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

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

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

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#### 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
\(S^1\)
\(S^1\)
\(\mathrm{SU}(3)\)
\(\mathrm{SU}(3)\)