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to the subgroup ai = δj yields
i
j


f i (X j , Xlk , um , unp ) = f i (X j , Xlk + ak X q , um + am urs , unp ).
lq rs


Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
43. The case of the second order tangent vectors 359


This gives f i = f i (X j , ukl ). Using the homotheties in i(G1 ) ‚ G3 , we obtain
m m
f i = f i (X j ). Example 24.14 then implies
Y i = kX i .
(7)
Consider further the di¬erence A ’ kT (2) with k taken from (7) and denote
by hi , hij its components. We evaluate easily
ai aj hkl (X m , Xp , Xrs , ut , uuv ) = hij (X m , Xp , Xrs , ut , uuv ).
¯ ¯n ¯q ¯ ¯
n q
(8) kl

Quite similarly as in the ¬rst step we deduce hij = hij (X k , ulm ). By homogene-
ity and the invariant tensor theorem, we then obtain
hij = cuij + aX i X j .
(9)
For hi , we ¬nd
(10) ai hj (X k , Xm , Xpq , ur , ust ) + cai ujk + aai X j X k =
l n
j jk jk
¯ ¯l ¯n ¯ ¯
= hi (X k , Xm , Xpq , ur , ust ).
By (3), hi is independent of Xjk . Then the homogeneity condition implies
i


hi = fj (Xlk )X j + gj (Xlk )uj .
i i
(11)
For X i = 0, the equivariance of (11) with respect to the subgroup ai = δj reads
i
j

gj (Xlk )uj + cai ujk = gj (Xlk )(uj + aj ukl ).
i i
(12) jk kl

Hence gj (Xlk ) = cδj . The remaining equivariance condition is
i i


fj (Xlk )X j + aai X j X k = fj (Xlk + ak X m )X j .
i i
(13) jk lm

This implies that all the ¬rst order partial derivatives of fj (Xlk ) are constant, so
i

that fj are at most linear in Xlk . By the invariant tensor theorem, fj (Xlk )X j =
i i

eX j Xj + bX i . Then (13) yields e = a, i.e.
i


hi = cui + bX i + aX j Xj .
i
(14)
This gives the coordinate expression of (1).
43.3. Remark. For a Weil functor TB , all natural operators T T TB are of
the form H —¦ TB , where H is a natural transformation T TB ’ T TB over the
identity of TB . For T (2) , one evaluates easily that all natural transformations
H : T T (2) ’ T T (2) over the identity of T (2) form the following 3-parameter
family
Y i =k1 Y i ,
U i =k1 U i + k2 Y i + k3 ui ,
U ij =k1 U ij + k3 uij ,

see [Doupovec, 90]. Hence the operators of the form H —¦T (2) form a 3-parameter
family only, in which the operator D is not included.

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


43.4. Remark. In the case of Weil bundles, theorem 42.12 implies that the
di¬erence between a natural operator T T TB and its associated absolute
operator is a linear operator. This is no more true for the non-product-preserving
functors, where the operator D is the simplest counter-example.

43.5. Remark. The operators T (2) , V and L transform every vector ¬eld on a
manifold M into a vector ¬eld on T (2) M tangent to the subbundle T M ‚ T (2) M ,
but D does not. With a little surprise we can express it by saying that the
T T (2) is not compatible with the natural inclusion
natural operator D : T
T M ‚ T (2) M .

43.6. Remark. Recently [Mikulski, to appear b], has solved the general prob-
T T (r) , r ∈ N. All such operators
lem of determining all natural operators T
form an (r + 2)-parameter family linearly generated by the ¬‚ow operator, by the
Liouville vector ¬eld of T (r) and by the analogies of the operator D from 43.1
de¬ned by f ’ X · · · X f , k = 1, . . . , r.
k-times



44. Induced vector ¬elds on jet bundles

44.1. Let F be a bundle functor on FMm,n . The idea of the ¬‚ow prolongation
of vector ¬elds can be applied to the projectable vector ¬elds on every object
p : Y ’ M of FMm,n . The ¬‚ow Fl· of a projectable vector ¬eld · on Y is
t
formed by the local isomorphisms of Y and we de¬ne the ¬‚ow operator F of F
by
FY · = ‚t 0 F (Fl· ).

t

The general concept of a natural operator A transforming every projectable
vector ¬eld on Y ∈ ObFMm,n into a vector ¬eld on F Y was introduced in
section 18. We shall denote such an operator brie¬‚y by A : Tproj TF.

44.2. Lemma. If F is an r-th order bundle functor on FMm,n , then the order
T F is ¤ r.
of every natural operator Tproj

Proof. This is quite similar to 42.5, see [Kol´ˇ, Slov´k, 90] for the details.
ar a

44.3. We shall discuss the case F is the functor J r of the r-th jet prolongation
of ¬bered manifolds. We remark that a simple evaluation leads to the following
coordinate formula for J 1 ·

‚· p ‚· p q ‚· j p
J 1 · = · i ‚xi + · p ‚yp +
‚ ‚ ‚

+ ‚y q yi ‚xi yj p
‚xi ‚yi


‚ ‚
provided · = · i (x) ‚xi + · p (x, y) ‚yp , see [Krupka, 84]. To evaluate J r ·, we
have to iterate this formula and use the canonical inclusion J r (Y ’ M ) ’
J 1 (J r’1 (Y ’ M ) ’ M ).

