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vector bundle if and only if the nilideal of A = F (R) is nilpotent of order
2, so A = F (R) = R · 1 • V , where the multiplication on V is 0. Then by
construction 35.11 we have F (M ) = V — T M . Finally by 35.17.(2) we have
N at(V — T, W — T ) = Hom(R · 1 • V, R · 1 • W ) ∼ L(V, W ).
=
37.3. The most important natural transformations. Let F , F1 , and F2
be multiplicative bundle functors (Weil functors by theorem 36.1) with Weil
algebras A = R • N , A1 = R • N1 , and A2 = R • N2 where the N ™s denote
the maximal nilpotent ideals. We will denote by N (F ) the nilpotent ideal in the
Weil algebra of a general functor F . By 36.13 we have Al(F2 —¦ F1 ) = A1 — A2 .
Using this and 35.17 we de¬ne the following natural transformations:
(1) The projections π1 : F1 ’ Id, π2 : F2 ’ Id induced by (».1 + n) ’
» ∈ R. In general we will write πF : F ’ Id. Thus we have also
F2 π1 : F2 —¦ F1 ’ F2 and π2 F1 : F2 —¦ F1 ’ F1 .
(2) The zero sections 01 : Id ’ F1 and 02 : Id ’ F2 induced by R ’ A1 ,
» ’ ».1. Then we have F2 01 : F2 ’ F2 —¦ F1 and 02 F1 : F1 ’ F2 —¦ F1 .
(3) The isomorphism A1 — A2 ∼ A2 — A1 , given by a1 — a2 ’ a2 — a1 induces
=
the canonical ¬‚ip mapping κF1 ,F2 = κ : F2 —¦ F1 ’ F1 —¦ F2 . We have
κF1 ,F2 = κ’1,F1 .
F2
(4) The multiplication m in A is a homomorphism A—A ’ A which induces
a natural transformation µ = µF : F —¦ F ’ F .
(5) Clearly the Weil algebra of the product F1 —Id F2 in the category of
bundle functors is given by R.1 • N1 • N2 . We consider the two natural
transformations

(π2 F1 , F2 π1 ), 0F1 —Id F2 —¦ πF2 —¦F1 : F2 —¦ F1 ’ (F1 —Id F2 ).

The equalizer of these two transformations will be denoted by vl : F2 —
F1 ’ F2 —¦ F1 and will be called the vertical lift. At the level of Weil
algebras one checks that the Weil algebra of F2 — F1 is given by R.1 •
(N1 — N2 ).
(6) The canonical ¬‚ip κ factors to a natural transformation κF2 —F1 : F2 —F1 ’
F1 — F2 with vl —¦ κF2 —F1 = κF2 ,F1 —¦ vl.
(7) The multiplication µ induces a natural transformation µ—¦vl : F —F ’ F .
It is clear that κ expresses the symmetry of higher derivatives. We will see that
the vertical lift vl expresses linearity of di¬erentiation.
The reader is advised to work out the Weil algebra side of all these natural
transformations.
37.4. The second tangent bundle. In the setting of 35.5 we let F1 = F2 = T
be the tangent bundle functor, and we let T 2 = T —¦ T be the second tangent
bundle. Its Weil algebra is D2 := Al(T 2 ) = D — D = R4 with generators
2 2
1, δ1 , and δ2 and with relations δ1 = δ2 = 0. Then (1, δ1 ; δ2 , δ1 δ2 ) is the
standard basis of R4 = T 2 R in the usual description, which we also used in 6.12.
From the list of natural transformations in 37.1 we get πT : (δ1 , δ2 ) ’ (δ, 0),
T π : (δ1 , δ2 ) ’ (0, δ), and µ = + —¦ (πT, T π) : T 2 ’ T, (δ1 , δ2 ) ’ (δ, δ). Then we

