On the Cylinder Theorem in $M^2\times \mathbb{R}^n $

TL;DR

This paper characterizes cylinders in the product space of a surface with constant Gaussian curvature and Euclidean space as hypersurfaces with zero intrinsic and extrinsic curvatures.

Contribution

It provides a geometric characterization of cylinders in $M^2\times \mathbb{R}^n$ based on curvature conditions, extending the classical cylinder theorem.

Findings

Cylinders are hypersurfaces with zero intrinsic and extrinsic curvatures in $M^2\times \mathbb{R}^n$.

The characterization holds for surfaces with Gaussian curvature less than or greater than zero.

The result generalizes the classical cylinder theorem to product spaces with curvature conditions.

Abstract

Consider a surface $M^2$ with Gaussian curvature either $< 0$ or $> 0$. We prove that in $M^2\times \mathbb{R}^n$ cylinders are characterized as the hypersurfaces with both the extrinsic and intrinsic curvatures equal to zero.