Graph immersions with parallel cubic form

Abstract

We consider non-degenerate graph immersions in R^(n+1) whose cubic form is parallel with respect to the Levi-Civita connection of the affine metric. There exists a correspondence between such graph immersions and pairs (J,g), where J is an n-dimensional real Jordan algebra and g is a non-degenerate trace form on J. Every graph immersion with parallel cubic form can be extended to an affine complete hypersurface immersion which is locally a graph immersion with parallel cubic form. This affine complete immersion is a homogeneous locally symmetric space covering the maximal connected component of zero in the set of quasi-regular elements in the algebra J. It is an improper affine hypersphere if and only if the corresponding Jordan algebra is nilpotent. In this case it is an affine complete, Euclidean complete graph immersion, with the defining function being a polynomial. Our approach can be used to study also other classes of hypersurfaces with parallel cubic form.