Скачать или смотреть Webinar - coclosed G2-structures - Viviana del Barco

Geometry Webinar AmSur/Am Sul 07
Title: (Purely) coclosed G2-structures on 2-step nilmanifolds
Speaker: Viviana del Barco
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 G2-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 G2-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 G2-structures, for which the defining 3-form
verifies d ?gφ φ = 0, and more specifically, purely coclosed structures, which are
defined as those which are coclosed and satisfy φ ∧ dφ = 0.

In this talk, there will be presented recent classification results regarding left-
invariant coclosed and purely coclosed G2-structures on 2-step nilpotent Lie

groups. Our techniques exploit the correspondence between left-invariant ten-
sors on the Lie group and their linear analogues at the Lie algebra level. In

particular, left-invariant G2-structures on a Lie group will be seen as alternat-
ing trilinear forms defined on the Lie algebra. The coclosed condition now refers

to the Chevalley-Eilenberg differential of the Lie algebra. We also rely on the
particular Lie algebraic structure of metric 2-step nilpotent Lie algebras.
Our goals are twofold. On the one hand we give the isomorphism classes of 2-step
nilpotent Lie algebras admitting purely coclosed G2-structures. The analogous
result for coclosed structures was obtained by Bagaglini, Fern ́andez 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 G2-structures. As a consequence, we obtain
new examples of compact nilmanifolds carrying purely coclosed G2-structures.