Abstract: The classical exponential map in Riemannian geometry has
the following very important implications: if an isometry f
fixes a point and has trivial derivative there, then f is trivial;
moreover, the differential gives a simple normal form for all
isometries fixing a given point. Conformal transformations
of a Riemannian manifold are required only to preserve
angles, not distances. These have no exponential map.
Nontrivial conformal transformations can have differential
equal the identity at a fixed point, but this occurrence has
very strong implications for the underlying manifold.
I will present this rigidity phenomonenon in conformal geometry and a wide range of
generalizations. The key to these results is the notion of Cartan geometry, which infinitesimally
models a manifold on a homogeneous space. This point of view leads to a normal forms
theorem for conformal Lorentzian flows. It also leads to a suite of results on a seemingly
widespread rigidity phenomenon for flows on parabolic geometries, a rich family of geometric
structures whose homogeneous models include flag varieties and boundaries of symmetric spaces.