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CONTACT CR-SUBMANIFOLDS OF A NEARLY SASAKIAN MANIFOLD

0. Introduction Among all submanifolds of Kahlerian manifolds, CR-submanifolds have been intensively studied from different points of view. Since A. Bejancu [1] introduced the concept, several important results have been obtained, some of them being brought together in [2]. The purpose of the present paper is to consider and study the concept of contact CR-submanifold in case of a nearly Sasakian manifold. Example of a nearly Sasakian manifold see [4], p. 90. In the first section we present some general formulae and basic results from theory of submanifolds in order to use them in the next sections. In the second section we obtain necessary and sufficient conditions for integrability of distributions defined on a contact CR-submanifold of a nearly Sasakian manifold. In the third section we are dealing with totally contact umbilical CR-submanifolds of a nearly Sasakian manifold M. More precisely, we prove that a totally contact umbilical CR-submanifold of M is totally contact geodesic provided the dimension of anti-invariant distribution is greater than 1 or the invariant distribution is autoparallel. Finally, in the fourth section we find theorems of decomposition for totally contact geodesic CR-submanifolds of M.