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recurrence procedure of similar type we further deduce

hj1 ...js (X1 , . . . , Xs’1 ) = 0
cs = c1 ,

for all s = 3, . . . , r.
This implies that the restriction of every natural transformation T (r) ’ T (r)
to each subcategory Mfm is a homothety with a coe¬cient km . Taking into
account the injection R ’ Rm , x ’ (x, 0, . . . , 0) we ¬nd km = k1 .
40.9. Remark. We remark that all natural tensors of type 1 on both T (r) M
1
and the so-called extended r-th order tangent bundle (J r (M, R))— are determined
in [Gancarzewicz, Kol´ˇ, to appear].
ar


41. Star bundle functors
The tangent functor T is a covariant functor on the category Mf , but its
dual T — can be interpreted as a covariant functor on the subcategory Mfm of
local di¬eomorphisms of m-manifolds only. In this section we explain how to
treat functors with a similar kind of contravariant character like T — on the whole
category Mf .
41.1. The category of star bundles. Consider a ¬bered manifold Y ’ M
and a smooth map f : N ’ M . Let us recall that the induced ¬bered manifold
f — Y ’ N is given by the pullback

w
fY
f —Y Y


u u
wM
f
N
The restrictions of the ¬bered morphism fY to individual ¬bers are di¬eomor-
phisms and we can write

f — Y = {(x, y); x ∈ N, y ∈ Yf (x) }, fY (x, y) = y.

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


Clearly (f —¦ g)— Y ∼ g — (f — Y ). Let us consider another ¬bered manifold Y ’ M
=
over the same base, and a base-preserving ¬bered morphism • : Y ’ Y . Given
a smooth map f : N ’ M , by the pullback property there is a unique ¬bered
morphism f — • : f — Y ’ f — Y such that
fY —¦ f — • = • —¦ fY .
(1)
The pullbacks appear in many well known constructions in di¬erential geome-
try. For example, given manifolds M , N and a smooth map f : M ’ N , the
cotangent mapping T — f transforms every form ω ∈ Tf (x) N into T — f ω ∈ Tx M .
— —

Hence the mapping f — (T — N ) ’ T — M is a morphism over the identity on M .
We know that the restriction of T — to manifolds of any ¬xed dimension and local
di¬eomorphisms is a bundle functor on Mfm , see 14.9, and it seems that the
construction could be functorial on the whole category Mf as well. However
the codomain of T — cannot be the category FM.
De¬nition. The category FM— of star bundles is de¬ned as follows. The ob-
jects coincide with those of FM, but morphisms • : (Y ’ M ) ’ (Y ’ M )
are couples (•0 , •1 ) where •0 : M ’ M is a smooth map and •1 : (•0 )— Y ’ Y
is a ¬bered morphism over idM . The composition of morphisms is given by
(ψ0 , ψ1 ) —¦ (•0 , •1 ) = (ψ0 —¦ •0 , •1 —¦ ((•0 )— ψ1 )).
(2)
Using the formulas (1) and (2) one veri¬es easily that this is a correct de¬nition
of a category. The base functor B : FM— ’ Mf is de¬ned by B(Y ’ M ) = M ,
B(•0 , •1 ) = •0 .
41.2. Star bundle functors. A star bundle functor on Mf is a covariant
functor F : Mf ’ FM— satisfying
(i) B —¦ F = IdMf , so that the bundle projections determine a natural trans-
formation p : F ’ IdMf .
(ii) If i : U ’ M is an inclusion of an open submanifold, then F U = p’1 (U )
M
and F i = (i, •1 ) where •1 : i— (F M ) ’ F U is the canonical identi¬cation
i— (F M ) ∼ p’1 (U ) ‚ F M .
=M
(iii) Every smoothly parameterized family of mappings is transformed into a
smoothly parameterized one.
Given a smooth map f : M ’ N we shall often use the same notation F f for
the second component • in F f = (f, •). We can also view the star bundle func-
tors as rules transforming any manifold M into a ¬ber bundle pM : F M ’ M and
any smooth map f : M ’ N into a base-preserving morphism F f : f — (F N ) ’
F M with F (idM ) = idF M and F (g —¦ f ) = F f —¦ f — (F g).
41.3. The associated maps. A star bundle functor F is said to be of order r
r r
if for every maps f , g : M ’ N and every point x ∈ M , the equality jx f = jx g
implies F f |(f — (F N ))x = F g|(g — (F N ))x , where we identify the ¬bers (f — (F N ))x
and (g — (F N ))x .
Let us consider an r-th order star bundle functor F : Mf ’ FM— . For every
r r
r-jet A = jx f ∈ Jx (M, N )y we de¬ne a map F A : Fy N ’ Fx M by
F A = F f —¦ (fF N |(f — (F N ))x )’1 ,
(1)

