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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 \((m,n)=(-1,0)\).

MSC:
53A05 Surfaces in Euclidean and related spaces
53C40 Global submanifolds
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[1] DOI: 10.1016/0167-8396(95)00046-1 · Zbl 0875.68873 · doi:10.1016/0167-8396(95)00046-1
[2] DOI: 10.1215/S0012-7094-45-01222-1 · Zbl 0063.00833 · doi:10.1215/S0012-7094-45-01222-1
[3] Delaunay C., J. Math. Pure Appl. 6 pp 309–
[4] Enneper A., Nach Königl Ges Wissensch Göttingen Math. Phys. Kl. 1 pp 243–
[5] Enneper A., Z. Math. Phys. 14 pp 393–
[6] DOI: 10.1007/s00605-002-0510-3 · Zbl 1056.53002 · doi:10.1007/s00605-002-0510-3
[7] DOI: 10.2307/2372698 · Zbl 0055.39601 · doi:10.2307/2372698
[8] DOI: 10.1002/mana.3210040122 · Zbl 0042.15703 · doi:10.1002/mana.3210040122
[9] DOI: 10.1007/s00605-005-0313-4 · Zbl 1093.53004 · doi:10.1007/s00605-005-0313-4
[10] López F. J., J. Differential Geom. 47 pp 376–
[11] DOI: 10.1007/s002220050241 · Zbl 0916.53004 · doi:10.1007/s002220050241
[12] DOI: 10.2991/jnmp.2004.11.s1.7 · Zbl 1362.33026 · doi:10.2991/jnmp.2004.11.s1.7
[13] Nitsche J. C. C., Nachr. Akad. Wiss. Gottingen Math. Phys. II 1 pp 1–
[14] Riemann B., Abh. Königl. Ges. Wissensch. Göttingen Mathema. Cl. 13 pp 329–
[15] DOI: 10.1215/S0012-7094-94-07314-6 · Zbl 0802.53002 · doi:10.1215/S0012-7094-94-07314-6
[16] DOI: 10.1007/BF01369665 · Zbl 0085.15902 · doi:10.1007/BF01369665
[17] Weingarten J., J. Reine Angew. Math. 59 pp 382–
[18] Weingarten J., J. Reine Angew. Math. 62 pp 61–
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