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
Published electronically: July 2, 2002
Erratum: Trans. Amer. Math. Soc. (recently posted)
MathSciNet review: 1926886
Full-text PDF Free Access

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. E. Marsden, and T. Ratiu, Manifolds, tensor analysis, and applications, 2nd ed., Applied Mathematical Sciences, vol. 75, Springer-Verlag, New York, 1988. MR 960687
2. Luis J. Alías, Alfonso Romero, and Miguel Sánchez, Spacelike hypersurfaces of constant mean curvature in certain spacetimes, Proceedings of the Second World Congress of Nonlinear Analysts, Part 1 (Athens, 1996), 1997, pp. 655–661. MR 1489832, 10.1016/S0362-546X(97)00246-0
3. Lars Andersson, Mattias Dahl, and Ralph Howard, Boundary and lens rigidity of Lorentzian surfaces, Trans. Amer. Math. Soc. 348 (1996), no. 6, 2307–2329. MR 1363008, 10.1090/S0002-9947-96-01688-1
4. John K. Beem, Paul E. Ehrlich, and Kevin L. Easley, Global Lorentzian geometry, 2nd ed., Monographs and Textbooks in Pure and Applied Mathematics, vol. 202, Marcel Dekker, Inc., New York, 1996. MR 1384756
5. Marcel Berger, Paul Gauduchon, and Edmond Mazet, Le spectre d’une variété riemannienne, Lecture Notes in Mathematics, Vol. 194, Springer-Verlag, Berlin-New York, 1971 (French). MR 0282313
6. Arthur L. Besse, Manifolds all of whose geodesics are closed, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 93, Springer-Verlag, Berlin-New York, 1978. With appendices by D. B. A. Epstein, J.-P. Bourguignon, L. Bérard-Bergery, M. Berger and J. L. Kazdan. MR 496885
7. Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987. MR 867684
8. Isaac Chavel, Riemannian geometry—a modern introduction, Cambridge Tracts in Mathematics, vol. 108, Cambridge University Press, Cambridge, 1993. MR 1271141
9. Marcos Dajczer and Katsumi Nomizu, On the boundedness of Ricci curvature of an indefinite metric, Bol. Soc. Brasil. Mat. 11 (1980), no. 1, 25–30. MR 607014, 10.1007/BF02584877
10. Paul E. Ehrlich and Seon-Bu Kim, From the Riccati inequality to the Raychaudhuri equation, Differential geometry and mathematical physics (Vancouver, BC, 1993), Contemp. Math., vol. 170, Amer. Math. Soc., Providence, RI, 1994, pp. 65–78. MR 1290565, 10.1090/conm/170/01745
11. Eduardo García-Río and Demir N. Kupeli, Singularity versus splitting theorems for stably causal spacetimes, Ann. Global Anal. Geom. 14 (1996), no. 3, 301–312. MR 1400291, 10.1007/BF00054475
12. Alfred Gray, Pseudo-Riemannian almost product manifolds and submersions, J. Math. Mech. 16 (1967), 715–737. MR 0205184
13. L. W. Green, Auf Wiedersehensflächen, Ann. of Math. (2) 78 (1963), 289–299 (German). MR 0155271
14. Steven G. Harris, A triangle comparison theorem for Lorentz manifolds, Indiana Univ. Math. J. 31 (1982), no. 3, 289–308. MR 652817, 10.1512/iumj.1982.31.31026
15. Steven G. Harris, A characterization of Robertson-Walker spaces by null sectional curvature, Gen. Relativity Gravitation 17 (1985), no. 5, 493–498. MR 789529, 10.1007/BF00761906
16. Dale Husemoller, Fibre bundles, 3rd ed., Graduate Texts in Mathematics, vol. 20, Springer-Verlag, New York, 1994. MR 1249482
17. Yoshinobu Kamishima, Completeness of Lorentz manifolds of constant curvature admitting Killing vector fields, J. Differential Geom. 37 (1993), no. 3, 569–601. MR 1217161
18. H. Karcher, Infinitesimale Charakterisierung von Friedmann-Universen, Arch. Math., 38, 58-64, 1982. MR 83:83055
19. Jerry L. Kazdan, An isoperimetric inequality and Wiedersehen manifolds, Seminar on Differential Geometry, Ann. of Math. Stud., vol. 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 143–157. MR 645734
20. Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol. II, Interscience Tracts in Pure and Applied Mathematics, No. 15 Vol. II, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969. MR 0238225
21. Lisa Koch-Sen, Infinitesimal null isotropy and Robertson-Walker metrics, J. Math. Phys. 26 (1985), no. 3, 407–410. MR 786395, 10.1063/1.526623
22. Ravi S. Kulkarni and Frank Raymond, 3-dimensional Lorentz space-forms and Seifert fiber spaces, J. Differential Geom. 21 (1985), no. 2, 231–268. MR 816671
23. Barrett O’Neill, Semi-Riemannian geometry, Pure and Applied Mathematics, vol. 103, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. With applications to relativity. MR 719023
24. Alfonso Romero and Miguel Sánchez, On completeness of certain families of semi-Riemannian manifolds, Geom. Dedicata 53 (1994), no. 1, 103–117. MR 1299888, 10.1007/BF01264047
25. Alfonso Romero and Miguel Sánchez, Completeness of compact Lorentz manifolds admitting a timelike conformal Killing vector field, Proc. Amer. Math. Soc. 123 (1995), no. 9, 2831–2833. MR 1257122, 10.1090/S0002-9939-1995-1257122-3
26. A. Romero and M. Sánchez, An integral inequality on compact Lorentz manifolds, and its applications, Bull. London Math. Soc. 28 (1996), no. 5, 509–513. MR 1396153, 10.1112/blms/28.5.509
27. Alfonso Romero and Miguel Sánchez, Bochner’s technique on Lorentzian manifolds and infinitesimal conformal symmetries, Pacific J. Math. 186 (1998), no. 1, 141–148. MR 1665060, 10.2140/pjm.1998.186.141
28. Rainer Kurt Sachs and Hung Hsi Wu, General relativity for mathematicians, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, Vol. 48. MR 0503498
29. Miguel Sánchez, Structure of Lorentzian tori with a Killing vector field, Trans. Amer. Math. Soc. 349 (1997), no. 3, 1063–1080. MR 1376554, 10.1090/S0002-9947-97-01745-5
30. Joseph A. Wolf, Spaces of constant curvature, 3rd ed., Publish or Perish, Inc., Boston, Mass., 1974. MR 0343214
31. H. Wu, On the de Rham decomposition theorem, Illinois J. Math. 8 (1964), 291–311. MR 0161280
32. Kentaro Yano and Shigeru Ishihara, Tangent and cotangent bundles: differential geometry, Marcel Dekker, Inc., New York, 1973. Pure and Applied Mathematics, No. 16. MR 0350650
33. Ulvi Yurtsever, Test fields on compact space-times, J. Math. Phys. 31 (1990), no. 12, 3064–3078. MR 1079255, 10.1063/1.528960