Geometric meaning of Sasakian space forms from the viewpoint of submanifold theory

Abstract

We show that M^2n-1 is a real hypersurface all of whose geodesics orthogonal to the characteristic vector ξ are mapped to circles of the same curvature 1 in an n-dimensional nonflat complex space form $\widetilde{M}_n$(c) (= CPⁿ(c) or CHⁿ(c)) if and only if M is a Sasakian manifold with respect to the almost contact metric structure from the ambient space $\widetilde{M}_n$(c). Moreover, this Sasakian manifold M is a Sasakian space form of constant φ-sectional curvature c + 1 for each c (≠0).