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48. Commuting with natural operators 381


47.13. In the end of this section we remark that the operations with linear vec-
tor ¬elds discussed here can be used to de¬ne, in a short way, the corresponding
operation with linear connections on vector bundles. We recall that a linear
connection “ on a vector bundle E ’ M is a section “ : E ’ J 1 E which is
a linear morphism from vector bundle E ’ M into vector bundle J 1 E ’ M .
Using local trivializations of E we ¬nd easily that this condition is equivalent to
the fact that the “-lift “ξ of every vector ¬eld ξ on M is a linear vector ¬eld on
E. By 47.9, the coordinate expression of a linear connection “ on E is

dy p = “p (x)y q dxi .
qi

¯ ¯
Let “ be another linear connection on a vector bundle E ’ M over the same
base with the equations
¯ bi
dz a = “a (x)z b dxi .
Using 47.10 and 47.11, we obtain immediately the following two assertions.
¯ ¯
47.14. Proposition. There is a unique linear connection “ — “ on E — E
satisfying
¯ ¯
(“ — “)(ξ) = (“ξ) — (“ξ)
for every vector ¬eld ξ on M .
47.15. Proposition. There is a unique linear connection “— on E — satisfying
“— (ξ) = (“ξ)— for every vector ¬eld ξ on M .
¯
Obviously, the equations of “ — “ are

dwpa = (“p (x)δb + δq “a (x))wqb dxi

a
bi
qi

and the coordinate expression of “— is

dvp = ’“q (x)vq dxi .
pi



48. Commuting with natural operators

48.1. The Lie derivative commutes with the exterior di¬erential, i.e. d(LX ω) =
LX (dω) for every exterior form ω and every vector ¬eld X, see 7.9.(5). Our
geometrical analysis of the concept of the Lie derivative leads to a general result,
which clari¬es that the speci¬c property of the exterior di¬erential used in the
above formula is its linearity.
Proposition. Let F and G be two natural vector bundles and A : F G be a
natural linear operator. Then

AM (LX s) = LX (AM s)
(1)

for every section s of F M and every vector ¬eld X on M .
In the special case of a linear natural transformation this is lemma 6.17.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
382 Chapter XI. General theory of Lie derivatives


Proof. The explicit meaning of (1) is AM (LF X s) = LGX (AM s). By the Peetre
theorem, AM is locally a di¬erential operator, so that AM commutes with limits.
Hence
1
AM F (FlX ) —¦ s —¦ FlX ’ AM s
AM (LF X s) = lim ’t t
t’0 t
1
= lim G(FlX —¦AM s —¦ FlX ’AM s = LGX (AM s)
’t t
t’0 t

by linearity and naturality.
48.2. A reasonable result of this type can be deduced even in the non linear case.
Let F and G be arbitrary natural bundles on Mfm , D : C ∞ (F M ) ’ C ∞ (GM )
be a local regular operator and s : M ’ F M be a section. The generalized
˜ ˜
Lie derivative LX s is a section of V F M , so that we cannot apply D to LX s.
However, we can consider the so called vertical prolongation V D : C ∞ (V F M ) ’
C ∞ (V GM ) of the operator D. This can be de¬ned as follows.
In general, let N ’ M and N ’ M be arbitrary ¬bered manifolds over the
same base and let D : C ∞ (N ) ’ C ∞ (N ) be a local regular operator. Every
local section q of V N ’ M is of the form ‚t 0 st , st ∈ C ∞ (N ) and we set



