the function β(p, l) = l(p).

Conclude that the forms ω1 = d±1 and ω2 = d±2 satisfy ω1 = ’ω2 .

8. Let F : V ’ R be strictly convex. Assume that F has quadratic growth

at in¬nity so that SF = V — . Let ΛF be the graph of the Legendre trans-

form LF . ΛF is a lagrangian submanifold of V — V — with respect to the

symplectic form ω1 ; why? Hence, ΛF is also lagrangian for ω2 .

Let pr1 : ΛF ’ V and pr2 : ΛF ’ V — be the restrictions of the projection

maps V — V — ’ V and V — V — ’ V — , and let i : ΛF ’ V — V — be the

inclusion map. Show that

i— ±1 = d(pr1 )— F .

Conclude that

i— ±2 = d(i— β ’ (pr1 )— F ) = d(pr2 )— F — ,

and from this conclude that the inverse of the Legendre transform associ-

ated with F is the Legendre transform associated with F — .

Part VIII

Moment Maps

The concept of a moment map12 is a generalization of that of a hamiltonian

function. The notion of a moment map associated to a group action on a

symplectic manifold formalizes the Noether principle, which states that to every

symmetry (such as a group action) in a mechanical system, there corresponds

a conserved quantity.

21 Actions

21.1 One-Parameter Groups of Di¬eomorphisms

Let M be a manifold and X a complete vector ¬eld on M . Let ρt : M ’ M ,

t ∈ R, be the family of di¬eomorphisms generated by X. For each p ∈ M , ρt (p),

t ∈ R, is by de¬nition the unique integral curve of X passing through p at time

0, i.e., ρt (p) satis¬es

±

ρ0 (p) = p

dρt (p)

= X(ρt (p)) .

dt

Claim. We have that ρt —¦ ρs = ρt+s .

Proof. Let ρs (q) = p. We need to show that (ρt —¦ ρs )(q) = ρt+s (q), for all

t ∈ R. Reparametrize as ρt (q) := ρt+s (q). Then

˜

±

ρ0 (q) = ρs (q) = p

˜

d˜t (q)

ρ dρt+s (q)

= = X(ρt+s (q)) = X(˜t (q)) ,

ρ

dt dt

i.e., ρt (q) is an integral curve of X through p. By uniqueness we must have

˜

ρt (q) = ρt (p), that is, ρt+s (q) = ρt (ρs (q)).

˜

Consequence. We have that ρ’1 = ρ’t .

t

In terms of the group (R, +) and the group (Di¬(M ), —¦) of all di¬eomor-

phisms of M , these results can be summarized as:

Corollary 21.1 The map R ’ Di¬(M ), t ’ ρt , is a group homomorphism.

12 Souriau invented the french name “application moment.” In the US, East and West

coasts could be distinguished by the choice of translation: moment map and momentum map,

respectively. We will stick to the more economical version.

127

128 21 ACTIONS

The family {ρt | t ∈ R} is then called a one-parameter group of di¬eo-

morphisms of M and denoted

ρt = exp tX .

21.2 Lie Groups

De¬nition 21.2 A Lie group is a manifold G equipped with a group structure

where the group operations

G — G ’’ G G ’’ G

and

a ’’ a’1

(a, b) ’’ a · b

are smooth maps.

Examples.

• R (with addition13 ).

• S 1 regarded as unit complex numbers with multiplication, represents ro-

tations of the plane: S 1 = U(1) = SO(2).

• U(n), unitary linear transformations of Cn .

• SU(n), unitary linear transformations of Cn with det = 1.

• O(n), orthogonal linear transformations of Rn .

• SO(n), elements of O(n) with det = 1.

• GL(V ), invertible linear transformations of a vector space V .

™¦

De¬nition 21.3 A representation of a Lie group G on a vector space V is a

group homomorphism G ’ GL(V ).

21.3 Smooth Actions

Let M be a manifold.

De¬nition 21.4 An action of a Lie group G on M is a group homomorphism

ψ : G ’’ Di¬(M )

g ’’ ψg .

13 The operation will be omitted when it is clear from the context.

129

21.4 Symplectic and Hamiltonian Actions

(We will only consider left actions where ψ is a homomorphism. A right ac-

tion is de¬ned with ψ being an anti-homomorphism.) The evaluation map

associated with an action ψ : G ’ Di¬(M ) is

evψ : M — G ’’ M

(p, g) ’’ ψg (p) .

The action ψ is smooth if evψ is a smooth map.

Example. If X is a complete vector ¬eld on M , then

ρ : R ’’ Di¬(M )

t ’’ ρt = exp tX

™¦

is a smooth action of R on M .

Every complete vector ¬eld gives rise to a smooth action of R on M . Con-

versely, every smooth action of R on M is de¬ned by a complete vector ¬eld.

1’1

{complete vector ¬elds on M } ←’ {smooth actions of R on M}

’’

X exp tX

dψt (p)

←’

Xp = ψ

dt t=0

21.4 Symplectic and Hamiltonian Actions

Let (M, ω) be a symplectic manifold, and G a Lie group. Let ψ : G ’’ Di¬(M )

be a (smooth) action.

De¬nition 21.5 The action ψ is a symplectic action if

ψ : G ’’ Sympl(M, ω) ‚ Di¬(M ) ,

i.e., G “acts by symplectomorphisms.”

1’1

{complete symplectic vector ¬elds on M} ←’ {symplectic actions of R on M }

Example. On R2n with ω = ‚

dxi § dyi , let X = ’ ‚y1 . The orbits of the

action generated by X are lines parallel to the y1 -axis,

{(x1 , y1 ’ t, x2 , y2 , . . . , xn , yn ) | t ∈ R} .

Since X = Xx1 is hamiltonian (with hamiltonian function H = x1 ), this is

™¦

actually an example of a hamiltonian action of R.

130 21 ACTIONS

‚

Example. On S 2 with ω = dθ § dh (cylindrical coordinates), let X = ‚θ . Each

orbit is a horizontal circle (called a “parallel”) {(θ + t, h) | t ∈ R}. Notice that

all orbits of this R-action close up after time 2π, so that this is an action of S 1 :

ψ : S 1 ’’ Sympl(S 2 , ω)

t ’’ rotation by angle t around h-axis .

