$G_2$-metrics arising from non-integrable special Lagrangian fibrations

TL;DR

This paper explores special Lagrangian fibrations in SU(3)-manifolds with non-integrable structures, decomposing them into geometric data, and applies this to describe certain G2-manifolds with group actions.

Contribution

It introduces a decomposition framework for SU(3)-structures with non-integrable fibrations and analyzes G2-manifolds with Lagrangian-type group actions using dynamical systems.

Findings

Decomposition of SU(3)-structures into solder forms, connection forms, and symmetric matrices.

Description of regular parts of G2-manifolds with T^3 and SO(3) symmetries.

Application of constrained dynamical systems to study these geometric structures.

Abstract

We study special Lagrangian fibrations of $\mathrm{SU}(3)$-manifolds, not necessarily torsion-free. In the case where the fiber is a unimodular Lie group $G$, we decompose such $\mathrm{SU}(3)$-structures into triples of solder 1-forms, connection 1-forms and equivariant $3\times3$ positive-definite symmetric matrix-valued functions on principal $G$-bundles over 3-manifolds. As applications, we describe regular parts of $G_2$-manifolds that admit Lagrangian-type 3-dimensional group actions by constrained dynamical systems on the spaces of the triples in the cases of $G=\mathrm{T}^3$ and $\mathrm{SO}(3)$.