Nearly Kahler geometry and (2,3,5)-distributions via projective holonomy

Gover, Rod; Panai, Roberto; Willse, Travis

doi:10.1512/iumj.2017.66.6089

Cited by 11 publications

(18 citation statements)

References 65 publications

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“…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%

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: Introductionmentioning

confidence: 99%

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

Willse

Universit²,

Wien³

2019

SIGMA

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“…Such distributions that are maximally nonintegrable in the sense that This geometry has proved interesting for numerous reasons: It comprises a first class of distributions with continuous local invariants, it is connected to the geometry of surfaces rolling on one another without slipping or twisting [5,8,10] and hence to natural problems in control theory [2], it is intimately linked (as Cartan showed in the aforementioned article 1 ) to the exceptional simple complex Lie group of type G C 2 and then via the natural inclusion G * 2 → SO(3, 4) to conformal geometry [26], and it is the appropriate structure for naturally projectively compactifying strictly nearly (para-)Kähler structures in dimension 6 [21]. (Here, G * 2 denotes the automorphism group of the algebra of split octonions, one of the split real forms of the complex, connected simple Lie group G C 2 .)…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, computing using the Ambrose-Singer Theorem gives that the holonomy group of g E is G * 2 , furnishing a new example a metric with that exceptional holonomy group. Finally, we apply some general facts relating ambient metrics to projective structures whose normal tractor connections enjoy orthogonal holonomy reductions [21] to produce from g E an explicit example of a projective structure with normal projective holonomy G * 2 . Many computations whose results are reported here were carried out with Ian Anderson's Maple Package DifferentialGeometry.…”

Section: Introductionmentioning

confidence: 99%

“…By [21] we may regard the real-analytic ambient manifold of a real-analytic conformal structure of odd dimension n 3 and signature ( p, q) as the so-called Thomas cone of a projective structure of dimension n + 1, which is defined on the space of orbits of the R + -action defined by the dilations δ s . This identifies the Levi-Civita connection of the ambient metric with the normal tractor connection-a natural, projectively invariant vector bundle connection on a particular vector bundle of rank n + 2 over and canonically associated to the projective manifold-whose holonomy is thus reduced to O( p + 1, q + 1).…”

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confidence: 99%

“…Example 7.1 of [21] also gives a 1-parameter family of nontrivial 6-dimensional projective structures with normal holonomy contained in G * 2 , but for every member of that family the containment is proper.…”

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Cartan’s incomplete classification and an explicit ambient metric of holonomy $$\mathrm{G}_2^*$$

Willse

2017

European Journal of Mathematics

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In his 1910 "Five Variables" paper, Cartan solved the equivalence problem for the geometry of (2, 3, 5) distributions and in doing so demonstrated an intimate link between this geometry and the exceptional simple Lie groups of type G 2 . He claimed to produce a local classification of all such (complex) distributions which have infinitesimal symmetry algebra of dimension at least 6 (and which satisfy a natural uniformity condition), but in 2013 Doubrov and Govorov showed that this classification misses a particular distribution E. We produce a closed form for the Fefferman-Graham ambient metric g E of the conformal class induced by (a real form of) E, expanding the small catalogue of known explicit, closed-form ambient metrics. We show that the holonomy group of g E is the exceptional group G * 2 and use that metric to give explicitly a projective structure with normal projective holonomy equal to that group. We also present some simple but apparently novel observations about ambient metrics of general left-invariant conformal structures that were used in the determination of the explicit formula for g E .

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Metric Projective Geometry, BGG Detour Complexes and Partially Massless Gauge Theories

Gover

Latini

Waldron

2015

Commun. Math. Phys.

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Abstract. A projective geometry is an equivalence class of torsion free connections sharing the same unparametrised geodesics; this is a basic structure for understanding physical systems. Metric projective geometry is concerned with the interaction of projective and pseudo-Riemannian geometry. We show that the BGG machinery of projective geometry combines with structures known as Yang-Mills detour complexes to produce a general tool for generating invariant pseudo-Riemannian gauge theories. This produces (detour) complexes of differential operators corresponding to gauge invariances and dynamics. We show, as an application, that curved versions of these sequences give geometric characterizations of the obstructions to propagation of higher spins in Einstein spaces. Further, we show that projective BGG detour complexes generate both gauge invariances and gauge invariant constraint systems for partially massless models: the input for this machinery is a projectively invariant gauge operator corresponding to the first operator of a certain BGG sequence. We also connect this technology to the log-radial reduction method and extend the latter to Einstein backgrounds.

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Nearly Kahler geometry and (2,3,5)-distributions via projective holonomy

Cited by 11 publications

References 65 publications

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

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

Cartan’s incomplete classification and an explicit ambient metric of holonomy $$\mathrm{G}_2^*$$

Metric Projective Geometry, BGG Detour Complexes and Partially Massless Gauge Theories

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