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

Author:
Sergey Orshanskiy

Journal:
Proc. Amer. Math. Soc. **139** (2011), 4475-4486

MSC (2010):
Primary 53C15, 53C23, 57Q25

DOI:
https://doi.org/10.1090/S0002-9939-2011-10861-X

Published electronically:
May 3, 2011

MathSciNet review:
2823093

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Abstract: Let be a PL-manifold of nonnegative curvature that is homeomorphic to a product of two spheres, . We prove that is a direct metric product of two spheres endowed with some polyhedral metrics. In other words, is a direct metric product of the surfaces of two convex polyhedra in .

The classical H. Hopf hypothesis states: for any Riemannian metric on 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:
https://doi.org/10.1090/S0002-9939-2011-10861-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.