Nilmanifolds with a calibrated G_2-structure

Abstract

We introduce obstructions to the existence of a calibrated G_2-structure on a Lie algebra g of dimension seven, not necessarily nilpotent. In particular, we prove that if there is a Lie algebra epimorphism from g to a six-dimensional Lie algebra h with kernel contained in the center of g, then h has a symplectic form. As a consequence, we obtain a classification of the nilpotent Lie algebras that admit a calibrated G_2-structure.

21 pages; v2: added some introductory details on G_2 structures in Section 2, exposition improved. To appear in Differential Geometry and its Applications

References in corpus (1)

• Half-flat nilmanifolds

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• Closed $\text{SL}(3,\mathbb{C})$-structures on nilmanifolds
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• Moduli spaces of (co)closed ${\mathrm{G}}_2$-structures on nilmanifolds
• Closed G${}_2$-structures on unimodular Lie algebras with non-trivial center