A Frobenius theorem for Cartan geometries, with applications

Abstract

We prove analogues for Cartan geometries of Gromov's major theorems on automorphisms of rigid geometric structures. The starting point is a Frobenius theorem, which says that infinitesimal automorphisms of sufficiently high order integrate to local automorphisms. Consequences include a stratification theorem describing the configuration of orbits for local Killing fields in a compact real-analytic Cartan geometry, and an open-dense theorem in the smooth case, which says that if there is a dense orbit, then there is an open, dense, locally homogeneous subset. Combining the Frobenius theorem with the embedding theorem of Bader, Frances, and the author gives a representation theorem that relates the fundamental group of the manifold with the automorphism group.