# Parallel transport of a vector along a closed curve in curvilinear coordinates

There is an expression indicating the change of the vector parallel translation along a closed infinitesimal curve in curvilinear coordinates (one way of introducing curvature tensor): $$\Delta A_{k} = \oint \Gamma^{i}_{kl}A_{i}dx^{l}.$$

Then, before using Stokes' theorem, there need to require the uniqueness of the integrand.

So, Christoffel symbol and vector's component expanded to a series near some point ${x^{\alpha}}^{0}$ on a curve:

$$\Gamma^{i}_{kl} = {\Gamma^{i}_{kl}}_{0} + (\partial_{\alpha}\Gamma^{i}_{kl})_{0}(x^{\alpha} - {x^{\alpha}}^{0}) + \frac{1}{2}(\partial_{p\alpha}\Gamma^{i}_{kl})_{0}(x^{\alpha} - {x^{\alpha}}^{0})(x^{p} - {x^{p}}^{0}) + ...,$$

$$A_{i} = (A_{i})_{0} + (\partial_{\alpha}A_{i})_{0}(x^{\alpha} - {x^{\alpha}}^{0}) + \frac{1}{2}(\partial_{p \alpha}A_{i})_{0}(x^{\alpha} - {x^{\alpha}}^{0})(x^{p} - {x^{p}}^{0}) + ...$$ After that there are a words about the fact that the ambiguity of the operation of parallel transport is covered in quadratic terms. What are some reasons to "believe" that?

-
Are you describing something in a book? What book? –  Ben Crowell May 7 '13 at 12:08
From Fock, "The theory of Space Time & Gravitation", or from Landau, "Field Theory". –  PhysiXxx May 7 '13 at 12:11

Anyway, I don't really see what is your problem -- just use Stokes' theorem from (6.19): $$\Delta A_i = \frac12\int df^{lm}\left(\partial_l(\Gamma_{km}^iA_i)-\partial_m(\Gamma_{kl}^iA_i)\right)$$ Then substitute your decompositions, account for the fact that everything with index "0" is constant, expand and you'll get something like (but note the absence of a linear term in $x-x^0$): $$\Delta A_i = \frac12 \int df^{lm}\left((R_{klm}^i)_0(A_i)_0+(\text{some other stuff}_{\alpha\beta lm})_0(x^\alpha-x^{\alpha0})(x^\beta-x^{\beta0})+...\right)$$ Finally, you just remember that integration is over infinitesimal volume $\Delta f^{lm}$, therefore $(x^\alpha-x^{\alpha0})(x^\beta-x^{\beta0})\sim\Delta f^{\alpha\beta}$ and this terms are, indeed, of the second order.
"...just use Stokes' theorem from..." I can't use it if integrand isn't single-valued function. Furthermore, the expression for the derivative $\partial_{l}A_{i} = \Gamma^{n}_{il}A_{n}$ can be used only after the requirements of "uniqueness" of $A_{n}$ along the curve. –  PhysiXxx May 7 '13 at 21:10
@PhysiXxx Have you noticed that quadratic terms (whatever they are) are, indeed, of the second order in $\Delta f^{lm}$? –  Kostya May 8 '13 at 8:46