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
41. Star bundle functors 347


where fF N : f — (F N ) ’ F N is the canonical map. Given another r-jet B =
r r
jy g ∈ Jy (N, P )z , we have

F (B —¦ A) = F f —¦ (f — (F g)) —¦ (fg— (F P ) |(f — g — (F P ))x )’1 —¦ (gF P |(g — (F P ))y )’1 .

Applying 41.1.(1) to individual ¬bers, we get

(fF N |(f — (F N ))x )’1 —¦ F g = f — (F g) —¦ (fg— (F P ) |(f — g — (F P ))x )’1

and that is why

F (B —¦ A) = F f —¦ (fF N |(f — F N )x )’1 —¦ F g —¦ (gF P |(g — F P )y )’1
(2)
= F A —¦ F B.

For any two manifolds M , N we de¬ne

FM,N : F N —N J r (M, N ) ’ F M, (q, A) ’ F A(q).
(3)

These maps are called the associated maps to F .
Proposition. The associated maps to any ¬nite order star bundle functor are
smooth.
Proof. This follows from the regularity and locality conditions in the way shown
in the proof of 14.4.
41.4. Description of ¬nite order star bundle functors. Let us consider
an r-th order star bundle functor F . We denote (Lr )op the dual category to
Lr , Sm = F0 Rm , m ∈ N0 , and we call the system S = {S0 , S1 , . . . } the system
of standard ¬bers of F , cf. 14.21. The restrictions m,n : Sn — Lr m,n ’ Sm ,
m,n (s, A) = F A(s), of the associated maps 41.3.(3) form the induced action of
(Lr )op on S. Indeed, given another jet B ∈ Lr (n, p) equality 41.3.(2) implies

—¦ A) =
m,p (s, B m,n ( n,p (s, B), A).

On the other hand, let be an action of (Lr )op on a system S = {S0 , S1 , . . . }
of smooth manifolds and denote m the left actions of Gr on Sm given by
m
’1
m (A, s) = m,m (s, A ). We shall construct a star bundle functor L from these
r
data. We put LM := P M [Sm ; m ] for all manifolds M and similarly to 14.22 we
also get the action on morphisms. Given a map f : M ’ N , x ∈ M , f (x) = y,
we de¬ne a map F A : Fy N ’ Fx M ,
’1
F A({v, s}) = {u, —¦ A —¦ u)},
m,n (s, v

r r r
where m = dimM , n = dimN , A = jx f , v ∈ Py N , s ∈ Sn , and u ∈ Px M is
chosen arbitrarily. The veri¬cation that this is a correct de¬nition of smooth
maps satisfying F (B —¦ A) = F A —¦ F B is quite analogous to the considerations in
14.22 and is left to the reader. Now, we de¬ne Lf |(f — (F N ))x = F A —¦ fF N and
it follows directly from 41.1.(1) that L(g —¦ f ) = Lf —¦ f — (Lg).

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


Theorem. There is a bijective correspondence between the set of all r-th order
star bundle functors on Mf and the set of all smooth actions of the category
(Lr )op on systems S of smooth manifolds.

Proof. In the formulation of the theorem we identify naturally equivalent func-
tors. Given an r-th order star bundle functor F , we have the induced action
of (Lr )op on the system of standard ¬bers. So we can construct the functor L.
Analogously to 14.22, the associated maps de¬ne a natural equivalence between
F and L.

