Geometric and Functional Analysis

, Volume 24, Issue 5, pp 1377–1405 | Cite as

Topological properties of positively curved manifolds with symmetry



Manifolds admitting positive sectional curvature are conjectured to have rigid homotopical structure and, in particular, comparatively small Euler charateristics. In this article, we obtain upper bounds for the Euler characteristic of a positively curved Riemannian manifold that admits a large isometric torus action. We apply our results to prove obstructions to symmetric spaces, products of manifolds, and connected sums admitting positively curved metrics with symmetry.

Mathematics Subject Classification

Primary 53C20 Secondary 57N65 

Keywords and phrases

Positive curvature Symmetry Euler characteristic Betti numbers Hopf conjecture Elliptic genus Symmetric spaces 


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Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Fakultät für MathematikInstitut für Algebra und Geometrie, Karlsruher Institut für TechnologieKarlsruheGermany
  2. 2.Department of MathematicsUniversity of CaliforniaSanta BarbaraUSA

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