which implies the desired result since for ω ∈ „¦k (M ) with dω = 0 we have g — ω ’

¯ ¯ ¯

f — ω = dhω ’ hdω = dhω.

34.3. Lemma. If a manifold is decomposed into a disjoint union M = M± of

±

open submanifolds, then H k (M ) = ± H k (M± ) for all k.

Proof. „¦k (M ) is isomorphic to ± „¦k (M± ) via • ’ (•|M± )± . This isomorphism

commutes with the exterior derivative d and induces the result.

34.4. The setting for the Mayer-Vietoris Sequence. Let M be a smooth

manifold, let U , V ‚ M be open subsets which cover M and admit a subordinated

smooth partition of unity {fU , fV } with supp(fU ) ‚ U and supp(fV ) ‚ V . We

consider the following embeddings:

‘ 99

e GA U ©V

”

‘ jU

j V

U

! heiu

eV

e

i U V

M.

Lemma. In this situation, the sequence

β

±

0 ’ „¦(M ) ’ „¦(U ) • „¦(V ) ’ „¦(U © V ) ’ 0

’ ’

is exact, where ±(ω) := (i— ω, i— ω) and β(•, ψ) = jU • ’ jV ψ. We also have

— —

U V

(d • d) —¦ ± = ± —¦ d and d —¦ β = β —¦ (d • d).

Proof. We have to show that ± is injective, ker β = im ±, and that β is surjective.

The ¬rst two assertions are obvious. For • ∈ „¦(U ©V ) we consider fV • ∈ „¦(U ©V ).

Note that supp(fV •) is closed in the closed subset supp(fV )©U of U and contained

in the open subset U © V of U , so we may extend fV • by 0 to a smooth form

•U ∈ „¦(U ). Likewise, we extend ’fU • by 0 to •V ∈ „¦(V ). Then we have

β(•U , •V ) = (fU + fV )• = •.

34.4

356 Chapter VII. Calculus on in¬nite dimensional manifolds 34.7

34.5. Theorem. Mayer-Vietoris sequence. Let U and V be open subsets in a

manifold M which cover M and admit a subordinated smooth partition of unity.

Then there is an exact sequence

β—

± δ

· · · ’ H k (M ) ’’ H k (U ) • H k (V ) ’ H k (U © V ) ’ H k+1 (M ) ’ · · ·

—

’ ’ ’

It is natural in the triple (M, U, V ). The homomorphisms ±— and β— are algebra

homomorphisms, but δ is not.

Proof. This follows from (34.4) and standard homological algebra.

Since we shall need it later we will now give a detailed description of the connecting

homomorphism δ. Let {fU , fV } be a partition of unity with supp fU ‚ U and

supp fV ‚ V . Let ω ∈ „¦k (U © V ) with dω = 0 so that [ω] ∈ H k (U © V ). Then

(fV .ω, ’fU .ω) ∈ „¦k (U ) • „¦k (V ) is mapped to ω by β, and we have

δ[ω] = [±’1 (d • d)(fV .ω, ’fU .ω)] = [±’1 (dfV § ω, ’dfU § ω)]

= [dfV § ω] = ’[dfU § ω)],

where we have used the following fact: fU + fV = 1 implies that on U © V we have

dfV = ’dfU , thus dfV § ω = ’dfU § ω, and o¬ U © V both are 0.

34.6. Theorem. Let M be a smooth manifold which is smoothly paracompact.

Then the De Rham cohomology of M coincides with the sheaf cohomology of M

with coe¬cients in the constant sheaf R on M .

Proof. Since M is smoothly paracompact it is also paracompact, and thus the

usual theory of sheaf cohomology using the notion of ¬ne sheafs is applicable. For

each k we consider the sheaf „¦k on M which associates to each c∞ -open set U ‚ M

M

the convenient vector space „¦k (U ). Then the following sequence of sheaves

d d

R ’ „¦0 ’ „¦1 ’ . . .

M’ M’

is a resolution of the constant sheaf R by the lemma of Poincar´ (33.20). Since

e

we have smooth partitions of unity on M , each sheaf „¦k is a ¬ne sheaf, so this

M

resolution is acyclic [Godement, 1958], [Hirzebruch, 1962, 2.11.1], and the sequence

of global sections may be used to compute the sheaf cohomology of the constant

sheaf R. But this is the De Rham cohomology.

34.7. Theorem. Let M be a smooth manifold which is smoothly paracompact.

Then the De Rham cohomology of M coincides with the singular cohomology with

coe¬cients in R via a canonical isomorphism which is induced by integration of

p-forms over smooth singular simplices.

k

Proof. Denote by S∞ the sheaf which is generated by the presheaf of singular

k

smooth cochains with real coe¬cients. In more detail: let us put S∞ (U, R) =

∞

R = RC (∆k ,U ) , where σ : ∆k ’ U is any mapping which extends to a smooth

σ

34.7

34.7 34. De Rham cohomology 357

mapping from a neighborhood of the standard k-simplex ∆k ‚ Rk+1 into U , where

U is c∞ -open in M . This de¬nes a presheaf. The associated sheaf is denoted by

k

S∞ . The sequence

— —

0δ 1δ 2

R’ S∞ ’’ S∞ ’’ S∞ ’ . . .

of sheafs is a resolution, because if U is a small open set, say di¬eomorphic to

a radial neighborhood of 0 in the modeling convenient vector space, then U is

—

smoothly contractible to a point. Smooth mappings induce mappings in the S∞ -

—

cohomology, thus H k (S∞ (U, R), ‚) = 0 for k > 0. This implies that the associated

sequence of stalks is exact, so the sequence above is a resolution. A standard

k

argument of sheaf theory shows that each sheaf S∞ is a ¬ne sheaf, so they form an

—

acyclic resolution, and H k (S∞ (M, R), ‚) coincides with the sheaf cohomology with

coe¬cients in the constant sheaf R.

Furthermore, integration of p-forms over smooth singular p-simplices de¬nes a map-

ping of resolutions

w„¦ w„¦ w ···

RRTR „¦0 1 2

R

¢ u

u u

wS wS w ···

0 1 2

S∞ ∞ ∞

which induces an isomorphism from the De Rham cohomology of M to the coho-

mology H — (S∞ (M, R), ‚).

—

Now we consider the resolution

δ— δ—

R ’ S0 ’ S1 ’ S2 ’ . . .

’ ’

of the constant sheaf R, where S k is the usual sheaf induced by the singular contin-

uous cochains. Since M is (even smoothly) paracompact and locally contractible,

this is an acyclic resolution, and the embedding of smooth singular chains into

continuous singular chains de¬nes a mapping of resolutions

wS wS w ···

RRTR S0 1 2

R

¢ u

u u

wS wS w ···

0 1 2

S∞ ∞ ∞

which induces an isomorphism from the singular cohomology of M to the cohomol-

ogy H — (S∞ (M, R), ‚).

—

34.7

358 Chapter VII. Calculus on in¬nite dimensional manifolds 35.2

35. Derivations on Di¬erential Forms

and the Fr¨licher-Nijenhuis Bracket

o

35.1. In this section let M be a smooth manifold. We consider the graded com-

mutative algebra

∞ ∞

∞

k

Lk (T M, M „¦k (M )

C (M ← — R)) =

„¦(M ) = „¦ (M ) = alt

k≥0 k=0 k=’∞

of di¬erential forms on M , see (33.22), where „¦0 (M ) = C ∞ (M, R), and where we

put „¦k (M ) = 0 for k < 0. We denote by Derk „¦(M ) the space of all (graded)

derivations of degree k, i.e., all bounded linear mappings D : „¦(M ) ’ „¦(M ) with

D(„¦l (M )) ‚ „¦k+l (M ) and D(• § ψ) = D(•) § ψ + (’1)kl • § D(ψ) for • ∈ „¦l (M ).

Convention. In general, derivations need not be of local nature. Thus, we consider

each derivation and homomorphism to be a sheaf morphism (compare (32.1) and

the de¬nition of modular 1-forms in (33.2)), or we assume that all manifolds in

question are again smoothly regular. This is justi¬ed by the obvious extension of

(32.4) and (33.3).

Lemma. Then the space Der „¦(M ) = k Derk „¦(M ) is a graded Lie algebra with

the graded commutator [D1 , D2 ] := D1 —¦ D2 ’ (’1)k1 k2 D2 —¦ D1 as bracket. This

means that the bracket is graded anticommutative and satis¬es the graded Jacobi

identity:

[D1 , D2 ] = ’(’1)k1 k2 [D2 , D1 ],

[D1 , [D2 , D3 ]] = [[D1 , D2 ], D3 ] + (’1)k1 k2 [D2 , [D1 , D3 ]]

(so that ad(D1 ) = [D1 , ] is itself a derivation of degree k1 ).

Proof. Plug in the de¬nition of the graded commutator and compute.

In section (33) we have already met some graded derivations: for a vector ¬eld X

on M the derivation iX is of degree ’1, LX is of degree 0, and d is of degree 1. In

(33.18) we already met some some graded commutators like LX = d iX + iX d =

[iX , d].

35.2. A derivation D ∈ Derk „¦(M ) is called algebraic if D | „¦0 (M ) = 0. Then

D(f.ω) = f.D(ω) for f ∈ C ∞ (M, R) and ω ∈ „¦(M ).

