﻿ JRD: Math 344

2020 December 10,

# Math 344: Differential Geometry

Carleton College, Fall 2020, Joshua R. Davis, , Anderson 238, x4095

## Introduction

If you studied geometry in high school, then you probably spent a lot of time talking about lengths and angles. Later, when you studied linear algebra or multivariable calculus, you might have learned that lengths, angles, and the rest of Euclidean geometry can be defined in terms of the dot product. This formulation is attractive because it works in any dimension — 1, 2, 3, 4, 19, etc. — and because it generalizes to the concept of an inner product.

In any fixed dimension n, there are infinitely many inner products, with each one engendering a "skewed" or "distorted" version of Euclidean geometry. Exploring this relationship is valuable, in that you improve your geometric understanding of linear algebra. However, after a little while you realize that all inner products are pretty similar to each other. They all obey the same theorems. There's not much for a mathematician to discover.

Here's a way to make inner products exciting. Instead of fixing one inner product on your whole space, let the inner product vary from point to point across your space. So, as you move from point to point, the geometry around you changes. In this way, your space is "curved". Sound good? Well, if you let the inner product vary too wildly, then you can't prove any theorems about it. Let's assume that the variation is "smooth", so that we can use calculus on it. Then we're doing differential geometry.

In short, differential geometry is the study of curved spaces using calculus and linear algebra. It relates to several other areas of math — particularly topology, complex analysis, and differential equations. It is used in computer graphics, robotic control, DNA folding, and other scientific fields. Its most famous application is general relativity, where the space in question is our universe, variation in the inner product is caused by the presence of matter and energy, and curvature corresponds to gravity.

This course is an introduction to differential geometry. We begin by studying curves, which are important in understanding higher-dimensional spaces. Then we study surfaces, which are richer and more nuanced than curves. I hope to prove the Gauss-Bonnet theorem, which says that a certain intricate question of geometry reduces to a simple question of topology. Toward the end of the course, I hope that we can glimpse higher-dimensional manifolds.

(Because this is my first time teaching differential geometry on a 10-week schedule, and because we are learning during a pandemic, it is hard to predict exactly what is going to happen. We all need to be ready to accommodate, adjust, and empathize. That said, I intend Math 344 to be a vigorous course in basic differential geometry.)

The official prerequisite for this course is Math 236: Mathematical Structures. We prove a lot of theorems. Compared to other upper-level "pure" math courses, we also do a lot of calculations, which are error-prone and sometimes tedious. To expedite the calculations, and to visualize weird geometric spaces, we use the software Mathematica. It is available in our computer labs, and Carleton students can install it for free on their own computers. I will supply you with Mathematica tutorials and instruction. No prior experience is expected, and I do not expect you to write programs. Talk to me if you are concerned about your background.

## Resources

My office hours are MonWedFri immediately after discussion, and TueThu 8:00 in our usual Zoom room. Just drop in. Going to office hours is a normal part of college. As you advance through your education, office hours become increasingly important, because the material gets harder, your interests become more specific, and you need advice and recommendations from faculty.

If you want to schedule an appointment out of office hours, then e-mail me, listing all times (for the intended day) at which you are available. Include evening times.

Here are some materials specifically for our course.

Here are some Mathematica notebooks specifically for our course, presented in the order in which we use them.

Here are Mathematica resources provided by other people.

• Set up Mathematica on your own computer. My impression is that the Online version is easier to set up, but installing the full version on your computer might yield better results. Getting a license might take a day, so get started early.
• The Department of Mathematics and Statistics offers Mathematica lab support. But I can't post the link on a public web site like this one. See the top of our Moodle page.
• Mathematica Tutorial Video from Wolfram
• Official Mathematica Documentation from Wolfram

Here are some other resources.

• Carleton LaTeX Workshop offers lots of guidance for LaTeX users. (In this course, typing your work is optional, but it is often faster than hand-writing your work, so consider it.)
• Information Technology Services has guides for the remote learning tools that we use.
• Academic Support Center encompasses many relevant resources: Math Skills Center, Quantitative Resource Center, Writing Center, Assistive Technologies, etc. Yes, they are available remotely.

Your numerical course grade is determined by your fulfillment of the following responsibilities, which are detailed in the sections below.

• Discussions: 5%.
• Lectures: 0%.
• Homework: 5%.
• Exams: 30% each for three of them.

Letter grades are assigned to numerical grades only at the end of the term. There are no predetermined percentages (90%, 80%, 70%, etc.) required for specific grades (A, B, C, etc.), because I cannot write problems that are so precisely and reliably tuned. Instead, I assign grades by comparing students' scores to the course goals. Roughly speaking, a student who meets the goals earns a B. A student who exceeds the goals earns an A. A student who demonstrates strong effort but does not meet the goals earns a C. Students, for whom I have insufficient evidence of learning and effort, might earn grades below C.

An advantage of this system is that students are not in competition with each other. Also, students don't suffer when I accidentally write a difficult exam. The disadvantage is that you cannot compute your own grade. Send me e-mail, if you want me to estimate your current grade for you.

If some medical condition affects your performance in this course, then let me know in the first week of class (or as soon as it arises). You may need to make official arrangements with Disability Services.

