Natural transformations between T

POLONICI MATHEMATICI LVI.1 (1991)

Natural transformations between T

T

M and T

T

M by Miroslav Doupovec (Brno)

Abstract. We determine all natural transformations T₁²T^∗→ T^∗T₁²where T_k^rM = J₀^r(R^k, M ). We also give a geometric characterization of the canonical isomorphism ψ2

defined by Cantrijn et al. [1] among such natural transformations.

The spaces T

M of one-dimensional velocities of order r are used in the geometric approach to higher-order mechanics. That is why several authors studied the relations between T

T

M and T

T

M . For example, Modugno and Stefani [7] introduced an intrinsic isomorphism s between the bundles T T

M and T

T M . Recently Cantrijn, Crampin, Sarlet and Saunders [1] constructed a canonical isomorphism ψ

: T

T

M → T

T

M , which coincides with s for r = 1. From the categorical point of view, ψ

is a natural equivalence between the functors T

T

and T

T

, defined on the category Mf

of m-dimensional manifolds and their local diffeomorphisms.

Starting from the isomorphism s, Kol´ aˇ r and Radziszewski [5] determined all natural transformations of T T

into T

T . In the present paper we determine all natural transformations T

T

→ T

T

and interpret them geometrically.

Further we show that the natural equivalence ψ

can be distinguished among all natural transformations by a simple geometric construction.

1. The equations of all natural transformations T

T

→ T

T

. We shall use the concept of a natural bundle in the sense of Nijenhuis [8].

Denote by Mf

the category of m-dimensional manifolds and their local dif-

feomorphisms, by F M the category of fibred manifolds and by B : F M →

Mf

the base functor. A natural bundle over m-manifolds is a covariant

functor F : Mf

→ F M satisfying B ◦ F = id and the localization condi-

tion: for every inclusion of an open subset i : U → M , F U is the restriction

to p

(U ) of p

: F M → M over U and F i is the inclusion p

(U ) → F M .

If we replace the category Mf

by the category Mf of all manifolds and

all smooth maps, we obtain the concept of bundle functor on the category

of all manifolds. A natural bundle F : Mf

→ F M is said to be of order r

if, for any local diffeomorphisms f, g : M → N and any x ∈ M , the relation j

f (x) = j

g(x) implies F f |F

M = F g|F

M , where F

M denotes the fibre of F M over x ∈ M .

A k-dimensional velocity of order r on a smooth manifold M is an r- jet of R

into M with source 0. The space T

M = J

(R

, M ) of all such velocities is a fibred manifold over M . Every smooth map f : M → N extends to an F M-morphism T

f : T

M → T

N defined by T

f (j

g) = j

(f ◦ g). Hence T

: Mf → F M is an rth order bundle functor. The simplest example is the functor T

, which coincides with the tangent func- tor T .

The cotangent bundle T

M is a vector bundle over the manifold M . Having a local diffeomorphism f : M → N , we define T

f : T

M → T

N by taking pointwise the inverse map to the dual map (T

f )

: T

N → T

M , x ∈ M . In this way the cotangent functor T

is a natural bundle over m-manifolds.

We are going to determine all natural transformations T

T

→ T

T

. The canonical coordinates x

on R

induce the additional coordinates p

on T

R

and ξ

= dx

/dt, X

= d

x

/dt

, π

= dp

/dt, P

= d

p

/dt

on T

T

R

. Further, if y

= dx

/dt, z

= d

x

/dt

are the induced coordinates on T

R

, then the expression σ

dx

+%

dy

+τ

dz

determines the additional coordinates σ

, %

, τ

on T

T

R

. Set

(1) I = p

ξ

, J = p

X

+ π

ξ

.

Let G

be the group of all invertible r-jets of R

into R

with source and target 0.

Proposition 1. All natural transformations T

T

→ T

T

are of the form

(2)

y

= F (I, J )ξ

,

z

= F

(I, J )X

+ H(I, J )ξ

, τ

= G(I, J )p

,

%

= 2F (I, J )G(I, J )π

+ M (I, J )p

,

σ

= F

(I, J )G(I, J )P

+ [F (I, J )M (I, J ) + H(I, J )G(I, J )]π

+ N (I, J )p

where F , G, H, M , N are arbitrary smooth functions of two variables and I, J are given by (1).

