  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 homogeneus 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.
