This talk is about smooth fiber bundles whose concrete fibers are each equipped with a Riemannian metric whose sectional curvatures are constrained to lie in a fixed interval I of real numbers. We will focus mostly on the case where I = (-infinity,0) called negatively curved bundles. In an attempt to find characteristic classes for such bundles, Bustamante, Jiang and I studied the tautological classes for bundles whose abstract fiber is non-positively curved. I will briefly state some results from this study.  This talk is a report on joint work with several people including Pedro Ontaneda, Andrey Gogolev, Mauricio Bustamante and Yi Jiang.

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