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A GL(2, R) structure on an (n+1)-dimensional manifold is a smooth pointwiseidentification of tangent vectors with polynomials in two variables homogeneousof 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 vanishingquadratic invariant. We focus on the case n=6 and show that the resultingconformal structure in seven dimensions is compatible with a conformal G_2structure or its non-compact analogue. If a GL(2, R) structure arises on amoduli 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 rationaland find the corresponding G_2 structures. In particular we show that Bryant'sweak G_2 holonomy metric on the homology seven-sphere SO(5)/SO(3) is the uniqueweak G_2 metric arising from a rational curve.

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