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

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 $a{K}_i+b{K}_e=c$ holds for some real constants $a\ne 0$, $b>0$ and $c$. The main result of this article state that when such a surface is a topological sphere it is rotational.