A Personal Perspective on Global Lorentzian Geometry

  • Paul E. Ehrlich
Part of the Lecture Notes in Physics book series (LNP, volume 692)

Abstract

A selected survey is given of aspects of global space-time geometry from a differential geometric perspective that were germane to the First and Second Editions of the monograph Global Lorentzian Geometry and beyond.

Keywords

Riemannian Manifold Conjugate Point Null Geodesic Complete Riemannian Manifold Geodesic Segment 
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.

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References

  1. 1.
    T. Aubin: Métriques riemannienes et courbure. J. Diff. Geom. 4, 383–424 (1970)MATHMathSciNetGoogle Scholar
  2. 2.
    R. Bartnik: Existence of maximal surfaces in asymptotically flat space-times. Commun. Math. Phys. 94, 155–175 (1984)MATHMathSciNetCrossRefADSGoogle Scholar
  3. 3.
    R. Bartnik: Remarks on cosmological space-times and constant mean curvature surfaces. Commun. Math. Phys. 117, 615–624 (1988)MATHMathSciNetCrossRefADSGoogle Scholar
  4. 4.
    J.K. Beem: Globally hyperbolic space-times which are timelike Cauchy complete. Gen. Rel. Grav. 7, 339–344 (1976)MATHMathSciNetCrossRefADSGoogle Scholar
  5. 5.
    J.K. Beem: Conformal changes and geodesic completeness. Commun. Math. Physics 49, 179–186 (1976)MATHMathSciNetCrossRefADSGoogle Scholar
  6. 6.
    J.K. Beem: Some examples of incomplete space-times. Gen. Rel. Grav. 7, 501–509 (1976)MATHMathSciNetCrossRefADSGoogle Scholar
  7. 7.
    J.K. Beem, P.E. Ehrlich: Distance lorentzienne finie et géodésiques f-causales incomplètes. C. R. Acad. Sci. Paris Ser. A 581, 1129–1131 (1977)MathSciNetADSGoogle Scholar
  8. 8.
    J.K. Beem, P.E. Ehrlich: Conformal deformations, Ricci curvature and energy conditions on globally hyperbolic space-times. Math. Proc. Camb. Phil. Soc. 84, 159–175 (1978)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    J.K. Beem, P.E. Ehrlich: The space-time cut locus. Gen. Rel. Grav. 11, 89–103 (1979)MATHMathSciNetCrossRefADSGoogle Scholar
  10. 10.
    J.K. Beem, P.E. Ehrlich: Cut points, conjugate points and Lorentzian comparison theorems. Math. Proc. Camb. Phil. Soc. 86, 365–384 (1979)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    J.K. Beem, P.E. Ehrlich: A Morse index theorem for null geodesics. Duke Math. J. 46, 561–569 (1979)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    J.K. Beem, P.E. Ehrlich: Global Lorentzian Geometry (Marcel Dekker, New York 1981)MATHGoogle Scholar
  13. 13.
    J.K. Beem, P.E. Ehrlich: Stability of geodesic incompleteness for Robertson–Walker space-times. Gen. Rel. Grav. 13, 239–255 (1981)MATHMathSciNetCrossRefADSGoogle Scholar
  14. 14.
    J.K. Beem, P.E. Ehrlich: Geodesic completeness and stability. Math. Proc. Camb. Phil. Soc. 102, 319–328 (1987)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    J.K. Beem, P.E. Ehrlich, K.L. Easley: Global Lorentzian Geometry, 2nd edn (Marcel Dekker, New York 1996)MATHGoogle Scholar
  16. 16.
    J.K. Beem, P.E. Ehrlich, S. Markvorsen, G. Galloway: Decomposition theorems for Lorentzian manifolds with nonpositive curvature. J. Diff. Geom. 22, 29–42 (1985)MATHMathSciNetGoogle Scholar
  17. 17.
    J.K. Beem, S.G. Harris: The generic condition is generic. Gen. Rel. Grav. 25, 939–962 (1993)MATHMathSciNetCrossRefADSGoogle Scholar
  18. 18.
    J.K. Beem, S.G. Harris: Nongeneric null vectors. Gen. Rel. Grav. 25, 963–973 (1993)MATHMathSciNetCrossRefADSGoogle Scholar
