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commutes for all maps f : M ’ M , g : N ’ N . So we have constructed a
natural transformation „ : Prod —¦ (F, F ) ’ F —¦ Prod, where Prod is the bifunctor
corresponding to the products of manifolds and maps. The projections p : M —
N ’ M , q : M — N ’ N determine the map (F p, F q) : F (M — N ) ’ F M — F N
and by the de¬nition of „M,N , we get (F p, F q) —¦ „M,N = idF M —F N . Now, given
a Lie group G with the operations µ : G — G ’ G, ν : G ’ G and e : pt ’ G, we
de¬ne µF G = F µ —¦ „G,G , νF G = F ν and eF G = F e = cG (e) where cG : G ’ F G
is the canonical section. By the de¬nition of „ , we get for every element (z, w) ∈
F G — F G over (x, y) ∈ G — G
µF G (z, w) = F (µ( , y))(z) + F (µ(x, ))(w)
and it is easy to check all axioms of Lie groups for the operations µF G , νF G and
eF G on F G. In particular, we have a canonical Lie group structure on the r-th
order tangent bundles T (r) G over any Lie group G and on all tensor bundles
over G.
Since „ is the identity if F equals to the tangent bundle T , we have generalized
the canonical Lie group structure on tangent bundles over Lie groups to all vector
bundle functors with the point property, cf. 37.2.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
38. The point property 333


38.11. Remark. Given a bundle functor F on Mf and a principal ¬ber bundle
(P, p, M, G) we might be interested in a natural principal bundle structure on
F p : F P ’ F M with structure group F G. If F is a Weil bundle, this structure
can be de¬ned by application of F to all maps in question, cf. 37.16. Though we
have found a natural Lie group structure on F G for vector bundle functors with
the point property which do not preserve products, there is still no structure of
principal ¬ber bundle (F P, F p, F M, F G) for dimension reasons, see 38.8.
38.12. Let us now consider a general bundle functor F on Mf and write
Q = F (pt). For every manifold M the unique map qM : M ’ pt induces
F qM : F M ’ Q and similarly to 38.3, every point a ∈ Q determines a canonical
natural section c(a)M (x) = F ix (a). Let G be the bundle functor on Mf de¬ned
by GM = M — Q on manifolds and Gf = f — idQ on maps.
Lemma. The maps σM (x, a) = c(a)M (x), (x, a) ∈ M — Q, and ρM (z) =
(pM (z), F qM (z)), z ∈ F M , de¬ne natural transformations σ : G ’ F and
ρ : F ’ G satisfying ρ —¦ σ = id. Moreover the σM are embeddings and the
ρM are submersions for all manifolds M . In particular, for every a ∈ Q the rule
Fa M = (F qM )’1 (a), Fa f = F f |Fa M determines a bundle functor on Mf with
the point property.
Proof. It is easy to verify that σ and ρ are natural transformations satisfying
ρ —¦ σ = id. This equality implies that σM is an embedding and also that ρM
is a surjective map which has maximal rank on a neighborhood U of the image
σM (M — Q). It su¬ces to prove that every ρRm is a submersion. Consider the
homotheties gt (x) = tx on Rm . Then F gt is a smoothly parameterized family
with F g1 = idRm and F g0 (F Rm ) = F i0 —¦ F qRm (F Rm ) ‚ σM (Rm — Q). Hence
every point of F Rm is mapped into U by some F gt with t > 0 and so ρRm has
maximal rank everywhere.
Since F qM is the second component of the surjective submersion ρM , all the
subsets Fa M ‚ F M are submanifolds and one easily checks all the axioms of
bundle functors.
38.13. Proposition. Every bundle functor on Mf transforms submersions
into submersions.
Proof. By the previous lemma, every value F f : F M ’ F N is a ¬bered mor-
phism of F qM : F M ’ Q into F qN : F N ’ Q over the identity on Q. If f
is a submersion, then every Fa f : Fa M ’ Fa N is a submersion according to
38.7.
38.14. Proposition. The dimensions of the standard ¬bers of every bundle
functor F on Mf satisfy km+n ≥ km + kn ’ dimF (pt). Equality holds if and
only if all bundle functors Fa preserve products in dimensions m and n.
38.15. Remarks. If the standard ¬bers of a bundle functor F on Mf are
compact, then all the functors Fa must coincide with the identity functor on
Mf according to 38.5. But then the natural transformations σ and ρ from
38.12 are natural equivalences.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
334 Chapter IX. Bundle functors on manifolds


