Session S23 - Group actions in Differential Geometry

Friday, July 16, 19:00 ~ 19:30 UTC-3

The classification of ERP $G_2$-structures on Lie groups

Marina Nicolini

Universidad Nacional de Córdoba, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

On a differentiable 7-manifold, a $G_2$-structure $\varphi$ is a differentiable 3-form satisfying certain positivity condition. A closed $G_2$-structure (i.e. $d\varphi=0$) is called extremally Ricci pinched (ERP) if $d\tau=\frac{1}{6}|\tau|^2\varphi+\frac{1}{6}\ast(\tau\wedge\tau)$, where $\tau$ is the torsion 2-form of $\varphi$. There were only two known examples of ERP $G_2$-structures, one given by Bryant and the other one by Lauret, both homogeneous.

We first proved that some strong structural conditions must hold on the Lie algebra for the existence of ERP structures. Secondly, by using such a structural theorem we have obtained a complete classification of left-invariant ERP structures on Lie groups, up to equivalence and scaling. There are five of them, they are defined on five different completely solvable Lie groups and the $G_2$-structure is exact in all cases except one, given by the only example in which the Lie group is unimodular.

Joint work with Jorge Lauret (Universidad Nacional de Córdoba).

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