Affine classification of points of multidimensional surfaces

Abstract

The classification of the points of a regular two-dimensional surface in Euclidean threespace is well known. One distinguishes three nondegenerate classes of points: hyperbolic points, at which the Gaussian curvature is negative; elliptic points, at which the Gaussian curvature is positive; and parabolic points, with Gaussian curvature zero; moreover, there is one degenerate class flat points, at which both principal curvatures vanish. In exactly the same way one can classify points on regular hypersurfaces in a Euclidean space: the class of a point is determined by the signs of the principal curvatures of the surface~