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.

Homework will be due at more-or-less weekly intervals. In addition, there will be a midterm exam and a 3 hour final exam. Graduate students will write a paper due at the end of semester, and will also sit for the final exam.

This is complicated material, and it’s worth the effort to read the book first before we go over the material in detail together. Hence, we’ll have reading assignments before we start each section of the book, and short quizzes in class to check reading comprehension before we start the lecture.

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

- Review of linear algebra and multivariable calculus.
- Arclength and Rectifiability.
- Framed curves and the Frenet frame.
- More on Curvature and Torsion.
- The Bishop Frame.
- 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.
- The Gauss Map and the Second Fundamental Form.
- The Second Fundamental Form (2)
- A (lengthy) Example.
- Classification of Points and Meusnier’s Formula.
- The Codazzi and Gauss Equations.
- Codazzi and Gauss Equations (2)
- Constructing Principal Coordinates
- Global Geometry of (Compact) Surfaces
- Recovering the Embedding of a Surface
- Cauchy’s uniqueness theorem. Polyhedral curvature.
- Domes, arches, and spans.
- Rigidity theory and the Maxwell-Cremona theorem.
- Geodesics from Calculus of Variations (shortest paths)
- The Tractrix
- Covariant Differentiation and Parallel Transport
- Geodesics as Straightest Paths
- Clairaut’s Relation
- Holonomy and Gauss-Bonnet (I).
- Holonomy and Gauss-Bonnet (II).

### Homework Assignments.

### 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.

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 .