A Frobenius theorem for Cartan geometries, with applications
Karin MelnickUniversity of Maryland, College Park, United States
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.
Cite this article
Karin Melnick, A Frobenius theorem for Cartan geometries, with applications. Enseign. Math. 57 (2011), no. 1/2, pp. 57–89DOI 10.4171/LEM/57-1-3