A G2-structure on a 7-manifold M is called a G2T-structure if M admits a G2-connection ∇^T with totally skew-symmetric torsion T_φ. If furthermore, T_φ is closed then it is called a strong G2T-structure. In this paper we investigate the geometry of (strong) G2T-manifolds in relation to its curvature, S1 action and almost Hermitian structures. In particular, we study the Ricci flatness condition of ∇^T and give an equivalent characterisation in terms of geometric properties of the G2 Lee form. Analogous results are also obtained for almost Hermitian 6-manifolds with skew-symmetric Nijenhuis tensor. Moreover, by considering the S1 reduction by the dual of the G2 Lee form, we show that Ricci-flat strong G2T-structures correspond to solutions of the 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 G2T-structures with ∇T not Ricci flat. Lastly, we classify G2-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.
Some remarks on strong G2-structures with torsion
A. Fino;
2026-01-01
Abstract
A G2-structure on a 7-manifold M is called a G2T-structure if M admits a G2-connection ∇^T with totally skew-symmetric torsion T_φ. If furthermore, T_φ is closed then it is called a strong G2T-structure. In this paper we investigate the geometry of (strong) G2T-manifolds in relation to its curvature, S1 action and almost Hermitian structures. In particular, we study the Ricci flatness condition of ∇^T and give an equivalent characterisation in terms of geometric properties of the G2 Lee form. Analogous results are also obtained for almost Hermitian 6-manifolds with skew-symmetric Nijenhuis tensor. Moreover, by considering the S1 reduction by the dual of the G2 Lee form, we show that Ricci-flat strong G2T-structures correspond to solutions of the 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 G2T-structures with ∇T not Ricci flat. Lastly, we classify G2-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.
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