Proof. This is obvious.

37.11. Proof of theorem 37.7. 1. It is easily checked that L is a group with

∞

multiplication ξ · = ξ•·+ξ+·, with unit 0, and with inverse ξ ’1 = i=1 (’ξ)•i

(recall that • is a nilpotent multiplication). As noted already at the of 37.8 the

mapping DF : XF (M ) ’ L is an isomorphism onto the subgroup of expansions,

because Id +C —¦ DF : XF (M ) ’ L ’ End(A — C ∞ (M, R)) is an isomorphism

onto the subgroup of automorphisms.

2. C —¦ DT —F : XT —F (M ) ’ End(A — C ∞ (M, R)) is a Lie algebra isomorphism

onto the sub Lie algebra of End(A — C ∞ (M, R)) of derivations.

F F

3. De¬ne exp : XT —F (M ) ’ XF (M ) by Dexp(X) = exp(DX ). The Baker-

Campbell-Hausdor¬ formula holds for

exp : Der(A — C ∞ (M, R)) ’ Aut(A — C ∞ (M, R)),

since the Lie algebra of derivations is nilpotent.

4. This is obvious since we used only natural constructions.

37.12. The Lie bracket. We come back to the tangent bundle functor T and

its iterates. For T the structures described in theorem 37.7 give just the addition

of vector ¬elds. In fact we have X Y = X + Y , and [X, Y ]T —T = 0.

But we may consider other structures here. We have by 37.1 Al(T ) = D =

R.1 • R.δ for δ 2 = 0. So N ∼ R with the nilpotent multiplication 0, but we still

=

have the usual multiplication, now called m, on R.

For X, Y ∈ XT (M ) we have DX ∈ L = LR (C ∞ (M, R), C ∞ (M, R)), a deriva-

tion given by f.1 + DX (f ).δ = T f —¦ X, see 37.6.(b) ” we changed slightly the

notation. So DX (f ) = X(f ) = df (X) in the usual sense. The space L has one

more structure now, composition, which is determined by specifying a generator

δ of the nilpotent ideal of Al(T ). The usual Lie bracket of vector ¬elds is now

given by D[X,Y ] := DX —¦ DY ’ DY —¦ DX .

37.13. Lemma. In the setting of 37.12 we have

(’T ) —¦ (T Y —¦ X, κT —¦ T X —¦ Y ) = (T +) —¦ (vl —¦ [X, Y ], 0T —¦ Y )

in terms of the natural transformations descibed in 37.4

This is a variant of lemma 6.13 and 6.19.(4). The following proof appears

to be more complicated then the earlier ones, but it demonstrates the use of

natural transformations, and we write out carefully the unusual notation.

Proof. For f ∈ C ∞ (M, R) and X, Y ∈ XT (M ) we compute as follows using

repeatedly the de¬ning equation for DX from 37.12:

T 2 f —¦ T Y —¦ X = T (T f —¦ Y ) —¦ X = T (f.1 • DY (f ).δ1 ) —¦ X

= (T f —¦ X).1 • (T (DY (f )) —¦ X).δ1 , since T preserves products,

= f.1 + DX (f ).δ2 + (DY (f ).1 + DX DY (f ).δ2 ).δ1

= f.1 + DY (f ).δ1 + DX (f ).δ2 + DX DY (f ).δ1 δ2 .

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326 Chapter VIII. Product preserving functors

Now we use the natural transformation and their commutation rules from 37.4

to compute:

T 2 f —¦ (’T ) —¦ (T Y —¦ X, κT —¦ T X —¦ Y ) =

= (’T ) —¦ (T 2 f —¦ T Y —¦ X, κT —¦ T 2 f —¦ T X —¦ Y )

= (’T ) —¦ f.1 + DY (f ).δ1 + DX (f ).δ2 + DX DY (f ).δ1 δ2 ,

κT (f.1 + DX (f ).δ1 + DY (f ).δ2 + DY DX (f ).δ1 δ2 )

