Homogeneous models for Levi degenerate CR manifolds

Abstract

We extend the notion of a fundamental negatively $\mathbb{Z}$-graded Lie algebra ${\mathfrak{m}}_x={\oplus }_{p\le -1}{\mathfrak{m}}_x^p$ associated to any point of a Levi nondegenerate Cauchy-Riemann (CR) manifold to the class of $k$-nondegenerate CR manifolds $(M,\mathcal{D},\mathcal{J})$ for all $k\ge 2$ and call this invariant the core at $x\in M$. It consists of a $\mathbb{Z}$-graded vector space ${\mathfrak{m}}_x={\oplus }_{p\le k-2}{\mathfrak{m}}_x^p$ 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 $\mathrm{Lie}\left(G\right)=\mathfrak{g}=\oplus {\mathfrak{g}}^p$ and subalgebra $\mathrm{Lie}\left(G_o\right)={\mathfrak{g}}_o=\oplus {\mathfrak{g}}_o^p$ of the nonnegative part ${\oplus }_{p\ge 0}{\mathfrak{g}}^p$. It generalizes the classical notion of Tanaka of homogeneous models 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.