In general relativity, a vector parallel along a curve on a manifold M with a connection field Γ can be expressed:(adsbygoogle = window.adsbygoogle || []).push({});

∂v+Γv=0

We know that if the curvature corresponding to Γ is non-zero, which means if we parallel transport a vector along different paths between two points, we will get different result, so we can say we can not get a globally parallel vector field on the manifold.

Does this conclusion means that to the parallel transport equation, if curvature of Γ is not zero, then we can not find a global vector field on manifold M to satisfy the parallel equation?

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# A Global solution to parallel transport equation?

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