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A GL(2, ℝ) structure on an ( n  + 1)-dimensional manifold is a smooth point-wise identification of tangent vectors with polynomials in two variables homogeneous of degree  n . This, for even  n  = 2 k , 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 seventh-order ordinary differential equations 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.

GL(2, ℝ) STRUCTURES, G 2 GEOMETRY AND TWISTOR THEORY