Math 5047 – Geometry and Topology III (Fall 2020)

Section information

  • Class time and location: M-W-F from 3:00PM to 3:50PM (Zoom meetings)
  • Office hours: Thursdays from 1:00PM to 3:00PM

Subject

This is an introduction to the local and global geometry of Riemannian manifolds. By the end of the course, you will become familiar with the basic geometric concepts of parallel translation, geodesics, curvatures, Laplacians; will develop your computational skills in tensor calculus; will learn about topological invariants called characteristic classes and how to relate the manifold’s cohomology and its local geometry. If all goes according to plan, the course will culminate with the proof of the generalized Gauss-Bonnet theorem, which gives the Euler characteristic of a compact Riemannian manifold of even dimension as an integral of a differential form obtained from the manifold’s local curvature.

Text

The main text for the course will be

Differential Geometry: Connections, Curvature, and Characteristic Classes by Loring W. Tu, Springer Graduate Texts in Mathematics, 2017.

You can download the book’s pdf from this Olin Library link.

Other books may be used occasionally as sources of homework exercises, since Tu’s selection seems to be a bit thin, but you don’t need to have them. Here are two that I may use for this purpose:

Riemannian Geometry by Manfredo do Carmo, Birkhauser, 1992.

Analysis and Algebra on Differentiable Manifolds by Gadea, Masque and Mykytyuk, Springer, 2009.

This last book can be downloaded from this Olin Library link.

Course plan

Given the exceptional circumstances in which we are all living, we’ll have to be flexible about plans and how we implement them (both on the instructor’s and on the students’ side). But the basic idea is this: We will meet via Zoom at the official times of the course (MWF from 3:00PM-3:50PM) for lectures and discussions. The lectures will be recorded and made available on Canvas. I expect that we will cover the first 5 chapters of Loring Tu’s book.

Coursework, exam and grades

Coursework will consist of weekly homework assigments (I plan to have 11 assignments; the lowest assignment grade will be droped) and a take home (open book) final exam. The exam will be posted sometime late December and will be due January 4.

The final will be worth twice as much as a single homework assignment. So each of the 10 assignments will be worth 100/12% and the final 100/6% of the final score.

In grading the assignments and final exam, I will consider not only correctness, but how well they are written. You may lose points if your writing is careless, disorganized, or difficult to read.

I hope and expect that cumulative scores will be such that letter grades will be assigned according to the following scale:

  • A (-, plain, +): cumulative score in [90%, 100%]

  • B (-, plain, +): cumulative score in [80%, 90%)

  • C (-, plain, +): cumulative score in [65%, 80%)

  • D: cumulative score in [50%, 65%)

  • F: cumulative score less than 50%.

The cut-offs for the letter grade sign (-, plain, +) will be set at the very end of the course, when all the scores have been computed.

I may change these cut-off scores if I find it necessary, although no changes will be made that would result in a tougher scale than the above.

Good writing

Please write your assignments clearly and succintly. You’ll make your instructor a happier person if you type them in Latex, but you won’t get extra points for that. Handwritten assignments are fine. Just remember the fundamental rule of good writing: Never submit your first draft!

Academic integrity

I will follow the University’s academic integrity policy. 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. You will find my email address on my Home page (see the link on the orange bar at the top of this page).

Homework assignments

In addition to the pdf file, I’m making available the TeX file of each homework assignment in case you’d like to type your solutions and use it as template.