Ex for each x ∈ M . Any vector bundle admits Riemannian metrics: local

existence is clear and we may glue with the help of a partition of unity on

M , since the positive de¬nite sections form an open convex subset. Now let

s = (s1 , . . . , sn ) ∈ C ∞ (GL(Rn , E)|U ) be a local frame ¬eld of the bundle E

over U ‚ M . Now we may apply the Gram-Schmidt orthonormalization pro-

cedure to the basis (s1 (x), . . . , sn (x)) of Ex for each x ∈ U . Since this proce-

dure is smooth (even real analytic), we obtain a frame ¬eld s = (s1 , . . . , sn )

of E over U which is orthonormal with respect to g. We call it an orthonor-

mal frame ¬eld. Now let (U± ) be an open cover of M with orthonormal frame

¬elds s± = (s± , . . . , s± ), where s± is de¬ned on U± . We consider the vector

n

1

bundle charts (U± , ψ± : E|U± ’ U± — Rn ) given by the orthonormal frame

’1

¬elds: ψ± (x, v 1 , . . . , v n ) = s± (x).v i =: s± (x).v. For x ∈ U±β we have

i

sj (x).gβ± i (x) for C -functions g±β j : U±β ’ R. Since s± (x) and

β j ∞

±

si (x) = i

s (x) are both orthonormal bases of Ex , the matrix g±β (x) = (g±β j (x)) is an

β

i

’1

± β

element of O(n). We write s = s .gβ± for short. Then we have ψβ (x, v) =

’1

’1

sβ (x).v = s± (x).g±β (x).v = ψ± (x, g±β (x).v) and consequently ψ± ψβ (x, v) =

(x, g±β (x).v). So the (g±β : U±β ’ O(n)) are the cocycle of transition functions

for the vector bundle atlas (U± , ψ± ). So we have constructed an O(n)-structure

on E. The corresponding principal ¬ber bundle will be denoted by O(Rn , (E, g));

it is usually called the orthonormal frame bundle of E. It is derived from the

linear frame bundle GL(Rn , E) by reduction of the structure group from GL(n)

to O(n). The phenomenon discussed here plays a prominent role in the theory

of classifying spaces.

10.12. Sections of associated bundles. Let (P, p, M, G) be a principal ¬ber

bundle and : G — S ’ S a left action. Let C ∞ (P, S)G denote the space

of all smooth mappings f : P ’ S which are G-equivariant in the sense that

f (u.g) = g ’1 .f (u) holds for g ∈ G and u ∈ P .

Theorem. The sections of the associated bundle P [S, ] correspond exactly

to the G-equivariant mappings P ’ S; we have a bijection C ∞ (P, S)G ∼ =

∞

C (P [S]).

Proof. If f ∈ C ∞ (P, S)G we get sf ∈ C ∞ (P [S]) by the following diagram:

w

(Id, f )

P —S

P

p

u u

q

(a)

w P [S]

sf

M

which exists by 10.9 since graph(f ) = (Id, f ) : P ’ P — S is G-equivariant:

(Id, f )(u.g) = (u.g, f (u.g)) = (u.g, g ’1 .f (u)) = ((Id, f )(u)).g.

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

10. Principal ¬ber bundles and G-bundles 95

If conversely s ∈ C ∞ (P [S]) we de¬ne fs ∈ C ∞ (P, S)G by fs := „ —¦(IdP —M s) :

P = P —M M ’ P —M P [S] ’ S. This is G-equivariant since fs (ux .g) =

„ (ux .g, s(x)) = g ’1 .„ (ux , s(x)) = g ’1 .fs (ux ) by 10.7. The two constructions are

inverse to each other since we have fs(f ) (u) = „ (u, sf (p(u))) = „ (u, q(u, f (u))) =

f (u) and sf (s) (p(u)) = q(u, fs (u)) = q(u, „ (u, s(p(u))) = s(p(u)).

The G-mapping fs : P ’ S determined by a section s of P [S] will be called

the frame form of the section s.

10.13. Theorem. Consider a principal ¬ber bundle (P, p, M, G) and a closed

subgroup K of G. Then the reductions of structure group from G to K corre-

¯

spond bijectively to the global sections of the associated bundle P [G/K, »] in a

¯

canonical way, where » : G—G/K ’ G/K is the left action on the homogeneous

space from 5.11.

Proof. By theorem 10.12 the section s ∈ C ∞ (P [G/K]) corresponds to fs ∈

¯

C ∞ (P, G/K)G , which is a surjective submersion since the action » : G—G/K ’

’1

G/K is transitive. Thus Ps := fs (¯) is a submanifold of P which is stable under

e

the right action of K on P . Furthermore the K-orbits are exactly the ¬bers of

the mapping p : Ps ’ M , so by lemma 10.3 we get a principal ¬ber bundle

(Ps , p, M, K). The embedding Ps ’ P is then a reduction of structure groups

as required.

If conversely we have a principal ¬ber bundle (P , p , M, K) and a reduction of

structure groups χ : P ’ P , then χ is an embedding covering the identity of M

and is K-equivariant, so we may view P as a sub ¬ber bundle of P which is stable

under the right action of K. Now we consider the mapping „ : P —M P ’ G

from 10.2 and restrict it to P —M P . Since we have „ (ux , vx .k) = „ (ux , vx ).k

for k ∈ K this restriction induces f : P ’ G/K by

wG

„

P —M P

p

u u

w G/K;

f

P = P —M P /K

and from „ (ux .g, vx ) = g ’1 .„ (ux , vx ) it follows that f is G-equivariant as re-

quired. Finally f ’1 (¯) = {u ∈ P : „ (u, Pp(u) ) ⊆ K } = P , so the two construc-

e

tions are inverse to each other.

10.14. The bundle of gauges. If (P, p, M, G) is a principal ¬ber bundle we

denote by Aut(P ) the group of all G-equivariant di¬eomorphisms χ : P ’ P .

Then p —¦ χ = χ —¦ p for a unique di¬eomorphism χ of M , so there is a group

homomorphism from Aut(P ) into the group Di¬(M ) of all di¬eomorphisms of

M . The kernel of this homomorphism is called Gau(P ), the group of gauge

transformations. So Gau(P ) is the space of all χ : P ’ P which satisfy p —¦ χ = p

and χ(u.g) = χ(u).g.

Theorem. The group Gau(P ) of gauge transformations is equal to the space

C ∞ (P, (G, conj))G ∼ C ∞ (P [G, conj]).

=

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96 Chapter III. Bundles and connections

Proof. We use again the mapping „ : P —M P ’ G from 10.2. For χ ∈

Gau(P ) we de¬ne fχ ∈ C ∞ (P, (G, conj))G by fχ := „ —¦ (Id, χ). Then fχ (u.g) =

„ (u.g, χ(u.g)) = g ’1 .„ (u, χ(u)).g = conjg’1 fχ (u), so fχ is indeed G-equivariant.

If conversely f ∈ C ∞ (P, (G, conj))G is given, we de¬ne χf : P ’ P by

χf (u) := u.f (u). It is easy to check that χf is indeed in Gau(P ) and that the

two constructions are inverse to each other.

10.15. The tangent bundles of homogeneous spaces. Let G be a Lie

group and K a closed subgroup, with Lie algebras g and k, respectively. We

recall the mapping AdG : G ’ AutLie (g) from 4.24 and put AdG,K := AdG |K :

K ’ AutLie (g). For X ∈ k and k ∈ K we have AdG,K (k)X = AdG (k)X =

AdK (k)X ∈ k, so k is an invariant subspace for the representation AdG,K of K

in g, and we have the factor representation Ad⊥ : K ’ GL(g/k). Then

0 ’ k ’ g ’ g/k ’ 0

(a)

is short exact and K-equivariant.

Now we consider the principal ¬ber bundle (G, p, G/K, K) and the associated

vector bundles G[g/k, Ad⊥ ] and G[k, AdK ].

Theorem. In these circumstances we have

T (G/K) = G[g/k, Ad⊥ ] = (G —K g/k, p, G/K, g/k).

¯

The left action g ’ T (»g ) of G on T (G/K) corresponds to the canonical left

action of G on G—K g/k. Furthermore G[g/k, Ad⊥ ]•G[k, AdK ] is a trivial vector

bundle.

Proof. For p : G ’ G/K we consider the tangent mapping Te p : g ’ Te (G/K)¯

which is linear and surjective and induces a linear isomorphism Te p : g/k ’

¯

Te (G/K). For k ∈ K we have p —¦ conjk = p —¦ »k —¦ ρk’1 = »k —¦ p and consequently

¯

¯

Te p—¦AdG,K (k) = Te p—¦Te (conjk ) = Te »k —¦Te p. Thus the isomorphism Te p : g/k ’

¯

Te (G/K) is K-equivariant for the representations Ad⊥ and Te » : k ’ Te »k .

¯ ¯

¯ ¯ ¯

¯

Now we consider the associated vector bundle G[Te (G/K), Te »] = (G —K

¯ ¯

Te (G/K), p, G/K, Te (G/K)), which is isomorphic to G[g/k, Ad⊥ ], since the rep-

¯ ¯

¯

resentation spaces are isomorphic. The mapping T2 » : G — Te (G/K) ’ T (G/K)

¯

(where T2 is the second partial tangent functor) is K-invariant and therefore

induces a mapping ψ as in the following diagram:

ee

q

G — Te (G/K)

¯

eg

e

¯

£

T»

ew T (G/K)

T (G/K)

ψ

(b) G —K

e

e

¯

ee

he π

e

p G/K

G/K.

This mapping ψ is an isomorphism of vector bundles.

