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 $φ$ such that $d φ= - θ\wedge φ$ for a closed non-vanishing 1-form $θ$. Moreover, we show that if $(M, φ)$ is a compact locally conformal calibrated $G_2$-manifold with $\mathcal{L}_{θ^{\#}} φ=0$, where ${θ^{\#}}$ is the dual of $θ$ with respect to the Riemannian metric $g_φ$ induced by $φ$, then $M$ is a fiber bundle over $S^1$ with a coupled $SU(3)$-manifold as fiber.

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