into itself. By the generalized invariant tensor theorem, it holds fs = cs Xs with

any cs ∈ R.

For r = 1 we have deduced fip = c1 Xi . For r = 2 consider the equivariance

p

with respect to the kernel of the jet projection G2 — G2 ’ G1 — G1 . Taking

m n m n

into account the coordinate form of the jet composition, we ¬nd that the action

p p ¯p p

of an element ((δj , ai ), (δq , bp )) on (Xi , Xij ) is Xi = Xi and

i p

˜jk qr

¯p p qr p

Xij = Xij + bp Xi Xj + Xk ak .

(4) ˜ij

qr

p

Then the equivariance condition for fij reads

p qr p p qr p

c2 Xij + (c1 )2 bp Xi Xj + c1 Xk ak = c2 (Xij + bp Xi Xj + Xk ak ).

(5) ˜ij ˜ij

qr qr

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

32. Jet functors 261

This implies c1 = c2 = 0 or c1 = c2 = 1. Assume by induction that our assertion

holds in the order r ’ 1. Consider the equivariance with respect to the kernel

of the jet projection Gr — Gr ’ Gr’1 — Gr’1 . The action of an element

m n m n

((δj , 0, . . . , 0, ai 1 ...jr ), (δq , 0, . . . , 0, bp1 ...qr )) leaves X1 , . . . , Xr’1 unchanged and

i p

˜j q

it holds

Xi1 ...ir = Xi1 ...ir + bp1 ...qr Xi11 . . . Xirr + Xj aj1 ...ir .

¯p p q q p

(6) ˜i

q

Then the equivariance condition for fip ...ir requires

1

(7) cr Xi1 ...ir + (c1 )r bp1 ...qr Xi11 . . . Xirr + c1 Xj aj1 ...ir

p q q p

˜i

q

= cr (Xi1 ...ir + bp1 ...qr Xi11 . . . Xirr + Xj aj1 ...ir ).

p q q p

˜i

q

This implies cr = c1 = 0 or 1.

For r = 1 we have a homothety f n : X ’ kn X, kn ∈ R, on each subcat-

egory Mfm — Mfn ‚ Mfm — Mf . If we take the value of the transforma-

tion (f 1 , . . . , f n , . . . ) on the product of idRm with the injection ia,b : Ra ’ Rb ,

(x1 , . . . , xa ) ’ (x1 , . . . , xa , 0, . . . , 0), a < b, and apply it to 1-jet at 0 of the map

x1 = t1 , x2 = 0, . . . , xa = 0, (t1 , . . . , tm ) ∈ Rm , we ¬nd ka = kb . For r ≥ 2 we

have on each subcategory either the identity or the contraction. Applying the

latter idea once again, we deduce that the same alternative must take place in

all cases.

32.2. The construction of the r-th jet prolongation J r Y of a ¬bered manifold

Y ’ X can be considered as a bundle functor on the category FMm . This

functor is also denoted by J r . However, in order to distinguish from 32.1, we

shall use J¬b for J r in the ¬bered case here.

r

r r

Proposition. The only natural transformation J¬b ’ J¬b is the identity.

Proof. The construction of product ¬bered manifolds de¬nes an injection Mfm

— Mf ’ FMm and the restriction of J¬b to Mfm — Mf is J r . For r = 1,

r

proposition 31.1 gives a one-parameter family

p p

(yi ) ’ (cyi )

(1)

1 1

of possible candidates for the natural transformation J¬b ’ J¬b . But the trans-

p

formation law of yi with respect to the kernel of the standard homomorphism

¯p p p

G1 1 1

m,n ’ Gm — Gn is yi = yi + ai . The equivariance condition for (1) reads

ap = cap , which implies c = 1.

i i

For r ≥ 2, proposition 32.1 o¬ers the contraction and the identity. But the

contraction is clearly not natural on the whole category FMm , so that only the

identity remains.

32.3. Natural transformations J 1 J 1 ’ J 1 J 1 . It is well known that the

canonical involution of the second tangent bundle plays a signi¬cant role in ap-

plications. A remarkable feature of the canonical involution on T T M is that it

exchanges both the projections pT M : T T M ’ T M and T pM : T T M ’ T M .

