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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Author: Sergey Orshanskiy
Journal: Proc. Amer. Math. Soc. 139 (2011), 4475-4486
MSC (2010): Primary 53C15, 53C23, 57Q25
Published electronically: May 3, 2011
MathSciNet review: 2823093
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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 classical H. Hopf 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. 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.


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Additional Information

Sergey Orshanskiy
Affiliation: Credit Suisse Derivatives IT, Credit Suisse Securities, 11 Madison Avenue, New York, New York 10010

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10861-X
PII: S 0002-9939(2011)10861-X
Received by editor(s): January 18, 2010
Received by editor(s) in revised form: September 16, 2010, and October 26, 2010
Published electronically: May 3, 2011
Communicated by: Jon G. Wolfson
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.