In the article I've referenced below, and many other articles for that matter, the notion of parallel transport along a line of latitude $\theta=\theta\_0$ on the unit 2-sphere is spoken about. What I can't understand is, intuitively, how there is any "rotation" after circuiting the full line-of-latitude path.

[![enter image description here](https://i.stack.imgur.com/OHi0t.png)](https://i.stack.imgur.com/OHi0t.png)

The above illustration is my understanding of what is meant by "parallel transport of a vector along the $\theta$ or $\phi$ direction". Clearly, after completing a full path, the rotation of the vectors is always exactly $2\pi$, and hence the parallely transported vectors are left unchanged. **What have I misunderstood? My understanding doesn't seem to match with the results derived in the article below.**

Reference: <http://www.physics.usu.edu/Wheeler/GenRel2013/Notes/Geodesics.pdf>