A PL-manifold of nonnegative curvature homeomorphic to 𝑆²×𝑆² is a direct metric product

Abstract

Let M 4 M^4 be a PL-manifold of nonnegative curvature that is homeomorphic to a product of two spheres, S 2 × S 2 S^2 \times S^2 . We prove that M M is a direct metric product of two spheres endowed with some polyhedral metrics. In other words, M M is a direct metric product of the surfaces of two convex polyhedra in R 3 \mathbb {R}^3 . The classical H. Hopf hypothesis states: for any Riemannian metric on S 2 × S 2 S^2 \times S^2 of nonnegative sectional curvature the curvature cannot be strictly positive at all points. The result of this paper can be viewed as a PL-version of Hopf’s hypothesis. It confirms the remark of M. Gromov that the condition of nonnegative curvature in the PL-case appears to be stronger than nonnegative sectional curvature of Riemannian manifolds and analogous to the condition of a nonnegative curvature operator.