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Gallot–Tanno theorem for closed incomplete pseudo-Riemannian manifolds and applications

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Abstract

We extend the Gallot–Tanno theorem to closed pseudo-Riemannian manifolds. It is done by showing that if the cone over a manifold admits a parallel symmetric (0, 2)-tensor then it is Riemannian. Applications of this result to the existence of metrics with distinct Levi-Civita connections but having the same unparametrized geodesics and to the projective Obata conjecture are given. We also apply our result to show that the holonomy group of a closed (O(p + 1, q), S p,q)-manifold does not preserve any nondegenerate splitting of \({\mathbb {R}^{p+1,q}}\).

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References

  1. Alekseevsky D.V., Cortes V., Galaev A.S., Leistner T.: Cones over pseudo-Riemannian manifolds and their holonomy. J. Reine Angew. Math. (Crelle’s journal) 635, 23–69 (2009) arXiv:0707.3063v2

    Article  MATH  MathSciNet  Google Scholar 

  2. Carrière Y., Rozoy L.: Complétude des métriques lorentziennes de T 2 et difféormorphismes du cercle. Bol. Soc. Brasil. Mat. (N.S.) 25(2), 223–235 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  3. Gallot S.: Équations différentielles caractéristiques de la sphère. Ann. scient. Éc. Norm. Sup. 4e série 12, 235–267 (1979)

    Google Scholar 

  4. Goldman W., Hirsch M.W.: The radiance obstruction and parallel forms on affine manifolds. Trans. Am. 286(2), 629–649 (1984)

    MATH  MathSciNet  Google Scholar 

  5. Hiramatu H.: Riemannian manifolds admitting a projective vector field. Kodai Math. J. 3(3), 397–406 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  6. Kiosak, V., Matveev, V.S.: Proof of projective Lichnerowicz conjecture for pseudo-Riemannian metrics with degree of mobility greater than two. Commun. Math. Phys. (2010). doi:10.1007/s00220-010-1037-4

  7. Kiosak V., Matveev V.S.: Complete Einstein metrics are geodesically rigid. Commun. Math. Phys. 289(1), 383–400 (2009) arXiv:0806.3169

    Article  MATH  MathSciNet  Google Scholar 

  8. Kiyohara K.: Compact Liouville surfaces. J. Math. Soc. Jpn. 43, 555–591 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  9. Klingler B.: Complétude des variétés lorentziennes à courbure constante. Math. Ann. 306(2), 353–370 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  10. Matveev V.S.: Die Vermutung von Obata für Dimension 2. Arch. Math. 82, 273–281 (2004)

    Article  MATH  Google Scholar 

  11. Matveev V.S.: Proof of the Lichnerowicz Conjecture. J. Diff. Geom. 75, 459–502 (2007) arXiv:math/0407337

    MATH  Google Scholar 

  12. Matveev, V.S.: Gallot–Tanno theorem for pseudo-Riemannian manifolds and a proof that decomposable cones over closed complete pseudo-Riemannian manifolds do not exist. J. Diff. Geom. Appl. 28(2), 236–240, (2010). doi:10.1016/j.difgeo.2009.10.009, arXiv:0906.2410

  13. Matveev, V.S.: Pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta, and proof of the projective Obata conjecture for two-dimensional pseudo-Riemannian metrics. arXiv:1002.3934 (submitted)

  14. Obata, M.: Riemannian manifolds admitting a solution of a certain system of differential equations. In: Proceedingds of the U.S.–Japan Seminar in Differential Geometry, pp. 101–114. Kyoto (1965)

  15. Obata M.: Certain conditions for a Riemannian manifold to be isometric with a sphere. J. Math. Soc. Jpn. 14, 333–340 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  16. Salein F.: Variétés anti-de Sitter de dimension 3 possédant un champ de Killing non trivial. C. R. Acad. Sci. Paris I Math. 324(5), 525–530 (1997)

    MATH  MathSciNet  Google Scholar 

  17. Solodovnikov A.S.: Projective transformations of Riemannian spaces. Uspehi Mat. Nauk (N.S.) 11(4(70)), 45–116 (1956)

    MATH  MathSciNet  Google Scholar 

  18. Tanno S.: Some differential equations on Riemannian manifolds. J. Math. Soc. Jpn. 30(3), 509–531 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  19. Thurston W.: Three dimensional Geometry and Topology vol 1. In: Silvio, L. (eds) Princeton Mathematical Series, vol. 35, Princeton University Press, Princeton, NJ, Princeton, NJ (1997)

    Google Scholar 

  20. Wu H.: On the de Rham decomposition theorem. Ill. J. Math. 8, 291–311 (1964)

    MATH  Google Scholar 

  21. Zeghib A.: On closed anti de Sitter spacetimes. Math. Ann. 310, 695–716 (1998)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, FSU Jena, 07737, Jena, Germany

    Vladimir S. Matveev

  2. Institut de Mathématiques de Bordeaux, Université Bordeaux 1, 351, cours de la libération, 33405, Talence, France

    Pierre Mounoud

Authors
  1. Vladimir S. Matveev
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  2. Pierre Mounoud
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Correspondence to Pierre Mounoud.

Additional information

Vladimir S. Matveev—partially supported by DFG (SPP 1154 and GK 1523).

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Matveev, V.S., Mounoud, P. Gallot–Tanno theorem for closed incomplete pseudo-Riemannian manifolds and applications. Ann Glob Anal Geom 38, 259–271 (2010). https://doi.org/10.1007/s10455-010-9211-7

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  • DOI: https://doi.org/10.1007/s10455-010-9211-7

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