Geodesic
The affine connection in the normal coordinates
Matematičeskie trudy, Tome 15 (2012) no. 1, pp. 27-54.

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We propose a method to calculate the derivatives of the Christoffel symbols in the normal coordinates on a smooth manifold with symmetric affine connection. As an application, we construct an algorithm for computing the Taylor series of the double exponential map. 
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A. V. Gavrilov. The affine connection in the normal coordinates. Matematičeskie trudy, Tome 15 (2012) no. 1, pp. 27-54. http://geodesic.mathdoc.fr/item/MT_2012_15_1_a1/

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