GL(2, R) structures, G_2 geometry and twistor theory

A GL(2, R) structure on an (n+1)-dimensional manifold is a smooth pointwise identification of tangent vectors with polynomials in two variables homogeneous of degree n. This, for even n=2k, defines a conformal structure of signature (k, k+1) by specifying the null vectors to be the polynomials with vanishing quadratic invariant. We focus on the case n=6 and show that the resulting conformal structure in seven dimensions is compatible with a conformal G_2 structure or its non-compact analogue. If a GL(2, R) structure arises on a moduli space of rational curves on a surface with self-intersection number 6, then certain components of the intrinsic torsion of the G_2 structure vanish. We give examples of simple 7th order ODEs whose solution curves are rational and find the corresponding G_2 structures. In particular we show that Bryant's weak G_2 holonomy metric on the homology seven-sphere SO(5)/SO(3) is the unique weak G_2 metric arising from a rational curve.