Geometry of horizontal bundles and connections

Abstract

An Ehresmann connection on a fiber bundle pi: E --> M is defined by prescribing a suitable horizontal subbundle H of the tangent bundle piT: TE --> E. For a horizontal bundle to be suitable, it must have a property called horizontal path lifting. This property ensures that the horizontal bundle determines a system of parallel transport between the fibers of E. The main result of this dissertation is a geometric characterization of the horizontal bundles on E that have horizontal path lifting, and hence are connections. In particular, it is shown that a horizontal bundle has horizontal path lifting if and only if its horizontal spaces are bounded away from the vertical spaces, uniformly along fibers of E. In order for a horizontal bundle to admit a system of parallel transport or have holonomy, it must be a connection. However, certain other geometric properties that are usually attributed to connections are actually properties of arbitrary horizontal bundles. These properties are studied in the case when E is either a vector bundle or tangent bundle, accordingly.