Dispersionless integrable hierarchies and GL(2,R) geometry

Abstract

Paraconformal or GL(2, ℝ) geometry on an <i>n</i>-dimensional manifold <i>M</i> is defined by a field of rational normal curves of degree <i>n</i> – 1 in the projectivised cotangent bundle <i>ℙT*M</i>. 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 <i>GL</i>(2, ℝ) structures, namely dispersionless integrable hierarchies of PDEs such as the dispersionless Kadomtsev–Petviashvili (dKP) hierarchy. In the latter context, <i>GL</i>(2, ℝ) structures coincide with the characteristic variety (principal symbol) of the hierarchy.<p>

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

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