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Geometriae Dedicata

, Volume 164, Issue 1, pp 319–349 | Cite as

The higher rank rigidity theorem for manifolds with no focal points

  • Jordan Watkins
Original Paper

Abstract

We say that a Riemannian manifold M has rank M ≥ k if every geodesic in M admits at least k parallel Jacobi fields. The Rank Rigidity Theorem of Ballmann and Burns–Spatzier, later generalized by Eberlein–Heber, states that a complete, irreducible, simply connected Riemannian manifold M of rank k ≥ 2 (the “higher rank” assumption) whose isometry group Γ satisfies the condition that the Γ-recurrent vectors are dense in SM is a symmetric space of noncompact type. This includes, for example, higher rank M which admit a finite volume quotient. We adapt the method of Ballmann and Eberlein–Heber to prove a generalization of this theorem where the manifold M is assumed only to have no focal points. We then use this theorem to generalize to no focal points a result of Ballmann–Eberlein stating that for compact manifolds of nonpositive curvature, rank is an invariant of the fundamental group.

Keywords

Rigidity No focal points Higher rank Duality condition Riemannian manifolds 

Mathematics Subject Classification (2000)

53C24 

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References

  1. 1.
    Ballmann W.: Axial isometries of manifolds of nonpositive curvature. Math. Ann. 259(1), 131–144 (1982)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Ballmann W.: Nonpositively curved manifolds of higher rank. Ann. Math. (2) 122(3), 597–609 (1985)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Ballmann, W.: Lectures on Spaces of Nonpositive Curvature, vol. 25 of DMV Seminar. Birkhäuser, Basel, With an Appendix by Misha Brin (1995)Google Scholar
  4. 4.
    Ballmann W., Brin M., Eberlein P.: Structure of manifolds of nonpositive curvature. I. Ann. Math. (2) 122(1), 171–203 (1985)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Ballmann W., Brin M., Spatzier R.: Structure of manifolds of nonpositive curvature. II. Ann. Math. (2) 122(2), 205–235 (1985)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Ballmann W., Eberlein P.: Fundamental groups of manifolds of nonpositive curvature. J. Differ. Geom. 25(1), 1–22 (1987)MathSciNetMATHGoogle Scholar
  7. 7.
    Burns K., Spatzier R. (1987) Manifolds of nonpositive curvature and their buildings Inst. Hautes Études Sci. Publ. Math. 65, 35–59MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Burns K., Spatzier R. (1987) On topological Tits buildings and their classification Inst. Hautes Études Sci. Publ. Math. 65, 5–34.MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Druetta M.J.: Clifford translations in manifolds without focal points. Geom. Dedicata 14(1), 95–103 (1983)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Eberlein P.: When is a geodesic flow of Anosov type? I, II. J. Differ. Geom. 8, 437–463 (1973)MathSciNetMATHGoogle Scholar
  11. 11.
    Eberlein P.: When is a geodesic flow of Anosov type? I,II. J. Differ. Geom. 8, 565–577 (1973)MathSciNetMATHGoogle Scholar
  12. 12.
    Eberlein P.: A canonical form for compact nonpositively curved manifolds whose fundamental groups have nontrivial center. Math. Ann. 260(1), 23–29 (1982)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Eberlein P.: Euclidean de Rham factor of a lattice of nonpositive curvature. J. Differ. Geom. 18(2), 209–220 (1983)MathSciNetMATHGoogle Scholar
  14. 14.
    Eberlein P., Heber J. (1990) A differential geometric characterization of symmetric spaces of higher rank Inst. Hautes Études Sci. Publ. Math. 71, 33–44.MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Eschenburg J.-H.: Horospheres and the stable part of the geodesic flow. Math. Z. 153(3), 237–251 (1977)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Eschenburg J.-H., O’Sullivan J.J.: Growth of Jacobi fields and divergence of geodesics. Math. Z. 150(3), 221–237 (1976)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Farb B., Weinberger S.: Isometries, rigidity and universal covers. Ann. Math. (2) 168(3), 915–940 (2008)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Goto, M.S.: The cone topology on a manifold without focal points. J. Differ. Geom. 14(4), 595–598 (1981) 1979Google Scholar
  19. 19.
    Goto M.S.: Manifolds without focal points. J. Differ. Geom. 13(3), 341–359 (1978)MathSciNetMATHGoogle Scholar
  20. 20.
    O’Sullivan J.J.: Manifolds without conjugate points. Math. Ann. 210, 295–311 (1974)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    O’Sullivan J.J.: Riemannian manifolds without focal points. J. Differ. Geom. 11(3), 321–333 (1976)MathSciNetMATHGoogle Scholar
  22. 22.
    Prasad G., Raghunathan M.S.: Cartan subgroups and lattices in semi-simple groups. Ann. Math. (2) 96, 296–317 (1972)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Yau S.-T.: On the fundamental group of compact manifolds of non-positive curvature. Ann. Math. (2) 93, 579–585 (1971)MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.University of MichiganAnn ArborUSA

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