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

262 Chapter VII. Further applications

Nowadays, in several problems of the ¬eld theory the role of the tangent bun-

dle of a smooth manifold is replaced by the ¬rst jet prolongation J 1 Y of a

¬bered manifold p : Y ’ M . On the second iterated jet prolongation J 1 J 1 Y =

J 1 (J 1 Y ’ M ) there are two analogous projections to J 1 Y , namely the target

jet projection β1 : J 1 J 1 Y ’ J 1 Y and the prolongation J 1 β : J 1 J 1 Y ’ J 1 Y of

the target jet projection β : J 1 Y ’ Y . Hence one can ask whether there exists

a natural transformation of J 1 J 1 Y into itself exchanging the projections β1 and

J 1 β, provided J 1 J 1 is considered as a functor on FMm,n . But the answer is

negative.

Proposition. The only natural transformation J 1 J 1 ’ J 1 J 1 is the identity.

This assertion follows directly from proposition 32.6 below, so that we shall

not prove it separately. It is remarkable that we have a di¬erent situation on

¯

the subspace J 2 Y = {X ∈ J 1 J 1 Y, β1 X = J 1 β(X)}, which is called the second

semiholonomic prolongation of Y . There is a one-parametric family of natural

¯ ¯

transformations J 2 ’ J 2 , see 32.5.

32.4. An exchange map. However, one can construct a suitable exchange

map eΛ : J 1 J 1 Y ’ J 1 J 1 Y by means of a linear connection Λ on the base man-

ifold M as follows. Interpreting Λ as a principal connection on the ¬rst order

frame bundle P 1 M of M , we ¬rst explain how Λ induces a map hΛ : J 1 J 1 Y •

QP 1 M ’ Tm (Tm Y ). Every X ∈ J 1 J 1 Y is of the form X = jx ρ(z), where ρ is

1 1 1

a local section of J 1 Y ’ M , and for every u ∈ Px M we have Λ(u) = jx σ(z),

1 1

where σ is a local section of P 1 M ‚ J0 (Rm , M ). Taking into account the canon-

1

ical inclusion J 1 Y ‚ J 1 (M, Y ), the jet composition ρ(z) —¦ σ(z) de¬nes a local

map M ’ J0 (Rm , Y ) = Tm Y , the 1-jet of which jx (ρ(z) —¦ σ(z)) ∈ Jx (M, Tm Y )

1 1 1 1 1

depends on X and Λ(u) only. Since u ∈ J0 (Rm , M ), we have hΛ (X, u) =

1

1 11

jx (ρ(z) —¦ σ(z)) —¦ u ∈ Tm Tm Y . Furthermore, there is a canonical exchange map

11 11

κ : Tm Tm Y ’ Tm Tm Y , the de¬nition of which will be presented in the frame-

work of the theory of Weil bundles in 35.18. Using κ and hΛ , we construct a

map eΛ : J 1 J 1 Y ’ J 1 J 1 Y .

Lemma. For every X ∈ (J 1 J 1 Y )y there exists a unique element eΛ (X) ∈

J 1 J 1 Y satisfying

(1) κ(hΛ (X, u)) = hΛ (eΛ (X), u)

˜

˜

1

for any frame u ∈ Px M , x = p(y), provided Λ means the conjugate connection

of Λ.

Proof consists in direct evaluation, for which the reader is referred to [Kol´ˇ,

ar

Modugno, 91]. The coordinate form of eΛ is

yi = Yip ,

p

Yip = yi ,

p p p p p

yij = yji + (yk ’ Yk )Λk

(2) ji

where Yip = ‚y p /‚xi , yij = ‚yi /‚xj are the additional coordinates on J 1 (J 1 Y

p p

’ M ).

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

32. Jet functors 263

32.5. Remark. The subbundle J 2 Y ‚ J 1 J 1 Y is characterized by yi = Yip .

p

¯

¯

Formula 32.4.(2) shows that the restriction of eΛ to J 2 Y does not depend on Λ,

¯ ¯ ¯

so that we have a natural map e : J 2 Y ’ J 2 Y . Since J 2 Y ’ J 1 Y is an a¬ne

¯ ¯

bundle, e generates a one-parameter family of natural transformations J 2 ’ J 2

X ’ kX + (1 ’ k)e(X), k ∈ R.

¯

One proves easily that this family represents all natural transformations J 2 ’

¯2

J , see [Kol´ˇ, Modugno, 91].

ar

32.6. The map eΛ was introduced by M. Modugno by another construction, in

which the naturality ideas were partially used. Hence it is interesting to study

the whole problem purely from the naturality point of view.

