Collapsing and Almost Nonnegative Curvature

Conference paper
Part of the Springer Proceedings in Mathematics book series (PROM, volume 17)


Almost nonnegatively curved manifolds are charming spaces for at least two reasons: From a classical point of view, they are natural generalizations of almost flat as well as nonnegatively and positively curved manifolds, and the study of all of the latter has a long tradition in Riemannian geometry. Secondly, almost nonnegatively curved manifolds are precisely the spaces which can be collapsed to a point under a fixed lower bound on sectional curvature, so that in degenerations and convergence of metrics under lower curvature bounds they play the same fundamental role that almost flat manifolds do in Cheeger–Fukaya–Gromov’s theory of collapse with curvature bounded in absolute value.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alexander, S., Kapovitch, V., Petrunin, A.: Alexandrov Geometry. Monograph. In preparation.Google Scholar
  2. 2.
    Anderson, M.T.: Hausdorff perturbations of Ricci-flat manifolds and the splitting theorem. Duke Math. J. 68(1):67–82 (1992)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Böhm, C., Wilking, B.: Manifolds with positive curvature operators are space forms. Ann. Math. (2) 167(3), 1079–1097 (2008)Google Scholar
  4. 4.
    Bredon, G.: Introduction to Compact Transformation Groups. Academic Press, New York (1972)MATHGoogle Scholar
  5. 5.
    Brieskorn, E.: Beispiele zur Differentialtopologie von Singularitäten. Invent. Math. 2, 1–14 (1966)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Browder, W.: The Kervaire invariant of framed manifolds and its generalization. Ann. Math. (2) 90, 157–186 (1969)Google Scholar
  7. 7.
    Burago, Y., Gromov, M., Perelman, G.: A. D. Alexandrov’s spaces with curvatures bounded from below I. Uspechi Mat. Nauk 47, 3–51 (1992)MathSciNetGoogle Scholar
  8. 8.
    Cheeger, J., Colding, T.: Lower bounds on Ricci curvature and the almost rigidity of warped products. Ann. Math. 144, 189–237 (1996)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Cheeger, J., Colding, T.: On the structure of spaces with Ricci curvature bounded below. I. J. Differ. Geom. 46, 406–480 (1997)MATHMathSciNetGoogle Scholar
  10. 10.
    Cheeger, J., Colding, T.: On the structure of spaces with Ricci curvature bounded below. II. J. Differ. Geom. 54(1), 13–35 (2000)MATHMathSciNetGoogle Scholar
  11. 11.
    Cheeger, J., Colding, T.: On the structure of spaces with Ricci curvature bounded below. III. J. Differ. Geom. 54(1), 37–74 (2000)MATHMathSciNetGoogle Scholar
  12. 12.
    Cheeger, J., Fukaya, K., Gromov, M.: Nilpotent structures and invariant metrics on collapsed manifolds. J. Am. Math. Soc. 5(2), 327–372 (1992)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Colding, T.: Ricci curvature and volume convergence. Ann. Math. (2) 145(3), 477–501 (1997)Google Scholar
  14. 14.
    Fukaya, K.: Metric Riemannian geometry. In: Handbook of Differential Geometry, vol. II, pp. 189–313. Elsevier, Amsterdam (2006)Google Scholar
  15. 15.
    Fukaya, K., Yamaguchi, T.: The fundamental groups of almost nonnegatively curved manifolds. Ann. Math. (2) 136(2), 253–333 (1992)Google Scholar
  16. 16.
    Gallot, S.: Inégalités isopérimétriques, courbure de Ricci et invariants géométriques. II. C. R. Acad. Sci. Paris Sér. I Math. 296(8), 365–368 (1983)MATHMathSciNetGoogle Scholar
  17. 17.
    Gromov, M.: Almost flat manifolds. J. Differ. Geom. 13, 231–241 (1978)MATHMathSciNetGoogle Scholar
  18. 18.
    Gromov, M.: Synthetic geometry in Riemannian manifolds. In: Proceedings of the International Congress of Mathematicians, Helsinki, pp. 415–419 (1978), Acad. Sci. Fennica, Helsinki, (1980)Google Scholar
  19. 19.
    Gromov, M.: Structures métriques pour les variétés riemanniennes. In: Lafontaine, J., Pansu, P. (eds.) Textes Mathématiques, vol. 1, CEDIC, Paris (1981)Google Scholar
  20. 20.
    Gromov, M.: Curvature, diameter and Betti numbers. Comment. Math. Helv. 56, 179–195 (1981)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Gromov, M.: Volume and bounded cohomology. Inst. Hautes Études Sci. Publ. Math. 56, 5–99 (1982)MATHMathSciNetGoogle Scholar
  22. 22.
    Gromov, M.: Metric Structures for Riemannian and Non-Riemannian Spaces. Progress in Mathematics, vol. 152. Birkhäuser, Boston (1999)Google Scholar
  23. 23.
    Grove, K.: Geometry of, and via, symmetries. In: Conformal, Riemannian and Lagrangian geometry, Knoxville, TN, 2000, University Lecture Series, vol. 27, pp. 31–53. American Mathematical Society, Providence, RI (2002)Google Scholar