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
320 Chapter VIII. Product preserving functors


have T — T = T , since N (T ) — N (T ) = N (T ), and the natural transformations
from 37.3 have the following form:
κ : T 2 ’ T 2,
κ(a1 + x1 δ1 + x2 δ2 + x3 δ1 δ2 ) = a1 + x2 δ1 + x1 δ2 + x3 δ1 δ2 .
vl : T ’ T 2 , vl(a1 + xδ) = a1 + xδ1 δ2 .
m» T : T 2 ’ T 2 ,
m» T (a1 + x1 δ1 + x2 δ2 + x3 δ1 δ2 ) = a1 + x1 δ1 + »x2 δ2 + »x3 δ1 δ2 .
T m» : T 2 ’ T 2 ,
T m» (a1 + x1 δ1 + x2 δ2 + x3 δ1 δ2 ) = a1 + »x1 δ1 + x2 δ2 + »x3 δ1 δ2 .
(+T ) : T 2 —T T 2 ’ T 2 ,
(+T )((a1 + x1 δ1 + x2 δ2 + x3 δ1 δ2 ), (a1 + x1 δ1 + y2 δ2 + y3 δ1 δ2 )) =
= a1 + x1 δ1 + (x2 + y2 )δ2 + (x3 + y3 )δ1 δ2 .
(T +)((a1 + x1 δ1 + x2 δ2 + x3 δ1 δ2 ), (a1 + y1 δ1 + x2 δ2 + y3 δ1 δ2 )) =
= a1 + (x1 + y1 )δ1 + x2 δ2 + (x3 + y3 )δ1 δ2 .
The space of all natural transformations Nat(T, T 2 ) ∼ Hom(D, D2 ) turns out to
=
2 2
be the real algebraic variety R ∪R R consisting of all homomorphisms δ ’ x1 δ1 +
x2 δ2 + x3 δ1 δ2 with x1 x2 = 0, since δ 2 = 0. The homomorphism δ ’ xδ1 + yδ1 δ2
corresponds to the natural transformation (+T ) —¦ (vl —¦ my , 0T —¦ mx ), and the
homomorphism δ ’ xδ2 + yδ1 δ2 corresponds to (T +) —¦ (vl —¦ my , T 0 —¦ mx ). So any
element in Nat(T, T 2 ) can be expressed in terms of the natural transformations
{0T, T 0, (T +), (+T ), T π, πT, vl, m» for » ∈ R}.
Similarly Nat(T 2 , T 2 ) ∼ Hom(D2 , D2 ) turns out to be the real algebraic vari-
=
ety (R ∪R R ) — (R ∪R R2 ) consisting of all
2 2 2


δ1 x1 δ1 + x2 δ2 + x3 δ1 δ2

δ2 y 1 δ 1 + y2 δ 2 + y 3 δ 1 δ 2
with x1 x2 = y1 y2 = 0. Again any element of Nat(T 2 , T 2 ) can be written in
terms of {0T, T 0, (T +), (+T ), T π, πT, κ, m» T, T m» for » ∈ R}. If for example
x2 = y1 = 0 then the corresponding transformation is
(+T ) —¦ (my2 T —¦ T mx1 , (T +) —¦ (vl —¦ + —¦ (mx3 —¦ πT, my3 —¦ T π), 0T —¦ mx1 —¦ πT )).
Note also the relations T π —¦ κ = πT , κ —¦ (T +) = (+T ) —¦ (κ — κ), κ —¦ vl = vl,
κ —¦ T m» = m» T ; so κ interchanges the two vector bundle structures on T 2 ’ T ,
namely ((+T ), m» T, πT ) and ((T +), T m» , T π), and vl : T ’ T 2 is linear for
both of them. The reader is advised now to have again a look at 6.12.
37.5. In the situation of 37.3 we let now F1 = F be a general Weil functor and
F2 = T . So we consider T —¦ F which is isomorphic to F —¦ T via κF,T . In general
we have (F1 —Id F2 ) — F = F1 — F —Id F2 — F , so + : T —Id T ’ T induces a ¬ber
addition (+ — F ) : T — F —Id T — F ’ T — F , and m» — F : T — F ’ T — F is a
¬ber scalar multiplication. So T — F is a vector bundle functor on the category
Mf which can be described in terms of lemma 37.2 as follows.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
37. Examples and applications 321


Lemma. In the notation of lemma 37.2 we have T — F ∼ N — T , where N is
=¯ ¯
the underlying vector spaces of the nilradical N (F ) of F .
Proof. The Weil algebra of T — F is R.1 • (N (F ) — N (T )) by 37.3.(5). We have
¯
N (F ) — N (T ) = N (F ) — R.δ = N as vector space, and the multiplication on
N (F ) — N (T ) is zero.
37.6. Sections and expansions. For a Weil functor F with Weil algebra
A = R.1 • N and for a manifold M we denote by XF (M ) the space of all smooth
sections of πF,M : F (M ) ’ M . Note that this space is in¬nite dimensional in
general. Recall from theorem 35.14 that
·M,A
F (M ) = TA (M ) ← ’ Hom(C ∞ (M, R), A)
’’

is an isomorphism. For f ∈ C ∞ (M, R) we can decompose F (f ) = TA (f ) :
F (M ) ’ F (R) = A = R.1 • N into

F (f ) = TA (f ) = (f —¦ π) • N (f ),
N (f ) : F (M ) ’ N.

