Horizontal bundles and connections
Ryan, Justin M. 2014. Horizontal Bundles and Connections. -- In Proceedings: 10th Annual Symposium on Graduate Research and Scholarly Projects. Wichita, KS: Wichita State University, p. 152
A manifold is a mathematical space that locally resembles standard Euclidean space. In order to study the geometry of such a space, it is necessary to prescribe a connection on the manifold. A connection describes how tangent spaces to the manifold change with respect to infinitesimal changes in the manifold. In 1950, Charles Ehresmann defined a connection to be an abstract object called a horizontal bundle, with a special property called horizontal path lifting. By including this additional property in his definition, Ehresmann implicitly acknowledged that it was nontrivial: not all horizontal bundles satisfy it. We give the first complete characterization of the horizontal bundles which have this property, and hence are connections.
Second place winner of poster presentations at 10th Annual Symposium on Graduate Research and Scholarly Projects (GRASP) held at the Heskett Center, Wichita State University, April 25, 2014.
Research completed at Department of Mathematics, Statistics, and Physics, College of Liberal Arts and Sciences