Geodesic
Algebraic Properties of Covariant Derivative and Composition of Exponential Maps
Matematičeskie trudy, Tome 9 (2006) no. 1, pp. 3-20 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider the problem of calculating the Taylor series for a function $h_x\colon T_xX\times T_xX\to T_xX$ defined by the composition of exponential maps, where $X$ is a smooth manifold with affine connection and $x\in X$. We show that the homogeneous summands of such a series can be derived by applying the Lie bracket and covariant derivative to the arguments of the function which are extended to vector fields. 
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A. V. Gavrilov. Algebraic Properties of Covariant Derivative and Composition of Exponential Maps. Matematičeskie trudy, Tome 9 (2006) no. 1, pp. 3-20. http://geodesic.mathdoc.fr/item/MT_2006_9_1_a0/

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