A Berger-Green type inequality for compact Lorentzian manifolds

Gutiérrez, Manuel; Palomo, Francisco; Romero, Alfonso

doi:10.1090/S0002-9947-02-03060-X

A Berger-Green type inequality for compact Lorentzian manifolds

Authors: Manuel Gutiérrez, Francisco J. Palomo and Alfonso Romero
Journal: Trans. Amer. Math. Soc. 354 (2002), 4505-4523
MSC (2000): Primary 53C50, 53C22; Secondary 53C20
DOI: https://doi.org/10.1090/S0002-9947-02-03060-X
Published electronically: July 2, 2002
Erratum: Trans. Amer. Math. Soc. (recently posted)
MathSciNet review: 1926886
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We give a Lorentzian metric on the null congruence associated with a timelike conformal vector field. A Liouville type theorem is proved and a boundedness for the volume of the null congruence, analogous to a well-known Berger-Green theorem in the Riemannian case, will be derived by studying conjugate points along null geodesics. As a consequence, several classification results on certain compact Lorentzian manifolds without conjugate points on its null geodesics are obtained. Finally, several properties of null geodesics of a natural Lorentzian metric on each odd-dimensional sphere have been found.

1. R. Abraham, J. Marsden, and T. Ratiu, Manifolds, Tensor Analysis, and Applications, Springer-Verlag, Applied Math. Sci. 1988. MR 89f:58001
2. L. J. Alías, A. Romero and M. Sánchez, Spacelike hypersurfaces of constant mean curvature in certain spacetimes, Nonlinear Anal., 30 655-661, 1997. MR 99c:53058
3. L. Andersson, M. Dahl and R. Howard, Boundary and Lens Rigidity of Lorentzian Surfaces, Trans. Amer. Math. Soc., 348 2307-2329, 1996. MR 97a:53105
4. J. K. Beem, P. E. Ehrlich and K. L. Easley, Global Lorentzian Geometry, Second edition, Pure and Applied Math. 202, Marcel Dekker, 1996. MR 97f:53100
5. M. Berger, P. Gauduchon et E. Mazet, Le spectre d'une variété riemanniene, Lecture Notes in Math. 194, Springer-Verlag, 1971. MR 43:8025
6. A. Besse, Manifolds all of whose geodesics are closed, Springer-Verlag, Ergeb. Math. Grenzgeb. 93, Berlin, 1978. MR 80c:53044
7. A. Besse, Einstein Manifolds, Springer-Verlag, Ergeb. Math. Grenzgeb. num. 10, Berlin, 1987. MR 88f:53087
8. I. Chavel, Riemannian Geometry: A Modern Introduction, Cambridge University Press, 1993. MR 95j:53001
9. M. Dajczer and K. Nomizu, On the boundedness of Ricci curvature of an indefinite metric, Bol. Soc. Brasil. Mat., 11 25-30, 1980. MR 82d:53039
10. P. E. Ehrlich and S-B Kim, From the Riccati Inequality to the Raychaudhuri Equation, Contemp. Math. 170 65-78, 1994. MR 95g:53083
11. E. Garcia-Rio and D. N. Kupeli, Singularity versus splitting theorems for stably causal spacetimes, Ann. Global Anal. Geom., 14 301-312, 1996. MR 97i:53077
12. A. Gray, Pseudo-Riemannian Almost Product Manifolds and Submersions, J. Math. and Mechanics, 16 715-737, 1967. MR 34:5018
13. L. W. Green, Auf Wiedersehensflächen, Ann. of Math., 78 289-299, 1963. MR 27:5206
14. S. G. Harris, A triangle comparison theorem for Lorentz manifolds, Indiana Univ. Math. J., 31 289-308, 1982. MR 83j:53064
15. S. G. Harris, A characterization of Robertson-Walker Spaces by null sectional curvature, Gen. Relativity Gravitation, 17 493-498, 1985. MR 86j:53099
16. D. Husemoller, Fibre Bundles, Third edition, Springer-Verlag, New York, 1994. MR 94k:55001
17. Y. Kamishima, Completeness of Lorentz manifolds of constant curvature admitting Killing vector fields, J. Differential Geom., 37 569-601, 1993. MR 94f:53116
18. H. Karcher, Infinitesimale Charakterisierung von Friedmann-Universen, Arch. Math., 38, 58-64, 1982. MR 83:83055
19. J. L. Kazdan, An isoperimetric inequality and wiedersehen manifolds, Seminar on Differential Geometry, edited by S.-T. Yau, Princenton Univ. Press, 143-157, 1982. MR 83h:53059a
20. S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Wiley Interscience Publ., New York, Vol. 2, 1969. MR 38:6501
21. L. Koch-Sen, Infinitesimal null isotropy and Robertson-Walker metrics, J. Math. Phys., 26 407-410, 1985. MR 86f:83036
22. R. Kulkarni and F. Raymond, -dimensional Lorentz space-forms and Seifert fiber spaces, J. Differential Geom., 21 231-268, 1985. MR 87h:53092
23. B. O'Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York, 1983. MR 85f:53002
24. A. Romero and M. Sánchez, On Completeness of certain families of Semi-Riemannian manifolds, Geom. Dedicata, 53 103-117, 1994. MR 95g:53049
25. A. Romero and M. Sánchez, Completeness of compact Lorentz manifolds admiting a timelike conformal-Killing vector field, Proc. Amer. Math. Soc., 123 2831-2833, 1995. MR 95k:53075
26. A. Romero and M. Sánchez, An integral inequality on compact Lorentz manifolds and its applications, Bull. London Math. Soc., 28 509-513, 1996. MR 97c:53106
27. A. Romero and M. Sánchez, Bochner's technique on Lorentz manifolds and infinitesimal conformal symmetries, Pacific J. Math., 186 141-148, 1998. MR 2000a:53121
28. R. Sachs and H. Wu, General Relativity for Mathematicians, Springer-Verlag, 1977. MR 58:20239a
29. M. Sánchez, Structure of Lorentzian tori with a Killing vector field, Trans. Amer. Math. Soc., 349 1063-1080, 1997. MR 97f:53108
30. J. A. Wolf, Spaces of constant curvature, Fourth edition, Publish or Perish, 1979. MR 49:7958
31. H. Wu, On the de Rham decomposition theorem, Illinois J. Math., 8 291-311, 1964. MR 28:4488
32. K. Yano and S. Ishihara, Tangent and Cotangent Bundles, Marcel Dekker, Inc. New York, 1973. MR 50:3142
33. U. Yurtsever, Test fields on compact space-times, J. Math. Phys., 31 3064-3078, 1990. MR 92a:53098