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In Carroll's there is a brief mention of how to get an idea about the curvature tensor using two infinitesimal vectors. Exercise 7 in Chapter 3 asks to compute the components of Riemann tensor by using the series expression for the parallel propagator. Can anyone please provide a sketch of the solution? When I use the parallel propagator on a vector to move from A to B and then from B to C the the vectors that make the infinitesimal loop do appear in the integrals but the indices of Christoffel symbol doesn't match with the equation 3.109. Is it correct to multiply four parallel propagators, each involving at least two terms in the series expression, to find the final vector components based on the initial components at A?

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# Using parallel propagator to derive Riemann tensor in Sean Carroll's

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