Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [Bou75] Jean-Pierre Bourguignon.
    Some constructions related to H. Hopf's Conjecture on product manifolds.
    Volume 27 of Proceedings of Symposia in Pure Mathematics, pages 33-37. American Mathematical Society, 1975. MR 0380906 (52:1803)
  • [Che86] Jeff Cheeger.
    A vanishing theorem for piecewise constant curvature spaces.
    In Curvature and topology of Riemannian manifolds, pages 33-40. Lecture Notes in Math., 1201, Springer, Berlin, 1986. MR 859575 (88a:58203)
  • [Cho06] P. Lu, L. Ni, B. Chow.
    Hamilton's Ricci flow.
    Volume 77 of Graduate Studies in Mathematics. American Mathematical Society, 2006. MR 2274812 (2008a:53068)
  • [Ham86] Richard S. Hamilton.
    Four-manifolds with positive curvature operator.
    Journal of Differential Geometry, 24(2):153-179, 1986. MR 862046 (87m:53055)
  • [Kur93] Masatake Kuranishi.
    On some metrics on $ {S}^2 \times {S}^2$.
    In R. Greene and S.T. Yau, editors, Differential Geometry, Part. 3: Riemannian Geometry, volume 54 of Proceedings of Symposia in Pure Mathematics, pages 439-450. American Mathematical Society, 1993. MR 1216636 (94b:53068)
  • [Pan09] Dmitri Panov.
    Polyhedral Kähler manifolds.
    Geometry and Topology, 13:2205-2252, 2009. MR 2507118 (2010f:53129)
  • [Wil07] Burkhard Wilking.
    Nonnegatively and positively curved manifolds.
    In Surveys in Differential Geometry, Vol. XI: Metric and Comparison Geometry, Int. Press, 2007. MR 2408263 (2009e:53048)
  • [Zvo08] Dimitri Zvonkine.
    Strebel differentials on stable curves and Kontsevich's proof of Witten's conjecture.
    arXiv:math/0209071v2 [math.AG].

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 53C15, 53C23, 57Q25

Retrieve articles in all journals with MSC (2010): 53C15, 53C23, 57Q25

Additional Information

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

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.

American Mathematical Society