Submanifolds of Sasakian Manifolds with Certain Parallel Operators

We study submanifolds of Sasakian manifolds and obtain a condition under which certain naturally defined symmetric tensor field on the submanifold is to be parallel and use this result to obtain conditions under which a submanifold of the Sasakian manifold is an invariant submanifold. 1. Introduction Differential geometry of submanifolds of Sasakian manifold is an interesting branch and has been subject of investigations of many mathematicians. There are four important types of submanifolds of a Sasakian manifold, namely, invariant submanifolds, anti-invariant submanifolds, CR submanifolds, and slant submanifolds. All these types of submanifolds have been studied quite extensively; for invariant and anti-invariant submanifolds one can refer to [1–5] and for slant submanifolds one can refer to [6–8] and references therein. A submanifold of a Sasakian manifold is said to be a CR-submanifold if there is a pair of orthogonal complementary distributions and defined on of which is invariant under and is totally real distribution so that is subbundle of the normal bundle. CR submanifolds of a Sasakian manifold have been studied in [9–13] getting various geometric properties of the CR-submanifold. A submanifold of a Sasakian manifold naturally carries four operators, , , , and , defined on this submanifold. In [14], it has been shown that a submanifold of a Sasakian manifold with parallel is essentially a CR-submanifold. In this paper, we are interested in finding conditions under which a submanifold of a Sasakian manifold with parallel operator is an invariant submanifold. First we show the existence of a symmetric tensor field on any submanifold of a Sasakian manifold and study its properties and use these properties to obtain conditions under which a submanifold is an invariant submanifold (cf. Theorem 5). 2. Preliminaries Let be a -dimensional almost contact metric manifold. Then the structure tensor satisfies (cf. [15]) where and are smooth vectors fields on . An almost contact metric manifold is said to be a Sasakian manifold if it satisfies for smooth vector fields and , where is the Riemannian connection on . Let be an -dimensional submanifold of the Sasakian manifold . Denote by the same the Riemannian metric induced on the submanifold and by the space of smooth sections of the normal bundle of . Then we define operators , , , and as follows: where is the Lie algebra of smooth vector fields on and ; (resp., ) denotes the tangent part (resp., normal part) of ; and (resp., ) denotes the tangent part (resp., normal part) of . Also for the structure vector