Invariant torsion and G_2-metrics

TL;DR

This paper introduces a new class of G_2-metrics derived from invariant intrinsic torsion geometries, providing a classification and establishing the uniqueness of Bryant-Salamon metrics with these properties.

Contribution

It defines invariant intrinsic torsion geometries related to G_2-metrics and classifies those arising from SO(3)-structures, proving the Bryant-Salamon metric's uniqueness.

Findings

Invariant intrinsic torsion geometries are linked to G_2-holonomy metrics.

The Bryant-Salamon metric is uniquely characterized among such geometries.

The geometries are shown to be locally homogeneous and classified accordingly.

Abstract

We introduce and study a notion of invariant intrinsic torsion geometry which appears, for instance, in connection with the Bryant-Salamon metric on the spinor bundle over S^3. This space is foliated by six-dimensional hypersurfaces, each of which carries a particular type of SO(3)-structure; the intrinsic torsion is invariant under SO(3). The last condition is sufficient to imply local homogeneity of such geometries, and this allows us to give a classification. We close the circle by showing that the Bryant-Salamon metric is the unique complete metric with holonomy G_2 that arises from SO(3)-structures with invariant intrinsic torsion.