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
$$D_{\mu} = \partial_{\mu}+iA_{\mu}$$

This post imported from StackExchange Physics at 2015-11-01 18:09 (UTC), posted by SE-user David Bar Moshe