Geometry Working seminar

Spiro Karigiannis, Pure Mathematics, University of Waterloo

"Holonomy, G-structures, and intrinsic torsion"

We will start with a quick review of principal bundles and connections on both vector bundles and principal bundles. Then we will define the holonomy group of a connection and relate it to the parallel tensors on the manifold. We will also mention the Ambrose-Singer Theorem which relates the holonomy to the curvature of the connection. Finally, we will discuss $G$-structures and the intrinsic torsion of a $G$-structure, and give several explicit examples, including $\mathrm{O}(n)$, $\mathrm{GL}(m,\mathbb C)$, and $\mathrm{U}(m)$ structures.