MTG 6257, Spring 2011

Differential Geometry II

Professor David Groisser

Office: Little 308

Phone: 392-0281 ext. 261

Email: groisser@ufl.edu

Office Hours:Tentatively Monday and Friday 7th period (1:55-2:45), and Tuesday 4th period (10:40-11:30). 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.

Textbook:None required. However, please see the MTG 6256 course announcement for information about two recommended books (Boothby,An Introduction to Differentiable Manifolds and Riemannian Geometry, and Spivak,A Comprehensive Introduction to Differential Geometry,volumes 1 and 2), and for a link to some other references.

- Connections on vector bundles. Subtopics include curvature, parallel transport, and holonomy.
- Chern-Weil theory and characteristic classes
- Lie groups (and, possibly, homogeneous spaces)
- Principal fiber bundles. Subtopics include:

- reduction and enlargement of structure group
- associated vector bundles
- connections on principal bundles, with sub-subtopics

- curvature, parallel transport, and holonomy, and the induced structures on associated vector bundles
- holonomy groups
- Spin and Spin
^{c}groups, structures, and bundles- Geometry of surfaces in
R^{3}. Subtopics include principal curvatures, Gaussian curvature, and the Gauss-Bonnet Theorem.- Further topics in Riemannian geometry. Among these are: geodesics, Jacobi fields, Hopf-Rinow Theorem, and curvature-comparison theorems.
- Selected topics in geometric PDEs:

- the Laplacian on differential forms; Hodge theory
- Heat equation and heat kernel on a Riemannian manifold
- Gauge theory and the Yang-Mills equations
- Dirac operators and the Dirac equation
- Seiberg-Witten equations
- Various curvature-related equations on Riemanian manifolds
- Symplectic geometry and its relation to classical mechanics
- Introduction to complex manifolds and Kaehler manifolds
- Selected topics in differential topology, such as the Poincare-Hopf Theorem, degree theory, Sard's theorem and some applications (such as embedding theorems), and the Lefschetz Fixed-Point Theorem.
- Morse Theory

Prerequisite: Grade of C or better in MTG 6256 (taken Fall 2010), or permission of the instructor.

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.)

- Homework. I expect to assign and collect approximately 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.

Attendance:I work hard to prepare my lectures, and I expect students to attend all of them (and to arriveon time), with the usual allowances for illness, emergency, conference-travel, etc. When you must miss a class, please obtain notes from a classmate.

Homework rules:The rules from last semester apply (see here, from the start of the section "More about homework" to the end of the section "Student Honor Code"). It seems that some of these rules were not written explicitly enough:

- "
Leave enough space for me to write comments" means, among other things, that you should leave ample margins.Please leave margins of at least 1.25 inches at the top, bottom, and both sides of each page.- All parts of the instruction "
Staplethe sheets together in the upper left-hand corner" should be taken literally. In addition, the request, "Please do not make this process [grading] more burdensome than it intrinsically needs to be," means, among other things, that when you staple your pages together, you should make sure that the staple is positioned so that when I open your booklet to any page, the booklet lies flat and I can easily see everything you've written on that page. (If you've followed the instructions toleave adequate marginsand to put the staple in theupper left-hand corner, this should happen automatically. Contrapositively, if this is not happening, then you have not followed one of these instructions.)- Last semester, some students apparently didn't read or didn't understand the section of the syllabus concerning the Student Honor Code, or didn't read or understand the points that I said should be self-evident but wrote out explicitly, or didn't understand what plagiarism is and that it's a serious violation of University of Florida rules, or understood all these things but chose to violate the rules knowingly. These students received only a grade penalty. (If I made no copying/plagiarism-related comment to you last semester, verbally or on your homework, then you are not one of these students.) This semester I will have
ABSOLUTELY ZERO TOLERANCEfor such behavior, or for not following academic-honesty ruleswhether or not you understand them, or for not seeing me when you are told that you should see me. If you violate the letter or spirit of my rules or the University's, the penalty will beIMMEDIATE FAILURE OF THE COURSE, WITH A GRADE OF E, whether or not you understand what you did wrong. If you plagiarized last semester as well, I may take even harsher action.

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.