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

Open Access
Author:
Orshanskiy, Sergey
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.