Symmetries of Parabolic Geometries

TL;DR

This paper extends the concept of symmetries to parabolic geometries, especially |1|-graded types, exploring curvature restrictions and the variety of symmetries possible at points with nonzero curvature.

Contribution

It introduces a generalized notion of symmetries for parabolic geometries and analyzes their implications on curvature and symmetry multiplicity.

Findings

Restrictions on curvature from symmetries

Existence of multiple symmetries at points with nonzero curvature

Use of Weyl structures to analyze symmetries

Abstract

We generalize the concept of affine locally symmetric spaces for parabolic geometries. We discuss mainly $|1|$--graded geometries and we show some restrictions on their curvature coming from the existence of symmetries. We use the theory of Weyl structures to discuss more interesting $|1|$--graded geometries which can carry a symmetry in a point with nonzero curvature. More concretely, we discuss the number of different symmetries which can exist at the point with nonzero curvature.