bundle T — E ’ E there is another vector bundle structure ρ : T — E ’ E — de¬ned

by the restriction of a linear map Ty E ’ R to the vertical tangent space, which

is identi¬ed with Ep(y) . This enables us to introduce a sum Y Z for every

Y ∈ Ty T M and Z ∈ Tπ(y) M as follows. We have (ρ(Y ), Z) ∈ T M —M T — M =

— — —

V T — M ’ T T — M and we can apply sM : T T — M ’ T — T M . Then Y Z is

de¬ned as the sum Y + sM (ρ(Y ), Z) with respect to the vector bundle structure

ρ.

26.12. For every X ∈ T T — M we write p ∈ T — M for its point of contact and

ξ = T q(X) ∈ T M . Taking into account both vector bundle structures on T — T M ,

we denote by Y ’ (k)1 Y or Y ’ (k)2 Y , k ∈ R, the scalar multiplication with

respect to the ¬rst or second one, respectively.

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

228 Chapter VI. Methods for ¬nding natural operators

Proposition. All natural transformations T T — ’ T — T are of the form

(1) F ( p, ξ ) G( p, ξ ) 2 sM (X) H( p, ξ )p

1

where F (t), G(t), H(t) are three arbitrary smooth functions of one variable.

Proof. Since T T — and T — T are second order bundle functors on Mfm , we have

to determine all G2 -equivariant maps of S := T T0 Rm into Z := T — T0 Rm . The

—

m

canonical coordinates xi on Rm induce the additional coordinates pi on T — Rm

and ξ i = dxi , πi = dpi on T T — Rm . If we evaluate the e¬ect of a di¬eomorphism

on Rm and pass to 2-jets, we ¬nd easily that the equations of the action of G2 m

on S are

pi = aj pj , πi = aj πj ’ al am aj pm ξ k .

¯

ξ i = ai ξ j ,

(2) ¯ ˜i ¯ ˜i jk ˜l ˜i

j

Further, if · i are the induced coordinates on T Rm , then the expression ρi dxi +

σi d· i determines the additional coordinates ρi , σi on T — T Rm . Similarly to (2)

we obtain the following action of G2 on Z

m

σi = aj σj , ρi = aj ρj ’ al am aj σm · k .

· i = ai · j ,

(3) ¯ ¯ ˜i ¯ ˜i jk ˜l ˜i

j

Any map • : S ’ Z has the form

· i = f i (p, ξ, π), σi = gi (p, ξ, π), ρi = hi (p, ξ, π).

The equivariance of f i is expressed by

ai f j (p, ξ, π) = f i (˜j pj , ai ξ j , aj πj ’ al am aj pm ξ k ).

(4) ai ˜i jk ˜l ˜i

j j

Setting ai = δj , we obtain f i (p, ξ, π) = f i (p, ξ, πj ’ al pl ξ k ). This implies that

i

j jk

i i

the f are independent of πj . Then (4) shows that f (p, ξ) is a GL(m)-equivariant

map Rm — Rm— ’ Rm . By proposition 26.9,

f i = F ( p, ξ )ξ i

(5)

where F is an arbitrary smooth function of one variable. Using the same pro-

cedure we obtain that the gi are independent of πj . Then proposition 26.10

yields

(6) gi = G( p, ξ )pi

where G is another smooth function of one variable.

Consider further the di¬erence ki = hi ’ F ( p, ξ )G( p, ξ )πi . Using the fact

that p, ξ is invariant, we express the equivariance of ki in the form

aj kj (p, ξ, π) = ki (˜j pj , ai ξ j , aj πj ’ al am aj pm ξ k ).

˜i ai ˜i jk ˜l ˜i

j

Quite similarly to (4) and (6) we then deduce ki = H( p, ξ )pi , i.e.

(7) hi = F ( p, ξ )G( p, ξ )πi + H( p, ξ )pi .

One veri¬es easily that (5), (6) and (7) is the coordinate form of (1).

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

26. The tensor evaluation theorem 229

26.13. To interpret all natural transformations of proposition 26.12 geometri-

cally, we ¬rst show that for any constant values F = f , G = g, H = h, 26.12.(1)

can be determined by a simple modi¬cation of the above mentioned construction

of s (s corresponds to the case f = 1, g = 1, h = 0). If A ∈ T T — M is tangent

to a curve γ(t), then f A is tangent to γ(f t). For every vector B ∈ Tf T q(A) T M ,

κB is tangent to a curve δ(t) : R ’ T M over the curve q(γ(f t)) on M . Then

we de¬ne an element s(f,g,h) A ∈ T — T M by

‚

(1) s(f,g,h) A, B = γ(f t), gδ(t) + h γ(0), δ(0) .

