Locally conformal calibrated $G_2$-manifolds

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.