  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.
