Math 401: Curves and Surfaces, Spring 2022

Prof. Jo Nelson

Email: jo [dot] nelson [at] rice [dot] edu
Lectures: Tues/Th 2.30-3.45pm HBH B21
Office Hours:
Prof Jo: HBH 402, T 12:30-1:30pm, R 1:15-2:15pm
Leo: HBH B40

Textbook

S. Montiel and A. Ros, Curves and Surfaces Second Edition, GSM, Vol. 69 (The first chapter is available for free)

An informal blog post explaining the Gauss-Bonnet Theorem, which will give you a flavor of what topics we will study rigorously in this course.

Teaching Assistant

The teaching assistant for this course is Leo DiGiosia. He will hold office hours, grade homework and the midterm.

Attendance and Collegiality

Attendance and participation will count for 15% of your final grade. Students are expected to be kind to each other and foster a friendly atmosphere both inside and outside of the classroom. This means abiding by the department's policy on Collegiality, Respect, and Sensitivity. Comportment not meeting these standards will result in a reduction from your attendance and participation grade.

Homework

Students should expect to spend 3-6 hours a week on homework in this course, and there will be approximately 10 homework sets in total. Homework will count for 30% of your final grade, and you must upload your homework to gradescope on Mondays by 11pm CST. Students are encouraged to work with classmates on homework, but the solutions must be written by you in your own words. Please indicate the students that you worked with. Two extensions per student will be given (with 3 days advance notice and my approval) and your lowest or nonexistent homework score will be dropped.

Take home midterm and final project

There will be one pledged take home midterm, worth 25% of your course grade, which you should spend no more than 6 hours actively working on. The midterm is open book and open note. It will be made available TBA and due one week later. You are not permitted to work with other students and you are not permitted to consult the internet. You are allowed to refer to canvas materials, the course textbook, and Folland's Advanced Calculus, but no other books.

The final project will count for 30% of your course grade. It is in the nonstandard format of Choose your own Differential Adventure! It is due Tuesday May 3 at 5pm. Collaboration with classmates during the proposal stage is encouraged. You may use outside resources with my permission.

In the event of illness or family emergency I must be notified ideally at least 24 hours in advance and documentation from your magister (or a doctor's note) must be provided to me in order to receive accommodations for the midterm or final project.

Help

If you find yourself confused, please seek help sooner rather than later. I will be available to answer questions during my office hours as will Leo. You can use Canvas to post questions about the course, including questions about topics covered in class or regarding the homework.

Outline

This course is about the geometry of curves and surfaces in three-dimensional space. We will also study the ``intrinsic" geometry of surfaces: that is, geometric notions which are described just in terms of the surface and not in terms of an embedding into higher dimensional euclidean space. A central theme of this course will be to study different kinds of curvature - defined locally on a curve (in chapter 1 of the book) or surface (in chapter 3) - and how curvature relates to global properties of the curve or surface (in chapters 4, 6, 7, and 9). One of the main results in this direction which we will prove near the end of the course is the Gauss-Bonnet theorem (chapter 8).

We will follow the modern point of view on differential geometry by emphasizing global aspects of the subject whenever possible. In order to do this, we will introduce the concept of Lebesgue measure and Lebesgue integrals and revisit multivariable calculus from this perspective (chapter 5). Time does not permit us to rigorously develop all these foundations, which are treated in Math 425/515 (not a pre-requisite).

This course is intended to be a precursor to graduate courses in differential geometry and topology. Thus more emphasis is placed on self-learning and lemmas and theorems will not typically be worked out in detail in lectures. I will state key lemmas and theorems and summarize the main points. You should expect to spend 1-2 hours a week reading the textbook each week.

