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.