Lemma.
(1) Each Xx ∈ F (M )x = π ’1 (x) for x ∈ M de¬nes an R-linear mapping

DXx : C ∞ (M, R) ’ N,
DXx (f ) := N (f )(Xx ) = F (f )(Xx ) ’ f (x).1,

which satis¬es

DXx (f.g) = DXx (f ).g(x) + f (x).DXx (g) + DXx (f ).DXx (g).

We call this the expansion property at x ∈ M .
(2) Each R-linear mapping ξ : C ∞ (M, R) ’ N which satis¬es the expansion
property at x ∈ M is of the form ξ = DXx for a unique Xx ∈ F (M )x .
(3) The R-linear mappings ξ : C ∞ (M, R) ’ C ∞ (M, N ) = N — C ∞ (M, R)
which have the expansion property

f, g ∈ C ∞ (M, R),
(a) ξ(f.g) = ξ(f ).g + f.ξ(g) + ξ(f ).ξ(g),

are exactly those induced (via 1 and 2) by the smooth sections of π :
F (M ) ’ M .

Linear mappings satisfying the expansion property 1 will be called expansions:
if N is generated by δ with δ k+1 = 0, so that F (M ) = J0 (R, M ), then these
k

are parametrized Taylor expansions of f to order k (applied to a k-jet of a
curve through each point). For X ∈ XF (M ) we will write DX : C ∞ (M, R) ’

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
322 Chapter VIII. Product preserving functors


C ∞ (M, N ) = N — C ∞ (M, R) for the expansion induced by X. Note the de¬ning
equation

F (f ) —¦ X = f.1 + DX (f ) = (f.1, DX (f )) or
(b)
’1
f (x).1 + DX (f )(x) = F (f )(X(x)) = ·M,A (X(x))(f ).

Proof. (1) and (2). For • ∈ Hom(C ∞ (M, R), A) = F (M ) we consider the foot
point π(·M,A (•)) = ·M,R (π(•)) = x ∈ M and ·M,A (•) = Xx ∈ F (M )x . Then
we have •(f ) = TA (f )(Xx ) and the expansion property for DXx is equivalent to
•(f.g) = •(f ).•(g).
(3) For each x ∈ M the mapping f ’ ξ(f )(x) ∈ N is of the form DX(x) for a
unique X(x) ∈ F (M )x by 1 and 2, and clearly X : M ’ F (M ) is smooth.
37.7. Theorem. Let F be a Weil functor with Weil algebra A = R.1 • N .
Using the natural transformations from 37.3 we have:
(1) XF (M ) is a group with multiplication X Y = µF —¦F (Y )—¦X and identity
0F .
(2) XT —F (M ) is a Lie algebra with bracket induced from the usual Lie bracket
on XT (M ) and the multiplication m : N —N ’ N by [a—X, b—Y ]T —F =
a.b — [X, Y ].
(3) There is a bijective mapping exp : XT —F (M ) ’ XF (M ) which expresses
the multiplication by the Baker-Campbell-Hausdor¬ formula.
(4) The multiplication , the Lie bracket [ , ]T —F , and exp are natural
in F (with respect to natural operators) and M (with respect to local
di¬eomorphisms).

Remark. If F = T , then XT (M ) is the space of all vector ¬elds on M , the
multiplication is X Y = X + Y , and the bracket is [X, Y ]T —T = 0, and exp is
the identity. So the multiplication in (1), which is commutative only if F is a
natural vector bundle, generalizes the linear structure on X(M ).
37.8. For the proof of theorem 37.7 we need some preparation. If a ∈ N and
X ∈ X(M ) is a smooth vector ¬eld on M , then by lemma 37.5 we have a — X ∈
XT —F (M ) and for f ∈ C ∞ (M, R) we use T f (X) = f.1 + df (X) to get

(T — F )(f )(a — X) = (IdN —T f )(a — X)
= f.1 + a.df (X) = f.1 + a.X(f )
= f.1 + Da—X (f ) by 37.6.(b). Thus
T —F
(a) Da—X (f ) = Da—X (f ) = a.X(f ) = a.df (X).

