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

p¯

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

y 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) =

r˜

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