# Left invariant connections on a Lie group

The exponential map associated with the (-) - connection on a Lie group is generally not surjective. This is because, for this connection, the one-parameter subgroups and geodesics coincide. If we drop the requirement that one-parameter subgroups are geodesics, is it possible to put a left invariant connection on a Lie group for which the geodesic exponential map is surjective?

• Thanks for the answer, Robert. One implication of this is that the $(0)$-connection on a non-compact Lie group will generally not be metric. Jul 29, 2018 at 23:15