C-totally real warped product submanifolds

Abstract

We obtain a basic inequality involving the Laplacian of the warping func- tion and the squared mean curvature of any warped product isometrically immersed in a Riemannian manifold (cf. Theorem 2.2). Applying this general theory, we obtain ba- sic inequalities involving the Laplacian of the warping function and the squared mean curvature of C-totally real warped product submanifolds of (κ, µ)-space forms, Sasakian space forms and non-Sasakian (κ, µ)-manifolds. Then we obtain obstructions to the exis- tence of minimal isometric immersions of C-totally real warped product submanifolds in (κ, µ)-space forms, non-Sasakian (κ, µ)-manifolds and Sasakian space forms. In the last, we obtain an example of a C-totally real warped product submanifold of a non-Sasakian (κ, µ)-manifold, which satisfies the equality case of the basic inequality. Mathematics Subject Classication 2010: 53C40, 53C25.