Translation surfaces of constant curvatures in a simply isotropic space

Abstract

In this paper we describe, up to a congruence, translation surfaces in a simply isotropic space having constant isotropic Gaussian or mean curvature. It turns out that, contrary to the Euclidean case, there exist translation surfaces with constant Gaussian curvature \(K\ne 0\) and translation surfaces with constant mean curvature \(H\ne 0\) that are not cylindrical. Furthermore, we investigate a class of Weingarten translation surfaces.