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
44. Induced vector ¬elds on jet bundles 361


T J r is a constant multiple of
Proposition. Every natural operator A : Tproj
the ¬‚ow operator J r .
Proof. Let V r be the space of all r-jets of the projectable vector ¬elds on
Rn+m ’ Rm with source 0 ∈ Rm+n , let V 0 ‚ V r be the space of all r-jets
of the constant vector ¬elds and V0 ‚ V 0 be the subset of all vector ¬elds with
zero component in Rn . Further, let S r or Z r be the ¬ber of J r (Rm+n ’ Rm )
or T J r (Rm+n ’ Rm ) over 0 ∈ Rm+n , respectively. By lemma 44.2 and by the
general theory, we have to determine all Gr+1 -maps A : V r — S r ’ Z r over the
m,n
r
identity of S . Analogously to section 42, every projectable vector ¬eld on Y
with non-zero projection to the base manifold can locally be transformed into

the vector ¬eld ‚x1 . Hence A is determined by its restriction A0 to V0 — S r .
However, in the ¬rst part of the proof we have to consider the restriction A0 of
A to V 0 — S r for technical reasons.
Having the canonical coordinates xi and y p on Rm+n , let X i , Y p be the
induced coordinates on V 0 , let y± , 1 ¤ |±| ¤ r, be the induced coordinates on
p

S r and Z i = dxi , Z p = dy p , Z± = dy± be the additional coordinates on Z r . The
p p

restriction A0 is given by some functions
Z i = f i (X j , Y q , yβ )
s

Z p = f p (X i , Y q , yβ )
s

Z± = f± (X i , Y q , yβ ).
p p s


Let us denote by g i , g p , g± the restrictions of the corresponding f ™s to V0 — S r .
p

The ¬‚ows of constant vector ¬elds are formed by translations, so that their r-jet
prolongations are the induced translations of J r (Rm+n ’ Rm ) identical on the
‚ ‚
standard ¬ber. Therefore J r ‚x1 = ‚x1 and it su¬ces to prove
g i = kX i , g p = 0, p
g± = 0.
We shall proceed by induction on the order r. It is easy to see that the ac-
tion of i(G1 — G1 ) ‚ Gr+1 on all quantities is tensorial. Consider the case
m n m,n
r = 1. Using the equivariance with respect to the homotheties in i(G1 ), we n
q q
obtain f i (X j , Y p , yl ) = f i (X j , kY p , kyl ), so that f i depends on X i only. Then
the equivariance of f i with respect to i(G1 ) yields f i = kX i by 24.7. The equiv-
m
ariance of f with respect to the homotheties in i(G1 ) gives kf p (X i , Y q , yj ) =
p s
n
q
f p (X i , kY q , kyj ). This kind of homogeneity implies f p = hp (X i )Y q + hpj (X i )yj
s
q q
with some smooth functions hp , hpj . Using the homotheties in i(G1 ), we fur-
q q m
ther obtain hp (kX) = hp (X) and hpj (kX) = khpj (X). Hence hp = const
q q q q q
pj i
and hq is linear in X . Then the generalized invariant tensor theorem yields
f p = aY p + byi X i , a, b ∈ R. Applying the same procedure to fip , we ¬nd
p

fip = cyi , c ∈ R.
p

Consider the injection G2 ’ G2 m,n determined by the products with the
n
identities on R . The action of an element (ap , ar ) of the latter subgroup is
m
q st
given by
¯p q
yi = ap yi
(2) q
Zi = ap yi Z t + ap Zi
¯p q q
(3) qt q


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


and V0 is an invariant subspace. In particular, (3) with ap = δq gives an equiv-
p
q
ariance condition
cyi = bap yi yj X j + cyi .
p qt p
qt

This yields b = 0, so that g p = 0. Further, the subspace V0 is invariant with the
respect to the inclusion of G1 into G2 . The equivariance of fip with respect
m,n m,n
to an element (δj , δq , ap ) ∈ G1 means cyi = c(yi + ap ). Hence c = 0, which
p p
ip
m,n
i i
completes the proof for r = 1.
For r ≥ 2 it su¬ces to discuss the g™s only. Using the homotheties in i(G1 ),
n
p q q
j
we ¬nd that gi1 ···is (X , yβ ), 1 ¤ |β| ¤ r, is linear in yβ . The homotheties in
i(G1 ) and the generalized invariant tensor theorem then yield
m

gi1 ···is = Wip ···is + cs yi1 ···is is+1 ···ir X is+1 . . . X ir
p p
(4) 1


where Wip ···is do not depend on yi1 ···ir , s = 1, . . . , r ’ 1, and
p
1


p p
(5) gi1 ···ir = cr yi1 ···ir
p p
g p = b1 yi X i + · · · + br yi1 ···ir X i1 . . . X ir .
(6)

