SCHWEITZER 1983 Problem 11

Let $M^n \subset \mathbb{R}^{n+1}$ be a complete, connected hypersurface embedded into the Euclidean space. Show that $M^n$ as a Riemannian manifold decomposes to a nontrivial global metric direct product if and only if it is a real cylinder, that is, $M^n$ can be decomposed to a direct product of the form $M^n=M^k \times \mathbb{R}^{n-k} \;(k<n)$ as well, where $M^k$ is a hypersurface in some $(k+1)$ -dimensional subspace $E^{k+1} \subset \mathbb{R}^{n+1} , \mathbb{R}^{n-k}$ is the orthogonal complement of $E^{k+1}$ .

Z. Szabo