38.16. Example. Taking any bundle functor G on Mf with the point property
and any manifold Q, we can de¬ne F M = GM — Q and F f = Gf — idQ
to get a bundle functor with F (pt) = Q. We present an example showing
that not all bundle functors on Mf are of this type. The basic idea is that
2
some of the individual ˜¬ber components™ Fa of F coincide with the functor T1
of 1-dimensional velocities of the second order while some other ones are the
Whitney sums T • T in dependence on the zero values of a smooth function
on Q. According to the general theory developed in section 14, it su¬ces to
construct a functor on the second order skeleton of Mf . So we take the system
of standard ¬bers Sn = Q — Rn — Rn , n ∈ N0 , and we have to de¬ne the action
of all jets from J0 (Rm , Rn )0 on Sm . Let us write ap , ap for the coe¬cients of
2
i ij
canonical polynomial representatives of the jets in question. Given any smooth
function f : Q ’ R we de¬ne a map J0 (Rm , Rn )0 — Sm ’ Sn by
2


(ap , ar )(q, y , z m ) = (q, ap y i , f (q)ar y i y j + ar z i ).
jk ij i
i i

One veri¬es easily that this is an action of the second order skeleton on the
system Sn . Obviously, the corresponding bundle functor F satis¬es F (pt) = Q
and the bundle functors Fq coincide with T • T for all q ∈ Q with f (q) = 0.
2
If f (q) = 0, then Fq is naturally equivalent to the functor T1 . Indeed, the
maps R2n ’ R2n , y i ’ y i , and z i ’ f (q)z i are invertible and de¬ne a natural
2
equivalence of T1 into Fq , see 18.15 for a help in a more detailed veri¬cation.
38.17. Consider a submersion f : Y ’ M and denote by µ : F Y ’ F M —M Y
the induced pullback map, cf. 2.19.
Proposition. The pullback map µ : F Y ’ F M —M Y of every submersion
f : Y ’ M is a submersion as well.
We remark that this property represents a special case of the so-called pro-
longation axiom which was introduced in [Pradines, 74b] for a more general
situation.
Proof. In view of 38.12 we may restrict ourselves to bundle functors with point
property (in general F qM : F M ’ F (pt) and F qY : F Y ’ F (pt) are ¬bered
manifolds and µ is a ¬bered morphism so that we can verify our assertion
¬berwise). Further we may consider the submersion f in its local form, i.e.
f : Rm+n ’ Rm , (x, y) ’ x, for then the claim follows from the locality of the
functors. Now we can easily choose a smoothly parametrized family of local
sections s : Y — M ’ Y with s(y, f (y)) = y, sy ∈ C ∞ (Y ), e.g. s(x,y) (¯) = (¯, y).
x x
Then we de¬ne a mapping ± : F M —M Y ’ F Y , ±(z, y) := F sy (z). Since locally
F f —¦F sy = idM and pF —¦F sy = sy —¦pF , we have constructed a section of µ. Since
Y M
the canonical sections cM : M ’ F M are natural, we get ±(cM (x), y) = cY (y).
Hence the section goes through the values of the canonical section cY and µ has
the maximal rank on a neighborhood of this section. Now the action of homo-
theties on Y = Rm+n and M = Rm commute with the canonical local form of f
and therefore the rank of µ is maximal globally.
In particular, given two bundle functors F , G on Mf , the natural transfor-
mation µ : F G ’ F — G de¬ned as the product of the natural transformations