  19. 19.
    J.K. Beem, P.E. Parker: Whitney stability of solvability. Pacific J. Math. 116, 11–23 (1985)MATHMathSciNetGoogle Scholar
  20. 20.
    J.K. Beem, P.E. Parker: Pseudoconvexity and geodesic connectedness. Annali Math. Pura Appl. 155, 137–142 (1989)MATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    J.K. Beem, P.E. Parker: Sectional curvature and tidal accelerations. J. Math. Phys. 31, 819–827 (1990)MATHMathSciNetCrossRefADSGoogle Scholar
  22. 22.
    J.K. Beem, T. Powell: Geodesic completeness and maximality in Lorentzian warped products. Tensor N.S. 39, 31–36 (1982)MATHMathSciNetGoogle Scholar
  23. 23.
    L. Bombelli, J. Noldus: The moduli space of isometry classes of globally hyperbolic spacetimes. Class. Quantum Grav. 21, 4429–4454 (2004). gr-qc/0402049MATHMathSciNetCrossRefADSGoogle Scholar
  24. 24.
    H. Busemann: Über die Geometrien, in denen die “Kreise mit unendlichem Radius” die kürzesten Linien sind. Math. Annalen 106, 140–160 (1932)MATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    H. Busemann: The Geometry of Geodesics (Academic Press, New York 1955)MATHGoogle Scholar
  26. 26.
    Y. Carrière: Autour de la conjecture de L. Markus sur les variétés affines. Invent. Math. 95, 615–628 (1989)MATHMathSciNetCrossRefADSGoogle Scholar
  27. 27.
    J. Cheeger, D. Ebin: Comparison Theorems in Riemannian Geometry (North–Holland, Amsterdam 1975)MATHGoogle Scholar
  28. 28.
    J. Cheeger, D. Gromoll: The splitting theorem for manifolds of nonnegative Ricci curvature. J. Diff. Geom. 6, 119–128 (1971)MATHMathSciNetGoogle Scholar
  29. 29.
    C.J.S. Clarke: On the geodesic completeness of causal space-times. Proc. Camb. Phil. Soc. 69, 319–324 (1971)MATHCrossRefGoogle Scholar
  30. 30.
    P. Eberlein, B. O'Neill: Visibility manifolds. Pacific J. Math. 46, 45–109 (1973)MATHMathSciNetGoogle Scholar
  31. 31.
    P.E. Ehrlich: Metric deformation of curvature. I: Local convex deformations. Geom. Dedicata 5, 1–23 (1976)MATHMathSciNetGoogle Scholar
  32. 32.
    P. Ehrlich: Astigmatic conjugacy and achronal boundaries. In: Geometry and Global Analysis, ed by T. Kotake, S. Nishikawa, and R. Schoen (Tohoku University, Sendai, Japan 1993) pp 197–208Google Scholar
  33. 33.
    P. Ehrlich, G. Emch: Gravitational waves and causality. Reviews in Mathematical Physics 4, 163–221 (1992). Errata 4, 501 (1992)MATHMathSciNetCrossRefADSGoogle Scholar
  34. 34.
    P. Ehrlich, G. Emch: Quasi-time functions in Lorentzian geometry. Lecture Notes in Pure and Applied Mathematics 144, 203–212 (1992)MathSciNetGoogle Scholar
  35. 35.
    P. Ehrlich, G. Emch: The conjugacy index and simple astigmatic focusing. Contemporary Mathematics 127, 27–39 (1992)MATHMathSciNetGoogle Scholar
  36. 36.
    P. Ehrlich, G. Emch: Geodesic and causal behavior of gravitational plane waves: astigmatic conjugacy. Proc. Symp. in Pure Mathematics (Amer. Math. Soc.) 54, Part 2, 203–209 (1993)MATHMathSciNetGoogle Scholar
  37. 37.
    J.-H. Eschenburg: The splitting theorem for space-times with strong energy condition. J. Diff. Geom. 27, 477–491 (1988)MATHMathSciNetGoogle Scholar
  38. 38.
    J.-H. Eschenburg, E. Heintze: An elementary proof of the Cheeger–Gromoll splitting theorem, Ann. Global Analysis Geometry 2, 141–151 (1984)MATHMathSciNetCrossRefGoogle Scholar
  39. 39.
    J.L. Flores, M. Sánchez: Causality and conjugate points in general plane waves. Class. Quantum Grav. 20, 2275–2291 (2003)MATHCrossRefGoogle Scholar
