Lectures on Symplectic Geometry

How does a student enter an exciting new field of mathematics? If they are lucky, they will happen upon a specialist in the field who is able to convey the beauty and the excitement of the topic. If they are a little less lucky, but still lucky, they will find a book on the subject that is able to transmit the same beauty and excitement, but through the medium of the written world.

When I began studying symplectic geometry, I was lucky enough to be in the former category. Were it only for the textbooks of the day, I’m not sure I would have been sucked in. Not that there weren’t some gems: Libermann and Marle’s classic Symplectic Geometry and Analytical Mechanics was the first of its kind and a go-to for me as a grad student, and the right chapters of Arnold’s Mathematical Method of Classical Mechanics could spark a flame.

I wish I had had access to Ana Cannas da Silva’s Lectures on Symplectic Geometry. Even today, 17 years after its original publication in 2001 (revised printing, 2008), I find this to be both the best introduction to symplectic geometry as well as a model for how to introduce any field of study.

What makes Cannas da Silva’s exposition such a good model? First, she quickly and efficiently identifies the “main players” in the field: symplectic manifolds as (necessarily even-dimensional) manifolds endowed with a closed, nondegenerate two-form, their automorphisms (the so-called symplectomorphisms), their local models as well as a “toy model” example (the cotangent bundle) which is simple but nontrivial. This part of the text (about the first 70 pages, nine chapters of 30) could have been titled, “What every beginner should know about symplectic geometry.” At the center of this is the proof of Darboux’s theorem, which states that every two symplectic manifolds of the same dimension are locally isomorphic. It is a pretty theorem, which she presents alongside its most natural generalizations. Cannas da Silva uses this section to showcase the so-called Moser method (or, as she says, the Moser “trick”) to prove the theorems. Here one witnesses the importance of the “closed” condition for a symplectic form.

The second ingredient in Cannas da Silva’s model text is the inclusion of carefully chosen examples. As I mentioned, her main example in the first part of the text is the cotangent bundle of a differentiable manifold. This is an excellent context to illustrate the notion of Lagrangian submanifolds, half-dimensional submanifolds which remarkably capture significant features of symplectic geometry and, as Cannas da Silva makes clear, are intimately related to symplectomorphisms.

In addition to this first example, Cannas da Silva spends Chapters 10 through 17 discussing what might on first reading seem to be tangential topics, namely contact, almost complex and complex structures. Contact manifolds are often described as odd-dimensional analogues to symplectic manifolds. As the author mentions, contact manifolds have their own local form, illustrating a pairing of coordinates with the exception of one distinguished one corresponding to a “normal” direction at the tangent space level. This gives contact geometry, and contact dynamics in particular, its own unique characteristics, and Cannas da Silva points out several research directions where these characteristics lead. But more to the central point of the text, contact manifolds offer a basis to construct new symplectic manifolds by a process known as “symplectization.”

Almost complex manifolds are those which have a “local” (i.e. at the level of the tangent space) complex structure; complex manifolds are almost complex manifolds whose local charts can be pieced together in a holomorphic way. Complex geometry is a huge topic of its own, for which Cannas da Silva admirably gives a short introduction. Her main goal, however, is to show off Kähler manifolds — complex manifolds which have an adapted symplectic form. Since Kähler manifolds arise naturally in algebraic geometry (complex projective spaces being prime examples), and since they have such rich structure, they provide important examples of symplectic manifolds with which any student of the field must become familiar.

A third important feature of Lectures on Symplectic Geometry is that the chapters build toward a non-trivial crescendo, in this case a collection of deep and profoundly geometric theorems related to symplectic quotients and moment maps. Before describing this part of the text, some explanation is in order. The quotient construction is ubiquitous in mathematics, and it is natural to try to consider the problem of constructing new symplectic manifolds as the quotient spaces (in a topological sense) of other manifolds. This project has some obstacles, however, in that the symplectic form should descend nicely to the quotient. In highly symmetric cases, such as when the symplectic manifold is acted upon by a Lie group in a Hamiltonian way (so-called Hamiltonian group actions are described in Chapter 21), these obstacles can be overcome. When considering the quotient of a manifold by a regular level set of the moment map associated to the Lie group action, the resulting “reduction” is a manifold — this is the content of the Marsden-Weinstein-Meyer theorem, which Cannas da Silva presents with full proof in Chapter 23. (The two previous chapters deal with the moment map construction, which the author describes as “a generalization of that of a Hamiltonian function.”

The final chapters concern toric symplectic manifolds — manifolds on which a torus acts in a Hamiltonian way. These special symplectic manifolds, as it turns out, can be completely classified by their “Delzant polytope” — this is the content of the 1988 Delzant Theorem. Toric manifolds and their associated polytopes are still active objects of study, and so Cannas da Silva’s text has rewarded the diligent reader by bringing them from an introductory level to within striking distance of active mathematical research. Any graduate student should be pleased.

One final ingredient in an excellent exposition is a wealth of relevant exercises. In this area, one feels the hand of a master in the text’s homework sets: concrete, illustrative, and enhancing the material presented. Cannas da Silva attributes several of the problem sets directly to Victor Guillemin. In fact, she credits Guillemin’s “masterful teaching of beautiful courses on topics related to symplectic geometry” as am inspiration for her own course and the resulting lecture notes.

Chapters 18 through 20 of the text cover “Hamiltonian mechanics,” and it would in fact be a disservice to the subject of symplectic geometry not to illustrate its roots in physics. Largely missing from the text is the subject of J-holomorphic curves, which, beginning with Gromov’s 1985 paper on the subject, revolutionized the field. This is not so much an omission as a choice of emphasis, since the texts that treat that subject require much more mathematics as background and require much more analysis to make progress. (Gromov’s paper is only mentioned in the last sentence of Homework Set 10!)

There are minor parts of this text where the reader clearly feels that they are reading lecture notes as opposed to polished prose. This is completely in keeping with the Springer Lecture Notes format, and it is to Cannas da Silva’s credit as an expositor that this provides no significant roadblock to understanding.

For an upper-level undergraduate or beginning graduate student, Lectures on Symplectic Geometry remains, in my opinion, an ideal starting point into an exciting, active and growing area of mathematics.

Andrew McInerney is Professor of Mathematics and Computer Science at Bronx Community College of the City University of New York. He is author of First Steps in Differential Geometry: Riemannian, Contact, Symplectic.