Pseudogroups of transformations of geometrical-differential structures and their invariants (Q1304127)

This is a survey paper devoted to exposition of the so-called Cartan-Laptev method allowing to construct invariants of general geometric structures. Let \(H^p(V)\) denote the principal fibre bundle of frames of order \(p\) over a smooth manifold \(V\) of dimension \(n\); its structure groups is denoted by \(L^p_n\). A reduction of \(H^p(V)\) to a Lie subgroup \(G\subset L^p_n\) is called a structure of order \(p\) of type \(G\) on \(V\). On any bundle \(H^p(V)\) the displacement 1-form \(\omega^p\) with values in \(\mathbb{R}^n\oplus T_e(L_n^{p-1})\) is defined. Let \(\Pi\) be a Lie pseudogroup of order \(q\) of local diffeomorphisms of \(V\), i.e., \(\varphi\in\Pi\) iff \(j^q_x\varphi\) belongs to a closed subgroupoid of the groupoid \(J^q(V)\) of all \(q\)-jets of diffeomorphisms on \(V\). Using the prolongation, one defines a submanifold \(H^p_{\Pi}(V)\subset H^p(V)\) for any \(p\geq 1\). It is proved that \(H^p_{\Pi}(V)\) is a structure of order \(p\) of type \(L^p_n(\Pi)\) for a Lie subgroup \(L^p_n(\Pi)\subset L^p_n\). These submanifolds are \(\Pi\)-invariant, and \(\Pi\) is the largest Lie pseudogroup on \(V\) preserving \(H^p_{\Pi}(V)\) for all \(p\geq q\). Denoting by \(\omega^p_{\Pi}\) the restriction of \(\omega^p\) onto \(H^p_{\Pi}(V)\), the author also proves that the prolongation of \(\Pi\) to \(H^p_{\Pi}(V), p\geq q\), is the largest Lie pseudogroup on \(H^p_{\Pi}(V)\) preserving the form \(\omega^p_{\Pi}\). Similar results are obtained for smooth fibre bundles with a structure transitive Lie pseudogroup.