1 Introduction

A \(\mathrm{GL}(2)\)-structure on a smooth \(4\)-manifold \(M\) is given by a smoothly varying family of twisted cubic curves, one in each projectivised tangent space of \(M\). Equivalently, a \(\mathrm{GL}(2)\)-structure is the same as \(G\)-structure \(\pi\colon B \to M\) on \(M\), where \(G\) is the image subgroup of the faithful irreducible \(4\)-dimensional representation of \(\mathrm{GL}(2,\mathbb{R})\) on the space of homogeneous polynomials of degree three with real coefficients in two real variables. A \(\mathrm{GL}(2)\)-structure is called torsion-free if its associated \(G\)-structure is torsion-free. Torsion-free \(\mathrm{GL}(2)\)-structures are of particular interest, as they provide examples of torsion-free connections with exotic holonomy group \(\mathrm{GL}(2,\mathbb{R})\). However, the local existence of torsion-free \(\mathrm{GL}(2)\)-structures is highly non-trivial, even when applying the Cartan–Kähler machinery, which is particularly well-suited for the construction of torsion-free connections with special holonomy. Adapting methods of Hitchin [10], Bryant [2] gave an elegant twistorial construction of real-analytic torsion-free \(\mathrm{GL}(2)\)-structures in dimension four, thus providing the first example of an irreducibly-acting holonomy group of a (non-metric) torsion-free connection missing from Berger’s list [1] of such connections.

A natural source for \(\mathrm{GL}(2)\)-structures are differential operators. Recall that the principal symbol \(\sigma(\mathrm{D})\) of a \(k\)-th order linear differential operator \(\mathrm{D} \colon C^{\infty}(M,\mathbb{R}^n) \to C^{\infty}(M,\mathbb{R}^m)\) assigns to each point \(p \in M\) a homogeneous polynomial of degree \(k\) on \(T^*_pM\), with values in \(\mathrm{Hom}(\mathbb{R}^n,\mathbb{R}^m)\). Therefore, in each projectivised cotangent space \(\mathbb{P}(T^*_pM)\) of \(M\) we obtain the so-called characteristic variety \(\Xi_p\) of \(\mathrm{D}\), consisting of those \([\xi] \in \mathbb{P}(T^*_pM)\), for which the linear mapping \(\sigma_{\xi}(\mathrm{D}) \colon \mathbb{R}^n \to \mathbb{R}^m\) fails to be injective. Given a (possibly non-linear) differential operator \(\mathrm{D}\) and a smooth \(\mathbb{R}^n\)-valued function \(u\) defined on some open subset \(U\subset M\) and which satisfies \(\mathrm{D}(u)=0\), we may ask that the linearisation \(\mathrm{L}_u(\mathrm{D})\) of \(\mathrm{D}\) around \(u\) has characteristic varieties all of which are the tangential variety of the twisted cubic curve. Consequently, one obtains a \(\mathrm{GL}(2)\)-structure on the domain of definition of each solution \(u\) of the PDE \(\mathrm{D}(u)=0\) for an appropriate class of differential operators. Various examples of such operators have recently been given by Ferapontov–Kruglikov [7]. In particular, they show that locally all torsion-free \(\mathrm{GL}(2)\)-structures arise in this fashion for some second order operator \(\mathrm{D}\), which furthermore has the property that the PDE \(\mathrm{D}(u)=0\) admits a dispersionless Lax representation. We also refer the reader to [8] for an application of similar ideas to the case of three-dimensional Einstein–Weyl structures.

Here we show that if a \(4\)-manifold \(M\) carries a torsion-free \(\mathrm{GL}(2)\)-structure \(\pi \colon B \to M\), then for every point \(p \in M\) there exists a \(p\)-neighbourhood \(U_p\), local coordinates \(x \colon U_p \to \mathbb{R}^4\) and a mapping \(h \colon U_p \to H\) into a certain \(4\)-dimensional subgroup \(H\subset \mathrm{SL}(4,\mathbb{R})\), so that the coframing \(\eta=h\, \mathrm{d}x\) is a local section of \(\pi \colon B \to M\). The group \(H\) is isomorphic to a semidirect product of the three-dimensional continuous Heisenberg group \(H_3(\mathbb{R})\) and the Abelian group \(\mathbb{R}\). Moreover, the mapping \(h\) satisfies a first order quasi-linear PDE system which admits a dispersionless Lax-pair. As in [7], linearising the PDE system around a solution \(h\) gives a linear first order differential operator whose characteristic variety is the tangential variety of the twisted cubic curve. Also, note that our result shows that \(4\)-dimensional torsion-free \(\mathrm{GL}(2)\)-structures are \(H\)-flat, that is, flat up to a coframe transformation with a mapping taking values in \(H\).

Along the way (see Theorem 2.4), we derive a first order PDE describing general \(H\)-flat torsion-free \(G\)-structures which may be of independent interest.

Acknowledgments

The authors would like to thank Maciej Dunajski and Evgeny Ferapontov for helpful conversations and correspondence. The authors are also grateful to the anonymous referees for their careful reading and useful suggestions.