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37. Examples and applications 325


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 .

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

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

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