# A PL-manifold of nonnegative curvature homeomorphic to S^2 × S^2 is a direct metric product

Open Access

- Graduate Program:
- Mathematics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- March 01, 2010
- Committee Members:
- Dmitri Yu Burago, Dissertation Advisor
- Dmitri Yu Burago, Committee Chair
- Anton Petrunin, Committee Member
- Yuri Zarhin, Committee Member
- Mark Levi, Committee Member
- Adam Smith, Committee Member

- Keywords:
- polyhedral space
- nonnegative curvature
- Hopf conjecture
- direct metric product

- Abstract:
- Let M^4 be a PL-manifold of nonnegative curvature that is homeomorphic to a product of two spheres, S^2 × 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 R^3. The background for the question is the following. The classical H.Hopf’s hypothesis states: for any Riemannian metric on S^2 × 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 × 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. This dissertation settles the PL-analog of the Hopf’s hypothesis as stated above.