A brief survey on singularities of geodesic flows in smooth signature changing metrics on 2-surfaces

TL;DR

This survey explores the singularities of geodesic flows on 2-surfaces with smooth signature-changing metrics, focusing on how geodesics behave near the degeneracy curve where the metric changes signature.

Contribution

It provides a comprehensive overview of the local properties and admissible directions of geodesics near signature-changing curves on 2-surfaces, highlighting new insights into their singularities.

Findings

Geodesics are well-behaved in R and L domains with unique passing directions.

Near the degeneracy curve, geodesics have limited admissible directions (1 to 3).

The study characterizes local geodesic behavior around signature-changing points.

Abstract

We present a survey on generic singularities of geodesic flows in smooth signature changing metrics (often called pseudo-Riemannian) in dimension 2. Generically, a pseudo-Riemannian metric on a 2-manifold $S$ changes its signature (degenerates) along a curve $S_0$, which locally separates $S$ into a Riemannian ($R$) and a Lorentzian ($L$) domain. The geodesic flow does not have singularities over $R$ and $L$, and for any point $q \in R \cup L$ and every tangential direction $p$ there exists a unique geodesic passing through the point $q$ with the direction $p$. On the contrary, geodesics cannot pass through a point $q \in S_0$ in arbitrary tangential directions, but only in some admissible directions; the number of admissible directions is 1 or 2 or 3. We study this phenomenon and the local properties of geodesics near $q \in S_0$.