Higher Rank Parabolic Geometries with Essential Automorphisms and Nonvanishing Curvature

Abstract

We construct infinite families of regular normal Cartan geometries with nonvanishing curvature and essential automorphisms on closed manifolds for many higher rank parabolic model geometries. To do this, we use particular elements of the kernel of the Kostant Laplacian to construct homogeneous Cartan geometries of the desired type, giving a global realization of an elegant local construction due to Kruglikov and The, and then modify these homogeneous geometries to make their base manifolds compact. As a demonstration, we apply the construction to quaternionic contact structures of mixed signature, among other examples.