These are the notes I made as I taught myself about holonomy, in preperation for my DPhil after the summer.
It is the aim of holonomy to take seriously the idea that a concept exsists to capture what happens when you follow a path, and each step you take produces only a little change, but when you loop back to where you started a big change has occured.
These notes aim to represent an advanced, research-oriented introduction to the theory of holonomy and connections on fibre bundles, with particular emphasis on Riemannian holonomy, exceptional geometries, and the differential geometry of calibrations. Beginning with a rigorous treatment of connections on vector and principal bundles, we move toward the classification of holonomy groups (Berger’s theorem), with the ambition of reaching the extraordinary geometries of $G_{2}$ and Spin(7), and glimpsing their place in spin geometry and twistor theory.
There are two principle references. The first is the book 'Riemannian Holonomy and Callibrated Geometry' by Joyce. The second contains intuition from a more physical interpretation, and that is 'Topology, Geometry and Gauge Fields: Foundations' by Naber. Additional readings will be drawn from classical literature, such as the work of Berger, where appropriate.
Vector bundles, principal bundles, the intuition of holonomy, defining holonomy, and the restricted holonomy.
More on the restricted holonomy, the link between curvature and holonomy, the Ambrose-Singer Theorem, and the proof of the Ambrose-Singer Theorem.