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Lie group geometry

  1. Jul 29, 2006 #1
    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.
  2. jcsd
  3. Aug 18, 2009 #2


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    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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