Syllabus
MTG 6256, Fall 2012
Differential Geometry I

Professor David Groisser
Office: Little 308
Phone: 392-0281 ext. 261
Email: groisser@ufl.edu
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:

• Homework. Depending on the number of students enrolled, I expect to assign and collect from five to eight problem-sets over the course of the semester. I will grade some subset of the problems. How large that subset is will depend on how many students handed in the assignment, how successful they were solving the problems, and how well-written their solutions are.
• Take-home final exam.

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:

• Linear algebra, including inner-product spaces.
  This is essential. By linear algebra I don't just mean matrix algebra (although the basics of matrix algebra are still essential) or numerical linear algebra (working with large linear systems on a computer); I mean the general concepts and theory of vector spaces, linear transformations, and inner products. Differential geometry uses linear algebra probably more than any other field does, and it's important that you be comfortable with it. By the end of the course you will know a lot more about linear algebra than when you started, but you have to start at a reasonable level. If you are not clear on the difference between a matrix and a linear transformation, or if you have trouble working with a vector space that is not given to you explicitly as a subspace of Rⁿ, you are not ready to learn differential geometry. For those of you familiar with UF's linear algebra courses, a course similar to MAS 4105 would be a sufficient prerequisite (provided you did well and have retained what you learned); a course similar to MAS 3114 or MAD 6406 would not be sufficient.

• Advanced calculus. Although the topics below will be reviewed briefly, students entering the class should already be familiar with:
  • the derivative viewed as a linear transformation
  • the chain rule
  • the Inverse Function Theorem and Implicit Function Theorem
  • the fundamental existence/uniqueness theorem for (systems of) ordinary differential equations

• Basic point-set topology
  • metric spaces (familiarity with more general topological spaces would be helpful as well)
  • open and closed sets; compactness; connectedness; completeness
  • continuous maps and homeomorphisms
  These topics are reviewed briefly in this Point-Set Topology: Glossary and Review, which students should go through before the course starts.

• Algebra: basics of groups, rings, homomorphisms, and quotient constructions. I hesitate to call all of this a prerequisite, since much of it will be used only occasionally. However, when it is used, if you're unfamiliar with the foundational material I'm using, it will be up to you to learn what you need on your own.

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:

• The homework you hand in must be neat, and must either be typed (in which cased TeX or LaTeX is preferred) or written in pen (not pencil!). Please do not turn homework that is messy or that has anything that's been erased and written over (or written over without erasing), making it harder to read. Anything that is difficult for me to read will be returned to you ungraded.

• Leave enough space for me to write comments.

• Staple the sheets together in the upper left-hand corner. Any other means of attachment makes more work for me. The staple should be close enough to the corner that when I turn pages, nothing that you've written is obscured. Also, don't use paper that's been ripped out of a spiral-bound notebook; it will make a mess on my floor.

• Write in complete, unambiguous, grammatically correct, and correctly punctuated sentences, just as you would find in (most) math journals and textbooks.

• Partial proofs. If a problem is of the form "Prove this" and you've been unable to produce a complete proof, but want to show me how far you got, tell me at the very start of the problem that your proof is not complete (before you start writing any part of your attempted proof). Do not just start writing a proof, and at some point say "This is as far as I got." Otherwise, when I start reading I will assume that you think you've written a complete and correct proof, and spend too long thinking about, and writing comments on, false statements and approaches or steps that were doomed to go nowhere.

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:

• I assign homework problems because I want you to figure them out, not to send you on a treasure-hunt through the literature. If I limit myself to assigning problems that I think are unlikely to have solutions somewhere in some book, you will not be getting the best education I can give you. When I know that something is a worthwhile problem for you to work on, and even struggle with, I don't want to have to worry about whether a solution exists in some textbook.

  That does not mean you are forbidden ever to look at textbooks. But solutions to homework problems should be your own. If you find yourself looking at a textbook while you are writing up a solution, that solution is not your own.

• You should first try all the problems yourself (alone). After attempting the problems, you may brainstorm with other students in the class for general ideas, but you may not completely work out problems together. You are also not permitted to split the workload with other students, with each student in a group writing up some solutions that all group-members hand in, or that all group-members work from in writing up what they're going to hand in.

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 http://www.dso.ufl.edu/drc.

Letter grades and their grade-point equivalents at UF: see www.registrar.ufl.edu/catalog/policies/regulationgrades.html.