Complete surfaces with zero curvatures in conformally flat spaces

Abstract

In this paper, we introduce a family of Riemannian manifolds E 3 F , which are Euclidean space R 3 endowed with conformally flat metrics. We characterize rotational surfaces with constant Gaussian and extrinsic curvatures in E 3 F . We present a particular space that is isometric to H 2 × S 1 , and, using a special parametrization, we construct a family of complete rotational surfaces with zero Gaussian and extrinsic curvatures in H 2 × S 1 . We have built a special space that is a warped product H 2 × f R, which is a complete space foliated by complete surfaces of constant Gaussian curvature −1; this shows that the hyperbolic space H 2 is isometrically immersed into the space H 2 × f R, and this space is isometric to neither H 3 nor H 2 × R, showing that in the ambient space, H 2 × f R Hilbert theorem does not hold.