The Berry Connection is $$A_\mu(R)=-i \langle \Psi(R) |\partial_\mu \Psi(R) \rangle$$ which allows us to parallel transport a state indexed by $R$. We can integrate the Berry Connection to get the Berry Phase, and we can differentiate the Berry Connection to get the Berry Curvature.
Can the Berry Connection be derived from a metric? As a prototypical example, I'm thinking about how the Christoffel Symbols in General Relativity (GR) can be derived from the metric tensor. Also, I think for every connection, there exists a metric for which the connection is a Levi-Civita connection. However, is there a natural and physical metric that induces the Berry Connection?
In this presentation, the great and powerful Haldane relates "quantum distance" to the Berry Curvature, but it doesn't look like you can derive the Berry Curvature from the quantum distance in the same way curvature and the metric are related in GR.