A generalization of Bonnet’s theorem on Darboux surfaces. (English. Russian original) Zbl 1315.53007

Bodrenko, I. I.

Math. Notes 95, No. 6, 760-767 (2014); translation from Mat. Zametki 95, No. 6, 812-820 (2014).

Summary: The well-known Bonnet theorem claims that, on a Darboux surface in three-dimensional Euclidean space, along each line of curvature, the corresponding principal curvature is proportional to the cube of another principal curvature. In the present paper, this theorem is generalized (with respect to dimension) to \(n\)-dimensional hypersurfaces of Euclidean spaces.

MSC:

53A07	Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
53A05	Surfaces in Euclidean and related spaces

Keywords:

Bonnet theorem; Darboux surface; Euclidean space; \(n\)-dimensional hypersurface; line of curvature; principal curvature; Darboux tensor; Gaussian curvature

References:

[1]	I. I. Bodrenko, Generalized Darboux Surfaces in the Spaces of Constant Curvature (LAP Lambert Acad. Publ., Saarbrücken, Germany, 2013). · Zbl 1366.53002
[2]	V. F. Kagan, The Fundamentals of the Theory of Surfaces in Tensor Presentation, Part 2: Surfaces in Space. Transformations and Deformations of Surfaces. Special Questions (Gostekhizdat, Moscow-Leningrad, 1948) [in Russian].
[3]	Yu. A. Aminov, “Condition of holonomicity of characteristic directions of a submanifold,” Mat. Zametki 41(4), 543-548 (1987) [Math. Notes 41 (4), 305-308 (1987)]. · Zbl 0625.53018