On pull-backs of the universal connection

Abstract Narasihman and Ramanan proved in [Amer. J. Math. 83 (1961), 563–572] that an arbitrary connection a vector bundle over base space B can be obtained as the pull-back (via correctly chosen classifying map from into appropriate Grassmannian) of universal Grassmannian. The purpose this paper is to relate geometric properties pulled-back connection. More specifically, we describe conditions on under which connection: (1) fat (in sphere bundle), (2) has parallel curvature tensor, (3) induces metric with nonnegative sectional (or positive bundle).