Invariant submanifolds of sasakian space forms

Abstract

In the present study, we consider isometric immersions f : M -> M (c) of (2n + 1)-dimensional invariant submanifold M2n+ 1 of (2m+ 1)dimensional Sasakian space form M (2m+ 1) of constant phi-sectional curvature c. We have shown that if f satisfies the curvature condition (R) over bar (X, Y) sigma = Q(g, sigma) then either M2n+ 1 is totally geodesic, or parallel to sigma parallel to(2) = 1/3 (2c+ n(c+ 1)), or parallel to sigma parallel to(2) (x) > 1/3 (2c + n(c + 1) at some point x of M2n+ 1. We also prove that R(X, Y).sigma = 1/2n Q(S, sigma) then either M2n+ 1 is totally geodesic, or parallel to sigma parallel to(2) = - 2/3 (1/2n T - 1/2 (n + 2)(c + 3) + 3), or parallel to sigma parallel to(2) (x) > - 2/3 (1/2n T (x) 1/2 (n + 2)(c + 3) + 3) at some point x of M2n+1.