# Locally conformal calibrated $$G_2$$ G 2 -manifolds

Research paper by Marisa FernÃ¡ndez, Anna Fino; Alberto Raffero

Indexed on: 31 Aug '16Published on: 01 Oct '16Published in: Annali di Matematica Pura ed Applicata (1923 -)

#### Abstract

Abstract We study conditions for which the mapping torus of a 6-manifold endowed with an $$\mathrm{SU}(3)$$ -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 nonvanishing 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 $$\mathrm{SU}(3)$$ -manifold as fiber.AbstractWe study conditions for which the mapping torus of a 6-manifold endowed with an $$\mathrm{SU}(3)$$ -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 nonvanishing 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 $$\mathrm{SU}(3)$$ -manifold as fiber. $$\mathrm{SU}(3)$$ $$\mathrm{SU}(3)$$ $$G_2$$ $$G_2$$ $$G_2$$ $$G_2$$ $$\varphi$$ $$\varphi$$ $$d \varphi = - \theta \wedge \varphi$$ $$d \varphi = - \theta \wedge \varphi$$ $$\theta$$ $$\theta$$ $$(M, \varphi )$$ $$(M, \varphi )$$ $$G_2$$ $$G_2$$ $$\mathcal {L}_{\theta ^{\#}} \varphi =0$$ $$\mathcal {L}_{\theta ^{\#}} \varphi =0$$ $${\theta ^{\#}}$$ $${\theta ^{\#}}$$ $$\theta$$ $$\theta$$ $$g_{\varphi }$$ $$g_{\varphi }$$ $$\varphi$$ $$\varphi$$M $$S^1$$ $$S^1$$ $$\mathrm{SU}(3)$$ $$\mathrm{SU}(3)$$