If the spaces Lk (Tx M ; R) are all re¬‚exive and have the bornological approximation

alt

property, then an algebraic derivation D induces for each x ∈ M a derivation

Dx ∈ Derk (L— (Tx M ; R)), by a method used in (33.5). It is not clear whether it

alt

su¬ces to assume that just the model spaces of M are all re¬‚exive and have the

bornological approximation property.

35.2

35.2 35. The Fr¨licher-Nijenhuis bracket

o 359

In the sequel, we will consider the space of all vector valued kinematic di¬erential

forms, which we will de¬ne by

C ∞ (M ← Lk (T M ; T M ))

„¦k (M ; T M ) =

„¦(M ; T M ) = alt

k≥0 k≥0

Note that „¦0 (M ; T M ) = X(M ) = C ∞ (M ← T M ). For simplicity™s sake, we will

not treat other kinds of vector valued di¬erential forms.

Theorem. (1) For K ∈ „¦k+1 (M ; T M ) the formula

(iK ω)(X1 , . . . , Xk+l ) =

1

= sign σ .ω(K(Xσ1 , . . . , Xσ(k+1) ), Xσ(k+2) , . . . )

(k+1)! (l’1)!

σ∈Sk+l

(k+1)(k+2)

(’1)i1 +···+ik+1 ’

= ω(K(Xi1 , . . . , Xik+1 ), X1 , . . . , Xi1 , . . . ),

2

i1 <···<ik+1

for f ∈ C ∞ (M, R) = „¦0 (M ),

iK f = 0

for ω ∈ „¦l (M ), Xi ∈ X(M ) de¬nes an algebraic graded derivation iK = i(K) ∈

Derk „¦(M ).

]§ on „¦—+1 (M ; T M ) for K ∈ „¦k+1 (M ; T M ), L ∈

(2) We de¬ne a bracket [ ,

„¦l+1 (M ; T M ) by

[K, L]§ := iK L ’ (’1)kl iL K,

where iK (L) is given by the same formula as in (1). This de¬nes a graded Lie

algebra structure with the grading as indicated, and we have i([K, L]§ ) = [iK , iL ] ∈

Der „¦(M ). Thus, i : „¦—+1 (M ; T M ) ’ Der— „¦(M ) is a homomorphism of graded

Lie algebras, which is injective under the assumptions of (35.1).

The concomitant [ , ]§ is called the algebraic bracket or the Nijenhuis-Richardson

bracket, compare [Nijenhuis, Richardson, 1967].

Proof. (1) We know that iX is a derivation of degree ’1 for a vector ¬eld X ∈

X(M ) = „¦0 (M ; T M ) by (33.11). By direct evaluation, one gets

(3) [iX , iK ] = i(iX K).

Using this and induction on the sum of the degrees of K ∈ „¦k (M ; T M ), • ∈ „¦(M ),

and ψ ∈ „¦(M ), one can then show that

iX iK (• § ψ) = iX (iK • § ψ + (’1)k deg• • § iK ψ)

holds, which implies that iK is a derivation of degree k.

35.2

360 Chapter VII. Calculus on in¬nite dimensional manifolds 35.4

(2) By induction on the sum of k = deg K ’ 1, l = deg L ’ 1, and p = deg •, and

by (3) we have

[iX , [iK , iL ]]• = [[iX , iK ], iL ]• + (’1)k [iK , [iX , iL ]]•

= [i(iX K), iL ]• + (’1)k [iK , i(iX L)]•

= i i(iX K)L ’ (’1)(k’1)l iL iX K •

+ (’1)k i iK iX L ’ (’1)k(l’1) i(iX L)K •

= i iX iK L ’ (’1)kl iX iL K • = i (iX [K, L]§ ) •,

iX [iK , iL ]• = [iX , [iK , iL ]]• + (’1)k+l [iK , iL ]iX •

= i (iX [K, L]§ ) • + (’1)k+l i([K, L]§ )iX •

= i (iX [K, L]§ ) • ’ i(iX [K, L]§ )• + iX i([K, L]§ )•

= iX i([K, L]§ )•.

This implies i([K, L]§ ) = [iK , iL ] since the iX for X ∈ T M separate points, in both

cases of the convention (35.1). From iK df = df —¦ K it follows that the mapping

i : „¦(M ; T M ) ’ Der(„¦(M )) is injective, so („¦—+1 (M ; T M ), [ , ]§ ) is a graded

Lie algebra.

35.3. The exterior derivative d is an element of Der1 „¦(M ). In view of the formula

LX = [iX , d] = iX d + d iX for vector ¬elds X (see (33.18.6)), we de¬ne for K ∈

„¦k (M ; T M ) the Lie derivative LK = L(K) ∈ Derk „¦(M ) by

LK := [iK , d] = iK d ’ (’1)k’1 d iK .

Since the 1-forms df for all local functions on M separate points on each Tx M , the

mapping L : „¦(M ; T M ) ’ Der „¦(M ) is injective, because LK f = iK df = df —¦ K

for f ∈ C ∞ (M, R).

From (35.2.1) it follows that i(IdT M )ω = kω for ω ∈ „¦k (M ). Hence, L(IdT M )ω =

i(IdT M )dω ’ d i(IdT M )ω = (k + 1)dω ’ kdω = dω, and thus L(IdT M ) = d.

35.4. Proposition. For K ∈ „¦k (M ; T M ) and ω ∈ „¦l (M ) the Lie derivative of ω

along K is given by the following formula, where the Xi are (local) vector ¬elds on

M.

(LK ω)(X1 , . . . , Xk+l ) =

1

sign σ LK(Xσ1 ,...,Xσk ) (ω(Xσ(k+1) , . . . , Xσ(k+l) ))

= k! l!

σ

+ (’1)k 1

sign σ ω([Xσ1 , K(Xσ2 , . . . , Xσ(k+1) )], Xσ(k+2) , . . . )

k! (l’1)!

σ

1

’ sign σ ω(K([Xσ1 , Xσ2 ], Xσ3 , . . . ), Xσ(k+2) , . . . )

(k’1)! (l’1)! 2!

σ

Proof. Consider LK ω = [iK , d]ω = iK dω ’ (’1)k’1 diK ω, and plug into this the

de¬nitions (35.2.1), second version, and (33.12.3). After computing some signs the

expression above follows.

35.4

35.5 35. The Fr¨licher-Nijenhuis bracket

o 361

35.5. De¬nition and theorem. For K ∈ „¦k (M ; T M ) and L ∈ „¦l (M ; T M ) we

de¬ne the Fr¨licher-Nijenhuis bracket [K, L] by the following formula, where the Xi

o

are vector ¬elds on M .

(1) [K, L](X1 , . . . , Xk+l ) =

1

= sign σ [K(Xσ1 , . . . , Xσk ), L(Xσ(k+1) , . . . , Xσ(k+l) )]

k! l!

σ

+ (’1)k 1

sign σ L([Xσ1 , K(Xσ2 , . . . , Xσ(k+1) )], Xσ(k+2) , . . . )

k! (l’1)!

σ

1

’ sign σ L(K([Xσ1 , Xσ2 ], Xσ3 , . . . ), Xσ(k+2) , . . . )

(k’1)! (l’1)! 2!

σ

’ (’1)kl+l 1

sign σ K([Xσ1 , L(Xσ2 , . . . , Xσ(l+1) )], Xσ(l+2) , . . . )

(k’1)! l!

σ

1

’ sign σ K(L([Xσ1 , Xσ2 ], Xσ3 , . . . ), Xσ(l+2) , . . . ) .

(k’1)! (l’1)! 2!

σ

Then [K, L] ∈ „¦k+l (M ; T M ), and we have

[L(K), L(L)] = L([K, L]) ∈ Der „¦(M ).

dim M

Therefore, the space „¦(M ; T M ) = k=0 „¦k (M ; T M ) with its usual grading is a

graded Lie algebra for the Fr¨licher-Nijenhuis bracket. So we have

o

[K, L] = ’(’1)kl [L, K]

[K1 , [K2 , K3 ]] = [[K1 , K2 ], K3 ] + (’1)k1 k2 [K2 , [K1 , K3 ]]

IdT M ∈ „¦1 (M ; T M ) is in the center, i.e., [K, IdT M ] = 0 for all K.

For vector ¬elds the Fr¨licher-Nijenhuis bracket coincides with the Lie bracket. The

o

—

mapping L : „¦ (M ; T M ) ’ Der— „¦(M ) is an injective homomorphism of graded

Lie algebras.

Proof. We ¬rst show that [K, L] ∈ „¦k+l (M ; T M ). By convention (35.1), this

is a local question in M , thus we may assume that M is a c∞ -open subset of a

convenient vector space E, that Xi : M ’ E, that K : M ’ Lk (E; E), and that

alt

l

L : M ’ Lalt (E; E). Then each expression in the formula is a kinematic vector

¬eld, and for such ¬elds Y1 , Y2 the Lie bracket is given by [Y1 , Y2 ] = dY2 .Y1 ’dY1 .Y2 ,

as shown in the beginning of the proof of (32.8). If we rewrite the formula in this

way, all terms containing the derivative of one Xi cancel, and the following local

expression for [K, L] remains, which is obviously an element of „¦k+l (M ; T M ).

[K, L](X1 , . . . , Xk+l ) =

1

= sign σ (dL.K(Xσ1 , . . . , Xσk ))(Xσ(k+1) , . . . )

k! l!

σ

’ (dK.L(Xσ(k+1) , . . . ))(Xσ1 , . . . , Xσk )

+ l L((dK.Xσ(k+1) )(Xσ1 , . . . , Xσk ), Xσ(k+2) , . . . )

’ k K((dL.Xσ1 )(Xσ(k+1) , . . . , Xσ(k+l) ), Xσ2 , . . . , Xσk ) .

