Classes of Weingarten Surfaces in S^2xR

Abstract

In this work we study surfaces in radial conformally flat spaces. We characterize surfaces of rotation with constant Gaussian and Extrinsic curvature in these radial 3-spaces. We prove that all the spheres in the conformal 3-space have constant Gaussian curvature $K=1$ if, and only if, the conformal factor is special. In this special case we study geometric properties of this ambient 3-space, and as an application we prove that it is isometric to the space ${\mathbb{S}}^2\times \mathbb{R}$, so we consider it as the {\em Radial Model} of ${\mathbb{S}}^2\times \mathbb{R}$. We obtain two classes of Weingarten surfaces in the {\em Radial Model}, which satisfy ${\tilde{K}}_E+{\tilde{H}}^2-\tilde{K}=0$ and $2{\tilde{K}}_E-\tilde{K}=0$, where $\tilde{K}$ is the Gaussian curvature, $\tilde{H}$ is the mean curvature and ${\tilde{K}}_E$ is the extrinsic curvature. Moreover, by using the relations between the curvatures of the {\em Radial Model} and the curvatures with respect to the euclidean metric ([CPS]), we prove that first class the Weingarten surfaces in {\em Radial Model} corresponds, up to isometries, to the minimal surfaces in ${\mathbb{R}}^3$, and second class corresponds to EDSGHW - surfaces in Euclidean space ${\mathbb{R}}^3$(\cite{DC}). Consequently these two classes of surfaces have a Weierstrass type representation depending on two holomorphic functions.