‚t 0

The coordinate expression of (1) is (f gπi +hpi )dxi +gpi d· i and our construction

implies · i = f ξ i . This gives 26.12.(1) with constant coe¬cients. Moreover, the

general case can also be interpreted in such a way. Let π : T T — M ’ T — M

be the bundle projection. Every A ∈ T T — M determines T q(A) ∈ T M and

π(A) ∈ T — M over the same base point in M . Then we take the values of F , G

and H at π(A), T q(A) and apply the latter construction.

We remark that the natural transformation s by Modugno and Stefani can be

distinguished among all natural transformations T T — ’ T — T by an interesting

geometric construction explained in [Kol´ˇ, Radziszewski, 88].

ar

26.14. The functor T — T — . The iterated cotangent functor T — T — is also a

second order bundle functor on Mfm . The problem of ¬nding of all natural

transformations between any two of the functors T T — , T — T and T — T — can be

reduced to proposition 26.12, if we take into account a classical geometrical con-

struction of a natural equivalence between T T — and T — T — . Consider the Liouville

1-form ω : T T — M ’ R de¬ned by ω(A) = π(A), T q(A) . The exterior di¬er-

ential dω = „¦ endows T — M with a natural symplectic structure. This de¬nes

a bijection between the tangent and cotangent bundles of T — M transforming

X ∈ T T — M into its inner product with „¦. Hence the natural transformations

between any two of the functors T T — , T — T and T — T — depend on three arbitrary

smooth functions of one variable. Their coordinate expressions can be found in

[Kol´ˇ, Radziszewski, 88].

ar

26.15. Non-existence of natural symplectic structure on the tangent

bundles. We shall see in 37.4 that the natural transformations of the iterated

tangent functor into itself depend on four real parameters. This is related with

the fact that T T is de¬ned on the whole category Mf and is product preserving.

Since the natural transformations of T T into itself are essentially di¬erent from

the natural transformations of T — T into itself, there is no natural equivalence

between T T and T — T . This implies that there is no natural symplectic structure

on the tangent bundles.

26.16. Remark. Taking into account the natural isomorphism s : T T — ’ T — T

and the canonical symplectic structure on the cotangent bundles, one sees easily

that any two of the third order functors T T T — , T T — T , T T — T — , T — T T , T — T T — ,

T — T — T and T — T — T — are naturally equivalent, but T T T is naturally equivalent

to none of them. All natural transformations T T T — ’ T T — T for manifolds of

dimension at least two are determined in [Doupovec, to appear].

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

230 Chapter VI. Methods for ¬nding natural operators

27. Generalized invariant tensors

To study the natural operators on FMm,n , we need a modi¬cation of the

Invariant tensor theorem.

27.1. Consider two vector spaces V and W . The tensor product of the standard

actions of GL(V ) on —p V — —q V — and of GL(W ) on —r W — —s W — de¬nes the

standard action of GL(V ) — GL(W ) on —p V — —q V — — —r W — —s W — . A tensor

B of the latter space is said to be a generalized invariant tensor, if aB = B for

all a ∈ GL(V ) — GL(W ). The invariance of B with respect to the homotheties

in GL(V ) or GL(W ) gives k p’q B = B or k r’s B = B, respectively. This implies

that for p = q or r = s the only generalized invariant tensor is the zero tensor.

Generalized invariant tensor theorem. Every generalized invariant tensor

B ∈ —q V — —q V — — —r W — —r W — is a linear combination of the tensor products

I — J, where I is an elementary GL(V )-invariant tensor of degree q and J is an

elementary GL(W )-invariant tensor of degree r.

Proof. Contracting B with q vectors of V and q covectors of V — , we obtain a

GL(W )-invariant tensor. By the invariant tensor theorem 24.4 and by multilin-

earity, B is of the form

with Bs ∈ —q V — —q V — ,

Bs — J s

(1) B=

s∈Sr

where J s are the elementary GL(W )-invariant tensors of degree r. If we con-

struct the total contraction of (1) with one tensor J σ , σ ∈ Sr , we obtain Bσ’1 .

Hence every Bs is a GL(V )-invariant tensor. Using theorem 24.4 once again, we

prove our assertion.

27.2. Example. We determine all smooth equivariant maps W — V — — W —

W — — V — ’ W — V — — V — . Let fij (xq , ysl ) be the coordinate expression of such

p r

k

1i

a map. The equivariance of f with respect to the homotheties k δj in GL(V )

gives

k 2 fij (xq , ysl ) = fij (kxq , kysl ).

p p

r r

k k

By the homogeneous function theorem, we have to discuss the condition 2 =

d1 + d2 . There are three possibilities: a) d1 = 2, d2 = 0, b) d1 = 1, d2 = 1, c)

p

d1 = 0, d2 = 2. In each case f is a polynomial map. The homotheties kδq in

GL(W ) yield

kfij (xq , ysl ) = fij (kxq , ysl ).

