Are There Special Properties of Geodesics in a Lie Group?

[Andre' Quanta]

Suppose to have a Lie group that is at the same time also a Riemannian manifold: is there a relation between Christoffel symbols and structure constants? What can i say about the geodesics in a Lie group? Do they have special properties?

 

[fzero]

It depends on what sort of metric the Lie group admits. http://www.seas.upenn.edu/~jean/diffgeom.pdf seems to have a lot of the explanation that you need. Chapter 17 discusses metrics on Lie groups. If the metric is bi-invariant (see the text for the definition), then the geodesics correspond to the exponential map. Section 17.6 discusses Cartan connections, which can be defined when the metric is not bi-invariant, for which the geodesics coincide with integral curves of left-invariant vector fields.

 

[WWGD]

Wow, "class notes" consisting of 807 pages. Some prof!