Closed geodesics in Lorentzian surfaces

Suhr, Stefan

doi:10.1090/S0002-9947-2012-05691-1

Closed geodesics in Lorentzian surfaces
HTML articles powered by AMS MathViewer

by Stefan Suhr PDF

Trans. Amer. Math. Soc. 365 (2013), 1469-1486 Request permission

Abstract:

We show that every closed Lorentzian surface contains at least two closed geodesics. Explicit examples show the optimality of this claim. Refining this result we relate the least number of closed geodesics to the causal structure of the surface and the homotopy type of the Lorentzian metric.

References

Leonardo Biliotti, Francesco Mercuri, and Paolo Piccione, On a Gromoll-Meyer type theorem in globally hyperbolic stationary spacetimes, Comm. Anal. Geom. 16 (2008), no. 2, 333–393. MR 2425470
M. Caponio, M. Javaloyes and A. Masiello. On the energy functional on Finsler manifolds and applications to stationary spacetimes, arXiv:math/0702323v3, (2009)
Yves Carrière and Luc Rozoy, Complétude des métriques lorentziennes de $\textbf {T}^2$ et difféormorphismes du cercle, Bol. Soc. Brasil. Mat. (N.S.) 25 (1994), no. 2, 223–235 (French, with English summary). MR 1306563, DOI 10.1007/BF01321310
José Luis Flores, Miguel Ángel Javaloyes, and Paolo Piccione, Periodic geodesics and geometry of compact Lorentzian manifolds with a Killing vector field, Math. Z. 267 (2011), no. 1-2, 221–233. MR 2772249, DOI 10.1007/s00209-009-0617-5
Gregory J. Galloway, Closed timelike geodesics, Trans. Amer. Math. Soc. 285 (1984), no. 1, 379–388. MR 748844, DOI 10.1090/S0002-9947-1984-0748844-6
Gregory J. Galloway, Compact Lorentzian manifolds without closed nonspacelike geodesics, Proc. Amer. Math. Soc. 98 (1986), no. 1, 119–123. MR 848888, DOI 10.1090/S0002-9939-1986-0848888-7
Robert P. Geroch, Topology in general relativity, J. Mathematical Phys. 8 (1967), 782–786. MR 213139, DOI 10.1063/1.1705276
Robert Geroch, Domain of dependence, J. Mathematical Phys. 11 (1970), 437–449. MR 270697, DOI 10.1063/1.1665157
Mohammed Guediri, On the existence of closed timelike geodesics in compact spacetimes, Math. Z. 239 (2002), no. 2, 277–291. MR 1888225, DOI 10.1007/s002090100295
S. W. Hawking and G. F. R. Ellis, The large scale structure of space-time, Cambridge Monographs on Mathematical Physics, No. 1, Cambridge University Press, London-New York, 1973. MR 0424186
Gilbert Hector and Ulrich Hirsch, Introduction to the geometry of foliations. Part A, 2nd ed., Aspects of Mathematics, vol. 1, Friedr. Vieweg & Sohn, Braunschweig, 1986. Foliations on compact surfaces, fundamentals for arbitrary codimension, and holonomy. MR 881799, DOI 10.1007/978-3-322-90115-6
Gustav A. Hedlund, Geodesics on a two-dimensional Riemannian manifold with periodic coefficients, Ann. of Math. (2) 33 (1932), no. 4, 719–739. MR 1503086, DOI 10.2307/1968215
Morris W. Hirsch and Stephen Smale, Differential equations, dynamical systems, and linear algebra, Pure and Applied Mathematics, Vol. 60, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0486784
A. Masiello. Variational methods in Lorentzian geometry Harlow: Longman Scientific & Technical (1994).
Ettore Minguzzi and Miguel Sánchez, The causal hierarchy of spacetimes, Recent developments in pseudo-Riemannian geometry, ESI Lect. Math. Phys., Eur. Math. Soc., Zürich, 2008, pp. 299–358. MR 2436235, DOI 10.4171/051-1/9
B. O’Neill. Semi-Riemannian geometry, Pure and Applied Mathematics, Vol. 103, New York-London: Academic Press (1983).
Roger Penrose, Techniques of differential topology in relativity, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 7, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1972. MR 0469146
Alfonso Romero and Miguel Sánchez, On completeness of certain families of semi-Riemannian manifolds, Geom. Dedicata 53 (1994), no. 1, 103–117. MR 1299888, DOI 10.1007/BF01264047
Miguel Sánchez, Structure of Lorentzian tori with a Killing vector field, Trans. Amer. Math. Soc. 349 (1997), no. 3, 1063–1080. MR 1376554, DOI 10.1090/S0002-9947-97-01745-5
Miguel Sánchez, On causality and closed geodesics of compact Lorentzian manifolds and static spacetimes, Differential Geom. Appl. 24 (2006), no. 1, 21–32. MR 2193746, DOI 10.1016/j.difgeo.2005.06.008
E. Schelling. Maximale Geodätische auf Lorentzmannigfaltigkeiten, Diplomarbeit, Freiburg, 1995.
Frank J. Tipler, Existence of closed timelike geodesics in Lorentz spaces, Proc. Amer. Math. Soc. 76 (1979), no. 1, 145–147. MR 534406, DOI 10.1090/S0002-9939-1979-0534406-6