The intrinsic torsion of SU(3) and G_2 structures

TL;DR

This paper analyzes how the intrinsic torsion of an $SU(3)$-structure on a 6-manifold governs a $G_2$-structure on a 7-manifold. It develops explicit decompositions of the torsion spaces ($\tau_1$ and $\tau_2$) and derives precise correspondences between seven- and six-dimensional data in static, product, and fibred settings, linking $SU(3)\subset G_2$ representations. Through static, dynamic, and fibred constructions—including half-flat evolutions and circle bundles—it shows how to realize $G_2$ holonomy metrics from $SU(3)$ geometries and provides concrete nilmanifold, conical, and homogeneous examples. The results offer systematic methods to build and recognize $G_2$-manifolds from lower-dimensional geometric data and clarify how torsion components transform under various geometric constructions.

Abstract

We analyse the relationship between the components of the intrinsic torsion of an SU(3) structure on a 6-manifold and a G_2 structure on a 7-manifold. Various examples illustrate the type of SU(3) structure that can arise as a reduction of a metric with holonomy G_2.