41.5 Remark. We clari¬ed in 14.24 that the actions of the category Lr on
systems of manifolds are in fact covariant functors Lr ’ Mf . In the same way,
actions of (Lr )op correspond to covariant functors (Lr )op ’ Mf or, equivalently,
to contravariant functors Lr ’ Mf , which will also be denoted by Finf . Hence
we can summarize: r-th order bundle functors correspond to covariant smooth
functors Lr ’ Mf while r-th order star bundle functors to the contravariant
ones.

41.6. Example. Consider a manifold Q and a point q ∈ Q. To any manifold M
±
we associate the ¬bered manifold F M = J r (M, Q)q ’ M and a map f : N ’ M

is transformed into a map F f : f — (F M ) ’ F N de¬ned as follows. Given a point
b ∈ f — (J r (M, Q)q ), b = (x, jf (x) g), we set F f (b) = jx (g —¦ f ) ∈ J r (N, Q)q . One
r r

veri¬es easily that F is a star bundle functor of order r. Let us mention the
corresponding contravariant functor Lr ’ Mf . We have Finf (m) = J0 (Rm , Q)q r

and for arbitrary jets j0 f ∈ Lr , j0 g ∈ Finf (m) it holds Finf (j0 f )(j0 g) =
r r r r
m,n
r
j0 (g —¦ f ).

41.7. Vector bundle functors and vector star bundle functors. Let F
be a bundle functor or a star bundle functor on Mf . By the de¬nition of the
induced action and by the construction of the (covariant or contravariant) func-
tor Finf : Lr ’ Mf , the values of the functor F belong to the subcategory of
vector bundles if and only if the functor Finf takes values in the category Vect of
¬nite dimensional vector spaces and linear mappings. But using the construction
of dual objects and morphisms in the category Vect, we get a duality between
covariant and contravariant functors Finf : Lr ’ Vect. The corresponding dual-
ity between vector bundle functors and vector star bundle functors is a source
of interesting geometric objects like r-th order tangent vectors, see 12.14 and
below.

41.8. Examples. Let us continue in example 41.6. If the manifold Q happens
to be a vector space and the point q its origin, we clearly get a vector star bundle
functor. Taking Q = R we get the r-th order cotangent functor T r— . If we set
Q = Rk , then the corresponding star bundle functor is the functor Tk of ther—

(k, r)-covelocities, cf. 12.14.
The dual vector bundle functor to T r— is the r-th order tangent functor. The
r
dual functor to the (k, r)-covelocities is the functor Tk , see 12.14.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
Remarks 349


Remarks
Most of the exposition concerning the bundle functors on Mf is based on
[Kol´ˇ, Slov´k, 89], but the prolongation of Lie groups was described in [Kol´ˇ,
ar a ar
83]. The generalization to bundle functors on FMm follows [Slov´k, 91].
a
The existence of the canonical ¬‚ow-natural transformation F T ’ T F was
¬rst deduced by A. Kock in the framework of the so called synthetic di¬erential
geometry, see e.g. [Kock, 81]. His unpublished note originated in a discussion
with the ¬rst author. Then the latter developed, with consent of the former, the
proof of that result dealing with classical manifolds only.
The description of all natural transformations with the source in a Weil bundle
by means of some special elements in the standard ¬ber is a generalization of
an idea from [Kol´ˇ, 86] due to [Mikulski, 89 b]. The natural transformations
ar
(r) (r)
T ’T were ¬rst classi¬ed in [Kol´ˇ, Vosmansk´, 89].
ar a




Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
350


CHAPTER X.
PROLONGATION OF VECTOR FIELDS
AND CONNECTIONS




This section is devoted to systematic investigation of the natural operators
transforming vector ¬elds into vector ¬elds or general connections into general
connections. For the sake of simplicity we also speak on the prolongations of vec-
tor ¬elds and connections. We ¬rst determine all natural operators transforming
vector ¬elds on a manifold M into vector ¬elds on a Weil bundle over M . In the
formulation of the result as well as in the proof we use heavily the technique of
Weil algebras. Then we study the prolongations of vector ¬elds to the bundle
of second order tangent vectors. We like to comment the interesting general
di¬erences between a product-preserving functor and a non-product-preserving
one in this case. For the prolongations of projectable vector ¬elds to the r-jet
prolongation of a ¬bered manifold, which play an important role in the varia-
tional calculus, we prove that the unique natural operator, up to a multiplicative
constant, is the ¬‚ow operator.
Using the ¬‚ow-natural equivalence we construct a natural operator transform-
ing general connections on Y ’ M into general connections on TA Y ’ TA M
for every Weil algebra A. In the case of the tangent functor we determine all
¬rst-order natural operators transforming connections on Y ’ M into connec-
tions on T Y ’ T M . This clari¬es that the above mentioned operator is not the
unique natural operator in general. Another class of problems is to study the
prolongations of connections from Y ’ M to F Y ’ M , where F is a functor
de¬ned on local isomorphisms of ¬bered manifolds. If we apply the idea of the
¬‚ow prolongation of vector ¬elds, we see that such a construction depends on an
r-th order linear connection on the base manifold, provided r means the horizon-
tal order of F . In the case of the vertical tangent functor we obtain the operator
de¬ned in another way in chapter VII. For the functor J 1 of the ¬rst jet prolon-
gation of ¬bered manifolds we deduce that all natural operators transforming
a general connection on Y ’ M and a linear connection on M into a general
connection on J 1 Y ’ M form a simple 4-parameter family. In conclusion we
study the prolongation of general connections from Y ’ M to V Y ’ Y . From
the general point of view it is interesting that such an operator exists only in the
case of a¬ne bundles (with vector bundles as a special sub case). But we can
consider arbitrary connections on them (i.e. arbitrary nonlinear connections in
the vector bundle case).




Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
42. Prolongations of vector ¬elds to Weil bundles 351


42. Prolongations of vector ¬elds to Weil bundles
Let F be an arbitrary natural bundle over m-manifolds. We ¬rst deduce
some general properties of the natural operators A : T T F , i.e. of the natural
operators transforming every vector ¬eld on a manifold M into a vector ¬eld on
F M . Starting from 42.7 we shall discuss the case that F is a Weil functor.
42.1. One general example of a natural operator T T F is the ¬‚ow operator
F of a natural bundle F de¬ned by
F (FlX )

FM X = t
‚t 0

where FlX means the ¬‚ow of a vector ¬eld X on M , cf. 6.19.
The composition T F = T —¦ F is another bundle functor on Mfm and the
bundle projection of T is a natural transformation T F ’ F . Assume we have
a natural transformation i : T F ’ T F over the identity of F . Then we can
construct further natural operators T T F by using the following lemma, the
proof of which consists in a standard diagram chase.
T F is a natural operator and i : T F ’ T F is a natural
Lemma. If A : T
transformation over the identity of F , then i —¦ A : T T F is also a natural
operator.
42.2. Absolute operators. This is another class of natural operators T
T F , which is related with the natural transformations F ’ F . Let 0M be the
zero vector ¬eld on M .
De¬nition. A natural operator A : T T F is said to be an absolute operator,
if AM X = AM 0M for every vector ¬eld X on M .
It is easy to check that, for every natural operator A : T T F , the operator

transforming every X ∈ C (T M ) into AM 0M is also natural. Hence this is an
absolute operator called associated with A.
Let LM be the Liouville vector ¬eld on T M , i.e. the vector ¬eld generated by
the one-parameter group of all homotheties of the vector bundle T M ’ M . The
rule transforming every vector ¬eld on M into LM is the simplest example of
an absolute operator in the case F = T . The naturality of this operator follows
from the fact that every homothety is a natural transformation T ’ T . Such a
construction can be generalized. Let •(t) be a smooth one-parameter family of
natural transformations F ’ F with •(0) = id, where smoothness means that
the map (•(t))M : R — F M ’ F M is smooth for every manifold M . Then

