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.