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 Vn. A submanifold X ⊂ Gr(d, n) gives rise to a differential system Σ(X) that governs d-dimensional submanifolds of Vn 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 P4 → Gr(3, 5)). The fourfolds corresponding to ‘generic’ integrable systems are not algebraic, and can be parametrised by generalised hypergeometric functions.