Can the Berry Connection be derived from a metric? 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.
 A: The answer is positive, except that the Berry connection being an Abelian connection, and the corresponding metric is not a metric on the tangent bundle as in the Riemannian case, but rather a metric on a line bundle , i.e., a one dimensional metric.
This line bundle was defined in In Barry Simon's  seminal work  , where he proved that the Berry phase is the holonomy of a (connection of) the Hermitian line bundle given by: $\{R, |\Psi (R)\rangle \} \in (\mathcal{M}, C^\times )$ subject to the constraint:
$$ H(R) |\Psi(R) \rangle = E(R) |\Psi (R)\rangle$$
Where $\mathcal{M}$ is the parameter space of the Hamiltonian $H(R)$. 
The line bundle is aligned at every point of $\mathcal{M}$ along the eigenvector $|\Psi (R)\rangle$ of the Schroedinger equation. This  bundle  possesses a metric on the space of sections which allows computing scalar products between two sections $x$ and $y$. ($x$ and $y$ are locally nonvanishing complex functions on $\mathcal{M}$):
$$ (x,y)(R) = \bar{x}(R) e^{-\langle \Psi(R)  |\Psi (R)\rangle } y(R)$$ 
This scalar product is invariant in the transition between patches of the manifold $\mathcal{M}$.
Now it is easy to show that the Berry connection is compatible with this metric, (just like the Levi-Civita connection is compatible with the Riemannian metric:
$$\partial_{\mu} (x,y) =  (D_{\mu} x,y) + (x, D_{\mu} y)$$
Where: $D_{\mu}$ is the covariant derivative corresponding to the Berry connection
$$D_{\mu} = \partial_{\mu}+iA_{\mu}$$
