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. / Thesis (Ph.D.)-- Wichita State University, Fairmount College of Liberal Arts and Sciences, Dept. of Mathematics, Statistics, and Physics
| Identifer | oai:union.ndltd.org:WICHITA/oai:soar.wichita.edu:10057/10941 |
| Date | 05 1900 |
| Creators | Ryan, Justin M. |
| Contributors | Parker, Phillip E. |
| Publisher | Wichita State University |
| Source Sets | Wichita State University |
| Language | en_US |
| Detected Language | English |
| Type | Dissertation |
| Format | ix, 60 p. |
| Rights | Copyright 2014 Justin M. Ryan |
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