  40. 40.
    T. Frankel: Gravitational Curvature (W.H. Freeman, San Francisco 1979)MATHGoogle Scholar
  41. 41.
    G. Galloway: Splitting theorems for spatially closed space-times. Commun. Math. Phys. 96, 423–429 (1984)MATHMathSciNetCrossRefADSGoogle Scholar
  42. 42.
    G. Galloway: The Lorentzian splitting theorem without completeness assumption. J. Diff. Geom. 29, 373–387 (1989)MATHMathSciNetGoogle Scholar
  43. 43.
    G. Galloway: Maximum principles for null hypersurfaces and splitting theorems. Ann. Henri Poincarè 1, 543–567 (2000)MATHMathSciNetCrossRefGoogle Scholar
  44. 44.
    G. Galloway, A. Horta: Regularity of Lorentzian Busemann functions. Trans. Amer. Math. Soc. 348, 2063–2084 (1996)MATHMathSciNetCrossRefGoogle Scholar
  45. 45.
    C. Gerhardt: Maximal H-surfaces in Lorentzian manifolds. Commun. Math. Phys. 96, 523–553 (1983)MathSciNetCrossRefADSGoogle Scholar
  46. 46.
    R.P. Geroch: What is a singularity in general relativity? Ann. Phys. (NY) 48, 526–540 (1968)MATHCrossRefADSGoogle Scholar
  47. 47.
    R.P. Geroch: Singularities in relativity. In: Relativity, ed by M. Carmeli, S. Fickler, L. Witten (Plenum, New York 1970) pp 259–291Google Scholar
  48. 48.
    R.P. Geroch: Domain of dependence. J. Math. Phys. 11, 437–449 (1970)MathSciNetCrossRefMATHADSGoogle Scholar
  49. 49.
    D. Gromoll, W. Klingenberg, W. Meyer: Riemannsche Geometrie im Großen, Lecture Notes in Mathematics 55 (Springer, Berlin 1968)Google Scholar
  50. 50.
    S.G. Harris: Some comparison theorems in the geometry of Lorentz manifolds. Ph.D. Thesis, University of Chicago (1979)Google Scholar
  51. 51.
    S.G. Harris: A triangle comparison theorem for Lorentz manifolds. Indiana Math. J. 31, 289–308 (1982)MATHCrossRefGoogle Scholar
  52. 52.
    S. Harris: Topology of the future chronological boundary: universality for space-like boundaries. Class. Quantum Grav. 17, 551–603 (2000)MATHCrossRefADSGoogle Scholar
  53. 53.
    S. Harris: Boundaries on spacetimes: an outline. In: Advances in differential geometry and general relativity. The Beemfest, Contemporary Mathematics 359, ed by S. Dostoglou, P.E. Ehrlich (American Mathematical Society, Providence 2004) pp 65–85Google Scholar
  54. 54.
    S.W. Hawking, G.F.R. Ellis: The Large Scale Structure of Space-Time (Cambridge University Press, Cambridge 1973)MATHCrossRefGoogle Scholar
  55. 55.
    H. Hopf, W. Rinow: Über den Begriff der vollständigen differentialgeometrischen Fläche. Comment. Math. Helv. 3, 209–225 (1931)MATHMathSciNetCrossRefGoogle Scholar
  56. 56.
    Y. Kamishima: Completeness of Lorentz manifolds of constant curvature admitting Killing vector fields. J. Diff. Geom. 37, 569–601 (1993)MATHMathSciNetGoogle Scholar
  57. 57.
    W. Kundt: Note on the completeness of space-times. Zeitschrift für Physik 172, 488–489 (1963)MATHMathSciNetCrossRefADSGoogle Scholar
  58. 58.
    D.E. Lerner: The space of Lorentz metrics. Commun. Math. Phys. 32, 19–38 (1973)MATHMathSciNetCrossRefADSGoogle Scholar
  59. 59.
    C. Margerin: General conjugate loci are not closed. Proc. Symp. in Pure Mathematics (Amer. Math. Soc.) 54, Part 3, 465–478 (1993)MATHMathSciNetGoogle Scholar
  60. 60.
    J.E. Marsden: On completeness of homogeneous pseudo-Riemannian manifolds. Indiana Univ. Math. J. 22, 1065–1066 (1973)MATHMathSciNetCrossRefGoogle Scholar
  61. 61.
