A Frobenius theorem for Cartan geometries, with applications

The classical result on local orbits in geometric manifolds is Singer’s homogeneity theorem for Riemannian manifolds [1]: given a Riemannian manifold M , there exists k, depending on dimM , such that if every x, y ∈ M are related by an infinitesimal isometry of order k, thenM is locally homogeneous. An open subset U ⊆ M of a geometric manifold is locally homogeneous if for every x, x ∈ U , there is a local automorphism f in U with f(x) = x. Such a local automorphism is a diffeomorphism from a neighborhood V of x in U to a neighborhood of x in U , with f an isomorphism between the geometric structures restricted to V and f(V ).