Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Annales Polonici Mathematici / All issues

Abstract

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.