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An Introduction to Certain Topics on Lorentzian Geometry

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Topics in Modern Differential Geometry

Part of the book series: Atlantis Transactions in Geometry ((ATLANTIS,volume 1))

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Abstract

These notes cover the content of a mini-course of three lectures I gave in March 17–20, 2008, to young researchers within the International Research School of the Simon Stevin Institute for Geometry at Katholieke Universiteit Leuven (Belgium). My main aim was providing to the students with an introduction to several research topics on Lorentzian Geometry, including background and a panoramic view of the developments throughout the time of some interesting problems. I would like to give my sincere thanks to the organizers Stefan Haesen and Johan Gielis, Simon Stevin Institute for Geometry, Netherlands, and Leopold Verstraelen, Katholieke Universiteit Leuven, Belgium, for giving me the opportunity to talk to a number of PhD students from several countries, and I hope that my lectures encourage them to face new challenges in the beautiful research area of Lorentzian Geometry.

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Notes

  1. 1.

    Unless otherwise is specified, a manifold will be assumed to be of class \(C^{\infty }\), connected and with a countable basis in its topology.

  2. 2.

    This section is based on a talk [10] given by the author in the Seminar of Geometry of Kyungpook National University, Taegu, Korea, in November, 1998.

References

  1. R. Abraham, J.E. Marsden, Foundations of Mechanics, 2nd edn. (Perseus Publishing, New York, 1978)

    MATH  Google Scholar 

  2. J.K. Beem, Some examples of incomplete spacetimes. Gen. Relat. Gravit. 7, 501–509 (1976)

    Article  MATH  Google Scholar 

  3. J.K. Beem, P.E. Ehrlich, K.L. Easley, Global Lorentzian Geometry, vol. 202, 2nd edn., Pure and Applied Mathematics (Marcel Dekker, New York, 1996)

    MATH  Google Scholar 

  4. G.S. Birman, K. Nomizu, The Gauss–Bonnet theorem for 2-dimensional spacetimes. Mich. Math. J. 31, 77–81 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  5. R.L. Bishop, R.J. Crittenden, Geometry of manifolds (Academic Press, Providence, 2001)

    Book  MATH  Google Scholar 

  6. E. Calabi, L. Markus, Relativistic space forms. Ann. Math. 75, 63–76 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  7. Y. Carriére, Autour de la conjeture de L. Markus sur las variétés affines. Invent. Math. 95, 615–628 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  8. Y. Carriére, L. Rozoy, Complétude des métriques lorentziennes. Bol. Soc. Bras. Mat. 25, 223–235 (1994)

    Article  Google Scholar 

  9. S.S. Chern, Pseudo-Riemannian geometry and the Gauss–Bonnet formula. An. Acad. Bras. Ci. 35, 17–26 (1963)

    MathSciNet  MATH  Google Scholar 

  10. Y.S. Choi, Y.J. Suh, Remarks on the topology of Lorentzian manifolds. Comm. Korean Math. Soc. 15, 641–648 (2000)

    MathSciNet  MATH  Google Scholar 

  11. A.A. Coley, B.O.J. Tupper, Space-times admitting special affine conformal vectors. J. Math. Phys. 31, 649–652 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  12. M. Dajczer, K. Nomizu, On the boundeness of ricci curvature of an indefinite metric. Bol. Soc. Brasil. Mat. 11, 25–30 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  13. R.P. Geroch, What is a singularity in general relativity? Ann. Phys. 48, 526–540 (1968)

    Article  MATH  Google Scholar 

  14. W.H. Greub, Line fields on lorentzian manifolds, differential geometric methods in mathematical physics. Lect. Notes Phys. 139, 290–309 (1981)

    Article  Google Scholar 

  15. W.H. Greub, S. Halperin, R. Vanstone, Connections, Curvature and Cohomology, vol. 1 (Academic Press, New York, 1972)

    Google Scholar 

  16. Y. Kamishima, Completeness of Lorentz manifolds of constant curvature admitting killing vector fields. J. Differ. Geom. 37, 569–601 (1993)

