Higher rank parabolic geometries with essential automorphisms and nonvanishing curvature

TL;DR

This paper constructs infinite families of higher rank parabolic geometries with nonvanishing curvature and essential automorphisms on compact manifolds, expanding the understanding of their global structure and examples.

Contribution

It introduces a method to produce compact, non-flat parabolic geometries with essential automorphisms using kernel elements of the Kostant Laplacian, extending local constructions to global examples.

Findings

Constructed infinite families of such geometries on closed manifolds.

Provided explicit examples including quaternionic contact structures of mixed signature.

Demonstrated the method's applicability to various higher rank parabolic geometries.

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 structrures of mixed signature, among other examples.