×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Doupovec, M.; Mikulski, W.M., Holonomic extensions of connections and symmetrization of jets, Rep. math. phys., 60, 299, (2007) · Zbl 1160.58001
[2] Doupovec, M.; Mikulski, W.M., Reduction theorems for principal and classical connections, Acta math. sinica English series, 26, 169-184, (2010) · Zbl 1186.53036
[3] Fatibene, L.; Francaviglia, M., ()
[4] Janyška, J., Higher-order utiyama invariant interaction, Rep. math. phys., 59, 63, (2007)
[5] Janyška, J.; Vondra, J., Natural principal connections on the principal gauge prolongation of a principal bundle, Rep. math. phys., 64, 395, (2009) · Zbl 1195.53040
[6] Kolář, I., Some gauge-natural operations on connections, (), 435 · Zbl 0806.53025
[7] Kolář, I., Torsion-free connections on higher order frame bundles, (), 233 · Zbl 0901.53016
[8] Kolář, I., Connections on principal prolongations of principal bundles, (), 279 · Zbl 1160.58003
[9] Kolář, I.; Michor, P.W.; Slovak, J., ()
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.