Lifting right-invariant vector fields and prolongation of connections

W. M. Mikulski¹

¹ Institute of Mathematics Jagiellonian University /Lojasiewicza 6 30-348 Kraków, Poland

Annales Polonici Mathematici, Tome 95 (2009) no. 3, pp. 243-252

Résumé

We describe all $\mathcal {P}\mathcal B_m(G)$-gauge-natural operators $\cal A$ lifting right-invariant vector fields $X$ on principal $G$-bundles $P\to M$ with $m$-dimensional bases into vector fields $\cal A(X)$ on the $r$th order principal prolongation $W^rP=P^rM\times_MJ^rP$ of $P\to M$. In other words, we classify all $\mathcal {P}\mathcal B_m(G)$-natural transformations $J^rLP\times_M W^rP\to TW^rP=LW^rP\times_MW^rP$ covering the identity of $W^rP$, where $J^rLP$ is the $r$-jet prolongation of the Lie algebroid $LP=TP/G$ of $P$, i.e. we find all $\mathcal {P}\mathcal B_m(G)$-natural transformations which are similar to the Kumpera–Spencer isomorphism $J^rLP=LW^rP$. We formulate axioms which characterize the flow operator of the gauge-bundle $W^rP\to M$. We apply the flow operator to prolongations of connections.

Détail

DOI : 10.4064/ap95-3-4

Keywords: describe mathcal mathcal gauge natural operators cal lifting right invariant vector fields principal g bundles m dimensional bases vector fields cal rth order principal prolongation times to other words classify mathcal mathcal natural transformations rlp times rp times covering identity where rlp r jet prolongation lie algebroid mathcal mathcal natural transformations which similar kumpera spencer isomorphism rlp formulate axioms which characterize flow operator gauge bundle apply flow operator prolongations connections