Geometric Structures Modeled on Affine Hypersurfaces and Generalizations of the Einstein–Weyl and Affine Sphere Equations

Abstract

Affine hypersurface structures (AH structures) simultaneously generalize Weyl structures and abstract geometric structures induced on a nondegenerate co-oriented hypersurface in flat affine space. The aim of this note is to define equations for AH structures, called Einstein which, for Weyl structures, specialize to the usual Einstein Weyl equations, and, in the case of the AH structure induced on a hypersurface in flat affine space, recover the equations for affine spheres. Additionally, we indicate the simplest constructions of Einstein AH structures that do not arise in either of these manners.