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I have learned Riemannian Geometry, so the only connection I have ever worked with is the Levi-Civita connection(covariant derivative of metric tensor vanishes and the Chrystoffel symbols are symmetric).

When performing a parallel transport with the L-C connection, angles and lengths are preserved.

Now, I am trying to learn about the Berry phase which is full of differential geometry. What I want to learn is whether or not the Berry connection is a Levi-Civita connection.

Thank you in advance.