(Dst ) ∈ C ∞ (V N ).
‚ ‚
(1) V Dq = V D( ‚t s) =
0t ‚t 0

We have to verify that this is a correct de¬nition. By the nonlinear Peetre
theorem the operator D is induced by a map D : J ∞ N ’ N . Moreover each
in¬nite jet has a neighborhood in the inverse limit topology on J ∞ N on which D
depends only on r-jets for some ¬nite r. Thus, there is neighborhood U of x in M
and a locally de¬ned smooth map Dr : J r N ’ N such that Dst (y) = Dr (jy st )
r

for y ∈ U and for t su¬ciently small. So we get
(Dr (jx st )) = T Dr ( ‚t
r
jr s ) = (T Dr —¦ κ)(jx q)
r
‚ ‚
(V D)q(x) = 0xt
‚t 0

where κ is the canonical exchange map, and thus the de¬nition does not depend
on the choice of the family st .
48.3. A local regular operator D : C ∞ (F M ) ’ C ∞ (GM ) is called in¬nitesi-
mally natural if it holds
˜ ˜
LX (Ds) = V D(LX s)
for all X ∈ X(M ), s ∈ C ∞ (F M ).
Proposition. If A : F G is a natural operator, then all operators AM are
in¬nitesimally natural.
Proof. By lemma 47.2, 48.2.(1) and naturality we have

˜ (F (FlX ) —¦ s —¦ FlX

V AM (LF X s) = V AM ’t t
‚t 0

AM F (FlX ) —¦ s —¦ FlX = G(FlX ) —¦ AM s —¦ FlX
‚ ‚
= ’t ’t
t t
‚t 0 ‚t 0
˜
= LGX AM s.



Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
48. Commuting with natural operators 383

+
48.4. Let Mfm be the category of oriented m-dimensional manifolds and ori-
entation preserving local di¬eomorphisms.
+
Theorem. Let F and G be two bundle functors on Mfm , M be an oriented m-
dimensional manifold and let AM : C ∞ (F M ) ’ C ∞ (GM ) be an in¬nitesimally
natural operator. Then AM is the value of a unique natural operator A : F G
on M.
We shall prove this theorem in several steps.
48.5. Let us ¬x an in¬nitesimally natural operator D : C ∞ (F Rm ) ’ C ∞ (GRm )
and let us write S and Q for the standard ¬bers F0 Rm and G0 Rm . Since each
local operator is locally of ¬nite order by the nonlinear Peetre theorem, there is
∞ ∞ ∞
the induced map D : Tm S ’ Q. Moreover, at each j0 s ∈ Tm S the application

of the Peetre theorem (with K = {0}) yields a smallest possible order r = χ(j0 s)
r r
such that for every section q with j0 q = j0 s we have Ds(0) = Dq(0), see 23.1.
˜ ∞ ∞
Let us de¬ne Vr ‚ Tm S as the subset of all jets with χ(j0 s) ¤ r. Let Vr be the
˜ ∞ r
interior of Vr in the inverse limit topology and put Ur := πr (Vr ) ‚ Tm S.

The Peetre theorem implies Tm S = ∪r Vr and so the sets Vr form an open

¬ltration of Tm S. On each Vr , the map D factors to a map Dr : Ur ’ Q.

w Qu  ’
  ’’



  ’ ’’ ’’
 ’’
D3
D2
D1


Uu Uu Uu
1 2 3

∞ ∞ ∞
π1 π2 π3
D



yy xw V 99999w V y
ˆy w ···
I
V1

xx 9 9
2 3



u x B9
x9

Tm S
Since there are the induced actions of the jet groups Gr+k on Tm S (here k is
r
m
the order of F ), we have the fundamental ¬eld mapping ζ (r) : gr+k ’ X(Tm S)
r
m
and we write ζ Q for the fundamental ¬eld mapping on Q. There is an analogy
to 34.3.
Lemma. For all X ∈ gr+k , j0 s ∈ Tm S it holds
r r
m

(r) r r˜
ζX (j0 s) = κ(j0 (L’X s)).

r
Proof. Write » for the action of the jet group on Tm S. We have
(r) r r r X X
‚ ‚
‚t 0 »(exptX)(j0 s) = ‚t 0 j0 (F (Flt ) —¦ s —¦ Fl’t )
ζX (j0 s) =

κ(j0 ( ‚t 0 (F (FlX ) —¦ s —¦ FlX ))) = κ(j0 (L’X s)).
r‚
= ’t
t




Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
384 Chapter XI. General theory of Lie derivatives

(r)
Q
48.6. Lemma. For all r ∈ N and X ∈ gr+k we have T Dr —¦ ζX = ζX —¦ Dr on
m
Ur .
Proof. Recall that (V D)q(0) = (T Dr —¦ κ)(j0 q) for all j0 q ∈ κ’1 (T Ur ). Using
r r

the above lemma and the in¬nitesimal naturality of D we compute
(r) r˜ ˜
r
(T Dr —¦ ζX )(j0 s) = T Dr (κ(j0 (L’X s))) = V D(L’X s)(0) =
Q Q
˜ r
= L’X (Ds)(0) = ζX (Ds(0)) = ζX (Dr (j0 s)).

