Research On Some Problems In Geometry Of Submanifolds In Almost Contact Manifolds

Almost contact metric manifolds are important branches of the differential ge-ometry and play an important role in the geometry. In this thesis, we study the some problems in geometry of Submanifolds in almost contact metric manifolds and get many interesting results.In chapter one, we introduce the background and the recent development, give an outline of main results of this dissertation.In chapter two, we study the parallelism of submanifolds in two class of almost contact metric manifolds, and obtain necessary and sufficient conditions under which certain operators on the submanifolds is to be parallel and use this result to obtain conditions under which a submanifolds of two class of almost contact metric manifold is an invariant submanifoldsIn chapter three, we consider an anti-invariant, minimal, pseudo-parallel sub-manifold M of CPn × R and CHn x R and also find a necessary condition for the submanifolds to be totally geodesic.In chapter four, we study invariant submanifolds of Kenmotsu S-manifolds and obtain some necessary and sufficient conditions for invariant submanifolds to be totally geodesic. Firstly, we prove that submanifolds is totally geodesic if and only if the second fundamental form is parallel or the second fundamental form is recurrent. Second, we obtain a necessary condition for a five dimensional invariant submanifold of a Kenmotsu S-manifold to be totally geodesic. Finally, we study invariant submanifolds of Kenmotsu iS-manifolds satisfying Q(h, R)= 0 or Q(S, h) = 0. where R, S are the curvature tensor and Ricci tensor respectively.