Einstein locally conformal calibrated $G_2$-structures

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.

16 pages, to appear in Math. Z

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