An Integrability Condition for Simple Lie Groups II

Maung Min-Oo

Department of Mathematics & Statistics, McMaster University, Hamilton, Canada

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].

Keywords: simple Lie groups and algebras; $G$-structure.

MSC: 53C10; 53C30

Received: December 17, 2014; in final form March 26, 2015; Published online April 1, 2015

Language: English

DOI: 10.3842/SIGMA.2015.027

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ArXiv: 1412.4721