The Journal of Geometric Analysis

, Volume 5, Issue 3, pp 419–426 | Cite as

Spaces on and beyond the boundary of existence

  • Peter Petersen
  • Frederick Wilhelm
  • Shun-hui Zhu
Article

Abstract

In this note we discuss various questions on whether or not quotients of Riemannian manifolds by Lie groups can be the Gromov-Hausdorff limits of manifolds with certain curvature bounds. In particular we show that any quotient of a manifold by a Lie group is a limit of manifolds with a lower curvature bound; this answers a question posed by Burago, Gromov, and Perelman. On the other hand, we prove that not all such spaces are limits of manifolds with absolute curvature bounds. We also give examples of spaces with curvature ≥1 that are not limits of manifolds with curvature ≥δ > 1/4.

Math Subject Classification

53C20 

Key Words and Phrases

Gromov-Hausdorff convergence Alexandrov spaces orbifolds Gromov-Hausdorff limits of Riemannian manifolds 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AW]
    Aloff, S., and Wallach, N. L. An infinite family of 7-manifolds admitting positively curved Riemannian structures.BAMS 81, 93–97 (1975).MathSciNetMATHGoogle Scholar
  2. [B]
    Browder, W. Higher torsion in H-spaces.TAMS 108, 353–375 (1963).CrossRefMathSciNetMATHGoogle Scholar
  3. [BGP]
    Burago, Yu., Gromov, M., and Perelman, G., A. D. Alexandrov’s spaces with curvatures bounded from below. To appear inUspekhi Mat. Nauk. Google Scholar
  4. [CG]
    Cheeger, J., and Gromov, M. Collapsing Riemannian manifolds while keeping their curvatures bounded I.J. Diff. Geom. 23, 309–346 (1986).MathSciNetMATHGoogle Scholar
  5. [F]
    Freedman, M. The topology of four-manifolds.J. Diff. Geom. 17, 357–453 (1982).MATHGoogle Scholar
  6. [F1]
    Fukaya, K. Hausdorff convergence of riemannian manifolds and its applications.Adv. Studies in Pure Math. 18-I, 143–238(1990).MathSciNetGoogle Scholar
  7. [F2]
    Fukaya, K. A boundary of the set of Riemannian manifolds with bounded curvatures and diameters.J. Diff. Geom. 28, 1–21 (1988).MathSciNetMATHGoogle Scholar
  8. [G]
    Gromov, M. Groups of polynomial growth and expanding maps.Publ. Math. IHES 53, 53–73 (1981).MathSciNetMATHGoogle Scholar
  9. [GP1]
    Grove, K., and Petersen, P. Manifolds near the boundary of existence.J. Diff. Geom. 33, 379–394 (1991).MathSciNetMATHGoogle Scholar
  10. [GP2]
    Grove, K., and Petersen, P. On the excess of metric spaces and manifolds. Preprint (see also A pinching theorem for homotopy spheres.JAMS 3(3), 671–677 (1990).CrossRefMathSciNetMATHGoogle Scholar
  11. [GPW]
    Grove, K., Petersen, P., and Wu, J. Y. Geometric finiteness theorems via geometric topology.Invt. Math. 99, 205–213 (1990).Erratum Invt. Math. 104, 221–222 (1991).CrossRefMathSciNetMATHGoogle Scholar
  12. [GS]
    Grove, K., and Shiohama, K. A generalized sphere theorem.Ann. Math. 106, 201–211 (1977).CrossRefMathSciNetGoogle Scholar
  13. [O’N]
    Neill, B. The fundamental equations of submersions.Mich. J. Math. 13, 459–469 (1966).CrossRefMATHGoogle Scholar
  14. [P]
    Perelman, G. Alexandrov’s spaces with curvatures bounded from below II. Preprint.Google Scholar
  15. [Pe]
    Petersen, P. Gromov-Hausdorff convergence of metric spaces. To appear inProc. AMS Summer Inst. in Diff. Geom. at UCLA.Google Scholar
  16. [PZ]
    Petersen, P., and Zhu, S.-h. An excess sphere theorem. To appear inAnn. Sci. Ec. Norm. Sup. Google Scholar
  17. [S]
    Smale, S. Generalized Poincare conjecture in dimensions greater than four.Ann. Math. 74, 391–406 (1961).CrossRefMathSciNetGoogle Scholar
  18. [T]
    Thurston, W. Geometry and topology of 3-manifolds. Preprint.Google Scholar
  19. [To]
    Toponogov, V. Riemannian spaces with curvature bounded below.Uspehi Mat. Nauk 14 (1959) (in Russian).Google Scholar
  20. [Y]
    Yamaguchi, T. Collapsing and pinching under a lower curvature bound.Ann. Math. 133, 317–357 (1991).CrossRefGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 1995

Authors and Affiliations

  • Peter Petersen
    • 1
    • 2
  • Frederick Wilhelm
    • 1
    • 2
  • Shun-hui Zhu
    • 1
    • 2
  1. 1.Department of MathematicsUniversity of CaliforniaLos Angeles
  2. 2.Department of MathematicsSUNY Stony BrookStony Brook

Personalised recommendations