Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Boundary and Lens Rigidity of
Lorentzian Surfaces

Authors: Lars Andersson, Mattias Dahl and Ralph Howard
Journal: Trans. Amer. Math. Soc. 348 (1996), 2307-2329
MSC (1991): Primary 53C50
MathSciNet review: 1363008
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $g$ be a Lorentzian metric on the plane $\r ^2$ that agrees with the standard metric $g_0=-dx^2+dy^2$ outside a compact set and so that there are no conjugate points along any time-like geodesic of $(\r ^2,g)$. Then $(\r ^2,g)$ and $(\r ^2,g_0)$ are isometric. Further, if $(M,g)$ and $(M^*,g^*)$ are two dimensional compact time oriented Lorentzian manifolds with space--like boundaries and so that all time-like geodesics of $(M,g)$ maximize the distances between their points and $(M,g)$ and $(M^*,g^*)$ are ``boundary isometric'', then there is a conformal diffeomorphism between $(M,g)$ and $(M^*,g^*)$ and they have the same areas. Similar results hold in higher dimensions under an extra assumption on the volumes of the manifolds.

References [Enhancements On Off] (What's this?)

  • 1. John K. Beem and Paul E. Ehrlich, Global Lorentzian geometry, Monographs and Textbooks in Pure and Applied Math., vol. 67, Marcel Dekker, Inc., New York, 1981. MR 619853
  • 2. 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
  • 3. D. Burago and S. Ivanov, Riemannian tori without conjugate points are flat, Geom. Funct. Anal. 4 (1994), no. 3, 259–269. MR 1274115,
  • 4. Christopher B. Croke, Rigidity for surfaces of nonpositive curvature, Comment. Math. Helv. 65 (1990), no. 1, 150–169. MR 1036134,
  • 5. Christopher B. Croke, Rigidity and the distance between boundary points, J. Differential Geom. 33 (1991), no. 2, 445–464. MR 1094465
  • 6. Leon Green and Robert Gulliver, Planes without conjugate points, J. Differential Geom. 22 (1985), no. 1, 43–47. MR 826423
  • 7. Mikhael Gromov, Filling Riemannian manifolds, J. Differential Geom. 18 (1983), no. 1, 1–147. MR 697984
  • 8. Florêncio F. Guimarães, The integral of the scalar curvature of complete manifolds without conjugate points, J. Differential Geom. 36 (1992), no. 3, 651–662. MR 1189499
  • 9. S. W. Hawking and G. F. R. Ellis, The large scale structure of space-time, Cambridge University Press, London-New York, 1973. Cambridge Monographs on Mathematical Physics, No. 1. MR 0424186
  • 10. E. Hopf, Closed surfaces without conjugate points, Proc. Nat. Acad. Sci. U.S.A., 34:47--51, 1948. MR 9:378d
  • 11. Hermann Karcher, Riemannian comparison constructions, Global differential geometry, MAA Stud. Math., vol. 27, Math. Assoc. America, Washington, DC, 1989, pp. 170–222. MR 1013810
  • 12. René Michel, Sur la rigidité imposée par la longueur des géodésiques, Invent. Math. 65 (1981/82), no. 1, 71–83 (French). MR 636880,
  • 13. L. A. Santaló, Integral geometry in general spaces. In Proc. Internat. Congr. Math. Cambridge, Mass., 1950, volume 1, American Math. Soc., Providence, R.I., pages 483--489, 1952. MR 13:377h
  • 14. Luis A. Santaló, Integral geometry and geometric probability, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. With a foreword by Mark Kac; Encyclopedia of Mathematics and its Applications, Vol. 1. MR 0433364
  • 15. T. Weinstein, An Introduction to Lorentz Surfaces, Preliminary Manuscript, 1994.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 53C50

Retrieve articles in all journals with MSC (1991): 53C50

Additional Information

Lars Andersson
Affiliation: Department of Mathematics, Royal Institute of Technology, S-100 44 Stockholm, Sweden

Mattias Dahl
Affiliation: Department of Mathematics, Royal Institute of Technology, S-100 44 Stockholm, Sweden

Ralph Howard
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208

Received by editor(s): January 18, 1995
Additional Notes: Lars Andersson supported in part by the Swedish Natural Sciences Research Council (SNSRC), contract no. F-FU 4873-307. Mattias Dahl supported in part by the Wallenberg foundation. Ralph Howard supported in part by the SNSRC, contract no. R-RA 4873-306, the Swedish Academy of Sciences and the Crafoord foundation.
Article copyright: © Copyright 1996 American Mathematical Society