Lie group, Riemannian metric, and connection

[marton]

hello, i have met with a problem. please help me.
A Lie group,with a left-invariant Riemannian metric, i want to compute the connection compatible with the Riemannian metric. C(ij, k) are the structure constants, g(ij) are the metric, then how to compute the Riemannian connection in terms of g(ij) and C(ij, k)?

thanks a lot.

 

[mathwonk]

there is a unique symmetric connexion compatible with a riemannian metric. see pages 48-49 of john milnors morse theory, for the formulas.

 

[Doodle Bob]

Also, check out p. 55 in Riemannian Geometry by Do Carmo. The equation in the middle of the page is precisely the way to find the answer to your question.