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The parallel transport of a vector $v_0^\alpha$ along the curve $\gamma$ is given by a vector field $v^\alpha$ which satisfies the equation $$ \frac{\mathrm d x^\mu}{\mathrm d \lambda}\frac{\partial v^\alpha}{\partial x^\mu} + v^\nu\Gamma^\alpha_{\mu\nu}\frac{\mathrm d x^\mu}{\mathrm d \lambda} = 0 $$ The first term of the LHS is, by chain rule, is $\mathrm d v^\alpha/\mathrm d \lambda$, so the equation becomes $$ \frac{\mathrm d v^\alpha}{\mathrm d \lambda} = - v^\nu\Gamma^\alpha_{\mu\nu}\frac{\mathrm d x^\mu}{\mathrm d \lambda} $$ Simplifying $\mathrm d\lambda$ from both sides, we get $$ \mathrm d v^\alpha = -v^\nu \Gamma_{\mu\nu}^\alpha\mathrm d x^\mu. $$ Now, if $\gamma$ is a closed curve, and we integrate this equation along the curve, we get $$ \oint \mathrm d v^\alpha = \oint -v^\nu\Gamma^\alpha_{\mu\nu} \mathrm d x^\mu $$ What is the result of this integral? I suppose it is zero if the space is flat and the the coordinates are Cartesian. But what in general? Could this integral be connected to the curvature of the space?

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The integral is known as the Holonomy. And yes, it is a measure the of the curvature. For a infinitesimal loops in the $\mu, \nu$ coordinate plane it is just a rotation matrix (a Lorentz transformation matrix in Minkowski signature) and is the definition of the curvature tensor ${R^a}_{b\mu\nu}$.

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  • $\begingroup$ Thank you very much for your prompt answer. May I ask you if you can suggest me a reference for this Holonomy? Thank you in advance for your help. $\endgroup$ – Logos Aug 21 '19 at 12:51
  • $\begingroup$ I'm not sure. I learned the word in a seminar long after I learned GR! It was not used in any of the books I had read. Mathematician use it all the time though, and they talk about "Holonomy groups" of a manifold --- this being the set of all rotation matrices abtained for all possible loops. It's generically the whole of ${\rm O}(n)$, but can be subgroup. I suggest Googling! $\endgroup$ – mike stone Aug 21 '19 at 12:56
  • $\begingroup$ Thank you anyway, @mikestone. I have just googled it: mathematicians seem to use it in the general abstract form, without using the coordinates, so I don’t quite recognise the integral anywhere. But I shall carry on searching: now at least I’ve got a name to start with :) $\endgroup$ – Logos Aug 21 '19 at 13:00
  • $\begingroup$ They tend to obscure the simplicity with fancy notations, but your integral is the same thing. Look at the Wikipedia picture of parallel tranport on a sphere for intuition. $\endgroup$ – mike stone Aug 21 '19 at 13:03
  • $\begingroup$ One more comment: the same integral in a general gauge theory is the Wilson Loop. There the Christoffel symbols $\oint {\Gamma^a}_{b\mu}dx^\mu$ are replaced by the gauge field so that we have $\oint \hat \lambda_a A^a_\mu dx^\mu $. Here $A_\mu$ is the connection on a principal bundle, rather than the Levi-Civita connection on the tangent bundle as in GR. $\endgroup$ – mike stone Aug 21 '19 at 13:38

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