Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    User Avatar
    Science Advisor

    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
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - Symmetry connection coefficients Date
I Connections on principal bundles Jan 22, 2018
Jet Prolongation Formulas for Lie Group Symmetries Dec 13, 2015
Symmetry Axis of Hyperbole Jun 13, 2014
Does spherical symmetry imply spherical submanifolds? Jun 4, 2014
Manifolds with symmetries Apr 1, 2014