Symmetry of connection coefficients? (simple question)

[pellman]

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

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

Just wanting some confirmation. Thanks.

 

[WannabeNewton]

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.

 

[pellman]

Thanks!