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## Main Question or Discussion Point

In a Stokes theorem, the integral of all curls of a vector field enclosed in some region is equal to the line integral around the boundary.

I'm wondering if a similar theorem exists for parallel transport. The Riemann curvature tensor gives a change in a vector when parallel transported around a small loop. If we were to parallel transport a vector around a much larger loop, would the change in the vector be proportional to the integral of the Riemannian tensor around small loops inside the larger loop?

I'm wondering if a similar theorem exists for parallel transport. The Riemann curvature tensor gives a change in a vector when parallel transported around a small loop. If we were to parallel transport a vector around a much larger loop, would the change in the vector be proportional to the integral of the Riemannian tensor around small loops inside the larger loop?