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
38. The point property 335


F (pG ) : F G ’ F and pF —¦ G : F G ’ G is formed by surjective submersions
µM : F (GM ) ’ F M —M GM .
38.18. At the end of this section, we shall indicate how the above results can
be extended to bundle functors on FMm . The point property still plays an
important role. Since any manifold M can be viewed as the ¬bered manifold
idM : M ’ M , we can say that a bundle functor F : FMm ’ FM has the point
property if F M = M for all m-dimensional manifolds. Bundle functors on FMm
with the point property do not admit canonical sections in general, but for every
¬bered manifold qY : Y ’ M in FMm we have the ¬bration F qY : F Y ’ M and
F qY = qY —¦ pY , where pY : F Y ’ Y is the bundle projection of F Y . Moreover,
the mapping C ∞ (qY : Y ’ M ) ’ C ∞ (F qY : F Y ’ M ), s ’ F s is natural with
respect to ¬bered isomorphisms. This enables us to generalize easily the proof
of proposition 38.5 to our more general situation, for we can use the image of
the section i : Rm ’ Rm+n , x ’ (x, 0) instead of the canonical sections cM from
38.5. So the standard ¬bers Sn = F Rm+n of a bundle functor with the point
property are di¬eomorphic to Rkn .
Proposition. The dimensions kn of standard ¬bers of every bundle functor
F : FMm ’ FM with the point property satisfy kn+p ≥ kn + kp and for every
¯ ¯
FMm -objects qY : Y ’ M , qY : Y ’ M the canonical map π : F (Y —M Y ) ’
¯
¯
F Y —M F Y is a surjective submersion. Equality holds if and only if F preserves
¬bered products in dimensions n and p of the ¬bers. So F preserves ¬bered
products if and only if k(n) = n.k(1) for all n ∈ N0 .
Proof. Consider the diagram

x
 xx
‘   xx x
¯
F (Y —M Y )
‘   xx xx x
‘ xx€

Fp
¯


‘ FY — FY x w FY
π


‘ pr2
¯ ¯
‘“
Fp M


u u
F qY
pr1 ¯



wM F qY
FY
¯
where p and p are the projections on Y —M Y .
¯
By locality of bundle functors it su¬ces to restrict ourselves to objects from
a local pointed skeleton. In particular, we shall deal with the values of F on
trivial bundles Y = M — S. In the special case m = 0, the proposition was
proved above.
For every point x ∈ M we write (F Y )x := (F qY )’1 (x) and we de¬ne a functor
G = Gx : Mf ’ FM as follows. We set G(Yx ) := (F Y )x and for every map
¯ ¯ ¯
f = idM — f1 : Y ’ Y , f1 : Yx ’ Yx we de¬ne Gf1 := F f |(F Y )x : GYx ’ GYx .
If we restrict all the maps in the diagram to the appropriate preimages, we get
pr1 pr2
¯ ¯ ¯
the product (F Y )x ← ’ (F Y )x — (F Y )x ’ ’ (F Y )x and πx : G(Yx — Yx ) ’
’ ’
¯
GYx — GYx . Since G has the point property, πx is a surjective submersion.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
336 Chapter IX. Bundle functors on manifolds


Hence π is a ¬bered morphism over the identity on M which is ¬ber wise
a surjective submersion. Consequently π is a surjective submersion and the
inequality kn+p ≥ kn + kp follows.
Now similarly to 38.8, if the equality holds, then π is a global isomorphism.

38.19. Vertical Weil bundles. Let A be a Weil algebra. We de¬ne a func-
tor VA : FMm ’ FM as follows. For every qY : Y ’ M , we put VA Y : =
¯
∪x∈M TA Yx and given f ∈ FMm (Y, Y ) we write fx = f |Yx , x ∈ M , and we set
m+n
’ Rm ) = Rm — TA Rn carries a canon-
VA f |(VA Y )x := TA fx . Since VA (R
ical smooth structure, every ¬bered atlas on Y ’ M induces a ¬bered atlas
on VA Y ’ Y . It is easy to verify that VA is a bundle functor which preserves
¬bered products. In the special case of the algebra D of dual numbers we get
the vertical tangent bundle V .
Consider a bundle functor F : FMm ’ FM with the point property which
preserves ¬bered products, and a trivial bundle Y = M — S. If we repeat the
construction of the product preserving functors G = Gx , x ∈ M , from the proof
of proposition 38.18 we have Gx = TAx for certain Weil algebras A = Ax . So
we conclude that F (idM — f1 )|(F Y )x = Gx (f1 ) = VAx (idM — f1 )|(F Y )x . At
the same time the general theory of bundle functors implies (we take A = A0 )
F Rm+n = Rm — Rn — Sn = Rm — An = VA Rm+n for all n ∈ N (including the
actions of jets of maps of the form idRm —f1 ). So all the algebras Ax coincide and
since the bundles in question are trivial, we can always ¬nd an atlas (U± , •± )
on Y such that the chart changings are over the identity on M . But a cocycle
de¬ning the topological structure of F Y is obtained if we apply F to these chart
changings and therefore the resulting cocycle coincides with that obtained from
the functor VA .
Hence we have deduced the following characterization (which is not a complete
description as in 36.1) of the ¬bered product preserving bundle functors on
FMm .

