Abstract

Many important geometric structures are geometrically rigid in the sense that their Lie algebra of infinitesimal automorphisms is finite-dimensional. Prominent examples of such structures are Riemannian and conformal manifolds, and in general all geometric structures admitting equivalent descriptions as so-called Cartan geometries. 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. In this talk I will discuss several such classification results and explain how Cartan connections can be used to study local and global questions of geometric rigidity. The talk will also provide an introduction to Cartan geometries with a focus on the subclass of parabolic geometries.