Geometry & Topology

Sur les surfaces lorentziennes compactes sans points conjugués

Christophe Bavard and Pierre Mounoud

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Abstract

Nous prouvons l’existence de métriques sans points conjugués dans toute composante connexe de l’espace des métriques lorentziennes du tore ou de la bouteille de Klein. En particulier, l’existence de tores lorentziens sans points conjugués non plats contraste avec la situation riemannienne (théorème de Hopf).

We prove the existence of metrics without conjugate points in any connected component of the space of Lorentzian metrics on the torus or on the Klein bottle. In particular, the existence of nonflat Lorentzian tori without conjugate points contrasts with the Riemannian case (the Hopf Theorem).

Article information

Source
Geom. Topol., Volume 17, Number 1 (2013), 469-492.

Dates
Received: 30 May 2011
Revised: 8 October 2012
Accepted: 3 July 2012
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732529

Digital Object Identifier
doi:10.2140/gt.2013.17.469

Mathematical Reviews number (MathSciNet)
MR3039767

Zentralblatt MATH identifier
1262.53059

Subjects
Primary: 53C50: Lorentz manifolds, manifolds with indefinite metrics

Keywords
Lorentzian surfaces conjugate points surfaces lorentziennes points conjugués

Citation

Bavard, Christophe; Mounoud, Pierre. Sur les surfaces lorentziennes compactes sans points conjugués. Geom. Topol. 17 (2013), no. 1, 469--492. doi:10.2140/gt.2013.17.469. https://projecteuclid.org/euclid.gt/1513732529


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References

  • L Andersson, M Dahl, R Howard, Boundary and lens rigidity of Lorentzian surfaces, Trans. Amer. Math. Soc. 348 (1996) 2307–2329
  • C Boubel, P Mounoud, C Tarquini, Lorentzian foliations on 3–manifolds, Ergodic Theory Dynam. Systems 26 (2006) 1339–1362
  • K S Brown, Cohomology of groups, Graduate Texts in Math. 87, Springer, New York (1982)
  • N Bruschlinsky, Stetige Abbildungen und Bettische Gruppen der Dimensionszahlen 1 und 3, Math. Ann. 109 (1934) 525–537
  • D Burago, S Ivanov, Riemannian tori without conjugate points are flat, Geom. Funct. Anal. 4 (1994) 259–269
  • Y Carrière, L Rozoy, Complétude des métriques lorentziennes de $\mathbold{T}\sp 2$ et difféormorphismes du cercle, Bol. Soc. Brasil. Mat. 25 (1994) 223–235
  • L Green, R Gulliver, Planes without conjugate points, J. Differential Geom. 22 (1985) 43–47
  • M Gutiérrez, F J Palomo, A Romero, Lorentzian manifolds with no null conjugate points, Math. Proc. Cambridge Philos. Soc. 137 (2004) 363–375
  • G H Halphen, Traité des fonctions elliptiques et de leurs applications. I, Gauthier-Villars, Paris (1886)
  • E Hopf, Closed surfaces without conjugate points, Proc. Nat. Acad. Sci. U. S. A. 34 (1948) 47–51
  • S-t Hu, Homotopy theory, Pure and Applied Mathematics 8, Academic Press, New York (1959)
  • W B R Lickorish, Homeomorphisms of non-orientable two-manifolds, Proc. Cambridge Philos. Soc. 59 (1963) 307–317
  • F Mercuri, P Piccione, D V Tausk, Stability of the conjugate index, degenerate conjugate points and the Maslov index in semi-Riemannian geometry, Pacific J. Math. 206 (2002) 375–400
  • B O'Neill, Semi-Riemannian geometry, with applications to relativity, Pure and Applied Math. 103, Academic Press, New York (1983)
  • B L Reinhart, Line elements on the torus, Amer. J. Math. 81 (1959) 617–631
  • R O Ruggiero, On the creation of conjugate points, Math. Z. 208 (1991) 41–55
  • N Steenrod, The Topology of Fibre Bundles, Princeton Math. Series 14, Princeton Univ. Press (1951)