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I'm currently studying parallel transport from Do Carmo's book (Differential geometry of curves and surfaces). And is a well known result that if a surface is such that the parallel transport of a vector from any point to another depends only on the two points then the Gaussian curvature of the surface must be zero everywhere. So if a surface $S\subset \mathbb{R}^3$ has a point where the Gaussian curvature is non-zero then must exist two points $p$ and $q$ and two paths (distinct) $\alpha$ and $\beta$ joining the two points and a vector $v\in T_pS$ such that the parallel transport of $v$ along $\alpha$ is different from along $\beta$. I want to find an explicity example of such things. Can we find two points on the sphere $S^2$ and two curves joining such points such that the parallel transport of a vector $v$ is different from one path to another?

Jacaré
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    Consider piecewise smooth loops based at the north pole of the sphere. Take a vector tangent to the north pole, and transport it along a great circle. Nothing happens to it. Now take the same vector, transport along a great circle till you reach the equator, then transport say halfway in the east-west direction, and then back up to north pole. You'll get a different answer. You can do this demonstration yourself with your fingers moving in the air. (Alternatively try running your fingers through the grooves of a basketball in different ways, you should roughly see the same thing). – peek-a-boo Jan 22 '23 at 15:35
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    Try a circle that is not a great circle. (See how does parallel transport work on the sphere.) Start at point $A$ on the circle and transport a vector all the way around the circle from $A$ to $A.$ It arrives in a different orientation than it started. Now choose any other point $B$ on the circle. You now have parallel transports in the "forward" and "backward" direction from $B$ to $A$ that come out with different results at $A.$ – David K Jan 22 '23 at 16:57

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