Parallelisms, prolongations of Lie algebras and rigid geometric structures (Q1882778)

Some results on rigid geometric structures are proved. A geometric structure of order \(k\) and type \(Q\) on a manifold is defined as an equivariant map from the \(k\)-th frame bundle of the manifold into \(Q\). Such a structure is rigid (in the sense of Gromov) if its automorphisms are determined by some jet of fixed order. Several results were previously known for \(H\)-structures, i.e., geometric structures of the type \(Q=Gl^{(k)}(n)/H\), where \(Gl^{(k)}(n)\) is the structure group of the \(k\)-th frame bundle and \(H\) is a closed subgroup. In this paper, the authors extend to general geometric structures several properties known for \(H\)-structures. The problem of associating ``natural'' parallelisms to geometric structures is also discussed. At the end of the paper the authors prove some results by Gromov (centralizer theorem) under more general hypotheses.