This article is cited in 3 papers

Conditions for a Two-Dimensional Surface in $E^5$ to Be Contained in a Hypersphere or a Hyperplane

Yu. A. Aminov, Ya. S. Nasedkina

B. Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine

Abstract: Two theorems on conditions under which a two-dimensional surface in Euclidean 5-space is contained in a hypersphere and one theorem on conditions under which such a surface is contained in a hyperplane are proved. The notion of hyperbolic and elliptic domains on a surface are introduced. The conditions in the theorems are expressed in terms of the behavior of the plane of the normal curvature ellipse of the surface and certain boundary conditions. An example which shows that the boundary conditions are essential is constructed.

Keywords: hyperspherical surface, hyperplanar surface, ellipse of normal curvature, hyperbolic domain, elliptic domain, parabolic domain.

UDC: 514.76

Received: 12.10.2008
Revised: 25.04.2012

DOI: 10.4213/mzm6347

 English version: Mathematical Notes, 2013, 94:2, 167–176

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