35.5

362 Chapter VII. Calculus on in¬nite dimensional manifolds 35.5

Next we show that L([K, L]) = [LK , LL ] holds, by the following purely algebraic

method, which is adapted from [Dubois-Violette, Michor, 1997]. The Chevalley

coboundary operator for the adjoint representation of the Lie algebra X(M ) is

given by [Koszul, 1950], see also [Cartan, Eilenberg, 1956]

1

‚K(X1 , . . . , Xk+1 ) = sign σ [Xσ1 , K(Xσ2 , . . . , Xσ(k+1) )]

k!

σ

1

’ sign σ K([Xσ1 , Xσ2 ], Xσ3 , . . . , Xσ(k+1) ),

(k’1)! 2!

σ

(’1)i [Xi , K(X0 , . . . , Xi , . . . , Xk )]

‚K(X0 , . . . , Xk ) =

0¤i¤k

(’1)i+j K([Xi , Xj ], X0 , . . . , Xi , . . . , Xj , . . . , Xk ),

+

0¤i<j¤k

and it is well known that ‚‚ = 0. The following computation and close relatives

will appear several times in the remainder of this proof, so we include it once.

(i‚K ω)(X1 , . . . , Xk+l ) =

1

= sign(σ)ω(‚K(Xσ1 , . . . ), Xσ(k+2) , . . . )

(k+1)! (l’1)!

σ∈Sk+l

k+1

(’1)i’1 ω([Xσi , K(Xσ1 , . . . , Xσi , . . . )], Xσ(k+2) , . . . )

1

= sign(σ)

(k+1)! (l’1)!

σ i=1

(’1)i+j ω(K([Xσi , Xσj ], Xσ1 , . . . , Xσi , . . . , Xσj , . . . ), Xσ(k+2) , . . . )

+

1¤i<j¤k+1

1

= sign(„ ) (k + 1)ω([X„ 1 , K(X„ 2 , . . . )], X„ (k+2) , . . . )

(k+1)! (l’1)!

„

k(k+1)

’ ω(K([X„ 1 , X„ 2 ], X„ 3 , . . . ), X„ (k+2) , . . . )

2

1

= sign(σ)ω([Xσ1 , K(Xσ2 , . . . , Xσ(k+1) )], Xσ(k+2) , . . . )

k! (l’1)!

σ

1

’ sign(σ)ω(K([Xσ1 , Xσ2 ], Xσ3 , . . . ), Xσ(k+2) , . . . ).

(k’1)! (l’1)! 2!

σ

Then the Fr¨licher-Nijenhuis bracket (1) is given by

o

[K, L] = [K, L]§ + (’1)k i(‚K)L ’ (’1)kl+l i(‚L)K,

(2)

where we have put

(3) [K, L]§ (X1 , . . . , Xk+l ) :=

1

= sign(σ)[K(Xσ1 , . . . , Xσk ), L(Xσ(k+1) , . . . , Xσ(k+l) )].

k! l!

σ

35.5

35.5 35. The Fr¨licher-Nijenhuis bracket

o 363

Formula (2) is the same as in [Nijenhuis, 1969, p. 100], where it is also stated that

from this formula ˜one can show (with a good deal of e¬ort) that this bracket de¬nes

a graded Lie algebra structure™. Similarly, we can write the Lie derivative (35.4) as

LK = L§ (K) + (’1)k i(‚K),

(4)

where the action L of X(M ) on C ∞ (M, R) is extended to L§ : „¦(M ; T M )—„¦(M ) ’

„¦(M ) by

(5) (L§ (K)ω)(X1 , . . . , Xq+k ) =

1

= sign(σ)L(K(Xσ1 , . . . , Xσk ))(ω(Xσ(k+1) , . . . , Xσ(k+q) )).

k! q!

σ

Using (4), we see that

[LK , LL ] = L§ (K)L§ (L) ’ (’1)kl L§ (L)L§ (K)

(6)

+ (’1)k i(‚K)L§ (L) ’ (’1)kl+k L§ (L)i(‚K)

’ (’1)kl+l i(‚L)L§ (K) + (’1)l L§ (K)i(‚L)

+ (’1)k+l i(‚K)i(‚L) ’ (’1)kl+k+l i(‚L)i(‚K),

and from (2) and (4) we get

L[K,L] = L[K,L]§ + (’1)k Li(‚K)L ’ (’1)kl+l Li(‚L)K

(7)

= L§ ([K, L]§ ) + (’1)k+l i(‚[K, L]§ )

+ (’1)k L§ (i(‚K)L) + (’1)k i(‚i(‚K)L)

’ (’1)kl+l L§ (i(‚L)K) ’ (’1)kl+k i(‚i(‚L)K).

By a straightforward direct computation, one checks that

L§ (K)L§ (L) ’ (’1)kl L§ (L)L§ (K) = L§ ([K, L]§ ).

(8)

The derivation iK of degree k is seeing the expression L§ (L)ω as a ˜wedge product™

L §L ω, as in (33.8). So we may apply theorem (35.2.1) and get

iK L§ (L)ω = L§ (iK L)ω + (’1)(k’1)l L§ (L)iK ω.

(9)

By a long but straightforward combinatorial computation, one can check directly

from the de¬nitions that the following formula holds:

‚(iK L) = i‚K L + (’1)k’1 iK ‚L + (’1)k [K, L]§ .

(10)

Moreover, it is well-known (and easy to check) that

‚[K, L]§ = [‚K, L]§ + (’1)k [K, ‚L]§ .

(11)

We have to show that (6) equals (7). This follows by using (8), twice (9), then the

¬rst three lines in (6) correspond to the ¬rst terms in the ¬rst three lines in (7).

For the remaining terms use twice (10), (11), and ‚‚ = 0.

That the Fr¨licher-Nijenhuis bracket de¬nes a graded Lie bracket now follows from

o

the fact that L : „¦(M ; T M ) ’ Der(„¦(M )) is injective, by convention (35.1).

Since we have [d, d] = 2d2 = 0, by the graded Jacobi identity we obtain 0 =

[iK , [d, d]] = [[iK , d], d] + (’1)k’1 [d, [iK , d]] = 2[LK , d] = 2L([K, IdT M ]).

35.5

364 Chapter VII. Calculus on in¬nite dimensional manifolds 35.8

35.6. Lemma. Moreover, the Chevalley coboundary operator is a homomorphism

from the Fr¨licher-Nijenhuis bracket to the Nijenhuis-Richardson bracket:

o

‚[K, L] = [‚K, ‚L]§ .

Proof. This follows directly from (35.5.2), (35.5.11), and twice (35.5.10), and from

(35.2.2):

‚[K, L] = ‚[K, L]§ + (’1)k ‚i(‚K)L ’ (’1)kl+l ‚i(‚L)K

= [‚K, L]§ + (’1)k [K, ‚L]§ + 0 + i(‚K)‚L ’ [‚K, L]§

’ 0 ’ (’1)kl i(‚L)‚K + (’1)kl [‚L, K]§

= [‚K, ‚L]§ .

35.7. Lemma. For K ∈ „¦k (M ; T M ) and L ∈ „¦l+1 (M ; T M ) we have

[LK , iL ] = i([K, L]) ’ (’1)kl L(iL K), or

[iL , LK ] = L(iL K) + (’1)k i([L, K]).

Proof. The two equations are obviously equivalent by graded skew symmetry, and

the second one follows by expanding the left hand side using (35.5.4), (35.5.9), and

(35.2.2), and by expanding the right hand side using (35.5.4), (35.5.2), and then

(35.5.10):

[iL , LK ] = [iL , L§ (K)] + (’1)k [iL , i‚K ]

= L§ (iL K) + (’1)k i(iL ‚K ’ (’1)(l’1)k i‚K L),

L(iL K) + (’1)k i([L, K]) = L§ (iL K) ’ (’1)k+l i(‚iL K)

+ (’1)k i([L, K]§ + (’1)l i‚L K ’ (’1)kl+k i‚K L).

35.8. The space Der „¦(M ) is a graded module over the graded algebra „¦(M ) with

the action (ω § D)• = ω § D(•), because „¦(M ) is graded commutative.

Theorem. Let the degrees of ω be q, of • be k, and of ψ be l. Let the other degrees

be given by the corresponding lower case letters. Then we have:

[ω § D1 , D2 ] = ω § [D1 , D2 ] ’ (’1)(q+k1 )k2 D2 (ω) § D1 .

(1)

i(ω § L) = ω § i(L)

(2)

ω § LK = L(ω § K) + (’1)q+k’1 i(dω § K).

(3)

[ω § L1 , L2 ]§ = ω § [L1 , L2 ]§ ’

(4)

’ (’1)(q+l1 ’1)(l2 ’1) i(L2 )ω § L1 .

35.8

35.10 35. The Fr¨licher-Nijenhuis bracket

o 365

[ω § K1 , K2 ] = ω § [K1 , K2 ] ’ (’1)(q+k1 )k2 L(K2 )ω § K1

(5)

+ (’1)q+k1 dω § i(K1 )K2 .

[• — X, ψ — Y ] = • § ψ — [X, Y ]

(6)

’ iY d• § ψ — X ’ (’1)kl iX dψ § • — Y

’ d(iY • § ψ) — X ’ (’1)kl d(iX ψ § •) — Y

= • § ψ — [X, Y ] + • § LX ψ — Y ’ LY • § ψ — X

+ (’1)k (d• § iX ψ — Y + iY • § dψ — X) .