= (’T ) —¦ f.1 + DY (f ).δ1 + DX (f ).δ2 + DX DY (f ).δ1 δ2 ,

f.1 + DY (f ).δ1 + DX (f ).δ2 + DY DX (f ).δ1 δ2 )

= f.1 + DY (f ).δ1 + (DX DY ’ DY DX )(f ).δ1 δ2

= (T +) —¦ (0T —¦ (f.1 + DY (f ).δ), vl —¦ (f.1 + D[X,Y ] (f ).δ))

= (T +) —¦ (0T —¦ T f —¦ Y, vl —¦ T f —¦ [X, Y ])

= (T +) —¦ (T 2 f —¦ 0T —¦ Y, T 2 f —¦ vl —¦ [X, Y ])

= T 2 f —¦ (T +) —¦ (0T —¦ Y, vl —¦ [X, Y ]).

37.14. Linear connections and their curvatures. Our next application

will be to derive a global formula for the curvature of a linear connection on a

vector bundle which involves the second tangent bundle of the vector bundle.

So let (E, p, M ) be a vector bundle. Recall from 11.10 and 11.12 that a linear

connection on the vector bundle E can be described by specifying its connector

K : T E ’ E. By lemma 11.10 and by 11.11 any smooth mapping K : T E ’ E

which is a (¬ber linear) homomorphism for both vector bundle structure on T E,

and which is a left inverse to the vertical lift, K —¦vlE = pr2 : E—M E ’ T E ’ E,

speci¬es a linear connection.

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

we have then the covariant derivative of s along X which is given by X s :=

K —¦ T s —¦ X : N ’ T N ’ T E ’ E, see 11.12.

For vector ¬elds X, Y ∈ X(M ) and a section s ∈ C ∞ (E) the curvature RE

of the connection is given by RE (X, Y )s = ([ X , Y ] ’ [X,Y ] )s, see 11.12.

37.15. Theorem.

(1) Let K : T E ’ E be the connector of a linear connection on a vector

bundle (E, p, M ). Then the curvature is given by

RE (X, Y )s = (K —¦ T K —¦ κE ’ K —¦ T K) —¦ T 2 s —¦ T X —¦ Y

for X, Y ∈ X(M ) and a section s ∈ C ∞ (E).

(2) If s : N ’ E is a section along f := p —¦ s : N ’ M then we have for

vector ¬elds X, Y ∈ X(N )

s’ ’

Xs [X,Y ] s =

X Y Y

= (K —¦ T K —¦ κE ’ K —¦ T K) —¦ T 2 s —¦ T X —¦ Y =

= RE (T f —¦ X, T f —¦ Y )s.

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

37. Examples and applications 327

(3) Let K : T 2 M ’ M be a linear connection on the tangent bundle. Then

its torsion is given by

Tor(X, Y ) = (K —¦ κM ’ K) —¦ T X —¦ Y.

Proof. (1) Let ¬rst mE : E ’ E denote the scalar multiplication. Then we have

t

‚ E

‚t 0 mt = vlE where vlE : E ’ T E is the vertical lift. We use then lemma

37.13 and the commutation relations from 37.4 and we get in turn:

mE —¦ K = K —¦ mT E

‚ ‚

vlE —¦ K = t t

‚t 0 ‚t 0

mT E = T K —¦ vl(T E,T p,T M ) .

‚

= TK —¦ t

‚t 0

s’ ’

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

X Y Y

= K —¦ T (K —¦ T s —¦ Y ) —¦ X ’ K —¦ T (K —¦ T s —¦ X) —¦ Y ’ K —¦ T s —¦ [X, Y ]

K —¦ T s —¦ [X, Y ] = K —¦ vlE —¦ K —¦ T s —¦ [X, Y ]

= K —¦ T K —¦ vlT E —¦ T s —¦ [X, Y ]

= K —¦ T K —¦ T 2 s —¦ vlT M —¦ [X, Y ]

= K —¦ T K —¦ T 2 s —¦ ((T Y —¦ X ’ κM —¦ T X —¦ Y ) (T ’) 0T M —¦ Y )

= K —¦ T K —¦ T 2 s —¦ T Y —¦ X ’ K —¦ T K —¦ T 2 s —¦ κM —¦ T X —¦ Y ’ 0.

