Syllabus and course information

Differential Geometry II, Spring 2022
MTG 6257—Section 1958 (30152)
MAT 4930—Section 9401 (31980)
MWF 8th period, LIT 219

Link to class home page

Changes may be made to this document, and linked pages or files, before the semester starts. Some links may not work until January 5, 2022.

Instructor : Dr. David Groisser

Textbook: None.

   Tentative, approximate weekly schedule of lectures. Click here.

  • Lecture modality: face-to-face lectures (barring a rule-change by UF after the writing of this document, or a need for the instructor to quarantine or isolate).

    I expect students who contract COVID-19, or who have had close contact (as defined by the CDC) with a COVID-infected person, to follow the recommendations at https://www.cdc.gov/coronavirus/2019-ncov/your-health/quarantine-isolation.html.

    Since I'm required to give face-to-face lectures, allowing students to choose between remote and in-person participation (or offering a Zoom option for quarantined/isolated students), would turn this class into a hybrid-delivery course, the worst of all modalities. I will not teach in a hybrid format unless legally forced to. For any lectures that you miss, please obtain notes from a classmate. I will not post my lecture notes (other than occasional handouts), and I do not plan to record my lectures.

    Consistent with UF's rules, I also expect students to wear a mask at all times in this class, and hope that all are fully vaccinated against COVID-19.

  • Syllabus (course content): Integration on oriented manifolds; Stokes's Theorem; de Rham cohomology; flows and Lie derivatives; Riemannian metrics; introduction to Riemannian geometry (including geodesics and Riemannian curvature); vector bundles and tensor bundles; connections and curvature in greater generality. Depending on students' interests and how much time remains, we may cover additional topics, for which some possibilities are:
    Surfaces in \( {\bf R}^3\) and the Gauss-Bonnet theorem

    Principal bundles; connections on principal bundles and associated vector bundles

    Further study in Riemannian geometry (conjugate points on geodesics, Hopf-Rinow Theorem, curvature-comparison theorems, Morse index, ...)

    Lie groups and Lie algebras

    Elliptic PDE on manifolds and Hodge Theory

    Curvature and characteristic classes Introduction to complex and Kaehler manifolds

  • Course-grade components: There will be no exams. Your final grade will be determined entirely by homework, assuming your attendance is good. If your attendance is poor a grade penalty may be imposed. (No such penalty will be imposed if your reason for poor attendance is COVID-19-related, and you keep up with the lectures and work conscientiosly.)

  • Homework: I expect to assign and collect from four to eight problem-sets over the course of the semester. The problem-sets will include some problems that are mandatory and some that are optional. I will grade some subset of the mandatory problems. The cardinality of that subset will depend on how many students handed in the assignment, how successful they were solving the problems, and how well-written their solutions are.

    Doing well on the graded subset of the mandatory homeworkk problems will be enough to earn an A in the class, and I don't think that you'll find the mandatory problems excessive (if you've met the prerequisites for the course). However, to get the most out of the course, you should do as many of the optional problems as you can. The more time you put in, the more you will learn. My intent is to give students who want to learn a great deal the opportunity to do so, without requiring any of you to do a lot more work than you'd have to do in other 6000-level courses in this department.

    See More about homework below.

  • How to get the most out of lectures: To keep up with the lectures and to learn the material really well, my best recommendation—assuming you are comfortable attending class—is that you 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. (Sadly, I will also no doubt make some comments that are unintentionally cryptic, and some mistakes that I can't claim are intentional ...) The deepest understanding will come only when you think about the material on your own. This will take you a good deal more time than the hours we spend together in class.

  • Prerequisite for MTG 6257: MTG 6256, taken last semester.

  • 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:

    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 stating them. But I've learned through experience that I need to say them explicitly, even in 6000-level classes:

    Student Honor Code. UF students are bound by The Honor Pledge, which states:

    Religious Holidays. The following is part of the University of Florida Policy on Religious Holidays. "Students, upon prior notification of their instructors, shall be excused from class or other scheduled academic activity to observe a religious holy day of their faith."

    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.

    Teaching-evaluations. Students are expected to provide feedback on the quality of instruction in this course based on 10 criteria. These evaluations are conducted online at https://evaluations.ufl.edu. Evaluations are typically open during the last two or three weeks of the semester, but students will be given specific times when they are open. Summary results of these assessments are available to students at https://evaluations.ufl.edu/results.

    U Matter, We Care initiative: Your well-being is important to the University of Florida. The U Matter, We Care initiative is committed to creating a culture of care on our campus by encouraging members of our community to look out for one another and to reach out for help if a member of our community is in need. If you or a friend is in distress, please contact umatter@ufl.edu so that the U Matter, We Care Team can reach out to the student in distress. A nighttime and weekend crisis counselor is available by phone at 352-392-1575. The U Matter, We Care Team can help connect students to the many other helping resources available including, but not limited to, Victim Advocates, Housing staff, and the Counseling and Wellness Center. Please remember that asking for help is a sign of strength. In case of emergency, call 911.

    Contact information for the Counseling and Wellness Center: http://www.counseling.ufl.edu/cwc/Default.aspx, 392-1575. For emergencies, call the University Police Department (392-1111) or 911.

    Goals of course: For the student to master the course-content.