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?


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 ${R^a}_{b\mu\nu}$.

  • $\begingroup$ 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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