Integrable Background Geometries

Calderbank, David M. J.

doi:10.3842/sigma.2014.034

Cited by 34 publications

(71 citation statements)

References 77 publications

(208 reference statements)

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“…Taking integral surfaces of the distribution spanned by X, Y in the extended four-space with coordinates x, y, t, λ, and projecting them down to the space of the independent variables x, y, t, we obtain the required two-parameter family of null totally geodesic surfaces. Let us mention that relations of dispersionless integrable systems in 3D to Einstein-Weyl geometry have been discussed previously in [9,16,24,52], see also references therein.…”

Section: Einstein-weyl Geometry In 3dmentioning

confidence: 96%

“…It turns out that the signature of g is always Lorentzian, and thus our PDE system is hyperbolic. Geometric aspects of conformal structures defined by the characteristic variety will play a key role in our characterisation of integrable systems: we will see that solutions to integrable equations carry integrable background geometry [9]. In 3D, this is the Einstein-Weyl geometry.…”

Section: Non-degeneracy Conditionmentioning

confidence: 99%

“…where the phases R i satisfy equation (6), and substituting this ansatz into (9), we obtain the relations…”

Section: Derivation Of the Integrability Conditionsmentioning

confidence: 99%

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On integrability in Grassmann geometries: integrable systems associated with fourfolds in Gr(3,5)

Doubrov

Ferapontov

Kruglikov

et al. 2018

Proc. London Math. Soc.

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Let Gr(d, n) be the Grassmannian of d-dimensional linear subspaces of an n-dimensional vector space V n . A submanifold X ⊂ Gr(d, n) gives rise to a differential system Σ(X) that governs d-dimensional submanifolds of V n whose Gaussian image is contained in X. Systems of the form Σ(X) appear in numerous applications in continuum mechanics, theory of integrable systems, general relativity and differential geometry. They include such well-known examples as the dispersionless Kadomtsev-Petviashvili equation, the Boyer-Finley equation, Plebańsky's heavenly equations and so on.In this paper we concentrate on the particularly interesting case of this construction where X is a fourfold in Gr(3, 5). Our main goal is to investigate differential-geometric and integrability aspects of the corresponding systems Σ(X). We demonstrate the equivalence of several approaches to dispersionless integrability such as• the method of hydrodynamic reductions;• the method of dispersionless Lax pairs; • integrability on solutions, based on the requirement that the characteristic variety of system Σ(X) defines an Einstein-Weyl geometry on every solution; • integrability on equation, meaning integrability (in twistor-theoretic sense) of the canonical GL(2, R) structure induced on a fourfold X ⊂ Gr(3, 5).All these seemingly different approaches lead to one and the same class of integrable systems Σ(X). We prove that the moduli space of such systems is six-dimensional. We give a complete description of linearisable systems (the corresponding fourfold X is a linear section of Gr(3, 5)) and linearly degenerate systems (the corresponding fourfold X is the image of a quadratic map P 4 Gr(3, 5)). The fourfolds corresponding to 'generic' integrable systems are not algebraic, and can be parametrised by generalised hypergeometric functions.

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Section: Einstein-weyl Geometry In 3dmentioning

confidence: 96%

Section: Non-degeneracy Conditionmentioning

confidence: 99%

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On integrability in Grassmann geometries: integrable systems associated with fourfolds in Gr(3,5)

Doubrov

Ferapontov

Kruglikov

et al. 2018

Proc. London Math. Soc.

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“…[22,23]), although the projective invariance is not noted there. (The fact that reductions of the selfduality equations for conformal structures result in gauge field equations given by reductions of selfdual Yang-Mills equations is a general phenomenon, which I have tried to systematize in my work on integrable background geometries [4], and that formalism gives another way to obtain some of the results of the present paper.) Theorem 3.2.…”

Section: The Inverse Construction From Projective Pairsmentioning

confidence: 99%

“…5 Hyperkähler, hypercomplex, and scalar-f lat Kähler structures There is a standard approach to this question in the context of reductions, using the theory of divisors (see also [3,4,5,9]). An antiselfdual (null-or pseudo-) complex structure J corresponds to a degree two divisor D in the twistor space Z, which locally has the form D = D 1 + D 2 with D 1 = D 2 in the null case, but disjoint otherwise.…”

Section: The Dunajski-west Constructionmentioning

confidence: 99%

Selfdual 4-Manifolds, Projective Surfaces, and the Dunajski-West Construction

Calderbank¹

2014

SIGMA

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Abstract. I present a construction of real or complex selfdual conformal 4-manifolds (of signature (2, 2) in the real case) from a natural gauge field equation on a real or complex projective surface, the gauge group being the group of diffeomorphisms of a real or complex 2-manifold. The 4-manifolds obtained are characterized by the existence of a foliation by selfdual null surfaces of a special kind. The classification by Dunajski and West of selfdual conformal 4-manifolds with a null conformal vector field is the special case in which the gauge group reduces to the group of diffeomorphisms commuting with a vector field, and I analyse the presence of compatible scalar-flat Kähler, hypercomplex and hyperkähler structures from a gauge-theoretic point of view. In an appendix, I discuss the twistor theory of projective surfaces, which is used in the body of the paper, but is also of independent interest.

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Second-Order PDEs in 3D with Einstein–Weyl Conformal Structure

Berjawi

Ferapontov

Kruglikov

et al. 2021

Ann. Henri Poincaré

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Einstein–Weyl geometry is a triple $$({\mathbb {D}},g,\omega )$$ ( D , g , ω ) where $${\mathbb {D}}$$ D is a symmetric connection, [g] is a conformal structure and $$\omega $$ ω is a covector such that $$\bullet $$ ∙ connection $${\mathbb {D}}$$ D preserves the conformal class [g], that is, $${\mathbb {D}}g=\omega g$$ D g = ω g ; $$\bullet $$ ∙ trace-free part of the symmetrised Ricci tensor of $${\mathbb {D}}$$ D vanishes. Three-dimensional Einstein–Weyl structures naturally arise on solutions of second-order dispersionless integrable PDEs in 3D. In this context, [g] coincides with the characteristic conformal structure and is therefore uniquely determined by the equation. On the contrary, covector $$\omega $$ ω is a somewhat more mysterious object, recovered from the Einstein–Weyl conditions. We demonstrate that, for generic second-order PDEs (for instance, for all equations not of Monge–Ampère type), the covector $$\omega $$ ω is also expressible in terms of the equation, thus providing an efficient ‘dispersionless integrability test’. The knowledge of g and $$\omega $$ ω provides a dispersionless Lax pair by an explicit formula which is apparently new. Some partial classification results of PDEs with Einstein–Weyl characteristic conformal structure are obtained. A rigidity conjecture is proposed according to which for any generic second-order PDE with Einstein–Weyl property, all dependence on the 1-jet variables can be eliminated via a suitable contact transformation.

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Integrable Background Geometries

Cited by 34 publications

References 77 publications

On integrability in Grassmann geometries: integrable systems associated with fourfolds in Gr(3,5)

On integrability in Grassmann geometries: integrable systems associated with fourfolds in Gr(3,5)

Selfdual 4-Manifolds, Projective Surfaces, and the Dunajski-West Construction

Second-Order PDEs in 3D with Einstein–Weyl Conformal Structure

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