So again by 37.5 we see that XT —F (M ) is isomorphic to the space of all R-linear
mappings ξ : C ∞ (M, R) ’ N — C ∞ (M, R) satisfying

ξ(f.g) = ξ(f ).g + f.ξ(g).

These mappings are called derivations.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
37. Examples and applications 323


Now we denote L := LR (C ∞ (M, R), N — C ∞ (M, R)) for short, and for ξ,
· ∈ L we de¬ne

(b) ξ • · := (m — IdC ∞ (M,R) ) —¦ (IdN —ξ) —¦ · : C ∞ (M, R) ’
’ N — C ∞ (M, R) ’ N — N — C ∞ (M, R) ’ N — C ∞ (M, R),

where m : N — N ’ N is the (nilpotent) multiplication on N . Note that
DF : XF (M ) ’ L and DT —F : XT —F (M ) ’ L are injective linear mappings.

37.9. Lemma. 1. L is a real associative nilpotent algebra without unit under
the multiplication •, and it is commutative if and only if m = 0 : N — N ’ N .
F F F F F
(1) For X, Y ∈ XF (M ) we have DX Y = DX • DY + DX + DY .
F F F F F
(2) For X, Y ∈ XT —F (M ) we have D[X,Y ]T —F = DX • DY ’ DY • DX .
(3) For ξ ∈ L de¬ne


1 •i
exp(ξ) := ξ
i!
i=1

(’1)i’1 •i
log(ξ) := ξ.
i
i=1


Then exp, log : L ’ L are bijective and inverse to each other. exp(ξ) is
an expansion if and only if ξ is a derivation.

Note that i = 0 lacks in the de¬nitions of exp and log, since L has no unit.

Proof. (1) We use that m is associative in the following computation.

ξ • (· • ζ) = (m — IdC ∞ (M,R) ) —¦ (IdN —ξ) —¦ (· • ζ)
= (m — Id) —¦ (IdN —ξ) —¦ (m — Id) —¦ (IdN —·) —¦ ζ
= (m — Id) —¦ (m — IdN — Id) —¦ (IdN —N —ξ) —¦ (IdN —·) —¦ ζ
= (m — Id) —¦ (IdN —m — Id) —¦ (IdN —N —ξ) —¦ (IdN —·) —¦ ζ
= (m — Id) —¦ (IdN — (m — Id) —¦ (IdN —ξ) —¦ · —¦ ζ
= (ξ • ·) • ζ.

So • is associative, and it is obviously R-bilinear. The order of nilpotence equals
that of N .
(2) Recall from 36.13 and 37.3 that

F (F (R)) = A — A = (R.1 — R.1) • (R.1 — N ) • (N — R.1) • (N — N )
∼ A • F (N ) ∼ F (R.1) — F (N ) ∼ F (R.1 — N ) = F (A).
= = =

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
324 Chapter VIII. Product preserving functors


We will use this decomposition in exactly this order in the following computation.

F
(f ) = F (f ) —¦ (X
f.1 + DX Y) by 37.6.(b)
Y
= F (f ) —¦ µF,M —¦ F (Y ) —¦ X by 37.7(1)
= µF,R —¦ F (F (f )) —¦ F (Y ) —¦ X since µ is natural
= m —¦ F (F (f ) —¦ Y ) —¦ X
F
= m —¦ F (f.1, DY (f )) —¦ X by 37.6.(b)
F
= m —¦ (1 — F (f ) —¦ X) • (F (DY (f )) —¦ X)
F F F F
= m —¦ 1 — (f.1 + DX (f )) + DY (f ) — 1 + (IdN —DX )(DY (f ))
F F F F
= f.1 + DX (f ) + DY (f ) + (DX • DY )(f ).

(3) For vector ¬elds X, Y ∈ X(M ) on M and a, b ∈ N we have

T —F
D[a—X,b—Y ]T —F (f ) = Da.b—[X,Y ] (f )
= a.b.[X, Y ](f ) by 37.8.(a)
= a.b.(X(Y (f )) ’ Y (X(f )))
T —F T —F
= (m — IdC ∞ (M,R) ) —¦ (IdN —Da—X ) —¦ Db—Y (f ) ’ . . .
T —F T —F T —F T —F
= (Da—X • Db—Y ’ Db—Y • Da—X )(f ).

