Symplectic and Contact Geometry - Summer Tutorial 2003

Here is Gromov's symplectic camel.

Can you squeeze it through the eye of the needle
using only symplectic transformations?


1. McDuff-Salamon, exercises 1.12, 1.13, 1.15, 3.10 (due Mon 7/7)

2. McDuff-Salamon, exercises 2.1, 2.13, 2.16, 2.25 (due Thu 7/10)


Symplectic manifolds are an intermediate case between real and complex (Kahler) manifolds. The original motivation for studying them comes from physics: the phase space of a mechanical system, describing both position and momentum, is in the most general case a symplectic manifold. Symplectic manifolds still play an important role in recent topics in physics, such as string theory. They have also proved useful in understanding the structure of four-dimensional real manifolds. Contact manifolds are the odd-dimensional analogues of symplectic manifolds, and they appear naturally when one is interested in symplectic structures on manifolds with boundary.

This tutorial will serve as an introduction to the study of symplectic and contact manifolds. We will discuss Hamiltonian mechanics and how symplectic manifolds arise in physics. Then we will define symplectic, complex, almost complex, and contact structures on a manifold, and give lots of examples of each of them. Two important results that we will prove are: Darboux's theorem on the local triviality of symplectic and contact manifolds, and Martinet's theorem that every closed orientable 3-manifold admits a contact structure.


A solid knowledge of manifolds, differential forms, and vector fields, at the level of Math 25/55 or 135.


We will meet on Mondays and Thursdays in Science Center 216, from 7:15 pm to 8:45 pm. The first class is on July 3 and the last on August 11. In the first four weeks of the course, there will be lectures and optional homeworks. By late July each student should choose a particular topic on which to write a short paper (5-7 pages) and give an in-class presentation. The papers are due on the last day of classes. They can be used to fulfill your junior paper requirement.

Topics to be covered in the first part of the course:

Suggestions for project topics:


D. McDuff and D. Salamon, "Introduction to symplectic topology," Oxford University Press, New York, 1998.

We will also use material from some additional sources, such as: