GL(2, R) structures, G2 geometry and twistor theory
Abstract
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 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 7th order ODEs 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.