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10. Principal ¬ber bundles and G-bundles 97

It remains to show the last assertion. The short exact sequence (a) induces a

sequence of vector bundles over G/K:

G/K — 0 ’ G[k, AdK ] ’ G[g, AdG,K ] ’ G[g/k, Ad⊥ ] ’ G/K — 0

This sequence splits ¬berwise thus also locally over G/K, so G[g/k, Ad⊥ ] •

G[k, AdK ] ∼ G[g, AdG,K ] and it remains to show that G[g, AdG,K ] is a trivial vec-

=

tor bundle. Let • : G—g ’ G—g be given by •(g, X) = (g, AdG (g)X). Then for

k ∈ K we have •((g, X).k) = •(gk, AdG,K (k ’1 )X) = (gk, AdG (g.k.k ’1 )X) =

(gk, AdG (g)X). So • is K-equivariant from the ˜joint™ K-action to the ˜on the

left™ K-action and therefore induces a mapping • as in the diagram:

¯

w G—g

•

G—g

u u

q

w G/K — g

ee •

¯

pr

(c) G —K g

ege £

p 1

G/K

The map • is a vector bundle isomorphism.

¯

10.16. Tangent bundles of Grassmann manifolds. From 10.5 we know

that (V (k, n) = O(n)/O(n ’ k), p, G(k, n), O(k)) is a principal ¬ber bundle.

Using the standard representation of O(k) we consider the associated vector

bundle (Ek := V (k, n)[Rk ], p, G(k, n)). It is called the universal vector bundle

over G(k, n). Recall from 10.5 the description of V (k, n) as the space of all linear

isometries Rk ’ Rn ; we get from it the evaluation mapping ev : V (k, n) — Rk ’

Rn . The mapping (p, ev) in the diagram

99

V (k, n) — Rk

99ev)

A

9

(p,

u

q

(a)

w G(k, n) — R

V (k, n) —O(k) Rk n

ψ

is O(k)-invariant for the action R and factors therefore to an embedding of

vector bundles ψ : Ek ’ G(k, n) — Rn . So the ¬ber (Ek )W over the k-plane W

in Rn is just the linear subspace W . Note ¬nally that the ¬ber wise orthogonal

complement Ek ⊥ of Ek in the trivial vector bundle G(k, n)—Rn with its standard

Riemannian metric is isomorphic to the universal vector bundle En’k over G(n’

k, n), where the isomorphism covers the di¬eomorphism G(k, n) ’ G(n ’ k, n)

given also by the orthogonal complement mapping.

Corollary. The tangent bundle of the Grassmann manifold is

T G(k, n) ∼ L(Ek , Ek ⊥ ).

=

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98 Chapter III. Bundles and connections

Proof. We have G(k, n) = O(n)/(O(k) — O(n ’ k)), so by theorem 10.15 we get

— (so(n)/(so(k) — so(n ’ k))).

T G(k, n) = O(n)

O(k)—O(n’k)

On the other hand we have V (k, n) = O(n)/O(n ’ k) and the right action of

O(k) commutes with the right action of O(n ’ k) on O(n), therefore

V (k, n)[Rk ] = (O(n)/O(n ’ k)) — Rk = O(n) Rk ,

—

O(k) O(k)—O(n’k)

where O(n ’ k) acts trivially on Rk . Finally

L(Ek , Ek ⊥ ) = L O(n) Rk , O(n) Rn’k

— —

O(k)—O(n’k) O(k)—O(n’k)

L(Rk , Rn’k ),

—

= O(n)

O(k)—O(n’k)

where the left action of O(k) — O(n ’ k) on L(Rk , Rn’k ) is given by (A, B)(C) =

B.C.A’1 . Finally we have an O(k) — O(n ’ k) - equivariant linear isomorphism

L(Rk , Rn’k ) ’ so(n)/(so(k) — so(n ’ k)), as follows:

so(n)/(so(k) — so(n ’ k)) =

skew 0 A

A ∈ L(Rk , Rn’k )

= :

’At 0

skew 0

0 skew

10.17. The tangent group of a Lie group. Let G be a Lie group with

Lie algebra g. We will use the notation from 4.1. First note that T G is

also a Lie group with multiplication T µ and inversion T ν, given by (see 4.2)

T(a,b) µ.(ξa , ·b ) = Ta (ρb ).ξa + Tb (»a ).·b and Ta ν.ξa = ’Te (»a’1 ).Ta (ρa’1 ).ξa .

Lemma. Via the isomomorphism T ρ : g — G ’ T G, T ρ.(X, g) = Te (ρg ).X, the

group structure on T G looks as follows: (X, a).(Y, b) = (X + Ad(a)Y, a.b) and

(X, a)’1 = (’ Ad(a’1 )X, a’1 ). So T G is isomorphic to the semidirect product

g G, see 5.16.

Proof. T(a,b) µ.(T ρa .X, T ρb .Y ) = T ρb .T ρa .X + T »a .T ρb .Y =

= T ρab .X + T ρb .T ρa .T ρa’1 .T »a .Y = T ρab (X + Ad(a)Y ).

Ta ν.T ρa .X = ’T ρa’1 .T »a’1 .T ρa .X = ’T ρa’1 . Ad(a’1 )X.

Remark. In the left trivialisation T » : G — g ’ T G, T ».(g, X) = Te (»g ).X,

the semidirect product structure is: (a, X).(b, Y ) = (ab, Ad(b’1 )X + Y ) and

(a, X)’1 = (a’1 , ’ Ad(a)X).

Lemma 10.17 is a special case of 37.16 and also 38.10 below.

10.18. Tangent bundles and vertical bundles. Let (E, p, M, S) be a ¬ber

bundle. The subbundle V E = { ξ ∈ T E : T p.ξ = 0 } of T E is called the

vertical bundle and is denoted by (V E, πE , E).

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

11. Principal and induced connections 99

Theorem. Let (P, p, M, G) be a principal ¬ber bundle with principal right ac-

tion r : P — G ’ P . Let : G — S ’ S be a left action. Then the following

assertions hold:

(1) (T P, T p, T M, T G) is again a principal ¬ber bundle with principal right

action T r : T P — T G ’ T P .

(2) The vertical bundle (V P, π, P, g) of the principal bundle is trivial as a

vector bundle over P : V P ∼ P — g.

=

(3) The vertical bundle of the principal bundle as bundle over M is again a

principal bundle: (V P, p —¦ π, M, T G).

(4) The tangent bundle of the associated bundle P [S, ] is given by

T (P [S, ]) = T P [T S, T ].

(5) The vertical bundle of the associated bundle P [S, ] is given by

V (P [S, ]) = P [T S, T2 ] = P —G T S, where T2 is the second partial

tangent functor.

Proof. Let (U± , •± : P |U± ’ U± — G) be a principal ¬ber bundle atlas with

cocycle of transition functions (•±β : U±β ’ G). Since T is a functor which

respects products, (T U± , T •± : T P |T U± ’ T U± — T G) is again a principal

¬ber bundle atlas with cocycle of transition functions (T •±β : T U±β ’ T G),

describing the principal ¬ber bundle (T P, T p, T M, T G). The assertion about

the principal action is obvious. So (1) follows. For completeness sake we include

here the transition formula for this atlas in the right trivialization of T G:

T (•± —¦ •’1 )(ξx , Te (ρg ).X) = (ξx , Te (ρ•±β (x).g ).(δ•±β (ξx ) + Ad(•±β (x))X)),

β

where δ•±β ∈ „¦1 (U±β ; g) is the right logarithmic derivative of •±β , see 4.26.

(2) The mapping (u, X) ’ Te (ru ).X = T(u,e) r.(0u , X) is a vector bundle iso-

morphism P — g ’ V P over P .

(3) Obviously T r : T P — T G ’ T P is a free right action which acts transitive on

the ¬bers of T p : T P ’ T M . Since V P = (T p)’1 (0M ), the bundle V P ’ M is

isomorphic to T P |0M and T r restricts to a free right action, which is transitive

on the ¬bers, so by lemma 10.3 the result follows.

(4) The transition functions of the ¬ber bundle P [S, ] are given by the expression

—¦(•±β —IdS ) : U±β —S ’ G—S ’ S. Then the transition functions of T (P [S, ])

are T ( —¦ (•±β — IdS )) = T —¦ (T •±β — IdT S ) : T U±β — T S ’ T G — T S ’ T S,

from which the result follows.

(5) Vertical vectors in T (P [S, ]) have local representations (0x , ·s ) ∈ T U±β —T S.

Under the transition functions of T (P [S, ]) they transform as T ( —¦ (•±β —

IdS )).(0x , ·s ) = T .(0•±β (x) , ·s ) = T ( •±β (x) ).·s = T2 .(•±β (x), ·s ) and this

implies the result.

11. Principal and induced connections

11.1. Principal connections. Let (P, p, M, G) be a principal ¬ber bundle.

Recall from 9.3 that a (general) connection on P is a ¬ber projection ¦ : T P ’

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

100 Chapter III. Bundles and connections

V P , viewed as a 1-form in „¦1 (P ; T P ). Such a connection ¦ is called a principal

connection if it is G-equivariant for the principal right action r : P — G ’ P , so

that T (rg ).¦ = ¦.T (rg ), i.e. ¦ is rg -related to itself, or (rg )— ¦ = ¦ in the sense

of 8.16, for all g ∈ G. By theorem 8.15.7 the curvature R = 1 .[¦, ¦] is then also

2

rg -related to itself for all g ∈ G.

