An Integrability Condition for Simple Lie Groups II

Abstract

It is shown that a simple Lie group $G$ ($\ne {\mathrm{SL}}_2$) can be locally characterised by an integrability condition on an $\mathrm{Aut}(\mathfrak{g})$ structure on the tangent bundle, where $\mathrm{Aut}(\mathfrak{g})$ is the automorphism group of the Lie algebra of $G$. The integrability condition is the vanishing of a torsion tensor of type $(1,2)$. This is a slight improvement of an earlier result proved in [Min-Oo M., Ruh E.A., in Differential Geometry and Complex Analysis, Springer, Berlin, 1985, 205-211].