(Purely) coclosed G2-structures on 2-step nilmanifolds - Viviana del Barco - Universidade Estadual de Campinas
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 eld. 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, de ned 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 de ning 3-form
veri es d ?g' ' = 0, and more speci cally, purely coclosed structures, which are
de ned as those which are coclosed and satisfy ' ^ d' = 0.
In this talk, there will be presented recent classi cation results regarding left-
invariant coclosed and purely coclosed G2-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 G2-structures. The
analogous result for coclosed structures was obtained by Bagaglini, Fernandez
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.
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