Connections on semisimple Lie groups

Abstract

The plus and minus connections of Cartan and Schouten, which exist on any Lie group, have the following three properties: (1) the connection is left invariant, (2) the curvature of the connection is zero, (3) the set of maximal geodesics through the identity of the Lie group is equal to the set of one-parameter subgroups of the Lie group. It is shown that the plus and minus connections are the only ones with these properties on a real simple Lie group. On a real semisimple Lie group the connections with these properties are in one-to-one correspondence with the ways of choosing an ideal of the Lie algebra and then choosing a complementary subspace to it.