Some remarks on strong $\mathrm{G}_2$-structures with torsion
Anna Fino, Udhav Fowdar
TL;DR
The paper develops a comprehensive local theory for strong $\mathrm{G}_2T$-structures by expressing curvature invariants entirely in terms of torsion data via Bryant's representation-theoretic approach. It establishes precise links between $\mathrm{Ric}^T=0$, the $\mathrm{G}_2$ Lee form, and the behavior of the characteristic connection, and extends these ideas to almost Hermitian $6$-manifolds with skew-symmetric Nijenhuis, including reductions to ${\rm SU}(3)$ heterotic systems under $S^1$ actions. It supplies explicit constructions and classifications of both characteristic Ricci-flat and non-Ricci-flat examples, including cohomogeneity-one and harmonic cases, and develops Gibbons–Hawking-type reductions. Finally, it proposes flow frameworks for strong $\mathrm{G}_2T$-geometries, including co-closed flows and gauge-fixed generalized Ricci flows, and studies short-time existence, constancy of torsion components, and potential connections to anomaly-type flows. These results deepen understanding of the geometry of torsion-ful $\mathrm{G}_2$ structures and open avenues for constructing new examples via reduction, symmetry, and flows.
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.