Geodesic
Algebraic Properties of Covariant Derivative and Composition of Exponential Maps
Matematičeskie trudy, Tome 9 (2006) no. 1, pp. 3-20 
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/
@article{MT_2006_9_1_a0,
     author = {A. V. Gavrilov},
     title = {Algebraic {Properties} of {Covariant} {Derivative} and {Composition} of {Exponential} {Maps}},
     journal = {Matemati\v{c}eskie trudy},
     pages = {3--20},
     year = {2006},
     volume = {9},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MT_2006_9_1_a0/}
}
TY  - JOUR
AU  - A. V. Gavrilov
TI  - Algebraic Properties of Covariant Derivative and Composition of Exponential Maps
JO  - Matematičeskie trudy
PY  - 2006
SP  - 3
EP  - 20
VL  - 9
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/MT_2006_9_1_a0/
LA  - ru
ID  - MT_2006_9_1_a0
ER  - 
%0 Journal Article
%A A. V. Gavrilov
%T Algebraic Properties of Covariant Derivative and Composition of Exponential Maps
%J Matematičeskie trudy
%D 2006
%P 3-20
%V 9
%N 1
%U http://geodesic.mathdoc.fr/item/MT_2006_9_1_a0/
%G ru
%F MT_2006_9_1_a0

Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Gavrilov A. V., “Dvoinoe eksponentsialnoe otobrazhenie i kovariantnoe differentsirovanie”, Sib. mat. zhurn., 48:1 (2007), 68–74 | MR

[2] Nomidzu K., Gruppy Li i differentsialnaya geometriya, Izd-vo inostr. lit., M., 1960 | MR

[3] Khelgason S., Differentsialnaya geometriya i simmetricheskie prostranstva, Mir, M., 1964 | Zbl

[4] Kobayashi Sh. and Nomizu K., Foundations of Differential Geometry, V. I; II, John Wiley Sons, New York; London, 1963 ; 1969 | Zbl

[5] Safarov Yu., “Pseudodifferential operators and linear connections”, Proc. London Math. Soc. (3), 74:2 (1997), 379–416 | DOI | MR | Zbl