2018/07/09 by Sabzevari, Masoud, Spiro, Andrea · 1 citation
#22F30 #22F50 #32V05 #32V40 #57S25 #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.1807.03076
A CR manifold M, with CR distribution \mathcal D10⊂ T^\mathbb C M, is called \it totally nondegenerate of depth μ 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 μ fields in \mathcal D10 + \mathcal 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 \mathcal D10 + \mathcal D10 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 \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, i.e. 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 gives a complete proof of a conjecture by Beloshapka on the group of automorphisms of homogeneous totally nondegenerate CR manifolds.