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Topology of non-negatively curved manifolds

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  • An important question in the study of Riemannian manifolds of positive sectional curvature is how to distinguish manifolds that admit a metric with non-negative sectional curvature from those that admit one of positive curvature. Surprisingly, if the manifolds are compact and simply connected, all known obstructions to positive curvature are already obstructions to non-negative curvature. On the other hand, there are very few known examples of manifolds with positive curvature. They consist, apart from the rank one symmetric spaces, of certain homogeneous spaces G/H in dimensions 6, 7, 12, 13 and 24 due to Berger [Be], Wallach [Wa], and Aloff-Wallach [AW], and of biquotients K G/H in dimensions 6, 7 and 13 due to Eschenburg [E1],[E2] and Bazaikin [Ba], see [Zi] for a survey. Recently, a new example of a positively curved 7-manifold was found which is homeomorphic but not diffeomorphic to the unit tangent bundle of S⁴, see [GVZ, De]. And in [PW] a method was proposed to construct a metric of positive curvature on the Gromoll-Meyer exotic 7-sphere.
  • Keywords: Eschenburg spaces, Non-negative curvature, Sphere bundles, Kreck–Stolz invariants
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  • Escher, C., & Ziller, W. (2014). Topology of non-negatively curved manifolds. Annals of Global Analysis and Geometry, 46(1), 23-55. doi:10.1007/s10455-013-9407-8
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  • 46
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  • 1
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  • C. Escher was supported by a grant from the Association for Women in Mathematics, by the University of Pennsylvania and by IMPA. W. Ziller was supported by a grant from the National Science Foundation, the Max Planck Institute in Bonn and CAPES.
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