We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.
MiS Preprint
79/2002
Nonnegatively and Positively Curved Invariant Metrics on Circle Bundles
Krishnan Shankar, Kristopher Tapp and Wilderich Tuschmann
Abstract
We derive and study necessary and sufficient conditions for an $S^1$ bundle to admit an invariant metric of nonnegative or positive 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.