Now we sum up and use T 2 s —¦ κM = κE —¦ T 2 s to get the result.

(2) The same proof as for (1) applies for the ¬rst equality, with some obvious

changes. To see that it coincides with RE (T f —¦ X, T f —¦ Y )s it su¬ces to write

out (1) and (T 2 s —¦ T X —¦ Y )(x) ∈ T 2 E in canonical charts induced from vector

bundle charts of E.

(3) We have in turn

’ X ’ [X, Y ]

Tor(X, Y ) = XY Y

= K —¦ T Y —¦ X ’ K —¦ T X —¦ Y ’ K —¦ vlT M —¦ [X, Y ]

K —¦ vlT M —¦ [X, Y ] = K —¦ ((T Y —¦ X ’ κM —¦ T X —¦ Y ) (T ’) 0T M —¦ Y )

= K —¦ T Y —¦ X ’ K —¦ κM —¦ T X —¦ Y ’ 0.

37.16. Weil functors and Lie groups. We have seen in 10.17 that the

tangent bundle T G of a Lie group G is again a Lie group, the semidirect product

g G of G with its Lie algebra g.

Now let A be a Weil algebra and let TA be its Weil functor. In the notation

of 4.1 the manifold TA (G) is again a Lie group with multiplication TA (µ) and

inversion TA (ν). By the properties 35.13 of the Weil functor TA we have a sur-

jective homomorphism πA : TA G ’ G of Lie groups. Following the analogy with

the tangent bundle, for a ∈ G we will denote its ¬ber over a by (TA )a G ‚ TA G,

likewise for mappings. With this notation we have the following commutative

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328 Chapter VIII. Product preserving functors

diagram:

w g—A

g—N

w (T wT wg w0

0 A )0 g Ag

u u u

expG

TA exp

(TA )0 exp

w (T wT wG we

πA

e A )e G AG

For a Lie group the structural mappings (multiplication, inversion, identity el-

ement, Lie bracket, exponential mapping, Baker-Campbell-Hausdor¬ formula,

adjoint action) determine each other mutually. Thus their images under the

Weil functor TA are again the same structural mappings. But note that the

canonical ¬‚ip mappings have to be inserted like follows. So for example

κ

g — A ∼ TA g = TA (Te G) ’ Te (TA G)

’

=

is the Lie algebra of TA G and the Lie bracket is just TA ([ , ]). Since the

bracket is bilinear, the description of 35.11 implies that [X — a, Y — b]TA g =

[X, Y ]g — ab. Also TA expG = expTA G . Since expG is a di¬eomorphism near

0 and since (TA )0 (expG ) depends only on the (invertible) jet of expG at 0, the

mapping (TA )0 (expG ) : (TA )0 g ’ (TA )e G is a di¬eomorphism. Since (TA )0 g is

a nilpotent Lie algebra, the multiplication on (TA )e G is globally given by the

Baker-Campbell-Hausdor¬ formula. The natural transformation 0G : G ’ TA G

is a homomorphism which splits the bottom row of the diagram, so TA G is the

semidirect product (TA )0 g G via the mapping TA ρ : (u, g) ’ TA (ρg )(u).

Since we will need it later, let us add the following ¬nal remark: If ω G : T G ’

Te G is the Maurer Cartan form of G (i.e. the left logarithmic derivative of IdG )

then

κ0 —¦ T A ω G —¦ κ : T T A G ∼ T A T G ’ T A T e G ∼ T e T A G

= =

is the Maurer Cartan form of TA G.

