Critical points of distance functions and applications to geometry

  • Jeff Cheeger
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1504)


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [A] U. Abresch: Lower curvature bounds, Toponogov’s theorem, and bounded topology I, II, Ann. scient. Ex. Norm. Sup. 18 (1985) 651–670 (4e série), and preprint MPI/SFB 84/41 &MathSciNetMATHGoogle Scholar
  2. [AG1] U. Abresch and D. Gromoll, “On complete manifolds with nonnegative Ricci curvature, J.A.M.S. (to appear).Google Scholar
  3. [An] M. Anderson, Short geodesics and gravitational instantons, J. Diff. Geom. V. 31 (1990) 265–275.MathSciNetMATHGoogle Scholar
  4. [AnKle] M. Anderson, P. Kronheimer and C. LeBrun Complete Ricci-flat Kahler manifolds of infinite topological type, Commun. Math. Phys. 125 (1989) 637–642.MathSciNetCrossRefMATHGoogle Scholar
  5. [Be] M. Berger. Les varietés 1/4-pincées, Ann. Scuola Norm. Pisa (111) 153 (1960) 161–170.MATHGoogle Scholar
  6. [BT] R. Bott and L. Tu, Differential forms in algebraic topology, Graduate Texts in Math., Springer-Verlag (1986).Google Scholar
  7. [Ca] E. Calabi, An extension of E. Hopf’s maximum principle with an application to Riemannian geometry, Duke Math. J. 25 (1958) 45–56.MathSciNetCrossRefMATHGoogle Scholar
  8. [C1] J. Cheeger, Comparison and finiteness theormes in Riemannian geometry, Ph.D. Thesis, Princeton Univ., 1967.Google Scholar
  9. [C2]-, The relation between the Laplacian and the diameter for manifolds of nonnegative curvature, Arch. der Math. 19 (1968) 558–560.MathSciNetCrossRefMATHGoogle Scholar
  10. [C3]-Finiteness theorems for Riemannian manifolds, Amer. J. Math. 92 (1970) 61–74.MathSciNetCrossRefMATHGoogle Scholar
  11. [CE] J. Cheeger and D. G. Ebin, Comparison Theorems in Riemannian Geometry, North-Holland Mathematical Library 9 (1975).Google Scholar
  12. [CGl1] J. Cheeger and D. Gromoll The splitting theorem for manifolds of nonnegative Ricci curvature, J. Diff. Geo. 6 (1971) 119–128.MathSciNetMATHGoogle Scholar
  13. [CGl2]-, On the structure of complete manifolds of nonnegative curvature, Ann. of Math. 96 (1972), 413–443.MathSciNetCrossRefMATHGoogle Scholar
  14. [EH] J. Eschenburg and E. Heintze, An elementary proof of the Cheeger-Gromoll splitting theorem, Ann. Glob. Anal. & Geom. 2 (1984) 141–151.MathSciNetCrossRefMATHGoogle Scholar
  15. [GrP] K. Grove and P. Petersen, Bounding homotopy types by geometry, Ann. of Math. 128 (1988) 195–206.MathSciNetCrossRefMATHGoogle Scholar
  16. [GrPW] K. Grove, P. Petersen J. Y. Wu, Controlled topology in geometry, Bull. AMS 20 (2) (1989) 181.MathSciNetCrossRefMATHGoogle Scholar
  17. [GrS] K. Grove and K. Shiohama, A generalized sphere theorem, Ann. of Math. 106 (1977) 201–211.MathSciNetCrossRefMATHGoogle Scholar
  18. [G] M. Gromov, Curvature, diameter and Betti numbers, Comm. Math. Helv. 56 (1981) 179–195.MathSciNetCrossRefMATHGoogle Scholar
  19. [GLP] M. Gromov, J Lafontaine, and P. Pansu, Structure Métrique pour les Variétiés Riemanniennes, Cedic/Fernand Nathan (1981).Google Scholar
  20. [GreWu] R. Greene and H. Wu, Lipschitz convergence of Riemannian manifolds, Pacific J. Math. 131 (1988) 119–141.MathSciNetCrossRefMATHGoogle Scholar
  21. [Liu] Z. Liu, Ball covering on manifolds with nonnegative Ricci curvature near infinity, Proc. A.M.S. (to appear).Google Scholar
  22. [M] J. Milnor, Morse Theory, Annals of Math. Studies, Princeton Univ. Press (1963).Google Scholar
  23. [Me] W. Meyer, Toponogov’s Theorem and Applications, Lecture Notes, Trieste, 1989.Google Scholar
  24. [Pe1] S. Peters, Cheeger’s, finiteness theorem for diffeomorphism classes of Riemannian manifolds, J. Reine Angew. Math. 394 (1984) 77–82.MATHGoogle Scholar
  25. [Pe2]-, Convergence of Riemannian manifolds, Comp. Math. 62 (1987) 3–16.MathSciNetMATHGoogle Scholar
  26. [ShY] J. P. Sha and D. G. Yang, Examples of manifolds of positive Ricci curvature, J. Diff. Geo. 29 (1) (1989) 95–104.MathSciNetMATHGoogle Scholar
  27. [Shen] Z. Shen, Finite topological type and vanishing theorems for Riemannian manifolds, Ph. D. Thesis, SUNY, Stony Brook, 1990.Google Scholar
  28. [We] A. Weinstein, On the homotopy type of positively pinched manifolds, Archiv. der Math. 18 (1967) 523–524.MathSciNetCrossRefMATHGoogle Scholar
  29. [Z] S. Zhu, Bounding topology by Ricci curvature in dimension three, Ph.D. Thesis, SUNY, Stony Brook, 1990. *** DIRECT SUPPORT *** A00J4422 00002Google Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • Jeff Cheeger

There are no affiliations available

Personalised recommendations