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Lie Derivative of one-form: an identity

  1. Mar 22, 2015 #1
    1. The problem statement, all variables and given/known data
    I am trying to prove an identity for the Lie derivative of a smooth one-form. The identity is: for X, Y smooth vector fields, alpha a smooth one-form, we have:
    $$L_{[X, Y]}\alpha = [L_X, L_Y]\alpha$$ For anyone familiar with the book, this is exercise 5.26 in the first edition of Nakahara: Geometry, Topology, and Physics.

    2. Relevant equations
    I am given the identity: for X, Y smooth vector fields, alpha a smooth one-form,
    $$(L_X\alpha)(Y)=L_X(\alpha(Y))-\alpha([X, Y])$$ ([X, Y] is the Lie Bracket of the vector fields X and Y, and $$L_XY=[X, Y])$$

    3. The attempt at a solution
    I keep trying to expand the lie derivatives and cancel terms but I think I am missing a property of the lie derivative; maybe I'm messing up the Leibniz rule or something? Any help would be greatly appreciated!
  2. jcsd
  3. Mar 27, 2015 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
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