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

bryant@math.duke.edu

This is an uno¬cial version of the notes and was last modi¬ed

on 20 September 1993. The .dvi ¬le for this preprint will be available

by anonymous ftp from publications.math.duke.edu in the directory

bryant until the manuscript is accepted for publication. You should get

the ReadMe ¬le ¬rst to see if the version there is more recent than this

one.

Please send any comments, corrections or bug reports to the above

e-mail address.

Introduction

These are the lecture notes for a short course entitled “Introduction to Lie groups and

symplectic geometry” which 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 di¬erential topology and rudimentary

di¬erential 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

which drove the development of the theory.

I deliberately tried to include a few topics which are not part of the mainstream

subject, such as Lie™s reduction of order for di¬erential 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 which 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, ¬nally

decided against this. Thus, I have contented myself with making only minor modi¬cations

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 which could be entered at several di¬erent

levels and so the exercises vary greatly in di¬culty. 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 ¬elds, and di¬erential

forms. All manifolds will be assumed to be both second countable and Hausdor¬. 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

de¬nition, 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 : T X ’ T Y denotes the induced mapping

on tangent bundles, with f (x) denoting its restriction to Tx X. (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 Ap (M). The algebra of all (smooth)

di¬erential forms on M is denoted by A— (M).

I generally reserve the letter d for the exterior derivative d: Ap (M) ’ Ap+1(M).

For any vector ¬eld 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 A— (M)

which satis¬es X (df) = Xf for all smooth functions f on M. For example, the Cartan

formula for the Lie derivative of di¬erential 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 speci¬ed order. No detailed knowledge about these objects will

be needed in these lectures, so the following comments should su¬ce:

If f and g are two smooth maps from a manifold X m to a manifold Y n , we say that

f and g agree to order k at x ∈ X if, ¬rst, 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 condition

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

seen to de¬ne an equivalence relation. Denote the ≡x,k -equivalence class of f by j k (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): Tx ’ Tf (x) Y .

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 j k (f): X ’ J k (X, Y ) is also smooth.

These jet spaces have various functorial properties which 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 : T X ’ T X is determined by its 1-jet”. Should the reader wish to

learn more about jets, I recommend the ¬rst 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 ∈ Ap(V ) is said to be π-basic if it can be written in

the form φ = π — (•) for some • ∈ Ap (X) and π-semi-basic if, for any π-vertical*vector

¬eld X, we have X φ = 0. When the map π is clear from context, the terms “basic” or

“semi-basic” are used.

It is an elementary result that if the ¬bers 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

Tx f ’1 (y) = ker(f (x): Tx X ’ Ty Y )

• Existence and Uniqueness of Solutions of ODE: If X is a vector ¬eld 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 Um = {t ∈ R | (t, m) ∈ U} is an open interval in R

(containing 0) and the smooth mapping φm : Um ’ M de¬ned by φm (t) = F (t, m) is

an integral curve of X.

iii. ( Maximality ) If φ: I ’ M is any integral curve of X where I ‚ R is an interval

containing 0, then I ‚ Uφ(0) and φ(t) = φφ(0)(t) for all t ∈ I.

The mapping F is called the (local) ¬‚ow of X and the open set U is called the domain

of the ¬‚ow of X. If U = R — M, then we say that X is complete.

Two useful properties of this ¬‚ow are easy consequences of this existence and unique-

ness theorem. First, the interval UF (t,m) ‚ R is simply the interval Um translated by ’t.

Second, F (s + t, m) = F (s, F (t, m)) whenever t and s + t lie in Um .

* A vector ¬eld X is π-vertical with respect to a map π: V ’ X if and only if π X(v) =

0 for all v ∈ V

I.3 4

• The Simultaneous Flow-Box Theorem: If X1 , X2 , . . ., Xr are smooth vector

¬elds on M which satisfy the Lie bracket identities

[Xi , Xj ] = 0

for all i and j, and if p ∈ M is a point where the r vectors X1 (p), X2 (p), . . . , Xr (p) are

linearly independent in Tp M, then there exists a local coordinate system x1 , x2 , . . . , xn on

an open neighborhood U of p so that, on U,

‚ ‚ ‚

X1 = , X2 = , ..., Xr = .

‚x1 ‚x2 ‚xr

The Simultaneous Flow-Box Theorem has two particularly useful consequences. Be-

fore describing them, we introduce an important concept.

Let M be a smooth manifold and let E ‚ T M be a smooth subbundle of rank p. We

say that E is integrable if, for any two vector ¬elds X and Y on M which are sections of

E, their Lie bracket [X, Y ] is also a section of E.

