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 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 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 total nondegeneracy condition. In this paper, we prove that, for any Tanaka symbol $\mathfrak {m}= \mathfrak {m}^{-\mu } + \cdots + \mathfrak {m}^{-1}$ 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 $\mathfrak {m}^{-\mu } + \cdots + \mathfrak {m}^{-1} + \mathfrak {g}^0$. 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.