On pull-backs of the universal connection

TL;DR

This paper explores how the geometric features of a classifying map influence the properties of the pulled-back universal connection, including conditions for fatness, parallel curvature, and nonnegative or positive sectional curvature.

Contribution

It establishes specific conditions on classifying maps that determine when the pulled-back connection exhibits desirable geometric properties.

Findings

Conditions for the pulled-back connection to be fat.

Criteria for the connection to have parallel curvature tensor.

Results on when the induced metrics have nonnegative or positive sectional curvature.

Abstract

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).