Classes of Weingarten Surfaces in S^2xR

Abstract

In this work we study surfaces in radial conformally flat 3-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 S 2 × R, so we consider it as the Radial Model of S 2 × R. We obtain two classes of Weingarten surfaces in the Radial Model, which satisfy KE + H2 − K = 0 and 2 KE − K = 0, where K is the Gaussian curvature, H is the mean curvature and KE is the extrinsic curvature. Moreover, by using the relations between the curvatures of the Radial Model and the curvatures with respect to the euclidean metric ([CPS]), we prove that first class the Weingarten surfaces in Radial Model corresponds, up to isometries, to the minimal surfaces in R 3 , and second class corresponds to EDSGHW -surfaces in Euclidean space R 3 ([11]). Consequently these two classes of surfaces have a Weierstrass type representation depending on two holomorphic functions.