Remarks

The material in section 35 is due to [Eck,86], [Luciano, 88] and [Kainz-Michor,

87], the original ideas are from [Weil, 51]. Section 36 is due to [Eck, 86] and

[Kainz-Michor, 87], 36.7 and 36.8 are from [Kainz-Michor, 87], under stronger

locality conditions also to [Eck, 86]. 36.14 is due to [Eck, 86]. The material in

section 37 is from [Kainz-Michor, 87].

¦

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

329

CHAPTER IX.

BUNDLE FUNCTORS

ON MANIFOLDS

The description of the product preserving bundle functors on Mf in terms

of Weil algebras re¬‚ects their general properties in a rather complete way. In

the present chapter we use some other procedures to deduce the basic geometric

properties of arbitrary bundle functors on Mf . Hence the basic subject of this

theory is a bundle functor on Mf that does not preserve products. Sometimes

we also contrast certain properties of the product-preserving and non-product-

preserving bundle functors on Mf . First we study the bundle functors with

the so-called point property, i.e. the image of a one-point set is a one-point

set. In particular, we deduce that their ¬bers are numerical spaces and that

they preserve products if and only if the dimensions of their values behave well.

Then we show that an arbitrary bundle functor on manifolds is, in a certain

sense, a ˜bundle™ of functors with the point property. For an arbitrary vector

bundle functor F on Mf with the point property we also derive a canonical Lie

group structure on the prolongation F G of a Lie group G.

Next we introduce the concept of a ¬‚ow-natural transformation of a bundle

functor F on manifolds. This is a natural transformation F T ’ T F with the

property that for every vector ¬eld X : M ’ T M its functorial prolongation

F X : F M ’ F T M is transformed into the ¬‚ow prolongation FX : F M ’

T F M . We deduce that every bundle functor F on manifolds has a canonical ¬‚ow-

natural transformation, which is a natural equivalence if and only if F preserves

products. Then we point out some special features of natural transformations

from a Weil functor into an arbitrary bundle functor on Mf . This gives a rather

e¬ective method for their description. We also deduce that the homotheties are

the only natural transformations of the r-th order tangent bundle T (r) into itself.

This demonstrates that some properties of T (r) are quite di¬erent from those of

Weil bundles, where such natural transformations are in bijection with a usually

much larger set of all endomorphisms of the corresponding Weil algebras. In the

last section we describe basic properties of the so-called star bundle functors,

which re¬‚ect some constructions of contravariant character on Mf .

38. The point property

38.1. Examples. First we mention some examples of vector bundle functors

which do not preserve products. In 37.2 we deduced that every product pre-

serving vector bundle functor on Mf is the ¬bered product of a ¬nite number

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330 Chapter IX. Bundle functors on manifolds

of copies of the tangent bundle T . In particular, every such functor is of order

one. Hence all tensor powers —p T , p > 1, their sub bundles like S p T , Λp T and

any combinations of them do not preserve products. This is also easily veri¬ed

by counting dimensions. An important example of an r-th order vector bundle

functor is the r-th tangent functor T (r) described in 12.14 and 41.8. Let us men-

tion that another interesting example of an r-th order vector bundle functor, the

bundle of sector r-forms, will be discussed in 48.4.

38.2. Proposition. Every bundle functor F : Mf ’ Mf transforms embed-

dings into embeddings and immersions into immersions.

Proof. According to 1.14, a smooth mapping f : M ’ N is an embedding if

and only if there is an open neighborhood U of f (M ) in N and a smooth map

g : U ’ M such that g —¦ f = idM . Hence if f is an embedding, then F U ‚ F N

is an open neighborhood of F f (F M ) and F g —¦ F f = idF M .

The locality of bundle functors now implies the assertion on immersions.

However this can be also proved easily considering the canonical local form

i : Rm ’ Rm+n , x ’ (x, 0), of immersions, cf. 2.6, and applying F to the

composition of i and the projections pr1 : Rm+n ’ Rm .

