Local Generalized Symmetries and Locally Symmetric Parabolic Geometries

TL;DR

This paper studies local automorphisms in parabolic geometries, revealing conditions under which they resemble symmetries in symmetric spaces and their relation to harmonic curvature.

Contribution

It characterizes when parabolic geometries admit unique generalized geodesic symmetries and links these to locally symmetric spaces.

Findings

Many parabolic geometries admit at most one symmetry at a point with non-zero harmonic curvature.

Unique symmetry at each point implies the geometry generalizes an affine symmetric space.

Abstract

We investigate (local) automorphisms of parabolic geometries that generalize geodesic symmetries. We show that many types of parabolic geometries admit at most one generalized geodesic symmetry at a point with non-zero harmonic curvature. Moreover, we show that if there is exactly one symmetry at each point, then the parabolic geometry is a generalization of an affine (locally) symmetric space.