We study locally conformal calibrated $G_2$-structures whose underlying Riemannian metric is Einstein, showing an integral formula for compact manifolds. We show that a compact homogeneous 7-manifold cannot admit an invariant Einstein locally conformal calibrated $G_2$-structure unless the underlying metric is flat. In contrast to the compact case we provide a non-compact example of homogeneous manifold endowed with a locally conformal calibrated $G_2$-structure whose associated Riemannian metric is Einstein and non Ricci-flat. The homogeneous Einstein metric is a rank-one extension of a Ricci soliton on the 3-dimensional complex Heisenberg group endowed with a left-invariant coupled half-flat SU(3)-structure $(\omega, \psi)$ such that $d \omega = - Re(\psi)$.

Einstein locally conformal calibrated G_2-structures

FINO, Anna Maria;RAFFERO, ALBERTO
2015

Abstract

We study locally conformal calibrated $G_2$-structures whose underlying Riemannian metric is Einstein, showing an integral formula for compact manifolds. We show that a compact homogeneous 7-manifold cannot admit an invariant Einstein locally conformal calibrated $G_2$-structure unless the underlying metric is flat. In contrast to the compact case we provide a non-compact example of homogeneous manifold endowed with a locally conformal calibrated $G_2$-structure whose associated Riemannian metric is Einstein and non Ricci-flat. The homogeneous Einstein metric is a rank-one extension of a Ricci soliton on the 3-dimensional complex Heisenberg group endowed with a left-invariant coupled half-flat SU(3)-structure $(\omega, \psi)$ such that $d \omega = - Re(\psi)$.
280
3-4
1093
1106
http://link.springer.com/article/10.1007/s00209-015-1468-x
A. Fino; A. Raffero
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2318/135182
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