Weingarten type surfaces in $\mathbb{H}^2\times\mathbb{R}$ and $\mathbb{S}^2\times\mathbb{R}$

TL;DR

This paper classifies complete, positively curved surfaces in hyperbolic and spherical product spaces satisfying a linear relation between intrinsic and extrinsic curvatures, showing that topological spheres with these properties are rotational.

Contribution

It proves that topological spheres with a specific linear curvature relation in these product spaces are necessarily rotational, extending understanding of Weingarten surfaces in these geometries.

Findings

Surfaces with the given curvature relation are rotational spheres when topologically spherical.

The study extends classification results of Weingarten surfaces in product spaces.

Provides conditions under which such surfaces are rotational in hyperbolic and spherical contexts.

Abstract

In this article, we study complete surfaces $\Sigma$, isometrically immersed in the product space $\mathbb{H}^2\times\mathbb{R}$ or $\mathbb{S}^2\times\mathbb{R}$ having positive extrinsic curvature $K_e$. Let $K_i$ denote the intrinsic curvature of $\Sigma$. Assume that the equation $aK_i+bK_e=c$ holds for some real constants $a\neq0$, $b>0$ and $c$. The main result of this article state that when such a surface is a topological sphere it is rotational.