In differential geometry many important geometric structures are geometrically rigid in the sense that their automorphism groups (in some natural topology) are finite-dimensional Lie groups. Prominent examples of such structures are Riemannian manifolds, conformal and projective structures and in general all geometric structures admitting equivalent description as so-called Cartan geometries, which comprise a huge variety of geometric structures. Generically these geometric structures have trivial automorphism groups and so the ones among them with large automorphism groups or special types of automorphisms are typically geometrically and topologically very constrained and hence can often be classified. Recall for instance that a Riemannian manifold (dim>2) whose group of conformal transformations is of largest possible dimension is conformally diffeomorphic to a sphere. Another further manifestation of the rigidity of conformal structures is the famous Ferrand-Obata Theorem, which proves the Lichnerowicz conjecture. It states that a compact conformal manifold with a non-compact automorphism group is conformally diffeomorphic to a sphere.

In this lecture series we will discuss various results along these lines, concerned with (local) automorphism groups of geometric structures and local and global phenomena of geometric rigidity. Since we will see that these phenomena are intimately related to the existence of Cartan connections for the discussed geometric structures, the lecture series can be also seen as providing an introduction to the concept of a Cartan geometry.