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)


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 \(q \in S_0\).


Pseudo-Riemannian metrics Geodesics Singular points Normal forms 

2010 Mathematics Subject classification

53C22 53B30 34C05 



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.


  1. 1.
    Aguirre, E., Fernandez, V., Lafuente, J.: On the conformal geometry of transverse Riemann-Lorentz manifolds. J. Geom. Phys. 57, 1541–1547 (2007)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Al’tshuler, B.L., Barvinski, A.O.: Quantum cosmology and physics of transitions with a change of spacetime signature. Uspekhi Fiz. Nauk 166(5), 459–492 (1996); English transl. in Physics-Uspekhi 39, 429Google Scholar
  3. 3.
    Aminova, A.V.: Projective transformations and symmetries of differential equations. Mat. Sb. 186(12), 21–36 (1995)MathSciNetMATHGoogle Scholar
  4. 4.
    Anosov, D.V., Arnold, V.I. (eds.): Dynamical systems I. Ordinary differential equations and smooth dynamical systems. Encyclopaedia of Mathematical Sciences 1. Springer, Berlin (1988)Google Scholar
  5. 5.
    Arnol’d, V.I.: Geometrical Methods in the Theory of Ordinary Differential Equations. Springer, New York (1988)CrossRefMATHGoogle Scholar
  6. 6.
    Bogaevsky, I.A.: Implicit ordinary differential equations: bifurcations and sharpening of equivalence. Izvestiya Math. 78(6), 1063–1078 (2014)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bolsinov, A.V., Matveev, V.S.: Local normal forms for geodesically equivalent pseudo-Riemannian metrics. Trans. Am. Math. Soc. 367, 6719–6749 (2015)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cibrario, M.: Sulla reduzione a forma delle equationi lineari alle derviate parziale di secondo ordine di tipo misto. Accademia di Scienze e Lettere, Instituto Lombardo Redicconti 65, 889–906 (1932)MATHGoogle Scholar
  9. 9.
    Dara, L.: Singularités générique des équations différentielles multiformes. Bol. Soc. Bras. Math. 6(2), 95–128 (1975)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Davydov, A.A.: The normal form of a differential equation, that is not solved with respect to the derivative, in the neighborhood of its singular point. Funktsional. Anal. i Prilozhen. 19, 1–10 (1985)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Davydov, A.A., Ishikawa, G., Izumiya, S., Sun, W.-Z.: Generic singularities of implicit systems of first order differential equations on the plane. Jpn. J. Math. 3, 93–119 (2008)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Genin, D., Khesin, B., Tabachnikov, S.: Geodesics on an ellipsoid in Minkowski space. Enseign. Math. 53, 307–331 (2007)MathSciNetMATHGoogle Scholar
  13. 13.
    Ghezzi, R., Remizov, A.O.: On a class of vector fields with discontinuities of divide-by-zero type and its applications to geodesics in singular metrics. J. Dyn. Control Syst. 18, 135–158 (2012)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Hartman, Ph: Ordinary Differential Equations. Birkhauser, Boston (1982)MATHGoogle Scholar
  15. 15.
    Hirsch, M.W., Pugh, C.C., Shub, M.: Invariant Manifolds. Lecture Notes in Mathematics, vol. 583. Springer, Berlin (1977)Google Scholar
  16. 16.
    Khesin, B., Tabachnikov, S.: Pseudo-Riemannian geodesics and billiards. Adv. Math. 221, 1364–1396 (2009)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Kossowski, M.: Pseudo-Riemannian metrics singularities and the extendability of parallel transport. Proc. Amer. Math. Soc. 99, 147–154 (1987)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Kossowski, M., Kriele, M.: Smooth and discontinuous signature type change in general relativity. Class. Quantum Grav. 10, 2363–2371 (1993)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Kossowski, M., Kriele, M.: Transverse, type changing, pseudo-Riemannian metrics and the extendability of geodesics. Proc. Roy. Soc. Lond. Ser. A Math. Phys. 444(1921), 297–306 (1994)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Kossowski, M., Kriele, M.: The Einstein equation for signature type changing spacetimes. Proc. Roy. Soc. Lond. Ser. A Math. Phys. 446(1926), 115–126 (1994)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Miernowski, T.: Formes normales d’une métrique mixte analytique réelle générique. Ann. Fac. Sci. Toulouse Math. 16, 923–946 (2007)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Pavlova, N.G., Remizov, A.O.: Geodesics on hypersurfaces in the Minkowski space: singularities of signature change. Russian Math. Surveys 66, 1201–1203 (2011)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Remizov, A.O.: Multidimensional Poincaré construction and singularities of lifted fields for implicit differential equations. J. Math. Sci. (N.Y.) 151(6), 3561–3602 (2008)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Remizov, A.O.: Geodesics on 2-surfaces with pseudo-Riemannian metric: singularities of changes of signature. Mat. Sb. 200(3), 75–94 (2009)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Remizov, A.O.: On the local and global properties of geodesics in pseudo-Riemannian metrics. Differ. Geom. Appl. 39, 36–58 (2015)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Remizov, A.O., Tari, F.: Singularities of the geodesic flow on surfaces with pseudo-Riemannian metrics. Geom. Dedicata. 185(1), 131–153 (2016)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Sakharov, A.D.: Cosmological transitions with changes in the signature of the metric. Zh. Eksper. Teor. Fiz. 87(2) (8) 375–383 (1984). English transl. in Soviet Phys. JETP 60(2), August 1984, 214–218Google Scholar
  28. 28.
    Steller, M.: A Gauss-Bonnet formula for metrics with varying signature. Z. Anal. Anwend. 25, 143–162 (2006)MathSciNetCrossRefMATHGoogle Scholar

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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