p p

r r

k k

This condition is compatible with the case b) only, so that f is bilinear in xq k

r

and ysl . Its total polarization corresponds to a generalized invariant tensor in

—2 V — —2 V — — —2 W — — —2 W — . By theorem 27.1, the coordinate form of f is

fij = aδq δs δi δj + bδs δq δi δj + cδq δs δj δi + dδs δq δj δi xq yrl ,

p prkl prkl prkl prkl s

k

a, b, c, d ∈ R. Hence all smooth equivariant maps W — V — — W — W — — V — ’

W — V — — V — form the following 4-parameter family

axp yqj + bxq yqj + cxp yqi + dxq yqi .

q p q p

i i j j

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

27. Generalized invariant tensors 231

27.3. Curvature like operators. Consider a general connection “ : Y ’ J 1 Y

on an arbitrary ¬bered manifold Y ’ BY , where B : FM ’ Mf denotes

the base functor. In 17.1 we have deduced that the curvature of “ is a map

CY “ : Y ’ V Y — Λ2 T — BY . The geometrical de¬nition of curvature implies

that C is a natural operator between two bundle functors J 1 and V — Λ2 T — B

de¬ned on the category FMm,n . In the following assertion we may replace the

second exterior power by the second tensor power (so that the antisymmetry of

the curvature operator is a consequence of its naturality).

V — —2 T — B are the constant multi-

Proposition. All natural operators J 1

ples kC of the curvature operator, k ∈ R.

Proof. We shall proceed in three steps as in the proof of proposition 25.2.

Step I. We ¬rst determine the ¬rst order operators. The canonical coordinates

p

on the standard ¬ber S1 = J0 (J 1 (Rn+m ’ Rm ) ’ Rn+m ) of J 1 J 1 are yi ,

1

p p p p

yij = ‚yi /‚xj , yiq = ‚yi /‚y q . Evaluating the e¬ect of the isomorphisms in

FMm,n and passing to 2-jets, we obtain the following action of G2 on S1

m,n

yi = ap yj aj + ap aj

¯p q

(1) ˜i j ˜i

q

yiq = ap yjs as aj + ap yj as aj + ap ar aj

¯p r r

(2) ˜q ˜i rs ˜q ˜i rj ˜q ˜i

r

yij = ap ykl ak al + ap ykr ar ak + ap yk ak al + ap yk ar ak

¯p q q q q

(3) ˜i ˜j ˜j ˜i ˜i ˜j ˜j ˜i

q q qr

ql

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

q

˜ij kl ˜i ˜j kq ˜j ˜i k ˜ij

q

On the other hand, the standard ¬ber of V — —2 T — B is Rn — —2 Rm— with

p

canonical coordinates zij and the following action

¯p q

zij = ap zkl ak al

˜i ˜j

q

We have to determine all G2 - equivariant maps S1 ’ Rn — —2 Rm— . Let

m,n

p p q r t

zij = fij (yk , y s , ymn ) be the coordinate expression of such a map. Consider the

canonical injection of GL(m)—GL(n) into G2 de¬ned by 2-jets of the products

m,n

m n

of linear transformations of R and R . The equivariance with respect to the

homotheties in GL(m) gives a homogeneity condition

p q p q

k 2 fij (yk , y rs , ymn ) = fij (kyk , ky rs , k 2 ymn ).

t t

When applying the homogeneous function theorem, we have to discuss the equa-

tion 2 = d1 + d2 + 2d3 . Hence fij is a sum gij + hp where gij is a linear map

p p p

ij

of Rn — Rm— — Rm— into itself and hp is a polynomial map Rn — Rm— — Rn —

ij

Rn— — Rm— ’ Rn — Rm— — Rm— . Then we see directly that both gij and hp are

p

ij

p

GL(m) — GL(n)-equivariant. For hij we have deduced in example 27.2

hp = ayi yjq + byi yjq + cyj yiq + dyj yiq

pq qp pq qp

ij

p

while for gij a direct use of theorem 27.1 yields

p p p

gij = eyij + f yji .

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

232 Chapter VI. Methods for ¬nding natural operators

Moreover, the equivariance with respect to the subgroup K ‚ G2 character-

m,n

ized by ai = δj , ap = δq leads to the relations a = 0 = c, e = ’f = ’b = d.

i p

q

j

p p p qp qp

Hence fij = e(yij ’ yji ’ yi yqj + yj yqi ), which is the coordinate expression of

eC, e ∈ R.

V ——2 T — B.