Recall from 10.18.2 that the vertical bundle of P is trivialized as a vector

bundle over P by the principal action. So we have ω(Xu ) := Te (ru )’1 .¦(Xu ) ∈ g

and in this way we get a g-valued 1-form ω ∈ „¦1 (P ; g), which is called the

(Lie algebra valued) connection form of the connection ¦. Recall from 5.13 the

fundamental vector ¬eld mapping ζ : g ’ X(P ) for the principal right action.

The de¬ning equation for ω can be written also as ¦(Xu ) = ζω(Xu ) (u).

Lemma. If ¦ ∈ „¦1 (P ; V P ) is a principal connection on the principal ¬ber

bundle (P, p, M, G) then the connection form has the following properties:

(1) ω reproduces the generators of fundamental vector ¬elds, so we have

ω(ζX (u)) = X for all X ∈ g.

(2) ω is G-equivariant, so ((rg )— ω)(Xu ) = ω(Tu (rg ).Xu ) = Ad(g ’1 ).ω(Xu )

for all g ∈ G and Xu ∈ Tu P .

(3) For the Lie derivative we have LζX ω = ’ ad(X).ω.

Conversely a 1-form ω ∈ „¦1 (P, g) satisfying (1) de¬nes a connection ¦ on P

by ¦(Xu ) = Te (ru ).ω(Xu ), which is a principal connection if and only if (2) is

satis¬ed.

Proof. (1) Te (ru ).ω(ζX (u)) = ¦(ζX (u)) = ζX (u) = Te (ru ).X. Since Te (ru ) :

g ’ Vu P is an isomorphism, the result follows.

(2) Both directions follow from

Te (rug ).ω(Tu (rg ).Xu ) = ζω(Tu (rg ).Xu ) (ug) = ¦(Tu (rg ).Xu )

Te (rug ). Ad(g ’1 ).ω(Xu ) = ζAd(g’1 ).ω(Xu ) (ug) =

= Tu (rg ).ζω(Xu ) (u) = Tu (rg ).¦(Xu ).

(3) is a consequence of (2).

11.2. Curvature. Let ¦ be a principal connection on the principal ¬ber bundle

(P, p, M, G) with connection form ω ∈ „¦1 (P ; g). We already noted in 11.1 that

the curvature R = 1 [¦, ¦] is then also G-equivariant, (rg )— R = R for all g ∈ G.

2

Since R has vertical values we may again de¬ne a g-valued 2-form „¦ ∈ „¦2 (P ; g)

by „¦(Xu , Yu ) := ’Te (ru )’1 .R(Xu , Yu ), which is called the (Lie algebra-valued)

curvature form of the connection. We also have R(Xu , Yu ) = ’ζ„¦(Xu ,Yu ) (u). We

take the negative sign here to get the usual curvature form as in [Kobayashi-

Nomizu I, 63].

We equip the space „¦(P ; g) of all g-valued forms on P in a canonical way

with the structure of a graded Lie algebra by

[Ψ, ˜]§ (X1 , . . . , Xp+q ) =

1

= signσ [Ψ(Xσ1 , . . . , Xσp ), ˜(Xσ(p+1) , . . . , Xσ(p+q) )]g

p! q! σ

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

11. Principal and induced connections 101

or equivalently by [ψ — X, θ — Y ]§ := ψ § θ — [X, Y ]g . From the latter description

it is clear that d[Ψ, ˜]§ = [dΨ, ˜]§ + (’1)deg Ψ [Ψ, d˜]§ . In particular for ω ∈

„¦1 (P ; g) we have [ω, ω]§ (X, Y ) = 2[ω(X), ω(Y )]g .

Theorem. The curvature form „¦ of a principal connection with connection

form ω has the following properties:

(1) „¦ is horizontal, i.e. it kills vertical vectors.

(2) „¦ is G-equivariant in the following sense: (rg )— „¦ = Ad(g ’1 ).„¦. Conse-

quently LζX „¦ = ’ ad(X).„¦.

(3) The Maurer-Cartan formula holds: „¦ = dω + 1 [ω, ω]§ .

2

Proof. (1) is true for R by 9.4. For (2) we compute as follows:

Te (rug ).((rg )— „¦)(Xu , Yu ) = Te (rug ).„¦(Tu (rg ).Xu , Tu (rg ).Yu ) =

= ’Rug (Tu (rg ).Xu , Tu (rg ).Yu ) = ’Tu (rg ).((rg )— R)(Xu , Yu ) =

= ’Tu (rg ).R(Xu , Yu ) = Tu (rg ).ζ„¦(Xu ,Yu ) (u) =

= ζAd(g’1 ).„¦(Xu ,Yu ) (ug) =

= Te (rug ). Ad(g ’1 ).„¦(Xu , Yu ), by 5.13.

(3) For X ∈ g we have iζX R = 0 by (1), and using 11.1.(3) we get

1 1 1

iζX (dω + [ω, ω]§ ) = iζX dω + [iζX ω, ω]§ ’ [ω, iζX ω]§ =

2 2 2

= LζX ω + [X, ω]§ = ’ad(X)ω + ad(X)ω = 0.

So the formula holds for vertical vectors, and for horizontal vector ¬elds X, Y ∈

C ∞ (H(P )) we have

R(X, Y ) = ¦[X ’ ¦X, Y ’ ¦Y ] = ¦[X, Y ] = ζω([X,Y ])

1

(dω + [ω, ω])(X, Y ) = Xω(Y ) ’ Y ω(X) ’ ω([X, Y ]) = ’ω([X, Y ]).

2

11.3. Lemma. Any principal ¬ber bundle (P, p, M, G) admits principal con-

nections.

Proof. Let (U± , •± : P |U± ’ U± — G)± be a principal ¬ber bundle atlas. Let

us de¬ne γ± (T •’1 (ξx , Te »g .X)) := X for ξx ∈ Tx U± and X ∈ g. An easy

±

computation involving lemma 5.13 shows that γ± ∈ „¦1 (P |U± ; g) satis¬es the

requirements of lemma 11.1 and thus is a principal connection on P |U± . Now

let (f± ) be a smooth partition of unity on M which is subordinated to the open

cover (U± ), and let ω := ± (f± —¦ p)γ± . Since both requirements of lemma 11.1

are invariant under convex linear combinations, ω is a principal connection on

P.

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102 Chapter III. Bundles and connections

11.4. Local descriptions of principal connections. We consider a principal

¬ber bundle (P, p, M, G) with some principal ¬ber bundle atlas (U± , •± : P |U± ’

U± — G) and corresponding cocycle (•±β : U±β ’ G) of transition functions.

We consider the sections s± ∈ C ∞ (P |U± ) which are given by •± (s± (x)) = (x, e)

and satisfy s± .•±β = sβ .

(1) Let ˜ ∈ „¦1 (G, g) be the left logarithmic derivative of the identity,

i.e. ˜(·g ) := Tg (»g’1 ).·g . We will use the forms ˜±β := •±β — ˜ ∈

„¦1 (U±β ; g).

Let ¦ = ζ —¦ ω ∈ „¦1 (P ; V P ) be a principal connection with connection form

ω ∈ „¦1 (P ; g). We may associate the following local data to the connection:

(2) ω± := s± — ω ∈ „¦1 (U± ; g), the physicists version of the connection.

(3) The Christo¬el forms “± ∈ „¦1 (U± ; X(G)) from 9.7, which are given by

(0x , “± (ξx , g)) = ’T (•± ).¦.T (•± )’1 (ξx , 0g ).

(4) γ± := (•’1 )— ω ∈ „¦1 (U± — G; g), the local expressions of ω.

±

Lemma. These local data have the following properties and are related by the

following formulas.

(5) The forms ω± ∈ „¦1 (U± ; g) satisfy the transition formulas

ω± = Ad(•’1 )ωβ + ˜β± ,

β±

and any set of forms like that with this transition behavior determines a

unique principal connection.

(6) We have γ± (ξx , T »g .X) = γ± (ξx , 0g ) + X = Ad(g ’1 )ω± (ξx ) + X.

(7) We have “± (ξx , g) = ’Te (»g ).γ± (ξx , 0g ) = ’Te (»g ). Ad(g ’1 )ω± (ξx ) =

’T (ρg )ω± (ξx ), so “± (ξx ) = ’Rω± (ξx ) , a right invariant vector ¬eld.

Proof. From the de¬nition of the Christo¬el forms we have

(0x , “± (ξx , g)) = ’T (•± ).¦.T (•± )’1 (ξx , 0g )

= ’T (•± ).Te (r•’1 (x,g) )ω.T (•± )’1 (ξx , 0g )

±

= ’Te (•± —¦ r•’1 (x,g) )ω.T (•± )’1 (ξx , 0g )

±

= ’(0x , Te (»g )ω.T (•± )’1 (ξx , 0g )) = ’(0x , Te (»g )γ± (ξx , 0g )).

This is the ¬rst part of (7). The second part follows from (6).

γ± (ξx , T »g .X) = γ± (ξx , 0g ) + γ± (0x , T »g .X)

= γ± (ξx , 0g ) + ω(T (•± )’1 (0x , T »g .X))

= γ± (ξx , 0g ) + ω(ζX (•’1 (x, g))) = γ± (ξx , 0g ) + X.

±

So the ¬rst part of (6) holds. The second part is seen from

γ± (ξx , 0g ) = γ± (ξx , Te (ρg )0e ) = (ω —¦ T (•± )’1 —¦ T (IdX —ρg ))(ξx , 0e ) =

= (ω —¦ T (rg —¦ •’1 ))(ξx , 0e ) = Ad(g ’1 )ω(T (•’1 )(ξx , 0e ))

± ±

= Ad(g ’1 )(s± — ω)(ξx ) = Ad(g ’1 )ω± (ξx ).

