(Purely) coclosed G$_2$-structures on 2-step nilmanifolds

Abstract: In Riemannian geometry, simply connected nilpotent Lie groups endowed with left-invariant metrics, and their compact quotients, have been the source of valuable examples in the field. This motivated several authors to study, in particular, left-invariant G$_2$-structures on 7-dimensional nilpotent Lie groups. These structures could also be induced to the associated compact quotients, also known as nilmanifolds.

Left-invariant torsion free G$_2$-structures, that is, defined by a simultaneously closed and coclosed positive $3$-form, do not exist on nilpotent Lie groups. But relaxations of this condition have been the subject of study on nilmanifolds lately. One of them are coclosed G$_2$-structures, for which the defining $3$-form verifies $\mathrm{d} \star_{g_\varphi}\varphi=0$, and more specifically, purely coclosed structures, which are defined as those which are coclosed and satisfy $\varphi\wedge \mathrm{d} \varphi=0$.

In this talk, there will be presented recent classification results regarding left-invariant coclosed and purely coclosed G$_2$-structures on 2-step nilpotent Lie groups.

Our results are twofold. On the one hand we give the isomorphism classes of 2-step nilpotent Lie algebras admitting purely coclosed G$_2$-structures. The analogous result for coclosed structures was obtained by Bagaglini, Fernández and Fino [Forum Math. 2018].

On the other hand, we focus on the question of which metrics on these Lie algebras can be induced by a coclosed or purely coclosed structure. We show that any left-invariant metric is induced by a coclosed structure, whereas every Lie algebra admitting purely coclosed structures admits metrics which are not induced by any such a structure. In the way of proving these results we obtain a method to construct purely coclosed G$_2$-structures. As a consequence, we obtain new examples of compact nilmanifolds carrying purely coclosed G$_2$-structures.

This is joint work with Andrei Moroianu and Alberto Raffero.

differential geometry

Audience: researchers in the topic

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Differential Geometry Seminar Torino

Series comments: This is a hybrid seminar organized by the Differential Geometry groups of Università and Politecnico di Torino.

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