• The Local Frobenius Theorem: If M n is a smooth manifold and E ‚ T M is

a smooth, integrable sub-bundle of rank r, then every p in M has a neighborhood U on

which there exist local coordinates x1 , . . . , xr , y 1 , . . . , y n’r so that the sections of E over

U are spanned by the vector ¬elds

‚ ‚ ‚

, , ..., .

‚x1 ‚x2 ‚xr

Associated to this local theorem is the following global version:

• The Global Frobenius Theorem: Let M be a smooth manifold and let E ‚ T M

be a smooth, integrable subbundle of rank r. Then for any p ∈ M, there exists a connected

r-dimensional submanifold L ‚ M which contains p, which satis¬es Tq L = Eq for all q ∈ S,

and which is maximal in the sense that any connected r -dimensional submanifold L ‚ M

which contains p and satis¬es Tq L ‚ Eq for all q ∈ L is a submanifold of L.

The submanifolds L provided by this theorem are called the leaves of the sub-bundle

E. (Some books call a sub-bundle E ‚ T M a distribution on M, but I avoid this since

“distribution” already has a well-established meaning in analysis.)

I.4 5

Contents

1. Introduction: Symmetry and Di¬erential Equations 7

First notions of di¬erential equations with symmetry, classical “integration methods.”

Examples: Motion in a central force ¬eld, linear equations, the Riccati equation, and

equations for space curves.

2. Lie Groups 12

Lie groups. Examples: Matrix Lie groups. Left-invariant vector ¬elds. The exponen-

tial mapping. The Lie bracket. Lie algebras. Subgroups and subalgebras. Classi¬ca-

tion of the two and three dimensional Lie groups and algebras.

3. Group Actions on Manifolds 38

Actions of Lie groups on manifolds. Orbit and stabilizers. Examples. Lie algebras

of vector ¬elds. Equations of Lie type. Solution by quadrature. Appendix: Lie™s

Transformation Groups, I. Appendix: Connections and Curvature.

4. Symmetries and Conservation Laws 61

Particle Lagrangians and Euler-Lagrange equations. Symmetries and conservation

laws: Noether™s Theorem. Hamiltonian formalism. Examples: Geodesics on Rie-

mannian Manifolds, Left-invariant metrics on Lie groups, Rigid Bodies. Poincar´

e

Recurrence.

5. Symplectic Manifolds, I 80

Symplectic Algebra. The structure theorem of Darboux. Examples: Complex Mani-

folds, Cotangent Bundles, Coadjoint orbits. Symplectic and Hamiltonian vector ¬elds.

Involutivity and complete integrability.

6. Symplectic Manifolds, II 100

Obstructions to the existence of a symplectic structure. Rigidity of symplectic struc-

tures. Symplectic and Lagrangian submanifolds. Fixed Points of Symplectomor-

phisms. Appendix: Lie™s Transformation Groups, II

7. Classical Reduction 116

Symplectic manifolds with symmetries. Hamiltonian and Poisson actions. The mo-

ment map. Reduction.

8. Recent Applications of Reduction 128

Riemannian holonomy. K¨hler Structures. K¨hler Reduction. Examples: Projective

a a

Space, Moduli of Flat Connections on Riemann Surfaces. HyperK¨hler structures and

a

reduction. Examples: Calabi™s Examples.

9. The Gromov School of Symplectic Geometry 147

The Soft Theory: The h-Principle. Gromov™s Immersion and Embedding Theorems.

Almost-complex structures on symplectic manifolds. The Hard Theory: Area esti-

mates, pseudo-holomorphic curves, and Gromov™s compactness theorem. A sample of

the new results.

I.1 6

Lecture 1:

Introduction: Symmetry and Di¬erential Equations

Consider the classical equations of motion for a particle in a conservative force ¬eld

x = ’grad V (x),

¨

where V : Rn ’ R is some function on Rn . If V is proper (i.e. the inverse image under V

of a compact set is compact, as when V (x) = |x|2 ), then, to a ¬rst approximation, V is the

potential for the motion of a ball of unit mass rolling around in a cup, moving only under

the in¬‚uence of gravity. For a general function V we have only the grossest knowledge of

how the solutions to this equation ought to behave.

Nevertheless, we can say a few things. The total energy (= kinetic plus potential) is

given by the formula E = 1 |x|2 + V (x) and is easily shown to be constant on any solution

2™

(just di¬erentiate E x(t) and use the equation). Since, V is proper, it follows that x

must stay inside a compact set V ’1 [0, E(x(0))] , and so the orbits are bounded. Without

knowing any more about V , one can show (see Lecture 4 for a precise statement) that

the motion has a certain “recurrent” behaviour: The trajectory resulting from “most”

initial positions and velocities tends to return, in¬nitely often, to a small neighborhood