The gap phenomenon in parabolic geometries

Kruglikov, Boris

doi:10.1515/crelle-2014-0072

Cited by 61 publications

(127 citation statements)

References 37 publications

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“…In the present paper we show that in this situation the canonical frame can be constructed in a unified way on a bundle of dimension 2n − 3 for all n ≥ 5 (Theorem 7.1, section 7), i.e. a gap phenomenon occurs (namely, the dimension of the pseudo-group of symmetries drops more than by 1 compared to the most symmetric model without the restriction on the generalized Wilczynski invariants) similar to one observed in [18,19]. We also describe all models with the pseudo-group of local symmetries of dimension 2n − 3, i.e.…”

Section: Introductionsupporting

confidence: 58%

Geometry of rank 2 distributions with nonzero Wilczynski invariants*

Doubrov

Zelenko

2021

JNMP

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In the famous 1910 "cinq variables" paper Cartan showed in particular that for maximally nonholonomic rank 2 distributions in R 5 with non-zero covariant binary biquadratic form the dimension of the pseudo-group of local symmetries does not exceed 7 and among such distributions he described the one-parametric family of distributions for which this pseudo-group is exactly 7-dimensional. Using the novel interpretation of the Cartan covariant binary biquadratic form via the classical Wilczynski invariant of curves in projective spaces associated with abnormal extremals of the distributions [4, 27, 28] one can generalize this Cartan result to rank 2 distributions in R n satisfying certain genericity assumption, called maximality of class, for arbitrary n ≥ 5.In the present paper for any rank 2 distribution of maximal class with at least one nonvanishing generalized Wilczynski invariants we construct the canonical frame on a (2n − 3)-dimensional bundle and describe explicitly the moduli spaces of the most symmetric models. The relation of our results to the divergence equivalence of Lagrangians of higher order is given as well.

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Section: Introductionsupporting

confidence: 58%

Geometry of rank 2 distributions with nonzero Wilczynski invariants*

Doubrov

Zelenko

2021

JNMP

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“…We give for each model a Lie algebra isomorphism h C r,s ∼ = aut(D r,s ) identifying E ↔ y∂ y + p∂ p + q∂ q + 2z∂ z and F ↔ ∂ x and identifying U with a constant multiple of ∂ z , and we set k := span{E, K} and d := span{F, L} ⊕ k for some complementary vectors K, L depending on (r, s). 4 As in footnote 2, the bound in [9] is established for distributions with constant Petrov type; this assumption was eliminated in [26]. 5 Here the factor 10 3 corrects a numerical error of Cartan [35].…”