In the proof of Proposition 1 we shall need the following result, which

comes from the book [6]. Let V denote the vector space R

with the stan-

dard action of the group G

and let V

= V × . . . × V

| {z }

k times

× V

× . . . × V

| {z }

l times

. Let h , i : V × V

→ R be the evaluation map hx, yi = y(x).

Lemma. (a) All G

-equivariant maps V

→ V are of the form

k

X

β=1

g

(hx

, y

i)x

with any smooth functions g

: R

→ R.

(b) All G

-equivariant maps V

→ V

are of the form

l

X

µ=1

g

(hx

, y

i)y

with any smooth functions g

: R

→ R.

P r o o f o f P r o p o s i t i o n 1. According to the general theory [3], if F and G are two rth order natural bundles, then the natural transforma- tions F → G are in a canonical bijection with the G

-equivariant maps F

R

→ G

R

. Hence we have to determine all G

-equivariant maps of S = (T

T

R

)

into Z = (T

T

R

)

. Using standard evaluations we find that the action of G

on S is

(3)

ξ

= a

ξ

, X

= a

ξ

ξ

+ a

X

, p

= e a

p

, π

= e a

π

− a

e a

e a

p

ξ

, P

= e a

P

− 2a

e a

e a

π

ξ

− a

e a

e a

ξ

ξ

p

− a

e a

e a

p

X

+ 2 e a

a

e a

a

e a

ξ

ξ

p

where a

, a

, a

are the canonical coordinates on G

and e a

is the inverse matrix of a

. Taking into account the natural equivalence ψ

: T

T

M → T

T

M of Cantrijn et al. with equations

(4) y

= ξ

, z

= X

, τ

= p

, %

= 2π

, σ

= P

,

we obtain from (3) the action of G

on Z. The coordinate form of any map S → Z is

y

= f

(p, ξ, X, π, P ), z

= g

(p, ξ, X, π, P ), σ

= h

(p, ξ, X, π, P ) ,

%

= l

(p, ξ, X, π, P ), τ

= t

(p, ξ, X, π, P ) .

First we discuss f

. The equivariance of f

with respect to the kernel of the jet projection G

→ G

leads to

f

(p

, ξ

, X

, π

, P

) = f

(p

, ξ

, X

, π

, P

− a

ξ

ξ

p

) .

This implies that f

is independent of P

. Now it will be useful to distinguish two cases according to the dimension m of the manifold M .

Consider first the case m ≥ 2. Taking into account the equivariance of f

with respect to the linear group G

⊂ G

we obtain

a

f

(p

, ξ

, X

, π

) = f

( e a

p

, a

ξ

, a

X

, e a

π

) ,

so that f

(p, ξ, X, π) is a G

-equivariant map R

×R

×R

×R

→ R

. By our Lemma,

f

(p, ξ, π, X) = ϕ(p

ξ

, p

X

, π

ξ

, π

X

)ξ

(5)

+ ψ(p

ξ

, p

X

, π

ξ

, π

X

)X

where ϕ and ψ are arbitrary two smooth functions of four variables. One calculates easily that the expressions I and J given by (1) are invariants with respect to the group G

. Replace (5) by

f

= ϕ(I, J, p

X

− π

ξ

, π

X

)ξ

+ ψ(I, J, p

X

− π

ξ

, π

X

)X

. Then the equivariance of f

with respect to the kernel of the jet projection G

→ G

reads

ϕ(I, J, p

X

− π

ξ

, π

X

)ξ

+ ψ(I, J, p

X

− π

ξ

, π

X

)X

(6)

= ϕ(I, J, p

X

− π

ξ

, π

X

)ξ

+ ψ(I, J, p

X

− π

ξ

, π

X

)X

where X

= X

+ a

ξ

ξ

and π

= π

− a

p

ξ

. Setting ξ = (1, 0, . . . , 0), X = (0, 1, 0, . . . , 0) and i = 1 in (6) we obtain

(7) ϕ(p

, p

+ π

, p

− π

, π

)

= ϕ(p

, p

+ π

, p

− π

+ 2a

p

, π

− a

p

+ π

a

− a

a

p

)

