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Einstein locally conformal calibrated G_2-structures

Research paper by Anna Fino, Alberto Raffero

Indexed on: 15 May '15Published on: 15 May '15Published in: Mathematische Zeitschrift



Abstract

We study locally conformal calibrated \(G_2\)-structures whose underlying Riemannian metric is Einstein, showing that in the compact case the scalar curvature cannot be positive. As a consequence, 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 \(\mathrm{SU}(3)\)-structure \((\omega , \Psi )\), i.e., such that \(d \omega = c \mathrm{Re}(\Psi )\), with \(c \in {\mathbb {R}}- \{ 0 \}\). Nilpotent Lie algebras admitting a coupled \(\mathrm{SU}(3)\)-structure are also classified.