Step II. Assume we have an r-th order natural operator A : J 1

It corresponds to a Gr+1 -equivariant map from the standard ¬ber Sr of J r J 1

m,n

p p

n m— m—

into R —R —R . Denote by yi±β the partial derivative of yi with respect to a

multi index ± in xi and β in y p . Any map f : Sr ’ Rn —Rm— —Rm— is of the form

p

f (yi±β ), ± + β ¤ r. Similarly to the ¬rst part of the proof, GL(m) — GL(n) can

be considered as a subgroup of Gr+1 . One veri¬es easily that the transformation

m,n

p

law of yi±β with respect to GL(m) — GL(n) is tensorial. Using the homotheties

p p

in GL(m), we obtain a homogeneity condition k 2 f (yi±β ) = f (k |±|+1 yi±β ). This

implies that f is a polynomial linear in the coordinates with |±| = 1 and bilinear

in the coordinates with |±| = 0. Using the homotheties in GL(n), we ¬nd

p p

kf (yi±β ) = f (k 1’|β| yi±β ). This yields that f is independent of all coordinates

with |±| + |β| > 1. Hence A is a ¬rst order operator.

V ——2 T — B

Step III. Using 23.7 we conclude that every natural operator J 1

has ¬nite order. This completes the proof.

27.4. Curvature-like operators on pairs of connections. The Fr¨licher- o

Nijenhuis bracket [“, ∆] =: κ(“, ∆) of two general connections “ and ∆ on Y is

a section Y ’ V Y — Λ2 T — BY , which may be called the mixed curvature of “

and ∆. Since the pair “, ∆ can be interpreted as a section Y ’ J 1 Y —Y J 1 Y ,

V — Λ2 T — B between two bundle functors

κ is a natural operator κ : J 1 • J 1

V — Λ2 T — B or C2 : J 1 • J 1 V — Λ2 T — B

on FMm,n . Let C1 : J 1 • J 1

denote the curvature operator of the ¬rst or the second connection, respectively.

The following assertion can be deduced in the same way as proposition 27.3, see

[Kol´ˇ, 87a].

ar

V — —2 T — B form the following

Proposition. All natural operators J 1 • J 1

3-parameter family

k1 , k2 , k3 ∈ R.

k1 C1 + k2 C2 + k3 κ,

From a general point of view, this result enlightens us on the fact that the

mixed curvature of two general connections can be de¬ned in an ˜essentially

unique™ way, i.e. the possibility of de¬ning the mixed curvature is limited by the

above 3-parameter family with trivial terms C1 and C2 .

27.5. Remark. [Kurek, 91] deduced that the only natural operator J 1

V — Λ3 T — B is the zero operator. This result presents an interesting point of

view to the Bianchi identity for general connections.

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

28. The orbit reduction 233

28. The orbit reduction

We are going to explain another general procedure used in the theory of nat-

ural operators. From the computational point of view, the orbit reduction is an

almost self-evident assertion about independence of the maps in question on some

variables. This was already used e.g. for the simpli¬cation of (4) in 26.12. But

the explicit formulation of such a procedure presented below is useful in several

problems. First we discuss a concrete example, in which we obtain a Utiyama-

like theorem for general connections. Then we present a complete treatment of

the ˜classical™ reduction theorems from the theory of linear connections and from

Riemannian geometry.

28.1. Let p : G ’ H be a Lie group homomorphism with kernel K, M be a G-

space, Q be an H-space and π : M ’ Q be a p-equivariant surjective submersion,

i.e. π(gx) = p(g)π(x) for all x ∈ M , g ∈ G. Having p, we can consider every

H-space N as a G-space by gy = p(g)y, g ∈ G, y ∈ N .

Proposition. If each π ’1 (q), q ∈ Q is a K-orbit in M , then there is a bijection

between the G-maps f : M ’ N and the H-maps • : Q ’ N given by f = • —¦ π.

Proof. Clearly, • —¦ π is a G-map M ’ N for every H-map • : Q ’ N . Con-

versely, let f : M ’ N be a G-map. Then we de¬ne • : Q ’ N by •(π(x)) =

f (x). This is a correct de¬nition, since π(¯) = π(x) implies x = kx with k ∈ K

x ¯

by the orbit condition, so that •(π(¯)) = f (kx) = p(k)f (x) = ef (x). We have

x

f = • —¦ π by de¬nition and • is smooth, since π is a surjective submersion.

28.2. Example. We continue in our study of the standard ¬ber

S1 = J0 (J 1 (Rm+n ’ Rm ) ’ Rm+n )

1

corresponding to the ¬rst order operators on general connections from 27.3. If

p

we replace the coordinates yij by

p p pq

(1) Yij = yij + yiq yj ,

we ¬nd easily that the action of G2 on S1 is given by 27.3.(1), 27.3.(2) and

m,n

Yij = ap Ykl ak al + ap yk yl ak al + ap yk al ak + ap yk ak al

¯p q q q

rs

˜i ˜j ˜i ˜j ˜i ˜j ˜i ˜j

q rs ql ql

(2)

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

q

˜ij kl ˜i ˜j k ˜ij

q