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The marked length spectrum of a projective manifold or orbifold

Authors: Daryl Cooper and Kelly Delp
Journal: Proc. Amer. Math. Soc. 138 (2010), 3361-3376
MSC (2010): Primary 57N16
Published electronically: April 6, 2010
MathSciNet review: 2653965
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Abstract: A strictly convex real projective orbifold is equipped with a natural Finsler metric called a Hilbert metric. In the case that the projective structure is hyperbolic, the Hilbert metric and the hyperbolic metric coincide. We prove that the marked Hilbert length spectrum determines the projective structure only up to projective duality. A corollary is the existence of non-isometric diffeomorphic strictly convex projective manifolds (and orbifolds) that are isospectral. This corollary follows from work of Goldman and Choi, and Benoist.

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

  • 1. Arthur Bartels and Wolfgang Lueck.
    The Borel Conjecture for hyperbolic and CAT(0)-groups.
    arXiv:0901.0442v1, 2009.
  • 2. Alan F. Beardon.
    The geometry of discrete groups, volume 91 of Graduate Texts in Mathematics.
    Springer-Verlag, New York, 1983. MR 698777 (85d:22026)
  • 3. Yves Benoist.
    Automorphismes des cônes convexes.
    Invent. Math., 141(1):149-193, 2000. MR 1767272 (2001f:22034)
  • 4. Yves Benoist.
    Convexes divisibles. I.
    In Algebraic groups and arithmetic, pages 339-374. Tata Inst. Fund. Res., Mumbai, 2004. MR 2094116 (2005h:37073)
  • 5. Yves Benoist.
    Convexes divisibles. III.
    Ann. Sci. École Norm. Sup. (4), 38(5):793-832, 2005. MR 2195260 (2007b:22011)
  • 6. G. Besson, G. Courtois, and S. Gallot.
    Entropies et rigidités des espaces localement symétriques de courbure strictement négative.
    Geom. Funct. Anal., 5(5):731-799, 1995. MR 1354289 (96i:58136)
  • 7. K. Burns and A. Katok.
    Manifolds with nonpositive curvature.
    Ergodic Theory Dynam. Systems, 5(2):307-317, 1985. MR 796758 (86j:53060)
  • 8. David A. Cox, John Little, and Donal O'Shea.
    Using algebraic geometry, volume 185 of Graduate Texts in Mathematics.
    Springer, New York, second edition, 2005. MR 2122859 (2005i:13037)
  • 9. C. Croke, A. Fathi, and J. Feldman.
    The marked length-spectrum of a surface of nonpositive curvature.
    Topology, 31(4):847-855, 1992. MR 1191384 (94b:58095)
  • 10. Pierre de la Harpe.
    On Hilbert's metric for simplices.
    In Geometric group theory, Vol. 1 (Sussex, 1991), of volume 181 of London Math. Soc. Lecture Note Ser., pages 97-119. Cambridge Univ. Press, Cambridge, 1993. MR 1238518 (94i:52006)
  • 11. M. Kapovich.
    Hyperbolic manifolds and discrete groups, volume 183 of Progress in Mathematics.
    Birkhäuser Boston, Inc., Boston, MA, 2001. MR 1792613 (2002m:57018)
  • 12. Inkang Kim.
    Rigidity and deformation spaces of strictly convex real projective structures on compact manifolds.
    J. Differential Geom., 58(2):189-218, 2001. MR 1913941 (2003g:53059)
  • 13. Serge Lang.
    Algebra, volume 211 of Graduate Texts in Mathematics.
    Springer-Verlag, New York, third edition, 2002. MR 1878556 (2003e:00003)
  • 14. John C. Loftin.
    Affine spheres and convex $ \mathbb{RP}\sp n$-manifolds.
    Amer. J. Math., 123(2):255-274, 2001. MR 1828223 (2002c:53018)
  • 15. G. D. Mostow.
    Strong rigidity of locally symmetric spaces.
    Annals of Mathematics Studies, No. 78,
    Princeton University Press, Princeton, NJ, 1973. MR 0385004 (52:5874)
  • 16. A. L. Onishchik and È. B. Vinberg.
    Lie groups and algebraic groups.
    Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1990.
    Translated from the Russian and with a preface by D. A. Leites. MR 1064110 (91g:22001)
  • 17. Jean-Pierre Otal.
    Sur les longueurs des géodésiques d'une métrique à courbure négative dans le disque.
    Comment. Math. Helv., 65(2):334-347, 1990. MR 1057248 (91i:53054)
  • 18. Joseph J. Rotman.
    An introduction to the theory of groups, volume 148 of Graduate Texts in Mathematics.
    Springer-Verlag, New York, fourth edition, 1995. MR 1307623 (95m:20001)
  • 19. T. A. Springer.
    Linear algebraic groups, volume 9 of Progress in Mathematics.
    Birkhäuser Boston, Boston, MA, 1981. MR 632835 (84i:20002)

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Additional Information

Daryl Cooper
Affiliation: Department of Mathematics, University of California at Santa Barbara, Santa Barbara, California 93106

Kelly Delp
Affiliation: Department of Mathematics, Buffalo State College, Buffalo, New York 14222

Received by editor(s): July 1, 2009
Received by editor(s) in revised form: December 12, 2009, and December 29, 2009
Published electronically: April 6, 2010
Communicated by: Richard A. Wentworth
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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