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SOLVED: Killing Vectors on Unit Sphere

  1. Jan 8, 2014 #1
    \bar{}1. The problem statement, all variables and given/known data
    Hi, I want to show that
    [itex]\frac{\partial}{\partial \phi}[/itex]
    is a Killing vector on the unit sphere with metric
    [itex] ds^2 = d\theta^2 + \sin^2 \theta d \phi^2 [/itex]


    2. Relevant equations
    I compute the Christoffel symbols to be
    [itex] \Gamma^\theta_{\phi \phi} = -\sin \theta \cos \theta [/itex]
    [itex] \Gamma^\phi_{\phi \theta} = \cot \theta [/itex]


    3. The attempt at a solution
    Then computing Killing's equation for the theta-phi component,
    [itex] \nabla_\phi X_\theta + \nabla_\theta X_\phi [/itex]

    This gives
    [itex] X_{\theta,\phi} + X_{\phi, \theta} - 2 \cot \theta X_\phi [/itex]

    But this doesn't give 0 since [itex]X_\phi \neq 0[/itex]. Where did I go wrong?

    EDIT:
    Nvm. Forgot about the lowered index.
     
    Last edited: Jan 8, 2014
  2. jcsd
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