Via (7) the transition formulas for the ω± are easily seen to be equivalent to the

transition formulas for the Christo¬el forms in lemma 9.7.

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11. Principal and induced connections 103

11.5. The covariant derivative. Let (P, p, M, G) be a principal ¬ber bundle

with principal connection ¦ = ζ —¦ ω. We consider the horizontal projection

χ = IdT P ’¦ : T P ’ HP , cf. 9.3, which satis¬es χ —¦ χ = χ, im χ = HP ,

ker χ = V P , and χ —¦ T (rg ) = T (rg ) —¦ χ for all g ∈ G.

If W is a ¬nite dimensional vector space, we consider the mapping χ— :

„¦(P ; W ) ’ „¦(P ; W ) which is given by

(χ— •)u (X1 , . . . , Xk ) = •u (χ(X1 ), . . . , χ(Xk )).

The mapping χ— is a projection onto the subspace of horizontal di¬erential forms,

i.e. the space „¦hor (P ; W ) := {ψ ∈ „¦(P ; W ) : iX ψ = 0 for X ∈ V P }. The notion

of horizontal form is independent of the choice of a connection.

The projection χ— has the following properties where in the ¬rst assertion one

of the two forms has values in R:

χ— (• § ψ) = χ— • § χ— ψ,

χ— —¦ χ— = χ— ,

χ— —¦ (rg )— = (rg )— —¦ χ— for all g ∈ G,

—

χ ω=0

χ— —¦ L(ζX ) = L(ζX ) —¦ χ— .

They follow easily from the corresponding properties of χ, the last property uses

ζ(X)

= rexp tX .

that Flt

Now we de¬ne the covariant exterior derivative dω : „¦k (P ; W ) ’ „¦k+1 (P ; W )

by the prescription dω := χ— —¦ d.

Theorem. The covariant exterior derivative dω has the following properties.

dω (• § ψ) = dω (•) § χ— ψ + (’1)deg • χ— • § dω (ψ) if • or ψ is real valued.

(1)

L(ζX ) —¦ dω = dω —¦ L(ζX ) for each X ∈ g.

(2)

(rg )— —¦ dω = dω —¦ (rg )— for each g ∈ G.

(3)

dω —¦ p— = d —¦ p— = p— —¦ d : „¦(M ; W ) ’ „¦hor (P ; W ).

(4)

(5)dω ω = „¦, the curvature form.

(6)dω „¦ = 0, the Bianchi identity.

dω —¦ χ— ’ dω = χ— —¦ i(R), where R is the curvature.

(7)

dω —¦ dω = χ— —¦ i(R) —¦ d.

(8)

Let „¦hor (P, g)G be the algebra of all horizontal G-equivariant g-valued

(9)

forms, i.e. (rg )— ψ = Ad(g ’1 )ψ. Then for any ψ ∈ „¦hor (P, g)G we have

dω ψ = dψ + [ω, ψ]§ .

(10) The mapping ψ ’ ζψ , where ζψ (X1 , . . . , Xk )(u) = ζψ(X1 ,... ,Xk )(u) (u), is

an isomorphism between „¦hor (P, g)G and the algebra „¦hor (P, V P )G of

all horizontal G-equivariant forms with values in the vertical bundle V P .

Then we have ζdω ψ = ’[¦, ζω ].

Proof. (1) through (4) follow from the properties of χ— .

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104 Chapter III. Bundles and connections

(5) We have

(dω ω)(ξ, ·) = (χ— dω)(ξ, ·) = dω(χξ, χ·)

= (χξ)ω(χ·) ’ (χ·)ω(χξ) ’ ω([χξ, χ·])

= ’ω([χξ, χ·]) and

’ζ(„¦(ξ, ·)) = R(ξ, ·) = ¦[χξ, χ·] = ζω([χξ,χ·]) .

(6) Using 11.2 we have

dω „¦ = dω (dω + 1 [ω, ω]§ )

2

= χ— ddω + 1 χ— d[ω, ω]§

2

= 2 χ— ([dω, ω]§ ’ [ω, dω]§ ) = χ— [dω, ω]§

1

= [χ— dω, χ— ω]§ = 0, since χ— ω = 0.

(7) For • ∈ „¦(P ; W ) we have

(dω χ— •)(X0 , . . . , Xk ) = (dχ— •)(χ(X0 ), . . . , χ(Xk ))

(’1)i χ(Xi )((χ— •)(χ(X0 ), . . . , χ(Xi ), . . . , χ(Xk )))

=

0¤i¤k

(’1)i+j (χ— •)([χ(Xi ), χ(Xj )], χ(X0 ), . . .

+

i<j

. . . , χ(Xi ), . . . , χ(Xj ), . . . )

(’1)i χ(Xi )(•(χ(X0 ), . . . , χ(Xi ), . . . , χ(Xk )))

=

0¤i¤k

(’1)i+j •([χ(Xi ), χ(Xj )] ’ ¦[χ(Xi ), χ(Xj )], χ(X0 ), . . .

+

i<j

. . . , χ(Xi ), . . . , χ(Xj ), . . . )

= (d•)(χ(X0 ), . . . , χ(Xk )) + (iR •)(χ(X0 ), . . . , χ(Xk ))

= (dω + χ— iR )(•)(X0 , . . . , Xk ).

(8) dω dω = χ— dχ— d = (χ— iR + χ— d)d = χ— iR d holds by (7).

(9) If we insert one vertical vector ¬eld, say ζX for X ∈ g, into dω ψ, we

get 0 by de¬nition. For the right hand side we use iζX ψ = 0 and LζX ψ =

ζX — exp tX —

‚ ‚ ‚

) ψ = ‚t 0 Ad(exp(’tX))ψ = ’ ad(X)ψ to get

‚t 0 (Flt ) ψ = ‚t 0 (r

iζX (dψ + [ω, ψ]§ ) = iζX dψ + diζX ψ + [iζX ω, ψ] ’ [ω, iζX ψ]

= LζX ψ + [X, ψ] = ’ ad(X)ψ + [X, ψ] = 0.

Let now all vector ¬elds ξi be horizontal, then we get

(dω ψ)(ξ0 , . . . , ξk ) = (χ— dψ)(ξ0 , . . . , ξk ) = dψ(ξ0 , . . . , ξk ),

(dψ + [ω, ψ]§ )(ξ0 , . . . , ξk ) = dψ(ξ0 , . . . , ξk ).

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11. Principal and induced connections 105

So the ¬rst formula holds.

(10) We proceed in a similar manner. Let Ψ be in the space „¦hor (P, V P )G

of all horizontal G-equivariant forms with vertical values. Then for each X ∈ g

we have iζX Ψ = 0; furthermore the G-equivariance (rg )— Ψ = Ψ implies that

LζX Ψ = [ζX , Ψ] = 0 by 8.16.(5). Using formula 8.11.(2) we have

iζX [¦, Ψ] = [iζX ¦, Ψ] ’ [¦, iζX Ψ] + i([¦, ζX ])Ψ + i([Ψ, ζX ])¦

= [ζX , Ψ] ’ 0 + 0 + 0 = 0.

Let now all vector ¬elds ξi again be horizontal, then from the huge formula 8.9

for the Fr¨licher-Nijenhuis bracket only the following terms in the third and ¬fth

o

line survive:

(’1)

[¦, Ψ](ξ1 , . . . , ξ +1 ) = sign σ ¦([Ψ(ξσ1 , . . . , ξσ ), ξσ( +1) ])

!

σ

1

+ sign σ ¦(Ψ([ξσ1 , ξσ2 ], ξσ3 , . . . , ξσ( +1) ).

( ’1)! 2!

σ

For f : P ’ g and horizontal ξ we have ¦[ξ, ζf ] = ζξ(f ) = ζdf (ξ) since it is

C ∞ (P, R)-linear in ξ. So the last expression becomes

’ζ(dψ(ξ0 , . . . , ξk )) = ’ζ(dω ψ(ξ0 , . . . , ξk )) = ’ζ((dψ + [ω, ψ]§ )(ξ0 , . . . , ξk ))

as required.

11.6. Theorem. Let (P, p, M, G) be a principal ¬ber bundle with principal

connection ω. Then the parallel transport for the principal connection is globally

de¬ned and G-equivariant.

In detail: For each smooth curve c : R ’ M there is a smooth mapping

Ptc : R — Pc(0) ’ P such that the following holds:

d

(1) Pt(c, t, u) ∈ Pc(t) , Pt(c, 0) = IdPc(0) , and ω( dt Pt(c, t, u)) = 0.

(2) Pt(c, t) : Pc(0) ’ Pc(t) is G-equivariant, i.e. Pt(c, t, u.g) = Pt(c, t, u).g

holds for all g ∈ G and u ∈ P . Moreover we have Pt(c, t)— (ζX |Pc(t) ) =

ζX |Pc(0) for all X ∈ g.

(3) For any smooth function f : R ’ R we have

Pt(c, f (t), u) = Pt(c —¦ f, t, Pt(c, f (0), u)).

Proof. By 11.4 the Christo¬el forms “± ∈ „¦1 (U± , X(G)) of the connection ω with

respect to a principal ¬ber bundle atlas (U± , •± ) are given by “± (ξx ) = Rω± (ξx ) ,

so they take values in the Lie subalgebra XR (G) of all right invariant vector

¬elds on G, which are bounded with respect to any right invariant Riemannian

metric on G. Each right invariant metric on a Lie group is complete. So the

connection is complete by the remark in 9.9.

