Bryant R. L. - Introduction to Lie groups and Symplectic Geometry.pdf - Geometry - lemoniada91

An Introduction to Lie Groups

and Symplectic Geometry

Aseriesofnine lectures on Lie groups and symplectic

geometry delivered at the Regional Geometry Institute in

Park City, Utah, 24 June–20 July 1991.

Robert L. Bryant

Duke University

Durham, NC

bryant@math.duke.edu

This is an uno cial version of the notes and was last modiﬁed

on 20 September 1993. The .dvi ﬁle 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

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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 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

which drove the development of the theory.

Ideliberately tried to include a few topics which 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 which are now becoming so importantinsymplectic geometry. However, a full

treatment of these topics in thespaceofnine lectures beginning at the elementary level

wasbeyond 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 anddelicate problems. My object was to

provide a path to understanding of the material which could be entered at several diﬀerent

levels and so the exercises vary greatly in diculty. 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.

Iwanttotake 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 myown,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

Background Material and Basic Terminology. In these lectures, I assume that

thereader 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.

TY denotes the induced mapping

on tangent bundles, with f ( x )denotingitsrestriction to T x X .(However,Ifollow tradition

when X = R and let f ( t )standfor f ( t )( ∂/∂t )forall t ∈ R .Itrustthat this abuse of

notation will not cause confusion.)

For any vector space V ,Igenerally use A p ( V )(insteadof, say, Λ p ( V ∗ )) to denote

the space of alternating (or exterior) p-forms on V .Forasmooth manifold M ,Idenote

the space of smooth, alternating p -forms on M by A

→

Y is a smooth mapping, then f : TX

→

p ( M ). The algebra of all (smooth)

diﬀerential forms on M is denoted by

A ∗ ( M ).

Igenerally reserve the letter d for the exterior derivative d : A

p +1 ( M ).

For any vector ﬁeld X on M ,Iwill denote left-hook with X (often called interior

product with X )bythesymbol X .Thisisthe graded derivation of degree

p ( M ) →A

A ∗ ( M )

which satisﬁes X ( df )= Xf for all smooth functions f on M .Forexample, the Cartan

formula for the Lie derivative of diﬀerential forms is written in the form

−

1of

L X 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 suce:

If f and g are two smooth maps from a manifold X m toamanifold Y n ,wesaythat

f and g agree toorderk at x

∈

X if, ﬁrst, f ( x )= g ( x )= y

∈

Y and, second, when

u : U → R

m and v : V → R

n are local coordinate systems centered on x and y respectively,

m up

to and including order k .UsingtheChain 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

◦

u − 1 and G = v

◦

u − 1 have the sameTaylorseries at 0

∈ R

≡ x,k g will mean that f and g agree to order k at x .Thisseasily

seen to deﬁne 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 )andthelinear map f ( x ): T x →

T f ( x ) Y .

I.2

The notation I use for smooth manifolds and mappings is fairly standard, but with a

few slight variations:

If f : X

the functions F = v

J k ( X, Y )isalsosmooth.

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 : TX

→

Y ,theobvious map j k ( f ): X

→

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

asmoothsubmersion, a p -form φ

∈A

p ( V )issaidtobe π-basic if it can be written in

p ( X )and π-semi-basic if, for any π -vertical*vector

ﬁeld X ,wehave X φ =0. Whenthe 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 x f − 1 ( y )=ker( f ( x ): T x X

→

T y Y )

Existence and Uniqueness of Solutions of ODE: If X is a vector ﬁeld on a

smooth manifold M ,thenthere exists an open neighborhood U of { 0 }×M in R × M and

asmoothmapping F : U

•

→

M with the following properties:

M .

ii. For each m ∈ M ,theslice U m =

∈

{t ∈ R | ( t, m ) ∈ U}

is an open interval in R

(containing 0) and the smoothmapping φ m : U m →

M deﬁned by φ m ( 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

⊂

U φ (0) and φ ( t )= φ φ (0) ( t )forall t

∈

I .

M ,thenwesaythat X is complete .

Two useful properties of this ﬂow are easy consequences of this existence and unique-

ness theorem. First, the interval U F ( t,m ) ⊂ R

R ×

is simply the interval U m translated by

−

t .

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

*Avector ﬁeld X is π -vertical with respect to a map π : V → X if and only if π X ( v ) =

0forall v

∈

I.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 )canbegiven a unique smooth manifold structure in such a way

that, for any smooth f : X

the form φ = π ∗ ( ϕ )forsome ϕ ∈A

i. F (0 ,m )= m for all m

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

of the ﬂow of X .If U =

The Simultaneous Flow-Box Theorem: If X 1 , X 2 , ... , X r are smooth vector

ﬁelds on M which satisfy the Lie bracket identities

•

[ X i ,X j ]=0

M is a point where the r vectors X 1 ( p ) ,X 2 ( p ) ,...,X r ( p )are

linearly independent in T p M ,thenthere exists a local coordinate system x 1 ,x 2 ,...,x n on

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

∈

X 1 = ∂

∂x 1 ,

2 = ∂

∂x 2 ,

...,

X r =

∂

∂x r .

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

fore describing them, we introduce an important concept.

Let M be a smoothmanifold and let E ⊂ TM be asmooth subbundle of rank p .We

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

E ,theirLiebracket[ X, Y ]isalsoasectionof E .

TM is

asmooth, integrable sub-bundle of rank r ,thenevery p in M has a neighborhood U on

which there exist local coordinates x 1 ,...,x r ,y 1 ,...,y n−r

•

The Local Frobenius Theorem: If M n

is a smooth manifold and E

⊂

so that the sections of E over

U are spanned by the vector ﬁelds

∂

∂x 1 ,

∂

∂x 2 ,

...,

∂

∂x r .

Associated to this local theorem is the following global version:

•

The Global Frobenius Theorem: Let M be a smoothmanifold and let E

⊂

be asmooth, integrable subbundle of rank r .Thenfor any p

∈

M ,thereexistsaconnected

r -dimensional submanifold L

⊂

M which contains p ,whichsatisﬁes T q L = E q for all q

∈

S ,

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

which contains p and satisﬁes T q L ⊂

E q 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

⊂

I.4

for all i and j ,andif p

TM a distribution on M , but I avoid this since

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

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