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Abstract

Consider the following problem:

Let Mj be a collapsing sequence (i,e, Vol Mj → 0) of complete Riemannian manifolds of bounded dimension, with sectional curvatures uniformly bounded below. What can be said about the Gromov-Hausdorff limit M of such a sequence? And what is the relation between the topology and geometry of M and that of manifolds Mj with large j?.

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References

  1. [Al]
    A. D. Alexandrov, Intrinsic geometry of convex surfaces, Moscow (1948).Google Scholar
  2. [A2]
    A. D. Alexandrov, Uber eine Verallgemeinerung der Riemannschen Geometric, Schriftenreihe Inst. Math. 1 (1957), 33–84.Google Scholar
  3. [BGP]
    Yu. Burago, M. Gromov, and G. Perelman, Alexandrov spaces with curvature bounded below, Uspekhi Mat. Nauk 47/2 (1992), 3–51.MathSciNetGoogle Scholar
  4. [B]
    S. Buyalo, Shortest lines on convex hypersurfaces in a Riemannian space, Zap. Nauchn. Sem. LOMI 66 (1976), 114–131.Google Scholar
  5. [BN]
    V. N. Berestovskii and I. G. Nikolaev, Multidimensional generalized Riemannian spaces, Encycl. Math. Sci. 70, Springer-Verlag, Berlin (1993), 165–243.Google Scholar
  6. [CFG]
    J. Cheeger, K. Fukaya, and M. Gromov, Nilpotent structures and invariant met-rics on collapsed manifolds, J. Amer. Math. Soc. 5/2 (1992), 327–372.MathSciNetCrossRefGoogle Scholar
  7. [CG]
    J. Cheeger and D. Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. of Math. (2) 96 (1972), 413–443.MathSciNetCrossRefGoogle Scholar
  8. [DC]
    L. Danzer and B. Grunbaum, Uber zwei Probleme beziiglich konvexer Körper von P. Erdés und von V. L. Klee, Math. Z. 79 (1962), 95–99.MathSciNetCrossRefGoogle Scholar
  9. [GI]
    M. Gromov, Synthetic geometry in Riemannian manifolds, Proc. Internat. Con-gress Math. 1978, Helsinki, Acad. Sci. Fennica (1980), 415–419.Google Scholar
  10. [G2]
    M. Gromov, Curvature, diameter and Betti numbers, Comment. Math. Helv. 56 (1982), 179–195.MathSciNetCrossRefGoogle Scholar
  11. [GLP]
    M. Gromov, J. Lafontaine, and P. Pansu, Structures métriques pour les variétés riemanniennes, Textes Math. no. 1, Cedic, Paris (1981).Google Scholar
  12. [GH]
    K. Grove and S. Halperin, Contributions of rational homotopy theory to global problems in geometry, Publ. IHES 56 (1983), 379–385.zbMATHGoogle Scholar
  13. [GP]
    K. Grove and P. Petersen, Bounding homotopy types by geometry, Ann. of Math. (2) 128 (1988), 195–206.MathSciNetCrossRefGoogle Scholar
  14. [GS]
    K. Grove and K. Shiohama, A generalized sphere theorem, Ann. of Math. (2) 106 (1977), 201–211.MathSciNetCrossRefGoogle Scholar
  15. [L]
    I. Liberman, Geodesic lines on convex surfaces, Dokl. Akad. Nauk SSSR 32/5 (1941), 310–313.MathSciNetzbMATHGoogle Scholar
  16. [P]
    G. Perelman, Elementary Morse theory on Alexandrov spaces, St. Petersburg Math. J. 5/1 (1994), 207–214; see also preprint, G. Perelman, Alexandrov spaces with curvature bounded below, II. Google Scholar
  17. [PP1]
    G. Perelman and A. Petrunin, Extremal subsets in Alexandrov spaces and a generalized Liberman theorem, St. Petersburg Math. J. 5/1 (1994), 215–227.MathSciNetGoogle Scholar
  18. [PP2]
    G. Perelman and A. Petrunin, Quasigeodesics and gradient curves in Alexandrov spaces, preprint.Google Scholar
  19. [Pet]
    A. Petrunin, Applications of quasigeodesics and gradient curves, preprint.Google Scholar
  20. [Pet2]
    A. Petrunin, Parallel transportation and second variation, preprint.Google Scholar
  21. [Pl]
    C. Plaut, Spaces of Wald curvature bounded below, preprint.Google Scholar
  22. [R]
    Yu. G. Reshetnyak, Two-dimensional manifolds of bounded curvature, Encycl. Math. Sci. 70, Springer-Verlag, Berlin (1993), 3–163.Google Scholar
  23. [5]
    L. Siebenmann, Deformations of homeomorphisms on stratified sets, Comment. Math. Helv. 47 (1972), 123–163.MathSciNetCrossRefGoogle Scholar
  24. [Y]
    T. Yamaguchi, Collapsing and pinching under a lower curvature bound, Ann. of Math. (2) 133 (1991), 317–357.MathSciNetCrossRefGoogle Scholar

Copyright information

© Birkhäser Verlag, Basel, Switzerland 1995

Authors and Affiliations

  • G. Perelman
    • 1
    • 2
  1. 1.Steklov InstituteSt. PetersburgRussia
  2. 2.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

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