On a generalized conjecture of Hopf with symmetry

Abstract

A famous conjecture of Hopf states that$\mathbb{S}^{2}\times \mathbb{S}^{2}$does not admit a Riemannian metric with positive sectional curvature. In this article, we prove that no manifold product$N\times N$can carry a metric of positive sectional curvature admitting a certain degree of torus symmetry.