Sean Carroll says that if we have metric compatibility then we may lower the index on a vector in a covariant derivative. As far as I know, metric compatibility means ##\nabla_\rho g_{\mu\nu}=\nabla_\rho g^{\mu\nu}=0##, so in that case ##\nabla_\lambda p^\mu=\nabla_\lambda p_\mu##. I can't see...
Hello,
Is the Berry connection compatible with the metric(covariant derivative of metric vanishes) in the same way that the Levi-Civita connection is compatible with the metric(as in Riemannanian Geometry and General Relativity)?
Also, does it have torsion? It must either have torsion or not be...
I am trying to learn GR, primarily from Wald. I understand that, given a metric, a unique covariant
derivative is picked out which preserves inner products of vectors which are parallel transported.
What I don't understand is the interpretation of the fact that, using this definition of the...