(4) After adjoining a unit to L we see that exp(ξ) = eξ ’ 1 and log(ξ) =
log(1 + ξ). So exp and log are inverse to each other in the ring of formal power
series of one variable. The elements 1 and ξ generate a quotient of the power
series ring in R.1 • L, and the formal expressions of exp and log commute with
’1
taking quotients. So exp = log . The second assertion follows from a direct
formal computation, or also from 37.10 below.

37.10. We consider now the R-linear mapping C of L in the ring of all R-linear
endomorphisms of the algebra A — C ∞ (M, R), given by

Cξ := m —¦ (IdA —ξ) : A — C ∞ (M, R) ’
’ A — N — C ∞ (M, R) ‚ A — A — C ∞ (M, R) ’ A — C ∞ (M, R),

where m : A — A ’ A is the multiplication. We have Cξ (a — f ) = a.ξ(f ).

Lemma.
(1) Cξ•· = Cξ —¦ C· , so C is an algebra homomorphism.
(2) ξ ∈ L is an expansion if and only if Id +Cξ is an automorphism of the
commutative algebra A — C ∞ (M, R).
(3) ξ ∈ L is a derivation if and only if Cξ is a derivation of the algebra
A — C ∞ (M, R).

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

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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 )
¯ ¯
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.

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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.

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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
˜
for every regular bundle functor F on C with locally ¬nite order, the maps F f ,
˜ k˜ k˜
F g : Q — F A ’ F B satisfy jq F f ( , z) = jq F g( , z) for all z ∈ F A.

39.8. Let us de¬ne ιRm+n (z) = j0 F µRm+n ( , z). If we repeat the considerations
from 39.4 we deduce that our maps ιRm+n determine a natural transformation
1
ι : F V ’ T F . But its values satisfy T pY —¦ ιY (z) = j0 pY —¦ F (µY )t (z) = pY (z) ∈
V Y and so ιY (z) ∈ V (F Y ’ BY ). So ι : F V ’ V F and similarly to 39.4 we
show that the upper triangle in 39.6.(1) commutes.
Every non-zero vertical vector ¬eld on Rm+n ’ Rm can be locally trans-
formed (by means of an FMm -morphism) into a constant one and for all con-
stant vertical vector ¬elds X on Rm+n we have FlX = (µRm+n —¦ (idRm+n — X)).
Hence we also have an analogue of lemma 39.5.
Theorem. For every bundle functor F : FMm ’ FM there is the canonical
¬‚ow-natural transformation ι : F V ’ V F . If F has the point property, then ι
is a natural equivalence if and only if F preserves ¬bered products.
We have to point out that we consider the ¬bered manifold structure F Y ’
BY for every object Y ’ BY ∈ ObFMm , i.e. ιY : F (V Y ’ BY ) ’ V (F Y ’
BY ).
Proof. We have proved that ι is ¬‚ow-natural. Assume F has the point property.
If ι is a natural equivalence, then proposition 38.18 implies that F preserves
¬bered products. On the other hand, F preserves ¬bered products if and only if
F Rm+n = VA Rm+n for a Weil algebra A and then also F f coincides with VA f for
morphisms of the form idRm —g : Rm+n ’ Rm+k , see 38.19. But each µRm+n (t, )
is of this form and any restriction of ιRm+n to a ¬ber (VA V Rm+n )x ∼ TA T Rm
=
coincides with the canonical ¬‚ow natural equivalence TA T ’ T TA , cf. 39.2.
Hence ι is a natural equivalence.

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


Let us remark that for F = J r we obtain the well known canonical natural
equivalence J r V ’ V J r , cf. [Goldschmidt, Sternberg, 73], [Mangiarotti, Mod-
ugno, 83].
39.9. The action of some bundle functors F : FMm ’ FM on morphisms can
be extended in such a way that the proof of theorem 39.8 might go through for
the whole tangent bundle. We shall show that this happens with the functors
J r : FMm ’ FM.
Since J r (Rm+n ’ Rm ) is a sub bundle in the bundle Km Rm+n of contact
r

elements of order r formed by the elements transversal to the ¬bration, the
action of J r f on a jet jx s extends to all local di¬eomorphisms transforming jx s
r r

