Math 401: Curves and Surfaces, Fall 2019

Prof. Jo Nelson

Email: jo [dot] nelson [at] rice [dot] edu
Lectures: MWF 2-2.50 pm HBH 423
Recitations: Tuesdays 2:30-3:30 HBH 40
Office Hours:
Prof Jo: M 3-4 pm and W 12-1pm in HBH 402
Leo: W 3-4pm and R 5:30-6:30pm in HBH 040



The official textbook for the course is:
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 grade homework and half of the exams.


Undergraduate students should expect to spend 6 hours a week on homework in this course. Homework will count for 30% of your final grade, and you must upload your homework to gradescope by 5pm on Wednesdays. Clearly print your first and last name on your assignment and indicate those students that you worked with. Late homework will not be accepted. Your lowest homework score will be dropped.


There will be one pledged take home midterm, worth 30% of your course grade, which you should spend no more than 5-6 hours actively working on. It will be made available on Monday September 30 and due by 5pm on Monday October 7. You are not permitted to work with other students and you are not permitted to consult the internet beyond the course Piazza page. You are allowed to refer to the course textbook but no other books.

The final exam will count for 40% of your course grade. It is in the nonstandard format of Choose your own Differential Adventure! It is due Monday December 16 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 must be provided to me in order to receive accommodations.


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 should use Piazza to post questions about the course, including questions about topics covered in class or regarding the homework.

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 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.


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 (Wednesdays)      
8/26 1: Curves and arclength
8/28 1: Plane curves
8/30 1: Space curves HW 1LaTeX
Due 9/4
9/2 Labor Day
9/4 2: Surfaces and parametrizations
9/6 2: Change of parameters HW 2LaTeX
Due 9/11
9/9 2: Differentiable functions and the tangent plane
9/11 2: Differentials Part I
9/13 2: Interlude on pringles and transversality HW 3LaTeX
Due 9/18
9/16 3: Normal fields and orientation
9/18 3: Gauss map
Short movie ``Outside In"
9/20 3: Second fundamental form (last section covered on Midterm) HW 4LaTeX
Due 9/25
9/23 3: Normal sections
9/25 3: Second fundamental form revisited
9/27 3: Continuity of Curvature
9/30 4: Separation
Midterm LaTeX
Due 10/7
10/2 4: Jordan-Brouwer separation theorem
10/4 4: Tubular neighborhoods
5: Integrable functions and integration
10/9 5: Change of variables and Fubini HW 5LaTeX
Due 10/16
10/11 5: Area formula
10/14 Fall Break
10/16 5: Divergence theorem and Brouwer fixed point theorem HW 6LaTeX
Due 10/23
10/18 6: Reflections on eggs and tube eggs
10/21 6: Positively curved surfaces
10/23 All the scattered things about parallel surfaces HW 7LaTeX
Due 10/30
10/25 6: Minkowski formulas
10/28 6: The Alexandrov theorem
10/30 Remarkable Pizza is seriously remarkable
7: Rigid motions and isometries
Due 11/6
11/1 7: Gauss' Theorema Egregium
Wikipedia animation
11/4 7: Geodesics
Informal blog post
Gory details for a sphere
11/6 7: Geodesics are trippy on an ellipsoid
On a donut
On a cone
Due 11/13
11/8 7: Geodesics
Maps with Least Distortion between surfaces
11/11 7: The exponential map
is the Azimuthal equidistant projection
11/13 7: The exponential map HW 10LaTeX
Due 11/25
11/15 8: Intro to Gauss-Bonnet and Euler Characteristic
11/18 8: Degree of maps between compact surfaces
11/20 8: Homotopies and Brower's furry cat
Differential Adventure Proposal
Due 12/2
11/22 8: Index of a critical point of a vector field
11/25 8: Gauss-Bonnet Theorem
11/27 No Class
12/2 8: Gradient vector fields and GB upgrade
12/4 Classification of Surfaces
12/6 Manifolds and more Completed Differential Adventure Write Up
Due Monday 12/16