Proposition. Let F : FMm ’ FM be a bundle functor with the point prop-
erty. The following conditions are equivalent.
(i) F preserves ¬bered products
(ii) For all n ∈ N it holds dimSn = n(dimS1 )
(iii) There is a Weil algebra A such that F Y = VA Y for every trivial bundle
¯
Y = M — S and for every mapping f1 : S ’ S we have F (idM — f1 ) =
¯
VA (idM — f1 ) : F (M — S) ’ F (M — S).



39. The ¬‚ow-natural transformation

39.1. De¬nition. Consider a bundle functor F : Mf ’ FM and the tangent
functor T : Mf ’ FM. A natural transformation ι : F T ’ T F is called a ¬‚ow-
natural transformation if the following diagram commutes for all m-dimensional

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
39. The ¬‚ow-natural transformation 337


manifolds M and all vector ¬elds X ∈ X(M ) on M .

u RRπ w F M
u
F M
FTM
RπT
R
ι M
(1) FX FM


F M FX w T F M
˜
39.2. Given a map f : Q — M ’ N , we have denoted by F f : Q — F M ’ F N
the ˜collection™ of F (f (q, )) for all q ∈ Q, see 14.1. Write (x, X) = Y ∈ T Rm =
Rm — Rm and de¬ne µRm : R — T Rm ’ Rm , µRm (t, Y ) = (x + tX) for t ∈ R,.
Theorem. Every bundle functor F : Mf ’ FM admits a canonical ¬‚ow-
natural transformation ι : F T ’ T F determined by

ιRm (z) = j0 F µRm ( , z).

If F has the point property, then ι is a natural equivalence if and only if F is a
Weil functor TA . In this case ι coincides with the canonical natural equivalence
TA T ’ T TA corresponding to the exchange homomorphism A — D ’ D — A
between the tensor products of Weil algebras.
39.3. The proof requires several steps. We start with a general lemma.
Lemma. Let M , N , Q be smooth manifolds and let f , g : Q — M ’ N be
k k
smooth maps. If jq f ( , y) = jq g( , y) for some q ∈ Q and all y ∈ M , then
˜ ˜
for every bundle functor F on Mf the maps F f , F g : Q — F M ’ F N satisfy
k˜ k˜
jq F f ( , z) = jq F g( , z) for all z ∈ F M .
Proof. It su¬ces to restrict ourselves to objects from the local skeleton (Rm ),
m = 0, 1, . . . , of Mf . Let r be the order of F valid for maps with source Rm ,
cf. 22.3, and write p for the bundle projection pRm . By the general theory of
bundle functors the values of F on morphisms f : Rm ’ Rn are determined by
the smooth associated map FRm ,Rn : J r (Rm , Rn ) —Rm F Rm ’ F Rn , see section
˜
14. Hence the map F f : Q — F Rm ’ F Rn is de¬ned by the composition of
FRm ,Rn with the smooth map f r : Q — F Rm ’ J r (Rm , Rn ) —Rm F Rm , (q, z) ’
(jp(z) f (q, ), z). Our assumption implies that f r ( , z) and g r ( , z) have the same
r

k-jet at q, which proves the lemma.
39.4. Now we deduce that the maps ιRm determine a natural transformation
ι : F T ’ T F such that the upper triangle in 39.1.(1) commutes. These maps
de¬ne a natural transformation between the bundle functors in question if they
obey the necessary commutativity with respect to the actions of morphisms
between the objects of the local skeleton Rm , m = 0, 1, . . . . Given such a
morphism f : Rm ’ Rn we have
1˜ 1
ιRn (F T f (z)) = j0 F µRn ( , F T f (z)) = j0 F ((µRn )t —¦ T f )(z)
˜
T F f (ιRm (z)) = T F f (j 1 F µRm ( , z)) = j 1 (F f —¦ F (µRm )t (z)) =
0 0
1
—¦ (µRm )t )(z).
= j0 F (f

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
338 Chapter IX. Bundle functors on manifolds


