The Journal of Geometric Analysis

, Volume 11, Issue 3, pp 519–560

An exotic sphere with positive curvature almost everywhere

Article

Abstract

In this article we show that there is an exotic sphere with positive sectional curvature almost everywhere.

In 1974 Gromoll and Meyer found a metric of nonnegative sectional on an exotic 7-sphere. They showed that the metric has positive curvature at a point and asserted, without proof, that the metric has positive sectional curvature almost everywhere [4]. We will show here that this assertion is wrong. In fact, the Gromoll-Meyer sphere has zero curvatures on an open set of points. Never the less, its metric can be perturbed to one that has positive curvature almost everywhere.

Math Subject Classifications

53C20 

Key Words and Phrases

exotic sphere positive curvature 

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References

  1. [1]
    Berger, M. On the diameter of some Riemannian manifolds, preprint, 1962.Google Scholar
  2. [2]
    Bourguignone, J.P., Deschamps, A., and Sentenac, P. Quelques variations particulieres d’un produit de metriques,Ann. Sci. Ecole Norm. Supper.,6, 1–16, (1973).Google Scholar
  3. [3]
    Cheeger, J. Some examples of manifolds of nonnegative curvature,J. Differential Geometry,8, 623–628, (1972).MathSciNetGoogle Scholar
  4. [4]
    Gromoll, D. and Meyer, W. An exotic sphere with nonnegative sectional curvature,Ann. of Math.,100, 401–406, (1974).CrossRefMathSciNetGoogle Scholar
  5. [5]
    Hatcher, A. A proof of the smale conjecture, Diff(S 3) ≃O(4),Ann. of Math.,117, 553–607, (1983).CrossRefMathSciNetGoogle Scholar
  6. [6]
    O’Neill, B. The fundamental equations of a submersion,Michigan Math. J.,13, 459–469, (1966).CrossRefMathSciNetMATHGoogle Scholar
  7. [7]
    Petersen, P. and Wilhelm, F. Examples of Riemannian manifolds with positive curvature almost everywhere,Geometry & Topology,3, 331–367, (1999).CrossRefMathSciNetMATHGoogle Scholar
  8. [8]
    Steenrod, N.Topology of Fiber Bundles, Princeton Mathematical Series, Princeton University Press, 1951.Google Scholar
  9. [9]
    Strake, M. Curvature increasing metric variations,Math. Ann.,276, 633–641, (1987).CrossRefMathSciNetMATHGoogle Scholar
  10. [10]
    Wilhelm, F. Exotic spheres with lots of positive curvatures,J. Geom. Anal.,11, 163–188, (2001).Google Scholar

Copyright information

© Mathematica Josephina, Inc. 2001

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaRiverside

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