|Jul29-06, 06:50 AM||#1|
Lie group geometry
I seem to be having difficulty proving something.
I hope you can help me.
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.
Thanks in advance.
|Aug18-09, 11:37 PM||#2|
The Levi-Civita connection can be expressed via the inner product (which is a fundamental result in metric differential geometry). Use this formula and bi-invariance to obtain the result.
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