So in view of lemma 39.3 it is su¬cient to prove for all Y ∈ T Rm , f : Rm ’ Rn
1 1
j0 ((f —¦ (µRm )t )(Y )) = j0 ((µRn )t —¦ T f )(Y ).
By the de¬nition of µ, the values of both sides are T f (Y ).
Since (µRm )0 = πRm : T Rm ’ Rm , we have πF Rm —¦ ιRm = F (µRm )0 = F πRm .
39.5. Let us now discuss the bottom triangle in 39.1.(1). Given a bundle functor
F on Mf , both the arrows FX and F X are values of natural operators and ι
is a natural transformation. If we ¬x dimension of the manifold M then these
operators are of ¬nite order. Therefore it su¬ces to restrict ourselves to the
¬bers over the distinguished points from the objects of a local pointed skeleton.
Moreover, if we verify ιRm —¦ F X = FX on the ¬ber (F T )0 Rm for a jet of a
suitable order of a ¬eld X at 0 ∈ Rm , then this equality holds on the whole
orbit of this jet under the action of the corresponding jet group. Further, the
operators in question are regular and so the equality follows for the closure of
the orbit.

Lemma. The vector ¬eld X = ‚x1 on (Rm , 0) has the following two properties.
(1) Its ¬‚ow satis¬es FlX = µRm —¦ (idR — X) : R — Rm ’ Rm .
(2) The orbit of the jet j0 X under the action of the jet group Gr+1 is dense
r
m
in the space of r-jets of vector ¬elds at 0 ∈ Rm .
Proof. We have FlX (x) = x+t(1, 0, . . . , 0) = µRm (t, X(x)). The second assertion
t
is proved in section 42 below.
By the lemma, the mappings ιRm determine a ¬‚ow-natural transformation
ι : FT ’ TF.
Assume further that F has the point property and write kn for the dimension
of the standard ¬ber of F Rn . If ι is a natural equivalence, then k2n = 2kn for all
n. Hence proposition 38.8 implies that F preserves products and so it must be
naturally equivalent to a Weil bundle. On the other hand, assume F = TA for
some Weil algebra A and denote 1 and e the generators of the algebra D of dual
numbers. For every jA f ∈ TA T R, with f : Rk ’ T R = D, f (x) = g(x) + h(x).e,
take q : R — Rk ’ R, q(t, x) = g(x) + th(x), i.e. f (x) = j0 q( , x). Then we get
1

1 1 1
ιR (jA f ) = j0 TA (µR )t (jA f ) = j0 jA (g( ) + th( )) = j0 jA q(t, ).
Hence ιR coincides with the canonical exchange homomorphism A — D ’ D — A
and so ι is the canonical natural equivalence TA T ’ T TA .
39.6. Let us now modify the idea from 39.1 to bundle functors on FMm .
De¬nition. Consider a bundle functor F : FMm ’ FM and the vertical tan-
gent functor V : FMm ’ FM. A natural transformation ι : F V ’ V F is called
a ¬‚ow-natural transformation if the diagram

u RR w u
F πY
FV Y FY
RT
R
ιY πF Y
(1) FX
FX w V F Y
FY
Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
39. The ¬‚ow-natural transformation 339


commutes for all ¬bered manifolds Y with m-dimensional basis and for all ver-
tical vector ¬elds X on Y .
For every ¬bered manifold q : Y ’ M in ObFMm , the ¬bration q—¦πY : V Y ’
M is an FMm -morphism. Further, consider the local skeleton (Rm+n ’ Rm )
of FMm and de¬ne

µRm+n : R — V Rm+n = R1+m+n+n ’ Rm+n , (t, x, y, X) ’ (x, y + tX).

Then every µRm+n (t, ) is a globally de¬ned FMm morphism and we have
1
j0 µRm+n ( , x, y, X) = (x, y, X).

39.7. The proof of 39.3 applies to general categories over manifolds. A bundle
functor on an admissible category C is said to be of a locally ¬nite order if for
every C-object A there is an order r such that for all C-morphisms f : A ’ B
r
the values F f (z), z ∈ F A, depend on the jets jpA (z) f only. Let us recall that all
bundle functors on FMm have locally ¬nite order, cf. 22.3.
Lemma. Let f , g : Q — mA ’ mB be smoothly parameterized families of C-
k k
morphisms with jq f ( , y) = jq g( , y) for some q ∈ Q and all y ∈ mA. Then
˜

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