The classification of ERP G2-structures on Lie groups

LAURET, JORGE; NICOLINI, MARINA

Torino

Seminario; Seminario di Geometria; 2019

Dipartimento di Matematica, Università di Torino

On a differentiable 7-manifold, a G2-structure is a differentiable 3-form satisfying certain positivity condition. A closed G2-structure is called extremally Ricci pinched (ERP) if the torsion 2-form of the G2-structure satisfies some equation. There were only two known examples of ERP G2-structures, one given by Bryant and the other one by Lauret, both homogeneous. Furthermore, Fino and Raffero gave a continuos family of examples that are all equivalent to Bryant's example but on pairwise nonisomorphic Lie groups. 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 classication of left-invariant ERP G2-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 G2-structure is exact in all cases except one, given by the only example in which the Lie group is unimodular.