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An affine hypersurface (AH) structure is a pair comprising a conformalstructure and a projective structure such that for any torsion-free connectionrepresenting the projective structure the completely trace-free part of thecovariant derivative of any metric representing the conformal structure iscompletely symmetric. AH structures simultaneously generalize Weyl structuresand abstract the geometric structure determined on a non-degenerate co-orientedhypersurface in flat affine space by its second fundamental form together witheither the projective structure induced by the affine normal or that induced bythe conormal Gauss map. There are proposed notions of Einstein equations for AHstructures which for Weyl structures specialize to the usual Einstein Weylequations and such that the AH structure induced on a non-degenerateco-oriented affine hypersurface is Einstein if and only if the hypersurface isan affine hypersphere. It is shown that a convex flat projective structureadmits a metric with which it generates an Einstein AH structure, and examplesare constructed on mean curvature zero Lagrangian submanifolds of certainpara-K\"ahler manifolds. The rough classification of Riemannian Einstein Weylstructures by properties of the scalar curvature is extended to this setting.Known estimates on the growth of the cubic form of an affine hypersphere arepartly generalized. The Riemannian Einstein equations are reformulated in termsof a given background metric as an algebraically constrained elliptic systemfor a cubic tensor. From certain commutative nonassociative algebras there areconstructed examples of exact Riemannian signature Einstein AH structures withself-conjugate curvature but which are not Weyl and are neither projectivelynor conjugate projectively flat.

Geometric structures modeled on affine hypersurfaces and generalizations of the Einstein Weyl and affine hypersphere equations