48.7. Lemma. The map D : Tm S ’ Q is G∞ + -equivariant.

m

Proof. Given a = j0 f ∈ G∞ + and y = j0 s ∈ Tm S we have to show D(a.y) =
∞ ∞ ∞
m
a.D(y). Each a is a composition of a jet of a linear map f and of a jet from
∞ ∞
the kernel B1 of the jet projection π1 . If f is linear, then there are linear
maps gi , i = 1, 2, . . . , l, lying in the image of the exponential map of G1 such
m

that f = g1 —¦ . . . —¦ gl . Since Tm S = ∪r Vr there must be an r ∈ N such that y
∞ ∞
and all elements (j0 gp —¦ . . . —¦ j0 gl ) · y are in Vr for all p ¤ l. Thus, D(a.y) =
r+k r+k
r r
Dr (j0 f.j0 s) = j0 f.Dr (j0 s) = a.D(y), for Dr preserves all the fundamental
¬elds.
r
Since the whole kernel B1 lies in the image of the exponential map for each
∞ ∞
r < ∞, an analogous consideration for j0 f ∈ B1 completes the proof of the
lemma.
+
48.8. Lemma. The natural operator A on Mfm which is determined by the
G∞ + -equivariant map D coincides on Rm with the operator D.
m

Proof. There is the associated map A : J ∞ F Rm ’ GRm to the operator ARm .
Let us write A0 for its restriction (J ∞ F )0 Rm ’ G0 Rm and similarly for the
map D corresponding to the original operator D. Now let tx : Rm ’ Rm be
the translation by x. Then the map A (and thus the operator A) is uniquely
determined by A0 since by naturality of A we have (t’x )— —¦ ARm —¦ (tx )— = ARm .
But tx is the ¬‚ow at time 1 of the constant vector ¬eld X. For every vector ¬eld
X and section s we have
˜ ˜
LX ((FlX )— s) = LX (F (FlX ) —¦ s —¦ FlX ) = ‚t (F (FlX ) —¦ s —¦ FlX )

’t ’t
t t t
˜ X— ˜
X X
= T (F (Fl’t )) —¦ LX s —¦ Flt = (Flt ) (LX s)
and so using in¬nitesimal naturality, for every complete vector ¬eld X we com-
pute
(FlX )— (D(FlX )— s) =

’t t
‚t
˜ ˜
= ’(FlX )— LX (D(FlX )— s) + (FlX )— (V D)((FlX )— LX s) =
’t ’t
t t
˜ ˜
= (FlX )— ’LX (D(FlX )— s) + (V D)(LX ((FlX )— s)) = 0.
’t t t

Thus (t’x )— —¦ D —¦ (tx )— = D and since A0 = D0 this completes the proof.
Lemmas 48.7 and 48.8 imply the assertion of theorem 48.4. Indeed, if M = Rm
we get the result immediately and it follows for general M by locality of the
operators in question.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
48. Commuting with natural operators 385


48.9. As we have seen, if F is a natural vector bundle, then V F is naturally
equivalent to F • F and the second component of our general Lie derivative is
just the usual Lie derivative. Thus, the condition of the in¬nitesimal naturality
becomes the usual form D —¦ LX = LX —¦ D if D : C ∞ (F M ) ’ C ∞ (GM ) is linear.
More generally, if F is a sum F = E1 • · · · • Ek of k natural vector bundles,
G is a natural vector bundle and D is k-linear, then we have

