Quasi-complete homogeneous contact manifold associated to a cubic form

TL;DR

This paper constructs a class of quasi-complete homogeneous contact manifolds from cubic forms and characterizes when they can be compactified into projective contact manifolds based on properties of cubic Jordan algebras.

Contribution

It introduces a general construction linking cubic forms to homogeneous contact manifolds and characterizes their compactification criteria via simple cubic Jordan algebras.

Findings

Construction of homogeneous contact manifolds from cubic forms

Characterization of compactification conditions using Jordan algebras

Identification of when the manifold can be compactified into a projective contact manifold

Abstract

Starting from a cubic form, we give a general construction of a quasi-complete homogeneous manifold endowed with a natural contact structure. We show that it can be compactified into a projective contact manifold if and only if the cubic form is the determinant of a simple cubic Jordan algebra.