Cartan Geometries

Abstract

We now move on to the main topic of this volume, the study of the structures on differentiable manifolds knows as Cartan geometries. We shall define a (finite) Cartan geometry as a special kind of fibre-morphism groupoid with a path connection, and use this to motivate a detailed investigation of infinitesimal Cartan geometries given in terms of Lie algebroids. In fact our main concern in subsequent chapters will be with the infinitesimal geometries, and so we shall spend some time defining them in an intrinsic way using Lie algebra actions; as we shall see, there are infinitesimal Cartan geometries which are ‘incomplete’ and which cannot be obtained as infinitesimal versions of finite Cartan geometries. We conclude the chapter with an introduction to the concept of a symmetry for a Cartan geometry.