Graph immersions with parallel cubic form

TL;DR

This paper classifies certain affine hypersurfaces with parallel cubic form using Jordan algebra theory, revealing their structure, completeness properties, and connections to Cayley hypersurfaces, up to dimension five.

Contribution

It introduces a novel algebraic framework linking graph immersions with parallel cubic form to Jordan algebras, enabling classification and analysis of these hypersurfaces.

Findings

Classified all such hyperspheres up to dimension 5.

Established correspondence between graph immersions and Jordan algebra pairs.

Connected Cayley hypersurfaces to polynomial quotient algebras.

Abstract

We consider non-degenerate graph immersions into affine space $\mathbb A^{n+1}$ 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 $(J,\gamma)$, where $J$ is an $n$-dimensional real Jordan algebra and $\gamma$ is a non-degenerate trace form on $J$. Every graph immersion with parallel cubic form can be extended to an affine complete 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 a polynomial as globally defining function. We classify all such hyperspheres up to dimension 5. As a special case we describe a connection between Cayley hypersurfaces and polynomial quotient algebras. Our algebraic approach can be used to study also other classes of hypersurfaces with parallel cubic form.