38.3. The point property. Let us write pt for a one-point manifold. A bundle

functor F on Mf is said to have the point property if F (pt) = pt. Given such

functor F let us consider the maps ix : pt ’ M , ix (pt) = x, for all manifolds

M and points x ∈ M . The regularity of bundle functors on Mf proved in 20.7

implies that the maps cM : M ’ F M , cM (x) = F ix (pt) are smooth sections of

pM : F M ’ M . By de¬nition, cN —¦f = F f —¦cM for all smooth maps f : M ’ N ,

so that we have found a natural transformation c : IdMf ’ F .

If F = TA for a Weil algebra A, this natural transformation corresponds to

the algebra homomorphism idR • 0 : R ’ R • N = A. The r-th order tangent

functor has the point property, i.e. we have found a bundle functor which does not

preserve the products in any dimension except dimension zero. The technique

from example 22.2 yields easily bundle functors on Mf which preserve products

just in all dimensions less then any ¬xed n ∈ N.

38.4. Lemma. Let S be an m-dimensional manifold and s ∈ S be a point.

If there is a smoothly parameterized system ht of maps, t ∈ R, such that all

ht are di¬eomorphisms except for t = 0, h0 (S) = {s} and h1 = idS , then S is

di¬eomorphic to RdimS .

Proof. Let us recall that if S = ∪∞ Sk where Sk are open submanifolds dif-

k=0

feomorphic to Rm and Sk ‚ Sk+1 for all k, then S is di¬eomorphic to Rm , see

[Hirsch, 76, Chapter 1, Section 2]. So let us choose an increasing sequence of

relatively compact open submanifolds Kn ‚ Kn+1 ‚ S with S = ∪∞ Kn and a

k=1

relatively compact neighborhood U of s di¬eomorphic to Rm . Put S0 = U . Since

S0 is relatively compact, there is an integer n1 with Kn1 ⊃ S0 and a t1 > 0 with

ht1 (Kn1 ) ‚ U . Then we de¬ne S1 = (ht1 )’1 (U ) so that we have S1 ⊃ Kn1 ⊃ S0

and S1 is relatively compact and di¬eomorphic to Rm . Iterating this procedure,

we construct sequences Sk and nk satisfying Sk ⊃ Knk ⊃ Sk’1 , nk > nk’1 .

Let us denote by km the dimensions of standard ¬bers Sm = F0 Rm .

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

38. The point property 331

38.5. Proposition. The standard ¬bers Sm of every bundle functor F on Mf

with the point property are di¬eomorphic to Rkm .

Proof. Let us write s = cRm (0), 0 ∈ Rm , and let gt : Rm ’ Rm be the homoth-

eties gt (x) = tx, t ∈ R. Since g0 (Rm ) = {0}, the smoothly parameterized family

ht = F gt |Sm : Sm ’ Sm satis¬es all assumptions of the previous lemma.

p q Fp Fq

For a product M ← M — N ’ N the values F M ←’ F (M — N ) ’’ F N

’ ’ ’ ’

determine a canonical map π : F (M — N ) ’ F M — F N .

38.6. Lemma. For every bundle functor F on Mf with the point property all

the maps π : F (M — N ) ’ F M — F N are surjective submersions.

Proof. By locality of F it su¬ces to discuss the case M = Rm , N = Rn . Write

0k = cRk (0) ∈ F Rk , k = 0, 1, . . . , and denote i : Rm ’ Rm+n , i(x) = (x, 0),

and j : Rn ’ Rm+n , j(y) = (0, y). In the tangent space T0m+n F Rm+n , there are

subspaces V = T F i(T0m F Rm ) and W = T F j(T0n F Rn ). We claim V © W =

0. Indeed, if A ∈ V © W , i.e. A = T F i(B) = T F j(C) with B ∈ T0m F Rm

and C ∈ T0n F Rn , then T F p(A) = T F p(T F i(B)) = B, but at the same time

T F p(A) = T F p —¦ T F j(C) = 0m , for p —¦ j is the constant map of Rn into 0 ∈ Rm ,

and A = T F i(B) = 0 follows.

