# how does parallel transport work on the sphere

I am starting to try to understand the concept of parallel transport from differential geometry and I have run into a problem. I have been attempting to compute parallel transport on a sphere (embedded into $\mathbb{R}^3$, with the Levi-Civita connection), along a circle (not necessarily the equator). I keep getting the result that in the general case, after a full rotation, the vector changes. My intuition tells me this can't be right, but the calculations say otherwise. Would someone be as kind as to explain to me how it works?

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