Conditions for a two-dimensional surface in \(E^5\) to be contained in a hypersphere or a hyperplane (Q382370)

The authors give conditions for a \(2\)-dimensional surface in the \(5\)-dimensional Euclidean space to be contained in a codimension-one sphere or in a hyperplane. The conditions are given in terms of the ellipse of normal curvature, which is, for any point of the surface, a curve in the normal subspace traced out by the normal curvature vectors. According to the position of a point with respect to its ellipse of normal curvature we may distinguish elliptic, parabolic and hyperbolic points. The authors, for each type of a surface, i.e., a surface with all points of given type, by the examination of some differential equations, give geometric conditions for a surface to be contained in a codimension-one sphere or a hyperplane.