A Class of Weingarten Surfaces in Euclidean 3-Space

The class of biconservative surfaces in Euclidean 3-space 3 are defined in (Caddeo et al., 2012) by the equation A(grad H)=-H grad H for the mean curvature function H and the Weingarten operator A. In this paper, we consider the more general case that surfaces in 3 satisfying A(grad H)=kH grad H for some constant k are called generalized bi-conservative surfaces. We show that this class of surfaces are linear Weingarten surfaces. We also give a complete classification of generalized bi-conservative surfaces in 3