A Morse theory for light rays on stably causal lorentzian manifolds

F. Giannoni; A. Masiello; P. Piccione

Annales de l'I.H.P. Physique théorique (1998)

  • Volume: 69, Issue: 4, page 359-412
  • ISSN: 0246-0211

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Giannoni, F., Masiello, A., and Piccione, P.. "A Morse theory for light rays on stably causal lorentzian manifolds." Annales de l'I.H.P. Physique théorique 69.4 (1998): 359-412. <http://eudml.org/doc/76805>.

@article{Giannoni1998,
author = {Giannoni, F., Masiello, A., Piccione, P.},
journal = {Annales de l'I.H.P. Physique théorique},
keywords = {Lorentzian manifolds; light rays; Fermat principle; Morse theory; gravitational lenses},
language = {eng},
number = {4},
pages = {359-412},
publisher = {Gauthier-Villars},
title = {A Morse theory for light rays on stably causal lorentzian manifolds},
url = {http://eudml.org/doc/76805},
volume = {69},
year = {1998},
}

TY - JOUR
AU - Giannoni, F.
AU - Masiello, A.
AU - Piccione, P.
TI - A Morse theory for light rays on stably causal lorentzian manifolds
JO - Annales de l'I.H.P. Physique théorique
PY - 1998
PB - Gauthier-Villars
VL - 69
IS - 4
SP - 359
EP - 412
LA - eng
KW - Lorentzian manifolds; light rays; Fermat principle; Morse theory; gravitational lenses
UR - http://eudml.org/doc/76805
ER -