into a jet of a section. Of course, we are not able to recover the whole theory
of bundle functors for this extended action of J r , but one veri¬es easily that
lemma 39.3 remains still valid.
So let us de¬ne µt : T Rm+n ’ Rm+n by µt (x, z, X, Z) = (x + tX, z + tZ).
For every section (x, z(x), X(x), Z(x)) of T Rm+n ’ Rm , its composition with
µt and the ¬rst projection gives the map x ’ x + tX(x). If we proceed in a
similar way as above, we deduce
Proposition. There is a canonical ¬‚ow-natural transformation ι : J r T ’ T J r
and its restriction J r V ’ V J r is the canonical ¬‚ow-natural equivalence.
39.10. Remark. Let us notice that ι : T J r ’ J r T cannot be an equivalence
for dimension reasons if m > 0. The ¬‚ow-natural transformations on jet bundles
were presented as a useful tool in [Mangiarotti, Modugno, 83].
It is instructive to derive the coordinate description of ιRm+n at least in the
β
±
case r = 1. Let us write a map f : (Rm+n ’ Rm ) ’ (Rm+n ’ Rm ) in the
’ ’
form z k = f k (xi , y p ), wq = f q (xi , y p ). In order to get the action of J 1 f in the
p ˜
extended sense on j0 s = (y p , yi ) we have to consider the map (β —¦ f —¦ s)’1 = f ,
1

˜ ˜ ˜ ˜
xi = f i (z). So z k = f k (f (z), y p (f (z))) and we evaluate that the matrix ‚ f i /‚z k
p
is the inverse matrix to ‚f k /‚xi + (‚f k /‚y p )yi (the invertibility of this matrix
is exactly the condition on j0 s to lie in the domain of J 1 f ). Now the coordinates
1
q
wk of J 1 f (j0 s) are
1

˜ ˜
‚f q ‚ f j ‚f q p ‚ f j
q
wk = + p yj q .
‚xj ‚z k ‚y ‚z
Consider the canonical coordinates xi , y p on Y = Rm+n and the additional
coordinates yi or X i , Y p or yi , Xj , Yip or yi , ξ i , · p , ·i on J 1 Y or T Y or J 1 T Y
p p p p
i

or T J 1 Y , respectively. If jx s = (xi , y p , X j , Y q , yk , Xm , Yn ), then
1 r s



¯q
J 1 (µY )t (j0 s) = (xi + tX i , y p + tY p , yj (t))
1


yi (t)(δj + tXj ) = (yi + tYip )δj .
¯p p
i i i


Di¬erentiating by t at 0 we get

ιRm+n (xi , y p , X j , Y q , yk , Xm , Yn ) = (xi , y p , yk , X j , Y q , Y s ’ ym X m ).
r s r s


This formula corresponds to the de¬nition in [Mangiarotti, Modugno, 83].

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
40. Natural transformations 341


40. Natural transformations

40.1. The ¬rst part of this section is concerned with natural transformations
with a Weil bundle as the source. In this case we get a result similar to the
Yoneda lemma well known from general category theory. Namely, each point in
a Weil bundle TA M is an equivalence class of mappings in C ∞ (Rn , M ) where n
is the width of the Weil algebra A, see 35.15, and the canonical projections yield
a natural transformation ± : C ∞ (Rn , ) ’ TA . Hence given any bundle functor
F on Mf , every natural transformation χ : TA ’ F gives rise to the natural
transformation χ —¦ ± : C ∞ (Rn , ) ’ F and this is determined by the value of
(χ —¦ ±)Rn (idRn ). So in order to classify all natural transformations χ : TA ’ F
we have to distinguish the possible values v := χRn —¦ ±Rn (idRn ) ∈ F Rn . Let
us recall that for every natural transformation χ between bundle functors on
Mf all maps χM are ¬bered maps over idM , see 14.11. Hence v ∈ F0 Rn and
another obvious condition is F f (v) = F g(v) for all maps f , g : Rn ’ M with
jA f = jA g. On the other hand, having chosen such v ∈ F0 Rn , we can de¬ne
χv (jA f ) = F f (v) and if all these maps are smooth, then they form a natural
M
transformation χv : TA ’ F .
So from the technical point of view, our next considerations consist in a
better description of the points v with the above properties. In particular, we
deduce that it su¬ces to verify F f (v) = F i(v) for all maps f : Rn ’ Rn+1 with
jA f = jA i where i : Rn ’ Rn+1 , x ’ (x, 0).
40.2. De¬nition. For every Weil algebra A of width n and for every bun-
dle functor F on Mf , an element v ∈ F0 Rn is called A-admissible if jA f =
jA i implies F f (v) = F i(v) for all f ∈ C ∞ (Rn , Rn+1 ). We denote by SA (F ) ‚
S = F0 Rn the set of all A-admissible elements.
40.3. Proposition. For every Weil algebra A of width n and every bundle
functor F on Mf , the map

