Geodesic
Conditions for a Two-Dimensional Surface in $E^5$ to Be Contained in a Hypersphere or a Hyperplane
Matematičeskie zametki, Tome 94 (2013) no. 2, pp. 163-174

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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
Mots-clés : elliptic domain, parabolic domain. 
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     author = {Yu. A. Aminov and Ya. S. Nasedkina},
     title = {Conditions for a {Two-Dimensional} {Surface} in $E^5$ to {Be} {Contained} in a {Hypersphere} or a {Hyperplane}},
     journal = {Matemati\v{c}eskie zametki},
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     number = {2},
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Yu. A. Aminov; Ya. S. Nasedkina. Conditions for a Two-Dimensional Surface in $E^5$ to Be Contained in a Hypersphere or a Hyperplane. Matematičeskie zametki, Tome 94 (2013) no. 2, pp. 163-174. http://geodesic.mathdoc.fr/item/MZM_2013_94_2_a0/