A PLmanifold of nonnegative curvature homeomorphic to is a direct metric product
Author:
Sergey Orshanskiy
Journal:
Proc. Amer. Math. Soc. 139 (2011), 44754486
MSC (2010):
Primary 53C15, 53C23, 57Q25
Published electronically:
May 3, 2011
MathSciNet review:
2823093
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Abstract: Let be a PLmanifold 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 PLversion of Hopf's hypothesis. It confirms the remark of M. Gromov that the condition of nonnegative curvature in the PLcase 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/S00029939201110861X
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.