Since X = Xh is hamiltonian (with hamiltonian function H = h), this is an

example of a hamiltonian action of S 1 . ™¦

De¬nition 21.6 A symplectic action ψ of S 1 or R on (M, ω) is hamiltonian

if the vector ¬eld generated by ψ is hamiltonian. Equivalently, an action ψ of

S 1 or R on (M, ω) is hamiltonian if there is H : M ’ R with dH = ±X ω,

where X is the vector ¬eld generated by ψ.

What is a “hamiltonian action” of an arbitrary Lie group?

For the case where G = Tn = S 1 — . . . — S 1 is an n-torus, an action ψ : G ’

Sympl(M, ω) should be called hamiltonian when each restriction

: S 1 ’’ Sympl(M, ω)

ψ i := ψ|ith S 1 factor

is hamiltonian in the previous sense with hamiltonian function preserved by the

action of the rest of G.

When G is not a product of S 1 ™s or R™s, the solution is to use an upgraded

hamiltonian function, known as a moment map. Before its de¬nition though (in

Lecture 22), we need a little Lie theory.

21.5 Adjoint and Coadjoint Representations

Let G be a Lie group. Given g ∈ G let

Lg : G ’’ G

a ’’ g · a

be left multiplication by g. A vector ¬eld X on G is called left-invariant if

(Lg )— X = X for every g ∈ G. (There are similar right notions.)

Let g be the vector space of all left-invariant vector ¬elds on G. Together

with the Lie bracket [·, ·] of vector ¬elds, g forms a Lie algebra, called the Lie

algebra of the Lie group G.

Exercise. Show that the map

g ’’ Te G

X ’’ Xe

™¦

where e is the identity element in G, is an isomorphism of vector spaces.

131

21.5 Adjoint and Coadjoint Representations

Any Lie group G acts on itself by conjugation:

G ’’ Di¬(G)

ψg (a) = g · a · g ’1 .

g ’’ ψg ,

The derivative at the identity of

ψg : G ’’ G

a ’’ g · a · g ’1

is an invertible linear map Adg : g ’’ g. Here we identify the Lie algebra g with

the tangent space Te G. Letting g vary, we obtain the adjoint representation

(or adjoint action) of G on g:

’’ GL(g)

Ad : G

’’ Adg .

g

Exercise. Check for matrix groups that

d

∀X, Y ∈ g .

Adexp tX Y = [X, Y ] ,

dt t=0

Hint: For a matrix group G (i.e., a subgroup of GL(n; R) for some n), we have

Adg (Y ) = gY g ’1 , ∀g ∈ G , ∀Y ∈ g

and

[X, Y ] = XY ’ Y X , ∀X, Y ∈ g .

™¦

Let ·, · be the natural pairing between g— and g:

·, · : g— — g ’’ R

(ξ, X) ’’ ξ, X = ξ(X) .

Given ξ ∈ g— , we de¬ne Ad— ξ by

g

Ad— ξ, X = ξ, Adg’1 X , for any X ∈ g .

g

The collection of maps Ad— forms the coadjoint representation (or coadjoint

g

—

action) of G on g :

Ad— : G ’’ GL(g— )

g ’’ Ad— . g

We take g ’1 in the de¬nition of Ad— ξ in order to obtain a (left) represen-

g

tation, i.e., a group homomorphism, instead of a “right” representation, i.e., a

group anti-homomorphism.

Ad— —¦ Ad— = Ad— .

Adg —¦ Adh = Adgh ™¦

Exercise. Show that and g h gh

Homework 16: Hermitian Matrices

Let H be the vector space of n — n complex hermitian matrices.

The unitary group U(n) acts on H by conjugation: A·ξ = AξA’1 , for A ∈

U(n) , ξ ∈ H.

For each » = (»1 , . . . , »n ) ∈ Rn , let H» be the set of all n — n complex

hermitian matrices whose spectrum is ».

1. Show that the orbits of the U(n)-action are the manifolds H» .

For a ¬xed » ∈ Rn , what is the stabilizer of a point in H» ?

Hint: If »1 , . . . , »n are all distinct, the stabilizer of the diagonal matrix is

the torus Tn of all diagonal unitary matrices.

2. Show that the symmetric bilinear form on H, (X, Y ) ’ trace (XY ) , is

nondegenerate.

For ξ ∈ H, de¬ne a skew-symmetric bilinear form ωξ on u(n) = T1 U(n) =

iH (space of skew-hermitian matrices) by

X, Y ∈ iH .

ωξ (X, Y ) = i trace ([X, Y ]ξ) ,

Check that ωξ (X, Y ) = i trace (X(Y ξ ’ ξY )) and Y ξ ’ ξY ∈ H.

Show that the kernel of ωξ is Kξ := {Y ∈ u(n) | [Y, ξ] = 0}.

3. Show that Kξ is the Lie algebra of the stabilizer of ξ.

Di¬erentiate the relation AξA’1 = ξ.

Hint:

Show that the ωξ ™s induce nondegenerate 2-forms on the orbits H» .

Show that these 2-forms are closed.

Conclude that all the orbits H» are compact symplectic manifolds.

4. Describe the manifolds H» .

When all eigenvalues are equal, there is only one point in the orbit.

Suppose that »1 = »2 = . . . = »n . Then the eigenspace associated with

»1 is a line, and the one associated with »2 is the orthogonal hyperplane.

CPn’1 . We have thus exhib-

Show that there is a di¬eomorphism H»

ited a lot of symplectic forms on CPn’1 , on for each pair of distinct real

numbers.

What about the other cases?

Hint: When the eigenvalues »1 < . . . < »n are all distinct, any element in

H» de¬nes a family of pairwise orthogonal lines in Cn : its eigenspaces.

5. Show that, for any skew-hermitian matrix X ∈ u(n), the vector ¬eld on H

#

generated by X ∈ u(n) for the U(n)-action by conjugation is Xξ = [X, ξ].

132

22 Hamiltonian Actions

22.1 Moment and Comoment Maps

Let

(M, ω) be a symplectic manifold,

G a Lie group, and

ψ : G ’ Sympl(M, ω) a (smooth) symplectic action, i.e., a group homomorphism

such that the evaluation map evψ (g, p) := ψg (p) is smooth.

Case G = R:

We have the following bijective correspondence:

1’1

{symplectic actions of R on M} ←’ {complete symplectic vector ¬elds on M}

dψt (p)

’’

ψ Xp = dt

←’

ψ = exp tX X

“¬‚ow of X” “vector ¬eld generated by ψ”

The action ψ is hamiltonian if there exists a function H : M ’ R such that

dH = ±X ω where X is the vector ¬eld on M generated by ψ.

