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

In Spring 2021, this is a somewhat flexibly-paced course taught in the “hybrid asynchronous” format. This webpage hosts a complete collection of course materials: readings, notes, videos, and related homework assignments. Each of these units corresponds roughly to a day or two of the old lecture-and-in-class work time class schedule. **The class is fully asynchronous, so there are no simultaneous Zoom sessions during class time; everything you need to know for the course is in the materials posted below.**

As a philosophical decision, since UGA is a public, taxpayer supported institution, I’ve decided to make all of these materials available freely on this webpage rather than putting them behind a university paywall.

**In-person course attendance is optional and requires advance booking because of limited classroom capacity.** The online materials and online office hours are intended to be just as good as the in-person sessions. In-person sessions will be offered Tuesdays and Thursdays from 9:35-10:50 in Marine Sciences/Dance Room 304. Because of social distancing rules at UGA, no more than 20 of the 40 students enrolled can attend any given session. Therefore, I need you to let me know that you plan to come by booking an in-person class session online before showing up in order to make sure there’s a (safe) seat for you.

**Online office hours require booking as well. **At the moment, there are a limited range of times available, and the sessions are limited to 5 people at a time. These will probably be adjusted as the semester progresses.

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, with some additional material on computational and discrete geometry. By the end of the semester, I hope to discuss some applications of differential geometry in machine learning and applied math.

Homework will be submitted (and exams returned) via Gradescope with course entry code 866J4X. **The schedule for completion of the material is in Gradescope:** each homework assignment has a due date (and a late date). There is no penalty for submitting homework by the “late” due date. However, we generally can’t grade or accept assignments after the late due date. It will help you pace yourself if you complete as many assignments as you can by the original due date. Please submit anything you have by the late date; if you are unable to start by that time, move on to a different assignment and contact me to make arrangements regarding grading. Please remember to select pages corresponding to each problem in Gradescope, including submitting a page which says “I didn’t do Problem 3.7” (or similar) for problems that you didn’t get to.

We also have a course syllabus.

### Course Material

- Parametrized curves, the dot product, cross product, and triple product.
- Reading: OpenStax physics chapter 2 (vectors and scalars)
- Optional video: 3Blue1Brown “Essence of linear algebra” series
- 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.

- Minihomework: Scalar and vector products.
- Minihomework: Getting comfortable again with linear algebra
- Video: A Tale of Two Matrices
- Minihomework: A tale of two matrices
- Video: Of symmetries, solids and coordinates
- Minihomework: The Dot Product, Point Groups, and the Regular Solids
- Optional reading: Coxeter, Regular Polytopes, Chapter 3.
- Optional reading: Verheyn, The complete set of Jitterbug transformers and the analysis of their motion.

- Constructing Parametrized Curves.
- Reading: Shifrin, p. 1-6.
- Video: The Tractrix
- Reading: The Tractrix (a different approach)
- Minihomework: Constructing curves.

- Arclength and Regular Curves.
- Reading: Shifrin, p. 6-8.
- Minihomework: Reparametrizing curves by arclength.
- Video: The square-wheeled car.
- Minihomework: The square-wheeled car

- Variations and curves.
- Video: The brachistochrone.
- Minihomework: Calculus of variations.

- Introduction to framings (video).
- Notes: Framed curves and the Frenet frame.
- Reading: Shifrin, p. 12-20.
- Minihomework: Curves and framings.

- The Frenet frame without an arclength parametrization.
- Reading: Shifrin, p. 12-20. (Note: This is the same reading as 5.)
- Optional video: The Gram-Schmidt Process (Khan Academy)
- Optional video: Gram-Schmidt Example (Khan Academy)
- Minihomework: Frenet without an arclength parametrization

- Why are curvature and torsion important?
- Reading: Shifrin, p. 23-34.
- Minihomework: Curvature and torsion theorems.
- Extra credit minihomework: Curve theory concluded.

- Intersection measure for plane curves. (Skipped in Spring 2021 course)
- Integralgeometric measure for space curves.
- Minihomework: Integralgeometric measure experiment.
- Video: Integralgeometric measure homework walkthrough.
- Software resource: FijI image analysis program.

- Geometric Inequalities for Curves. (Skipped in Spring 2021 course)
- The Fabricius-Bjerre Theorem. (Skipped in Spring 2021 course)
- Linear Functions and Quadratic Forms. The Gradient and Hessian.
- Minihomework: The Gradient and the Hessian.

- Eigenvalues and Eigenvectors. Paraboloids: elliptic, hyperbolic, and cylindrical.
- Video: 3Blue1Brown “Essence of Linear Algebra”
- Minihomework: Understanding quadratic functions of several variables.
- Extra credit: Quadratic surface projects.

- Inner products, positive-definite matrices, and quadratic forms.
- Surfaces and the First Fundamental Form.
- Reading. Shifrin 2.1. (all pages)
- Minihomework: First Fundamental Form.
- Graduate material and minihomework: The intrinsic gradient.
- Note: Graduate assignments are assigned to everyone in Gradescope. For students in 6250, they are required. For students in 4250, they are extra credit.

- The Gauss Map and the Second Fundamental Form.
- Reading. Shifrin 2.2. pages 46-48
- Minihomework: Shape operator identity implies sphere.

