Mengxue Yang, Department of Pure Mathematics, University of Waterloo
“Curvature of Surfaces”
Given a very concrete surface in R3, we can use the curvature of a surface curve to define the curvature of the surface at points of the curve in the directions the curve travels in. Although the motivation for the definition relies on curves, the curvature of a surface does not actually depend on specific surface curves. I will formally introduce the notions of normal curvature, principal curvatures, Gaussian curvature, and mean curvature of a surface and show you how to compute them. In the process, I will talk about things like the Gauss map, the shape operator, and the second fundamental form, which lead to the list of curvatures in the previous sentence. I will also discuss some geometric interpretations.