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Dispersionless integrable hierarchies and GL(2, ℝ) geometry

Part of: Equations of mathematical physics and other areas of application Local differential geometry Global differential geometry Infinite-dimensional Hamiltonian systems Classical differential geometry

Published online by Cambridge University Press:  08 October 2019

EVGENY FERAPONTOV  and
BORIS KRUGLIKOV
EVGENY FERAPONTOV
Affiliation:
Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire LE11 3TU. Institute of Mathematics, Ufa Federal Research Centre, Russian Academy of Sciences, Ufa 450008, Russia. e-mail: [email protected]
BORIS KRUGLIKOV
Affiliation:
Department of Mathematics and Statistics, UiT the Arctic University of Norway, Tromsø 9037, Norway. e-mail: [email protected]
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Abstract

Paraconformal or GL(2, ℝ) geometry on an n-dimensional manifold M is defined by a field of rational normal curves of degree n – 1 in the projectivised cotangent bundle ℙT*M. Such geometry is known to arise on solution spaces of ODEs with vanishing Wünschmann (Doubrov–Wilczynski) invariants. In this paper we discuss yet another natural source of GL(2, ℝ) structures, namely dispersionless integrable hierarchies of PDEs such as the dispersionless Kadomtsev–Petviashvili (dKP) hierarchy. In the latter context, GL(2, ℝ) structures coincide with the characteristic variety (principal symbol) of the hierarchy.

Dispersionless hierarchies provide explicit examples of particularly interesting classes of involutive GL(2, ℝ) structures studied in the literature. Thus, we obtain torsion-free GL(2, ℝ) structures of Bryant [5] that appeared in the context of exotic holonomy in dimension four, as well as totally geodesic GL(2, ℝ) structures of Krynski [33]. The latter possess a compatible affine connection (with torsion) and a two-parameter family of totally geodesic α-manifolds (coming from the dispersionless Lax equations), which makes them a natural generalisation of the Einstein–Weyl geometry.

Our main result states that involutive GL(2, ℝ) structures are governed by a dispersionless integrable system whose general local solution depends on 2n – 4 arbitrary functions of 3 variables. This establishes integrability of the system of Wünschmann conditions.

MSC classification

Primary: 35Q51: Soliton-like equations 37K10: Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies (KdV, KP, Toda, etc.) 37K25: Relations with differential geometry 53A40: Other special differential geometries 53B05: Linear and affine connections 53B50: Applications to physics 53C26: Hyper-Kähler and quaternionic Kähler geometry, “special” geometry 53C80: Applications to physics
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 170 , Issue 1 , January 2021 , pp. 129 - 154
Copyright
© Cambridge Philosophical Society 2019

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