In do Carmo's book, we are told that the angle between any unit vector $w_0$ and the vector $w$ resulting from its parallel transport around a closed path on a surface, back to the location of $w_0$ is
$ \displaystyle \Delta \phi = \int K d\sigma $
where $K$ is the Gaussian curvature on the surface and the integral is taken over the portion of the surface enclosed by the closed path.
Now, let's say we have a unit sphere $S^2$ with a unit vector $w_0$ sitting on the equator, and we parallel transport it around the equator. Since $K=1$, we should get $\Delta \phi = 2 \pi$.
However, what if $w_0=\hat{z}$? I try and try to see it otherwise, but it seems to me that the parallel transport of $\hat{z}$ around the equator must be simply $\hat{z}$ and therefore $\Delta \phi$ must be zero, not $2 \pi$.
Someone please tell me where I am going wrong! This seems like it should be completely straightforward!