An Integrability Condition for Simple Lie Groups II

TL;DR

This paper establishes a local characterization of simple Lie groups (excluding SL_2) through an integrability condition involving the automorphism group of their Lie algebra, refining previous results by linking torsion tensor vanishing to group structure.

Contribution

It introduces a refined integrability condition based on torsion tensor vanishing that characterizes simple Lie groups, improving upon earlier characterizations.

Findings

Characterizes simple Lie groups via torsion tensor vanishing

Excludes SL_2 from the characterization

Provides a more precise integrability condition than previous work

Abstract

It is shown that a simple Lie group $G$ ($ \neq {\rm SL}_2$) can be locally characterised by an integrability condition on an $\operatorname{Aut}(\mathfrak{g})$ structure on the tangent bundle, where $\operatorname{Aut}(\mathfrak{g})$ 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].