G_2-structures on Einstein solvmanifolds

TL;DR

This paper investigates the existence of special $G_2$-structures on 7-dimensional Einstein solvmanifolds, proving non-existence results for certain structures and providing examples of Ricci-soliton structures.

Contribution

It establishes that non-flat Einstein solvmanifolds cannot admit certain calibrated or cocalibrated $G_2$-structures, and provides an example of a Ricci-soliton $G_2$-structure.

Findings

No non-flat Einstein solvmanifold admits a calibrated $G_2$-structure with Einstein metric.

A 7-dimensional solvmanifold can admit a Ricci-soliton $G_2$-structure.

Non-flat Einstein solvmanifolds do not admit cocalibrated $G_2$-structures with the same metric.

Abstract

We study the $G_2$ analogue of the Goldberg conjecture on non-compact solvmanifolds. In contrast to the almost-K\"ahler case we prove that a 7-dimensional solvmanifold cannot admit any left-invariant calibrated $G_2$-structure $\varphi$ such that the induced metric $g_{\varphi}$ is Einstein, unless $g_{\varphi}$ is flat. We give an example of 7-dimensional solvmanifold admitting a left-invariant calibrated $G_2$-structure $\varphi$ such that $g_{\varphi}$ is Ricci-soliton. Moreover, we show that a 7-dimensional (non-flat) Einstein solvmanifold $(S,g)$ cannot admit any left-invariant cocalibrated $G_2$-structure $\varphi$ such that the induced metric $g_{\varphi} = g$.