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

In Spring 2022, this is an in-person class 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.

**Due to COVID, office hours are online or outdoors at the turtle **pond in from of the Ecology school**. **I will be announcing the office hours each week by email (since the weather will make it impossible to hold outdoor office hours every week, I’ll decide on Tuesday whether we’re going to be outside or on Zoom.) The office hours are drop-in, drop-out in either case, so please log into the Zoom or join us at the turtle pond and join the discussion at any time.

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 with course entry code 3Y55YE. **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.

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.

There are a few different kinds of homework assignments.

- “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 10 or so questions. You generally have 1-2 weeks to complete a problem set.
- “After class” assignments. These are small collections of short problems for you to think about in-between classes. The due date is generally the next class period, so you may only have a few days to complete an “after class” assignment.
- “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.

We also have a course syllabus.

### Course Material

- (1/13, no reading quiz) 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.

- Problem Set. Scalar and Vector Products. (Gradescope).
- After class homework. Getting comfortable again with Linear Algebra (Gradescope).

- (1/18, no reading quiz) Video: A Tale of Two Matrices
- Active Learning. A Tale of Two Matrices (Gradescope).

- (1/20, no reading quiz) Video: Of symmetries, solids and coordinates
- Active Learning. The Dot Product, Point Groups, and Regular Solids (Gradescope).
- Optional reading: Coxeter, Regular Polytopes, Chapter 3.
- Optional reading: Verheyn, The complete set of Jitterbug transformers and the analysis of their motion.

- (1/25, no reading quiz) Constructing Parametrized Curves.
- Reading: Shifrin, p. 1-6.

- (1/27, reading quiz) Video: The Tractrix
- Reading: The Tractrix (a different approach)
- Problem Set. Hyperbolic Trigonometric Functions (Gradescope).

- (2/1-2/3 reading quiz) Arc-length and Regular Curves.
- Reading: Shifrin, p. 6-8.
- Problem Set. Reparametrizing curves by arclength (Gradescope).

- (2/8, no reading quiz) Video: The square-wheeled car.
- Active Learning. The Square-Wheeled car (Gradescope).

- (2/10 and 2/15, reading quiz) Variations and curves.
- After class. Separation of variables, the multivariable chain rule, Lagrange multipliers. (Gradescope).
- Video: The brachistochrone.
- Problem Set. Calculus of Variations (Gradescope)

- (2/17, reading quiz.) Introduction to framings (video).
- Notes: Framed curves and the Frenet frame.
- Reading: Shifrin, p. 12-20.
- After class: Curves and Framings (Gradescope).

- (2/22 and 2/24, reading quiz.) The Frenet frame without an arclength parametrization.
- Reading: Shifrin, p. 12-20. (Note: This is the same reading as last time.)
- Optional video: The Gram-Schmidt Process (Khan Academy)
- Optional video: Gram-Schmidt Example (Khan Academy)
- Problem Set. The Frenet Frame without an arclength parametrization. (Gradescope).

- (3/1 and 3/3, reading quiz) Why are curvature and torsion important?
- Reading: Shifrin, p. 23-34.
- Problem Set. Curvature and Torsion Theorems . (Gradescope).
- Problem Set for 6250 Students. Support Functions, Convexity and the Isoperimetric Inequality. (Gradescope).

- Intersection measure for plane curves. (Skipped in Spring 2022 course)
- Integralgeometric measure for space curves. (Skipped in Spring 2022 course)
- Minihomework: Integralgeometric measure experiment.
- Video: Integralgeometric measure homework walkthrough.
- Software resource: FijI image analysis program.

- Geometric Inequalities for Curves. (Skipped in Spring 2022 course)
- The Fabricius-Bjerre Theorem. (Skipped in Spring 2022 course)
- (3/15, reading quiz) Linear Functions and Quadratic Forms. The Gradient and Hessian.
- After class. The Gradient and the Hessian. (Gradescope).

- (3/17.)
**Midterm Exam.**Covers material from start of course through 3/3: “Curvature and Torsion Theorems”. Open book (see description below.) - (3/22, reading quiz) Eigenvalues and Eigenvectors. Paraboloids: elliptic, hyperbolic, and cylindrical.
- Video: 3Blue1Brown “Essence of Linear Algebra”
- Problem Set. Understanding Paraboloids. (Gradescope).
- Extra Credit. Quadratic Surface Projects. (Gradescope).

- (3/24) Inner products, positive-definite matrices, and quadratic forms.
- (3/29, reading quiz) Surfaces and the First Fundamental Form.
- Reading. Shifrin 2.1. (all pages)
- Problem Set. The First Fundamental Form. (Gradescope).
- Problem Set for 6250 Students. The Intrinsic Gradient. (Gradescope).

- (3/31, reading quiz) The Gauss Map and the Second Fundamental Form.
- Reading. Shifrin 2.2. pages 46-48
- After class: If the shape operator is the identity, the surface is a sphere. (Gradescope).

- (4/5, reading quiz) The Second Fundamental Form (2)
- Reading. Shifrin 2.2 pages 47-49.
- After class. Self-adjoint maps and symmetric matrices. (Gradescope).

- (Homework only.) A (lengthy) example.
- Reading. Shifrin 2.2 Example 6. pages 49-50.
- Problem Set. The fundamental example of surface theory. (Gradescope).
- Graduate reading: The Implicit and Inverse Function Theorems: Easy Proofs, Oliveira.

- (4/7, 4/12, reading quiz) Classification of Points and Meusnier’s Formula.
- Reading. Shifrin 2.2 pages 50-53.

- (4/14) The Codazzi and Gauss Equations.
- Reading. Shifrin 2.3 pages 57-59.

- (4/14) Codazzi and Gauss Equations (2)
- Reading. Shifrin 2.3 pages 59-60.

- (4/19, reading quiz) Global Geometry of (Compact) Surfaces
- Reading. Shifrin 2.3 pages 61-63.

- Recovering the Embedding of a Surface (skipped in Spring 2022 course)
- Reading. Shifrin 2.3 pages 63-64.

- (4/21, reading quiz) Covariant Differentiation and Parallel Transport
- Reading. Shifrin 2.4 pages 66-69

- (4/26, reading quiz) Geodesics as Straightest Paths
- Reading. Shifrin 2.4 pages 70-72

- Clairaut’s Relation (skipped in Spring 2022 course)
- Reading. Shifrin 2.4 pages 73-75.

- (4/28, reading quiz) Holonomy and Gauss-Bonnet (I).
- Reading. Shifrin 3.1 pages 79-82.

- (5/3, reading quiz) Holonomy and Gauss-Bonnet (II).
- Reading. Shifrin 3.1 pages 82-89.

### Course Evaluation.

The midterm exam will be held **on Thursday, March 17** in class. 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 course final will be held on **Tuesday, May 10 **at 3:30pm.

### 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.