Universal Models For Real Submanifolds

Beloshapka, V. K.

doi:10.1023/b:matn.0000023331.50692.87

Cited by 30 publications

(79 citation statements)

References 16 publications

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“…Totally nondegenerate models and the maximum conjecture. In the seminal work [4], Beloshapka showed that after appropriate weight assignment to complex coordinates (z 1 , . .…”

Section: 2mentioning

confidence: 99%

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On the maximum conjecture

Sabzevari

2018

Forum Mathematicum

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We verify the maximum conjecture on the rigidity of totally nondegenerate model CR manifolds in the following two cases: (i) for all models of CR dimension one (ii) for the so-called full-models, namely those in which their associated symbol algebras are free CR. In particular, we discover that in each arbitrary CR dimension and length ≥ 3, there exists at least one totally nondegenerate model, enjoying this conjecture. Our proofs rely upon some recent results in the Tanaka theory of transitive prolongation of fundamental algebras.

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“…Totally nondegenerate models and the maximum conjecture. In the seminal work [4], Beloshapka showed that after appropriate weight assignment to complex coordinates (z 1 , . .…”

Section: 2mentioning

confidence: 99%

“…He also established a practical way to construct the associated defining polynomials Φ • (see §2, below). We refer the reader to [4,Theorem 14] for various nice features of these models.…”

Section: 2mentioning

confidence: 99%

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On the maximum conjecture

Sabzevari

2018

Forum Mathematicum

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“…As it was explained above, the key point in the obtained classification theorems is the main trichotomy, enabling us to assume that the homogeneity of a non-degenerate 4-dimensional CR-manifold is provided by a 4-dimensional local transitively acting Lie group, which is a priori not clear at all (see, for example, [13]). The main trichotomy, as well as the sphericity criterion for non-degenerate 4-dimensional CR-manifolds in C 3 (see Section 3), are based on the model surface method (see [4,3]). Thus the model surface method for dimension 4 CR-manifolds enables to classify all (sensible) classes of CR-manifolds with symmetries in the dimension under consideration.…”

Section: Casementioning

confidence: 99%

“…The cubic is the main example of a homogeneous totally non-degenerate CR-manifold. The cubic is a particular case of a model manifold (see, for example, [4]) and has many remarkable properties (see [5][6][7]9] for details), the main one is given by the following trichotomy for a 4-dimensional locally homogeneous CR-submanifold M in C 3 :…”

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confidence: 99%

Classification of homogeneous CR-manifolds in dimension 4

Beloshapka

Kossovskiy

2011

Journal of Mathematical Analysis and Applications

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On the geometric order of totally nondegenerate CR manifolds

Sabzevari

Spiro

2019

Math. Z.

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A CR manifold M , with CR distribution D 10 ⊂ 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 D 10 +D 10 ; (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 D 10 + 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 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 + 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.2010 Mathematics Subject Classification. 32V05, 32V40, 22F30, 22F50, 57S25.

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Universal Models For Real Submanifolds

Cited by 30 publications

References 16 publications

On the maximum conjecture

On the maximum conjecture

Classification of homogeneous CR-manifolds in dimension 4

On the geometric order of totally nondegenerate CR manifolds

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