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

Viviana Del Barco

created by raffero on 14 Apr 2021
modified on 08 Jun 2021

23 jun 2021 -- 17:00

Differential Geometry Seminar Torino (online)


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.


The link of the virtual room will be sent one day before the talk to all registered participants. To register, please send an e-mail to

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