χ ’ χRn (jA idRn )

is a bijection between the natural transformations χ : TA ’ F and the subset of
A-admissible elements SA (F ) ‚ F0 Rn .
The proof consists in two steps. First we have to prove that each v ∈ SA (F )
de¬nes the transformation χv : TA ’ F at the level of sets, cf. 40.1, and then
we have to verify that all maps χv are smooth.
M

40.4. Lemma. Let F : Mf ’ FM be a bundle functor and A be a Weil
algebra of width n. For each point v ∈ SA (F ) and for all mappings f , g : Rn ’
M the equality jA f = jA g implies F f (v) = F g(v).
Proof. The proof is a straightforward generalization of the proof of theorem 22.3
with m = 0. Therefore we shall present it in a rather condensed form.
During the whole proof, we may restrict ourselves to mappings f , g : Rn ’ Rk
of maximal rank. The reason lies in the regularity of all bundle functors on Mf ,
cf. 22.3 and 20.7.

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


The canonical local form of a map f : Rn ’ Rn+1 of maximal rank is i and
therefore the assertion is trivial for the dimension k = n + 1.
Since the equivalence on the spaces C ∞ (Rn , Rk ) determined by A is compat-
ible with the products of maps, we can complete the proof as in 22.3.(b) and
r
22.3.(e) with m = 0, j0 replaced by jA and Sn replaced by SA (F ).

Let us remark that for m = 0 theorem 22.3 follows easily from this lemma.
r
Indeed, we can take the Weil algebra A corresponding to the bundle Tnn+1 of
r
n-dimensional velocities of order rn+1 . Then jA f = jA g if and only if j0 n+1 f =
r
j0 n+1 g and according to the assumptions in 22.3, SA (F ) = Sn . By the general
theory, the order rn+1 extends from the standard ¬ber Sn to all objects of
dimension n.

40.5. Lemma. For every Weil algebra A of width n and every smooth curve
c : R ’ TA Rk there is a smoothly parameterized family of maps γ : R—Rn ’ Rk
such that jA γt = c(t).

Proof. There is an ideal A in the algebra of germs En = C0 (Rn , R), cf. 35.5,
such that A = En /A. Write Dn = Mr+1 where M is the maximal ideal in
r

En , and Dr = En /Dn , i.e. TDr = Tn . Then A ⊃ Dn for suitable r and so we
r r r
n n
get the linear projection Dr ’ A, j0 f ’ jA f . Let us choose a smooth section
r
n
s : A ’ Dr of this projection. Now, given a curve c(t) = jA ft in TA Rk there
n
are the canonical polynomial representatives gt of the jets s(jA ft ). If c(t) is
smooth, then gt is a smoothly parameterized family of polynomials and so jA gt
is a smooth curve with jA gt = c(t).

Proof of proposition 40.3. Given a natural transformation χ : TA ’ F , the value
χRn (jA idRn ) is an A-admissible element in F0 Rn . On the other hand, every
A-admissible element v ∈ SA (F ) determines the maps χv k : TA Rk ’ F Rk ,
R
χv k (jA f ) = F f (v) and all these maps are smooth. By the de¬nition, χv n obey
R
R
the necessary commutativity relations and so they determine the unique natural
transformation χv : TA ’ F with χv n (v) = v.
R

40.6. Let us apply proposition 40.3 to the case F = T (r) , the r-th order tangent
functor. The elements in the standard ¬ber of T (r) Rn are the linear forms on
the vector space J0 (Rn , R)0 and for every Weil algebra A of width n one veri¬es
r

easily that such a form ω lies in SA (T (r) ) if and only if ω(j0 g) = 0 for all g with
r

jA g = jA 0.
q
As a simple illustration, we ¬nd all natural transformations T1 ’ T (r) . Every
r r— r
element j0 f ∈ T0 R = J0 (R, R)0 has the canonical representative f (x) = a1 x +
(r)
a2 x2 +· · ·+ar xr . Let us de¬ne 1-forms vi ∈ T0 R by vi (j0 f ) = ai , i = 1, 2, . . . , r.
r
q q
Since jDq f = j0 f , the forms vi are D1 -admissible if and only if i ¤ q. So
1
q
the linear space of all natural transformations T1 ’ T (r) is generated by the
linearly independent transformations χvi , i = 1, . . . , min{q, r}. The maps χvi M
q q
can be described as follows. Every j0 g ∈ T1 M determines a curve g : R ’ M
r r
through x = g(0) up to the order q and given any jx f ∈ Jx (M, R)0 the value
q
χvi (j0 g)(jx f ) is obtained by the evaluation of the i-th order term in f —¦g : R ’ R
r
M


Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
40. Natural transformations 343

q
at 0 ∈ R. So χvi (j0 g) might be viewed as the i-th derivative on Jx (M, R)0 in
r
M
q
the direction j0 g.
In general, given any vector bundle functor F on Mf , the natural transfor-
mations TA ’ F carry a vector space structure and the corresponding set SA (F )
is a linear subspace in F0 Rn . In particular, the space of all natural transforma-
tions TA ’ F is a ¬nite dimensional vector space with dimension bounded by
the dimension of the standard ¬ber F0 Rn .
As an example let us consider the two natural vector bundle structures given
by πT M : T T M ’ T M and T πM : T T M ’ T M which form linearly indepen-
dent natural transformations T T ’ T . For dimension reasons these must form
a basis of the linear space of all natural transformations T T ’ T . Analogously
the products T πM § πT M : T T M ’ Λ2 T M generate the one-dimensional space
of all natural transformations T T ’ Λ2 T and there are no non-zero natural
transformations T T ’ Λp T for p > 2.

40.7. Remark. [Mikulski, to appear a] also determined the natural operators
transforming functions on a manifold M of dimension at least two into functions
on F M for every bundle functor F : Mf ’ FM. All of them have the form
f ’ h —¦ F f , f ∈ C ∞ (M, R), where h is any smooth function h : F R ’ R.

40.8. Natural transformations T (r) ’ T (r) . Now we are going to show that
there are no other natural transformations T (r) ’ T (r) beside the real multiples
of the identity. Thus, in this direction the properties of T (r) are quite di¬erent
from the higher order product preserving functors where the corresponding Weil
algebras have many endomorphisms as a rule. Let us remark that from the
technical point of view we shall prove the proposition in all dimensions separately
and only then we ˜join™ all these partial results together.

Proposition. All natural transformations T (r) ’ T (r) form the one-parameter
family
X ’ kX, k ∈ R.


Proof. If xi are local coordinates on a manifold M , then the induced ¬ber co-
r—
ordinates ui , ui1 i2 , . . . , ui1 ...ir (symmetric in all indices) on T1 M correspond
to the polynomial representant ui xi + ui1 i2 xi1 xi2 + · · · + ui1 ...ir xi1 . . . xir of a
jet from T1 M . A linear functional on (T1 M )x with the ¬ber coordinates X i ,
r— r—

X i1 i2 , . . . , X i1 ...ir (symmetric in all indices) has the form

X i ui + X i1 i2 ui1 i2 + · · · + X i1 ...ir ui1 ...ir .
(1)

Let y p be some local coordinates on N , let Y p , Y p1 p2 , . . . , Y p1 ...pr be the induced
¬ber coordinates on T (r) N and y p = f p (xi ) be the coordinate expression of a
map f : M ’ N . If we evaluate the jet composition from the de¬nition of the
action of the higher order tangent bundles on morphisms, we deduce by (1) the

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


coordinate expression of T (r) f

‚f p i 1 ‚2f p ‚rf p
1
p i1 i2
X i1 ...ir
+ ··· +
Y= X+ X
i i1 ‚xi2 i1 . . . ‚xir
‚x 2! ‚x r! ‚x
.
.
.
‚f p1 ‚f ps i1 ...is
Y p1 ...ps =
(2) ... X + ...
‚xi1 ‚xis
.
.
.
‚f p1 ‚f pr i1 ...ir
p1 ...pr
Y = ... X
‚xi1 ‚xir
where the dots in the middle row denote a polynomial expression, each term of
which contains at least one partial derivative of f p of order at least two.
Consider ¬rst T (r) as a bundle functor on the subcategory Mfm ‚ Mf .
(r)
According to (2), its standard ¬ber S = T0 Rm is a Gr -space with the following
m
action
¯
X i = ai X j + ai 1 j2 X j1 j2 + · · · + ai 1 ...jr X j1 ...jr
j j j
.
.
.
¯
X i1 ...is = ai1 . . . ais X j1 ...js + . . .
(3) j1 js
.
.
.
¯
X i1 ...ir = ai1 . . . air X j1 ...jr
j1 jr

where the dots in the middle row denote a polynomial expression, each term of
which contains at least one of the quantities ai 1 j2 , . . . , ai 1 ...jr . Write
j j

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