Noninvariant Hypersurfaces of a Nearly Trans-Sasakian Manifolds

The present paper focuses on the study of noninvariant hypersurfaces of a nearly trans-Sasakian manifold equipped with -structure. Initially some properties of this structure have been discussed. Further, the second fundamental forms of noninvariant hypersurfaces of nearly trans-Sasakian manifolds and nearly cosymplectic manifolds with -structure have been calculated provided is parallel. In addition, the eigenvalues of have been found and proved that a noninvariant hypersurface with -structure of nearly cosymplectic manifold with contact structure becomes totally geodesic. Finally the paper has been concluded by investigating the necessary condition for totally geodesic or totally umbilical noninvariant hypersurface with -structure of a nearly trans-Sasakian manifold. 1. Introduction A hypersurface such that transform of a tangent vector of the hypersurface by the structure tensor field defining the almost contact structure is never tangent to the hypersurface was studied by Goldberg and Yano in 1970 [1]. Yano and Okumura [2] introduced -structure and termed it as a noninvariant hypersurface of almost contact metric manifold. In this paper, they showed that there always exists a -structure and gave the results that there does not exist an invariant hypersurface of a contact manifold. As a consequence, an invariant hypersurface of (almost) cosymplectic manifold is (almost) K？hler manifold. Further, they proved that there is no noninvariant hypersurfaces of a Sasakian manifold. In 1990, Chen [3] introduced slant submanifold which is a generalization of invariant and anti-invariant submanifold of almost Hermite manifold. Recently, Prasad [4] studied the noninvariant hypersurface of trans-Sasakian manifold. In this paper, we study noninvariant hypersurface to the setting of nearly trans-Sasakian manifold. 2. Preliminaries Let be an almost contact metric manifold with almost contact metric structure ; that is, is tensor field, is -form, and is a compatible Riemannian metric such that for all . An almost contact metric manifold is called a Sasakian manifold if there exists a killing vector field of unit length on so that the tensor field of type , defined by , satisfies the condition for any pair of vector fields and on . The notion of trans-Sasakian structure is given by Oubina [5]. In 1990, Blair and Oubina [6] find the condition for all vector fields , on and where and are smooth functions. In this case we say that the trans-Sasakian structure is of type . Gherghe introduced a nearly trans-Sasakian structure of type . An almost contact metric manifold