PSEUDO-SLANT SUBMANIFOLD IN KENMOTSU SPACE FORMS

In this paper, we study geometry of the pseudo-slant submanifold of a Kenmotsu space form. Necessary and sufficient conditions are given for a submanifold to be a pseudo-slant submanifold in Kenmotsu manifolds. Finally, we give some results for totally umbilical pseudo-slant submanifold in a Kenmotsu manifold and Kenmotsu space form.


INTRODUCTION
The differential geometry of slant submanifolds has shown an incresing development since B-Y. Chen [9,10] defined slant submanifolds in complex manifolds as a natural generalization of both the holomorphic and totally real submanifolds. Many research articles have been appeared on the existence of these submanifolds in different knows spaces. The slant submanifols of an almost contact metric manifolds were defined and studied by A. Lotta [14]. After, these submanifolds were studied by J. L Cabrerizo et. al [6] of Sasakian manifolds. Recently, in [3]. M. Atçeken studied slant and pseudoslant submanifold in () n LCS -manifolds.
The notion of semi-slant submanifolds of an almost Hermitian manifold was introduced by N. Papagiuc [15]. Cabrerizo et al. studied and characterized slant submanifolds of K-contact and Sasakian manifolds and have given sereval examples of such submanifolds. Recently, Carizzo [7,6] defined and studied bi-slant immersions in almost Hermityen manifolds and simultaneously gave the notion of pseudo-slant submanifold in almost Hermityen manifolds. The contact version of pseudo-slant submanifolds has been defined and studied by V. A. Khan and M. A Khan [12]. The present paper is organized as follows.
In this paper, we study pseudo-slant submanifolds of a Kenmotsu manifold. In section 2, we review basic formulas and definitions for a Kenmotsu manifold and their submanifolds. In section 3, we recall the definition and some basic results of a pseudo-slant submanifold of almost contact metric manifold. In section 4, we will give same results for totally umbilical pseudo-slant submanifold in a Kenmotsu manifold and Kenmotsu space form ( ).

Mc  2. PRELIMINARIES
In this section, we give some notations used throughout this paper. We recall some necessary fact and formulas from the theory of Kenmotsu manifolds and their submanifols.

Let
M  be a (2 1) m  dimensional almost contact metric manifold together with a metric tensor g a tensor field  of type (1,1) , a vector field  and a 1- and for any vector fields , XY on M  If in addition to above relations and the covariant derivative The normal and tangent components of (12) are, respectively, given by and

PSEUDO-SLANT SUBMANIFOLD OF KENMOTSU MANIFOLD
In this section, we will study pseudo-slant submanifolds in a Kenmotsu manifold, give some characterization and submanifold is characterized.
Furthermore, the covariant derivatives of the tensor field ,, T N t and n are, respectively, defined by and ( ) .
(26) J a n u a r y 1 6 , 2 0 1 6 Furthermore, for any , These Show that T and n are also skew-symmetric tensor fields. Moreover, for any () which gives the relation between N and . t Taking into account (6) and (17), we have By using (6) and (12), the Riemannian curvature tensor From (6) and (12), for a submanifold, Codazzi equation is given by By an easy computation, we obtain the following formulas and Similarly, for any and Since M is tangent to ,  making use of (5) and (9) we obtain ).

X TM 
In contact geometry, A. Lotta introduced slant immersions as follows [14]. J a n u a r y 1 6 , 2 0 1 6 The distribution D  is a slant, that is, the slant angle between of D  and () D   is a constant.   (4) and (9), we have .
Thus we have This equivalent to J a n u a r y 1 6 , 2 0 1 6 2 sec . Proof. If the shape operator V A of M is parallel, then by using (5) and (14), we have Here, choosing Y   and taking into account (14) and (35), we obtain ( ( ) ) 0.   This proves our assertion.
This proves our assertion.

Theorem 3. 8. Let
M be a proper pseudo-slant submanifold of a Kenmotsu manifold . M  The derivation of T is skew-symmetric, that is J a n u a r y 1 6 , 2 0 1 6 This complete of the proof. .
This proves our assertion.   (1) and (35), we obtain J a n u a r y 1 6 , 2 0 1 6 On the other hand, also by using (9) .