Ralf Spatzier University of Michigan Department of Mathematics  


Time and Place: 1-2 pm MWF in EH 2866
Instructor: Ralf Spatzier, 5850 East Hall, 763-2192, spatzier@umich.edu
Office Hours: TBD and by appointment. Besides individual office hours, there will also be a discussion session, Thursdays at 6 pm in EH 3866.
Prerequisites Differential topology (basic theory of manifolds): Math 591 or equivalent
Course Outline: This course is an introduction to modern differential geometry. The main problem is how to describe the shape of a space.
We will discuss the following basic tools: affine connections and covariant derivatives, Riemannian metrics and the Levi-Civita connection, geodesics, curvature, Jacobi fields, Riemannian submanifolds, spaces of constant curvature, variational formulas, comparison theorems, Morse index theorem.
Later, we will discuss global problems: how is the shape of a space related to its topology? Can you have a metric of negative curvature on a sphere for example? This leads to structural results both in positive and negative curvature such as the classification of spaces of constant curvature, the Cartan-Hadamard Theorem, sphere theorems. Most exciting are various rigidity theorems that completely identify a Riemannian manifold up to isometry under mild assumptions. If time permits we will pursue more advanced topics.
Homework and Discussion Sessions: Weekly homework with discussion sessions in which you will present and discuss your solutions to the homework problems. You are welcome to work with others in class on the homework problems.
Final: The Final will be an oral exam based on a set of problems handed out at least two days before your exam.
Grading Policy: Your final grade will redetermined in one of two ways:
Choice Final: Your grade will entirely be determined by your performance on an oral Final Exam.
Choice Discussion: Your grade will be determined equally by:
          1: the oral Final Exam
          2: participation (both discussion and presentation) in the discussions sessions. You are expected to attend at least 8 discussion sessions.
          3: written solutions to selected problems from the homework. I will indicate those on each homework set.
Book: Do Carmo, Riemannian Geometry, Birkhauser, 2nd edition. The book is optional but highly recommended.
Disabilities Accommodations: Please ket me know as soon as possible if you need such. In particular, a Verified Individualized Services and Accommodations (VISA) form must be provided to me at least two weeks prior to the need for an accommodation for a graded event. The Services for Students with Disabilities (SSD) Office (G664 Haven Hall; http://ssd.umich.edu/) issues VISA forms.
Homeworks : Homework 1            Homework 2            Homework 3            Homework 4            Homework 5