GL(2, ℝ) STRUCTURES, G2 GEOMETRY AND TWISTOR THEORY

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 = 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 G2 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 G2 structure vanish. We give examples of simple seventh-order ordinary differential equations whose solution curves are rational and find the corresponding G2 structures. In particular we show that Bryant's weak G2 holonomy metric on the homology seven-sphere SO(5)/SO(3) is the unique weak G2 metric arising from a rational curve.