Properties (1) and (3) follow from theorem 9.8, and (2) is seen as follows:

ω( dt Pt(c, t, u).g) = Ad(g ’1 )ω( dt Pt(c, t, u)) = 0

d d

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106 Chapter III. Bundles and connections

implies that Pt(c, t, u).g = Pt(c, t, u.g). For the second assertion we compute for

u ∈ Pc(0) :

Pt(c, t)— (ζX |Pc(t) )(u) = T Pt(c, t)’1 ζX (Pt(c, t, u)) =

= T Pt(c, t)’1 ds |0 Pt(c, t, u). exp(sX) =

d

= T Pt(c, t)’1 ds |0 Pt(c, t, u. exp(sX)) =

d

’1

d

ds |0 Pt(c, t)

= Pt(c, t, u. exp(sX))

d

ds |0 u. exp(sX) = ζX (u).

=

11.7. Holonomy groups. Let (P, p, M, G) be a principal ¬ber bundle with

principal connection ¦ = ζ —¦ ω. We assume that M is connected and we ¬x

x0 ∈ M .

In 9.10 we de¬ned the holonomy group Hol(¦, x0 ) ‚ Di¬(Px0 ) as the group

of all Pt(c, 1) : Px0 ’ Px0 for c any piecewise smooth closed loop through

x0 . (Reparametrizing c by a function which is ¬‚at at each corner of c we may

assume that any c is smooth.) If we consider only those curves c which are

nullhomotopic, we obtain the restricted holonomy group Hol0 (¦, x0 ).

Now let us ¬x u0 ∈ Px0 . The elements „ (u0 , Pt(c, t, u0 )) ∈ G form a subgroup

of the structure group G which is isomorphic to Hol(¦, x0 ); we denote it by

Hol(ω, u0 ) and we call it also the holonomy group of the connection. Considering

only nullhomotopic curves we get the restricted holonomy group Hol0 (ω, u0 ) a

normal subgroup of Hol(ω, u0 ).

Theorem. The main results for the holonomy are as follows:

(1) We have Hol(ω, u0 .g) = conj(g ’1 ) Hol(ω, u0 ) and

Hol0 (ω, u0 .g) = conj(g ’1 ) Hol0 (ω, u0 ).

(2) For each curve c in M with c(0) = x0 we have Hol(ω, Pt(c, t, u0 )) =

Hol(ω, u0 ) and Hol0 (ω, Pt(c, t, u0 )) = Hol0 (ω, u0 ).

(3) Hol0 (ω, u0 ) is a connected Lie subgroup of G and the quotient group

Hol(ω, u0 )/ Hol0 (ω, u0 ) is at most countable, so Hol(ω, u0 ) is also a Lie

subgroup of G.

(4) The Lie algebra hol(ω, u0 ) ‚ g of Hol(ω, u0 ) is linearly generated by

{„¦(Xu , Yu ) : Xu , Yu ∈ Tu P }, and it is isomorphic to the holonomy Lie

algebra hol(¦, x0 ) we considered in 9.10.

(5) For u0 ∈ Px0 let P (ω, u0 ) be the set of all Pt(c, t, u0 ) for c any (piecewise)

smooth curve in M with c(0) = x0 and for t ∈ R. Then P (ω, u0 ) is

a sub ¬ber bundle of P which is invariant under the right action of

Hol(ω, u0 ); so it is itself a principal ¬ber bundle over M with structure

group Hol(ω, u0 ) and we have a reduction of structure group, cf. 10.6 and

10.13. The pullback of ω to P (ω, u0 ) is then again a principal connection

form i— ω ∈ „¦1 (P (ω, u0 ); hol(ω, u0 )).

(6) P is foliated by the leaves P (ω, u), u ∈ Px0 .

(7) If the curvature „¦ = 0 then Hol0 (ω, u0 ) = {e} and each P (ω, u) is a

covering of M .

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11. Principal and induced connections 107

(8) If one uses piecewise C k -curves for 1 ¤ k < ∞ in the de¬nition, one gets

the same holonomy groups.

In view of assertion 5 a principal connection ω is called irreducible if Hol(ω, u0 )

equals the structure group G for some (equivalently any) u0 ∈ Px0 .

Proof. 1. This follows from the properties of the mapping „ from 10.2 and from

the G-equivariance of the parallel transport.

The rest of this theorem is a compilation of well known results, and we refer

to [Kobayashi-Nomizu I, 63, p. 83¬] for proofs.

11.8. Inducing connections on associated bundles. Let (P, p, M, G) be a

principal bundle with principal right action r : P — G ’ P and let : G — S ’

S be a left action of the structure group G on some manifold S. Then we

consider the associated bundle P [S] = P [S, ] = P —G S, constructed in 10.7.

Recall from 10.18 that its tangent and vertical bundle are given by T (P [S, ]) =

T P [T S, T ] = T P —T G T S and V (P [S, ]) = P [T S, T2 ] = P —G T S.

Let ¦ = ζ —¦ ω ∈ „¦1 (P ; T P ) be a principal connection on the principal bundle

¯

P . We construct the induced connection ¦ ∈ „¦1 (P [S], T (P [S])) by the following

diagram:

w w

¦ — Id =

TP — TS TP — TS T (P — S)

u u u

Tq = q q Tq

w TP — w T (P —

¯

¦ =

T P —T G T S TS S).

TG G

Let us ¬rst check that the top mapping ¦ — Id is T G-equivariant. For g ∈ G and

X ∈ g the inverse of Te (»g )X in the Lie group T G is denoted by (Te (»g )X)’1 ,

see lemma 10.17. Furthermore by 5.13 we have

T r(ξu , Te (»g )X) = Tu (rg )ξu + T r((0P — LX )(u, g))

= Tu (rg )ξu + Tg (ru )(Te (»g )X)

= Tu (rg )ξu + ζX (ug).

We may compute

(¦ — Id)(T r(ξu , Te (»g )X), T ((Te (»g )X)’1 , ·s ))

= (¦(Tu (rg )ξu + ζX (ug)), T ((Te (»g )X)’1 , ·s ))

= (¦(Tu (rg )ξu ) + ¦(ζX (ug)), T ((Te (»g )X)’1 , ·s ))

= ((Tu (rg )¦ξu ) + ζX (ug), T ((Te (»g )X)’1 , ·s ))

= (T r(¦(ξu ), Te (»g )X), T ((Te (»g )X)’1 , ·s )).

¯

So the mapping ¦ — Id factors to ¦ as indicated in the diagram, and we have

¯¯ ¯ ¯

¦ —¦ ¦ = ¦ from (¦ — Id) —¦ (¦ — Id) = ¦ — Id. The mapping ¦ is ¬berwise linear,

¯

since ¦ — Id and q = T q are. The image of ¦ is

q (V P — T S) = q (ker(T p : T P — T S ’ T M ))

= ker(T p : T P —T G T S ’ T M ) = V (P [S, ]).

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

108 Chapter III. Bundles and connections

¯

Thus ¦ is a connection on the associated bundle P [S]. We call it the induced

connection.

From the diagram it also follows, that the vector valued forms ¦ — Id ∈

¯

„¦ (P — S; T P — T S) and ¦ ∈ „¦1 (P [S]; T (P [S])) are (q : P — S ’ P [S])-related.

1

So by 8.15 we have for the curvatures

R¦—Id = 1 [¦ — Id, ¦ — Id] = 1 [¦, ¦] — 0 = R¦ — 0,

2 2

1¯ ¯

R¦ = [¦, ¦],

¯ 2

that they are also q-related, i.e. T q —¦ (R¦ — 0) = R¦ —¦ (T q —M T q).

¯

By uniqueness of the solutions of the de¬ning di¬erential equation we also get

that

Pt¦ (c, t, q(u, s)) = q(Pt¦ (c, t, u), s).

¯

11.9. Recognizing induced connections. We consider again a principal

¬ber bundle (P, p, M, G) and a left action : G — S ’ S. Suppose that Ψ ∈

„¦1 (P [S]; T (P [S])) is a connection on the associated bundle P [S] = P [S, ]. Then

the following question arises: When is the connection Ψ induced from a principal

connection on P ? If this is the case, we say that Ψ is compatible with the G-

bundle structure on P [S]. The answer is given in the following

Theorem. Let Ψ be a (general) connection on the associated bundle P [S]. Let

us suppose that the action is in¬nitesimally e¬ective, i.e. the fundamental

vector ¬eld mapping ζ : g ’ X(S) is injective.

Then the connection Ψ is induced from a principal connection ω on P if and

only if the following condition is satis¬ed:

In some (equivalently any) ¬ber bundle atlas (U± , ψ± ) of P [S] belong-

ing to the G-bundle structure of the associated bundle the Christo¬el

forms “± ∈ „¦1 (U± ; X(S)) have values in the sub Lie algebra Xf und (S) of

fundamental vector ¬elds for the action .

Proof. Let (U± , •± : P |U± ’ U± — G) be a principal ¬ber bundle atlas for P .

Then by the proof of theorem 10.7 it is seen that the induced ¬ber bundle atlas

(U± , ψ± : P [S]|U± ’ U± — S) is given by

ψ± (x, s) = q(•’1 (x, e), s),

’1

(1) ±

(ψ± —¦ q)(•’1 (x, g), s) = (x, g.s).

