On the local and global properties of geodesics in pseudo-Riemannian metrics

TL;DR

This paper investigates the local and global behaviors of geodesics in two-dimensional pseudo-Riemannian metrics, revealing unique properties near parabolic points and exploring geodesic behavior in symmetric and discontinuous metrics.

Contribution

It provides new insights into geodesic flow near parabolic points and examines global properties in symmetric and discontinuous pseudo-Riemannian metrics.

Findings

Geodesics cannot pass through parabolic points in arbitrary directions.

Number of admissible directions at parabolic points is typically 1 or 3.

Global properties of geodesics are characterized in metrics with symmetry groups.

Abstract

The paper is a study of geodesic in two-dimensional pseudo-Riemannian metrics. Firstly, the local properties of geodesics in a neighborhood of generic parabolic points are investigated. The equation of the geodesic flow has singularities at such points that leads to a curious phenomenon: geodesics cannot pass through such a point in arbitrary tangential directions, but only in certain directions said to be admissible (the number of admissible directions is generically 1 or 3). Secondly, we study the global properties of geodesics in pseudo-Riemannian metrics possessing differentiable groups of symmetries. At the end of the paper, two special types of discontinuous metrics are considered.