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arXiv · 1207.3616

G_2-structures on Einstein solvmanifolds

Marisa Fernández·Anna Fino·Victor Manero
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Abstract

We study the $G_2$ analogue of the Goldberg conjecture on non-compact solvmanifolds. In contrast to the almost-Kähler case we prove that a 7-dimensional solvmanifold cannot admit any left-invariant calibrated $G_2$-structure $φ$ such that the induced metric $g_φ$ is Einstein, unless $g_φ$ is flat. We give an example of 7-dimensional solvmanifold admitting a left-invariant calibrated $G_2$-structure $φ$ such that $g_φ$ 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 $φ$ such that the induced metric $g_φ = g$.

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Marisa Fernández, Anna Fino, Victor Manero. 2013-12-30. G_2-structures on Einstein solvmanifolds. https://arxiv.org/abs/1207.3616

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