Case G = S 1 :

An action of S 1 is an action of R which is 2π-periodic: ψ2π = ψ0 . The

S 1 -action is called hamiltonian if the underlying R-action is hamiltonian.

General case:

Let

(M, ω) be a symplectic manifold,

G a Lie group,

g the Lie algebra of G,

g— the dual vector space of g, and

ψ : G ’’ Sympl(M, ω) a symplectic action.

De¬nition 22.1 The action ψ is a hamiltonian action if there exists a map

µ : M ’’ g—

satisfying:

1. For each X ∈ g, let

133

134 22 HAMILTONIAN ACTIONS

• µX : M ’ R, µX (p) := µ(p), X , be the component of µ along X,

• X # be the vector ¬eld on M generated by the one-parameter subgroup

{exp tX | t ∈ R} ⊆ G.

Then

dµX = ±X # ω

i.e., µX is a hamiltonian function for the vector ¬eld X # .

2. µ is equivariant with respect to the given action ψ of G on M and the

coadjoint action Ad— of G on g— :

µ —¦ ψg = Ad— —¦ µ , for all g ∈ G .

g

The vector (M, ω, G, µ) is then called a hamiltonian G-space and µ is a mo-

ment map.

For connected Lie groups, hamiltonian actions can be equivalently de¬ned

in terms of a comoment map

µ— : g ’’ C ∞ (M ) ,

with the two conditions rephrased as:

1. µ— (X) := µX is a hamiltonian function for the vector ¬eld X # ,

2. µ— is a Lie algebra homomorphism:

µ— [X, Y ] = {µ— (X), µ— (Y )}

where {·, ·} is the Poisson bracket on C ∞ (M ).

These de¬nitions match the previous ones for the cases G = R, S 1 , torus,

where equivariance becomes invariance since the coadjoint action is trivial.

Case G = S 1 (or R):

R, g— R. A moment map µ : M ’’ R satis¬es:

Here g

1. For the generator X = 1 of g, we have µX (p) = µ(p) · 1, i.e., µX = µ, and

X # is the standard vector ¬eld on M generated by S 1 . Then dµ = ±X # ω.

2. µ is invariant: LX # µ = ±X # dµ = 0.

Case G = Tn = n-torus:

R n , g— Rn . A moment map µ : M ’’ Rn satis¬es:

Here g

1. For each basis vector Xi of Rn , µXi is a hamiltonian function for Xi# .

2. µ is invariant.

135

22.2 Orbit Spaces

22.2 Orbit Spaces

Let ψ : G ’ Di¬(M ) be any action.

De¬nition 22.2 The orbit of G through p ∈ M is {ψg (p) | g ∈ G}

The stabilizer (or isotropy) of p ∈ M is the subgroup Gp := {g ∈ G |

ψg (p) = p}.

Exercise. If q is in the orbit of p, then Gq and Gp are conjugate subgroups. ™¦

De¬nition 22.3 We say that the action of G on M is . . .

• transitive if there is just one orbit,

• free if all stabilizers are trivial {e},

• locally free if all stabilizers are discrete.

Let ∼ be the orbit equivalence relation; for p, q ∈ M ,

p∼q ⇐’ p and q are on the same orbit.

The space of orbits M/ ∼ = M/G is called the orbit space. Let

π : M ’’ M/G

p ’’ orbit through p

be the point-orbit projection.

Topology of the orbit space:

We equip M/G with the weakest topology for which π is continuous, i.e.,

U ⊆ M/G is open if and only if π ’1 (U) is open in M . This is called the

quotient topology. This topology can be “bad.” For instance:

Example. Let G = R act on M = R by

t ’’ ψt = multiplication by et .

There are three orbits R+ , R’ and {0}. The point in the three-point orbit space

corresponding to the orbit {0} is not open, so the orbit space with the quotient

™¦

topology is not Hausdor¬.

Example. Let G = C\{0} act on M = Cn by

» ’’ ψ» = multiplication by » .

The orbits are the punctured complex lines (through non-zero vectors z ∈ Cn ),

plus one “unstable” orbit through 0, which has a single point. The orbit space

is

M/G = CPn’1 {point} .

136 22 HAMILTONIAN ACTIONS

The quotient topology restricts to the usual topology on CPn’1 . The only open

set containing {point} in the quotient topology is the full space. Again the

quotient topology in M/G is not Hausdor¬.

However, it su¬ces to remove 0 from Cn to obtain a Hausdor¬ orbit space:

CPn’1 . Then there is also a compact (yet not complex) description of the orbit

space by taking only unit vectors:

C\{0} = S 2n’1 /S 1 .

CPn’1 = Cn \{0}

™¦

22.3 Preview of Reduction

i

dzi § d¯i = dxi § dyi = ri dri § dθi be the standard symplectic

Let ω = 2 z

form on C . Consider the following S 1 -action on (Cn , ω):

n

t ∈ S 1 ’’ ψt = multiplication by eit .

ψ is hamiltonian with moment map

µ : Cn ’’ R

2

’ |z| + constant

’’

z 2

since

1 2

dµ = ’ 2 d( ri )

‚ ‚ ‚

X# = + +...+

‚θ1 ‚θ2 ‚θn

1 2

=’ ri dri = ’ 2

±X # ω dri .

If we choose the constant to be 1 , then µ’1 (0) = S 2n’1 is the unit sphere. The

2

orbit space of the zero level of the moment map is

µ’1 (0)/S 1 = S 2n’1 /S 1 = CPn’1 .

CPn’1 is thus called a reduced space. Notice also that the image of the

moment map is half-space.

These particular observations are related to major theorems:

Under assumptions (explained in Lectures 23-29),

• [Marsden-Weinstein-Meyer] reduced spaces are symplectic manifolds;

• [Atiyah-Guillemin-Sternberg] the image of the moment map is a convex

polytope;

• [Delzant] hamiltonian Tn -spaces are classi¬ed by the image of the moment

map.

137

22.4 Classical Examples

22.4 Classical Examples

Example.

Let G = SO(3) = {A ∈ GL(3; R) | A t A = Id and detA = 1}. Then

g = {A ∈ gl(3; R) | A + At = 0} is the space of 3 — 3 skew-symmetric matrices

and can be identi¬ed with R3 . The Lie bracket on g can be identi¬ed with the

exterior product via

®

’a3 a2

0

’a1 » ’’ ’ = (a1 , a2 , a3 )

’

A = ° a3 0 a

’a2 a1 0

’ — ’.

