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Let $\mathcal{M}$ be an n-dimensional manifold endowed with an affine connection $\nabla$. Let $\gamma_1:[a,b]\rightarrow M$ and $\gamma_2:[c,d]\rightarrow \mathcal{M}$ be two curves with the same initial and final points, that is, $p=\gamma_1(a)=\gamma_2(c), q=\gamma_1(b)=\gamma_2(d)$. Take $X\in T_p\mathcal{M}$. Parallelly propagating $X$ along $\gamma_1$ and $\gamma_2$ we obtain two vectors $X_1, X_2\in T_q\mathcal{M}$, respectively. Let $R$ be the curvature tensor of the connection, $R(X,Y)Z=\nabla_X\nabla_Y Z - \nabla_Y\nabla_X Z -\nabla_{[X,Y]}Z$, and $\tau$ its torsion, $\tau(X,Y)=\nabla_X Y-\nabla_Y X -[X,Y]$.

The question is: How can I compare the two vectors $X_1$ and $X_2$? Can I write the difference $(X_2-X_1)$ in terms of $R,\tau$ and the curves?

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2 Answers 2

up vote 3 down vote accepted

If your two paths are homotopic you can make a comparison between your two vectors. And yes the comparison involves an integral over the homotopy of a function of curvature.

See for example Theorem 13.6.4 in Pressley's "Elementary Differential Geometry" (Google books will bring up the statement of the theorem).

If your paths are not homotopic you're out of luck, as T describes. There are things you can say of course but it's not clear what you're looking for. You should think of an example of a Riemann manifold you're interested in, to get a sense for how bad it can get.

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Don't you need to integrate the induced curvature from the manifold onto the 2-dimensional "worldsheet" given by the homotopy? In other words, it's a sort of additional data that is not pure local information about the manifold but also about the placement of an additional gadget inside the manifold (here, the graph of the homotopy). –  T.. Sep 12 '10 at 22:58
Isn't your question answered in my 2nd sentence? –  Ryan Budney Sep 13 '10 at 20:14
Thank you both for the answers and references. I don't know anything about holonomy, but now I know where to start. –  Ronaldo Sep 14 '10 at 0:42

The difference depends on the homotopy classes of the paths, not only local data like the curvature and torsion. I don't think there is a simpler answer than "the difference in the parallel transport along the two paths". But see the (superb) Wikipedia page on holonomy, especially the sections on the Ambrose-Singer theorem and affine connections.


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