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Maslov index in semi-Riemannian submersions

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Abstract

We study focal points and Maslov index of a horizontal geodesic γ : I → M in the total space of a semi-Riemannian submersion π : M → B by determining an explicit relation with the corresponding objects along the projected geodesic \({\pi\circ\gamma:I\to B}\) in the base space. We use this result to calculate the focal Maslov index of a (spacelike) geodesic in a stationary spacetime which is orthogonal to a timelike Killing vector field.

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References

  1. Beem J.K., Ehrlich P.E., Easley K.: Global Lorentzian Geometry. 2nd edn. Marcel Dekker Inc., NY (1996)

    MATH  Google Scholar 

  2. Bourguignon, J.P.: A mathematician’s visit to Kaluza-Klein theory. Rend. Sem. Mat. Univ. Politec. Torino, special issue, pp. 143–163 (1989)

  3. de Gosson M.: La définition de l’indice de Maslov sans hypothèse de transversalité, C. R. Acad. Sci., Paris, Sér. I Math. 310(5), 279–282 (1990)

    MATH  Google Scholar 

  4. de Gosson M.: The structure of q-symplectic geometry. J. Math. Pures Appl. 71, 429–453 (1992)

    MATH  MathSciNet  Google Scholar 

  5. Ehresmann, C.: Les connexions infinitésimales dans un espace fibré différentiable, Colloque de topologie (espaces fibrés), Bruxelles, 1950, Georges Thone, Liège (1951), pp. 29–55

  6. Gray A.: Pseudo-Riemannian almost product manifolds and submersions. J. Math. Mech. 16, 715–737 (1967)

    MATH  MathSciNet  Google Scholar 

  7. Helfer A.D.: Conjugate points on spacelike geodesics or pseudo-selfadjoint Morse-Sturm-Liouville systems. Pac. J. Math. 164, 321–350 (1994)

    MATH  MathSciNet  Google Scholar 

  8. Hermann R.: A sufficient condition that a mapping of Riemannian manifolds be a fibre bundle. Proc. Am. Math. Soc. 11, 236–242 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  9. Hogan P.A.: Kaluza-Klein theory derived from a Riemannian submersion. J. Math. Phys. 25, 2301–2305 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  10. Javaloyes M.A., Masiello A., Piccione P.: Pseudo focal points along Lorentzian geodesics and Morse index. Adv. Nonlinear Stud. 10, 53–82 (2041)

    MathSciNet  Google Scholar 

  11. Javaloyes M.A., Piccione P.: Comparison results for conjugate and focal points in semi-Riemannian geometry via Maslov index. Pac. J. Math. 243(1), 43–56 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  12. Javaloyes, M.A., Sánchez, M.: A note on the existence of standard splittings for conformally stationary spacetimes. Class. Quantum Gravity 25(16) (2008), 168001, 7 pp

    Google Scholar 

  13. Mercuri F., Piccione P., Tausk D.V.: Stability of the conjugate index, degenerate conjugate points and the Maslov index in semi-Riemannian geometry. Pac. J. Math. 206, 375–400 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  14. Morvan Maslov, J.-M.: Duistermaat, Conley-Zehnder invariants in Riemannian Geometry. In: Geometry and Topology of Submanifold, V (Leuven/Brussels, 1992), pp. 174–200

  15. O’Neill B.: The fundamental equations of a submersion. Michigan Math. J. 13, 459–469 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  16. O’Neill B.: Submersions and geodesics. Duke Math. J. 34, 363–373 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  17. O’Neill, B.: Semi-Riemannian geometry, vol. 103 of Pure and Applied Mathematics, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York (1983). With applications to relativity

  18. Piccione P., Portaluri A., Tausk D.V.: Spectral flow, Maslov index and bifurcation of semi-Riemannian geodesics. Ann. Global Anal. Geom. 25, 121–149 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  19. Piccione P., Tausk D.V.: A note on the Morse index theorem for geodesics between submanifolds in semi-Riemannian geometry. J. Math. Phys. 40, 6682–6688 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  20. Piccione P., Tausk D.V.: An index theorem for non periodic solutions of Hamiltonian systems. Proc. Lond. Math. Soc. 83, 351–389 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  21. Piccione P., Tausk D.V.: The Morse index theorem in semi-Riemannian geometry. Topology 41, 1123–1159 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  22. Piccione P., Tausk D.V.: On the distribution of conjugate points along semi-Riemannian geodesics. Commun. Anal. Geom. 11, 33–48 (2003)

    MATH  MathSciNet  Google Scholar 

  23. Piccione P., Tausk D.V. A Students’ Guide to Symplectic Spaces, Grassmannians and Maslov Index, Publicações Matemática do IMPA, Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2008. ISBN: 978-85-244-0283-8

  24. Robbin J., Salamon D.: The Maslov index for paths. Topology 32(4), 827–844 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  25. Salamon D., Zehnder E.: Morse theory for periodic solutions of Hamiltonian systems and the Maslov index. Commun. Pure Appl. Math. 45(10), 1303–1360 (1992)

    Article  MATH  MathSciNet  Google Scholar 

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Author notes

  1. Paolo Piccione

    Present address: Department of Mathematics, University of Murcia, Campus de Espinardo, 30100, Espinardo, Murcia, Spain

Authors and Affiliations

  1. Dipartimento di Matematica, Politecnico di Bari, Via Orabona 4, 70125, Bari, Italy

    Erasmo Caponio

  2. Departamento de Geometría y Topología, Facultad de Ciencias, Universidad de Granada, Campus Fuentenueva s/n, 18071, Granada, Spain

    Miguel Angel Javaloyes

  3. Departamento de Matemática, Universidade de São Paulo, Rua do Matão 1010, São Paulo, SP, 05508-900, Brazil

    Paolo Piccione

Authors
  1. Erasmo Caponio
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  2. Miguel Angel Javaloyes
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  3. Paolo Piccione
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Correspondence to Erasmo Caponio.

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Caponio, E., Javaloyes, M.A. & Piccione, P. Maslov index in semi-Riemannian submersions. Ann Glob Anal Geom 38, 57–75 (2010). https://doi.org/10.1007/s10455-010-9200-x

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  • DOI: https://doi.org/10.1007/s10455-010-9200-x

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