[[iL , LK ], d] = [iL , [LK , d]] ’ (’1)k [LK , [iL , d]] =

= 0 ’ (’1)k L([K, L]) = (’1)k [i([L, K]), d].

Since [ , d] kills the L™s and is injective on the i™s, the algebraic part of [iL , LK ]

is (’1)k i([L, K]).

8.7. 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 degree of ω be q, of • be k, and of ψ be . Let the other

degrees be as indicated. 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 ’1)( 2 ’1)

’ (’1)(q+ i(L2 )ω § L1 .

[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)k iX dψ § • — Y

’ d(iY • § ψ) — X ’ (’1)k 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) .

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8. Derivations on the algebra of di¬erential forms and the Fr¨licher-Nijenhuis bracket 71

o

8.8. Theorem. For K ∈ „¦k (M ; T M ) and ω ∈ „¦ (M ) the Lie derivative of ω

along K is given by the following formula, where the Xi are vector ¬elds on M .

(LK ω)(X1 , . . . , Xk+ ) =

1

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

= k! !

σ

’1

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

k! ( ’1)!

σ

k’1

(’1)

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

(k’1)! ( ’1)! 2!

σ

Proof. It su¬ces to consider K = • — X. Then by 8.7.3 we have L(• — X) =

• § LX ’ (’1)k’1 d• § iX . Now use the global formulas of section 7 to expand

this.

8.9. Theorem. For K ∈ „¦k (M ; T M ) and L ∈ „¦ (M ; T M ) we have for the

Fr¨licher-Nijenhuis bracket [K, L] the following formula, where the Xi are vector

o

¬elds on M .

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

1

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

k! !

σ

’1

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

k! ( ’1)!

σ

k

(’1)

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

(k’1)! !

σ

k’1

(’1)

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

(k’1)! ( ’1)! 2!

σ

(k’1)

(’1)

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

(k’1)! ( ’1)! 2!

σ

Proof. It su¬ces to consider K = • — X and L = ψ — Y , then for [• — X, ψ — Y ]

we may use 8.7.6 and evaluate that at (X1 , . . . , Xk+ ). After some combinatorial

computation we get the right hand side of the above formula for K = • — X and

L=ψ—Y.

There are more illuminating ways to prove this formula, see [Michor, 87].

8.10. Local formulas. In a local chart (U, u) on the manifold M we put

Lj dβ — ‚j , and ω | U =

K± d± — ‚i , L | U =

i

ωγ dγ , where

K|U= β

± = (1 ¤ ±1 < ±2 < · · · < ±k ¤ dim M ) is a form index, d± = du±1 § . . . § du±k ,

‚

‚i = ‚ui and so on.

Plugging Xj = ‚ij into the global formulas 8.2, 8.8, and 8.9, we get the

following local formulas:

K±1 ...±k ωi±k+1 ...±k+ ’1 d±

i

iK ω | U =

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

72 Chapter II. Di¬erential forms

K±1 ...±k Lj k+1 ...±k+

[K, L]§ | U = i

i±

j

’1)

’ (’1)(k’1)( Li 1 ...± Ki± +1 ...±k+ d± — ‚j

±

i

LK ω | U = K±1 ...±k ‚i ω±k+1 ...±k+

+ (’1)k (‚±1 K±2 ...±k+1 ) ωi±k+2 ...±k+ d±

i

K±1 ...±k ‚i Lj k+1 ...±k+

i

[K, L] | U = ±

’ (’1)k Li 1 ...± ‚i K± +1 ...±k+

j

±

j

’ kK±1 ...±k’1 i ‚±k Li k+1 ...±k+

±

+ (’1)k Lj 1 ...± ’1 i ‚± K± +1 ...±k+

i

d± — ‚j

±

8.11. 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. We have instead for

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

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

(2)

’ (’1)k1 i([K1 , L])K2 ’ (’1)(k1 + )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 the relations of this theorem and its consequences in

group theory have been investigated in [Michor, 89]. The corresponding product

of groups is well known to algebraists under the name ˜Zappa-Szep™-product.

Proof. Equation (1) is an immediate consequence of 8.6. Equations (2) and (3)

follow from (1) by writing out the graded Jacobi identity, or as follows: Consider

L(iL [K1 , K2 ]) and use 8.6 repeatedly to obtain L of the right hand side of (2).

Then consider i([K, [L1 , L2 ]§ ]) and use again 8.6 several times to obtain i of the

right hand side of (3).

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

8. Derivations on the algebra of di¬erential forms and the Fr¨licher-Nijenhuis bracket 73

o

8.12. Corollary (of 8.9). 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 ].

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

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

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

of the connection. If P is of constant rank, then both are sub vector bundles of

T M . If im P is some primarily ¬xed sub vector bundle or (tangent bundle of) a

foliation, P can be called a connection for it. Special cases of this will be treated

extensively later on. The following result is immediate from 8.12.

