Singularities of the geodesic flow on surfaces with pseudo-Riemannian metrics

TL;DR

This paper investigates the behavior of geodesics on surfaces with pseudo-Riemannian metrics that change signature along a curve, focusing on singularities at points where isotropic directions are tangent to this curve.

Contribution

It provides a detailed analysis of the local behavior and singularities of geodesic flow at points where the metric signature changes on a surface.

Findings

Characterization of geodesic singularities at the discriminant curve

Analysis of isotropic directions tangent to the discriminant

Insights into the local structure of geodesic flow near signature change points

Abstract

We consider a pseudo-Riemannian metric that changes signature along a smooth curve on a surface, called the discriminant curve. The discriminant curve separates the surface locally into a Riemannian and a Lorentzian domain. We study the local behaviour and properties of geodesics at a point on the discriminant where the isotropic direction is tangent to the discriminant curve.