Spring 2003 : MAT 569 Differential Geometry II

Instructor : Justin Sawon

Monday, Wednesday, and Friday 10:30-11:25am in PHY P128

My office is Math Tower 4-104 and you can either come to my office hours (Wednesday and Friday 9:20-10:20am, and Wednesday 1:30-2:30pm in MLC) or email me on sawon@math.sunysb.edu to make an appointment.

About the course

This course will aim to cover the `standard' results in Riemannian geometry. It is a continuation of the course MAT 568 : Differential Geometry I, taught by Santiago Simanca last semester. As such, we will assume familiarity with the material presented in that course.

This semester we will aim to cover the following:

• geodesics : the exponential map, distance minimizing properties, Jacobi fields, conjugate points,
• completeness, the Hopf-Rinow theorem, Hadamard's theorem,
• the energy functional, first and second variation, Bonnet-Myers theorem, Synge-Weinstein theorem, the Rauch comparison theorem and applications,
• isometric immersions, second fundamental form, the Gauss, Ricci, and Codazzi equations,
• space forms : spherical, Euclidean, and hyperbolic space,
• negative curvature : the fundamental group, Preissman's theorem,
• positive curvature : Toponogov's theorem, the sphere theorem,
• the Gauss map and the Gauss-Bonnet theorem,
• the Bochner method : Killing fields, Hodge theory, harmonic forms, Weitzenboeck formula,
• holonomy and symmetric spaces.

Prerequisites : you should have taken MAT 530 and MAT 531 (Geometry and Topology I and II) and MAT 568 Differential Geometry I. Please come to see me at the beginning of the semester if you haven't taken these courses but you'd still like to attend MAT 569 (for instance, if you've taken similar courses elsewhere).

The text-book for this course is :

Manfredo Perdigao do Carmo "Riemannian geometry" Birkhauser 1994.

It will be available from the campus bookstore. A copy will also be placed on two-hour loan in the Math/Physics/Astronomy Library, along with the following two reference books :

Peter Petersen "Riemannian geometry" Springer 1998.

Sigurdur Helgason "Differential geometry, Lie groups, and symmetric spaces" Academic Press 1978.

There won't be an examination for this course, but instead I will set about five problem sheets. These will be designed to supplement what is covered in class and will consist of both examples and general results. I strongly urge you to have a look at all of the problems : some of them may contain results which will be important in later lectures, and of course a wide collection of examples is essential to a good understand of the subject.

Note: If you have a physical, psychological, medical or learning disability that may impact on your ability to carry out assigned course work, I would urge that you contact the staff in the Disabled Student Services office (DSS), Room 133 Humanities, 632-6748/TDD. DSS will review your concerns and determine, with you, what accommodations are necessary and appropriate. All information and documentation of disability is confidential.

Summary of Chapter 4 - Curvature: postscript, pdf.

Exercise sheet one: postscript, pdf.

Please attempt four or five problems including problems six and seven, and submit your work during the class on Monday 17th February.

Exercise sheet two: postscript, pdf.

Please attempt five or six problems and submit your work during the class on Wednesday 5th March.

Exercise sheet three: postscript, pdf.

Please attempt four or five problems and submit your work during the class on Monday 24th March.

Exercise sheet four and suggested presentation topics: postscript, pdf.

Please attempt a total of ten problems from all four problem sheets and submit your work before the end of the semester. Alternatively, instead of five problems you may present something in class; this could be one of the presentation topics, another topic related to differential geometry, or even a solution to one of the problems.

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This page last modified by Justin Sawon
Sunday, 20-Apr-2003 10:09:42 MDT
Email corrections and comments to sawon@math.sunysb.edu