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Math Help - Lie Groups and riemannian metrics

  1. #1
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    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.

    Thanks in advance.
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  2. #2
    Super Member Rebesques's Avatar
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    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.
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