Cartan geometries, Lie groupoids, and subriemannian manifolds

par Francesco Cattafi (Université de Wurtzbourg)

mardi 20 janv. 2026, 11:00 → 13:00 Europe/Paris

Connections on principal bundles are an ubiquitous tool in differential geometry and mathematical physics. The standard approach to connections, due to Ehresmann, describes them in terms of horizontal subbundles or of Lie algebra-valued differential forms. However, a different (although closely related) approach to connections originated from the pioneering works of Felix Klein and Élie Cartan.

Indeed, in his famous Erlangen program, Klein introduced the idea that each (possibly non-Euclidean) “geometry” should be described by groups of transformations. Cartan took these geometries as "standard model" and used them to give rise to his "espaces généralisés". Such spaces become “locally Klein” under a suitable flatness condition, in the same way that a Riemannian manifold whose curvature vanishes is locally the flat Euclidean model.

In this talk I will review the modern formulation of Cartan geometries (principal bundles endowed with Cartan connections) and their role in obtaining invariants of geometric structures. I will then sketch how Lie groupoids (a natural generalisation of Lie groups) provide us with a clearer framework to understand these topics and to tackle new problems. If time permits, at the end I will briefly mention the main ideas behind a joint work in progress with Ivan Beschastnyi and João Nuno Mestre, where we associate a "non-transitive" Cartan geometry to a subriemannian manifold.