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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 \(\mathbb E^3\).

MSC:
53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
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[1] Chen, B.-Y., Some open problems and conjectures on submanifolds of finite type, Soochow Journal of Mathematics, 17, 2, 169-188, (1991) · Zbl 0749.53037
[2] Hasanis, T.; Vlachos, T., Hypersurfaces in \(\mathbb{E}^4\) with harmonic mean curvature vector field, Mathematische Nachrichten, 172, 145-169, (1995) · Zbl 0839.53007
[3] Defever, F., Hypersurfaces of \(\mathbb{E}^4\) with harmonic mean curvature vector, Mathematische Nachrichten, 196, 61-69, (1998) · Zbl 0944.53005
[4] Balmuş, A.; Montaldo, S.; Oniciuc, C., Classification results for biharmonic submanifolds in spheres, Israel Journal of Mathematics, 168, 201-220, (2008) · Zbl 1172.58004
[5] Balmuş, A.; Montaldo, S.; Oniciuc, C., Biharmonic hypersurfaces in 4-dimensional space forms, Mathematische Nachrichten, 283, 12, 1696-1705, (2010) · Zbl 1210.58013
[6] Caddeo, R.; Montaldo, S.; Oniciuc, C., Biharmonic submanifolds of \(S^3,\) International Journal of Mathematics, 12, 8, 867-876, (2001) · Zbl 1111.53302
[7] Caddeo, R.; Montaldo, S.; Oniciuc, C., Biharmonic submanifolds in spheres, Israel Journal of Mathematics, 130, 109-123, (2002) · Zbl 1038.58011
[8] Chen, B.-Y., Pseudo-Riemannian Geometry, δ-Invariants and Applications, (2011), Hackensack, NJ, USA: World Scientific, Hackensack, NJ, USA
[9] Dimitrić, I., Submanifolds of \(\mathbb{E}^m\) with harmonic mean curvature vector, Bulletin of the Institute of Mathematics. Academia Sinica, 20, 1, 53-65, (1992) · Zbl 0778.53046
[10] López, R., On linear Weingarten surfaces, International Journal of Mathematics, 19, 4, 439-448, (2008) · Zbl 1151.53005
[11] Montaldo, S.; Oniciuc, C., A short survey on biharmonic maps between Riemannian manifolds, Revista de la Unión Matemática Argentina, 47, 2, 1-22, (2006) · Zbl 1140.58004
[12] Oniciuc, C., Biharmonic maps between Riemannian manifolds, Analele Stiintifice ale Universitatii, 48, 2, 237-248, (2002) · Zbl 1061.58015
[13] Ou, Y.-L., Biharmonic hypersurfaces in Riemannian manifolds, Pacific Journal of Mathematics, 248, 1, 217-232, (2010) · Zbl 1205.53066
[14] Caddeo, R.; Montaldo, S.; Oniciuc, C.; Piu, P., Surfaces in the three-dimensional space forms with divergence-free stress-bienergy tensor, Annali di Matematica Pura ed Applicata, (2012) · Zbl 1294.53006
[15] Chen, B.-Y., Geometry of Submanifolds, (1973), New York, NY, USA: Marcel Dekker, New York, NY, USA
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