Math 4250: Differential Geometry

Welcome to the homepage for Differential Geometry (Math 4250/6250)!

In Spring 2025, this is an in-person class with required attendance consisting (mostly) of active lecture with some in-class active learning components. This webpage hosts a complete collection of course materials: readings, notes, videos, and related homework assignments. We won’t use eLC. The course is proof-based and you will be need to be able to create proofs on exams in order to pass the class. Students with a grade of “C” (not ready to proceed to subsequent courses) in MATH 3200 should reconsider enrolling in this course; I recommend re-enrolling in MATH 3200 instead.

The course textbook is by Ted Shifrin, which is available for free online here. The course will cover the geometry of smooth curves and surfaces in 3-dimensional space. An alternate book is Do Carmo, Differential Geometry of Curves and Surfaces. This book is on reserve for you at the Science Library in Boyd.

Homework will be submitted (and exams returned) via Gradescope. The schedule for completion of the material, and the templates for the homework assignments are in Gradescope. You should download and print out the template for each homework assignment, complete the work, and then scan and upload your work. Note that Math 6250 is a completely different Gradescope course with very different homework assignments. 

Each homework assignment has a due date and time (and a late date a few hours later in case you have trouble uploading or something). We generally can’t grade or accept assignments after the late due date. Please submit anything you have on time; if you are unable to start by that time, move on to a different assignment and contact me to see if we can accept later work. 

There are a few different kinds of homework assignments.

  1.  “Problem Set” assignments. These are complete problem sets for you to take home and think about on your own or with your study group, usually with 5 or so questions. You generally have 1-2 weeks to complete a problem set, and they are mostly graded by the course TA, with some questions graded by me.
  2. “Active Learning” assignments. These problem sets are designed to be worked on in class, and often introduce new material in between the problems. You should complete most of an active learning assignment during the class period, but you’ll generally have a week to write up a final version of your work before it’s turned in for grading. The active learning assignments should be completed (and turned in) in groups of 2 with your study partner.

We also have a course syllabus for MATH 4250  and a course syllabus for MATH 6250.

