Differential Geometry I (MATH 622)

Homework 1 Homework 2 Homework 3 Homework 4 Homework 5

Lecture 1 Lecture 2 Lecture 3 Lecture 4 Lecture 5 Lecture 6 Lecture 7 Lecture 8 Lecture 9 Lecture 10 Lecture 11 Lecture 12 Lecture 13 Lecture 14 Lecture 15 Lecture 16 Lecture 17 Lecture 18 Lecture 19 Lecture 20 Lecture 21 Lecture 22 Lecture 23 Lecture 24 Lecture 25

An Unofficial Syllabus (updated 01/18/2021)

The administrative purpose of this course (together with Topology I MATH 636) is to prepare the students for the Geometry/Topology qualification exam. However, you will learn many mathematics beyond the scope of the qualification exam. We would like get you ready for learning more advanced courses and for discussing more serious mathematics with senior people.
The main focus of this course is to know the basics of "smooth manifolds." Roughly speaking, these are curved spaces on which one can do multivariable calculus. This course is very different from the undergraduate differential geometry course which considers curves and surfaces in the three-dimensional Euclidean space. Familiarity with point-set topology, multivariable calculus, and linear algebra is expected.
To meet the school's minimal requirement of formaity, there is the official website on Canvas where you can find the official syllabus. Feel free to make suggestions on utilizing the functions of Canvas.

Basic Information

  • Office Hour: Monday 2PM-3PM over Zoom (see Zoom meeting information in syllabus)

    Basic Coverage

  • The concept of smooth manifolds
  • Tensors and differential forms
  • de Rham cohomology
  • Vector bundles and connections
  • (optional) Riemannian metric

    • Textbooks and References

    This course does not have a canonical choice of textbook. The absolute necessary materials will be provided in the lecture notes. We will follow the narrative of Chern's book (see below). The other two books serve as references and source of exercises.
  • Chern, Chen, and Lam Lectures on Differential Geometry
    This was the book I used when I first learned differential geometry. Very concise but also elegant.
  • John Lee Introduction to Smooth Manifolds
    It is a very detailed, and in some sense, gentle, introduction to smooth manifolds. Also contains many exercises.
  • Michael Spivak A comprehensive introduction to differential geometry, volume 1
    More concise than Lee's book but follows the same narrative.

    • Expectation

    I expect students to commit a substantial amount of time and effort to self-study, especially reading the text (either one above or other appropriate ones) before the class. I expect students to have a reasonable understanding of point-set topology, a solid foundation on calculus (mathematical analysis) and linear algebra.

    About the Pandemic

    The course will be delivered over Zoom (or unlikely, but possibly some other platform). It is a challenge to both students and the instructor as online learning significantly limits the interaction among students and between students and the instructor. I strongly suggest students use all chances to talk to me or other professors in Geometry/Topology for advice (especially if you do not have an adivsor yet). I will try my best to provide opportunities to enhance the interaction.