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

Início: 19/06/2020 17:00

Término: 19/06/2020 18:00

Palestrante: Viviana del Barco (Université Paris-Sud)

E-mail do Palestrante:

Resumo: 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 {em 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 $d star_{varphi}varphi=0$, and more specifically, purely coclosed structures, which are defined as those which are coclosed and satisfy $varphiwedge d varphi=0$. 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 techniques exploit the correspondence between left-invariant tensors on the Lie group and their linear analogues at the Lie algebra level. In particular, left-invariant G$_2$-structures on a Lie group will be seen as alternating 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 G$_2$-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 {em 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.