On the kernel of holonomy (Q677837)

Let \(M\) be a connected manifold, \(p\) a fixed point in \(M\), and denote by \({\mathcal G} {\mathcal L}^\infty (M)\) the group of equivalence classes of loops on \(M\). Two loops are equivalent if they are linked by a ``one-rank-homotopy''. Let \(G\) be a Lie group and \(\pi: P\to M\) a principal \(G\)-bundle over \(M\). A holonomy is defined as a smooth group morphism \({\mathcal H} :{\mathcal G} {\mathcal L}^\infty (M)\to G\). The kernel of \({\mathcal H}\) consists of the classes of loops along which the parallel transport is trivial. Moreover, holonomies are in a one-to-one correspondence with the triples consisting of a principal \(G\)-bundle over \(M\), a connection on this bundle and a point on the fiber over \(p\), up to isomorphism. The author establishes a formula expressing the gauge potential as a suitable derivative of the holonomy and using such a formula, he gives the main result of the paper, i.e., a different proof of a theorem of Lewandowski, which states that the kernel of the holonomy contains all information about the corresponding connection. Finally, he defines a generalized holonomy as a group morphism \({\mathcal H}: {\mathcal G} {\mathcal L}^\infty (M)\to G\) and proves that if \({\mathcal H}\) is a continuous map, then its image is a Lie subgroup of \(G\).