Advertisement

Some Variational Problems in Semi-Riemannian Geometry

  • Antonio Masiello
Chapter
  • 1.1k Downloads
Part of the Lecture Notes in Physics book series (LNP, volume 692)

Abstract

In this contribution we are concerned with global properties of geodesics on semi-Riemannian manifolds obtained by studying the variational properties of the action functional. Applications to physically meaningful spacetimes in General Relativity will be presented.

Keywords

Riemannian Manifold Morse Index Morse Theory Complete Riemannian Manifold Critical Point Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. Abbondandolo: Morse Theory for Hamiltonian systems, CRC Research Notes in Mathematics 425 (Chapman and Hall, London 2001)Google Scholar
  2. 2.
    A. Abbondandolo, V. Benci, D. Fortunate, A. Masiello: Math. R.es. Lett. 10, 435 (2003)zbMATHGoogle Scholar
  3. 3.
    D.E. Allison, B. Ünal: J. Geom. Phys. 46, 193 (2003)zbMATHMathSciNetCrossRefADSGoogle Scholar
  4. 4.
    J.K. Beem, P. E. Ehrlich, K. L. Easley: Global Lorentzian Geometry, 2nd edn (Marcel Dekker, New York 1996)zbMATHGoogle Scholar
  5. 5.
    V. Benci, D. Fortunate, F. Giannoni: Ann. Inst. H. Poincaré, Analyse Non-linéaire 8, 79 (1991)zbMATHGoogle Scholar
  6. 6.
    V. Benci, D. Fortunate, F. Giannoni: Ann. Sc. Norm. Sup. (IV) XIX, 255 (1992)Google Scholar
  7. 7.
    V. Benci, D. Fortunate, A. Masiello: Math. Z. 217, 73 (1994)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    V. Benci, F. Giannoni, A. Masiello: J. Geom. Phys. 27, 267 (1998)zbMATHMathSciNetCrossRefADSGoogle Scholar
  9. 9.
    V. Benci, A. Masiello: Math. Ann. 293, 433 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    A.N. Bernal, M. Sánchez: Comm. Math. Phys. 243, 461 (2003)zbMATHMathSciNetCrossRefADSGoogle Scholar
  11. 11.
    R. Bott: Bull. Am. Math. Soc 7, 331 (1982)zbMATHMathSciNetCrossRefADSGoogle Scholar
  12. 12.
    E. Calabi, L. Markus: Ann. Math. 75, 63 (1962)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    A.M. Candela, F. Giannoni, A. Masiello: J. Diff. Eq. 155, 203 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    A.M. Candela, M. Sánchez: Diff. Geom. Appl. 12, 105 (2000)zbMATHCrossRefGoogle Scholar
  15. 15.
    E. Caponio, A. Masiello: Class. Quantum Grav. 19, 2229 (2002)zbMATHMathSciNetCrossRefADSGoogle Scholar
  16. 16.
    E. Caponio, A. Masiello: J. Math. Phys. 45, 4134 (2004)zbMATHMathSciNetCrossRefADSGoogle Scholar
  17. 17.
    E. Caponio, A. Masiello, P. Piccione: Math. Z. 244, 457 (2003)zbMATHMathSciNetGoogle Scholar
  18. 18.
    E. Caponio, A. Masiello, P. Piccione: Manuscr. Math. 113, 471 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    K.C. Chang: Infinite Dimensional Morse Theory and Multiple Solutions Problems (Birkhäuser, Boston 1993)Google Scholar
  20. 20.
    C. Conley, E. Zehnder: Comm. Pure Appl. Math. 37, 207 (1984)zbMATHMathSciNetADSCrossRefGoogle Scholar
  21. 21.
    M. Degiovanni, A. Giacomini: Nonlinear Anal. T. M. A. 47, 5041 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    E. Fadell, S. Husseini: Nonlinear Anal. T. M. A 17, 1153 (1991)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    E. Fadell, S. Husseini: Rend. Sem. Mat. Fis. Univ. Milano, LXIV, 99 (1994)MathSciNetCrossRefGoogle Scholar
  24. 24.
    J.L. Flores, M. Sánchez: J. Geom. Phys. 36, 285 (2000)zbMATHMathSciNetCrossRefADSGoogle Scholar
  25. 25.
    J.L. Flores, M. Sánchez: J. Math. Phys. 43, 4861 (2002)zbMATHMathSciNetCrossRefADSGoogle Scholar
  26. 26.
    J.L. Flores, M. Sánchez: this volumeGoogle Scholar
  27. 27.
    D. Fortunate, F. Giannoni, A. Masiello: J. Geom. Phys. 15, 159 (1995)MathSciNetCrossRefADSGoogle Scholar
  28. 28.
    G. Fournier and M. Willem: Relative category and the calculus of variations. In Variational Problems, ed by H. Beresticki, J.M. Coron, I. Ekeland (Birkhäuser, Basel 1990) pp 95–104Google Scholar
  29. 29.
    G.J. Galloway: Trans. Am. Math. Soc. 285, 379 (1984)zbMATHMathSciNetCrossRefGoogle Scholar
  30. 30.
    G.J. Galloway: Proc. Am. Math. Soc. 98, 119 (1986)zbMATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    R. Geroch: J. Math. Phys. 11, 437 (1970)MathSciNetCrossRefzbMATHADSGoogle Scholar
  32. 32.
    F. Giannoni, A. Masiello: J. Funct. Anal. 101, 340 (1991)zbMATHMathSciNetCrossRefGoogle Scholar
  33. 33.
