Some remarks on strong $\mathrm{G}_2$-structures with torsion

TL;DR

This paper explores the geometry of strong G2-structures with torsion on 7-manifolds, relating their curvature, symmetries, and reductions to almost Hermitian structures, and provides explicit examples and classifications.

Contribution

It introduces new characterizations of strong G2T-structures, constructs the first examples with non-Ricci-flat torsion, and classifies G2-flows related to generalized Ricci flow.

Findings

Ricci-flatness characterized by G2 Lee form properties

Explicit examples of strong G2T-structures with non-Ricci-flat torsion

Classification of G2-flows inducing solutions to generalized Ricci flow

Abstract

A $\mathrm{G}_2$-structure on a $7$-manifold $M$ is called a $\mathrm{G}_2T$-structure if $M$ admits a $\mathrm{G}_2$-connection $\nabla^T$ with totally skew-symmetric torsion $T_\varphi$. If furthermore, $T_\varphi$ is closed then it is called a strong $\mathrm{G}_2T$-structure. In this paper we investigate the geometry of (strong) $\mathrm{G}_2T$-manifolds in relation to its curvature, $S^1$ action and almost Hermitian structures. In particular, we study the Ricci flatness condition of $\nabla^T$ and give an equivalent characterisation in terms of geometric properties of the $\mathrm{G}_2$ Lee form. Analogous results are also obtained for almost Hermitian $6$-manifolds with skew-symmetric Nijenhuis tensor. Moreover, by considering the $S^1$ reduction by the dual of the $\mathrm{G}_2$ Lee form, we show that Ricci-flat strong $\mathrm{G}_2T$-structures correspond to solutions of the $\mathrm{SU}(3)$ heterotic system on certain almost Hermitian half-flat $6$-manifolds. Many explicit examples are described and in particular, we construct the first examples of strong $\mathrm{G}_2T$-structures with $\nabla^T$ not Ricci flat. Lastly, we classify $\mathrm{G}_2$-flows inducing gauge fixed solutions to the generalised Ricci flow akin to the pluriclosed flow in complex geometry. The approach is this paper is based on the representation theoretic methods due to Bryant.