On complete submanifolds with parallel mean curvature in product spaces

TL;DR

This paper derives a Simons type formula for submanifolds with parallel mean curvature in product spaces and uses it to characterize specific submanifolds, advancing understanding in differential geometry.

Contribution

It introduces a new Simons type formula for submanifolds with parallel mean curvature in product spaces and applies it to characterize these submanifolds.

Findings

Derived a Simons type formula for the setting

Characterized certain submanifolds with parallel mean curvature

Enhanced understanding of submanifold geometry in product spaces

Abstract

We prove a Simons type formula for submanifolds with parallel mean curvature vector field in product spaces of type $M^n(c)\times\mathbb{R}$, where $M^n(c)$ is a space form with constant sectional curvature $c$, and then we use it to characterize some of these submanifolds.