Plane and space curves (chapter 1)
Surfaces in Euclidean space (chapter 2)
The second fundamental form (chapter 3)
Separation and Orientability (chapter 4)
Integration on surfaces (chapter 5)
Global extrinsic geometry (chapter 6)
Intrinsic geometry of surfaces (chapter 7)
The Gauss-Bonnet Theorem(chapter 8)
Global geometry of curves (chapter 9)

Schedule & Assignments

Date	Material Covered	Homework (usually Mondays)
1/11	1: Curves and arclength	HW 1 LaTeX Due Monday 1/17
1/13	1: Plane and space curves
1/18	2: Torsion and curvature characterize space curves	HW 2 LaTeX Due Monday 1/24
1/20	2: Surfaces and parametrizations,
1/25	2: Inverse and Implicit function theorems. Change of parameters	HW 3 LaTeX Monday 1/31
1/27	2: Differentiable functions, tangent plane
2/1	2: The tangenet plane, differentials	HW 4 LaTeX Due Monday 2/7
2/3	2: Chain rule. Transversality
2/8	3: Normal fields, orientation, and the Gauss map Short movie ``Outside In"
2/10	Spring Recess, no classes
2/15	First fundamental form and surface area	MIDTERM HANDED OUT Midterm LaTeX Due Monday 2/21
2/17	3: Second fundamental form
2/21	Midterm due today!
2/22	3: Normal sections	HW 5 LaTeX Due Monday 2/28
2/24	3: The Hessian
3/1	3: Continuity of curvature	HW 6 LaTeX Due 3/7
3/3	Remarkable Pizza is seriously remarkable 7: Rigid motions and isometries
3/8	7: Gauss' Theorema Egregium Wikipedia animation All world maps are wrong World map projection comparisions	HW 7 LaTeX REVISED: Due 3/28
3/10	7: Gauss' Theorema Egregium Claire Saffitz and the art of making Pringles Bon Appetit did not grant equal opportunities to chefs of color
3/15	Spring Break
3/17	Spring Break
3/22	7: Geodesics Informal blog post Gory details for a sphere
3/24	7: Geodesics are trippy on an ellipsoid On a donut On a cone
3/29	7: The exponential map is the Azimuthal equidistant projection Maps with Least Distortion between surfaces	HW 8 LaTeX Due 4/4
3/31	5: Integration over surfaces
4/5	8: Intro to Gauss-Bonnet and Euler Characteristic	Diff Adventure Proposal Due 4/11
4/7	8: Degree of maps between compact surfaces, homotopies
4/12	8: Index of a critical point of a vector field Brower's furry cat
4/14	8: Gauss-Bonnet Theorem
4/19	8: Gradient vector fields and GB upgrade
4/21	Classification of Surfaces	Diff Write Up Due Tuesday 5/3

Additional Course Policies

Comportment Expectations. The Department of Mathematics supports an inclusive learning environment where diversity and individual differences are understood, respected, and recognized as a source of strength. Racism, discrimination, harassment, and bullying will not be tolerated. We expect all participants in mathematics courses (students and faculty alike) to treat each other with courtesy and respect, and to adhere to the Mathematics Department Standards of Collegiality, Respect, and Sensitivity as well as the Rice Student Code of Conduct. If you think you have experienced or witnessed unprofessional or antagonistic behavior, then the matter should be brought to the attention of the instructor and/or department chair. The Ombudsperson is also available as an intermediate, informal option, and contacting them will not necessarily trigger a formal inquiry.

Title IX Responsible Employee Notification. Rice University cares about your wellbeing and safety. Rice encourages any student who has experienced an incident of harassment, pregnancy discrimination or gender discrimination or relationship, sexual, or other forms interpersonal violence to seek support through The SAFE Office. Students should be aware when seeking support on campus that most employees, including myself, as the instructor/TA, are required by Title IX to disclose all incidents of non-consensual interpersonal behaviors to Title IX professionals on campus who can act to support that student and meet their needs. For more information, please visit safe.rice.edu or email titleixsupport@rice.edu.

Disability-related Academic Accommodations. In order to receive disability-related academic accommodations, students must first be registered with the Disability Resource Center (DRC). Students who may need accommodations in this course should give me a written letter from the DRC within the first two weeks. More information on the DRC registration process is available online at https://drc.rice.edu/. Registered students must present an accommodation letter to the professor before exams or other accommodations can be provided. Students who have, or think they may have, a disability are invited to contact DRC for a confidential discussion.

Midterm or Final Project Accommodations. In the event of illness or family emergency I must be notified ideally at least 24 hours in advance and documentation from your magister (or a doctor's note) must be provided to me in order to receive accommodations for the midterm or final project.