Similarly to the ¬rst order case, we have an inclusion Gr+1 ’ Gr+1 determined
n m,n
by the products of di¬eomorphisms on Rn with the identity of Rm . One ¬nds
easily the following transformation law

yi1 ···is = ap yi1 ···is + Fip ···is + ap1 ···qs yi1 . . . yis
¯p q q1 qs
(7) q q
1


where Fip ···is is a polynomial expression linear in ap with 2 ¤ |±| ¤ s ’ 1 and
±
1
p
independent of yi1 ···is . This implies

Zi1 ···is = ap Zi1 ···is + Gp1 ···is + ap1 ···qs qs+1 yi1 . . . yis Z qs+1
¯p q q1 qs
(8) q q
i

where Gp1 ···is is a polynomial expression linear in ap with 2 ¤ |±| ¤ s and linear
±
i
p
in Z± , 0 ¤ |±| ¤ s ’ 1.
p p
We deduce that every gi1 ···is , 0 ¤ s ¤ r ’ 1 , is independent of yi1 ···ir . On the
kernel of the jet projection Gr+1 ’ Gr , (8) for r = s gives
n n

q1 qr
0 = ap1 ···qr qr+1 yi1 . . . yir g qr+1 .
q


Hence g p = 0. On the kernel of the jet projection Gr ’ Gr’1 , (8) with s =
n n
1, . . . , r ’ 1, implies
q1 qr
0 = cs ap1 ...qr yi1 . . . yir X is+1 . . . X ir ,
q

i.e. cs = 0. By projectability, g i and g± , 0 ¤ |±| ¤ r ’ 1, correspond to a
p

Gr -equivariant map V0 — S r’1 ’ Z r’1 . By the induction hypothesis, g± = 0
p
m,n
for all 0 ¤ |±| ¤ r ’ 1. Then on the kernel of the jet projection Gr+1 ’ Gr’1
n n
q1 qr p
(8) gives 0 = cr ap1 ...qr yi1 . . . yir , i.e. gi1 ···ir = 0.
q



Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
45. Prolongations of connections to F Y ’ M 363

r
44.4. Bundles of contact elements. Consider the bundle functor Kn on
Mfm of the n-dimensional contact elements of order r de¬ned in 12.15.
r
Proposition. Every natural operator A : T T Kn is a constant multiple of
r
the ¬‚ow operator Kn .
Proof. It su¬ces to discuss the case M = Rm . Consider the canonical ¬bration
Rm = Rn — Rm’n ’ Rn . As remarked at the end of 12.16, there is an identi¬-
cation of an open dense subset in Kn Rm with J r (Rm ’ Rn ). By de¬nition, on
r

this subset it holds J r ξ = Kn ξ for every projectable vector ¬eld ξ on Rm ’ Rn .
r

Since the operator A commutes with the action of all di¬eomorphisms preserving
‚ ‚
¬bration Rm ’ Rn , the restriction of A to ‚x1 is a constant multiple of Kn ( ‚x1 )
r

by proposition 44.3. But every vector ¬eld on Rm can be locally transformed

into ‚x1 in a neighborhood of any point where it does not vanish.
We ¬nd it interesting that we have ¬nished our investigation of the basic
properties of the natural operators T T F for di¬erent bundle functors on
Mfm by an example in which the constant multiples of the ¬‚ow operator are
the only natural operators T TF.
T T — and
44.5. Remark. [Kobak, 91] determined all natural operators T
T (T T — ) for manifolds of dimension at least two. Let T — be the ¬‚ow operator
T
of the cotangent bundle, LM : T — M ’ T T — M be the vector ¬eld generated by
the homotheties of the vector bundle T — M and ωM : T M —M T — M ’ R be the
T T — are of the form f (ω)T — +
evaluation map. Then all natural operators T
g(ω)L, where f , g ∈ C ∞ (R, R) are any smooth functions of one variable. In
the case F = T T — the result is of similar character, but the complete list is
somewhat longer, so that we refer the reader to the above mentioned paper.


45. Prolongations of connections to F Y ’ M

45.1. In 31.1 we deduced that there is exactly one natural operator transforming
every general connection on Y ’ M into a general connection on V Y ’ M .
However, one meets a quite di¬erent situation when replacing ¬bered manifold
V Y ’ M e.g. by the ¬rst jet prolongation J 1 Y ’ M of Y . Pohl has observed in
the vector bundle case, [Pohl, 66], that one needs an auxiliary linear connection
on the base manifold M to construct an induced connection on J 1 Y ’ M . Our
¬rst goal is to clarify this di¬erence from the conceptual point of view.
45.2. Bundle functors of order (r, s). We recall that two maps f , g of a
¬bered manifold p : Y ’ M into another manifold determine the same (r, s)-jet
r,s r,s r r
jy f = jy g at y ∈ Y , s ≥ r, if jy f = jy g and the restrictions of f and g to the
s s
¬ber Yp(y) satisfy jy (f |Yp(y) ) = jy (g|Yp(y) ), see 12.19.
De¬nition. A bundle functor on a category C over FM is said to be of order
¯
(r, s), if for any two C-morphisms f , g of Y into Y
r,s r,s
jy f = jy g implies (F f )|(F Y )y = (F g)|(F Y )y .

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

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