Math 5047 – Geometry/Topology III: Differential Geometry (Fall 2024)

Section information

  • Class time and location: Monday-Wednesday-Friday from 3:00PM to 3:50PM in January Hall 20
  • Office hours: Monday-Wednesday from 4:00PM to 5:00PM, Friday from 11:00AM to 1:00PM

Subject

This third semester in the Geometry/Topology qualifying sequence is dedicated to differential geometry, with a focus on Riemannian Geometry.

Text

I plan to follow

  • Introduction to Riemannian Manifolds by John M. Lee, 2nd Edition, 2018.

The electronic copy is available for download through the Olin library catalog.

In addition, I’ll make lecture notes available in some form.

A set of notes about tensor calculus and Riemannian geometry, still in rather rough but possibly useful state, can be found at the following address: first go to https://www.math.wustl.edu/~feres/Math5047Fall24/ (you will be greeted by the message ‘Forbidden.’) Now complete the address by adding ‘book_rough’ at the end: *****Math505Fall24/book_rough. Sorry about this roundabout procedure instead of simply providing a link. As this text is still in very preliminary form, I’m trying to avoid it being found by web scraping bots.

Topics we hope to cover.

We will cover, more or less systematically, up to chapter 10 of Lee’s textbook. The more fundamental topics will be covered in detail while others may be surveyed is less detail. If time allows, we will cover the Chern-Gauss-Bonnet theorem.

  1. Review of differentiable manifolds
    • Tangent and cotangent bundles
    • Vector fields
    • Immersions and embeddings
    • Orientation
    • Examples (including an overview of Lie groups and Lie algebras)
  2. Riemannian metrics
    • Definition and examples
  3. Connections
    • Affine connections
    • Parallel transport
    • Riemannian connections
    • Geodesics and the geodesic flow
    • Variational characterization of geodesics
  4. Curvature
    • The curvature tensor
    • Sectional curvature
    • The Gauss-Bonnet theorem
  5. Jacobi fields
    • The Jacobi equation
    • Conjugate points
    • The second variation formula
  6. Isometric immersions
    • The second fundamental form
    • The equations of Gauss, Ricci, Codazzi
  7. Complete manifolds (survey of results)
    • The Hopf-Rinow theorem
    • The Hadamard theorem
    • The fundamental group of manifolds of negative curvature
  8. Spaces of constant curvature (overview)
  9. The Chern-Gauss-Bonnet theorem (overview)

Coursework and grades

Coursework will consist of homework and reading assignments, one take-home mid-term test and one in-class final exam. Your final grade is determined as follows: homework 60%, midterm 20%, final exam 20%. The lowest grade you receive on homework will be dropped. Assignments will be handled through Gradescope and grades will be posted on Canvas.

Grade intervals may vary a bit but it will not be harsher than the following scale:

A+ A A- B+ B B- C+ C C- D F
TBA 90+ [85,90) [80,85) [75,80) [70,75) [65,70) [60,65) [55,60) [50,55) [0,50)

If you are a graduate student, a letter grade of B is required to pass; if you are an undergraduate taking this class Pass/Fail, you need a C- to pass.

In writing the tests, I will rely strongly on the homework assignments. To feel confident, you should master the topics and exercises covered by the assignments.

Schedule of tests and final

Test date
MT October 28 (due date for the take-home test)
Final December 16, 03:30PM-05:30PM (same place as classes)

Recording

I plan to record the in-person lectures using Zoom and make them available in Canvas.

Academic integrity

I will follow the University’s academic integrity policy, which you can read here. Work submitted under your name is expected to be your own. You are permitted, and encouraged, to collaborate on assignments. You may research broadly over the internet and in books when working on assignments, but please indicate your sources. (You won’t lose points for doing this kind of research!)

If you have any concerns or questions about this policy or academic integrity in class, please contact me.

Email

Please include Math 5047 in the subject line of any email message that pertains to this course.

Homework assignments

Weekly homework assignments will be posted on this course web page, in the below links, and the solutions will be posted in Canvas (under “Pages”).