Syllabus and Course Information

MTG 6257, Spring 2013
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 Wednesday 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 syllabus
 for information about a recommended book (Boothby, An
Introduction to Differentiable Manifolds and Riemannian Geometry),
and for a link to some other references.

Course description: MTG 6257 is
the second semester of a year-long graduate sequence that introduces
the tools of differential geometry and differential topology. The
first part of the semester will be devoted to some basics that we
didn’t cover, or covered incompletely, including some basic Riemannian
geometry. The topics for the rest of the semester will be selected
based on input from students, with higher priority given to
the registered students. Some of the possible topics, several
of which are interdependent, are listed below.

  • Riemannian geometry (a more in-depth study of Riemannian
    manifolds): covariant derivatives, curvature, geodesics, and other
    features
  • Geometry of surfaces in R3. 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.
  • Connections on vector bundles. Subtopics include
    curvature, parallel transport, and holonomy.
  • 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
  • Chern-Weil theory and characteristic classes
  • 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, or permission of the instructor. Students who did not attend the first
semester of this course should obtain and read notes from someone who
did attend. Students who were not registered in the first semester
(whether or not they attended) should read the fall homework
assignments; in the spring I may make use of facts asserted in these
assignments. All the materials for the fall semester can be
found here.


Exams, Homework, and Grading: There will be no exams.
Your final grade will be determined by homework and attendance. I
expect to assign and collect approximately four 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.

Attendance: I work hard to prepare my lectures, and I
expect students to attend all of them (and to arrive on 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”). To emphasize some of these rules more
explicitly:

  • “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 “Staple the 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 to leave
    adequate margins     and to put the staple in the upper left-hand
    corner    , this should happen automatically. Contrapositively, if
    this is not happening, then you have not followed one of these
    instructions.)
  •  

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 this page.

Last update made by D. Groisser Fri Jan 4 18:44:34 EST 2013