(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