Our goal is to ¬nd all natural transformations J 1 J 1 Y • QP 1 M ’ J 1 J 1 Y .

Since J 1 Y ’ Y is an a¬ne bundle with associated vector bundle V Y — T — M ,

we can de¬ne a map

δ : J 1 J 1 Y ’ V Y — T — M, A ’ β1 (A) ’ J 1 β(A).

(1)

On the other hand, proposition 25.2 implies directly that all natural operators

T M — T — M — T — M form the 3-parameter family

N : QP 1 M

ˆ ˆ

N : Λ ’ k1 S + k2 I — S + k3 S — I

(2)

ˆ

where S is the torsion tensor of Λ, S is the contracted torsion tensor and I is

the identity of T M . Using the contraction with respect to T M , we construct a

3-parameter family of maps

δ, N (Λ) : J 1 J 1 Y ’ V Y — T — M — T — M.

(3)

The well known exact sequence of vector bundles over J 1 Y

Vβ

0 ’ V Y — T —M ’ V J 1Y ’’ V Y ’ 0

’

(4)

shows that V Y —T — M —T — M can be considered as a subbundle in V J 1 Y —T — M ,

which is the vector bundle associated with the a¬ne bundle β1 : J 1 J 1 Y ’ J 1 Y .

Proposition. All natural transformations f : J 1 J 1 Y ’ J 1 J 1 Y depending on

a linear connection Λ on the base manifold form the two 3-parameter families

(5) I. f = id + δ, N (Λ) , II. f = eΛ + δ, N (Λ) .

Proof. The standard ¬bers V = (yi , Yip , yij ) and Z = (Λi ) are G2 -spaces

p p

m,n

jk

2

and we have to ¬nd all Gm,n -equivariant maps f : V — Z ’ V . The action of

G2 on V is

m,n

yi = ap yj aj + ap aj ,

¯p q

Yip = ap Yjq aj + ap aj

¯

˜i j ˜i ˜i j ˜i

q q

yij = ap ykl ak al + ap yk Ylr ak al + ap Ylq ak al +

¯p q q

(6) ˜i ˜j ˜i ˜j ˜i ˜j

q qr qk

+ ap yk ak al + ap yk ak + ap ak al + ap ak

q q

˜i ˜j ˜ij kl ˜i ˜j k ˜ij

q

ql

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

264 Chapter VII. Further applications

while the action of G2 on Z is given by 25.2.(3).

m,n

The coordinate form of an arbitrary map f : V — Z ’ V is

y = F (y, Y, y2 , Λ)

(7) Y = G(y, Y, y2 , Λ)

y2 = H(y, Y, y2 , Λ)

where y = (yi ), Y = (Yip ), y2 = (yij ), Λ = (Λi ). Considering equivariance of

p p

jk

(7) with respect to the base homotheties we ¬nd

kF (y, Y, y2 , Λ) = F (ky, kY, k 2 y2 , kΛ)

kG(y, Y, y2 , Λ) = G(ky, kY, k 2 y2 , kΛ)

(8)

k 2 H(y, Y, y2 , Λ) = H(ky, kY, k 2 y2 , kΛ).

By the homogeneous function theorem, F and G are linear in y, Y , Λ and

independent of y2 , while H is linear in y2 and bilinear in y, Y , Λ. The ¬ber

homotheties then yield

kF (y, Y, Λ) = F (ky, kY, Λ)

(9) kG(y, Y, Λ) = G(ky, kY, Λ)

kH(y, Y, Λ) = H(ky, kY, ky2 , Λ).

Comparing (9) with (8) we ¬nd that F and G are independent of Λ and H is

linear in y2 and bilinear in (y, Λ) and in (Y, Λ).

Since f is GL(m)—GL(n)-equivariant, we can apply the generalized invariant

tensor theorem. This yields

Fip = ayi + bYip

p

Gp = cyi + dYip

p

i

p p p

Hij = eyij + f yji +

(10)

p p p p p p

gyi Λk + hyi Λk + iyj Λk + jyj Λk + kyk Λk + lyk Λk +

jk kj ik ki ij ji

mYip Λk + nYip Λk + pYjp Λk + qYjp Λk + rYk Λk + sYk Λk .

p p

jk kj ik ki ij ji

The last step consists in expressing the equivariance of (10) with respect to the

subgroup of G2 characterized by ai = δj , ap = δq . This leads to certain simple

i p

m,n q

j

algebraic identities, which are equivalent to (5).

