Local Generalized Symmetries and Locally Symmetric Parabolic Geometries

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.

References in corpus (5)

• Projectively flat surfaces, null parallel distributions, and conformally symmetric manifolds
• Parabolic Symmetric Spaces
• Symmetries of parabolic contact structures
• Local reflexion spaces
• Classification of invariant AHS--structures on semisimple locally symmetric spaces