An exotic sphere with positive curvature almost everywhere
- Frederick Wilhelm
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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 . 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.
- Berger, M. On the diameter of some Riemannian manifolds, preprint, 1962.
- 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).
- Cheeger, J. Some examples of manifolds of nonnegative curvature,J. Differential Geometry,8, 623–628, (1972).
- Gromoll, D. and Meyer, W. An exotic sphere with nonnegative sectional curvature,Ann. of Math.,100, 401–406, (1974). CrossRef
- Hatcher, A. A proof of the smale conjecture, Diff(S 3) ≃O(4),Ann. of Math.,117, 553–607, (1983). CrossRef
- O’Neill, B. The fundamental equations of a submersion,Michigan Math. J.,13, 459–469, (1966). CrossRef
- Petersen, P. and Wilhelm, F. Examples of Riemannian manifolds with positive curvature almost everywhere,Geometry & Topology,3, 331–367, (1999). CrossRef
- Steenrod, N.Topology of Fiber Bundles, Princeton Mathematical Series, Princeton University Press, 1951.
- Strake, M. Curvature increasing metric variations,Math. Ann.,276, 633–641, (1987). CrossRef
- Wilhelm, F. Exotic spheres with lots of positive curvatures,J. Geom. Anal.,11, 163–188, (2001).
- An exotic sphere with positive curvature almost everywhere
The Journal of Geometric Analysis
Volume 11, Issue 3 , pp 519-560
- Cover Date
- Print ISSN
- Online ISSN
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- exotic sphere
- positive curvature
- Author Affiliations
- 1. Department of Mathematics, University of California, 92521-0135, Riverside, CA