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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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up vote 12 down vote accepted

If the circle you're transporting along is not a great circle, then you should not expect parallel transport all the way around the circle to be the identity.

Imagine a conic surface that touches your circle everywhere. If you cut up the cone, you can unroll it into part of a flat plane, but then you will find that the two ends of the circle that meet the cut will point in different directions after the cone is unfolded. Therefore, if you start parallel transporting a vector from the cut point an all the way around the circle, it will not point int he same direction relative to the circle locally when you reach the other side of the cut.

(Parallel transport in the plane keeps the vector actually parallel at all times, and since the cone is tangent to your spherical circle everywhere, parallel transport in the cone agrees with parallel transport in the sphere).

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Thank you! This was very helpful. Also, your reasoning allows to find the parallel transport without doing the mundane computations and solving differential equations. – Jakub Konieczny Nov 8 '11 at 22:18
Beware: This works only when the curve you're transporting along happens to be a circle (or something else that you can easily cover by paper strips). – Henning Makholm Nov 9 '11 at 2:31

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