## Discussions

Our class meets for discussion MonWed 1:00-2:10, Fri 1:50-2:50. These discussions correspond to the interactive parts of a normal class meeting. Usually these discussions are conducted in Zoom. Occasionally they are conducted in person in small groups and preferably outdoors. Office hours are optional extra discussion time, in a sense, and they are conducted in the same way.

Students who are in distant time zones need other arrangements. They are required to meet with me occasionally at some other time to be determined, and they are required to meet with their assigned study group frequently.

You are required to participate in these discussions. Discussions are not just for English and history classes; communication skills are essential to every academic discipline, and they are prized by employers. During a discussion, you are expected to maintain a distraction-free environment: a quiet place (if possible), no phone, etc. You are required to take notes, just as in a normal class meeting. The material involves many diagrams and matrices, so you need to use paper or an electronic device that supports rapid drawing. Typing on a laptop is probably not adequate.

When technical problems prevent you from participating in a discussion, send me e-mail about your technical problems (if possible). Do not suffer in silence.

You do not have permission to redistribute any images, video, audio, etc. from any of these discussions. The discussions should be a safe space, where no one feels that they're on display. Thank you for respecting the rights of everyone.

## Lectures

I am preparing pre-recorded video lectures. They correspond to the non-interactive parts of a normal class meeting. There might be 40 or 45 minutes of video per class meeting, broken into chunks by topic.

You are required to watch each video on its assigned day. You are again required to take notes, because the act of taking notes helps get the information into your brain. On the other hand, you are free to choose the time of day of your viewing. You are able to pause, rewind, speed up, take breaks, etc. I hope that the viewing is not unpleasant.

I try to make the lectures self-contained, so that you don't have to do a lot of other reading. But I don't always succeed. So I also give you textbook sections and sometimes my personal notes or tutorials. They can fill in details or give you another perspective. You should at least skim them, to make sure you're not missing anything. Conversely, not everything we do is in the textbook.

When technical problems prevent you from viewing a video, then do the reading strenuously. Also send me e-mail about your technical problems (if possible). Do not suffer in silence.

The videos are my intellectual property. I retain copyright for them. You do not have permission to redistribute them. Thank you for respecting my rights to my work.

## Homework

Homework is the core of the course. It's where you do most of your learning and where you prepare for the exams. I plan to assign a few problems per day. You are expected to attempt problems promptly, confer with your assigned study group, visit office hours if you want to discuss with me, and submit your solutions by the due date — typically two class days (four or five calendar days) after the problems are assigned.

Although you are encouraged to work with others, you should always write/type your solutions individually in your own words. You may not copy someone else's work or allow them to copy yours. Presenting someone else's work as your own is a violation of Carleton's Academic Integrity standards.

Writing is not just for English and history majors; communication skills are essential to every academic discipline, and they are prized by employers. In this course, your written work is evaluated both for correctness and for presentation. Use your skills from Math 236. Compose your solutions as if the intended audience is your fellow students. By doing so, you show enough detail that the grader can ascertain whether you yourself understand the material.

If you type your homework, then submit a PDF file (not Microsoft Word or Google Docs). If you write your homework, then submit a high-quality scan or photograph. Occasionally I might ask you to submit a Mathematica notebook instead. If you need to submit multiple files, then put them into a single ZIP archive.

If you need to submit homework late, then there is a two-step process. At the due date, submit what you have completed. Later, if you want to submit more problems, then submit them in a separate file. It may or may not be graded. Also, depending on time constraints in any given week, perhaps not all of your homework is graded. In order to ensure full credit, submit all of the assigned problems on time.

You are expected to spend about 10 hours per week on this course outside lecture/discussion. Some students need to spend more than 10 hours. If you find yourself spending more than 15 hours, then talk to me.

## Exams

We have three exams.

• Exam A: The first exam is given around 1/3 of the way through the course.
• Exam B: The second exam is given around 2/3 of the way through the course. It is cumulative but focused on recent material.
• Exam C: The third exam is given at the end of the course. Probably it is given during the official finals period. It is cumulative but focused on recent material.

The collaboration that is encouraged on homework is forbidden on exams. Academic Integrity standards still apply. Communication skills are still valued.

## Schedule

Here is an outline of the course, labeled by sections in our textbook. Probably we can't cover all of this material, so we'll have to revise.

1. local theory of curves
1. arc length, curvature (1.1-1.5)
2. Frenet frame (1.7-1.8)
2. one topic from the global theory of curves
1. rotation index (1.6)
2. Hopf's Umlaufsatz (2.1)
3. local theory of surfaces
1. basics (3.1-3.4)
2. surface area, metric (3.6-3.7, 3.9)
3. curvature (4.1-4.5)
4. geodesics, Theorema Egregium (5.1-5.3)
5. covariant derivative, parallel transport, holonomy (5.5-5.7)
4. global theory of surfaces
1. orientable surfaces (3.5)
2. complete surfaces (5.4)
3. Gauss-Bonnet theorem (6.1-6.2)
5. higher dimensions
6. other topics
1. more global curve theory (2.2-2.6)
2. equal-area and equal-angle surface parametrizations (3.8)
3. minimal surfaces (4.6)
4. more global surface theory (4.7, 6.3-6.4)