The Hopf conjecture for manifolds with low cohomogeneity or high symmetry rank

TL;DR

This paper proves the Hopf conjecture for certain manifolds with high symmetry, showing positive Euler characteristic under specific symmetry and curvature conditions, advancing understanding of geometric topology.

Contribution

It establishes new conditions involving symmetry rank and cohomogeneity under which the Hopf conjecture holds for positively curved manifolds.

Findings

Euler characteristic is positive for manifolds with symmetry rank ≥ l.

Euler characteristic is positive for manifolds with cohomogeneity ≤ 5.

Results apply to manifolds with positive sectional curvature and high symmetry.

Abstract

We prove that the Euler characteristic of an even-dimensional compact manifold with positive (nonnegative) sectional curvature is positive (nonnegative) provided that the manifold admits an isometric action of a compact Lie group $G$ with principal isotropy group $H$ and cohomogeneity $k$ such that $k - (\rank G - \rank H)\le 5$. Moreover, we prove that the Euler characteristic of a compact Riemannian manifold $M^{4l+4}$ or $M^{4l+2}$ with positive sectional curvature is positive if $M$ admits an effective isometric action of a torus $T^l$, i.e., if the symmetry rank of $M$ is $\ge l$.