This course is intended to be an introduction to differential geometry, more particularly the differential geometry of embedded manifolds. Roughly speaking, we will study curves, surfaces, and their higher dimensional analogs in terms of notions like distance, length, and curvature, using differentiation and integration. In this context, the word differential refers to the fact that we will be making use of the tools and techniques of calculus. Geometry roughly refers to the fact that we will be using concepts like distance and curvature (this stands in contrast to the subject of differential topology where distances are not necessarily well-defined). Finally, embedded means we will only be considering those spaces which are subspaces of n-dimensional real Euclidean space, \(\mathbb{R}^n\) .
This course will be proofs-based, meaning that you will be expected to be able to read, write, and understand mathematical proofs. Throughout the course I will assume familiarity with basic concepts from multivariable calculus, differential equa- tions, and linear algebra. Some material from real analysis will also be used, though for the purposes of this course some such theorems can be treated as black boxes.
The aim of the course will be to reach a pair of profound statements about the nature of curved surfaces in \(\mathbb{R}^n\) . The Gauß-Bonnet Theorem relates several notions of curvature to an invariant called the Euler characteristic. The Theorema Egregium — also proven by Gauß — shows that the curvature of a surface depends only on only a few types of information about that surface. Along the way to these two theorems, we will encounter concepts like Frenet frames, arc length, curvature, tangent spaces, and fundamental forms. Time permitting, we may also discuss additional topics such as Minkowski space, manifolds, differential forms, and tensors.
Below is a list of resources which may be of help in the course. I will periodically update the list throughout the semester