Proof. For (1), (2), (3) write out the de¬nitions. For (4) compute i([ω § L1 , L2 ]§ ).

For (5) compute L([ω § K1 , K2 ]). For (6) use (5).

35.9. Theorem. For Ki ∈ „¦ki (M ; T M ) and Li ∈ „¦ki +1 (M ; T M ) we have

[LK1 + iL1 , LK2 + iL2 ] =

(1)

= L [K1 , K2 ] + iL1 K2 ’ (’1)k1 k2 iL2 K1

+ i [L1 , L2 ]§ + [K1 , L2 ] ’ (’1)k1 k2 [K2 , L1 ] .

Each summand of this formula looks like a semidirect product of graded Lie algebras,

but the mappings

i : „¦(M ; T M ) ’ End(„¦(M ; T M ), [ , ])

]§ )

ad : „¦(M ; T M ) ’ End(„¦(M ; T M ), [ ,

do not take values in the subspaces of graded derivations. Instead we have for

K ∈ „¦k (M ; T M ) and L ∈ „¦l+1 (M ; T M ) the following relations:

iL [K1 , K2 ] = [iL K1 , K2 ] + (’1)k1 l [K1 , iL K2 ]

(2)

’ (’1)k1 l i([K1 , L])K2 ’ (’1)(k1 +l)k2 i([K2 , L])K1 .

[K, [L1 , L2 ]§ ] = [[K, L1 ], L2 ]§ + (’1)kk1 [L1 , [K, L2 ]]§ ’

(3)

’ (’1)kk1 [i(L1 )K, L2 ] ’ (’1)(k+k1 )k2 [i(L2 )K, L1 ] .

The algebraic meaning of these relations and its consequences in group theory have

been investigated in [Michor, 1989a]. The corresponding product of groups is well

known to algebraists under the name ˜Zappa-Szep™-product.

Proof. Equation (1) is an immediate consequence of (35.7). Equations (2) and (3)

follow from (1) by writing out the graded Jacobi identity.

35.10. Corollary of (28.6). For K, L ∈ „¦1 (M ; T M ) we have

[K, L](X, Y ) = [KX, LY ] ’ [KY, LX] ’ L([KX, Y ] ’ [KY, X])

’ K([LX, Y ] ’ [LY, X]) + (LK + KL)([X, Y ]).

35.10

366 Chapter VII. Calculus on in¬nite dimensional manifolds 35.13

35.11. Curvature. Let P ∈ „¦1 (M ; T M ) satisfy P —¦ P = P , i.e., P is a projection

in each ¬ber of T M . This is the most general case of a (¬rst order) connection.

We call ker P the horizontal space and im P the vertical space of the connection. If

im P is some ¬xed sub vector bundle or (tangent bundle of) a foliation, P can be

called a connection for it. The following result is immediate from (35.10).

Lemma. We have

¯

[P, P ] = 2R + 2R,

¯

where R, R ∈ „¦2 (M ; T M ) are given by R(X, Y ) = P [(Id ’P )X, (Id ’P )Y ] and

¯

R(X, Y ) = (Id ’P )[P X, P Y ].

If im(P ) is a sub vector bundle, then R is an obstruction against integrability of

¯

the horizontal bundle ker P , and R is an obstruction against integrability of the

¯

vertical bundle im P . Thus, we call R the curvature and R the cocurvature of the

connection P .

35.12. Lemma. Bianchi identity. If P ∈ „¦1 (M ; T M ) is a connection (¬ber

¯

projection) with curvature R and cocurvature R, then we have

¯

[P, R + R] = 0

¯

[R, P ] = iR R + iR R.

¯

¯

Proof. We have [P, P ] = 2R + 2R by (35.11), and [P, [P, P ]] = 0 by the graded

Jacobi identity. So the ¬rst formula follows. We have 2R = P —¦ [P, P ] = i[P,P ] P .

By (35.9.2) we get i[P,P ] [P, P ] = 2[i[P,P ] P, P ] ’ 0 = 4[R, P ]. Therefore, [R, P ] =

¯ ¯ ¯

1

4 i[P,P ] [P, P ] = i(R + R)(R + R) = iR R + iR R since R has vertical values and kills

¯

¯

vertical vectors, so iR R = 0; likewise for R.

35.13. f -relatedness of the Fr¨licher-Nijenhuis bracket. Let f : M ’ N be

o

a smooth mapping between manifolds. Two vector valued forms K ∈ „¦k (M ; T M )

and K ∈ „¦k (N ; T N ) are called f -related or f -dependent, if for all Xi ∈ Tx M we

have

Kf (x) (Tx f · X1 , . . . , Tx f · Xk ) = Tx f · Kx (X1 , . . . , Xk ).

(1)

Theorem.

(2) If K and K as above are f -related then iK —¦ f — = f — —¦ iK : „¦(N ) ’ „¦(M ).

(3) If iK —¦ f — | B 1 (N ) = f — —¦ iK | B 1 (N ), then K and K are f -related, where

B 1 denotes the space of exact 1-forms.

(4) If Kj and Kj are f -related for j = 1, 2, then iK1 K2 and iK1 K2 are f -related,

and also [K1 , K2 ]§ and [K1 , K2 ]§ are f -related.

(5) If K and K are f -related then LK —¦ f — = f — —¦ LK : „¦(N ) ’ „¦(M ).

(6) If LK —¦ f — | „¦0 (N ) = f — —¦ LK | „¦0 (N ), then K and K are f -related.

(7) If Kj and Kj are f -related for j = 1, 2, then their Fr¨licher-Nijenhuis

o

brackets [K1 , K2 ] and [K1 , K2 ] are also f -related.

35.13

35.14 35. The Fr¨licher-Nijenhuis bracket

o 367

Proof. (2) By (35.2), we have for ω ∈ „¦q (N ) and Xi ∈ Tx M :

(iK f — ω)x (X1 , . . . , Xq+k’1 ) =

sign σ (f — ω)x (Kx (Xσ1 , . . . , Xσk ), Xσ(k+1) , . . . )

1

= k! (q’1)!

σ

1

sign σ ωf (x) (Tx f · Kx (Xσ1 , . . . ), Tx f · Xσ(k+1) , . . . )

= k! (q’1)!

σ

1

sign σ ωf (x) (Kf (x) (Tx f · Xσ1 , . . . ), Tx f · Xσ(k+1) , . . . )

= k! (q’1)!

σ

= (f — iK ω)x (X1 , . . . , Xq+k’1 ).

(3) follows from this computation, since the dg, g ∈ C ∞ (M, R) separate points, by

convention (35.1).

(4) follows from the same computation for K2 instead of ω, the result for the bracket

then follows by (35.2.2).

(5) The algebra homomorphism f — intertwines the operators iK and iK by (2), and

f — commutes with the exterior derivative d. Thus, f — intertwines the commutators

[iK , d] = LK and [iK , d] = LK .

(6) For g ∈ „¦0 (N ) we have LK f — g = iK d f — g = iK f — dg, and on the other hand

f — LK g = f — iK dg. By (3) the result follows.

(7) The algebra homomorphism f — intertwines LKj and LKj , thus also their graded

commutators, which are equal to L([K1 , K2 ]) and L([K1 , K2 ]), respectively. Then

use (6).

35.14. Let f : M ’ N be a local di¬eomorphism. Then we can consider the

pullback operator f — : „¦(N ; T N ) ’ „¦(M ; T M ), given by

(f — K)x (X1 , . . . , Xk ) = (Tx f )’1 Kf (x) (Tx f · X1 , . . . , Tx f · Xk ).

(1)

This is a special case of the pullback operator for sections of natural vector bundles.

Clearly, K and f — K are then f -related.

Theorem. In this situation we have:

f — [K, L] = [f — K, f — L].

(2)

f — iK L = if — K f — L.

(3)

f — [K, L]§ = [f — K, f — L]§ .

(4)

For a vector ¬eld X ∈ X(M ) admitting a local ¬‚ow FlX and K ∈ „¦(M ; T M )

(5) t

X—

‚

the Lie derivative LX K = ‚t 0 (Flt ) K is de¬ned. Then we have LX K =

[X, K], the Fr¨licher-Nijenhuis-bracket.

o

This is sometimes expressed by saying that the Fr¨licher-Nijenhuis bracket, the

o

bracket [ , ]§ , etc., are natural bilinear concomitants.

Proof. (2) “ (4) are obvious from (35.13). They also follow directly from the

geometrical constructions of the operators in question.

35.14

368 Chapter VII. Calculus on in¬nite dimensional manifolds 35.15

(5) By inserting Yi ∈ X(M ) we get from (1) the following expression which we can

di¬erentiate using (32.15) repeatedly.

(FlX )— (K(Y1 , . . . , Yk )) = ((FlX )— K)((FlX )— Y1 , . . . , (FlX )— Yk )

’t ’t ’t

t

(FlX )— (K(Y1 , . . . , Yk ))

‚

[X,K(Y1 , . . . , Yk )] = ’t

‚t 0

((FlX )— K)((FlX )— Y1 , . . . , (FlX )— Yk )

‚

= ’t ’t

t

‚t 0

(FlX )— K)(Y1 , . . . , Yk ) ’ (FlX )— Yi , . . . , Yk )

‚ ‚

= ( ‚t K(Y1 , . . . , ’t

t ‚t 0

0

1¤i¤k

(FlX )— K)(Y1 , . . . , Yk ) ’

‚

= ( ‚t K(Y1 , . . . , [X, Yi ], . . . , Yk ).

t

0

1¤i¤k

This leads to

(FlX )— K)(Y1 , . . . , Yk ) = [X, K(Y1 , . . . , Yk )]

‚

( ‚t t

0

+ K([X, Yi ], Y1 , . . . Yi . . . , Yk )

1¤i¤k

= [X, K](Y1 , . . . , Yk ), by (35.5.1).