(2) ±

¯

Let ¦ = ζ —¦ω be a principal connection on P and let ¦ be the induced connection

on the associated bundle P [S]. By 9.7 its Christo¬el symbols are given by

¯ ’1

(0x , “± (ξx , s)) = ’(T (ψ± ) —¦ ¦ —¦ T (ψ± ))(ξx , 0s )

¯

¦

¯

= ’(T (ψ± ) —¦ ¦ —¦ T q —¦ (T (•’1 ) — Id))(ξx , 0e , 0s ) by (1)

±

= ’(T (ψ± ) —¦ T q —¦ (¦ — Id))(T (•’1 )(ξx , 0e ), 0s ) by 11.8

±

= ’(T (ψ± ) —¦ T q)(¦(T (•’1 )(ξx , 0e )), 0s )

±

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11. Principal and induced connections 109

= (T (ψ± ) —¦ T q)(T (•’1 )(0x , “± (ξx , e)), 0s ) by 11.4.(3)

± ¦

= ’T (ψ± —¦ q —¦ (•’1 — Id))(0x , ω± (ξx ), 0s ) by 11.4.(7)

±

= ’Te ( s )ω± (ξx ) by (2)

= ’ζω± (ξx ) (s).

So the condition is necessary. Now let us conversely suppose that a connection

Ψ on P [S] is given such that the Christo¬el forms “± with respect to a ¬ber

Ψ

bundle atlas of the G-structure have values in Xf und (S). Then unique g-valued

forms ω± ∈ „¦1 (U± ; g) are given by the equation “± (ξx ) = ζ(ω± (ξx )), since the

Ψ

action is in¬nitesimally e¬ective. From the transition formulas 9.7 for the “± Ψ

follow the transition formulas 11.4.(5) for the ω ± , so that they give a unique

principal connection on P , which by the ¬rst part of the proof induces the given

connection Ψ on P [S].

11.10. Inducing connections on associated vector bundles.

Let (P, p, M, G) be a principal ¬ber bundle and let ρ : G ’ GL(W ) be a

representation of the structure group G on a ¬nite dimensional vector space W .

We consider the associated vector bundle (E := P [W, ρ], p, M, W ), from 10.11.

Recall from 6.11 that T (E) = T P —T G T W has two vector bundle structures

with the projections

πE : T (E) = T P —T G T W ’ P —G W = E,

T p —¦ pr1 : T (E) = T P —T G T W ’ T M.

Now let ¦ = ζ —¦ ω ∈ „¦1 (P ; T P ) be a principal connection on P . We consider

¯

the induced connection ¦ ∈ „¦1 (E; T (E)) from 11.8. Inserting the projections

of both vector bundle structures on T (E) into the diagram in 11.8 we get the

following diagram

xw

&& ¦ — Id

TP — TW TP — TW TP — W — W

&& x

xxπ

(

&

x

π

P —W

u

q

Tq Tq

x

x

P —G W = E

u xu

π πE

E

xw T P —

TP — TW &

&& TW TE

x

TG TG

¯

¦

&& xxT p

Tp (

x

TM

and from it one easily sees that the induced connection is linear in both vector

bundle structures. We say that it is a linear connection on the associated bundle.

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

110 Chapter III. Bundles and connections

Recall now from 6.11 the vertical lift vlE : E —M E ’ V E, which is an

isomorphism, pr1 “πE “¬berwise linear and also p“T p“¬berwise linear.

¯

Now we de¬ne the connector K of the linear connection ¦ by

¯

K := pr2 —¦ (vlE )’1 —¦ ¦ : T E ’ V E ’ E —M E ’ E.

Lemma. The connector K : T E ’ E is a vector bundle homomorphism for

both vector bundle structures on T E and satis¬es K —¦ vlE = pr2 : E —M E ’

T E ’ E.

So K is πE “p“¬berwise linear and T p“p“¬berwise linear.

Proof. This follows from the ¬berwise linearity of the components of K and from

its de¬nition.

11.11. Linear connections. If (E, p, M ) is a vector bundle, a connection

Ψ ∈ „¦1 (E; T E) such that Ψ : T E ’ V E ’ T E is also T p“T p“¬berwise linear

is called a linear connection. An easy check with 11.9 or a direct construction

shows that Ψ is then induced from a unique principal connection on the linear

frame bundle GL(Rn , E) of E (where n is the ¬ber dimension of E).

Equivalently a linear connection may be speci¬ed by a connector K : T E ’ E

with the three properties of lemma 11.10. For then HE := {ξu : K(ξu ) = 0p(u) }

is a complement to V E in T E which is T p“¬berwise linearly chosen.

11.12. Covariant derivative on vector bundles. Let (E, p, M ) be a vector

bundle with a linear connection, given by a connector K : T E ’ E with the

properties in lemma 11.10.

For any manifold N , smooth mapping s : N ’ E, and vector ¬eld X ∈ X(N )

we de¬ne the covariant derivative of s along X by

:= K —¦ T s —¦ X : N ’ T N ’ T E ’ E.

(1) Xs

∞

If f : N ’ M is a ¬xed smooth mapping, let us denote by Cf (N, E) the vector

space of all smooth mappings s : N ’ E with p —¦ s = f “ they are called sections

of E along f . From the universal property of the pullback it follows that the

vector space Cf (N, E) is canonically linearly isomorphic to the space C ∞ (f — E)

∞

of sections of the pullback bundle. Then the covariant derivative may be viewed

as a bilinear mapping

∞ ∞

: X(N ) — Cf (N, E) ’ Cf (N, E).

(2)

Lemma. This covariant derivative has the following properties:

(3) X s is C ∞ (N, R)-linear in X ∈ X(N ). So for a tangent vector Xx ∈

∞

Tx N the mapping Xx : Cf (N, E) ’ Ef (x) makes sense and we have

( X s)(x) = X(x) s.

∞

(4) X s is R-linear in s ∈ Cf (N, E).

(5) X (h.s) = dh(X).s + h. X s for h ∈ C ∞ (N, R), the derivation property

of X .

(6) For any manifold Q and smooth mapping g : Q ’ N and Yy ∈ Ty Q we

have T g.Yy s = Yy (s —¦ g). If Y ∈ X(Q) and X ∈ X(N ) are g-related,

then we have Y (s —¦ g) = ( X s) —¦ g.

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

11. Principal and induced connections 111

Proof. All these properties follow easily from the de¬nition (1).

For vector ¬elds X, Y ∈ X(M ) and a section s ∈ C ∞ (E) an easy computation

shows that

RE (X, Y )s : = s’ ’

Xs [X,Y ] s

X Y Y

]’

= ([ X, [X,Y ] )s

Y

is C ∞ (M, R)-linear in X, Y , and s. By the method of 7.3 it follows that RE is a 2-

form on M with values in the vector bundle L(E, E), i.e. RE ∈ „¦2 (M ; L(E, E)).

It is called the curvature of the covariant derivative.

∞

For f : N ’ M , vector ¬elds X, Y ∈ X(N ) and a section s ∈ Cf (N, E)

along f one may prove that

= (f — RE )(X, Y )s = RE (T f.X, T f.Y )s.

s’ ’

Xs [X,Y ] s

X Y Y

We will do this in 37.15.(2).

11.13. Covariant exterior derivative. Let (E, p, M ) be a vector bundle with

a linear connection, given by a connector K : T E ’ E.

For a smooth mapping f : N ’ M let „¦(N ; f — E) be the vector space of all

forms on N with values in the vector bundle f — E. We can also view them as

forms on N with values along f in E, but we do not introduce an extra notation

for this.

The graded space „¦(N ; f — E) is a graded „¦(N )-module via

(• § ¦)(X1 , . . . , Xp+q ) =

1

= sign(σ) •(Xσ1 , . . . , Xσp )¦(Xσ(p+1) , . . . , Xσ(p+q) ).

p! q!

σ

It can be shown that the graded module homomorphisms H : „¦(N ; f — E) ’

„¦(N ; f — E) (so that H(• § ¦) = (’1)deg H. deg • • § H(¦)) are exactly the map-

pings µ(A) for A ∈ „¦p (N ; f — L(E, E)), which are given by

(µ(A)¦)(X1 , . . . , Xp+q ) =

1

= sign(σ) A(Xσ1 , . . . , Xσp )(¦(Xσ(p+1) , . . . , Xσ(p+q) )).

p! q!

σ

The covariant exterior derivative d : „¦p (N ; f — E) ’ „¦p+1 (N ; f — E) is de¬ned

by (where the Xi are vector ¬elds on N )

p

(’1)i

(d ¦)(X0 , . . . , Xp ) = Xi ¦(X0 , . . . , Xi , . . . , Xp )

i=0

(’1)i+j ¦([Xi , Xj ], X0 , . . . , Xi , . . . , Xj , . . . , Xp ).

+

0¤i<j¤p

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112 Chapter III. Bundles and connections

Lemma. The covariant exterior derivative is well de¬ned and has the following

properties.

(1) For s ∈ C ∞ (f — E) = „¦0 (N ; f — E) we have (d s)(X) = X s.

(2) d (• § ¦) = d• § ¦ + (’1)deg • • § d ¦.

(3) For smooth g : Q ’ N and ¦ ∈ „¦(N ; f — E) we have d (g — ¦) = g — (d ¦).

(4) d d ¦ = µ(f — RE )¦.

Proof. It su¬ces to investigate decomposable forms ¦ = • — s for • ∈ „¦p (N )

and s ∈ C ∞ (f — E). Then from the de¬nition we have d (• — s) = d• — s +

(’1)p • § d s. Since by 11.12.(3) d s ∈ „¦1 (N ; f — E), the mapping d is well

de¬ned. This formula also implies (2) immediately. (3) follows from 11.12.(6).

(4) is checked as follows:

d d (• — s) = d (d• — s + (’1)p • § d s) by (2)

= 0 + (’1)2p • § d d s

= • § µ(f — RE )s by the de¬nition of RE

= µ(f — RE )(• — s).

11.14. Let (P, p, M, G) be a principal ¬ber bundle and let ρ : G ’ GL(W ) be

a representation of the structure group G on a ¬nite dimensional vector space

W.