Section: The Flat Model Omentioning

confidence: 99%

Homogeneous Real (2,3,5) Distributions with Isotropy

Willse

Universit²,

Wien³

2019

SIGMA

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We classify multiply transitive homogeneous real (2, 3, 5) distributions up to local diffeomorphism equivalence.Proposition 2.1 ([18, Section 76]). Let (M, D) be a real (complex) (2, 3, 5) distribution and fix a point u ∈ M . There is a neighborhood U ⊆ M of u, a diffeomorphism (biholomorphism) Ψ : U → V ⊂ J 2,0 (F, F 2 ), and a smooth (complex-analytic) Homogeneous distributions Infinitesimal symmetriesAn infinitesimal symmetry of a (2, 3, 5) distribution (M, D) is a vector field ξ ∈ Γ(D) whose (local) flow preserves D, or equivalently for which L ξ η ∈ Γ(D) for all η ∈ Γ(D). We denote the Lie algebra of infinitesimal symmetries, called the (infinitesimal) symmetry algebra, by aut(D), and we say that (M, D) is infinitesimally homogeneous if aut(D) acts infinitesimally transitively, that is, if {ξ u : ξ ∈ aut(D)} = T u M for all u ∈ M . This article concerns (infinitesimally) multiply transitive homogeneous distributions, that is, infinitesimally homogeneous distributions for which the isotropy subalgebra k u < aut(D) of infinitesimal symmetries vanishing at any u ∈ M is nontrivial, or equivalently, for which dim aut(D) ≥ 6. Algebraic models for homogeneous distributionsFix a homogeneous (2, 3, 5) (real or complex) distribution (M, D) with transitive symmetry algebra h := aut(D), fix a point u ∈ M , and denote by k < h the subalgebra of vector fields in h vanishing at u and by d ⊂ h the subspace d := {ξ ∈ h : ξ u ∈ D u }. Then, k ⊆ d, [k, d] ⊆ d (we call this property k-invariance), and dim(d/k) = 2. The fact that D is a (2, 3, 5) distribution implies the genericity condition d + [d, d] + [d, [d, d]] = h. We call the triple (h, k; d) a (real or complex) algebraic model (of (M, D)).Given an algebraic model, we can reconstruct D up to local equivalence: For any groups H, K with K < H and respectively realizing h, k (for the groups that occur in the classification, we can choose K to be a closed subgroup of H), invoke the canonical identification T id ·K (H/K) ∼ = h/k to take D ⊂ T (H/K) to be the distribution with fibers D h·K = T id ·K L h · (d/k), where L h : H/K → H/K is the map L h : g · K → (hg) · K; by k-invariance this definition is independent of the coset representative h, and by genericity D is an H-invariant (2, 3, 5) distribution. Via the above identification, [D, D] id ·K = (d + [d, d])/k. 1 We declare two algebraic models (h, k; d), (h , k ; d ) to be equivalent iff there is a Lie algebra isomorphism α : h → h satisfying α(k) = k and α(d) = d . Unwinding definitions shows that equivalent algebraic models determine locally equivalent distributions. Multiply transitive homogeneous complex distributionsCartan showed that for all (2, 3, 5) distributions D, dim aut(D) ≤ 14, and that equality holds iff it is locally equivalent to the so-called flat distribution ∆ [9]; his argument applies to both the real and complex settings. We call the corresponding (complex) algebraic model O (see Section 4.1). In this case, aut(D) is isomorphic to the simple complex Lie algebra of type G 2 -we denote it by g 2 (C) -a...

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“…Let us note that the symmetry gap problem, to determine the difference between the maximal and submaximal symmetry dimensions, has recently been in focus for many geometric structures, see e.g. [9,11] and the references therein. Usually the gaps for dimensions of group and algebra of symmetries are different.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Submaximally Symmetric CR-Structures

Kruglikov

2015

J Geom Anal

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Abstract. Hypersurface type CR-structures with non-degenerate Levi form on a manifold of dimension (2n + 1) have maximal symmetry dimension n 2 + 4n + 3. We prove that the next (submaximal) possible dimension for a (local) symmetry algebra is n 2 + 4 for Levi-indefinite structures and n 2 + 3 for Levi-definite structures when n > 1. In the exceptional case of CR-dimension n = 1, the submaximal symmetry dimension 3 was computed by E. Cartan. Introduction and Main ResultLevi-nondegenerate Cauchy-Riemann (CR) structure of hypersurface type on a manifold M of dim = 2n+1 consists of a contact distribution C ⊂ T M and a complex structure J on it that preserves the canonical conformal symplectic structure ω on C associated to the natural tensor). We will assume J integrable, which is a condition analogous to vanishing of the Nijenhuis tensor:. In this case under additional assumption of analyticity the CR-structure can be locally realized by a real hypersurface in C n+1 . Moreover the internal symmetries of CR-structures (vector fields preserving both C and J) are bijective with external symmetries (holomorphic vector fields tangent to the hypersurface). We refer to these statements and other details to [BER].Levi form of CR-structure (C, J) is the conformal quadric q(X, Y ) = ω(X, JY ). It is non-degenerate (this is equivalent to nondegeneracy of ω, i.e. to C being contact), and so has signature (2k, 2n − 2k) (J is qorthogonal; we fix k ≤ n 2 using a choice of the conformal factor). In the case k = 0 we say q is a Levi-definite form, elsewise Q is Levi-indefinite. E. Cartan [Ca] for n = 1 and N. Tanaka [T] for n > 1 showed (in the analytic case) that the symmetry (automorphism) group of a CRstructure of the considered type has maximal dimension (n + 2) 2 − 1. The same estimate applies to the Lie algebra of (local) symmetries.2010 Mathematics Subject Classification. 32V20, 32M25, 57S20, 58J70.

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The gap phenomenon in parabolic geometries

Cited by 61 publications

References 37 publications

Geometry of rank 2 distributions with nonzero Wilczynski invariants*

Geometry of rank 2 distributions with nonzero Wilczynski invariants*

Homogeneous Real (2,3,5) Distributions with Isotropy

Submaximally Symmetric CR-Structures

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