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On O. Bonnet III-isometry of surfaces in three dimensional Euclidean space

Part of: Classical differential geometry Local differential geometry

Published online by Cambridge University Press:  09 April 2009

Wenmao Yang
Wenmao Yang
Affiliation:
Wuhan UniversityWuhan, HubeiPeople's Republic of China
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Abstract

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In this paper we consider O. Bonnet III-isometry (or BIII-isometry) of surfaces in 3-dimensional Euclidean space E3 Suppose a map F: M → M* is a diffeomorphism, and F* (III*) = III, ki(m) = k*i (m*), i = 1, 2, where m ∈ M, m* ∈ M*, m* = F (m), ki and k*i are the principal curvatures of surfaces M and M* at the points m and m*, respectively, III and III* are the third fundmental forms of M and M*, respectively. In this case, we call F an O. Bonnet III-isometry from M to M*. O. Bonnet I-isometries were considered in references [1]-[5].

We distinguish three cases about BIII-surfaces, which admits a non-trivial BIII-ismetry. We obtain some geometric properties of BIII-surfaces and BIII-isometries in these three cases; see Theorems 1, 2, 3 (in Section 2). We study some special BIII-surfaces: the minimal BIII-surfaces; BIII-surfaces of revolution; and BIII-surfaces with constant Gaussian curvature; see Theorems 4, 5, 6 (in Section 3).

MSC classification

Secondary: 53A05: Surfaces in Euclidean space 53B20: Local Riemannian geometry
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 49 , Issue 1 , August 1990 , pp. 90 - 110
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Bonnet, O., ‘Memoire sur la theorie des surfaces applicables’, J. Ecole Polytechnique 42 (1867), 72–92.Google Scholar
[2]Cartan, E., ‘Couples des surfaces applicables avec conservation des courbures principales’, Bull. Sci. Math. 66 (1942), 55–85.Google Scholar
[3]Chern, S. S., ‘Deformation of surfaces preserving principal curvatures’, Differential geometry and complex analysis, Springer Verlag, New York, 1985, pp. 155–163.CrossRefGoogle Scholar
[4]Graustein, W. C., ‘Applicability with preservation of both curvatures’, Bull. Amer. Math. Soc. 30 (1924), 19–27.CrossRefGoogle Scholar
[5]Roussos, I. M., ‘Mean-curvature-preserving isometries of surfaces in ordinary space’, to appear.Google Scholar
[6]Yang, Wenmao, ‘Infinitesimal O. Bonnet-deformations of surfaces in E3’, Differential Geometry and Topology, Proceedings, Tianjin, 1986–1987, edited by Jiang, Boju, Peng, Chia-Kuei and Hou, Zixin, pp. 306–321. (Lecture Notes in Mathematics, no. 1369, Springer Verlag, New York, 1989).Google Scholar