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

Instead of long homework sets, there are a series of “minihomework” assignments and readings which go with each class meeting. The purpose of minihomework is to give you an immediate couple of questions to try after each class (sometimes we’ll work on them in class) to get comfortable with the material.

The class schedule is updated on the MATH 4250 Google calendar. Please subscribe to get information on reading assignments, quiz dates, office hours, and the midterm exam.

We also have a course syllabus.

### Lecture Notes

- Parametrized curves, the dot product, cross product, and triple product.
- Reading: OpenStax physics chapter 2 (vectors and scalars)
- Minihomework: Vectors and products.
- Graduate material and homework

- Constructing Parametrized Curves.
- Arclength and Rectifiability.
- Minihomework:

- Finding the equation of a curve from an optimality condition.
- Introduction to the calculus of variations.
- Shortest paths.
- The tautochrone, isochrone, and brachistochrone.

- Framed curves and the Frenet frame.
- How to find the Frenet frame of a curve.
- How to compute curvature and torsion.

- More on Curvature and Torsion.
- The Tangent Indicatrix.
- Integralgeometric measure.
- Geometric Inequalities for Curves.
- The Fabricius-Bjerre Theorem.
- Review of linear algebra. Gradient and Hessian. Quadratic forms. Eigenvalues and Eigenvectors.
- Surfaces and the First Fundamental Form.
- Reading. Shifrin 2.1. (all pages)
- Minihomework: First Fundamental Form.
- Graduate material and minihomework: The intrinsic gradient.

- The Gauss Map and the Second Fundamental Form.
- Reading. Shifrin 2.2. pages 44-47
- Minihomework: Shape operator identity implies sphere.
- Graduate material and minihomework: None.

- 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.

### Homework Assignments.

### Grading Policy for Spring 2020.

In ordinary years, we’d go by the grading policy on the syllabus, and have a final exam and projects. This year, with the transition to online education amid the pandemic, that’s clearly not a reasonable expectation. So we’ll do it this way: I’ve posted minihomeworks and we have the original homework assignments, and the first midterm as well. I’ll grade from that and turn in grades (you’re welcome to submit more work until this Friday). Afterwards, if you don’t feel that the grade reflects your knowledge of the material, please contact me and we’ll arrange for an additional assessment by Zoom. (I’m also emailing this to everyone, but it doesn’t seem like everyone is getting the emails, so I’m putting it here too.)

~~Midterm and Final Exam.~~

~~The midterm and final exams will be open book timed exams during the scheduled finals time for the course. Here are some rules. You may bring up to 350 pages of text with you (that’s your copy of Shifrin, plus all the lecture notes, put about 100 pages of your handwritten notes or other material as you like).~~

~~I think that there has to be ~~**some** page limit in order to keep you from doing something painful and silly like photocopying Gradshteyn and Ryzhik in case you need an integral, but you’re welcome to petition me for a (classwide) increase of the page limit if you have a reasonable case that you need more space. In addition, you can bring any calculator permitted for the SAT Math Subject test, though I don’t think you’ll really need one.

~~Final paper (for students in 6250).~~

~~Students in 6250 have to complete a final paper as well as attend the final exam. The idea of reading a paper is that you want to work through the argument and fill in any missing pieces or any parts that seem unclear to you. This can require outside reading, consulting Wikipedia, working through examples, and talking with me and your other classmates. ~~*When you actually write things up, you must do so alone, and use only your own handwritten notes from your conversations with other students.* (It’s very easy to tell when you haven’t done this; if there is significant text overlap between student papers, this will be an academic honesty problem.) I think your writeup should be 5-10 pages long; if it seems like it’s going to be much longer or shorter, please come see me.

- 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.

### Additional Lecture 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.

Practice Exams.

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)

## Electronic Resources

- Poonen, List of errata in DoCarmo .
- Bishop, There is more than one way to frame a curve .
- Ghomi, h-Principles for Curves and Knots of Constant Curvature .