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

Abstract

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.

Integrability of Dispersionless Hirota-Type Equations and the Symplectic Monge–Ampère Property

“…This point of view has been developed in [24,57] leading to remarkable connections with integrable GL(2, R) geometry. Integrability aspects of dispersionless systems related to Grassmann geometries were recently studied in [15,16].…”

“…One can verify that dω = 0. Equations (15) and (17) constitute an involutive differential system for f which characterises Monge-Ampère equations. It remains to point out that equations (17) can be obtained from the consistency of equations (15) alone, without invoking (16).…”

Three dimensional near-horizon metrics that are Einstein-Weyl

“…Einstein-Weyl geometries in 3 dimensions play an important role because of its relationship with the geometry of third order ODEs (see [20] and [21]), twistor theory (see [12] and [15]) and integrable systems (see [1], [2], [4] and [11]). Let (M 3 , [g]) be a smooth conformal manifold equipped with a conformal class of metrics [g] of (pseudo)-Riemannian signature.…”

“…Let M 3 be a three dimensional smooth manifold equipped with a Lorentzian metric. A near-horizon metric on M 3 is a Lorentzian metric of the form (1) g N H = 2dν dr + rh(x)dx + r 2 2 F (x)dν + dxdx, where x, ν and r are local coordinates and h(x), F (x) are arbitrary functions of x. Near-horizon geometries in higher dimensions are studied in relation to the existence of extremal black holes [5], [14], [16], [17], [18]. The near-horizon metric (1) is derived as follows.…”

On a class of integrable systems of Monge-Ampère type

“…, x d ). In paper [5] we have initiated the study of integrability of first-order systems of the form F (u 1 , . .…”