So I asked a similar question here and even though I still believe it's a valid question, the formulation may have been a bit too complicated to pique people's interest, so let me try to break it up into parts. Indeed, let us first consider the non-degenerate case of the (abelian) Berry phase, so that using the adiabatic theorem, one can find the connection $$ iA = \langle \psi(s)| \frac{\partial}{\partial s} \psi (s) \rangle $$
Conversely, one would think that this comes "naturally" from some metric. More specifically, it should be the unique connection compatible with some "natural" metric and (possibly some other "natural conditions" since compatibility usually isn't enough for uniqueness). In this question, the answer seems to provide partial resolution and shows that the Berry connection is indeed compatible with some metric. With that said, the answer itself is a bit confusing to me. Indeed, (1) why would the "natural metric" $h$ $$ h(x,y) = \bar{x} y \exp(-\langle \psi|\psi \rangle) $$ Have the exponential term since $\langle \psi|\psi \rangle$ is constant.
And (2) what other condition would one need so that the Berry connection is uniquely defined? Any insight towards questions (1) and (2) would be appreciated!
