Uniqueness of torsion free connection on some invariant structures on Lie groups

Abstract:

Let $\mathcal {G}$ be a connected Lie group with Lie algebra $\mathfrak {g}$. Let $\operatorname {Int}(\mathfrak {g})$ be the group of inner automorphisms of $\mathfrak {g}$. The group $\mathcal {G}$ is naturally equipped with $\operatorname {Int}(\mathfrak {g})$-reductions of the bundle of linear frames on $\mathcal {G}$. We investigate for what kind of Lie group the $0$-connection of E. Cartan is the unique torsion free connection adapted to any of those $\operatorname {Int}(\mathfrak {g})$-reductions.