The history of Cartan geometries dates back to the pioneering studies on non-euclidean geometries of the XIX century. The slogan of the so-called “Erlangen program” by Felix Klein was that any “geometry” should be described by a group of transformations. Later, Élie Cartan took these geometries as “standard model” and used them to give rise to his espaces généralisés. Such curved spaces become “locally Klein” under a suitable integrability condition, in the same way that Riemannian manifolds whose curvature vanish are locally the flat Euclidean model.
These concepts are now formalised in the modern language of principal bundles and vector-valued differential forms. During my stay at ESI I will interact with Andreas Cap (University of Vienna) and investigate an alternative formulation of Cartan geometries in terms of Lie groupoids and of multiplicative forms. More precisely, starting from the well-known correspondence between principal bundles and transitive Lie groupoids, one can relate Cartan geometries with the recent notion of Pfaffian groupoid. This object consists of a Lie groupoid together with a multiplicative structure, and originates from the study of differential equations on jet bundles. The recent advancements in the theory of Pfaffian groupoids will allow me to explore several aspects of Cartan geometries, dealing e.g. with Morita equivalences and prolongations, as well as applications to G-structures.