    F. Giannoni, A. Masiello: Manuscr. Math. 78, 381 (1993)zbMATHMathSciNetCrossRefGoogle Scholar
  34. 34.
    F. Giannoni, A. Masiello: Ann. Inst. H. Poincaré, Analyse Nonlinéaire 12, 27 (1995)zbMATHMathSciNetGoogle Scholar
  35. 35.
    F. Giannoni, A. Masiello: Top. Meth. Nonlinear Anal. 6, 1 (1995)zbMATHMathSciNetGoogle Scholar
  36. 36.
    F. Giannoni, A. Masiello, P. Piccione: Comm. Math. Phys. 187, 1375 (1997)MathSciNetCrossRefGoogle Scholar
  37. 37.
    F. Giannoni, A. Masiello, P. Piccione: Ann. Inst. H. Poincaré, Phys. Theor. 69, 359 (1998)zbMATHMathSciNetGoogle Scholar
  38. 38.
    F. Giannoni, A. Masiello, P. Piccione: Class. Quantum Grav. 69, 731 (1999)MathSciNetCrossRefADSGoogle Scholar
  39. 39.
    F. Giannoni, A. Masiello, P. Piccione: J. Geom. Phys. 35, 1 (2000)zbMATHMathSciNetCrossRefADSGoogle Scholar
  40. 40.
    F. Giannoni, A. Masiello, P. Piccione: Gen. Rel. Grav. 33, 491 (2001)zbMATHMathSciNetCrossRefADSGoogle Scholar
  41. 41.
    F. Giannoni, A. Masiello, P. Piccione: J. Math. Phys. 43, 563 (2002)zbMATHMathSciNetCrossRefADSGoogle Scholar
  42. 42.
    F. Giannoni, A. Masiello, P. Piccione, D. Tausk: Asian J. Math. 3, 441 (2001)zbMATHMathSciNetGoogle Scholar
  43. 43.
    F. Giannoni, P. Piccione: Comm. Anal. Geom. 7, 157 (1999)zbMATHMathSciNetGoogle Scholar
  44. 44.
    F. Giannoni, P. Piccione, R. Sampalmieri: J. Math. Anal. Appl. 252, 444 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  45. 45.
    M. Guediri: Math. Z. 239, 277 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  46. 46.
    S.W. Hawking, G.F.R. Ellis: The Large Scale Structure of Spacetime, (Cambridge University Press, Cambridge 1973)CrossRefGoogle Scholar
  47. 47.
    A. Heifer: Pac. J. Math. 164, 321 (1994)Google Scholar
  48. 48.
    W. Klingenberg: Riemannian Geometry, 2nd edition (W. De Gruyter, Berlin 1995)zbMATHGoogle Scholar
  49. 49.
    A. Masiello: J. Diff. Eq. 104, 48 (1993)zbMATHMathSciNetCrossRefGoogle Scholar
  50. 50.
    A. Masiello: Ann. Mat. Pura Appl. (IV) CLXVII, 299 (1994)MathSciNetCrossRefGoogle Scholar
  51. 51.
    A. Masiello: Variational Methods in Lorentzian Geometry, Pitman Research Notes in Mathematics 309 (Longman, London 1994)Google Scholar
  52. 52.
    J. Mawhin, M. Willem: Critical Point Theory and Hamiltonian Systems, (Springer, Berlin 1989)zbMATHGoogle Scholar
  53. 53.
    F. Mercuri, P. Piccione, D. Tausk: Pac. J. Math. 206, 375 (2002)zbMATHMathSciNetCrossRefGoogle Scholar
  54. 54.
    J. Milnor: Morse Theory, Ann. of Math. Studies 51 (Princeton Univ. Press, Princeton 1963)Google Scholar
  55. 55.
    B. O'Neill: Semi-Riemannian Geometry with Applications to Relativity, (Academic Press, New York 1983)zbMATHGoogle Scholar
  56. 56.
    R. Palais: Topology 2, 299 (1963)zbMATHMathSciNetCrossRefGoogle Scholar
  57. 57.
    R. Penrose: Techniques of Differential Topology in Relativity, Regional Conference Series in Applied Math. 7 (Society for Industrial and Applied Mathematics, Philadelphia 1972)Google Scholar
  58. 58.
    V. Perlick: Ray Optics, Fermat's Principle, and Applications to General Relativity, Lecture Notes in Physics m61 (Springer, Heidelberg 2000)Google Scholar
  59. 59.
    V. Perlick: Living Rev. Relativity 7 (2004), 9. http://www.livingreviews.org/lrr-2004-9
  60. 60.
    P. Piccione, D. Tausk: Proc. London Math. Soc. (3) 83, 351 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  61. 61.
    P. Piccione, D. Tausk: Comm. Anal. Geom. 11, 33 (2003)zbMATHMathSciNetGoogle Scholar
  62. 62.
    J.P. Serre: Ann. Math. 54, 425 (1951)zbMATHMathSciNetCrossRefGoogle Scholar
  63. 63.
    E.H. Spanier: Algebraic Topology (McGraw Hill, New York 1966)zbMATHGoogle Scholar
  64. 64.
    M. Struwe: Variational Methods, 3rd edn (Springer, Heidelberg 2000)zbMATHGoogle Scholar
  65. 65.
    A. Szulkin: Math. Z. 209, 375 (1992)zbMATHMathSciNetCrossRefGoogle Scholar
  66. 66.
    F.J. Tipler: Proc. Am. Math. Soc. 76, 145 (1979)zbMATHMathSciNetCrossRefGoogle Scholar
  67. 67.
    K. Uhlenbeck: Topology 14, 69 (1975)zbMATHMathSciNetCrossRefGoogle Scholar
  68. 68.
    R. Wald: General Relativity (University of Chicago Press, Chicago 1984)zbMATHGoogle Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  • Antonio Masiello
    • 1
  1. 1.Dipartimento di MatematicaBariItaly

Personalised recommendations