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 $\phi$ such that $d \varphi = - \theta \wedge \varphi$foraclosednon-vanishing1-form$\theta$. Moreover, we show that if $(M,\phi )$ is a compact locally conformal calibrated $G_2$-manifold with ${\mathcal{L}}_{{\theta }^{\mathrm{\#}}}\phi =0$, where ${\theta }^{\mathrm{\#}}$ is the dual of $\theta$ with respect to the Riemannian metric $g_{\phi }$ induced by $\phi$, then $M$ is a fiber bundle over $S^1$ with a coupled $SU(3)$-manifold as fiber.