Natural principal connections on the principal gauge prolongation of a principal bundle
For a principal bundle \(P\) over a manifold \(M\) with structure group \(G\) and an integer \(r\), one can form the \(r\)-th principal prolongation \(W^rP\), which essentially consists of \(r\)-jets of local principal bundle charts. This is a principal bundle with structure group the \(r\)-th prolongation \(W^r_mG\), where \(m\) is the dimension of \(M\). The aim of this article is to determine all ways to naturally associate a principal connection on \(W^rP\) to a principal connection on \(P\) and an affine connection on \(M\). Using the Utiyama theorem, the authors arrive at a description of such connections for general \(r\) in terms of a number of tensor fields depending naturally on the two given connections. They also give a description of such tensor fields in terms of Ad-invariant elements in \(\mathfrak g\) and Ad-equivariant linear maps \(\mathfrak g\to\mathfrak g\), where \(\mathfrak g\) denotes the Lie algebra of the structure group \(G\). For \(r=1\) and \(r=2\), the classification is carried out completely. Finally, the Ad-invariant elements and Ad-equivariant linear maps needed in the description are determined in the case \(\mathfrak g=\mathfrak{gl}(n,\mathbb R)\).
scientific article; zbMATH DE number 5689970
scientific article