References

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  1. [1] R.R. Adams, Sobolev spaces. Ac. Press. New York, 1975. Zbl0314.46030MR450957
  2. [2] F. Antonacci and P. Piccione, A Fermat principle on Lorentzian manifolds and applications, Appl. Math. Lett., Vol. 9, 1996, pp. 91-96. Zbl0855.53038MR1383689
  3. [3] J.K. Beem and P.H. Ehrlich and K.L. Easley, Global Lorentzian Geometry. Marcel Dekker. New York, 1996. Zbl0846.53001MR1384756
  4. [4] V. Benci, A new approach to Morse-Conley theory and some applications, Ann. Mat. Pura ed Appl., Vol. 158, 1991, pp. 231-305. Zbl0778.58011MR1131853
  5. [5] R. Bott, Lectures on Morse Theory old and new, Bull. Am. Math. Soc., Vol. 7, 1982, pp. 331-358. Zbl0505.58001MR663786
  6. [6] A. Capozzi, D. Fortunato and C. Greco, Null geodesics on Lorentz manifolds, in Nonlinear variational problems and partial differential equations, Isola d'Elba 1990 (A. MARINO and M.K.V. MURTHY eds.), pp. 81-84. Pitman research notes in Mathematics, Vol. 320. Longman, London1995. Zbl0890.53042MR1330004
  7. [7] K. Deimiling, Nonlinear Functional Analysis. Springer-Verlag, Berlin1985. Zbl0559.47040MR787404
  8. [8] D. Fortunato, F. Giannoni and A. Masiello, A Fermat principle for stationary space-times with applications to light rays, J. Geom. Phys., Vol. 15, 1995, pp. 159-188. Zbl0819.53037MR1310949
  9. [9] A. Germinario, Morse Theory for light rays without nondegeneration assumptions, Nonlinear World, Vol. 4, 1997, pp. 173-206. Zbl0911.58006MR1485197
  10. [10] F. Giannoni and A. Masiello, Morse Relations for geodesics on stationary Lorentzian manifolds with boundary, Top. Meth. in Nonlinear Anal., Vol. 6, 1995, pp. 1-30. Zbl0852.58016MR1391942
  11. [11] F. Giannoni and A. Masiello, On a Fermat principle in General Relativity. A Ljustemik–Schnirelmann theory for light rays, Ann. Mat. Pura Appl., in press. Zbl0983.58008
  12. [12] F. Giannoni and A. Masiello, On a Fermat principle in General Relativity. A Morse Theory for light rays, Gen. Rel. Grav., Vol. 28, 1996, pp. 855-897. Zbl0855.53039MR1398288
  13. [13] F. Giannoni, A. Masiello and P. Piccione, A variational theory for light rays on causally stable Lorentzian manifolds: Regularity and multiplicity results, Comm. Math. Phys., Vol. 187, 1997, pp. 375-415. Zbl0884.53048MR1463834
  14. [14] F. Giannoni, A. Masiello and P. Piccione, A variational theory for light rays on causally stable Lorentzian manifolds II: Existence and multiplicity results, preprint n. 16/96 Dip. Mat. Univ. Bari, 1996. 
  15. [15] S.W. Hawking and G.F. Ellis, The Large Scale Structure of Space-Time. Cambridge University Press, London/New York, 1973. Zbl0265.53054MR424186
  16. [16] L.L. Kelley, General Topology. Van Nostrand, Princeton1955. Zbl0066.16604MR70144
  17. [17] W. Klingenberg, Riemannian Geometry. W. de Gruyter, Berlin/New York, 1982. Zbl0495.53036MR666697
  18. [18] I. Kovner, Fermat principles for arbitrary space-times, Astrophys. J., Vol. 351, 1990, pp. 114-120. 
  19. [19] T. Levi-Civita, Fondamenti di Meccanica Relativistica. Zanichelli, Bologna1928. JFM54.0939.01
  20. [20] A. Masiello, Variational Methods in Lorentzian Geometry. Pitman Research Notes in Mathematics, 309. Longman, London1994. Zbl0816.58001MR1294140
  21. [21 ] A. Masiello and P. Piccione, Shortening null geodesics in stationary Lorentzian manifolds. Applications to closed light rays, Diff. Geom. Appl., Vol. 8, 1998, pp. 47-70. Zbl0901.58010MR1601534
  22. [22] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems. Springer–Verlag, Berlin, 1989. Zbl0676.58017MR982267
  23. [23] R. Mckenzie, A gravitational lens produces an odd number of images, J. Math. Phys., Vol. 26, 1985, pp. 1592-1596. Zbl0569.53043MR793300
  24. [24] J. Milnor, Morse Theory. Princeton University Press, Princeton, 1963. Zbl0108.10401MR163331
  25. [25] M. Morse, The Calculus of Variations in the Large. Coll. Lect. Am. Math. Soc., Vol. 18, 1934. Zbl0011.02802JFM60.0450.01
  26. [26] B. O'Neill, Semi-Riemannian Geometry with applications to Relativity. Acad. Press, New-York-London, 1983. Zbl0531.53051
  27. [27] R. Palais, Morse Theory on Hilbert manifolds, Topology, Vol. 2, 1963, pp. 299-340. Zbl0122.10702MR158410
  28. [28] V. Perlick, On Fermat's principle in General Relativity: I. The general case, Class. Quantum Grav., Vol. 7, 1990, pp. 1319-1331. Zbl0707.53054MR1064182
  29. [29] V. Perlick, Infinite dimensional Morse Theory and Fermat's principle in general relativity. I, J. Math. Phys., Vol. 36, 1995, pp. 6915-6928. Zbl0854.58014MR1359671
  30. [30] A. Petters, Morse Theory and gravitational microlensing, J. Math. Phys., 1992, Vol. 33, pp. 1915-1931. MR1159012
  31. [3 1 ] A. Petters, Multiplane gravitational lensing. I. Morse Theory and image counting, J. Math. Phys., Vol. 36, 1995, pp. 4263-4275. Zbl0854.57027MR1341990
  32. [32] A. Petters, Multiplane gravitational lensing. II. Global Geometry of caustics, J. Math. Phys., Vol. 36, 1995, pp. 4276-4295. Zbl0854.57028MR1341991
  33. [33] P. Schneider, J. Ehlers and E. Falco, Gravitational lensing. Springer, Berlin, 1992. 
  34. [34] J.P. Serre, Homologie singuliere des espaces fibres, Ann. Math., Vol. 54, 1951, pp. 425-505. Zbl0045.26003MR45386
  35. [35] E.H. Spanier, Algebraic Topology. Mc Graw Hill. New York, 1966. Zbl0145.43303MR210112
  36. [36] K. Uhlenbeck, A Morse Theory for geodesics on a Lorentz manifold, Topology, Vol. 14, 1975, pp. 69-90. Zbl0323.58010MR383461
  37. [37] H. Weyl, Zur Gravitationstheorie, Annln. Phys., Vol. 54, 1917, pp. 117-145. JFM46.1303.01

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