# Thread: Lie Groups and riemannian metrics

1. ## Lie Groups and riemannian metrics

Hello,
I will write del_X(Y) when I refer to the levi-chivita connection (used on Y in the direction of X).

Let G be a lie group, with a bi-invariant metric , g , on G.
I want to prove that del_X(Y) = 0.5 [X,Y] (Lie brackets) , whenever X,Y are left-invariant vector fields on G.

2. There is a fundamental formula expressing the connection in terms of the metric and Lie products, try it out using the two additional properties the Levi-Civita connection has, cf. Do Carmo's book (and I have found the formula nowhere online ). Don't know any other textbooks equally famous.