Introduction to Cartan Connections and Applications: parte 3

In this mini-course we will give an introduction to Cartan geometries, which pro- vide a uniform approach to a large variety of differential geometric structures. We will focus on parabolic geometries which are Cartan geometries infinitesimally modelled on flag varieties. Among the most prominent examples of geometric structures admit- ting descriptions as parabolic geometries are conformal manifolds (dim>2), projective structures, non-degenerate CR-structures of hypersurface type and various types of bracket-generating distributions. After having introduced the basic concepts and hav- ing studied some examples, we will discuss some applications of Cartan connections to classical problems in differential geometry. On the one hand, we will see how Cartan connections can be applied to questions of geometric rigidity such as, which Lie groups can act on manifolds preserving a given geometric structure or to which extent does the group of automorphisms determine the geometric structure. On the other hand, we will study applications of Cartan connections to compactifications of geometric structures.