35.15. Remark. Finally, we mention the best known application of the Fr¨licher-

o

Nijenhuis bracket, which also led to its discovery. A vector valued 1-form J ∈

„¦1 (M ; T M ) with J —¦ J = ’ Id is called an almost complex structure. If it exists, J

√

can be viewed as a ¬ber multiplication with ’1 on T M . By (35.10) we have

[J, J](X, Y ) = 2([JX, JY ] ’ [X, Y ] ’ J[X, JY ] ’ J[JX, Y ]).

1

The vector valued form 2 [J, J] is also called the Nijenhuis tensor of J. In ¬nite

dimensions an almost complex structure J comes from a complex structure on the

manifold if and only if the Nijenhuis tensor vanishes.

35.15

369

Chapter VIII

In¬nite Dimensional Di¬erential Geometry

36. Lie Groups . . . . . . . . . . . . . .... . . . . . . . . . . 369

37. Bundles and Connections . . . . . . . .... . . . . . . . . . . 375

38. Regular Lie Groups . . . . . . . . . .... . . . . . . . . . . 404

39. Bundles with Regular Structure Groups .... . . . . . . . . . . 422

40. Rudiments of Lie Theory for Regular Lie Groups . . . . . . . . . . 426

The theory of in¬nite dimensional Lie groups can be pushed surprisingly far: Ex-

ponential mappings are unique if they exist. In general, they are neither locally

surjective nor locally injective. A stronger requirement (leading to regular Lie

groups) is to assume that smooth curves in the Lie algebra integrate to smooth

curves in the group in a smooth way (an ˜evolution operator™ exists). This is due

to [Milnor, 1984] who weakened the concept of [Omori et al., 1982]. It turns out

that regular Lie groups have strong permanence properties. In fact, up to now

all known Lie groups are regular. Connections on smooth principal bundles with

a regular Lie group as structure group have parallel transport (39.1), and for ¬‚at

connections the horizontal distribution is integrable (39.2). So some (equivariant)

partial di¬erential equations in in¬nite dimensions are very well behaved, although

in general there are counter-examples in every possible direction (some can be found

in (32.12)).

The actual development is quite involved. We start with general in¬nite dimensional

Lie groups in section (36), but for a detailed study of the evolution operator of

regular Lie groups (38.4) we need in (38.10) the Maurer-Cartan equation for right

(or left) logarithmic derivatives of mappings with values in the Lie group (38.1),

and this we can only get by looking at principal connections. Thus, in the second

section (37) bundles, connections, principal bundles, curvature, associated bundles,

and all results of principal bundle geometry which do not involve parallel transport

are developed. Finally, we then prove the strong existence results mentioned above

and treat regular Lie groups in section (38), and principal bundles with regular

structure groups in section (39). The material in this chapter is an extended version

of [Kriegl, Michor, 1997].

36. Lie Groups

36.1. De¬nition. A Lie group G is a smooth manifold and a group such that the

multiplication µ : G — G ’ G and the inversion ν : G ’ G are smooth. If not

36.1

370 Chapter VIII. In¬nite dimensional di¬erential geometry 36.3

stated otherwise, G may be in¬nite dimensional. If an implicit function theorem is

available, then smoothness of ν follows from smoothness of µ.

We shall use the following notation:

µ : G — G ’ G, multiplication, µ(x, y) = x.y.

µa : G ’ G, left translation, µa (x) = a.x.

µa : G ’ G, right translation, µa (x) = x.a.

ν : G ’ G, inversion, ν(x) = x’1 .

e ∈ G, the unit element.

36.2. Lemma. The kinematic tangent mapping T(a,b) µ : Ta G — Tb G ’ Tab G is

given by

T(a,b) µ.(Xa , Yb ) = Ta (µb ).Xa + Tb (µa ).Yb ,

and Ta ν : Ta G ’ Ta’1 G is given by

’1 ’1

Ta ν = ’Te (µa ).Ta (µa’1 ) = ’Te (µa’1 ).Ta (µa ).

Proof. Let insa : G ’ G — G, insa (x) = (a, x) be the right insertion, and let

insb : G ’ G — G, insb (x) = (x, b) be the left insertion. Then we have

T(a,b) µ.(Xa , Yb ) = T(a,b) µ.(Ta (insb ).Xa + Tb (insa ).Yb ) =

= Ta (µ —¦ insb ).Xa + Tb (µ —¦ insa ).Yb = Ta (µb ).Xa + Tb (µa ).Yb .

Now we di¬erentiate the equation µ(a, ν(a)) = e; this gives in turn

’1

0e = T(a,a’1 ) µ.(Xa , Ta ν.Xa ) = Ta (µa ).Xa + Ta’1 (µa ).Ta ν.Xa ,

’1 ’1

Ta ν.Xa = ’Te (µa )’1 .Ta (µa ).Xa = ’Te (µa’1 ).Ta (µa ).Xa .

36.3. Invariant vector ¬elds and Lie algebras. Let G be a (real) Lie group. A

(kinematic) vector ¬eld ξ on G is called left invariant, if µ— ξ = ξ for all a ∈ G, where

a

µa ξ = T (µa’1 )—¦ξ—¦µa as in (32.9). Since by (32.11) we have µ— [ξ, ·] = [µ— ξ, µ— ·], the

—

a a a

space XL (G) of all left invariant vector ¬elds on G is closed under the Lie bracket,

so it is a sub Lie algebra of X(G). Any left invariant vector ¬eld ξ is uniquely

determined by ξ(e) ∈ Te G, since ξ(a) = Te (µa ).ξ(e). Thus, the Lie algebra XL (G)

of left invariant vector ¬elds is linearly isomorphic to Te G, and the Lie bracket on

XL (G) induces a Lie algebra structure on Te G, whose bracket is again denoted by

[ , ]. This Lie algebra will be denoted as usual by g, sometimes by Lie(G).

We will also give a name to the isomorphism with the space of left invariant vector

¬elds: L : g ’ XL (G), X ’ LX , where LX (a) = Te µa .X. Thus, [X, Y ] =

[LX , LY ](e).

A vector ¬eld · on G is called right invariant, if (µa )— · = · for all a ∈ G. If ξ

is left invariant, then ν — ξ is right invariant, since ν —¦ µa = µa’1 —¦ ν implies that

(µa )— ν — ξ = (ν —¦µa )— ξ = (µa’1 —¦ν)— ξ = ν — (µa’1 )— ξ = ν — ξ. The right invariant vector

¬elds form a sub Lie algebra XR (G) of X(G), which also is linearly isomorphic to

Te G and induces a Lie algebra structure on Te G. Since ν — : XL (G) ’ XR (G) is an

isomorphism of Lie algebras by (32.11), Te ν = ’ Id : Te G ’ Te G is an isomorphism

between the two Lie algebra structures. We will denote by R : g = Te G ’ XR (G)

the isomorphism discussed, which is given by RX (a) = Te (µa ).X.

36.3

36.7 36. Lie groups 371

36.4. Remark. It would be tempting to apply also other kinds of tangent bundle

functors like D and D[1,∞) , where one gets Lie algebras of smooth sections, see

(32.8). Some results will stay true like (36.3), (36.5). In general, one gets strictly

larger Lie algebras for Lie groups, see (28.4). But the functors D and D[1,∞) do not

respect products in general, see (28.16), so e.g. (36.2) is wrong for these functors.

36.5. Lemma. If LX is a left invariant vector ¬eld and RY is a right invariant

one, then [LX , RY ] = 0. So if the ¬‚ows of LX and RY exist, they commute.

Proof. We consider the vector ¬eld 0 — LX ∈ X(G — G), given by (0 — LX )(a, b) =

(0a , LX (b)). Then T(a,b) µ.(0a , LX (b)) = Ta µb .0a + Tb µa .LX (b) = LX (ab), so 0 — LX

is µ-related to LX . Likewise, RY — 0 is µ-related to RY . But then 0 = [0 —

LX , RY — 0] is µ-related to [LX , RY ] by (32.10). Since µ is surjective, [LX , RY ] = 0

follows.

36.6. Lemma. Let • : G ’ H be a smooth homomorphism of Lie groups. Then

• := Te • : g = Te G ’ h = Te H is a Lie algebra homomorphism.

Proof. For X ∈ g and x ∈ G we have

Tx •.LX (x) = Tx •.Te µx .X = Te (• —¦ µx ).X

= Te (µ•(x) —¦ •).X = Te (µ•(x) ).Te •.X = L• (X) (•(x)).

So LX is •-related to L• (X) . By (32.10), the ¬eld [LX , LY ] = L[X,Y ] is •-related

to [L• (X) , L• (Y ) ] = L[• (X),• (Y )] . So we have T • —¦ L[X,Y ] = L[• (X),• (Y )] —¦ •. If

we evaluate this at e the result follows.

36.7. One parameter subgroups. Let G be a Lie group with Lie algebra g. A

one parameter subgroup of G is a Lie group homomorphism ± : (R, +) ’ G, i.e. a

smooth curve ± in G with ±(s + t) = ±(s).±(t), and hence ±(0) = e.