- The Second Fundamental Form (2)
- Reading. Shifrin 2.2 pages 47-49.
- Minihomework: Symmetric Linear Maps and Matrices

- A (lengthy) Example.
- Reading. Shifrin 2.2 Example 6. pages 49-50.
- Minihomework: Quadratic surfaces.
- Graduate material and minihomework: General form for surfaces.
- Graduate reading: The Implicit and Inverse Function Theorems: Easy Proofs, Oliveira.

- Classification of Points and Meusnier’s Formula.
- Reading. Shifrin 2.2 pages 50-53.

- The Codazzi and Gauss Equations.
- Reading. Shifrin 2.3 pages 57-59.

- Codazzi and Gauss Equations (2)
- Reading. Shifrin 2.3 pages 59-60.

- Global Geometry of (Compact) Surfaces
- Reading. Shifrin 2.3 pages 61-63

- Recovering the Embedding of a Surface
- Reading. Shifrin 2.3 pages 63-64.

- Covariant Differentiation and Parallel Transport
- Reading. Shifrin 2.4 pages 66-69

- Geodesics as Straightest Paths
- Reading. Shifrin 2.4 pages 70-72

- Clairaut’s Relation
- Reading. Shifrin 2.4 pages 73-75.

- Holonomy and Gauss-Bonnet (I).
- Reading. Shifrin 3.1 pages 79-82.

- Holonomy and Gauss-Bonnet (II).
- Reading. Shifrin 3.1 pages 82-89.

### Course Evaluation.

The midterm exam will be held **on Thursday, March 18. **There are three options for for taking the exam: during the regular class time and room (book using the usual link), or in the evening from 6:00pm to 7:30pm or 7:30pm to 9:00pm (book using this link). We will have an **in person final on Tues., May 11, from 8:00 – 11:00 am. **(If you’re ill or in quarantine on exam day, we’ll find a solution, but this means you should plan to be physically present on campus at least twice during the semester.)

You may bring up to 350 pages of text with you (on actual paper, not on a device), plus any bound books that you like. In addition, you can bring any calculator permitted for the SAT Math Subject test. The midterm exam will cover units 1-7 in the course materials above.

Practice Midterms. 2007 Take-home test 1. 2012 Take-home test 1. Wikipedia page on Gram-Schmidt . This was ok to refer to while doing the exam. 2007 Take-home test 2

Practice Final Exams. 2007 Final Exam (take-home) 2012 Final Exam (3 hour open book final)

Note: Solutions are not available for the practice exams. Also, the syllabus has shifted over the years, so old tests may cover material that is no longer present in the course.

### 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!)

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

### Some optional additional notes.

Here are links to lecture notes for the course on additional material, or on Do Carmo’s book. I’m leaving them up as a source of other perspectives for you, and also to draw from for extra credit assignments.

- Crofton’s Formula and Buffon’s Needle.
- Crofton’s Formula and the Indicatrices.
- The Four Vertex Theorem.
- The Bishop Frame.

Do Carmo Notes.

- Introduction and Overview .
- The Frenet Frame .

## Homework 1. - Curves of constant curvature and torsion .
- The Bishop Frame .
- Link, Twist, and Writhe . Correction: The integrals for
*Link*and*Writhe*should be multiplied by 1/(4 pi) in these notes. Thanks to Matt Mastin for pointing this out! - The Four-Vertex Theorem .

## Homework 2. - The Fabricius-Bjerre Theorem .
- An Introduction to Integral Geometry .
- Integral Geometry II .
- Stuff turning inside out .

## Homework 3. - Introduction to Regular Surfaces .
- Regular Surfaces as Level Sets of Smooth Functions .
- Tangent planes and differentials. Review of quadratic forms. .
- The first fundamental form. How to measure lengths, angles, and areas in the uv plane.

## Homework 4. - The Gauss map and the second fundamental form. Defining the form.
- The geometric meaning of the second fundamental form. The definition of Gauss and Mean curvature.

## Homework 5. - The meaning of the second fundamental form, part II. Umbilic points, Asymptotic and Conjugate directions, the Dupin indicatrix. <br /> Sorry about the scan quality– these came out really light.
- Computing with the second fundamental form in local coordinates. e, f, g, formulas for Gauss and Mean curvature.
- Extracting geometric information from the Second Fundamental Form. Differential equations for asymptotic curves and lines of curvature. Surfaces of revolution. Graphs.

## Homework 6. - Isometries, proof that Helicoid and Catenoid are isometric.
*There is no scan of these notes. Read 4.1 in DoCarmo.* - The Christoffel symbols, proof of the Theorema Egregium, Mainardi-Codazzi equations and Gauss Formula, compatibility equations and theorem of Bonnet.
- Theory of Geodesics, Geodesics on Surfaces of Revolution. Note: This is really different than the corresponding chapter in DoCarmo, so you won’t be able to match it up as easily as you did the previous lecture notes.

## Some examples of geodesics near a black hole

## (pdf version of the relativity examples)

## Homework 7. - (Signed) geodesic curvature and the local Gauss-Bonnet theorem. (These notes have everything there, but could make more sense. Read with a bit of caution. The reference here is McCleary, Geometry from the Differentiable Viewpoint, p. 173-177.)
- The global Gauss Bonnet theorem, Euler characteristic, applications of G-B, conclusion.