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

  • N. G. Pavlova
  • A. O. RemizovEmail author
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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle 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)CrossRefzbMATHGoogle Scholar
  6. 6.
    Bogaevsky, I.A.: Implicit ordinary differential equations: bifurcations and sharpening of equivalence. Izvestiya Math. 78(6), 1063–1078 (2014)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)zbMATHGoogle Scholar
  9. 9.
    Dara, L.: Singularités générique des équations différentielles multiformes. Bol. Soc. Bras. Math. 6(2), 95–128 (1975)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Genin, D., Khesin, B., Tabachnikov, S.: Geodesics on an ellipsoid in Minkowski space. Enseign. Math. 53, 307–331 (2007)MathSciNetzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hartman, Ph: Ordinary Differential Equations. Birkhauser, Boston (1982)zbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kossowski, M.: Pseudo-Riemannian metrics singularities and the extendability of parallel transport. Proc. Amer. Math. Soc. 99, 147–154 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kossowski, M., Kriele, M.: Smooth and discontinuous signature type change in general relativity. Class. Quantum Grav. 10, 2363–2371 (1993)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Miernowski, T.: Formes normales d’une métrique mixte analytique réelle générique. Ann. Fac. Sci. Toulouse Math. 16, 923–946 (2007)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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

Personalised recommendations