ON NORMAL AND CONORMAL MAPS FOR AFFINE HYPERSURFACES

Abstract

We prove two main results in affine differential geometry that characterize ellipsoids among the ovaloids. The first theorem states that an ovaloid in the 3-dimensional affine space is an ellipsoid if and only if the Laplacian of the normal map is proportional to the normal map. The second theorem says that a hyperovaloid in an affine space of any dimension is a hyperellipsoid if and only if the conormal image (or the normal image) is a hyperellipsoid with center at the origin.