A classification theorem of nondegenerate equiaffine symmetric hypersurfaces

TL;DR

This paper establishes a complete classification of nondegenerate equiaffine symmetric hypersurfaces with nonzero affine mean curvature by linking them to semi-simple real Jordan algebras, extending previous geometric classifications.

Contribution

It introduces a novel correspondence between equiaffine symmetric hypersurfaces and semi-simple real Jordan algebras, enabling a comprehensive classification.

Findings

Classified all such hypersurfaces using Jordan algebra theory.

Connected geometric properties with algebraic structures.

Provided a complete classification of hypersurfaces with parallel Fubini-Pick forms.

Abstract

Motivated by the ideas and methods used by Naitoh in the consideration of parallel totally real submanifolds in complex space forms, the author of the present paper successfully makes use of the so called Jordan triple and (restricted) structure Lie algebra associated with a given Jordan algebra to establish a one-to-one correspondence between the set of equivalence classes of connected, simply connected and nondegenerate equiaffine symmetric hypersurfaces with a given nonzero affine mean curvature and that of the equivalence classes of semi-simple real Jordan algebras. Then, via the existing classification theorem of the semi-simple real Jordan algebras with unity, a complete classification for the nondegenerate and locally equiaffine symmetric hypersurfaces with nonzero affine mean curvatures is established. As an direct application of the main theorems, we prove at the end of the paper a complete classification of nondegenerate hypersurfaces with parallel Fubini-Pick forms and nonzero affine mean curvatures.