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 P has constant rank, then R is the obstruction against integrability of the

¯

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

¯

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

connection P . We will see later, that for a principal ¬ber bundle R is just the

negative of the usual curvature.

8.14. 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 8.13 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 8.11.2 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.

8.15. f -relatedness of the Fr¨licher-Nijenhuis bracket. Let f : M ’

o

N be 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)

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74 Chapter II. Di¬erential forms

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.

Proof. (2) By 8.2 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 df , f ∈ C ∞ (M, R) separate

points.

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

bracket then follows 8.2.2.

(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 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 ]), respec-

tively. Then use (6).

8.16. 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)

Note that this is a special case of the pullback operator for sections of natural

vector bundles in 6.15. Clearly K and f — K are then f -related.

Theorem. In this situation we have:

(2) f — [K, L] = [f — K, f — L].

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

Remarks 75

(3) f — iK L = if — K f — L.

(4) f — [K, L]§ = [f — K, f — L]§ .

(5) For a vector ¬eld X ∈ X(M ) and K ∈ „¦(M ; T M ) by 6.15 the Lie

derivative LX K = ‚t 0 (FlX )— K is de¬ned. Then we have LX K =

‚

t

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

o

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

o

§

[ , ] , etc. are natural bilinear concomitants.

Proof. (2) “ (4) are obvious from 8.15. They also follow directly from the geo-

metrical constructions of the operators in question. (5) Obviously LX is R-linear,

so it su¬ces to check this formula for K = ψ — Y , ψ ∈ „¦(M ) and Y ∈ X(M ).

But then

LX (ψ — Y ) = LX ψ — Y + ψ — LX Y by 6.16

= LX ψ — Y + ψ — [X, Y ]

= [X, ψ — Y ] by 8.7.6.

8.17. Remark. At last 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 a almost complex structure; if it exists,

√

dim M is even and J can be viewed as a ¬ber multiplication with ’1 on T M .

By 8.12 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, because

we have the following result:

A manifold M with an almost complex structure J is a complex

manifold (i.e., there exists an atlas for M with holomorphic chart-

change mappings) if and only if [J, J] = 0. See [Newlander-Nirenberg,

57].

Remarks

The material on the Lie derivative on natural vector bundles 6.14“6.20 appears

here for the ¬rst time. Most of section 8 is due to [Fr¨licher-Nijenhuis, 56], the

o

formula in 8.9 was proved by [Mangiarotti-Modugno, 84] and [Michor, 87]. The

Bianchi identity 8.14 is from [Michor, 89a].

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

76

CHAPTER III.

BUNDLES AND CONNECTIONS

We begin our treatment of connections in the general setting of ¬ber bundles

(without structure group). A connection on a ¬ber bundle is just a projection

onto the vertical bundle. Curvature and the Bianchi identity is expressed with

the help of the Fr¨licher-Nijenhuis bracket. The parallel transport for such a

o

general connection is not de¬ned along the whole of the curve in the base in

general - if this is the case for all curves, the connection is called complete. We

show that every ¬ber bundle admits complete connections. For complete con-

nections we treat holonomy groups and the holonomy Lie algebra, a subalgebra

of the Lie algebra of all vector ¬elds on the standard ¬ber.

Then we present principal bundles and associated bundles in detail together

with the most important examples. Finally we investigate principal connections

by requiring equivariance under the structure group. It is remarkable how fast

the usual structure equations can be derived from the basic properties of the

Fr¨licher-Nijenhuis bracket. Induced connections are investigated thoroughly -

o

we describe tools to recognize induced connections among general ones.

If the holonomy Lie algebra of a connection on a ¬ber bundle is ¬nite dimen-

sional and consists of complete vector ¬elds on the ¬ber, we are able to show,

that in fact the ¬ber bundle is associated to a principal bundle and the connec-

tion is induced from an irreducible principal connection (theorem 9.11). This is

a powerful generalization of the theorem of Ambrose and Singer.

Connections will be treated once again from the point of view of jets, when

we have them at our disposal in chapter IV.

We think that the treatment of connections presented here o¬ers some di-

dactical advantages besides presenting new results: the geometric content of a

connection is treated ¬rst, and the additional requirement of equivariance under

a structure group is seen to be additional and can be dealt with later - so the

reader is not required to grasp all the structures at the same time. Besides that

it gives new results and new insights. There are naturally appearing connec-

tions in di¬erential geometry which are not principal or induced connections:

The universal connection on the bundle J 1 P/G of all connections of a principal

bundle, and also the Cartan connections.