¦(M ) = (•(t))M
‚t 0

is a vertical vector ¬eld on F M . The rule X ’ ¦(M ) for every X ∈ C ∞ (T M )
is an absolute operator T T F , which is said to be generated by •(t).
42.3. Lemma. For an absolute operator A : T T F every AM 0M is a vertical
vector ¬eld on F M .
Proof. Let J : U ’ F M , U ‚ R — F M , be the ¬‚ow of AM 0M and let Jt be
its restriction for a ¬xed t ∈ R. Assume there exists W ∈ Fx M and t ∈ R such

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
352 Chapter X. Prolongation of vector ¬elds and connections


that pM Jt (W ) = y = x, where pM : F M ’ M is the bundle projection. Take
f ∈ Di¬(M ) with the identity germ at x and f (y) = y, so that the restriction
of F f to Fx M is the identity. Since AM 0M is a vector ¬eld F f -related with
itself, we have F f —¦ Jt = Jt —¦ F f whenever both sides are de¬ned. In particular,
pM (F f )Jt (W ) = f pM Jt (W ) = f (y) and pM Jt (F f )(W ) = pM Jt (W ) = y,
which is a contradiction. Hence the value of AM 0M at every W ∈ F M is a
vertical vector.
42.4. Order estimate. It is well known that every vector ¬eld X on a manifold
M with non-zero value at x ∈ M can be expressed in a suitable local coordinate
system centered at x as the constant vector ¬eld

(1) X= ‚x1 .
This simple fact has several pleasant consequences for the study of natural oper-
ators on vector ¬elds. The ¬rst of them can be seen in the proof of the following
lemma.
r r
Lemma. Let X and Y be two vector ¬elds on M with X(x) = 0 and jx X = jx Y .
Then there exists a local di¬eomorphism f transforming X into Y such that
r+1 r+1
jx f = jx idM .
Proof. Take a local coordinate system centered at x such that (1) holds. Then
the coordinate functions Y i of Y have the form Y i = δ1 + g i (x) with j0 g i = 0.
i r

Consider the solution f = (f i (x)) of the following system of equations
‚f i (x)
δ1 + g i (f 1 (x), . . . , f m (x)) =
i
‚x1
determined by the initial condition f = id on the hyperplane x1 = 0. Then f
is a local di¬eomorphism transforming X into Y . We claim that the k-th order
partial derivatives of f at the origin vanish for all 1 < k ¤ r + 1. Indeed, if
there is no derivative along the ¬rst axis, all the derivatives of order higher than
one vanish according to the initial condition, and all other cases follow directly
from the equations. By the same argument we ¬nd that the ¬rst order partial
derivatives of f at the origin coincide with the partial derivatives of the identity
map.
This lemma enables us to derive a simple estimate of the order of the natural
operators T TF.
42.5.Proposition. If F is an r-th order natural bundle, then the order of every
natural operator A : T T F is less than or equal to r.
r r
Proof. Assume ¬rst X(x) = 0 and jx X = jx Y , x ∈ M . Taking a local di¬eomor-
phism f of lemma 42.4, we have locally AM Y = (T F f ) —¦ AM X —¦ (F f )’1 . But
r+1 r+1
T F is an (r + 1)-st order natural bundle, so that jx f = jx idM implies that
the restriction of T F f to the ¬ber of T F M ’ M over x is the identity. Hence
AM Y |Fx M = AM X|Fx M . In the case X(x) = 0 we take any vector ¬eld Z with
Z(x) = 0 and consider the one-parameter families of vector ¬elds X + tZ and
Y + tZ, t ∈ R. For every t = 0 we have AM (X + tZ)|Fx M = AM (Y + tZ)|Fx M
by the ¬rst part of the proof. Since A is regular, this relation holds for t = 0 as
well.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
42. Prolongations of vector ¬elds to Weil bundles 353


42.6. Let S be the standard ¬ber of an r-th order bundle functor F on Mfm ,
let Z be the standard ¬ber of T F and let q : Z ’ S be the canonical projection.
Further, let Vm = J0 T Rm be the space of all r-jets at zero of vector ¬elds on Rm

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