Theorem. There is a canonical isomorphism from the space of P [W, ρ]-valued

di¬erential forms on M onto the space of horizontal G-equivariant W -valued

di¬erential forms on P :

q : „¦(M ; P [W, ρ]) ’ „¦hor (P ; W )G := {• ∈ „¦(P ; W ) : iX • = 0

for all X ∈ V P, (rg )— • = ρ(g ’1 ) —¦ • for all g ∈ G}.

In particular for W = R with trivial representation we see that

p— : „¦(M ) ’ „¦hor (P )G = {• ∈ „¦hor (P ) : (rg )— • = •}

is also an isomorphism. The isomorphism

q : „¦0 (M ; P [W ]) = C ∞ (P [W ]) ’ „¦0 (P ; W )G = C ∞ (P, W )G

hor

is a special case of the one from 10.12.

Proof. Recall the smooth mapping „ G : P —M P ’ G from 10.2, which satis¬es

r(ux , „ G (ux , vx )) = vx , „ G (ux .g, ux .g ) = g ’1 .„ G (ux , ux ).g , and „ G (ux , ux ) = e.

Let • ∈ „¦k (P ; W )G , X1 , . . . , Xk ∈ Tu P , and X1 , . . . , Xk ∈ Tu P such that

hor

Tu p.Xi = Tu p.Xi for each i. Then we have for g = „ G (u, u ), so that ug = u :

q(u, •u (X1 , . . . , Xk )) = q(ug, ρ(g ’1 )•u (X1 , . . . , Xk ))

= q(u , ((rg )— •)u (X1 , . . . , Xk ))

= q(u , •ug (Tu (rg ).X1 , . . . , Tu (rg ).Xk ))

= q(u , •u (X1 , . . . , Xk )), since Tu (rg )Xi ’ Xi ∈ Vu P.

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

11. Principal and induced connections 113

By this prescription a vector bundle valued form ¦ ∈ „¦k (M ; P [W ]) is uniquely

determined.

For the converse recall the smooth mapping „ W : P —M P [W, ρ] ’ W

from 10.7, which satis¬es „ W (u, q(u, w)) = w, q(ux , „ W (ux , vx )) = vx , and

„ W (ux g, vx ) = ρ(g ’1 )„ W (ux , vx ).

For ¦ ∈ „¦k (M ; P [W ]) we de¬ne q ¦ ∈ „¦k (P ; W ) as follows. For Xi ∈ Tu P

we put

(q ¦)u (X1 , . . . , Xk ) := „ W (u, ¦p(u) (Tu p.X1 , . . . , Tu p.Xk )).

Then q ¦ is smooth and horizontal. For g ∈ G we have

((rg )— (q ¦))u (X1 , . . . , Xk ) = (q ¦)ug (Tu (rg ).X1 , . . . , Tu (rg ).Xk )

= „ W (ug, ¦p(ug) (Tug p.Tu (rg ).X1 , . . . , Tug p.Tu (rg ).Xk ))

= ρ(g ’1 )„ W (u, ¦p(u) (Tu p.X1 , . . . , Tu p.Xk ))

= ρ(g ’1 )(q ¦)u (X1 , . . . , Xk ).

Clearly the two constructions are inverse to each other.

11.15. Let (P, p, M, G) be a principal ¬ber bundle with a principal connection

¦ = ζ —¦ ω, and let ρ : G ’ GL(W ) be a representation of the structure group G

on a ¬nite dimensional vector space W . We consider the associated vector bundle

¯

(E := P [W, ρ], p, M, W ), the induced connection ¦ on it and the corresponding

covariant derivative.

Theorem. The covariant exterior derivative dω from 11.5 on P and the co-

variant exterior derivative for P [W ]-valued forms on M are connected by the

mapping q from 11.14, as follows:

q —¦ d = dω —¦ q : „¦(M ; P [W ]) ’ „¦hor (P ; W )G .

Proof. Let ¬rst f ∈ „¦0 (P ; W )G = C ∞ (P, W )G , then we have f = q s for s ∈

hor

∞

C (P [W ]). Moreover we have f (u) = „ W (u, s(p(u))) and s(p(u)) = q(u, f (u))

by 11.14 and 10.12. Therefore T s.T p.Xu = T q(Xu , T f.Xu ), where T f.Xu =

(f (u), df (Xu )) ∈ T W = W — W . If χ : T P ’ HP is the horizontal projection

as in 11.5, we have T s.T p.Xu = T s.T p.χ.Xu = T q(χ.Xu , T f.χ.Xu ). So we get

(q d s)(Xu ) = „ W (u, (d s)(T p.Xu ))

= „ W (u, T p.Xu s) by 11.13.(1)

= „ W (u, K.T s.T p.Xu ) by 11.12.(1)

= „ W (u, K.T q(χ.Xu , T f.χ.Xu )) from above

= „ W (u, pr2 .vl’1 .¦.T q(χ.Xu , T f.χ.Xu ))

¯ by 11.10

P [W ]

’1

„ W (u, pr2 .vlP [W ] .T q.(¦ — Id)(χ.Xu , T f.χ.Xu )))

= by 11.8

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

114 Chapter III. Bundles and connections

’1

= „ W (u, pr2 .vlP [W ] .T q(0u , T f.χ.Xu ))) since ¦.χ = 0

’1

= „ W (u, q.pr2 .vlP —W .(0u , T f.χ.Xu ))) since q is ¬ber linear

= „ W (u, q(u, df.χ.Xu )) = (χ— df )(Xu )

= (dω q s)(Xu ).

Now we turn to the general case. It su¬ces to check the formula for a decom-

posable P [W ]-valued form Ψ = ψ — s ∈ „¦k (M, P [W ]), where ψ ∈ „¦k (M ) and

s ∈ C ∞ (P [W ]). Then we have

dω q (ψ — s) = dω (p— ψ · q s)

= dω (p— ψ) · q s + (’1)k χ— p— ψ § dω q s by 11.5.(1)

= χ— p— dψ · q s + (’1)k p— ψ § q d s from above and 11.5.(4)

= p— dψ · q s + (’1)k p— ψ § q d s

= q (dψ — s + (’1)k ψ § d s)

= q d (ψ — s).

11.16. Corollary. In the situation of theorem 11.15 above we have for the cur-

vature form „¦ ∈ „¦2 (P ; g) and the curvature RP [W ] ∈ „¦2 (M ; L(P [W ], P [W ]))

hor

the relation

qL(P [W ],P [W ]) RP [W ] = ρ —¦ „¦,

where ρ = Te ρ : g ’ L(W, W ) is the derivative of the representation ρ.

Proof. We use the notation of the proof of theorem 11.15. By this theorem we

have for X, Y ∈ Tu P

(dω dω qP [W ] s)u (X, Y ) = (q d d s)u (X, Y )

= (q RP [W ] s)u (X, Y )

= „ W (u, RP [W ] (Tu p.X, Tu p.Y )s(p(u)))

= (qL(P [W ],P [W ]) RP [W ] )u (X, Y )(qP [W ] s)(u).

On the other hand we have by theorem 11.5.(8)

(dω dω q s)u (X, Y ) = (χ— iR dq s)u (X, Y )

= (dq s)u (R(X, Y )) since R is horizontal

= (dq s)(’ζ„¦(X,Y ) (u)) by 11.2

ζ

‚ „¦(X,Y )

= (q s)(Fl’t (u))

‚t 0

„ W (u. exp(’t„¦(X, Y )), s(p(u. exp(’t„¦(X, Y )))))

‚

= ‚t 0

„ W (u. exp(’t„¦(X, Y )), s(p(u)))

‚

= ‚t 0

ρ(exp t„¦(X, Y ))„ W (u, s(p(u)))

‚

= by 10.7

‚t 0

= ρ („¦(X, Y ))(q s)(u).

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

Remarks 115

Remarks

The concept of connections on general ¬ber bundles was formulated at about

1970, see e. g. [Libermann, 73]. The theorem 9.9 that each ¬ber bundle admits

a complete connection is contained in [Wolf, 67], with an incorrect proof. It is

an exercise in [Greub-Halperin-Vanstone I, 72, p 314]. The proof given here and

the generalization 9.11 of the Ambrose Singer theorem are from [Michor, 88], see

also [Michor, 91], which are also the source for 11.8 and 11.9. The results 11.15

and 11.16 appear here for the ¬rst time.

¦

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

116

CHAPTER IV.

JETS AND NATURAL BUNDLES

In this chapter we start our systematic treatment of geometric objects and

operators. It has become commonplace to think of geometric objects on a man-

ifold M as forming ¬ber bundles over the base M and to work with sections

of these bundles. The concrete objects were frequently described in coordinates

by their behavior under the coordinate changes. Stressing the change of coor-

dinates, we can say that local di¬eomorphisms on the base manifold operate on

the bundles of geometric objects. Since a further usual assumption is that the

resulting transformations depend only on germs of the underlying morphisms,

we actually deal with functors de¬ned on all open submanifolds of M and local

di¬eomorphisms between them (let us recall that local di¬eomorphisms are glob-

ally de¬ned locally invertible maps), see the preface. This is the point of view

introduced by [Nijenhuis, 72] and worked out later by [Terng, 78], [Palais, Terng,

77], [Epstein, Thurston, 79] and others. These functors are fully determined by

their restriction to any open submanifold and therefore they extend to the whole

category Mfm of m-dimensional manifolds and local di¬eomorphisms. An im-

portant advantage of such a de¬nition of bundles of geometric objects is that we

get a precise de¬nition of geometric operators in the concept of natural opera-

tors. These are rules transforming sections of one natural bundle into sections of

another one and commuting with the induced actions of local di¬eomorphisms

between the base manifolds.

