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Why is there maximally 1/2 n(n+1) killing vectors?

  1. Sep 24, 2014 #1
    It is often stated that there are maximum number of n/2 (n+1) linearly independent killing vectors in an n-dimensional Riemannian manifold.

    How is this fact derived?
  2. jcsd
  3. Sep 27, 2014 #2
    I'd posted on that in the thread "Riemannian curvature of maximally symmetric spaces".

    One first finds an equation for the second derivative of Killing vector ξ. This means that ξ anywhere is a function of ξ and its first derivative at some point. The defining equation for ξ constrains the first derivative of ξ to be antisymmetric or sort-of antisymmetric where ξ itself is 0. For n space dimensions:
    • Possible values of ξ: n
    • Possible values of the antisymmetrized first derivative of ξ: n(n-1)/2
    Total: n(n+1)/2
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