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

TL;DR

This paper explores the relationship between GL(2, R) structures, G_2 geometry, and twistor theory, focusing on seven-dimensional conformal structures derived from polynomial identifications and their connection to rational curves and special holonomy metrics.

Contribution

It establishes a link between GL(2, R) structures and conformal G_2 structures in seven dimensions, providing explicit examples from rational curves and 7th order ODEs.

Findings

Identification of conformal G_2 structures from GL(2, R) structures

Examples of G_2 structures from rational curves and ODEs

Uniqueness of Bryant's weak G_2 metric on SO(5)/SO(3)

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 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.