’’

[A, B] = AB ’ BA ’’ a b

Exercise. Under the identi¬cations g, g— R3 , the adjoint and coadjoint ac-

tions are the usual SO(3)-action on R3 by rotations. ™¦

Therefore, the coadjoint orbits are the spheres in R3 centered at the origin.

™¦

Homework 17 shows that coadjoint orbits are symplectic.

The name “moment map” comes from being the generalization of linear and

angular momenta in classical mechanics.

Translation: Consider R6 with coordinates x1 , x2 , x3 , y1 , y2 , y3 and symplectic

form ω = dxi § dyi . Let R3 act on R6 by translations:

’ ∈ R3 ’’ ψ’ ∈ Sympl(R6 , ω)

’ ’

a a

ψ’ (’, ’) = (’ + ’, ’) .

’’ ’ ’ ’’

axy x ay

Then X # = a1 ‚x1 + a2 ‚x2 + a3 ‚x3 for X = ’, and

’

‚ ‚ ‚

a

µ(’, ’) = ’

’’ ’

µ : R6 ’’ R3 , xy y

is a moment map, with

’’ ’

’

µ a (’, ’) = µ(’, ’), ’ = ’ · ’ .

’’ ’ ’’

xy xy a ya

Classically, ’ is called the momentum vector corresponding to the position

’

y

’, and the map µ is called the linear momentum.

’

vector x

Rotation: The SO(3)-action on R3 by rotations lifts to a symplectic action ψ

on the cotangent bundle R6 . The in¬nitesimal version of this action is

’ ∈ R3

’ ’’ dψ(’) ∈ χsympl (R6 )

’

a a

dψ(’)(’, ’) = (’ — ’, ’ — ’) .

’ ’’ ’ ’’ ’

a xy a xa y

138 22 HAMILTONIAN ACTIONS

Then

µ(’, ’) = ’ — ’

’’ ’’

µ : R6 ’’ R3 , xy x y

is a moment map, with

’’ ’

’

µ a (’, ’) = µ(’, ’), ’ = (’ — ’) · ’.

’’ ’ ’’’

xy xy a x y a

The map µ is called the angular momentum.

Homework 17: Coadjoint Orbits

Let G be a Lie group, g its Lie algebra and g— the dual vector space of g.

1. Let g X # be the vector ¬eld generated by X ∈ g for the adjoint represen-

tation of G on g. Show that

#

∀Y ∈g.

g

XY = [X, Y ]

2. Let X # be the vector ¬eld generated by X ∈ g for the coadjoint represen-

tation of G on g— . Show that

#

∀Y ∈g.

Xξ , Y = ξ, [Y, X]

3. For any ξ ∈ g— , de¬ne a skew-symmetric bilinear form on g by

ωξ (X, Y ) := ξ, [X, Y ] .

Show that the kernel of ωξ is the Lie algebra gξ of the stabilizer of ξ for

the coadjoint representation.

4. Show that ωξ de¬nes a nondegenerate 2-form on the tangent space at ξ to

the coadjoint orbit through ξ.

5. Show that ωξ de¬nes a closed 2-form on the orbit of ξ in g— .

Hint: The tangent space to the orbit being generated by the vector ¬elds

X # , this is a consequence of the Jacobi identity in g.

This canonical symplectic form on the coadjoint orbits in g— is also

known as the Lie-Poisson or Kostant-Kirillov symplectic structure.

6. The Lie algebra structure of g de¬nes a canonical Poisson structure on g— :

{f, g}(ξ) := ξ, [dfξ , dgξ ]

for f, g ∈ C ∞ (g— ) and ξ ∈ g— . Notice that dfξ : Tξ g— g— ’ R is identi¬ed

with an element of g g—— .

Check that {·, ·} satis¬es the Leibniz rule:

{f, gh} = g{f, h} + h{f, g} .

7. Show that the jacobiator

J(f, g, h) := {{f, g}, h} + {{g, h}, f } + {{h, f }, g}

is a trivector ¬eld, i.e., J is a skew-symmetric trilinear map C ∞ (g— ) —

C ∞ (g— ) — C ∞ (g— ) ’ C ∞ (g— ), which is a derivation in each argument.

Being a derivation amounts to the Leibniz rule from exercise 6.

Hint:

8. Show that J ≡ 0, i.e., {·, ·} satis¬es the Jacobi identity.

Hint: Follows from the Jacobi identity for [·, ·] in g. It is enough to check on

coordinate functions.

139

Part IX

Symplectic Reduction

The phase space of a system of n particles is the space parametrizing the position

and momenta of the particles. The mathematical model for the phase space is

a symplectic manifold. Classical physicists realized that, whenever there is a

symmetry group of dimension k acting on a mechanical system, then the number

of degrees of freedom for the position and momenta of the particles may be

reduced by 2k. Symplectic reduction formulates this feature mathematically.

23 The Marsden-Weinstein-Meyer Theorem

23.1 Statement

Theorem 23.1 (Marsden-Weinstein-Meyer [76, 84]) Let (M, ω, G, µ) be

a hamiltonian G-space for a compact Lie group G. Let i : µ’1 (0) ’ M be the

inclusion map. Assume that G acts freely on µ’1 (0). Then

• the orbit space Mred = µ’1 (0)/G is a manifold,

• π : µ’1 (0) ’ Mred is a principal G-bundle, and

• there is a symplectic form ωred on Mred satisfying i— ω = π — ωred .

De¬nition 23.2 The pair (Mred , ωred ) is called the reduction of (M, ω) with

respect to G, µ, or the reduced space, or the symplectic quotient, or the

Marsden-Weinstein-Meyer quotient, etc.

Low-brow proof for the case G = S 1 and dim M = 4.

In this case the moment map is µ : M ’ R. Let p ∈ µ’1 (0). Choose local

coordinates:

• θ along the orbit through p,

• µ given by the moment map, and

• ·1 , ·2 pullback of coordinates on µ’1 (0)/S 1 .

Then the symplectic form can be written

ω = A dθ § dµ + Bj dθ § d·j + Cj dµ § d·j + D d·1 § d·2 .

‚

Since dµ = ± ω, we must have A = 1, Bj = 0. Hence,

‚θ

ω = dθ § dµ + Cj dµ § d·j + D d·1 § d·2 .

Since ω is symplectic, we must have D = 0. Therefore, i— ω = D d·1 § d·2 is the

pullback of a symplectic form on Mred.

