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

Research paper by Marisa Fernández, 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 \(S^1\) \(S^1\) \(\mathrm{SU}(3)\) \(\mathrm{SU}(3)\)