Gauge Theory, Connections, and Holonomy: Background for the Ambrose-Singer Theorem

In this thesis, we discuss some of the elementary ideas in gauge theory that allow us to describe the notion of holonomy on bundles. We seek to exploit the important relationship between vector bundles and $G$-principal bundles, which will be defined below. We begin by covering the background information in geometry and algebra at the level of first year graduate classes, which will be assumed in the rest of the thesis. The main part of the thesis begins with the chapter ``Bundles, which discusses fiber bundles and special cases of fiber bundles, including the vector bundles and the $G$-principal bundles we just mentioned. After describing bundles, we move on to connections on these bundles, and finally parallel transport and holonomy in the last chapter. The section on connections includes a couple of charts meant to relate several different notions of the term ``connection found in the mathematical literature dealing with geometry and gauge theory. The thesis ends with a theorem, the Ambrose-Singer (Holonomy) Theorem, that ties together all the concepts introduced in the thesis