# Introduction to Differential Geometry

MATH 4720 (UVA)

### Course information

- Lectures: Mondays, Wednesdays, and Fridays 12:00pm — 12:50pm
- First Lecture: Wednesday 24. August, 2022.
- Office: 311
- Office hours:
- Syllabus
- Course notes

### About

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 equations, 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.

### Exercise Sheets

- Exercise Sheet 1
- Exercise Sheet 2
- Exercise Sheet 3
- Exercise Sheet 4
- Exercise Sheet 5
- Exercise Sheet 6
- Exercise Sheet 7
- Exercise Sheet 8
- Exercise Sheet 9

### Course References

We will not be using a fixed textbook, instead working from my notes (posted above). However, I will be primarily be making reference to the following resources:

- Wilhelm Klingenberg.
*A Course in Differential Geometry*. Springer Graduate Texts in Mathematics, Volume 51. ISBN: 978-1461299257 - Wolfgang Kühnel.
*Differential Geometry: Curves — Surfaces — Manifolds*. AMS. ISBN: 978-1470423209 - Manfredo P. Do Carmo.
*Differential Geometry of Curves and Surfaces: Revised and Updated Second Edition**. Dover Books on Mathematics. ISBN: 978-0486806990*