# metric compatibility

1. ### I Metric compatibility and covariant derivative

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...
2. ### A Is the Berry connection compatible with the metric?

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...
3. ### A Interpretation of the derivative of the metric = 0 ?

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...