In the theory of natural bundles and operators, a prominent role is played

by jets. Roughly speaking, jets are certain equivalence classes of smooth maps

between manifolds, which are represented by Taylor polynomials. We start this

chapter with a thorough treatment of jets and jet bundles, and we investigate the

so called jet groups. Then we give the de¬nition of natural bundles and deduce

that the r-th order natural bundles coincide with the associated ¬bre bundles to

r-th order frame bundles. So they are in bijection with the actions of the r-th

order jet group Gr on manifolds. Moreover, natural transformations between

m

the r-th order natural bundles bijectively correspond to Gr -equivariant maps.

m

Let us note that in chapter V we deduce a rather general theory of functors on

categories over manifolds and we prove that both the ¬niteness of the order and

the regularity of natural bundles are consequences of the other axioms, so that

actually we describe all natural bundles here. Next we treat the basic properties

of natural operators. In particular, we show that k-th order natural operators

are described by natural transformations of the k-th order jet prolongations of

the bundles in question. This reduces even the problem of ¬nding ¬nite order

natural operators to determining Gr -equivariant maps, which will be discussed

m

in chapter VI.

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

12. Jets 117

Further we present the procedure of principal prolongation of principal ¬ber

bundles based on an idea of [Ehresmann, 55] and we show that the jet prolonga-

tions of associated bundles are associated bundles to the principal prolongations

of the corresponding principal bundles. This fact is of basic importance for the

theory of gauge natural bundles and operators, the foundations of which will be

presented in chapter XII. The canonical form on ¬rst order principal prolonga-

tion of a principal bundle generalizes the well known canonical form on an r-th

order frame bundle. These canonical forms are used in a formula for the ¬rst jet

prolongation of sections of arbitrary associated ¬ber bundles, which represents a

common basis for several algorithms in di¬erent branches of di¬erential geome-

try. At the end of the chapter, we reformulate a part of the theory of connections

from the point of view of jets, natural bundles and natural operators. This is

necessary for our investigation of natural operations with connections, but we

believe that this also demonstrates the power of the jet approach to give a clear

picture of geometric concepts.

12. Jets

12.1. Roughly speaking, two maps f , g : M ’ N are said to determine the

same r-jet at x ∈ M , if they have the r-th order contact at x. To make this idea

precise, we ¬rst de¬ne the r-th order contact of two curves on a manifold. We

recall that a smooth function R ’ R is said to vanish to r-th order at a point,

if all its derivatives up to order r vanish at this point.

De¬nition. Two curves γ, δ : R ’ M have the r-th contact at zero, if for every

smooth function • on M the di¬erence • —¦ γ ’ • —¦ δ vanishes to r-th order at

0 ∈ R.

In this case we write γ ∼r δ. Obviously, ∼r is an equivalence relation. For

r = 0 this relation means γ(0) = δ(0).

Lemma. If γ ∼r δ, then f —¦ γ ∼r f —¦ δ for every map f : M ’ N .

Proof. If • is a function on N , then • —¦ f is a function on M . Hence • —¦ f —¦ γ ’

• —¦ f —¦ δ has r-th order zero at 0.

12.2. De¬nition. Two maps f , g : M ’ N are said to determine the same

r-jet at x ∈ M , if for every curve γ : R ’ M with γ(0) = x the curves f —¦ γ and

g —¦ γ have the r-th order contact at zero.

In such a case we write jx f = jx g or j r f (x) = j r g(x).

r r

An equivalence class of this relation is called an r-jet of M into N . Obviously,

r

jx f depends on the germ of f at x only. The set of all r-jets of M into N is

denoted by J r (M, N ). For X = jx f ∈ J r (M, N ), the point x =: ±X is the

r

r

source of X and the point f (x) =: βX is the target of X. We denote by πs ,

r s r

0 ¤ s ¤ r, the projection jx f ’ jx f of r-jets into s-jets. By Jx (M, N ) or

J r (M, N )y we mean the set of all r-jets of M into N with source x ∈ M or

target y ∈ N , respectively, and we write Jx (M, N )y = Jx (M, N ) © J r (M, N )y .

r r

The map j r f : M ’ J r (M, N ) is called the r-th jet prolongation of f : M ’ N .

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118 Chapter IV. Jets and natural bundles

¯

12.3. Proposition. If two pairs of maps f , f : M ’ N and g, g : N ’ Q ¯

r¯ ¯(x), then j r (g —¦ f ) = j r (¯ —¦ f ).

¯

r r r

satisfy jx f = jx f and jy g = jy g , f (x) = y = f

¯ xg

x

r¯ ¯

r

Proof. Take a curve γ on M with γ(0) = x. Then jx f = jx f implies f —¦γ ∼r f —¦γ,

¯¯ r r

lemma 12.1 gives g —¦ f —¦ γ ∼r g —¦ f —¦ γ and jy g = jy g yields g —¦ f —¦ γ ∼r g —¦ f —¦ γ.

¯ ¯ ¯

¯¯

Hence g —¦ f —¦ γ ∼r g —¦ f —¦ γ.

In other words, r-th order contact of maps is preserved under composition. If

r r r r

X ∈ Jx (M, N )y and Y ∈ Jy (N, Q)z are of the form X = jx f and Y = jy g, we

r

can de¬ne the composition Y —¦ X ∈ Jx (M, Q)z by

r

Y —¦ X = jx (g —¦ f ).

By the above proposition, Y —¦ X does not depend on the choice of f and g. We

remark that we ¬nd it useful to denote the composition of r-jets by the same

symbol as the composition of maps. Since the composition of maps is associative,

the same holds for r-jets. Hence all r-jets form a category, the units of which

r

are the r-jets of the identity maps of manifolds. An element X ∈ Jx (M, N )y

is invertible, if there exists X ’1 ∈ Jy (N, M )x such that X ’1 —¦ X = jx (idM )

r r

and X —¦ X ’1 = jy (idN ). By the implicit function theorem, X ∈ J r (M, N ) is

r

r

invertible if and only if the underlying 1-jet π1 X is invertible. The existence of

such a jet implies dim M = dim N . We denote by invJ r (M, N ) the set of all

invertible r-jets of M into N .

¯ ¯

12.4. Let f : M ’ M be a local di¬eomorphism and g : N ’ N be a map.

¯¯

Then there is an induced map J r (f, g) : J r (M, N ) ’ J r (M , N ) de¬ned by

J r (f, g)(X) = (jy g) —¦ X —¦ (jx f )’1

r r

where x = ±X and y = βX are the source and target of X ∈ J r (M, N ). Since

the jet composition is associative, J r is a functor de¬ned on the product category

Mfm —Mf . (We shall see in 12.6 that the values of J r lie in the category FM.)

12.5. We are going to describe the coordinate expression of r-jets. We recall

that a multiindex of range m is a m-tuple ± = (±1 , . . . , ±m ) of non-negative

integers. We write |±| = ±1 + · · · + ±m , ±! = ±1 ! · · · ±m ! (with 0! = 1), x± =

(x1 )±1 . . . (xm )±m for x = (x1 , . . . , xm ) ∈ Rm . We denote by

‚ |±| f

D± f =

(‚x1 )±1 . . . (‚xm )±m

the partial derivative with respect to the multiindex ± of a function f : U ‚

Rm ’ R.

Proposition. Given a local coordinate system xi on M in a neighborhood of x

and a local coordinate system y p on N in a neighborhood of f (x), two maps f ,

r r

g : M ’ N satisfy jx f = jx g if and only if all the partial derivatives up to order

r of the components f p and g p of their coordinate expressions coincide at x.

Proof. We ¬rst deduce that two curves γ(t), δ(t) : R ’ N satisfy γ ∼r δ if and

only if

dk (y p —¦ γ)(0) dk (y p —¦ δ)(0)

(1) =

dtk dtk

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

12. Jets 119

k = 0, 1, . . . , r, for all coordinate functions y p . On one hand, if γ ∼r δ, then

y p —¦ γ ’ y p —¦ δ vanishes to order r, i.e. (1) is true. On the other hand, let (1)

hold. Given a function • on N with coordinate expression •(y 1 , . . . , y n ), we ¬nd

by the chain rule that all derivatives up to order r of • —¦ δ depend only on the

partial derivatives up to order r of • at γ(0) and on (1). Hence • —¦ γ ’ • —¦ δ

vanishes to order r at 0.

If the partial derivatives up to the order r of f p and g p coincide at x, then

r r

the chain rule implies f —¦ γ ∼r g —¦ γ by (1). This means jx f = jx g. Conversely,

assume jx f = jx g. Consider the curves xi = ai t with arbitrary ai . Then the

r r

coordinate condition for f —¦ γ ∼r g —¦ γ reads

(D± f p (x))a± = (D± g p (x))a±

(2)

|±|=k |±|=k

k = 0, 1, . . . , r. Since ai are arbitrary, (2) implies that all partial derivatives up

to order r of f p and g p coincide at x.

Now we can easily prove that the auxiliary relation γ ∼r δ can be expressed

in terms of r-jets.

r r

Corollary. Two curves γ, δ : R ’ M satisfy γ ∼r δ if and only if j0 γ = j0 δ.

Proof. Since xi —¦ γ and xi —¦ δ are the coordinate expressions of γ and δ, (1) is

r r

equivalent to j0 γ = j0 δ.

12.6. Write Lr r m n r

m,n = J0 (R , R )0 . By proposition 12.5, the elements of Lm,n

can be identi¬ed with the r-th order Taylor expansions of the generating maps,

i.e. with the n-tuples of polynomials of degree r in m variables without absolute

term. Such an expression

ap x±

±