Complete surfaces with zero curvatures in conformally flat spaces

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