Hypersurfaces of Product Spaces with a Canonical Direction

Abstract

Consider a complete Riemannian manifold $M^n$ and let ${\Sigma }^n$ be an orientable hypersurface of the product manifold $M\times \mathbb{R}$ endowed with its standard product metric $⟨,⟩.$ Let $\nabla \xi$ denote the gradient of the height function $\xi$ of $\Sigma .$ In this note, we characterize the hypersurfaces $\Sigma$ which have $\nabla \xi$ as a principal direction. Our approach is based on the work of R. Tojeiro, who considered the case where $M$ is a constant sectional curvature space form.

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