Affine normal curvature of hypersurfaces from the point of view of singularity theory

Abstract

We consider smoothly embedded hypersurfaces \({M \subset \mathbb{R}^{n+1}}\) under the action of the special affine group \({{\rm SL}(n+1,\mathbb{R}) < imes \mathbb{R}^{n+1}}\). We construct a differential invariant, called affine normal curvature, which assigns to a point and a tangent direction a number. We prove some of its nice properties which connect it with affine principal directions, affine umbilics, and affine mean curvature.