Asian Journal of Mathematics

Volume 19 (2015)

Number 2

$G_2$-structures on Einstein solvmanifolds

Pages: 321 – 342

DOI: http://dx.doi.org/10.4310/AJM.2015.v19.n2.a7

Authors

Marisa Fernández (Departamento de Matemáticas, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Bilbao, Spain)

Anna Fino (Dipartimento di Matematica, Università di Torino, Italy)

Víctor Manero (Departamento de Matemáticas, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Bilbao, Spain)

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 $\varphi$ 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 $g_\varphi = g$.

Keywords

calibrated $G_2$-structures, cocalibrated $G_2$-structures, Einstein metrics, Riccisolitons, Kähler-Einstein metrics, solvable Lie groups

2010 Mathematics Subject Classification

Primary 53C25, 53C38. Secondary 22E25, 53C55.

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