On complete submanifolds with parallel normalized mean curvature in product spaces

Abstract

A Simons type formula for submanifolds with parallel normalized mean curvature vector field (pnmc submanifolds) in the product spaces $M^{n}(c)\times \mathbb {R}$ , where $M^{n}(c)$ is a space form with constant sectional curvature $c\in \{-1,1\}$ , it is shown. As an application is obtained rigidity results for submanifolds with constant second mean curvature.