Parabolic geodesics as parallel curves in parabolic geometries

TL;DR

This paper provides a unified and simplified characterization of parabolic geodesics across all parabolic geometries by introducing a natural connection on Weyl structures, linking geodesics to parallel curves.

Contribution

It introduces a natural connection on the space of Weyl structures and characterizes parabolic geodesics as parallel curves with respect to this connection.

Findings

Unified characterization of parabolic geodesics across geometries

Connection on Weyl structures simplifies understanding of geodesics

Geodesics are curves with parallel Weyl structures along them

Abstract

We give a simple characterization of the parabolic geodesics introduced by Cap, Slovak and Zadnik for all parabolic geometries. This goes through the definition of a natural connection on the space of Weyl structures. We then show that parabolic geodesics can be characterized as the following data: a curve on the manifold and a Weyl structure along the curve, so that the curve is a geodesic for its companion Weyl structure and the Weyl structure is parallel along the curve and in the direction of the tangent vector of the curve.