Locally conformal calibrated $G_2$-manifolds

TL;DR

This paper investigates conditions under which certain 7-manifolds with a special $G_2$-structure are constructed as mapping tori of 6-manifolds with $SU(3)$-structure, revealing a fiber bundle structure under specific symmetry conditions.

Contribution

It characterizes when mapping tori of 6-manifolds with $SU(3)$-structure form locally conformal calibrated $G_2$-manifolds and describes their fiber bundle structure under symmetry assumptions.

Findings

Mapping tori of 6-manifolds with $SU(3)$-structure can form locally conformal calibrated $G_2$-manifolds.

Such $G_2$-manifolds are fiber bundles over $S^1$ with coupled $SU(3)$-manifold fibers.

Additional symmetry conditions imply a fiber bundle structure for these manifolds.

Abstract

We study conditions for which the mapping torus of a 6-manifold endowed with an $SU(3)$-structure is a locally conformal calibrated $G_2$-manifold, that is, a 7-manifold endowed with a $G_2$-structure $\varphi$ such that $d \varphi = - \theta \wedge \varphi$ for a closed non-vanishing 1-form $\theta$. Moreover, we show that if $(M, \varphi)$ is a compact locally conformal calibrated $G_2$-manifold with $\mathcal{L}_{\theta^{\#}} \varphi =0$, where ${\theta^{\#}}$ is the dual of $\theta$ with respect to the Riemannian metric $g_{\varphi}$ induced by $\varphi$, then $M$ is a fiber bundle over $S^1$ with a coupled $SU(3)$-manifold as fiber.