A Note on Nearly Sasakian Manifolds

A class of nearly Sasakian manifolds is considered in this paper. We discuss the geometric effects of some symmetries on such manifolds and show, under a certain condition, that the class of Ricci semi-symmetric nearly Sasakian manifolds is a subclass of Einstein manifolds. We prove that a Codazzi-type Ricci nearly Sasakian space form is either a Sasakian manifold with a constant f-holomorphic sectional curvature H=1 or a 5-dimensional proper nearly Sasakian manifold with a constant f-holomorphic sectional curvature H>1. We also prove that the spectrum of the operator H2 generated by the nearly Sasakian space form is a set of a simple eigenvalue of 0 and an eigenvalue of multiplicity 4, and we induce that the underlying space form carries a Sasaki-Einstein structure. We show that there exist integrable distributions with totally geodesic leaves on the same manifolds, and we prove that there are no proper nearly Sasakian space forms with constant sectional curvature.