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

摘要

Let M4 be a PL-manifold of nonnegative curvature that is homeomorphic to a product of two spheres, S 2 x S2. 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 R3 .;The background for the question is the following. The classical H.Hopf's hypothesis states: for any Riemannian metric on S 2 x S2 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 S2 x S2 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. This dissertation settles the PL-analog of the Hopf's hypothesis as stated above.