MTG 6256, Fall 2012
Differential Geometry I

Professor David Groisser
Office: Little 308
Phone: 392-0281 ext. 261
Office Hours: Tentatively Tuesday 4th period (10:40-11:30), Wednesday 9th period (4:05-4:55), and Friday 5th period (11:45-12:35). Please come early in the period or let me know to expect you later; otherwise I may not stay in my office for the whole period. See my schedule for updates. Students who can't make scheduled office hours may see me by appointment on most weekdays (but never on a Thursday).

Textbook: None required. Recommended: William M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, either the second edition (1986) or the revised second edition (2003). The differences between the second edition and the revised second edition are minor.

Boothby is not a perfect book for MTG 6256–6257 (I've never found one); the ratio of elementary to advanced material is too large, and I may deviate quite a bit from the Boothby's order of topics. But the book is generally well-written, and gives a good review of the relevant prerequisites from advanced calculus.

Here is a longer list of references on manifolds.

Course description: MTG 6256–6257 is a year-long graduate sequence that introduces the tools of differential geometry and differential topology. In the first semester, we will focus primarily on the foundations of manifold theory. Topics will include a brief review of advanced calculus from a geometric viewpoint; definition and examples of manifolds (including Lie groups and submanifolds of Euclidean space); maps of manifolds; critical points and the Regular Value Theorem; Sard's Theorem (possibly deferred to spring); vector fields, flows, and Lie derivatives; exterior algebra and differential forms; Stokes' theorem and de Rham cohomology; vector bundles and tensor bundles (possibly deferred to spring); Riemannian metrics; introduction to Riemannian geometry; and introduction to connections and curvature.

Riemannian geometry and the theory of connections are large topics; we will not have time to go into depth in these areas in the first semester. They are potential topics for the second semester, along with several other possibilities mentioned in the course announcement.

Exams, Homework, and Grading: Your final grade will be determined by the following:

The weighting of each of these two components will be between 25% and 75%, and may vary from student to student. Generally I will give more than 50% weight to the component on which you've been more successful. (But that does not mean I will give 75% weight to it.)

See More about homework below.

Miscellaneous: One of the most important things you must do to learn the material is to go through your notes from each class before the next class, filling in any gaps, trying figuring out anything you didn't understand at the time, and determining what you still don't understand and should ask me about. Time permitting, the best thing you can do is rewrite your notes. Do not expect to understand everything I say in class at the time I say it. I will sometimes make comments that are intentionally cryptic, will sometimes deliberately omit some steps in proofs, etc., to force you to think more about something. The deepest understanding will come only when you think about the material on your own. This will take you far more time than the hours we spend together in class.

I work hard to prepare my lectures, and I expect all enrolled students to attend all of them, with the usual allowances for illness, emergency, etc. When you must miss a class, please obtain notes from a classmate.

Prerequisites for MTG 6256: Essentially, these can be summarized as "Everything you would learn as an undergraduate (pure-)math major". Differential geometry pulls together strands of all three major divisions of mathematics: analysis, algebra, and topology. Here are some particulars:

More about homework: Even when homework is well-written, reading and grading it is very time-consuming and physically difficult for your instructor. Please do not make this process more burdensome than it intrinsically needs to be. So:

Also, I think the following points should be self-evident, and I apologize to anyone who agrees that they're self-evident and is offended by my saying them. But past experience has taught me that I need to say them explicitly, even in 6000-level classes:

Student Honor Code: Students are expected to abide by the the Honor Code:

We, the members of the University of Florida community, pledge to hold ourselves and our peers to the highest standards of honesty and integrity.
On all work submitted for credit by students at the university, the following pledge is either required or implied: "On my honor, I have neither given nor received unauthorized aid in doing this assignment."

Accommodations for students with disabilities: Students requesting classroom accommodation must first register with the Dean of Students Office. The Dean of Students Office will provide documentation to the student who must then provide this documentation to the instructor when requesting accommodation. See

Letter grades and their grade-point equivalents at UF: see

Last update made by D. Groisser Sun Aug 5 19:36:18 EDT 2012