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Dispersionless integrable hierarchies and GL(2, R) geometry

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posted on 29.08.2019, 13:53 by Evgeny FerapontovEvgeny Ferapontov, Boris Kruglikov
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.

Funding

LMS

History

School

  • Science

Department

  • Mathematical Sciences

Published in

Mathematical Proceedings of the Cambridge Philosophical Society

Volume

170

Issue

1

Pages

129-154

Publisher

Cambridge University Press

Version

AM (Accepted Manuscript)

Rights holder

© Cambridge Philosophical Society

Publisher statement

This article has been published in a revised form in Mathematical Proceedings of the Cambridge Philosophical Society https://doi.org/10.1017/S0305004119000355. This version is published under a Creative Commons CC-BY-NC-ND. No commercial re-distribution or re-use allowed. Derivative works cannot be distributed. © Cambridge Philosophical Society.

Acceptance date

28/08/2019

Publication date

2019-10-08

Copyright date

2021

ISSN

0305-0041

eISSN

1469-8064

Language

en

Depositor

Prof Evgeny Ferapontov