Gauge interpretation of characteristic classes (Q5953001)

Let \(p:CP\to M\) be the connection bundle of a principal \(G\)-bundle \(P\to M\). The canonical contact form on the first jet prolongation \(J^1P\) of \(P\) can be considered as a connection on \(J^1P\to CP\). Let \(\Theta\) be its curvature form. It is well known that for every Weil polynomial \(f\), the form \(f (\Theta)\) is projectable onto \(CP\). The main result of the paper reads: If \(G\) is connected, then for every gauge invariant differential form \(\Omega\) on \(P\) there exist differential forms \(\omega_1, \dots, \omega_k\) on \(M\) and Weil polynomials \(f_1,\dots,f_k\) such that \(\Omega=p^* \omega_1\wedge f_1(\Theta)+ \cdots+p^* \omega_k \wedge f_k(\Theta)\).