The Center Map of an Affine Immersion

Abstract

We study the center map of an equiaffine immersion which is introduced using the equiaffine support function. The center map is a constant map if and only if the hypersurface is an equiaffine sphere. We investigate those immersions for which the center map is affine congruent with the original hypersurface. In terms of centroaffine geometry, we show that such hypersurfaces provide examples of hypersurfaces with vanishing centroaffine Tchebychev operator. We also characterize them in equiaffine differential geometry using a curvature condition involving the covariant derivative of the shape operator. From both approaches, assuming the dimension is 2 and the surface is definite, a complete classification follows.