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|Paper IPM / M / 16045||
A CR manifold M , with CR distribution D10 ⊂ T\mathbb C M , is called totally nondegenerate of depth μ if:(a) the complex tangent space T\mathbb CM is generated by all complex vector fields that might be determined by iterated Lie brackets between at most μ fields in D10+―D10 ;(b) for each integer 2 ≤ k ≤ μ−1, the families of all vector fields that might be determined by iterated Lie brackets between at most k fields in D10+―D10 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 \frak m=\frak m−μ+…+\frak m−1 of a totally nondegenerate CR manifold of depth μ ≥ 4, the full Tanaka prolongation of \frak m has trivial subspaces of degree k ≥ 1, ie it has the form \frak m−μ+…+\frak m−1+\frak g0. 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 give also a complete proof of a conjecture by Beloshapka on the group of automorphisms of homogeneous totally nondegenerate CR manifolds.
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