Homogeneous models for Levi-degenerate CR manifolds

TL;DR

This paper extends the concept of fundamental Lie algebra invariants to higher nondegeneracy levels in Levi-degenerate CR manifolds, proposing homogeneous models and classifying cores for 7-dimensional cases.

Contribution

It introduces the core invariant for $k$-nondegenerate CR manifolds and constructs homogeneous models generalizing classical Tanaka models.

Findings

Classified cores of 7-dimensional 2-nondegenerate CR manifolds.

Constructed homogeneous models for seven classes of these manifolds.

Proved the uniqueness of the core and model in the 3-nondegenerate case.

Abstract

We extend the notion of a fundamental negatively $\mathbb Z$-graded Lie algebra $\mathfrak{m}_x=\bigoplus_{p\leq -1}\mathfrak{m}_x^p$ associated to any point of a Levi nondegenerate CR manifold to the class of $k$-nondegenerate CR manifolds $(M,\mathcal D,\mathcal J)$ for all $k\geq 2$ and call this invariant the core at $x\in M$. It consists of a $\mathbb Z$-graded vector space $\mathfrak{m}_x=\bigoplus_{p\leq k-2}\mathfrak{m}_x^p$ of height $k-2$ endowed with the natural algebraic structure induced by the Tanaka and Freeman sequences of $(M,\mathcal D,\mathcal J)$ and the Levi forms of higher order. In the case of CR manifolds of hypersurface type we propose a definition of a homogeneous model of type $\mathfrak m$, that is, a homogeneous $k$-nondegenerate CR manifold $M=G/G_o$ with core $\mathfrak m$ associated with an appropriate $\mathbb Z$-graded Lie algebra $Lie(G)=\mathfrak g=\bigoplus\mathfrak g^p$ and subalgebra $Lie(G_o)=\mathfrak g_o=\bigoplus\mathfrak g_o^p$ of the nonnegative part $\bigoplus_{p\geq 0}\mathfrak g^p$. It generalizes the classical notion of Tanaka of homogeneous model for Levi nondegenerate CR manifolds and the tube over the future light cone, the unique (up to local CR diffeomorphisms) maximally homogeneous $5$-dimensional $2$-nondegenerate CR manifold. We investigate the basic properties of cores and models and study the $7$-dimensional CR manifolds of hypersurface type from this perspective. We first classify cores of $7$-dimensional $2$-nondegenerate CR manifolds up to isomorphism and then construct homogeneous models for seven of these classes. We finally show that there exists a unique core and homogeneous model in the $3$-nondegenerate class.