  Narasihman and Ramanan proved that an arbitrary connection in a vector bundle
over a base space B can be obtained as the pull-back (via a correctly chosen
classifying map from B into the appropriate Grassmannian) of the universal
connection in the universal bundle over the Grassmannian. The purpose of this
paper is to relate geometric properties of the classifying map to geometric
properties of the pulled-back connection. More specifically, we describe
conditions on the classifying map under which the pulled-back connection: (1)
is fat (in the sphere bundle), (2) has a parallel curvature tensor, and (3)
induces a connection metric with nonnegative sectional curvature on the vector
bundle (or positive sectional curvature on the sphere bundle).
