Parallelisms, Prolongations Of Lie Algebras And Rigid Geometric Structures

For geometric structures of type Q, we prove that being rigid depends only on the stabilizers for the action on Q. We also prove that to any rigid structure we can associate a ‘‘natural’’ parallelism. Moreover, if the rigid structure is analytic, then the parallelism can be taken to be analytic as well. This implies an extension theorem for infinitesimal Killing fields. As an application we obtain Gromov’s centralizer theorem for arbitrary analytic rigid unimodular structures.