φ .
For any non-zero vector field V ∈ Γ(T ⊥ M), we have where I is the identity operator.
Proof. The proof is the same as in [16].

Proposition 5. Let M be a quasi-hemi-slant submanifold of a Sasakian manifold M, then
h(X, ξ ) = −NX and ∇ X ξ = −T X for all X ∈ Γ (T M) .

INTEGRABILITY OF DISTRIBUTIONS AND DECOMPOSITION THEOREMS
We now examine the integrability conditions for invariant distribution D, slant distribution    for any X,Y ∈ Γ(D⊕ < ξ >) and Z ∈ Γ(D θ ⊕ D ⊥ ).
From above theorem we have the following sufficient conditions for the slant distribution to be integrable: for any Z,W ∈ Γ(D θ ⊕ < ξ >), then the distribution D θ ⊕ < ξ > is integrable. and Lemma 2, we obtain Hence the proof.
We now obtain a necessary and sufficient condition for a quasi-hemi-slant submanifold to be totally geodesic. for any X,Y ∈ Γ(T M) and U ∈ Γ(T ⊥ M).

EXAMPLE
Now, we construct an example of quasi hami-slant submanifold of a Sasakian manifold.
By direct computation, it is easy to check that the F defines 7-dimensional submanifold M in Sasakian manifold in R 11 (as defined above). The set {Z 1 , Z 2 , Z 3 , Z 4 , Z 5 , Z 6 , Z 7 } is a base field on M, where We define the distributions D =< Z 1 , Z 2 >, D θ =< Z 3 , Z 4 > and D ⊥ =< Z 5 , Z 6 > . Then, it is clear that T M = D ⊕ D θ ⊕ D ⊥ ⊕ < ξ > and it can be easily prove that D θ is a slant distribution with angle θ = α, where α is a non zero constant and D ⊥ is anti-invariant distribution.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.