Positively curved Riemannian metrics with logarithmic symmetry rank bounds

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8 Scopus citations


We prove an obstruction at the level of rational cohomology to the existence of positively curved metrics with large symmetry rank. The symmetry rank bound is logarithmic in the dimension of the manifold. As one application, we provide evidence for a generalized conjecture of H. Hopf, which states that no symmetric space of rank at least two admits a metric with positive curvature. Other applications concern product manifolds, connected sums, and manifolds with nontrivial fundamental group.

Original languageEnglish (US)
Pages (from-to)937-962
Number of pages26
JournalCommentarii Mathematici Helvetici
Issue number4
StatePublished - 2014
Externally publishedYes


  • Hopf conjecture
  • Positive sectional curvature
  • Symmetric spaces
  • Symmetry

ASJC Scopus subject areas

  • General Mathematics


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