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Differential Geometry: Lie derivative of tensor fields.

  1. Nov 1, 2011 #1

    B L

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    1. The problem statement, all variables and given/known data
    Let M be a differentiable manifold. Let X and Y be two vector fields on M, and let t be a tensor field on M. Prove
    [itex] \mathcal{L}_{[X,Y]}t = \mathcal{L}_X\mathcal{L}_Yt -\mathcal{L}_Y\mathcal{L}_Xt[/itex]

    2. Relevant equations
    All is fair game, though presumably a coordinate-free description is superior to one in local coordinates.

    3. The attempt at a solution
    I've tried working it out in local coordinates, but, because we need to show that the statement holds for a general tensor field of any type, the algebra gets very hairy very quickly. Any suggestions are much appreciated.
     
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