Note that a smooth mapping β : (’µ, µ) ’ G satisfying β(t)β(s) = β(t + s) for

|t|, |s|, |t + s| < µ is the restriction of a one parameter subgroup. Namely, choose

0 < t0 < µ/2. Any t ∈ R can be uniquely written as t = N.t0 + t for 0 ¤ t < t0

and N ∈ Z. Put ±(t) = β(t0 )N β(t ). The required properties are easy to check.

Lemma. Let ± : R ’ G be a smooth curve with ±(0) = e. Let X ∈ g. Then the

following assertions are equivalent.

‚

(1) ± is a one parameter subgroup with X = ‚t 0 ±(t).

(2) ±(t) is an integral curve of the left invariant vector ¬eld LX and also an

integral curve of the right invariant vector ¬eld RX .

(3) FlLX (t, x) := x.±(t) (or FlLX = µ±(t) ) is the (unique by (32.16)) global ¬‚ow

t

of LX in the sense of (32.13).

(4) FlRX (t, x) := ±(t).x (or FlRX = µ±(t) ) is the (unique) global ¬‚ow of RX .

t

Moreover, each of these properties determines ± uniquely.

Proof. (1) ’ (3) We have

d d d

|0 x.±(t + s) = ds |0 x.±(t).±(s)

dt x.±(t) = ds

d d

|0 µx.±(t) ±(s) = Te (µx.±(t) ). ds |0 ±(s)

= = LX (x.±(t)).

ds

36.7

372 Chapter VIII. In¬nite dimensional di¬erential geometry 36.8

Since it is obviously a ¬‚ow, we have (3).

ν—ξ

= ν ’1 —¦ Flξ —¦ν by (32.16). Therefore, we have by (36.3)

(3) ” (4) We have Flt t

—

(FlRX (x’1 ))’1 = (ν —¦ FlRX —¦ν)(x) = Flν RX

(x)

t t t

= Fl’LX (x) = FlLX (x) = x.±(’t).

’t

t

So FlRX (x’1 ) = ±(t).x’1 , and FlRX (y) = ±(t).y.

t t

(3) and (4) together clearly imply (2).

(2) ’ (1) This is a consequence of the following result.

Claim. Consider two smooth curves ±, β : R ’ G with ±(0) = e = β(0) which

satisfy the two di¬erential equations

d

dt ±(t) = LX (±(t))

d

dt β(t) = RX (β(t)).

Then ± = β, and it is a 1-parameter subgroup.

We have ± = β since

= T µβ(’t) .LX (±(t)) ’ T µ±(t) .RX (β(’t))

d

dt (±(t)β(’t))

= T µβ(’t) .T µ±(t) .X ’ T µ±(t) .T µβ(’t) .X = 0.

Next we calculate for ¬xed s

’ s)β(s)) = T µβ(s) .RX (β(t ’ s)) = RX (β(t ’ s)β(s)).

d

dt (β(t

Hence, by the ¬rst part of the proof β(t ’ s)β(s) = ±(t) = β(t).

The statement about uniqueness follows from (32.16), or from the claim.

36.8. De¬nition. Let G be a Lie group with Lie algebra g. We say that G admits

an exponential mapping if there exists a smooth mapping exp : g ’ G such that

t ’ exp(tX) is the (unique by (36.7)) 1-parameter subgroup with tangent vector

X at 0. Then we have by (36.7)

(1) FlLX (t, x) = x. exp(tX).

(2) FlRX (t, x) = exp(tX).x.

(3) exp(0) = e and T0 exp = Id : T0 g = g ’ Te G = g since T0 exp .X =

|0 exp(0 + t.X) = dt |0 FlLX (t, e) = X.

d d

dt

(4) Let • : G ’ H be a smooth homomorphism between Lie groups admitting

exponential mappings. Then the diagram

wh

•

g

u u

expG expH

wH

•

G

commutes, since t ’ •(expG (tX)) is a one parameter subgroup of H, and

d G G H

dt |0 •(exp tX) = • (X), so •(exp tX) = exp (t• (X)).

36.8

36.10 36. Lie groups 373

36.9. Remarks. [Omori et al., 1982, 1983, etc.] gave conditions under which a

smooth Lie group modeled on Fr´chet spaces admits an exponential mapping. We

e

shall elaborate on this notion in (38.4) below. They called such groups ˜regular

Fr´chet Lie groups™. We do not know any smooth Fr´chet Lie group which does not

e e

admit an exponential mapping.

If G admits an exponential mapping, it follows from (36.8.3) that exp is a di¬eo-

morphism from a neighborhood of 0 in g onto a neighborhood of e in G, if a suitable

inverse function theorem is applicable. This is true, for example, for smooth Banach

Lie groups, also for gauge groups, see (42.21) but it is wrong for di¬eomorphism

groups, see (43.3).

If E is a Banach space, then in the Banach Lie group GL(E) of all bounded linear

automorphisms of E the exponential mapping is given by the series exp(X) =

∞1 i

i=0 i! X .

If G is connected with exponential mapping and U ‚ g is open with 0 ∈ U , then

one may ask whether the group generated by exp(U ) equals G. Note that this is a

normal subgroup. So if G is simple, the answer is yes. This is true for connected

components of di¬eomorphism groups and many of their important subgroups, see

[Epstein, 1970], [Thurston, 1974], [Mather, 1974, 1975, 1984, 1985], [Banyaga, 1978].

Results on weakened versions of the Baker-Campbell-Hausdor¬ formula can be

found in [Wojty´ski, 1994].

n

36.10. The adjoint representation. Let G be a Lie group with Lie algebra

g. For a ∈ G we de¬ne conja : G ’ G by conja (x) = axa’1 . It is called the

conjugation or the inner automorphism by a ∈ G. This de¬nes a smooth action of

G on itself by automorphisms.

The adjoint representation Ad : G ’ GL(g) ‚ L(g, g) is given by Ad(a) =

(conja ) = Te (conja ) : g ’ g for a ∈ G. By (36.6), Ad(a) is a Lie algebra ho-

momorphism, moreover

’1 ’1

Ad(a) = Te (conja ) = Ta (µa ).Te (µa ) = Ta’1 (µa ).Te (µa ).

Finally, we de¬ne the (lower case) adjoint representation of the Lie algebra g, ad :

g ’ gl(g) := L(g, g) by ad := Ad = Te Ad.

We shall also use the right Maurer-Cartan form κr ∈ „¦1 (G, g), given by κr = g

’1

Tg (µg ) : Tg G ’ g; similarly the left Maurer-Cartan form κl ∈ „¦1 (G, g) is given

by κl = Tg (µg’1 ) : Tg G ’ g.

g

Lemma.

(1) LX (a) = RAd(a)X (a) for X ∈ g and a ∈ G.

(2) ad(X)Y = [X, Y ] for X, Y ∈ g.

(3) dAd = (ad —¦ κr ).Ad = Ad.(ad —¦ κl ) : T G ’ L(g, g).

’1

Proof. (1) LX (a) = Te (µa ).X = Te (µa ).Te (µa —¦ µa ).X = RAd(a)X (a).

36.10

374 Chapter VIII. In¬nite dimensional di¬erential geometry 36.10

Proof of (2). We need some preparation. Let V be a convenient vector space. For

f ∈ C ∞ (G, V ) we de¬ne the left trivialized derivative Dl f ∈ C ∞ (G, L(g, V )) by

(4) Dl f (x).X := df (x).Te µx .X = (LX f )(x).

For f ∈ C ∞ (G, R) and g ∈ C ∞ (G, V ) we have

(5) Dl (f.g)(x).X = d(f.g)(Te µx .X)

= df (Te µx .X).g(x) + f (x).dg(Te µx .X)

= (f.Dl g + Dl f — g)(x).X.

From the formula

Dl Dl f (x)(X)(Y ) = Dl (Dl f ( ).Y )(x).X

= Dl (LY f )(x).X = LX LY f (x)

follows

Dl Dl f (x)(X)(Y ) ’ Dl Dl f (x)(Y )(X) = L[X,Y ] f (x) = Dl f (x).[X, Y ].

(6)

We consider now the linear isomorphism L : C ∞ (G, g) ’ X(G) given by Lf (x) =

Te µx .f (x) = Lf (x) (x) for f ∈ C ∞ (G, g). If h ∈ C ∞ (G, V ) we get (Lf h)(x) =

Dl h(x).f (x). For f, g ∈ C ∞ (G, g) and h ∈ C ∞ (G, R) we get in turn, using (6) and

(5), generalized to the bilinear pairing L(g, R) — g ’ R,

(Lf Lg h)(x) = Dl (Dl h( ).g( ))(x).f (x)

= Dl Dl h(x)(f (x))(g(x)) + Dl h(x).Dl g(x).f (x)

2

([Lf , Lg ]h)(x) = Dl h(x).(f (x), g(x)) + Dl h(x).Dl g(x).f (x)’

2

’ Dl h(x).(g(x), f (x)) ’ Dl h(x).Dl f (x).g(x)

= Dl h(x). [f (x), g(x)]g + Dl g(x).f (x) ’ Dl f (x).g(x)

[Lf , Lg ] = L [f, g]g + Dl g.f ’ Dl f.g .

