A Brief Survey on Singularities of Geodesic Flows in Smooth Signature Changing Metrics on 2-Surfaces

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 222)

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 \(p \in {\mathbb R}{\mathbb P}\) 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 \(q \in S_0\).

Keywords

Pseudo-Riemannian metrics Geodesics Singular points Normal forms 

2010 Mathematics Subject classification

53C22 53B30 34C05 

Notes

Acknowledgements

The publication was supported by the Russian Foundation for Basic Research (research projects 16-01-00766, 17-01-00849) and the Laboratory of Dynamical Systems NRU  HSE.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Nonlinear Analysis and OptimizationRUDN UniversityMoscowRussia
  2. 2.V.A. Trapeznikov Institute of Control Sciences of Russian Academy of SciencesMoscowRussia
  3. 3.Laboratory of Dynamical SystemsNational Research University Higher School of EconomicsMoscowRussia

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