On linear weingarten surfaces

In this paper, we study surfaces in Euclidean 3-space that satisfy a Weingarten condition of linear type as k(1) = mk(2) + n, where m and n are real numbers and k(1) and k(2) denote the principal curvatures at each point of the surface. We investigate the existence of such surfaces parametrized by a uniparametric family of circles. We prove that the only surfaces that exist are surfaces of revolution and the classical examples of minimal surfaces discovered by Riemann. The latter situation only occurs in the case ( m, n) = (-1, 0).