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On the gauge version of exponential map. (English) Zbl 1210.53031
The aim of the paper is to extend the definition of the exponential map of a linear connection to a new setting. Using a classical connection $$\Lambda$$ on a manifold $$M$$ and a principal connection $$\Gamma$$ on a principal bundle $$P=P(M,G)$$, the author constructs an exponential map $$\exp _{u}^{\Gamma ,\Lambda }:T_{u}P\rightarrow P$$, $$u\in P$$, and a reduction of the $$(r+1)$$-th principal gauge prolongation $$W^{r+1}P$$ to the group $$GL(m)\times G$$. This allows to find all natural induced connections on $$W^{r}P$$ as natural transformations of couples $$(\Gamma ,\Lambda )$$ in the line of the well-known monograph [I. Kolár, P. W. Michor and J. Slovák, Natural operations in differential geometry. Berlin: Springer-Verlag (corrected electronic version) (1993; Zbl 0782.53013)]. Finally, the author proves that the exponential map constructed in the paper coincides with the exponential map of a classical connection on $$P$$ considered previously by the author in [in: Differential geometry and its applications. Proceedings of a colloquium, held in Eger, Hungary, August 20-25, 1989, organized by the János Bolyai Mathematical Society. Amsterdam: North- Holland Publishing Company. Colloq. Math. Soc. János Bolyai. 56, 435–445 (1992; Zbl 0806.53025)].
##### MSC:
 53C05 Connections (general theory) 58A20 Jets in global analysis 58A32 Natural bundles
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##### References:
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