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.
 [Bou75]
Jean
Pierre Bourguignon, Some constructions related to H. Hopf’s
conjecture on product manifolds, Differential geometry (Proc. Sympos.
Pure Math., Vol. XXVII, Part 1, Stanford Univ., Stanford, Calif., 1973),
Amer. Math. Soc., Providence, R.I., 1975, pp. 33–37. MR 0380906
(52 #1803)
 [Che86]
Jeff
Cheeger, A vanishing theorem for piecewise constant curvature
spaces, Curvature and topology of Riemannian manifolds (Katata, 1985)
Lecture Notes in Math., vol. 1201, Springer, Berlin, 1986,
pp. 33–40. MR 859575
(88a:58203), http://dx.doi.org/10.1007/BFb0075646
 [Cho06]
Bennett
Chow, Peng
Lu, and Lei
Ni, Hamilton’s Ricci flow, Graduate Studies in
Mathematics, vol. 77, American Mathematical Society, Providence, RI,
2006. MR
2274812 (2008a:53068)
 [Ham86]
Richard
S. Hamilton, Fourmanifolds with positive curvature operator,
J. Differential Geom. 24 (1986), no. 2,
153–179. MR
862046 (87m:53055)
 [Kur93]
Masatake
Kuranishi, On some metrics on
𝑆²×𝑆², Differential geometry:
Riemannian geometry (Los Angeles, CA, 1990), Proc. Sympos. Pure Math.,
vol. 54, Amer. Math. Soc., Providence, RI, 1993,
pp. 439–450. MR 1216636
(94b:53068)
 [Pan09]
Dmitri
Panov, Polyhedral Kähler manifolds, Geom. Topol.
13 (2009), no. 4, 2205–2252. MR 2507118
(2010f:53129), http://dx.doi.org/10.2140/gt.2009.13.2205
 [Wil07]
Burkhard
Wilking, Nonnegatively and positively curved manifolds,
Surveys in differential geometry. Vol. XI, Surv. Differ. Geom.,
vol. 11, Int. Press, Somerville, MA, 2007, pp. 25–62. MR 2408263
(2009e:53048)
 [Zvo08]
Dimitri Zvonkine.
Strebel differentials on stable curves and Kontsevich's proof of Witten's conjecture. 2004. arXiv:math/0209071v2 [math.AG].
 [Bou75]
 JeanPierre Bourguignon.
Some constructions related to H. Hopf's Conjecture on product manifolds. Volume 27 of Proceedings of Symposia in Pure Mathematics, pages 3337. 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 3340. 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.
Fourmanifolds with positive curvature operator. Journal of Differential Geometry, 24(2):153179, 1986. MR 862046 (87m:53055)
 [Kur93]
 Masatake Kuranishi.
On some metrics on . In R. Greene and S.T. Yau, editors, Differential Geometry, Part. 3: Riemannian Geometry, volume 54 of Proceedings of Symposia in Pure Mathematics, pages 439450. American Mathematical Society, 1993. MR 1216636 (94b:53068)
 [Pan09]
 Dmitri Panov.
Polyhedral Kähler manifolds. Geometry and Topology, 13:22052252, 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. 2004. arXiv:math/0209071v2 [math.AG].
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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
PII:
S 00029939(2011)10861X
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.
