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Nonnegatively and Positively Curved Invariant Metrics on Circle Bundles
Krishnan Shankar, Kristopher Tapp and Wilderich Tuschmann
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.