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According to a classical theorem of Cheeger and Gromoll, an open complete manifold of nonnegative sectional curvature is diffeomorphic to the normal bundle of a compact totally geodesic submanifold. A natural and difficult problem is to decide to what extent the converse holds, namely, what vector bundles admit nonnegatively curved metrics. No obstruction are known when the base of the bundle has finite fundamental group. By contrast, if the fundamental group is infinite there are many obstruction (this is a joint work with Vitali Kapovitch). In particular, we show that for a "generic" base B with infinite fundamental group, "most" vector bundles over B admit no complete nonnegatively curved metric.

Refreshments will be served prior to talk in room 551, Kerr Hall