Projective bundles and projective connections
Let \(E(M)\) be a smooth vector bundle of rank \(n+1\) over a manifold \(M\). Denoting by \(E_0(M)\) its open subbundle of nonzero sections and by \(\Lambda\) the fibre homotopy group, then \({\mathcal E}(M)= E_0(M)/\Lambda\) is a projective bundle over \(M\) with structure group \(PG(n)=GL(n+1)/\Lambda\). The link between \(E_0(M)\) and \({\mathcal E}(M)\) is given by the morphism \(\pi:E_0(M)\to{\mathcal E}(M)\) over \(M\), which is also a principal \(\Lambda\)-bundle over \({\mathcal E}(M)\). The author, referring also to some of his previous papers, studies the \(\pi\)-projectability of fields of objects such as scalar fields and tensor fields. With the same projecting technique, he constructs a projective connection on \({\mathcal E}(M)\) and states a relationship between curvature tensors and holonomy groups of the involved bundles. Computations are in local coordinates.
scientific article; zbMATH DE number 857518