# On Parallel Transport

Let's consider the issue of parallel transport in relation to the figure on the following Wikipedia link: http://en.wikipedia.org/wiki/Parallel_transport

With reference to the Figure on the link:

Instead of parallel transporting the vector from A to N lets (parallel) transport it from A to N' along the meridian where N' is a point just below N [say latitude=89.9999 degrees]. Now we move the vector parallel to itself along the line of latitude passing through N'to reach the corresponding point on the meridian NB.The vector is now almost parallel to the meridian NB[Since the concerned line of latitude is not a geodesic I have used the word almost]. The vector is moved down and then moved back to A along the equator. It turns by a very small amount.The exclusion of a tiny[you could make it microscopic] spherical triangle is causing so much of difference.Why?

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As Ron tells you, you are doing parallel transport incorrectly. It's good to have a particular example and a general enough answer. Consider the surface of a sphere and do parallel transport around a curve C. Then the transformation induced on the tangent space is nothing else than the rotation by X radians where X is the solid angle of the encircle area in steradians. For example, a parallel transport on the equator is the rotation by $2\pi$ because hemisphere is $2\pi$ steradians. If you add a tiny triangle, tiny solid angle, the change in the parallel transport will be tiny. – Luboš Motl Dec 2 '12 at 8:14
I am referring to the change between the initial and the final positions of the vector when it goes round a loop on a curved surface(by parallel transport). You may connect the point N'(in the original posting) with some point on the meridian NB by a $small{\;\;}$ curve so that the transported vector on landing on the meridian becomes tangential parallel or nearly tangential to the meridian NB. – Anamitra Palit Dec 3 '12 at 4:41
Does "parallel transport" move a vector parallel to itself on a curved surface even in the infinitesimal sense? You may think of two adjacent tangent planes on a curved surface.Is it always possible to have parallel vectors at the points of contact(one vector being preassigned) even if the planes are awkwardly inclined? – Anamitra Palit Dec 4 '12 at 13:23