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On linear Weingarten surfaces. (English) Zbl 1151.53005
In the present paper the author studies surfaces in Euclidean three-space which fulfil two properties. One the one hand a linear Weingarten condition is satisfied that means \(\kappa_1=m\kappa_2+n\), where \(\kappa_1, \kappa_2\) are the main curvatures and \(m,n\) are constant real numbers (\(m\neq0\)) . On the other hand the surfaces are supposed to carry a smooth one parameter family of (pieces of) circles. Surfaces fulfilling these two properties are called cyclic LW-surfaces. In the special case of rotational LW-surfaces there are many investigations (see for instance H. Hopf [Math. Nachr. 4, 232–249 (1951; Zbl 0042.15703)] and W. Kühnel, M. Steller [Monatsh. Math. 146, No. 2, 113–126 (2005; Zbl 1093.53004)]). Calling a cyclic surface of Riemannian type if the planes containing the circles of the foliation are parallel, the author proves the following two results. The first one is, that a cyclic LW-surface with \((m,n)=(m,0)\) must be of Riemannian type. The second result is: Besides the surfaces of revolution the only LW-surfaces of Riemannian type with arbitrary pair \((m,n)\) are the classical Riemannian examples of minimal surfaces, that is, if \((m,n)=(-1,0)\).

53A05 Surfaces in Euclidean and related spaces
53C40 Global submanifolds
Full Text: DOI arXiv
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