A Frobenius theorem for Cartan geometries, with applications

TL;DR

This paper extends Gromov's theorems to Cartan geometries by proving a Frobenius theorem that links infinitesimal automorphisms to local automorphisms, with applications to orbit structure and automorphism groups.

Contribution

It introduces a Frobenius theorem for Cartan geometries, enabling new insights into automorphism groups and orbit structures, and connects fundamental groups with automorphisms.

Findings

Infinitesimal automorphisms integrate to local automorphisms in high order.

Stratification of orbits for local Killing fields in compact geometries.

Dense orbits imply open, dense, locally homogeneous subsets.

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 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.