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Positive biorthogonal curvature on $ S^2\times S^2$


Author: Renato G. Bettiol
Journal: Proc. Amer. Math. Soc. 142 (2014), 4341-4353
MSC (2010): Primary 53C20, 53C21; Secondary 53B21
DOI: https://doi.org/10.1090/S0002-9939-2014-12173-3
Published electronically: August 14, 2014
MathSciNet review: 3267002
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Abstract: We prove that $ S^2\times S^2$ satisfies an intermediate condition between $ \operatorname {Ric}>0$ and $ \operatorname {sec}>0$. Namely, there exist metrics for which the average of the sectional curvatures of any two planes tangent at the same point, but separated by a minimum distance in the $ 2$-Grassmannian, is strictly positive. This can be done with an arbitrarily small lower bound on the distance between the planes considered. Although they have positive Ricci curvature, these metrics do not have nonnegative sectional curvature. Such metrics also have positive biorthogonal curvature, meaning that the average of sectional curvatures of any two orthogonal planes is positive.


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  • [1] Jeff Cheeger, Some examples of manifolds of nonnegative curvature, J. Differential Geometry 8 (1973), 623-628. MR 0341334 (49 #6085)
  • [2] E. A. Costa, A modified Yamabe invariant and a Hopf conjecture, preprint, arXiv:1207.7107.
  • [3] Karsten Grove and Wolfgang Ziller, Curvature and symmetry of Milnor spheres, Ann. of Math. (2) 152 (2000), no. 1, 331-367. MR 1792298 (2001i:53047), https://doi.org/10.2307/2661385
  • [4] Martin Kerin, On the curvature of biquotients, Math. Ann. 352 (2012), no. 1, 155-178. MR 2885580, https://doi.org/10.1007/s00208-011-0634-7
  • [5] Mario J. Micallef and McKenzie Y. Wang, Metrics with nonnegative isotropic curvature, Duke Math. J. 72 (1993), no. 3, 649-672. MR 1253619 (94k:53052), https://doi.org/10.1215/S0012-7094-93-07224-9
  • [6] M. Müter, Krümmungserhöhende Deformationen mittels Gruppenaktionen, PhD thesis, University of Münster, 1987.
  • [7] P. Petersen and F. Wilhelm, Some principles for deforming nonnegative curvature, arXiv:0908.3026.
  • [8] P. Petersen and F. Wilhelm, An exotic sphere with positive sectional curvature, arXiv:0805.0812.
  • [9] Walter Seaman, On manifolds with nonnegative curvature on totally isotropic 2-planes, Trans. Amer. Math. Soc. 338 (1993), no. 2, 843-855. MR 1123458 (93j:53053), https://doi.org/10.2307/2154431
  • [10] Walter Seaman, Orthogonally pinched curvature tensors and applications, Math. Scand. 69 (1991), no. 1, 5-14. MR 1143470 (92k:53067)
  • [11] M. Strake, Curvature increasing metric variations, Math. Ann. 276 (1987), no. 4, 633-641. MR 879541 (88b:53054), https://doi.org/10.1007/BF01456991
  • [12] Martin Strake, Variationen von Metriken nichtnegativer Schnittkrümmung, Schriftenreihe des Mathematischen Instituts der Universität Münster, 2. Serie [Series of the Mathematical Institute of the University of Münster, Series 2], vol. 41, Universität Münster Mathematisches Institut, Münster, 1986 (German). MR 870002 (87k:53097)
  • [13] W. Ziller, On M. Mueter's Ph.D. Thesis on Cheeger deformations, arXiv:0909.0161.
  • [14] Wolfgang Ziller, Examples of Riemannian manifolds with non-negative sectional curvature, Surveys in differential geometry. Vol. XI, Surv. Differ. Geom., vol. 11, Int. Press, Somerville, MA, 2007, pp. 63-102. MR 2408264 (2009h:53080)

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Additional Information

Renato G. Bettiol
Affiliation: Department of Mathematics, 255 Hurley Building, University of Notre Dame, Notre Dame, Indiana 46556-4618
Email: rbettiol@nd.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-12173-3
Received by editor(s): September 28, 2012
Received by editor(s) in revised form: January 29, 2013
Published electronically: August 14, 2014
Additional Notes: The author was partially supported by the NSF grant DMS-0941615.
Communicated by: Lei Ni
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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