The actual proof of the Marsden-Weinstein-Meyer theorem requires the follow-

ing ingredients.

141

142 23 THE MARSDEN-WEINSTEIN-MEYER THEOREM

23.2 Ingredients

1. Let gp be the Lie algebra of the stabilizer of p ∈ M . Then dµp : Tp M ’ g—

has

ker dµp = (Tp Op )ωp

im dµp = g0 p

where Op is the G-orbit through p, and g0 = {ξ ∈ g— | ξ, X = 0, ∀X ∈

p

gp } is the annihilator of gp .

#

Proof. Stare at the expression ωp (Xp , v) = dµp (v), X , for all v ∈ Tp M

and all X ∈ g, and count dimensions.

Consequences:

• The action is locally free at p

⇐’ gp = {0}

⇐’ dµp is surjective

⇐’ p is a regular point of µ.

• G acts freely on µ’1 (0)

=’ 0 is a regular value of µ

=’ µ’1 (0) is a closed submanifold of M

of codimension equal to dim G.

• G acts freely on µ’1 (0)

=’ Tp µ’1 (0) = ker dµp (for p ∈ µ’1 (0))

=’ Tp µ’1 (0) and Tp Op are symplectic orthocomplements in Tp M .

In particular, the tangent space to the orbit through p ∈ µ’1 (0) is

an isotropic subspace of Tp M . Hence, orbits in µ’1 (0) are isotropic.

Since any tangent vector to the orbit is the value of a vector ¬eld gen-

erated by the group, we can con¬rm that orbits are isotropic directly by

computing, for any X, Y ∈ g and any p ∈ µ’1 (0),

ωp (Xp , Yp# )

#

= hamiltonian function for [Y # , X # ] at p

= hamiltonian function for [Y, X]# at p

= µ[Y,X] (p) = 0 .

2. Lemma 23.3 Let (V, ω) be a symplectic vector space. Suppose that I

is an isotropic subspace, that is, ω|I ≡ 0. Then ω induces a canonical

symplectic form „¦ on I ω /I.

Proof. Let u, v ∈ I ω , and [u], [v] ∈ I ω /I. De¬ne „¦([u], [v]) = ω(u, v).

• „¦ is well-de¬ned:

∀i, j ∈ I .

ω(u + i, v + j) = ω(u, v) + ω(u, j) + ω(i, v) + ω(i, j) ,

0 0 0

143

23.2 Ingredients

• „¦ is nondegenerate:

Suppose that u ∈ I ω has ω(u, v) = 0, for all v ∈ I ω .

Then u ∈ (I ω )ω = I, i.e., [u] = 0.

3. Theorem 23.4 If a compact Lie group G acts freely on a manifold M ,

then M/G is a manifold and the map π : M ’ M/G is a principal G-

bundle.

Proof. We will ¬rst show that, for any p ∈ M , the G-orbit through p is

a compact embedded submanifold of M di¬eomorphic to G.

Since the action is smooth, the evaluation map ev : G—M ’ M , ev(g, p) =

g · p, is smooth. Let evp : G ’ M be de¬ned by evp (g) = g · p. The map

evp provides the embedding we seek:

The image of evp is the G-orbit through p. Injectivity of evp follows from

the action of G being free. The map evp is proper because, if A is a

compact, hence closed, subset of M , then its inverse image (ev p )’1 (A),

being a closed subset of the compact Lie group G, is also compact. It

remains to show that evp is an immersion. For X ∈ g Te G, we have

#

d(evp )e (X) = 0 ⇐’ Xp = 0 ⇐’ X = 0 ,

as the action is free. We conclude that d(ev p )e is injective. At any other

point g ∈ G, for X ∈ Tg G, we have

d(evp )g (X) = 0 ⇐’ d(evp —¦ Rg )e —¦ (dRg’1 )g (X) = 0 ,

where Rg : G ’ G is right multiplication by g. But evp —¦ Rg = evg·p has

an injective di¬erential at e, and (dRg’1 )g is an isomorphism. It follows

that d(evp )g is always injective.

Exercise. Show that, even if the action is not free, the G-orbit through

p is a compact embedded submanifold of M . In that case, the orbit is

di¬eomorphic to the quotient of G by the isotropy of p: Op G/Gp . ™¦

Let S be a transverse section to Op at p; this is called a slice. Choose a

coordinate system x1 , . . . , xn centered at p such that

Op G : x1 = . . . = xk =0

S : xk+1 = . . . = xn = 0.

Let Sµ = S © Bµ (0, Rn ) where Bµ (0, Rn ) is the ball of radius µ centered

at 0 in Rn . Let · : G — S ’ M , ·(g, s) = g · s. Apply the following

equivariant tubular neighborhood theorem.

144 23 THE MARSDEN-WEINSTEIN-MEYER THEOREM

Theorem 23.5 (Slice Theorem) Let G be a compact Lie group acting

on a manifold M such that G acts freely at p ∈ M . For su¬ciently small

µ, · : G — Sµ ’ M maps G — Sµ di¬eomorphically onto a G-invariant

neighborhood U of the G-orbit through p.

The proof of this slice theorem is sketched further below.

Corollary 23.6 If the action of G is free at p, then the action is free on

U.

Corollary 23.7 The set of points where G acts freely is open.

Corollary 23.8 The set G — Sµ U is G-invariant. Hence, the quotient

U/G Sµ is smooth.

Conclusion of the proof that M/G is a manifold and π : M ’ M/G is

a smooth ¬ber map.

For p ∈ M , let q = π(p) ∈ M/G. Choose a G-invariant neighborhood U of

p as in the slice theorem: U G — S (where S = Sµ for an appropriate µ).

Then π(U) = U/G =: V is an open neighborhood of q in M/G. By the slice

theorem, S ’ V is a homeomorphism. We will use such neighborhoods

V as charts on M/G. To show that the transition functions associated

with these charts are smooth, consider two G-invariant open sets U1 , U2 in

M and corresponding slices S1 , S2 of the G-action. Then S12 = S1 © U2 ,

S21 = S2 © U1 are both slices for the G-action on U1 © U2 . To compute the

transition map S12 ’ S21 , consider the diagram

’’ id — S12 ’ G — S12

S12

U1 © U 2 .

’’ id — S21 ’ G — S21

S21

Then the composition

pr

S12 ’ U1 © U2 ’’ G — S21 ’’ S21

is smooth.

