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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Nonnegatively and positively curved invariant metrics on circle bundles
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by Krishnan Shankar, Kristopher Tapp and Wilderich Tuschmann PDF
Proc. Amer. Math. Soc. 133 (2005), 2449-2459 Request permission

Abstract:

We derive and study necessary and sufficient conditions for an $S^1$-bundle to admit an invariant metric of positive or nonnegative sectional curvature. In case the total space has an invariant metric of nonnegative curvature and the base space is odd dimensional, we prove that the total space contains a flat totally geodesic immersed cylinder. We provide several examples, including a connection metric of nonnegative curvature on the trivial bundle $S^1\times S^3$ that is not a product metric.
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Additional Information
  • Krishnan Shankar
  • Affiliation: Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019
  • Email: shankar@math.ou.edu
  • Kristopher Tapp
  • Affiliation: Department of Mathematics, Bryn Mawr College, Philadelphia, Pennsylvania 19010
  • MR Author ID: 630309
  • Email: ktapp@brynmawr.edu
  • Wilderich Tuschmann
  • Affiliation: Mathematisches Institut, Westfälische Wilhelms-Universität Münster, Einstein- strasse 62, D-48149 Münster, Germany
  • MR Author ID: 350718
  • Email: wtusch@math.uni-muenster.de
  • Received by editor(s): October 15, 2002
  • Received by editor(s) in revised form: April 14, 2003
  • Published electronically: March 21, 2005
  • Additional Notes: The first author was supported in part by NSF grant DMS–0103993.
    The third author’s research was supported by a DFG Heisenberg Fellowship.
  • Communicated by: Wolfgang Ziller
  • © Copyright 2005 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 133 (2005), 2449-2459
  • MSC (2000): Primary 53C20
  • DOI: https://doi.org/10.1090/S0002-9939-05-08135-9
  • MathSciNet review: 2138888