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Symmetry of connection coefficients? (simple question)

  1. May 22, 2013 #1
    If we have a metric connection with no other restrictions (torsion may be non-zero), do the connection coefficients [itex]{\Gamma^{\alpha}}_{\mu\nu}[/itex] have any symmetries among the indeces? I'm thinking not.

    Or.. for a Levi-Civita connection, the only fixed symmetry condition is [itex]{\Gamma^{\alpha}}_{\mu\nu}={\Gamma^{\alpha}}_{\nu\mu}[/itex], right? There is no relation involving the alpha index

    Just wanting some confirmation. Thanks.
     
  2. jcsd
  3. May 22, 2013 #2

    WannabeNewton

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    For the Levi-Civita connection, the Christoffel symbols have symmetry in the lower two indices and that's it. For a general connection, there is no symmetry in the indices of the connection coefficients.
     
  4. May 22, 2013 #3
    Thanks!!
     
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