Hence T π|(V • W ) : V • W ’ T0m F Rm — T0n F Rn is invertible and so π is a

submersion at 0m+n and consequently on a neighborhood U ‚ F Rm+n of 0m+n .

Since the actions of R de¬ned by the homotheties gt on Rm , Rn and Rm+n

commute with the product projections p and q, the induced actions on F Rm ,

F Rn , F Rm+n commute with π as well (draw a diagram if necessary). The family

F gt is smoothly parameterized and F g0 (F Rm+n ) = {0m+n }, so that every point

of F Rm+n is mapped into U by a suitable F gt , t > 0. Further all F gt with t > 0

are di¬eomorphisms and so π is a submersion globally. Therefore the image

π(F Rm+n ) is an open neighborhood of (0m , 0n ) ∈ F Rm — F Rn . But similarly

as above, every point of F Rm — F Rn can be mapped into this neighborhood by

a suitable F gt , t > 0. This implies that π is surjective.

It should be an easy exercise for the reader to extend the lemma to arbitrary

¬nite products of manifolds.

38.7. Corollary. Every bundle functor F on Mf with the point property

transforms submersions into submersions.

Proof. The local canonical form of any submersion is p : Rn —Rk ’ Rn , p(x, y) =

x, cf. 2.2. Then F p = pr1 —¦ π is a composition of two submersions π : F (Rn —

Rk ) ’ F Rn — F Rk and pr1 : F Rn — F Rk ’ F Rn . Since every bundle functor is

local, this concludes the proof.

38.8. Proposition. If a bundle functor F on Mf has the point property, then

the dimensions of its standard ¬bers satisfy km+n ≥ km + kn for all 0 ¤ m + n <

∞. Equality holds if and only if F preserves products in dimensions m and n.

Proof. By lemma 38.6, we have the submersions π : F (Rm — Rn ) ’ F Rm — F Rn

which implies km+n ≥ km + kn . If the equality holds, then π is a local di¬eomor-

phism at each point. Since π commutes with the action of the homotheties, it

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

332 Chapter IX. Bundle functors on manifolds

must be bijective on each ¬ber over Rm+n , and therefore π must be a global dif-

feomorphism. Given arbitrary manifolds M and N of the proper dimensions, the

locality of bundle functors and a standard diagram chasing lead to the conclusion

that

Fp Fq

F M ←’ F (M — N ) ’’ F N

’ ’

is a product.

In view of the results of the previous chapter we get

38.9. Corollary. For every bundle functor F on Mf with the point property

the dimensions of its values satisfy dimF Rm = mdimF R if and only if there is

a Weil algebra A such that F is naturally equivalent to the Weil bundle TA .

38.10. For every Weil algebra A and every Lie group G there is a canonical Lie

group structure on TA G obtained by the application of the Weil bundle TA to

all operations on G, cf. 37.16. If we replace TA by an arbitrary bundle functor

on Mf , we are not able to repeat this construction. However, in the special case

of a vector bundle functor F on Mf with the point property we can perform

another procedure.

For all manifolds M , N the inclusions iy : M ’ M — N , iy (x) = (x, y),

jx : N ’ M — N , jx (y) = (x, y), (x, y) ∈ M — N , form smoothly parameterized

families of morphisms and so we can de¬ne a morphism „M,N : F M — F N ’

F (M — N ) by „M,N (z, w) = F ipN (w) (z) + F jpM (z) (w), where pM : F M ’ M are

the canonical projections. One veri¬es easily that the diagram

w

„M,N

FM — FN F (M — N )

u u

Ff — Fg F (f — g)

w

„M ,N

¯¯

¯ ¯ ¯ ¯

FM — FN F (M — N )

¯ ¯