An exotic sphere with positive curvature almost everywhere. (English) Zbl 1039.53037
Since Milnor’s discovery of exotic spheres, an interesting problem in Riemannian geometry has been whether there are exotic spheres with positive curvature. The Gromoll-Meyer sphere gives an example of a nonnegatively curved exotic \(7\)-dimensional sphere [D. Gromoll, and W. Meyer, Ann. Math. (2) 100, 401–406 (1974; Zbl 0293.53015)]. This was the first example of an exotic sphere with such a property. In the present paper, the author shows that the Gromoll-Meyer metric has zero curvature on an open set of points. Moreover, the Gromoll-Meyer metric can be perturbed to one with positive curvature almost everywhere and admitting an effective \(SO(3)\)-action.
Reviewer: Sergio Console (Torino)
MSC:
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
| 53C20 | Global Riemannian geometry, including pinching |
| 57R55 | Differentiable structures in differential topology |
Citations:
Zbl 0293.53015References:
| [1] | Berger, M. On the diameter of some Riemannian manifolds, preprint, 1962. |
| [2] | Bourguignone, J. P.; Deschamps, A.; Sentenac, P., Quelques variations particulieres d’un produit de metriques, Ann. Sci. Ecole Norm. Supper., 6, 1-16 (1973) · Zbl 0257.53040 |
| [3] | Cheeger, J., Some examples of manifolds of nonnegative curvature, J. Differential Geometry, 8, 623-628 (1972) · Zbl 0281.53040 |
| [4] | Gromoll, D.; Meyer, W., An exotic sphere with nonnegative sectional curvature, Ann. of Math., 100, 401-406 (1974) · Zbl 0293.53015 · doi:10.2307/1971078 |
| [5] | Hatcher, A., A proof of the smale conjecture, Diff(S^3) ≃O(4), Ann. of Math., 117, 553-607 (1983) · Zbl 0531.57028 · doi:10.2307/2007035 |
| [6] | O’Neill, B., The fundamental equations of a submersion, Michigan Math. J., 13, 459-469 (1966) · Zbl 0145.18602 · doi:10.1307/mmj/1028999604 |
| [7] | Petersen, P.; Wilhelm, F., Examples of Riemannian manifolds with positive curvature almost everywhere, Geometry & Topology, 3, 331-367 (1999) · Zbl 0963.53020 · doi:10.2140/gt.1999.3.331 |
| [8] | Steenrod, N.Topology of Fiber Bundles, Princeton Mathematical Series, Princeton University Press, 1951. · Zbl 0054.07103 |
| [9] | Strake, M., Curvature increasing metric variations, Math. Ann., 276, 633-641 (1987) · Zbl 0616.53036 · doi:10.1007/BF01456991 |
| [10] | Wilhelm, F., Exotic spheres with lots of positive curvatures, J. Geom. Anal., 11, 163-188 (2001) · Zbl 1023.53024 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