Finally, we need to show that π : M ’ M/G is a smooth ¬ber map.

For p ∈ M , q = π(p), choose a G-invariant neighborhood U of the G-

orbit through p of the form · : G — S ’ U. Then V = U/G S is the

corresponding neighborhood of q in M/G:

·

M⊇ U G—S G—V

“π “

M/G ⊇ V V

=

145

23.3 Proof of the Marsden-Weinstein-Meyer Theorem

Since the projection on the right is smooth, π is smooth.

Exercise. Check that the transition functions for the bundle de¬ned by

™¦

π are smooth.

Sketch for the proof of the slice theorem. We need to show that,

for µ su¬ciently small, · : G — Sµ ’ U is a di¬eomorphism where U ⊆ M

is a G-invariant neighborhood of the G-orbit through p. Show that:

(a) d·(id,p) is bijective.

(b) Let G act on G — S by the product of its left action on G and trivial

action on S. Then · : G — S ’ M is G-equivariant.

(c) d· is bijective at all points of G — {p}. This follows from (a) and (b).

(d) The set G — {p} is compact, and · : G — S ’ M is injective on

G — {p} with d· bijective at all these points. By the implicit function

theorem, there is a neighborhood U0 of G — {p} in G — S such that

· maps U0 di¬eomorphically onto a neighborhood U of the G-orbit

through p.

(e) The sets G — Sµ , varying µ, form a neighborhood base for G — {p} in

G — S. So in (d) we may take U0 = G — Sµ .

23.3 Proof of the Marsden-Weinstein-Meyer Theorem

Since

G acts freely on µ’1 (0) =’ dµp is surjective for all p ∈ µ’1 (0)

=’ 0 is a regular value

=’ µ’1 (0) is a submanifold of codimension = dim G

for the ¬rst two parts of the Marsden-Weinstein-Meyer theorem it is enough to

apply the third ingredient from Section 23.2 to the free action of G on µ’1 (0).

At p ∈ µ’1 (0) the tangent space to the orbit Tp Op is an isotropic subspace

of the symplectic vector space (Tp M, ωp ), i.e., Tp Op ⊆ (Tp Op )ω .

(Tp Op )ω = ker dµp = Tp µ’1 (0) .

The lemma (second ingredient) gives a canonical symplectic structure on the

quotient Tp µ’1 (0)/Tp Op . The point [p] ∈ Mred = µ’1 (0)/G has tangent space

Tp µ’1 (0)/Tp Op . Thus the lemma de¬nes a nondegenerate 2-form

T[p] Mred

ωred on Mred. This is well-de¬ned because ω is G-invariant.

146 23 THE MARSDEN-WEINSTEIN-MEYER THEOREM

By construction i— ω = π — ωred where

i

µ’1 (0) ’M

“π

Mred

Hence, π — dωred = dπ — ωred = d±— ω = ±— dω = 0. The closedness of ωred follows

from the injectivity of π — .

Remark. Suppose that another Lie group H acts on (M, ω) in a hamiltonian

way with moment map φ : M ’ h— . If the H-action commutes with the G-

action, and if φ is G-invariant, then Mred inherits a hamiltonian action of H,

with moment map φred : Mred ’ h— satisfying φred —¦ π = φ —¦ i. ™¦

24 Reduction

24.1 Noether Principle

Let (M, ω, G, µ) be a hamiltonian G-space.

Theorem 24.1 (Noether) If f : M ’ R is a G-invariant function, then µ

is constant on the trajectories of the hamiltonian vector ¬eld of f .

Proof. Let vf be the hamiltonian vector ¬eld of f . Let X ∈ g and µX =

µ, X : M ’ R. We have

L v f µX = ±vf dµX = ±vf ±X # ω

= ’±X # ±vf ω = ’±X # df

= ’LX # f = 0

because f is G-invariant.

De¬nition 24.2 A G-invariant function f : M ’ R is called an integral of

motion of (M, ω, G, µ). If µ is constant on the trajectories of a hamiltonian

vector ¬eld vf , then the corresponding one-parameter group of di¬eomorphisms

{exp tvf | t ∈ R} is called a symmetry of (M, ω, G, µ).

The Noether principle asserts that there is a one-to-one correspondence

between symmetries and integrals of motion.

24.2 Elementary Theory of Reduction

Finding a symmetry for a 2n-dimensional mechanical problem may reduce it

to a (2n ’ 2)-dimensional problem as follows: an integral of motion f for a

2n-dimensional hamiltonian system (M, ω, H) may enable us understand the

trajectories of this system in terms of the trajectories of a (2n ’ 2)-dimensional

hamiltonian system (Mred , ωred , Hred ). To make this precise, we will describe

this process locally. Suppose that U is an open set in M with Darboux coordi-

nates x1 , . . . , xn , ξ1 , . . . , ξn such that f = ξn for this chart, and write H in these

coordinates: H = H(x1 , . . . , xn , ξ1 , . . . , ξn ). Then

±

the trajectories of vH lie on the

hyperplane ξ = constant

n

ξn is an integral of motion =’ ‚H

{ξn , H} = 0 = ’ ‚xn

=’ H = H(x1 , . . . , xn’1 , ξ1 , . . . , ξn ) .

147

148 24 REDUCTION

If we set ξn = c, the motion of the system on this hyperplane is described

by the following Hamilton equations:

±

dx1 ‚H

= (x1 , . . . , xn’1 , ξ1 , . . . , ξn’1 , c)

dt ‚ξ1

.

.

.

dxn’1 ‚H

= (x1 , . . . , xn’1 , ξ1 , . . . , ξn’1 , c)

dt ‚ξn’1

dξ

‚H

1

’

= (x1 , . . . , xn’1 , ξ1 , . . . , ξn’1 , c)

dt ‚x1

.

.

.

dξn’1 ‚H

=’ (x1 , . . . , xn’1 , ξ1 , . . . , ξn’1 , c)

dt ‚xn’1

dxn ‚H

=

dt ‚ξn

dξn ‚H

=’ =0.

dt ‚xn

The reduced phase space is

Ured = {(x1 , . . . , xn’1 , ξ1 , . . . , ξn’1 ) ∈ R2n’2 |

(x1 , . . . , xn’1 , a, ξ1 , . . . , ξn’1 , c) ∈ U for some a} .

The reduced hamiltonian is

Hred : Ured ’’ R ,

Hred (x1 , . . . , xn’1 , ξ1 , . . . , ξn’1 ) = H(x1 , . . . , xn’1 , ξ1 , . . . , ξn’1 , c) .