    C.W. Misner: Taub–NUT space as a counterexample to almost anything. In: Relativity and Astrophysics I: Relativity and Cosmology, ed by J. Ehlers (American Mathematical Society, Providence 1967) pp 160–169Google Scholar
  62. 62.
    C.W. Misner, K. Thorne, J. A. Wheeler: Gravitation (W.H. Freeman, San Francisco 1973)Google Scholar
  63. 63.
    R.P.A.C. Newman: A proof of the splitting conjecture of S.-T. Yau. J. Diff. Geom. 31, 163–184 (1990)MATHGoogle Scholar
  64. 64.
    J. Noldus: A new topology on the space of Lorentzian metrics on a fixed manifold. Class. Quantum Grav. 19, 6075–6107 (2002)MATHMathSciNetCrossRefADSGoogle Scholar
  65. 65.
    J. Noldus: A Lorentzian Gromov Hausdorff notion of distance. Class. Quantum Grav. 21, 839–850 (2004). gr-qc/0308074MATHMathSciNetCrossRefADSGoogle Scholar
  66. 66.
    J. Noldus: The limit space of a Cauchy sequence of globally hyperbolic space-times. Class. Quantum Grav. 21, 851–874 (2004). gr-qc/0308075MATHMathSciNetCrossRefADSGoogle Scholar
  67. 67.
    J. Noldus: Lorentzian Gromov Hausdorff theory as a tool for quantum gravity kinematics. PhD Thesis, University of Genf (2004). gr-qc/0401126Google Scholar
  68. 68.
    K. Nomizu, H. Ozeki: The existence of complete Riemannian metrics. Proc. Amer. Math. Soc. 12, 889–891 (1961)MATHMathSciNetCrossRefGoogle Scholar
  69. 69.
    B. O'Neill: Semi-Riemannian Geometry with Applications to Relativity (Academic Press, New York 1983)MATHGoogle Scholar
  70. 70.
    P. Parker: Geometry of Bicharacteristics. In: Advances in differential geometry and general relativity. The Beemfest, Contemporary Mathematics 359, ed by S. Dostoglou, P.E. Ehrlich (American Mathematical Society, Providence 2004) pp 31–40Google Scholar
  71. 71.
    R. Penrose: A remarkable property of plane waves in general relativity. Rev. Mod. Phys. 37, 215–220 (1965)MATHMathSciNetCrossRefADSGoogle Scholar
  72. 72.
    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
  73. 73.
    A. Romero, M. Sánchez: On completeness of certain families of semi-Rie-mannian manifolds. Geom. Dedicata 53, 103–117 (1994)MATHMathSciNetCrossRefGoogle Scholar
  74. 74.
    R.K. Sachs, H. Wu: General Relativity for Mathematicians (Springer, New York 1977)MATHGoogle Scholar
  75. 75.
    M. Sanchez: Structure of Lorentzian tori with a Killing vector field. Trans. Amer. Math. Soc. 349, 1063–1080 (1997)MATHMathSciNetCrossRefGoogle Scholar
  76. 76.
    H.-J. Seifert: Global connectivity by timelike geodesics. Zeitschrift für Natur-forschung 22a, 1356–1360 (1967)MathSciNetADSGoogle Scholar
  77. 77.
    H.-J. Seifert: The causal boundary of space-times. Gen. Rel. Grav. 1, 247–259 (1971)MATHMathSciNetCrossRefADSGoogle Scholar
  78. 78.
    K. Uhlenbeck: A Morse theory for geodesics on a Lorentz manifold. Topology 14, 69–90 (1975)MATHMathSciNetCrossRefGoogle Scholar
  79. 79.
    P.M. Williams: Completeness and its stability on manifolds with connection. Ph.D. Thesis, University of Lancaster (1984)Google Scholar
  80. 80.
    N.M.J. Woodhouse: An application of Morse theory to space-time geometry. Commun. Math. Phys. 46, 135–152 (1976)MATHMathSciNetCrossRefADSGoogle Scholar
  81. 81.
    S.T. Yau: Problem Section in Seminar on differential geometry. Ann. of Math. Studies 102, ed by S.T. Yau (Princeton University Press, Princeton 1982) pp 669–706Google Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  • Paul E. Ehrlich
    • 1
  1. 1.Department of MathematicsUniversity of FloridaGainesvilleUSA

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