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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)].
53C05 Connections (general theory)
58A20 Jets in global analysis
58A32 Natural bundles
Full Text: DOI
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