An exotic sphere with positive sectional curvature

Source: arXiv

ABSTRACT We show that there is a metric on the Gromoll-Meyer sphere with positive sectional curvature.

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    • "By constructing invariant metrics of non-negative sectional curvature on cohomogeneity one manifolds with codimension two singular orbits, and applying this to the associated principle bundles of the Milnor spheres, Grove and Ziller [14] proved that all 10 Milnor spheres admit metrics with non-negative sectional curvature. In 2008, Petersen and Wilhelm [35] showed that there is a metric on the Gromoll–Meyer sphere with positive sectional curvature. "
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    • "Further, until the recent example due to Grove, Verdiani, and Ziller [17] and Dearicott [7], all known examples of compact manifolds with positive sectional curvature were diffeomorphic to biquotients. See [5] [1] [37] [11] [12] [4] [31]. Furthermore, all known examples of manifolds with almost or quasipositive curvature are diffeomorphic to biquotients. "
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    ABSTRACT: We classify all compact simply connected biquotients of dimension 6 and 7.
    Differential Geometry and its Applications 03/2014; 34. DOI:10.1016/j.difgeo.2014.04.002 · 0.69 Impact Factor
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    • "In fact, besides the rank one symmetric spaces S n , CP n , HP n and Ca P 2 , there are no known simply connected manifolds in dimensions above 24 that admit metrics with positive sectional curvature. See Ziller [30] for a survey of known examples and [7] [14] [22] for recent examples in dimension seven. "
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    ABSTRACT: Manifolds admitting positive sectional curvature are conjectured to have a very rigid homotopical structure and, in particular, comparatively small Euler charateristics. However, this structure is still highly speculative and best results in this direction are known under the assumption of large isometric torus actions. In this article, we obtain upper bounds for Euler characteristics of closed manifolds that admit metrics with positive curvature and isometric torus actions. We apply our results to prove obstructions to symmetric spaces and manifold products and connected sums admitting positively curved metrics with symmetry, providing evidence for a conjecture of Hopf. We also derive vanishing properties of the elliptic genus of positively curved manifolds.
    Geometric and Functional Analysis 02/2013; 24(5). DOI:10.1007/s00039-014-0300-9 · 1.64 Impact Factor
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