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Math Help - Lie Derivative of a Vector Field

  1. #1
    Senior Member slevvio's Avatar
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    Lie Derivative of a Vector Field

    Let X,Y be vector fields on a smooth manifold M and let \phi_t be the flow of X.

    I was trying to write the lie derivative of Y w.r.t. X in local co-ordinates, so I have that

    \phi_{-t}^* \left( \frac{\partial}{\partial x_j} \Big|_{\phi_t(p)} \right) = \displaystyle\sum_{i,j} \frac{\partial \phi_i}{\partial x_j} (-t, \phi_t (p)) \frac{\partial}{\partial x_j} \Big|_p.

    I got this because I know that

    \phi_t^* \left( \frac{\partial}{\partial x_j} \Big|_p \right) =  \displaystyle\sum_{i,j} \frac{\partial \phi_ i}{\partial x_j} (t,p)  \frac{\partial}{\partial x_j} \Big|_{\phi_t(p)},

    So I just swapped p for \phi_t(p) and t for -t.

    But I have been told this calculation should give

    \phi_{-t}^* \left( \frac{\partial}{\partial x_j} \Big|_{\phi_t(p)}  \right) = \displaystyle\sum_{i,j} \frac{\partial \phi_i}{\partial x_j}  (-t, p) \frac{\partial}{\partial x_j} \Big|_p, and it has to for various proofs to work.

    Can anyone tell me why I am wrong? I would appreciate it. Thanks.
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  2. #2
    Senior Member slevvio's Avatar
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    Re: Lie Derivative of a Vector Field

    The stars should be on the bottom and represent pushforwards
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