J Austral Math Soc Ser A 49 pp90--110, 1990.

On O. Bonnet III-Isometry of Surfaces in Three Dimensional Euclidean Space

Wenmao Yang

Abstract

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, k_i(m) = k_i*(m*), i = 1, 2, where m Î M, m* Î M*, m* = F(m), k_i 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 fundamental 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-isometry. 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 Section3).

1980 AMS Subject Classification (1985 Revision): 53A05, 53B20

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Authors

Wenmao Yang

Wuhan University, Wuhan, Hubei, People's Republic of China.