JournalscmhVol. 89, No. 4pp. 937–962

Positively curved Riemannian metrics with logarithmic symmetry rank bounds

  • Lee Kennard

    University of California, Santa Barbara, USA
Positively curved Riemannian metrics with logarithmic symmetry rank bounds cover

Abstract

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.

Cite this article

Lee Kennard, Positively curved Riemannian metrics with logarithmic symmetry rank bounds. Comment. Math. Helv. 89 (2014), no. 4, pp. 937–962

DOI 10.4171/CMH/340