In order to ¬nd the trajectories of the original system on the hypersurface

ξn = c, we look for the trajectories

x1 (t), . . . , xn’1 (t), ξ1 (t), . . . , ξn’1 (t)

of the reduced system on Ured . We integrate the equation

dxn ‚H

(t) = (x1 (t), . . . , xn’1 (t), ξ1 (t), . . . , ξn’1 (t), c)

dt ‚ξn

to obtain the original trajectories

t ‚H

xn (t) = xn (0) + 0 ‚ξn (. . .)dt

ξn (t) = c .

149

24.3 Reduction for Product Groups

24.3 Reduction for Product Groups

Let G1 and G2 be compact connected Lie groups and let G = G1 — G2 . Then

g— = g— • g— .

g = g 1 • g2 and 1 2

Suppose that (M, ω, G, ψ) is a hamiltonian G-space with moment map

ψ : M ’’ g— • g— .

1 2

Write ψ = (ψ1 , ψ2 ) where ψi : M ’ g— for i = 1, 2. The fact that ψ is

i

equivariant implies that ψ1 is invariant under G2 and ψ2 is invariant under G1 .

Now reduce (M, ω) with respect to the G1 -action. Let

’1

Z1 = ψ1 (0) .

Assume that G1 acts freely on Z1 . Let M1 = Z1 /G1 be the reduced space

and let ω1 be the corresponding reduced symplectic form. The action of G2

on Z1 commutes with the G1 -action. Since G2 preserves ω, it follows that

G2 acts symplectically on (M1 , ω1 ). Since G1 preserves ψ2 , G1 also preserves

ψ2 —¦ ι1 : Z1 ’ g— , where ι1 : Z1 ’ M is inclusion. Thus ψ2 —¦ ι is constant on

2

p1

¬bers of Z1 ’ M1 . We conclude that there exists a smooth map µ2 : M1 ’ g— 2

such that µ2 —¦ p = ψ2 —¦ i.

Exercise. Show that:

(a) the map µ2 is a moment map for the action of G2 on (M1 , ω1 ), and

(b) if G acts freely on ψ ’1 (0, 0), then G2 acts freely on µ’1 (0), and there is a

2

natural symplectomorphism

µ’1 (0)/G2 ψ ’1 (0, 0)/G .

2

™¦

This technique of performing reduction with respect to one factor of a prod-

uct group at a time is called reduction in stages. It may be extended to

reduction by a normal subgroup H ‚ G and by the corresponding quotient

group G/H.

24.4 Reduction at Other Levels

Suppose that a compact Lie group G acts on a symplectic manifold (M, ω) in a

hamiltonian way with moment map µ : M ’ g— . Let ξ ∈ g— .

To reduce at the level ξ of µ, we need µ’1 (ξ) to be preserved by G, or else

take the G-orbit of µ’1 (ξ), or else take the quotient by the maximal subgroup

of G which preserves µ’1 (ξ).

150 24 REDUCTION

Since µ is equivariant,

G preserves µ’1 (ξ) ⇐’ G preserves ξ

Ad— ξ = ξ, ∀g ∈ G .

⇐’ g

Of course the level 0 is always preserved. Also, when G is a torus, any level

is preserved and reduction at ξ for the moment map µ, is equivalent to reduction

at 0 for a shifted moment map φ : M ’ g— , φ(p) := µ(p) ’ ξ.

Let O be a coadjoint orbit in g— equipped with the canonical symplec-

tic form (also know as the Kostant-Kirillov symplectic form or the Lie-

Poisson symplectic form) ωO de¬ned in Homework 17. Let O ’ be the orbit

O equipped with ’ωO . The natural product action of G on M — O ’ is hamil-

tonian with moment map µO (p, ξ) = µ(p) ’ ξ. If the Marsden-Weinstein-Meyer

hypothesis is satis¬ed for M — O ’ , then one obtains a reduced space with

respect to the coadjoint orbit O.

24.5 Orbifolds

Example. Let G = Tn be an n-torus. For any ξ ∈ (tn )— , µ’1 (ξ) is preserved by

the Tn -action. Suppose that ξ is a regular value of µ. (By Sard™s theorem, the

singular values of µ form a set of measure zero.) Then µ’1 (ξ) is a submanifold

of codimension n. Note that

dµp is surjective at all p ∈ µ’1 (ξ)

ξ regular =’

gp = 0 for all p ∈ µ’1 (ξ)

=’

the stabilizers on µ’1 (ξ) are ¬nite

=’

µ’1 (ξ)/G is an orbifold [90, 91] .

=’

Let Gp be the stabilizer of p. By the slice theorem (Lecture 23), µ’1 (ξ)/G

is modeled by S/Gp , where S is a Gp -invariant disk in µ’1 (ξ) through p and

transverse to Op . Hence, locally µ’1 (ξ)/G looks indeed like Rn divided by a

™¦

¬nite group action.

Example. Consider the S 1 -action on C2 given by eiθ · (z1 , z2 ) = (eikθ z1 , eiθ z2 )

for some ¬xed integer k ≥ 2. This is hamiltonian with moment map

C2 ’’ R

µ:

1

(z1 , z2 ) ’’ ’ 2 (k|z1 |2 + |z2 |2 ) .

Any ξ < 0 is a regular value and µ’1 (ξ) is a 3-dimensional ellipsoid. The stabi-

2π

lizer of (z1 , z2 ) ∈ µ’1 (ξ) is {1} if z2 = 0, and is Zk = ei k | = 0, 1, . . . , k ’ 1

if z2 = 0. The reduced space µ’1 (ξ)/S 1 is called a teardrop orbifold or cone-

head; it has one cone (also known as a dunce cap) singularity of type k (with

cone angle 2π ). ™¦

k

151

24.5 Orbifolds

Example. Let S 1 act on C2 by eiθ · (z1 , z2 ) = (eikθ z1 , ei θ z2 ) for some integers

k, ≥ 2. Suppose that k and are relatively prime. Then

(z1 , 0) has stabilizer Zk (for z1 = 0) ,

(0, z2 ) has stabilizer Z (for z2 = 0) ,

(z1 , z2 ) has stabilizer {1} (for z1 , z2 = 0) .

µ’1 (ξ)/S 1 is called a football orbifold. It has two cone singularities, one of

™¦

type k and another of type .

