Geometric phase of a parallel transport over the surface of a sphere

[lld212]

I have this question on the calculation of the geometric phase (Berry phase) of a parallel transporting vector over the surface of a sphere, illustrated by Prof. Berry for example in the attached file starting on page 2.
The vector performing parallel transport is defined as ψ=(e+ie')/√2,
satisfying the parallel transport law, Imψ*・dψ=0.
Then another local basis was defined, n(r)=(u(r)+iv(r))/√2,
and ψ=n(r)exp(-iα).
Together the geometric phase (or so called anholonomy) is given as
α(C)=Im∫_Cn*・dn.

I can't see the difference between n and ψ here, except for a phase factor α. I think both of them performing the same parallel transport with α being constant. But why
Imψ*・dψ=0 while Imn*・dn≠0, even with the latter being a gauge of the geometric phase?

Thanks in advance.

 

[fzero]

The phase is local, expressed as $\alpha=\alpha(\mathbf{t})$, where $t$ is a parameter on the path. You should be able to compute that $\mathrm{Im}\mathbf{n}^* \cdot d\mathbf{n} = d\alpha$.