Vertical symmetries of Cartan geometries

Abstract

Élie Cartan's “éspaces généralizés” are, intuitively, curved geometries where the geometrical structure is that of a flat Klein geometry (a homogeneous space of a group) being rolled around a curved manifold without slipping or twisting.In modern terminology we may think of such a Cartan geometry as a fibre bundle with a means of lifting curves in the base manifold to curves in the Lie groupoid of structure-preserving fibre maps. The infinitesimal geometry will then be the Lie algebroid of certain projectable vector field on the fibre bundle, together with a horizontal lift to represent the connection.This paper considers symmetries of these structures, and explains why any vertical symmetry (projecting to the identity on the base manifold of the bundle) or any vertical infinitesimal symmetry (projecting to zero) must necessarily be trivial.