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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 compatible with the metric, otherwise it is a Levi-Civita connection, which I think it is not.

Thank you!