An Introduction to Lie Groups and Symplectic Geometry A series of nine lectures on Lie groups and symplectic geometry delivered at the Regional Geometry Institute in Park City, Utah, 24 June–20 July 1991. by Robert L. Bryant Duke University Durham, NC [email protected] This is an unofficial version of the notes and was last modified on 23 July 2018. Introduction These are the lecture notes for a short course entitled “Introduction to Lie groups and symplectic geometry” that I gave at the 1991 Regional Geometry Institute at Park City, Utah starting on 24 June and ending on 11 July. The course really was designed to be an introduction, aimed at an audience of stu- dents who were familiar with basic constructions in differential topology and rudimentary differential geometry, who wanted to get a feel for Lie groups and symplectic geometry. My purpose was not to provide an exhaustive treatment of either Lie groups, which would have been impossible even if I had had an entire year, or of symplectic manifolds, which has lately undergone something of a revolution. Instead, I tried to provide an introduction to what I regard as the basic concepts of the two subjects, with an emphasis on examples that drove the development of the theory. I deliberately tried to include a few topics that are not part of the mainstream subject, such as Lie’s reduction of order for differential equations and its relation with the notion of a solvable group on the one hand and integration of ODE by quadrature on the other. I also tried, in the later lectures to introduce the reader to some of the global methods that are now becoming so important in symplectic geometry. However, a full treatment of these topics in the space of nine lectures beginning at the elementary level was beyond my abilities. After the lectures were over, I contemplated reworking these notes into a comprehen- sive introduction to modern symplectic geometry and, after some soul-searching, finally decided against this. Thus, I have contented myself with making only minor modifications and corrections, with the hope that an interested person could read these notes in a few weeks and get some sense of what the subject was about. An essential feature of the course was the exercise sets. Each set begins with elemen- tary material and works up to more involved and delicate problems. My object was to provide a path to understanding of the material that could be entered at several different levels and so the exercises vary greatly in difficulty. Many of these exercise sets are obvi- ously too long for any person to do them during the three weeks the course, so I provided extensive hints to aid the student in completing the exercises after the course was over. I want to take this opportunity to thank the many people who made helpful sugges- tions for these notes both during and after the course. Particular thanks goes to Karen Uhlenbeck and Dan Freed, who invited me to give an introductory set of lectures at the RGI, and to my course assistant, Tom Ivey, who provided invaluable help and criticism in the early stages of the notes and tirelessly helped the students with the exercises. While the faults of the presentation are entirely my own, without the help, encouragement, and proofreading contributed by these folks and others, neither these notes nor the course would never have come to pass. I.1 2 Background Material and Basic Terminology. In these lectures, I assume that the reader is familiar with the basic notions of manifolds, vector fields, and differential forms. All manifolds will be assumed to be both second countable and Hausdorff. Also, unless I say otherwise, I generally assume that all maps and manifolds are C∞. Since it came up several times in the course of the course of the lectures, it is probably worth emphasizing the following point: A submanifold of a smooth manifold X is, by definition, a pair (S, f) where S is a smooth manifold and f: S X is a one-to-one immersion. In particular, f need not be an embedding. → The notation I use for smooth manifolds and mappings is fairly standard, but with a few slight variations: If f: X Y is a smooth mapping, then f ′: TX TY denotes the induced mapping → ′ → on tangent bundles, with f (x) denoting its restriction to TxX. (However, I follow tradition when X = R and let f ′(t) stand for f ′(t)(∂/∂t) for all t R. I trust that this abuse of notation will not cause confusion.) ∈ For any vector space V , I generally use Ap(V ) (instead of, say, Λp(V ∗)) to denote the space of alternating (or exterior) p-forms on V . For a smooth manifold M, I denote the space of smooth, alternating p-forms on M by p(M). The algebra of all (smooth) differential forms on M is denoted by ∗(M). A A I generally reserve the letter d for the exterior derivative d: p(M) p+1(M). A →A For any vector field X on M, I will denote left-hook with X (often called interior product with X) by the symbol X . This is the graded derivation of degree 1 of ∗(M) that satisfies X (df) = Xf for all smooth functions f on M. For example,− theA Cartan formula for the Lie derivative of differential forms is written in the form LX φ = X dφ + d(X φ). Jets. Occasionally, it will be convenient to use the language of jets in describing certain constructions. Jets provide a coordinate free way to talk about the Taylor expansion of some mapping up to a specified order. No detailed knowledge about these objects will be needed in these lectures, so the following comments should suffice: If f and g are two smooth maps from a manifold Xm to a manifold Y n, we say that f and g agree to order k at x X if, first, f(x) = g(x) = y Y and, second, when u: U Rm and v: V Rn are local∈ coordinate systems centered∈ on x and y respectively, the functions→ F = v →f u−1 and G = v g u−1 have the same Taylor series at 0 Rm up to and including order◦ k◦. Using the Chain◦ ◦ Rule, it is not hard to show that this conditi∈ on is independent of the choice of local coordinates u and v centered at x and y respectively. The notation f x,k g will mean that f and g agree to order k at x. This is easily ≡ k seen to define an equivalence relation. Denote the x,k-equivalence class of f by j (f)(x), and call it the k-jet of f at x. ≡ For example, knowing the 1-jet at x of a map f: X Y is equivalent to knowing both f(x) and the linear map f ′(x): T T Y . → x → f(x) I.2 3 The set of k-jets of maps from X to Y is usually denoted by J k(X,Y ). It is not hard to show that J k(X,Y ) can be given a unique smooth manifold structure in such a way that, for any smooth f: X Y , the obvious map jk(f): X J k(X,Y ) is also smooth. → → These jet spaces have various functorial properties that we shall not need at all. The main reason for introducing this notion is to give meaning to concise statements like “The critical points of f are determined by its 1-jet”, “The curvature at x of a Riemannian metric g is determined by its 2-jet at x”, or, from Lecture 8, “The integrability of an almost complex structure J: TX TX is determined by its 1-jet”. Should the reader wish to learn more about jets, I recommend→ the first two chapters of [GG]. Basic and Semi-Basic. Finally, I use the following terminology: If π: V X is a smooth submersion, a p-form φ p(V ) is said to be π-basic if it can be written→ in the form φ = π∗(ϕ) for some ϕ ∈ Ap(X) and π-semi-basic if, for any π-vertical*vector field X, we have X φ = 0. When∈ the A map π is clear from context, the terms “basic” or “semi-basic” are used. It is an elementary result that if the fibers of π are connected and φ is a p-form on V with the property that both φ and dφ are π-semi-basic, then φ is actually π-basic. At least in the early lectures, we will need very little in the way of major theorems, but we will make extensive use of the following results: The Implicit Function Theorem: If f: X Y is a smooth map of manifolds and y• Y is a regular value of f, then f −1(y) X →is a smooth embedded submanifold of X, with∈ ⊂ T f −1(y) = ker(f ′(x): T X T Y ) x x → y Existence and Uniqueness of Solutions of ODE: If X is a vector field on a smooth• manifold M, then there exists an open neighborhood U of 0 M in R M and a smooth mapping F : U M with the following properties: { } × × → i. F (0, m) = m for all m M. ∈ ii. For each m M, the slice U = t R (t, m) U is an open interval in R ∈ m { ∈ | ∈ } (containing 0) and the smooth mapping φm: Um M defined by φm(t) = F (t, m) is an integral curve of X.