    MathSciNet  MATH  Google Scholar 

  17. B. Klinger, Completude des variétes lorentziennes à courbure constante. Math. Ann. 306, 353–370 (1996)

    Google Scholar 

  18. S. Kobayashi, K. Nomizu, Foundations of Differential Geometry (Wiley Interscience Publishing, New York, 1963)

    MATH  Google Scholar 

  19. R. Kulkarni, The values of sectional curvature in indefinite metrics. Comment. Math. Helv. 54, 173–176 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  20. W. Kundt, Note on the completeness of spacetimes. Zs. F. Phys. 172, 488–499 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  21. J. Lafuente-López, A geodesic completeness theorem for locally symmetric Lorentz manifolds. Rev. Mat. Univ. Complut. Madr. 1, 101–110 (1988)

    MathSciNet  MATH  Google Scholar 

  22. K.B. Marathe, A condition for paracompacteness of a manifold. J. Differ. Geom. 7, 571–573 (1972)

    MATH  Google Scholar 

  23. J.E. Marsden, On completeness of pseudo-riemannian manifolds. Indiana Univ. Math. J. 22, 1065–1066 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  24. K. Nomizu, Remarks on sectional curvature of an indefinite metric. Proc. Am. Math. Soc. 89, 473–476 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  25. K. Nomizu, H. Ozeki, The existence of complete riemannian metrics. Proc. Am. Math. Soc. 12, 889–891 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  26. B. O’Neill, Semi-Riemannian Geometry with Applications to Relativity (Academic Press, New York, 1983)

    MATH  Google Scholar 

  27. F.J. Palomo, A. Romero, Certain actual topics on modern Lorentzian geometry, in Handbook of Differential Geometry, vol. 2, ed. by F. Dillen, L. Verstraelen, (Elsevier, 2006), pp. 513–546

    Google Scholar 

  28. A. Romero, The introduction of bochner’s technique on Lorentzian manifolds. Nonlinear Analysis TMA 47, 3047–3059 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  29. A. Romero, M. Sánchez, On completeness of geodesics obtained as a limit. J. Math. Phys. 34, 3768–3774 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  30. A. Romero, M. Sánchez, New properties and examples of incomplete Lorentzian tori. J. Math. Phys. 35, 1992–1997 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  31. A. Romero, M. Sánchez, On Completeness of Compact Lorentzian Manifolds, Geometry and Topology of Submanifolds VI (World Scientific Publishing, River Edge, 1994)

    MATH  Google Scholar 

  32. A. Romero, M. Sánchez, On completeness of certain families of semi-riemannian manifolds. Geom. Dedic. 53, 103–117 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  33. A. Romero, M. Sánchez, Completeness of compact Lorentz manifolds admiting a timelike conformal-killing vector field. Proc. Am. Math. Soc. 123, 2831–2833 (1995)

    Article  MATH  Google Scholar 

  34. R. Sachs, H. Wu, General Relativity for Mathematicians, vol. 48, Graduate Texts in Mathematics (Springer, New York, 1977)

    MATH  Google Scholar 

  35. T. Sakai, Riemannian Geometry, vol. 149, Translations of Mathematical Monographs (American Mathematical Society, Providence, 1997)

    Google Scholar 

  36. M. Spivak, Differential Geometry, vol. 2 (Publish or Perish, Berkeley, 1979)

    MATH  Google Scholar 

  37. M. Wang, Some examples of homogeneous Einstein manifolds in dimension seven. Duke Math. J. 49, 23–28 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  38. F. Warner, Foundations of Differentiable Manifolds and Lie Groups (Scott Foresman, Glenview, 1971)

    MATH  Google Scholar 

  39. K. Yano, On harmonic and killing vector fields. Ann. Math. 55, 38–45 (1952)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Geometry and Topology, University of Granada, 18071, Granada, Spain

    Alfonso Romero

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  1. Alfonso Romero
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Correspondence to Alfonso Romero .

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Editors and Affiliations

  1. Department of Teacher Education, Thomas More University College Department of Teacher Education, Vorselaar, Belgium

    Stefan Haesen

  2. Department of Mathematics, University of Leuven Department of Mathematics, Leuven, Belgium

    Leopold Verstraelen

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Romero, A. (2017). An Introduction to Certain Topics on Lorentzian Geometry. In: Haesen, S., Verstraelen, L. (eds) Topics in Modern Differential Geometry. Atlantis Transactions in Geometry, vol 1. Atlantis Press, Paris. https://doi.org/10.2991/978-94-6239-240-3_10

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