(7)

Now we are able to prove the second assertion of the lemma. For X, Y ∈ g we will

apply (7) to f (x) = X and g(x) = Ad(x’1 ).Y . We have Lg = RY by (1), and

[Lf , Lg ] = [LX , RY ] = 0 by (36.5). So

0 = [LX , RY ](x) = [Lf , Lg ](x)

= L([X, (Ad —¦ ν)Y ]g + Dl ((Ad —¦ ν)( ).X).Y ’ 0)(x)

[X, Y ] = [X, Ad(e)Y ] = ’Dl ((Ad —¦ ν)( ).X)(e).Y

= d(Ad( ).X)(e).Y = ad(X)Y.

Proof of (3). Let X, Y ∈ g and g ∈ G, and let c : R ’ G be a smooth curve with

c(0) = e and c (0) = X. Then we have

‚ ‚

‚t |0 Ad(c(t).g).Y ‚t |0 Ad(c(t)).Ad(g).Y

(dAd(RX (g))).Y = =

(ad —¦ κr )(RX (g)).Ad(g).Y,

= ad(X)Ad(g)Y =

and similarly for the second formula.

36.10

37.1 37. Bundles and connections 375

36.11. Let : G — M ’ M be a smooth left action of a Lie group G, so ∨ : G ’

Di¬(M ) is a group homomorphism. Then we have partial mappings a : M ’ M

and x : G ’ M , given by a (x) = x (a) = (a, x) = a.x.

M

For any X ∈ g we de¬ne the fundamental vector ¬eld ζX = ζX ∈ X(M ) by ζX (x) =

Te ( x ).X = T(e,x) .(X, 0x ).

Lemma. In this situation, the following assertions hold:

ζ : g ’ X(M ) is a linear mapping.

(1)

(2) Tx ( a ).ζX (x) = ζAd(a)X (a.x).

RX — 0M ∈ X(G — M ) is -related to ζX ∈ X(M ).

(3)

[ζX , ζY ] = ’ζ[X,Y ] .

(4)

Proof. (1) is clear.

(b) = abx = aba’1 ax =

x ax

(2) We have conja (b), so

a

Tx ( a ).ζX (x) = Tx ( a ).Te ( x ).X = Te ( x

—¦ ).X

a

ax

= Te ( ).Ad(a).X = ζAd(a)X (ax).

(3) We have —¦ (Id — a ) = —¦ (µa — Id) : G — M ’ M , so

ζX ( (a, x)) = T(e,ax) .(X, 0ax ) = T .(Id —T ( a )).(X, 0x )

= T .(T (µa ) — Id).(X, 0x ) = T .(RX — 0M )(a, x).

(4) [RX — 0M , RY — 0M ] = [RX , RY ] — 0M = ’R[X,Y ] — 0M is -related to [ζX , ζY ]

by (3) and by (32.10). On the other hand, ’R[X,Y ] — 0M is -related to ’ζ[X,Y ] by

(3) again. Since is surjective we get [ζX , ζY ] = ’ζ[X,Y ] .

36.12. Let r : M — G ’ M be a right action, so r∨ : G ’ Di¬(M ) is a group anti

homomorphism. We will use the following notation: ra : M ’ M and rx : G ’ M ,

given by rx (a) = ra (x) = r(x, a) = x.a.

M

For any X ∈ g we de¬ne the fundamental vector ¬eld ζX = ζX ∈ X(M ) by ζX (x) =

Te (rx ).X = T(x,e) r.(0x , X).

Lemma. In this situation the following assertions hold:

ζ : g ’ X(M ) is a linear mapping.

(1)

Tx (ra ).ζX (x) = ζAd(a’1 )X (x.a).

(2)

0M — LX ∈ X(M — G) is r-related to ζX ∈ X(M ).

(3)

(4) [ζX , ζY ] = ζ[X,Y ] .

37. Bundles and Connections

37.1. De¬nition. A (¬ber) bundle (p : E ’ M, S) = (E, p, M, S) consists of

smooth manifolds E, M , S, and a smooth mapping p : E ’ M . Furthermore, each

37.1

376 Chapter VIII. In¬nite dimensional di¬erential geometry 37.2

x ∈ M has an open neighborhood U such that E | U := p’1 (U ) is di¬eomorphic to

U — S via a ¬ber respecting di¬eomorphism:

w

‘ ψ

E|U U —S

‘p“ &

‘ &pr

)

& 1

U.

E is called total space, M is called base space or basis, p is a ¬nal surjective smooth

mapping, called projection, and S is called standard ¬ber. (U, ψ) as above is called

a ¬ber chart.

A collection of ¬ber charts (U± , ψ± ) such that (U± ) is an open cover of M , is called

a ¬ber bundle atlas. If we ¬x such an atlas, then ψ± —¦ ψβ ’1 (x, s) = (x, ψ±β (x, s)),

where ψ±β : (U± ©Uβ )—S ’ S is smooth, and where ψ±β (x, ) is a di¬eomorphism of

S for each x ∈ U±β := U± © Uβ . These mappings ψ±β are called transition functions

of the bundle. They satisfy the cocycle condition: ψ±β (x) —¦ ψβγ (x) = ψ±γ (x) for

x ∈ U±βγ and ψ±± (x) = IdS for x ∈ U± . Therefore, the collection (ψ±β ) is called a

cocycle of transition functions.

Given an open cover (U± ) of a manifold M and a cocycle of transition functions

(ψ±β ) we may construct a ¬ber bundle (p : E ’ M, S), as in ¬nite dimensions.

37.2. Let (p : E ’ M, S) be a ¬ber bundle. We consider the ¬ber linear tangent

mapping T p : T E ’ T M and its kernel ker T p =: V E, which is called the vertical

bundle of E. It is a locally splitting vector subbundle of the tangent bundle T E,

by the following argument: E looks locally like UM — US , where UM is c∞ -open in

a modeling space WM of M and US in a modeling space WS of S. Then T E looks

locally like UM — WM — US — WS , and the mapping T p corresponds to (x, v, y, w) ’

(x, v), so that V E looks locally like UM — 0 — US — WS .

De¬nition. A connection on the ¬ber bundle (p : E ’ M, S) is a vector valued

1-form ¦ ∈ „¦1 (E; V E) with values in the vertical bundle V E such that ¦ —¦ ¦ = ¦

and im¦ = V E; so ¦ is just a projection T E ’ V E.

The kernel ker ¦ is a sub vector bundle of T E, it is called the space of horizontal

vectors or the horizontal bundle, and it is denoted by HE. Clearly, T E = HE •V E

and Tu E = Hu E • Vu E for u ∈ E.

Now we consider the mapping (T p, πE ) : T E ’ T M —M E. Then by de¬nition

(T p, πE )’1 (0p(u) , u) = Vu E, so (T p, πE ) | HE : HE ’ T M —M E is a ¬ber linear

isomorphism, which may be checked in a chart. Its inverse is denoted by

C := ((T p, πE ) | HE)’1 : T M —M E ’ HE ’ T E.

So C : T M —M E ’ T E is ¬ber linear over E and a right inverse for (T p, πE ). C

is called the horizontal lift associated to the connection ¦.

Note the formula ¦(ξu ) = ξu ’ C(T p.ξu , u) for ξu ∈ Tu E. So we can equally well

describe a connection ¦ by specifying C. Then we call ¦ vertical projection and

χ := idT E ’ ¦ = C —¦ (T p, πE ) will be called horizontal projection.

37.2

37.5 37. Bundles and connections 377

37.3. Curvature. If ¦ : T E ’ V E is a connection on the bundle (p : E ’ M, S),

then as in (35.11) the curvature R of ¦ is given by

2R = [¦, ¦] = [Id ’¦, Id ’¦] ∈ „¦2 (E; V E).

¯

The cocurvature R vanishes since the vertical bundle V E is integrable. We have

1

R(X, Y ) = 2 [¦, ¦](X, Y ) = ¦[χX, χY ] by (35.11), so R is an obstruction against

involutivity of the horizontal subbundle in the following sense: If the curvature

R vanishes, then horizontal kinematic vector ¬elds on E also have a horizontal

Lie bracket. Note that for vector ¬elds ξ, · ∈ X(M ) and their horizontal lifts

Cξ, C· ∈ X(E) we have R(Cξ, C·) = [Cξ, C·] ’ C([ξ, ·]). Since the vertical bundle

V E is even integrable, by (35.12) we have the Bianchi identity [¦, R] = 0.

37.4. Pullback. Let (p : E ’ M, S) be a ¬ber bundle, and consider a smooth

mapping f : N ’ M . Let us consider the pullback N —(f,M,p) E := {(n, e) ∈

N — E : f (n) = p(e)}; we will denote it by f — E. The following diagram sets up

w

some further notation for it: p— f

—

fE E

p

f —p

u u

w M.

f

N

Proposition. In the situation above we have:

(1) (f — E, f — p, N, S) is a ¬ber bundle, and p— f is a ¬berwise di¬eomorphism.

(2) If ¦ ∈ „¦1 (E; T E) is a connection on the bundle E, then the vector valued

form f — ¦, given by (f — ¦)u (X) := Tu (p— f )’1 .¦.Tu (p— f ).X for X ∈ Tu E, is

a connection on the bundle f — E. The forms f — ¦ and ¦ are p— f -related in

the sense of (35.13).

(3) The curvatures of f — ¦ and ¦ are also p— f -related.

Proof. (1) If (U± , ψ± ) is a ¬ber bundle atlas of (p : E ’ M, S) in the sense

of (37.1), then (f ’1 (U± ), (f — p, pr2 —¦ψ± —¦ p— f )) is visibly a ¬ber bundle atlas for

the pullback bundle (f — E, f — p, N, S). (2) is obvious. (3) follows from (2) and

(35.13.7).

