Another proof of Bianchi's identity in arbitrary bundles (Q1891973)

The purpose of this paper is to prove the following theorem: Let \(P\) be a principal fibre bundle, \(\omega\) a connection form on \(P\), and \(\Omega^ \omega\) its curvature form. Then the following holds: \(f^* \Omega^ \omega = \Omega^{f^* \omega}\) for all \(\omega\)-horizontal automorphisms \(f\) of \(P\), i.e. the validity of the identity \(L_ X \Omega^ \omega = D^ \omega (L_ x \omega)\) for all \(\omega\)- horizontal and \(G\)-invariant vector fields \(X\) on \(P\), is equivalent to Bianchi's identity \(D^ \omega \Omega^ \omega = 0\). This theorem generalizes the result obtained by \textit{J. L. Kazdan} [Proc. Am. Math. Soc. 81, 341-342 (1981; Zbl 0459.53033)] for the case of a Riemannian manifold.