Canonical example for Abelian Bery phase is a spin in a magnetic filed, e.g.. Usually authors calculate spin eigenstates, conclude that they don't depend on B in spherical components and so deduce that the sphere $S^2$ is the parameter space manifold. First question - shouldn't the parameter space actually be $S^2 \times (0,\infty) \cong B^2/O$, with $O$ being a ball center, as |B| $\in (0,\infty)$?
I would like to obtain more a priori understanding of the parameter space manifold identification. Working in Cartesian coordinates, for example, this $S^2$ identification is not transparent. My reasoning is along these lines - Fixing |B| one would naively first conclude that the $SO(3)$ is actually this manifold as we can orient B as we wish. However, we should ''subtract'' rotations around B and so we are lead to the coset manifold $SO(3)/SO(2)\cong S^2$. Letting |B| change, we finally arrive at $B^2/O$. Is this correct reasoning? Thing that puzzles me here is that we have spins here so $SU(2)$ should be more appropriate but how do we arrive to $S^2$ then?