Hypersurfaces of Product Spaces with a Canonical Direction

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.