On the geometric order of totally nondegenerate CR manifolds

Abstract

A CR manifold $M$, with CR distribution ${\mathcal{D}}^{10}\subset T^{\mathbb{C}}M$, is called {\it totally nondegenerate of depth $\mu$} if: (a) the complex tangent space $T^{\mathbb{C}}M$ is generated by all complex vector fields that might be determined by iterated Lie brackets between at most $\mu$ fields in ${\mathcal{D}}^{10}+\overline{{\mathcal{D}}^{10}}$; (b) for each integer $2\le k\le \mu -1$, the families of all vector fields that might be determined by iterated Lie brackets between at most $k$ fields in ${\mathcal{D}}^{10}+\overline{{\mathcal{D}}^{10}}$ generate regular complex distributions; (c) the ranks of the distributions in (b) have the {\it maximal values} that can be obtained amongst all CR manifolds of the same CR dimension and satisfying (a) and (b) -- this maximality property is the {\it total nondegeneracy} condition. In this paper, we prove that, for any Tanaka symbol $\mathfrak{m}={\mathfrak{m}}^{-\mu }+\dots +{\mathfrak{m}}^{-1}$ of a totally nondegenerate CR manifold of depth $\mu \ge 4$, the full Tanaka prolongation of $\mathfrak{m}$ has trivial subspaces of degree $k\ge 1$, i.e. it has the form ${\mathfrak{m}}^{-\mu }+\dots +{\mathfrak{m}}^{-1}+{\mathfrak{g}}^0$. This result has various consequences. For instance it implies that any (local) CR automorphism of a regular totally nondegenerate CR manifold is uniquely determined by its first order jet at a fixed point of the manifold. It also gives a complete proof of a conjecture by Beloshapka on the group of automorphisms of homogeneous totally nondegenerate CR manifolds.

A faulty argument in the proof of Proposition 5.6 is replaced by a correct one, the presentation of the proof of the main theorem (Section 5.4, p. 25) is improved and several misprints are corrected. To appear on Math. Z