A PL-manifold of nonnegative curvature homeomorphic to $S^2 \times S^2$ is a direct metric product (Preliminary version)

TL;DR

This paper proves that a piecewise-linear manifold homeomorphic to S^2 × S^2 with nonnegative curvature must be a direct product of two convex polyhedral surfaces, confirming a PL-version of Hopf's hypothesis.

Contribution

It establishes that such PL-manifolds are necessarily metric products of convex polyhedral surfaces, advancing understanding of curvature conditions in PL-geometry.

Findings

PL-manifold is a direct metric product of two convex polyhedra surfaces

Supports the PL-version of Hopf's hypothesis for nonnegative curvature

Links PL-curvature conditions to classical Riemannian curvature concepts

Abstract

Let $M^4$ be a PL-manifold of nonnegative curvature that is homeomorphic to a product of two spheres, $S^2 \times S^2$. We prove that $M$ is a direct metric product of two spheres endowed with some polyhedral metrics. In other words, $M$ is a direct metric product of the surfaces of two convex polyhedra in $\mathbb{R}^3$. The background for the question is the following. The classical H.Hopf's hypothesis states: for any Riemannian metric on $S^2 \times S^2$ of nonnegative sectional curvature the curvature cannot be strictly positive at all points. There is no quick answer to this question: it is known that a Riemannian metric on $S^2 \times S^2$ of nonnegative sectional curvature need not be a product metric. However, M.Gromov has pointed out that the condition of nonnegative curvature in the PL-case appears to be stronger than nonnegative sectional curvature of Riemannian manifolds and analogous to some condition on the curvature operator. So the motivation for the question addressed in this text is to settle the PL-version of the Hopf's hypothesis.