The Cylinder Theorem in ${\cal H}^2\times R$

TL;DR

This paper proves that in the product space ${ m H}^2 imes m R$, any complete, connected surface with zero Gauss and extrinsic curvatures must be a cylinder, extending the classical Cylinder Theorem to this setting.

Contribution

It establishes a characterization of cylinders in ${ m H}^2 imes m R$ based on curvature conditions, generalizing the classical theorem to this non-Euclidean space.

Findings

Surfaces with zero Gauss and extrinsic curvatures are cylinders in ${ m H}^2 imes m R$.

The result extends the classical Cylinder Theorem to hyperbolic product spaces.

Abstract

We consider cylinders in ${\cal H}^2\times R$ (see definitions in the introduction) and prove that a complete and connected surface in ${\cal H}^2\times R$ with the vanishing of the Gauss and extrinsic curvatures is a cylinder.