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Proceedings of the American Mathematical Society

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Nonnegatively and positively curved invariant metrics on circle bundles


Authors: Krishnan Shankar, Kristopher Tapp and Wilderich Tuschmann
Journal: Proc. Amer. Math. Soc. 133 (2005), 2449-2459
MSC (2000): Primary 53C20
Published electronically: March 21, 2005
MathSciNet review: 2138888
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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
Email: ktapp@brynmawr.edu

Wilderich Tuschmann
Affiliation: Mathematisches Institut, Westfälische Wilhelms-Universität Münster, Einstein- strasse 62, D-48149 Münster, Germany
Email: wtusch@math.uni-muenster.de

DOI: https://doi.org/10.1090/S0002-9939-05-08135-9
Keywords: Nonnegative sectional curvature, principal circle bundles, connection metrics.
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
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.