The dancing metric, ${G}_2$-symmetry and projective rolling

Bor, Gil; Lamoneda, Luis Hern Andez; Nurowski, Paweł

doi:10.1090/tran/7277

Cited by 19 publications

(29 citation statements)

References 25 publications

(131 reference statements)

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“…Our initial motivation for this article is twofold. Firstly, it is an extension of the observation made in [7] where, inspired by the rolling problem of Riemannian surfaces [6], a notion of projective rolling was defined which gives rise to (2,3,5)-distributions. Consequently, it was observed that the (2,3,5)-distributions whose algebra of infinitesimal symmetries is maximal i.e.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Proposition 2. 7 An almost para-Hermitian structure (M, g, K ) is para-Kähler if and only if 1 4 = 0 and 4 1 = 0, in one (and therefore any) adapted coframe. As a result, the Levi-Civita connection form of g is reduced to…”

Section: Pk Structures In An Adapted Coframementioning

confidence: 99%

“…The potential V 1 corresponds to the homogeneous para-Kähler-Einstein structure referred to as the dancing metric in [7] which is the unique homogeneous model that is self-dual, i.e. W eyl − = 0, and not anti-self-dual for which 2 = 1.…”

Section: Straightforward Computation Givesmentioning

confidence: 99%

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Para-Kähler-Einstein 4-manifolds and non-integrable twistor distributions

2022

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We study the local geometry of 4-manifolds equipped with a para-Kähler-Einstein (pKE) metric, a special type of split-signature pseudo-Riemannian metric, and their associated twistor distribution, a rank 2 distribution on the 5-dimensional total space of the circle bundle of self-dual null 2-planes. For pKE metrics with non-zero scalar curvature this twistor distribution has exactly two integral leaves and is ‘maximally non-integrable’ on their complement, a so-called (2,3,5)-distribution. Our main result establishes a simple correspondence between the anti-self-dual Weyl tensor of a pKE metric with non-zero scalar curvature and the Cartan quartic of the associated twistor distribution. This will be followed by a discussion of this correspondence for general split-signature metrics which is shown to be much more involved. We use Cartan’s method of equivalence to produce a large number of explicit examples of pKE metrics with non-zero scalar curvature whose anti-self-dual Weyl tensor have special real Petrov type. In the case of real Petrov type D, we obtain a complete local classification. Combined with the main result, this produces twistor distributions whose Cartan quartic has the same algebraic type as the Petrov type of the constructed pKE metrics. In a similar manner, one can obtain twistor distributions with Cartan quartic of arbitrary algebraic type. As a byproduct of our pKE examples we naturally obtain para-Sasaki-Einstein metrics in five dimensions. Furthermore, we study various Cartan geometries naturally associated to certain classes of pKE 4-dimensional metrics. We observe that in some geometrically distinguished cases the corresponding Cartan connections satisfy the Yang-Mills equations. We then provide explicit examples of such Yang-Mills Cartan connections.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Pk Structures In An Adapted Coframementioning

confidence: 99%

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Para-Kähler-Einstein 4-manifolds and non-integrable twistor distributions

2022

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“…This construction encapsulates the configuration space of 2 surfaces rolling along one another without slipping and twisting. The authors in [5] then found new examples of flat (2, 3, 5)-distributions that arise from rolling bodies, prompting further search in [8]. The solutions to (1.2) give examples of 4-dimensional split signature metrics that have their An-Nurowski twistor distributions having split G 2 as its group of symmetries and we exhibit them in Section 7.…”

Section: Introductionmentioning

confidence: 91%

Flat (2,3,5)-Distributions and Chazy's Equations

Randall

2016

SIGMA

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Abstract. In the geometry of generic 2-plane fields on 5-manifolds, the local equivalence problem was solved by Cartan who also constructed the fundamental curvature invariant. For generic 2-plane fields or (2, 3, 5)-distributions determined by a single function of the form F (q), the vanishing condition for the curvature invariant is given by a 6 th order nonlinear ODE. Furthermore, An and Nurowski showed that this ODE is the Legendre transform of the 7 th order nonlinear ODE described in Dunajski and Sokolov. We show that the 6 th order ODE can be reduced to a 3 rd order nonlinear ODE that is a generalised Chazy equation. The 7 th order ODE can similarly be reduced to another generalised Chazy equation, which has its Chazy parameter given by the reciprocal of the former. As a consequence of solving the related generalised Chazy equations, we obtain additional examples of flat (2, 3, 5)-distributions not of the form F (q) = q m . We also give 4-dimensional split signature metrics where their twistor distributions via the An-Nurowski construction have split G 2 as their group of symmetries.

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“…If ε = 1, CL 4 → L 4 can be identified with the subbundle of the twistor bundle whose fiber over a point x ∈ L 4 comprises all self-dual 2-planes in T x CL 4 except the eigenspaces of the endomorphismK x . The total space CL 4 carries a tautological rank 2-distribution obtained by lifting each self-dual totally isotropic 2-plane horizontally to its point in the fiber, and it was observed [2,6] that, provided the self-dual Weyl tensor of the metric on L 4 vanishes nowhere, this distribution is (2, 3, 5) almost everywhere. This suggests a relation of the present work to the An-Nurowski twistor construction (and recent work of Bor and Nurowski).…”

Section: The ε-Kähler-einstein Fef Ferman Constructionmentioning

confidence: 99%

The Geometry of Almost Einstein (2,3,5) Distributions

Sagerschnig¹,

Willse²

2017

SIGMA

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Abstract. We analyze the classic problem of existence of Einstein metrics in a given conformal structure for the class of conformal structures inducedf Nurowski's construction by (oriented) (2, 3, 5) distributions. We characterize in two ways such conformal structures that admit an almost Einstein scale: First, they are precisely the oriented conformal structures c that are induced by at least two distinct oriented (2, 3, 5) distributions; in this case there is a 1-parameter family of such distributions that induce c. Second, they are characterized by the existence of a holonomy reduction to SU(1, 2), SL(3, R), or a particular semidirect product SL(2, R) Q + , according to the sign of the Einstein constant of the corresponding metric. Via the curved orbit decomposition formalism such a reduction partitions the underlying manifold into several submanifolds and endows each ith a geometric structure. This establishes novel links between (2, 3, 5) distributions and many other geometries -several classical geometries among them -including: Sasaki-Einstein geometry and its paracomplex and null-complex analogues in dimension 5; Kähler-Einstein geometry and its paracomplex and null-complex analogues, Fefferman Lorentzian conformal structures, and para-Fefferman neutral conformal structures in dimension 4; CR geometry and the point geometry of second-order ordinary differential equations in dimension 3; and projective geometry in dimension 2. We describe a generalized Fefferman construction that builds from a 4-dimensional Kähler-Einstein or para-Kähler-Einstein structure a family of (2, 3, 5) distributions that induce the same (Einstein) conformal structure. We exploit some of these links to construct new examples, establishing the existence of nonflat almost Einstein (2, 3, 5) conformal structures for which the Einstein constant is positive and negative.

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The dancing metric, ${G}_2$-symmetry and projective rolling

Cited by 19 publications

References 25 publications

Para-Kähler-Einstein 4-manifolds and non-integrable twistor distributions

Para-Kähler-Einstein 4-manifolds and non-integrable twistor distributions

Flat (2,3,5)-Distributions and Chazy's Equations

The Geometry of Almost Einstein (2,3,5) Distributions

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