37.5. Local description. Let ¦ be a connection on (p : E ’ M, S). Let us ¬x

a ¬ber bundle atlas (U± ) with transition functions (ψ±β ), and let us consider the

connection ((ψ± )’1 )— ¦ ∈ „¦1 (U± — S; U± — T S), which may be written in the form

(((ψ± )’1 )— ¦)(ξx , ·y ) =: ’“± (ξx , y) + ·y for ξx ∈ Tx U± and ·y ∈ Ty S,

since it reproduces vertical vectors. The “± are given by

(0x , “± (ξx , y)) := ’T (ψ± ).¦.T (ψ± )’1 .(ξx , 0y ).

We consider “± as an element of the space „¦1 (U± ; X(S)), a 1-form on U ± with

values in the Lie algebra X(S) of all kinematic vector ¬elds on the standard ¬ber.

The “± are called the Christo¬el forms of the connection ¦ with respect to the

bundle atlas (U± , ψ± ).

37.5

378 Chapter VIII. In¬nite dimensional di¬erential geometry 37.6

Lemma. The transformation law for the Christo¬el forms is

)).“β (ξx , y) = “± (ξx , ψ±β (x, y)) ’ Tx (ψ±β (

Ty (ψ±β (x, , y)).ξx .

The curvature R of ¦ satis¬es

(ψ± )— R = d“± + 1 [“± , “± ]§ .

’1

X(S)

2

Here d“± is the exterior derivative of the 1-form “± ∈ „¦1 (U± , X(S)) with values in

the convenient vector space X(S).

The formula for the curvature is the Maurer-Cartan formula which in this general

setting appears only on the level of local description.

Proof. From (ψ± —¦ (ψβ )’1 )(x, y) = (x, ψ±β (x, y)) we get that

T (ψ± —¦ (ψβ )’1 ).(ξx , ·y ) = (ξx , T(x,y) (ψ±β ).(ξx , ·y )), and thus

’1 ’1

T (ψβ ).(0x , “β (ξx , y)) = ’¦(T (ψβ )(ξx , 0y )) =

’1

’1

= ’¦(T (ψ± ).T (ψ± —¦ ψβ ).(ξx , 0y )) =

’1

= ’¦(T (ψ± )(ξx , T(x,y) (ψ±β )(ξx , 0y ))) =

’1 ’1

= ’¦(T (ψ± )(ξx , 0ψ±β (x,y) )) ’ ¦(T (ψ± )(0x , T(x,y) ψ±β (ξx , 0y ))) =

’1 ’1

= T (ψ± ).(0x , “± (ξx , ψ±β (x, y))) ’ T (ψ± )(0x , Tx (ψ±β ( , y)).ξx ).

This implies the transformation law.

For the curvature R of ¦ we have by (37.3) and (37.4.3)

(ψ± )— R ((ξ 1 , · 1 ), (ξ 2 , · 2 )) =

’1

= (ψ± )— ¦ [(Id ’(ψ± )— ¦)(ξ 1 , · 1 ), (Id ’(ψ± )— ¦)(ξ 2 , · 2 )] =

’1 ’1 ’1

= (ψ± )— ¦[(ξ 1 , “± (ξ 1 )), (ξ 2 , “± (ξ 2 ))] =

’1

= (ψ± )— ¦ [ξ 1 , ξ 2 ], ξ 1 “± (ξ 2 ) ’ ξ 2 “± (ξ 1 ) + [“± (ξ 1 ), “± (ξ 2 )] =

’1

= ’“± ([ξ 1 , ξ 2 ]) + ξ 1 “± (ξ 2 ) ’ ξ 2 “± (ξ 1 ) + [“± (ξ 1 ), “± (ξ 2 )] =

= d“± (ξ 1 , ξ 2 ) + [“± (ξ 1 ), “± (ξ 2 )]X(S) .

37.6. Parallel transport. Let ¦ be a connection on a bundle (p : E ’ M, S),

and let c : (a, b) ’ M be a smooth curve with 0 ∈ (a, b), c(0) = x. The parallel

transport along c is a smooth mapping Ptc : U ’ E, where U is a neighborhood

of Diag(a, b) —(c—¦pr2 ,M,p) E in (a, b) — (a, b) —(c—¦pr2 ,M,p) E, such that the following

properties hold:

(1) U © ((a, b) — {s} — {uc(s) }) is connected for each s ∈ (a, b) and each uc(s) ∈

Ec(s) .

(2) p(Pt(c, t, s, uc(s) )) = c(t) if de¬ned, and Pt(c, s, s, uc(s) ) = uc(s) .

d

(3) ¦( dt Pt(c, t, s, uc(s) )) = 0 if de¬ned.

37.6

37.7 37. Bundles and connections 379

(4) If Pt(c, t, s, uc(s) ) exists then Pt(c, r, s, uc(s) ) = Pt(c, r, t, Pt(c, t, s, uc(s) )) in

the sense that existence of both sides is equivalent and we have equality.

(5) U is maximal for properties (1) to (4).

(6) Pt also depends smoothly on c in the Fr¨licher space C ∞ ((a, b), M ), see

o

(23.1), in the following sense: For any smooth mapping c : R — (a, b) ’ M

we have: For each s ∈ (a, b), each r ∈ R, and each u ∈ Ec(r,s) there are

a neighborhood Uc,r,s,u of (r, s, s, u) in R — (a, b) — (a, b) —(c—¦pr1,3 ,M,p) E ‚

R — (a, b) — (a, b) — E such that Uc,r,s,u (r , t, s , u ) ’ Pt(c(r , ), t, s , u )

is de¬ned and smooth.

(7) Reparameterization invariance: If f : (a , b ) ’ (a, b) is smooth, then we

have Pt(c, f (t), f (s), uc(f (s)) ) = Pt(c —¦ f, t, s, uc(f (s)) )

Requirements (1) “ (5) are essential. (6) is a further requirement which is not nec-

essary for the uniqueness result below, and (7) is a consequence of this uniqueness

result.

Proposition. The parallel transport along c is unique if it exists.

Proof. Consider the pullback bundle (c— E, c— p, (a, b), S) and the pullback con-

nection c— ¦ on it. We shall need the horizontal lift C : T (a, b) —(a,b) c— E ’

T (c— E) = (T c)— (T E) associated to c— ¦, from (37.2). Consider the constant vec-

tor ¬eld ‚ ∈ X((a, b)) and its horizontal lift C(‚) ∈ X(c— E) which is given by

C(‚)(us ) = C(‚|s , us ) ∈ Tus (c— E). Now from the properties of the parallel trans-

port we see that t ’ Pt(c(s+ ), t, s, us ) is a ¬‚ow line of the horizontal vector

¬eld C(‚) with initial value us = (s, uc(s) ) ∈ (c— E)s ∼ {s} — Ec(s) . (3) says that

=

it has the ¬‚ow property, so that by uniqueness of the ¬‚ow (32.16) we see that

C(‚)

Pt(c, t) = Flt is unique if it exists.

At this place one could consider complete connections (those whose parallel trans-

port exists globally), which then give rise to holonomy groups, even for ¬ber bundles

without structure groups. In ¬nite dimensions some deep results are available, see

[Kol´ˇ, Michor, Slov´k, 1993, pp81].

ar a

37.7. De¬nition. Let G be a Lie group, and let (p : E ’ M, S) be a ¬ber bundle

as in (37.1). A G-bundle structure on the ¬ber bundle consists of the following

data:

(1) A left action : G — S ’ S of the Lie group on the standard ¬ber.

(2) A ¬ber bundle atlas (U± , ψ± ) whose transition functions (ψ±β ) act on S

via the G-action: There is a family of smooth mappings (•±β : U±β ’ G)

which satis¬es the cocycle condition •±β (x)•βγ (x) = •±γ (x) for x ∈ U±βγ

and •±± (x) = e, the unit in the group, such that ψ±β (x, s) = (•±β (x), s) =

•±β (x).s.

A ¬ber bundle with a G-bundle structure is called a G-bundle. A ¬ber bundle

atlas as in (2) is called a G-atlas, and the family (•±β ) is also called a cocycle of

transition functions, but now for the G-bundle.

37.7

380 Chapter VIII. In¬nite dimensional di¬erential geometry 37.9

To be more precise, two G-atlas are said to be equivalent (to describe the same

G-bundle), if their union is also a G-atlas. This translates to the two cocycles of

transition functions as follows, where we assume that the two coverings of M are

the same (by passing to the common re¬nement, if necessary): (•±β ) and (•±β )

are called cohomologous if there is a family („± : U± ’ G) such that •±β (x) =

„± (x)’1 .•±β (x).„β (x) holds for all x ∈ U±β .

In (2) one should specify only an equivalence class of G-bundle structures or only

a cohomology class of cocycles of G-valued transition functions. From any open

cover (U± ) of M , some cocycle of transition functions (•±β : U±β ’ G) for it,

and a left G-action on a manifold S, we may construct a G-bundle, which depends

only on the cohomology class of the cocycle. By some abuse of notation, we write

(p : E ’ M, S, G) for a ¬ber bundle with speci¬ed G-bundle structure.

37.8. De¬nition. A principal (¬ber) bundle (p : P ’ M, G) is a G-bundle with

typical ¬ber a Lie group G, where the left action of G on G is just the left translation.