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Covariant derivative from connections

  1. Oct 22, 2011 #1

    lavinia

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    On a 2 dimensional Riemannian manifold how does one derive the covariant derivative from the connection 1 form on the tangent unit circle bundle?
     
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  3. Oct 22, 2011 #2

    Ben Niehoff

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    Its the same formula in any dimension. Let [itex]X = X^a e_a[/itex] be a vector field, where [itex]e_a[/itex] is an orthonormal frame. Then

    [tex]\nabla X = (D X^a) \otimes e_a = (d X^a + \omega^a{}_b X^b) \otimes e_a[/tex]
     
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