Locally conformal calibrated $G_2$-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 $\phi$ such that $d\phi =-\theta \wedge \phi$ for a closed non-vanishing 1-form $\theta$. Moreover, we show that if $(M,\phi )$ is a compact locally conformal calibrated $G_2$-manifold with ${\mathcal{L}}_{{\theta }^{\#}}\phi =0$, where ${\theta }^{\#}$ 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.

17 pages; to appear in Annali di Matematica Pura ed Applicata

References in corpus (2)

• Laplacian Solitons and Symmetry in G_2-geometry
• Einstein locally conformal calibrated $G_2$-structures

Cited by in corpus (1)

• The Weitzenböck formula for the Fueter-Dirac operator