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A class of Weingarten surfaces in Euclidean 3-space. (English) Zbl 1293.53007
Summary: The class of biconservative surfaces in Euclidean 3-space \(\mathbb E^3\) was defined by R. Caddeo et al. in [Ann. Mat. Pura Appl. (4) 193, No. 2, 529–550 (2014; Zbl 1294.53006)] by the equation \(A(\mathrm{grad}H)=-H\mathrm{grad}H\) for the mean curvature function \(H\) and the Weingarten operator \(A\). In this paper, we consider the more general case that surfaces in \(\mathbb E^3\) satisfying \(A(\mathrm{grad}H)=kH\mathrm{grad}H\) 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 \(\mathbb E^3\).

53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
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