Hypersurfaces of Product Spaces with a Canonical Direction

TL;DR

This paper characterizes hypersurfaces in product manifolds where the gradient of the height function is a principal direction, extending previous work from space forms to more general manifolds.

Contribution

It generalizes the classification of hypersurfaces with a principal direction of the height function beyond space forms to arbitrary complete Riemannian manifolds.

Findings

Characterization of hypersurfaces with a principal height direction in product manifolds.

Extension of Tojeiro's results to more general base manifolds.

New geometric conditions for hypersurfaces in product spaces.

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 $\langle \,,\, \rangle.$ 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.