On the geometric order of totally nondegenerate CR manifolds
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Mathematische Zeitschrift
On the geometric order of totally nondegenerate CR manifolds Masoud Sabzevari1,2 · Andrea Spiro3 Received: 31 July 2018 / Accepted: 29 September 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract A CR manifold M, with CR distribution D10 ⊂ T C M, is called totally nondegenerate of depth μ if: (a) the complex tangent space T C M 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 m = m−μ + · · · + m−1 of a totally nondegenerate CR manifold of depth μ ≥ 4, the full Tanaka prolongation of m has trivial subspaces of degree k ≥ 1, i.e. it has the form m−μ +· · ·+m−1 +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. Keywords Totally nondegenerate CR manifold · Maximum conjecture · Tanaka structure · Tanaka symbol Mathematics Subject Classification 32V05 · 32V40 · 22F30 · 22F50 · 57S25
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Masoud Sabzevari [email protected] Andrea Spiro [email protected]
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Department of Mathematics, Shahrekord University, Shahrekord 88186-34141, Iran
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School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P. O. Box: 19395-5746 Tehran, Iran
3
Scuola di Scienze e Tecnologie, Università di Camerino, Via Madonna delle Carceri, 62032 Camerino, Macerata, Italy
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M. Sabzevari, A. Spiro
1 Introduction An (abstract) CR manifold is a real manifold M, equipped with a C ∞ complex distribution D10 ⊂ T C M such that: (i) D10 ∩ D10 = {0}; (ii) for all vector fields X , Y ∈ D10 the Lie bracket [X , Y ] is also in D10 . The rank of the distribution D10 is called CR dimension. The most natural and studied examples are the embedded CR manifolds, which are the smooth real submanifolds M ⊂ C N , N ≥ 2, satisfying appropriate constant rank conditions that guarantee that the family of all complex vector fields of the holomorphic distribution of C N with real and imaginary parts tangent to M, generate a complex distribution D10 ⊂ T C M of constant rank. In this paper, we study the geometric structures of the totally nondegenerate CR manifolds of depth μ, a class of CR manifolds introduced by Beloshapka in [2]. For motivation and main reasons of interest for such important class of CR structures we refer to the or
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