On Nurowski’s conformal structure associated to a generic rank two distribution in dimension five

Čap, Andreas; Sagerschnig, Katja

doi:10.1016/j.geomphys.2009.04.001

Cited by 15 publications

(23 citation statements)

References 9 publications

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“…, which we denote O R . The real flat model can be described as the rolling distribution (see Section 7) determined by a pair of spheres, one whose radius is thrice that of the other [5], 7 and also as a canonical distribution determined by the algebra O of split octonions on the null quadric in the projectivization P Im O of the space of purely imaginary split octonions [5, Section 6], [31]; see also [4].…”

Section: The Real Flat Model O Rmentioning

confidence: 99%

“…That article (1) resolved the equivalence problem for this geometry, (2) in doing so revealed a surprising connection with the exceptional complex Lie algebra of type G 2 , and (3) (nearly) locally classified complex (2,3,5) distributions D whose infinitesimal symmetry algebra aut(D) has dimension at least 6. Besides its historical significance and its connection with G 2 , which mediates its relationship with other geometries [7,19,20,21,22,27,28], [29,Section 5], this geometry is significant because of its appearance in simple, nonholonomic kinematic systems [5,6]. It has enjoyed heightened attention in the last decade or so [2,3,11,12,14,15,24,25,30,33,34,35,36,37], owing in part to its realization in the class of parabolic geometries [8,Section 4.3.2], [31,32], a broad family of Cartan geometries for which many powerful results are available.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: The Real Flat Model O Rmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Willse

Universit²,

Wien³

2019

SIGMA

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“…As ψ is parallel, it is at each point annihilated by hol x (M, c) under Clifford multiplication, i.e. [48] that D canonically defines a conformal structure c D of signature (2, 3) on M 5 whose conformal holonomy is reduced to g 2 ⊂ so (3,4), see also [19].…”

Section: 33mentioning

confidence: 99%

The ambient obstruction tensor and conformal holonomy

Leistner

Lischewski

2017

Pacific J. Math.

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Many institutions now require or encourage their researchers to deposit published articles in their institutional repositories. MSP cooperates with this requirement by sending authors or librarians, upon demand, a version of the published file suitable for the purpose. We request that you use the final published version and not a preprint or postprint version, which are not considered definitive. October 2017 Pacific Journal of Mathematics THE AMBIENT OBSTRUCTION TENSOR AND CONFORMAL HOLONOMY THOMAS LEISTNER AND ANDREE LISCHEWSKIFor a conformal manifold, we describe a new relation between the ambient obstruction tensor of Fefferman and Graham and the holonomy of the normal conformal Cartan connection. This relation allows us to prove several results on the vanishing and the rank of the obstruction tensor, for example for conformal structures admitting twistor spinors or normal conformal Killing forms. As our main tool we introduce the notion of a conformal holonomy distribution and show that its integrability is closely related to the exceptional conformal structures in dimensions five and six that were found by Nurowski and Bryant.

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“…The leading term is not invariant, but instead under a change of Weyl connection s 7 In this formula we regard φ ∈ Γ(T * M ) as an adjoint tractor via the canonical inclusion T * M ֒→ A. 8 The upper index A here is just an index on the standard tractor bundle; we use it here only for consistency. 9 In fact, we can perform this computation with a little less effort: Since we project ∇ V L 0 (σ) onto the space D * [1] of (appropriately weighted) symmetric tensors, we only need to compute the symmetric part of the second-lowest slot of the right-hand side in the first formula in Proposition 5(3), namely, ∇ (α τ β) − σP (αβ) .…”

Section: Proof Of Propositionmentioning

confidence: 99%

“…4 Weyl structures arising that way are called exact Weyl structures. In [23] (see also [8]), the Weyl connection, the soldering form, and the Rho tensor determined by a scale were constructed.…”

Section: Introductionmentioning

confidence: 99%

The almost Einstein operator for $(2, 3, 5)$ distributions

Sagerschnig¹,

Willse²

2017

Arch. Math. (Brno)

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For the geometry of oriented (2, 3, 5) distributions (M, D), which correspond to regular, normal parabolic geometries of type (G 2 , P ) for a particular parabolic subgroup P < G 2 , we develop the corresponding tractor calculus and use it to analyze the first BGG operator Θ 0 associated to the 7-dimensional irreducible representation of G 2 . We give an explicit formula for the normal connection on the corresponding tractor bundle and use it to derive explicit expressions for this operator. We also show that solutions of this operator are automatically normal, yielding a geometric interpretation of ker Θ 0 : For any (M, D), this kernel consists precisely of the almost Einstein scales of the Nurowski conformal structure on M that D determines.We apply our formula for Θ 0 (1) to recover efficiently some known solutions, (2) to construct a distribution with root type [3, 1] with a nonzero solution, and (3) to show efficiently that the conformal holonomy of a particular (2, 3, 5) conformal structure is equal to G 2 .

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On Nurowski’s conformal structure associated to a generic rank two distribution in dimension five

Cited by 15 publications

References 9 publications

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

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

The ambient obstruction tensor and conformal holonomy

The almost Einstein operator for $(2, 3, 5)$ distributions

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