˜ D F (FlX ) —¦ (s1 , . . . , sk ) —¦ FlX

pr2 —¦ V D(LX (s1 , . . . , sk )) = ’t t
‚t 0
k
D(s1 , . . . , LX si , . . . , sk ).
=
i=1

Hence for the k-linear operators we have
Corollary. Every natural k-linear operator A : E1 • · · · • Ek F satis¬es
k
LX AM (s1 , . . . , sk ) = AM (s1 , . . . , LX si , . . . , sk )
(1) i=1

for all s1 ∈ C ∞ E1 M ,. . . ,sk ∈ C ∞ Ek M , X ∈ C ∞ T M .
Formula (1) covers, among others, the cases of the Fr¨licher-Nijenhuis bracket
o
and the Schouten bracket discussed in 30.10 and 8.5.
48.10. The converse implication follows immediately for vector bundle functors
+
on Mfm . But we can prove more.
Let E1 , . . . , Ek be r-th order natural vector bundles corresponding to actions
»i of the jet group Gr on standard ¬bers Si , and assume that with the re-
m
1
stricted actions »i |Gm the spaces Si are invariant subspaces in spaces of the
form •j (—pj Rm — —qj Rm— ). In particular this applies to all natural vector bun-
dles which are constructed from the tangent bundle. Given any natural vector
bundle F we have
Theorem. Every local regular k-linear operator

AM : C ∞ (E1 M ) • · · · • C ∞ (Ek M ) ’ C ∞ (F M )

over an m-dimensional manifold M which satis¬es 48.9.1 is a value of a unique
natural operator A on Mfm .
The theorem follows from the theorem 48.4 and the next lemma
+
Lemma. Every k-linear natural operator A : E1 • · · · • Ek F on Mfm
extends to a unique natural operator on Mfm .
Let us remark, the proper sense of this lemma is that every operator in ques-
tion obeys the necessary commutativity properties with respect to all local di¬eo-
morhpisms between oriented m-manifolds and hence determines a unique natural
operator over the whole Mfm .
Proof. By the multilinear Peetre theorem A is of some ¬nite order . Thus A is
determined by the associated k-linear (Gr+ )+ -equivariant map A : Tm S1 — . . . —
m


Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
386 Chapter XI. General theory of Lie derivatives


Tm Sk ’ Q. Recall that the jet group Gr+ is the semidirect product of GL(m)
m
r+
and the kernel B1 , while (Gr+ )+ is the semidirect product of the connected
m
r+
+
component GL (m) of the unit and the same kernel B1 . Thus, in particular
the map A : Tm S1 — . . . — Tm Sk ’ Q is k-linear and GL+ (m)-equivariant. By
the descriptions of (Gr+ )+ and Gr+ above we only have to show that any such
m m
map is GL(m) equivariant, too. Using the standard polarization technique we
can express the map A by means of a GL+ (m) invariant tensor. But looking
at the proof of the Invariant tensor theorem one concludes that the spaces of
GL+ (m) invariant and of GL(m) invariant tensors coincide, so the map A is
GL(m) equivariant.
48.11. Lie derivatives of sector forms. At the end of this section we present
an original application of proposition 48.1. This is related with the di¬erentiation
of a certain kind of r-th order forms on a manifold M . The simplest case is
the ˜ordinary™ di¬erential of a classical 1-form on M . Such a 1-form ω can be
considered as a map ω : T M ’ R linear on each ¬ber. Beside its exterior
di¬erential dω : §2 T M ’ R, E. Cartan and some other classical geometers used
another kind of di¬erentiating ω in certain concrete geometric problems. This
was called the ordinary di¬erential of ω to be contrasted from the exterior one.
We can de¬ne it by constructing the tangent map T ω : T T M ’ T R = R — R,
which is of the form T ω = (ω, δω). The second component δω : T T M ’ R is
said to be the (ordinary) di¬erential of ω . In an arbitrary order r we consider
the r-th iterated tangent bundle T r M = T (· · · T (T M ) · · · ) (r times) of M .
The elements of T r M are called the r-sectors on M. Analogously to the case
r = 2, in which we have two well-known vector bundle structures pT M and

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