Example. More generally, the reduced spaces of S 1 acting on Cn by

eiθ · (z1 , . . . , zn ) = (eik1 θ z1 , . . . , eikn θ zn ) ,

™¦

are called weighted (or twisted) projective spaces.

Homework 18: Spherical Pendulum

This set of problems is from [52].

The spherical pendulum is a mechanical system consisting of a massless

rigid rod of length l, ¬xed at one end, whereas the other end has a plumb bob

of mass m, which may oscillate freely in all directions. Assume that the force

of gravity is constant pointing vertically downwards, and that this is the only

external force acting on this one-particle system.

Let •, θ (0 < • < π, 0 < θ < 2π) be spherical coordinates for the bob. For

simplicity assume that m = l = 1.

1. Let ·, ξ be the coordinates along the ¬bers of T — S 2 induced by the spher-

ical coordinates •, θ on S 2 . Show that the function H : T — S 2 ’ R given

by

ξ2

1

·2 +

H(•, θ, ·, ξ) = + cos • ,

(sin •)2

2

is an appropriate hamiltonian function to describe the spherical pendulum.

2. Compute the critical points of the function H. Show that, on S 2 , there

are exactly two critical points: s (where H has a minimum) and u. These

points are called the stable and unstable points of H, respectively. Jus-

tify this terminology, i.e., show that a trajectory whose initial point is

close to s stays close to s forever, and show that this is not the case for u.

What is happening physically?

3. Show that the group of rotations about the vertical axis is a group of

symmetries of the spherical pendulum.

Show that, in the coordinates above, the integral of motion associated

with these symmetries is the function

J(•, θ, ·, ξ) = ξ .

Give a more coordinate-independent description of J, one that makes sense

also on the cotangent ¬bers above the North and South poles.

152

153

HOMEWORK 18

4. Locate all points p ∈ T — S 2 where dHp and dJp are linearly dependent:

(a) Clearly, the two critical points s and u belong to this set. Show that

these are the only two points where dHp = dJp = 0.

(b) Show that, if x ∈ S 2 is in the southern hemisphere (x3 < 0), then

there exist exactly two points, p+ = (x, ·, ξ) and p’ = (x, ’·, ’ξ),

in the cotangent ¬ber above x where dHp and dJp are linearly de-

pendent.

(c) Show that dHp and dJp are linearly dependent along the trajectory

of the hamiltonian vector ¬eld of H through p+ .

Conclude that this trajectory is also a trajectory of the hamiltonian

vector ¬eld of J, and, hence, that its projection onto S 2 is a latitu-

dinal circle (of the form x3 = constant).

Show that the projection of the trajectory through p’ is the same

latitudinal circle traced in the opposite direction.

5. Show that any nonzero value j is a regular value of J, and that S 1 acts

freely on the level set J = j. What happens on the cotangent ¬bers above

the North and South poles?

6. For j = 0 describe the reduced system and sketch the level curves of the

reduced hamiltonian.

7. Show that the integral curves of the original system on the level set J = j

can be obtained from those of the reduced system by “quadrature”, in

other words, by a simple integration.

8. Show that the reduced system for j = 0 has exactly one equilibrium point.

Show that the corresponding relative equilibrium for the original system

is one of the horizontal curves in exercise 4.

9. The energy-momentum map is the map (H, J) : T — S 2 ’ R2 . Show

that, if j = 0, the level set (H, J) = (h, j) of the energy-momentum map is

either a circle (in which case it is one of the horizontal curves in exercise 4),

or a two-torus. Show that the projection onto the con¬guration space of

the two-torus is an annular region on S 2 .

Part X

Moment Maps Revisited

Moment maps and symplectic reduction have been ¬nding in¬nite-dimensional

incarnations with amazing consequences for di¬erential geometry. Lecture 25

sketches the symplectic approach of Atiyah and Bott to Yang-Mills theory.

Lecture 27 describes the convexity of the image of a torus moment map, one

of the most striking geometric characteristics of moment maps.

25 Moment Map in Gauge Theory

25.1 Connections on a Principal Bundle

Let G be a Lie group and B a manifold.

De¬nition 25.1 A principal G-bundle over B is a manifold P with a smooth

map π : P ’ B satisfying the following conditions:

(a) G acts freely on P (on the left),

(b) B is the orbit space for this action and π is the point-orbit projection, and

(c) there is an open covering of B, such that, to each set U in that covering

corresponds a map •U : π ’1 (U) ’ U — G with

∀p ∈ π ’1 (U) .

sU (g · p) = g · sU (p) ,

•U (p) = (π(p), sU (p)) and

The G-valued maps sU are determined by the corresponding •U . Condition (c)

is called the property of being locally trivial.

If P with map π : P ’ B is a principal G-bundle over B, then the manifold

B is called the base, the manifold P is called the total space, the Lie group G

is called the structure group, and the map π is called the projection. This

principal bundle is also represented by the following diagram:

EP

G ‚

π

c

B

Example. Let P be the 3-sphere regarded as unit vectors in C2 :

P = S 3 = {(z1 , z2 ) ∈ C2 : |z1 |2 + |z2 |2 = 1} .

155

156 25 MOMENT MAP IN GAUGE THEORY

Let G be the circle group, where eiθ ∈ S 1 acts on S 3 by complex multiplication,

(z1 , z2 ) ’’ (eiθ z1 , eiθ z2 ) .

Then the quotient space B is the ¬rst complex projective space, that is, the two-

sphere. This data forms a principal S 1 -bundle, known as the Hopf ¬bration:

E S3

S1 ‚

π

c

S2

™¦

An action ψ : G ’ Di¬(P ) induces an in¬nitesimal action

g ’’ χ(P )

dψ :

X ’’ X # = vector ¬eld generated by the

one-parameter group {exp tX(e) | t ∈ R} .

From now on, ¬x a basis X1 , . . . , Xk of g.

Let P be a principal G-bundle over B. Since the G-action is free, the vector

# #

¬elds X1 , . . . , Xk are linearly independent at each p ∈ P . The vertical bundle

# #

V is the rank k subbundle of T P generated by X1 , . . . , Xk .

Exercise. Check that the vertical bundle V is the set of vectors tangent to P

which lie in the kernel of the derivative of the bundle projection π. (This shows

™¦

that V is independent of the choice of basis for g.)

De¬nition 25.2 A (Ehresmann) connection on a principal bundle P is a

choice of a splitting

TP = V • H ,

where H is a G-invariant subbundle of T P complementary to the vertical bundle