32.7. Remark. The only map in 32.6.(5) independent of Λ is the identity.

This proves proposition 32.3.

If we consider a linear symmetric connection Λ, then the whole family N (Λ)

vanishes identically. This implies

Corollary. The only two natural transformations J 1 J 1 Y ’ J 1 J 1 Y depending

on a linear symmetric connection Λ on the base manifold are the identity and

eΛ .

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

33. Topics from Riemannian geometry 265

¯

32.8. Remark. The functors J 2 and J 1 J 1 restricted to the category Mfm —

Mf ‚ FMm de¬ne the so called semiholonomic and non-holonomic 2-jets in the

sense of [Ehresmann, 54]. We remark that all natural transformations of each of

those restricted functors into itself are determined in [Kol´ˇ, Vosmansk´, 87].

ar a

Further we remark that [Kurek, to appear b] described all natural transfor-

mations T r— ’ T s— between any two one-dimensional covelocities functors from

12.8. He also determined all natural tensors of type 1 on T r— M , [Kurek, to

1

appear c].

33. Topics from Riemannian geometry

33.1. Our aim is to outline the application of our general procedures to the

study of geometric operations on Riemannian manifolds. Since the Riemannian

metrics are sections of a natural bundle (a subbundle in S 2 T — ), we can always

add the metrics to the arguments of the operation in question instead of spe-

cializing our general approach to categories over manifolds for the category of

Riemannian manifolds and local isometries. In this way, we reduce the problem

to the study of some equivariant maps between the standard ¬bers, in spite of

the fact that the Riemannian manifolds are not locally homogeneous in the sense

of 18.4. However, at some stage we mostly have to ¬x the values of the metric

entry by restricting ourselves to the invariance with respect to the isometries and

so we need description of all tensors invariant under the action of the orthogonal

group.

Let us write S+ T — for the natural bundle of elements of Riemannian metrics.

2

33.2. O(m)-invariant tensors. An O(m)-invariant tensor is a tensor B ∈

—p Rm — —q Rm— satisfying aB = B for all a ∈ O(m). The canonical scalar

product on Rm de¬nes an O(m)-equivariant isomorphism Rm ∼ Rm— . This

=

p+q m—

identi¬es B with an element from — R , i.e. with an O(m)-invariant linear

map —p+q Rm ’ R. Let us de¬ne a linear map •σ : —2s Rm ’ R, by

•σ (v1 — · · · — v2s ) = (vσ(1) , vσ(2) ).(vσ(3) , vσ(4) ) · · · (vσ(2k’1) , vσ(2s) ),

where ( , ) means the canonical scalar product de¬ned on Rm and σ ∈ Σ2s

is a permutation. The maps •σ are called the elementary invariants. The

fundamental result due to [Weyl, 46] is

Theorem. The linear space of all O(m)-invariant linear maps —k Rm ’ R is

spanned by the elementary invariants for k = 2s and is the zero space if k is

odd.

Proof. We present a proof based on the Invariant tensor theorem (see 24.4),

following the lines of [Atiyah, Bott, Patodi, 73]. The idea is to involve explicitly

all metrics gij ∈ S+ Rm— and then to look for GL(m)-invariant maps. So together

2

with an O(m)-invariant map • : —k Rm ’ R we consider the map • : S+ Rm— —

¯2

—k Rm ’ R, de¬ned by •(Im , x) = •(x) and •(G, x) = •((A’1 )T GA’1 , Ax)

¯ ¯ ¯

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

266 Chapter VII. Further applications

for all A ∈ GL(m), G ∈ S+ Rm— . By de¬nition, • is GL(m)-invariant. With

2

¯

the help of the next lemma, we shall be able to extend the map • to the whole

¯

2 m— km

S R —— R .

Let us write V = —k Rm . The map • induces a map GL(m) — V ’ R,

¯

T

(A, x) ’ •(A A, x) = •(Ax) and this map is extended by the same formula to

¯

a polynomial map f : gl(m) — V ’ R, linear in V . So fx (A) = f (A, x) = •(Ax)

is polynomial and O(m)-invariant for all x ∈ V , and f (A, x) = •(AT A, x) if A

¯

invertible.

Lemma. Let h : gl(m) ’ R be a polynomial map such that h(BA) = h(A)

for all B ∈ O(m). Then there is a polynomial F on the space of all symmetric