  24. 24.
    Grove, K., Halperin, S.: Contributions of rational homotopy theory to global problems in geometry. Inst. Hautes Études Sci. Publ. Math. 56, 171–178 (1982)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Grove, K., Verdiani, L., Wilking, B., Ziller, W.: Non-negative curvature obstructions in cohomogeneity one and the Kervaire spheres. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 5(2), 159–170 (2006)Google Scholar
  26. 26.
    Grove, K., Ziller, W.: Curvature and symmetry of Milnor spheres. Ann. Math. (2) 152, 331–36 (2000)Google Scholar
  27. 27.
    Grove, K., Ziller, W.: Cohomogeneity one manifolds with positive Ricci curvature. Invent. Math. 149, 619–646 (2002)CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Hirzebruch, F., Mayer, K.: O(n)-Mannigfaltigkeiten, exotische Sphären und Singularitäten. Lecture Notes in Mathematics, vol. 57, Springer, Berlin-New York (1968)Google Scholar
  29. 29.
    Hsiang, W.-C., Hsiang, W.-Y.: On compact subgroups of the diffeomorphism groups of Kervaire spheres. Ann. Math. (2) 85, 359–369 (1967)Google Scholar
  30. 30.
    Kapovitch, V.: Perelman’s stability theorem. In: Surveys in Differential Geometry, vol. XI, pp. 103–136. International Press, Somerville, MA (2007)Google Scholar
  31. 31.
    Kapovitch, V., Petrunin, A., Tuschmann, W.: Nilpotency, almost nonnegative curvature, and the gradient flow on Alexandrov spaces. Ann. Math. (2) 171(1), 343–373 (2010)Google Scholar
  32. 32.
    LeBrun, C.: Kodaira dimension and the Yamabe problem. Comm. Anal. Geom. 7(1), 133–156 (1999)MATHMathSciNetGoogle Scholar
  33. 33.
    LeBrun, C.: Ricci curvature, minimal volumes, and Seiberg–Witten theory. Invent. Math. 145(2), 279–316 (2001)CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Mercuri, F., Noronha, M.H.: On the topology of complete Riemannian manifolds with nonnegative curvature operator. Rend. Sem. Fac. Sci. Univ. Cagliari 63(2), 149–171(1993)Google Scholar
  35. 35.
    Milnor, J.: Singular Points of Complex Hypersurfaces. Annals of Mathematics Studies, No. 61. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo (1968)Google Scholar
  36. 36.
    Perelman, G.: Alexandrov spaces with curvatures bounded from below II. Preprint (1991)Google Scholar
  37. 37.
    Petrunin, A.: Semiconcave functions in Alexandrov’s geometry. In: Surveys in Differential Geometry, vol. XI, pp. 137–201. International Press, Somerville, MA (2007)Google Scholar
  38. 38.
    Paternain, G., Petean, J.: Zero entropy and bounded topology. Comment. Math. Helv. 81(2), 287–304 (2006)CrossRefMATHMathSciNetGoogle Scholar
  39. 39.
    Rong, X.: On the fundamental groups of manifolds of positive sectional curvature. Ann. Math., II. Ser. 143(2), 397–411 (1996)Google Scholar
  40. 40.
    Rong, X.: The almost cyclicity of the fundamental groups of positively curved manifolds. Invent. Math. 126(1), 47–64 (1996)CrossRefMATHMathSciNetGoogle Scholar
  41. 41.
    Schoen, R.M.: Variational theory for the total scalar curvature functional for Riemannian metrics and related topics. In: Topics in Calculus of variations, Montecatini Terme, 1987, Lecture Notes in Mathematics, vol. 1365, pp. 120–154. Springer, Berlin (1989)Google Scholar
  42. 42.
    Schwachhöfer, L. Tuschmann, W.: Almost Nonnegative Curvature and Cohomogeneity One. Max Planck Institute for Mathematics in the Sciences Preprint Series, Preprint no. 62/2001,
  43. 43.
    Schwachhöfer, L.J., Tuschmann,W.: Metrics of positive Ricci curvature on quotient spaces. Math. Ann. 330(1), 59–91 (2004)CrossRefMATHMathSciNetGoogle Scholar
  44. 44.
    Sebastian, D.: Konstruktionen von Mannigfaltigkeiten mit fast nichtnegativem Krümmungsoperator. Dissertationsschrift, Karlsruher Institut für Topologie (2011)Google Scholar
  45. 45.
    Sebastian, D., Tuschmann, W.: Manifolds with almost nonnegative curvature operator. In preparation.Google Scholar
  46. 46.
    Yamaguchi, T.: Collapsing and pinching under a lower curvature bound. Ann. Math. 133, 317–357 (1991)CrossRefMATHGoogle Scholar
  47. 47.
    Yamaguchi, T.: A convergence theorem in the geometry of Alexandrov spaces. In: Actes de la Table Ronde de Géométrie Differentielle, Séminaires & Congrès, vol. 1, pp. 601–642. Soc. Math. France, Paris, (1996)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Fakultät für Mathematik Karlsruher Institut für Technologie (KIT)KarlsruheGermany

Personalised recommendations