Course Material

  1.  (1/7, 1/9) Parametrized curves, the dot product, cross product, and triple product.
    1. Reading: OpenStax physics chapter 2 (vectors and scalars) 
    2. Optional video: 3Blue1Brown “Essence of linear algebra” series
      1. Note: The video playlist is basically an entire course on linear algebra. So I certainly don’t expect anyone to watch the whole thing. However, it’s a really good resource– if you feel like you want to refresh your memory on a particular topic– you can just watch that particular video.
    3. Problem Set. Scalar and Vector Products. 
  2.  (1/14) Constructing  Parametrized  Curves.
    1. Reading: Shifrin, p. 1-6. 
  3. (1/16) Video: The Tractrix
    1. Reading: The Tractrix
    2. Before class: The tractrix.
    3. Problem Set. Hyperbolic Trigonometric Functions 
  4. (1/21) Arc-length and Regular Curves.
    1. Reading: Shifrin, p. 6-8. 
    2. Problem Set. Reparametrizing curves by arclength 
  5. (1/23) Curvature and Framings for plane curves
    1. Problem Set. Curvature of plane curves.
    2. Problem Set for 6250. Support Functions, Cams, and Convexity
  6. (1/28) Framed curves in space and the Frenet frame.
    1. Video:  Introduction to framings (video).
    2. Reading: Shifrin, p. 12-20.
    3. Problem Set: Curves and Framings.
  7. (1/30) Examples of curvature, torsion and frames.
    1. Reading: Shifrin, p. 12-20. (Note: Same as last time.) 
    2. Problem Set. The Frenet Frame. 
  8. (2/4) Why are curvature and torsion important?
    1. Reading: Shifrin, p. 23-34.
    2. Problem Set. The Frenet Frame.
  9. (2/6) Linear Functions and Quadratic Forms. The Gradient and Hessian.
    1. Problem Set. The Gradient and the Hessian.  
  10. (2/11) Quadratic approximating surfaces.
    1. Video: 3Blue1Brown “Essence of Linear Algebra”
      1. Eigenvalues and eigenvectors 
    2. Problem Set. Parabolas and Paraboloids.
  11. (2/13) Midterm Exam. Covers material from start of course through “2/4: Why are curvature and torsion important?”.  With notes and calculator, see description below.
  12. (2/18-2/20) Surfaces and the First Fundamental Form.
    1. Reading. Shifrin 2.1. (all pages)
    2. Problem Set. The First Fundamental Form. 
    3. Problem Set for 6250 Students. The Intrinsic Gradient. 
  13. (2/25-2/27) The Gauss Map and the Second Fundamental Form.
    1. Reading. Shifrin 2.2. pages 46-48
    2. Problem Set: The Second Fundamental Form. 
  14. (3/11) More about the second fundamental form and curvature
    1. Reading. Shifrin 2.2 pages 47-49.
  15. (3/13) A (lengthy) example.
    1. Reading. Shifrin 2.2 Example 6. pages 49-50.
    2. Problem Set. The fundamental example of surface theory. 
    3. Graduate reading: The Implicit and Inverse Function Theorems: Easy Proofs, Oliveira.
  16. (3/18) Surfaces of Revolution and the Space Forms
    1. Reading. Shifrin 2.2 pages 50-53.
  17. (3/20) The Meaning of Mean Curvature. 
  18. (3/25) Gauss-Bonnet and the Meaning of Gauss Curvature.
  19. (3/27-4/1) Geodesics.
    1. Reading. Shifrin 2.4 pages 68-77.
    2. Problem Set. Geodesics on surfaces of revolution.
    3. Problem Set for 6250 Students. Geodesics in Surfaces of Revolution and Space Forms.
  20. (4/3) Geodesics and Abstract Geometries.
    1. Reading. Shifrin 3.2 pages 93-99.
    2. Problem Set. Spherical and Hyperbolic Geometry.
  21. (4/8) Discrete Differential Geometry: Triangulated Surfaces
  22. (4/10) Discrete Differential Geometry: Surface Curvature
  23. (4/15) Discrete Differential Geometry: The Exterior Calculus and Forms
  24. (4/17) Discrete Differential Geometry: The Discrete Gauss-Bonnet Theorem
  25. (4/22) Discrete Differential Geometry: Geodesics and Distances
  26. (4/24) Discrete Differential Geometry: Geodesics and Heat Flow

Course Evaluation.

The midterm exam will be held in class. You may bring up to 10 pages of notes with you. In addition, you can bring any calculator permitted for the SAT Math Subject test. The course final will be held on Tuesday, May 6 at 12:00pm.

Some optional additional reading.

These papers should be readable after you’ve taken the class. Especially if you’re graduate-school bound, you may enjoy reading them (and I’ll be happy to discuss them with you!)

  1. An Inequality for Closed Space Curves. G.D. Chakerian, 1962.
  2. Curves and Surfaces in Euclidean Space. S.S. Chern, 1967. Section 1 or Section 5.
  3. A Geometric Inequality for Plane Curves with Restricted Curvature. G.D. Chakerian, H.H. Johnson, A. Vogt. 1976.
  4. A Spherical Fabricius-Bjerre Formula with Applications to Closed Space Curves. J. Weiner. 1987.
  5. Curves of Constant Precession. P. Scofield. 1995.
  6. Tantrices of Spherical Curves. B. Solomon. 1996.
  7. A Four Vertex Theorem for Polygons. S. Tabachnikov. 2000.
  8. There is more than one way to frame a curve . R. Bishop. 
  9. h-Principles for Curves and Knots of Constant Curvature . M. Ghomi.

Some optional additional notes.

Optional additional material for the course can be found on the course archive page. This is a collection of material incorporated into previous versions of the class. Here are some other additional topics which may be interesting to you.

  1. Intersection measure for plane curves. 
  2. Integralgeometric measure for space curves.
    1. Active Learning: Integralgeometric measure experiment.
    2. Software resource: FijI image analysis program.
  3. Geometric Inequalities for Curves. 
  4. The Fabricius-Bjerre Theorem.
    1. Video: An amazing theorem for tangents (Matt Warren)

Material on this page is a work-for-hire produced for the University of Georgia.