+ ψ(p

, p

+ π

, p

− π

+ 2a

p

, π

− a

p

+ π

a

− a

a

p

)a

. If all a

except a

and a

are zero, then (7) reads

(8) ϕ(p

, p

+ π

, p

− π

, π

)

= ϕ(p

, p

+ π

, p

− π

+ 2a

p

, π

− a

p

+ π

a

− a

a

p

) . Putting a

= 0 we get

ϕ(p

, p

+ π

, p

− π

, π

) = ϕ(p

, p

+ π

, p

− π

, π

− a

p

) . This implies that ϕ does not depend on the fourth variable . Then (8) with arbitrary a

gives ϕ = ϕ(I, J ).

Further, let a

= 1 and let the other a’s in (7) be zero. Then (9) 0 = ψ(p

, p

+ π

, p

− π

+ 2p

, π

+ π

− p

) .

The components of ψ in (9) are linearly independent functions, so that ψ = 0. We have thus deduced that

(10) f

= F (I, J )ξ

with an arbitrary smooth function F : R

→ R.

Quite analogously one can prove that

(11) t

= G(I, J )p

where G is another smooth function of two variables.

Now write

g

(p, ξ, X, π, P ) = F

(I, J )X

+ g

(p, ξ, X, π, P )

with F taken from (10). Applying the equivariance of g

with respect to the whole group G

we find

a

F

(I, J )ξ

ξ

+ a

F

(I, J )X

+ a

g

(p, ξ, X, π, P )

= F

(I, J )(a

ξ

ξ

+ a

X

) + g

(p, ξ, X, π, P ) . We see that g

has the same transformation law as f

, so that g

(p, ξ, X, π, P )

= H(I, J )ξ

and

(12) g

= F

(I, J )X

+ H(I, J )ξ

. Consider now the map l

and set

l

(p, ξ, X, π, P ) = 2F (I, J )G(I, J )π

+ l

(p, ξ, X, π, P ) . Using equivariance we get

2 e a

F (I, J )G(I, J )π

+ e a

l

(p, ξ, X, π, P ) − 2a

e a

e a

F (I, J )G(I, J )p

ξ

= 2F (I, J )G(I, J )( e a

π

− a

e a

e a

p

ξ

) + l

(p, ξ, X, π, P ) . Quite similarly to (10) and (11) we then deduce l

(p, ξ, X, π, P ) = M (I, J )p

, so that

(13) l

= 2F (I, J )G(I, J )π

+ M (I, J )p

. Finally, assume h

has the form

h

(p, ξ, X, π, P ) = F

(I, J )G(I, J )P

+ [F (I, J )M (I, J ) + H(I, J )G(I, J )]π

+ h

(p, ξ, X, π, P ) .

Applying the same procedure as for g

and l

we obtain h

(p, ξ, X, π, P ) = N (I, J )p

, i.e.

h

= F

(I, J )G(I, J )P

+ [F (I, J )M (I, J ) + H(I, J )G(I, J )]π

(14)

+ N (I, J )p

.

Thus, if the dimension m of the manifold M is ≥ 2, then (10)–(14) prove our proposition.

It remains to discuss the case of one-dimensional manifolds. The fact

that the map f (p, ξ, X, π, P ) does not depend on P can be derived in the

same way as above. Denote by (a

, a

, a

) the coordinates on G

. We shall only need the following equations of the action of G

on S and Z:

ξ = a

ξ, p = 1 a

p, X = a

ξ

+ a

X, y = a

y , π = 1

a

π − a

a

pξ . Take any u ∈ R

, so that u =

1

u. Then f (p, ξ, X, π)u is a G

-invariant function. Let I = pξ, J = pX + πξ and K = uξ.

For any G

-invariant function F (p, ξ, π, X, u) define a smooth function ψ(x, y, z) = F (x, 1, y, 0, z). We claim that

(15) F (p, ξ, π, X, u) = ψ(I, J, K) .

Indeed, since F (p, ξ, π, X, u) is G

-invariant, in the case ξ 6= 0 for a

= ξ, a

= 0 we have

ψ(ξp, ξπ, ξu) = F (ξp, 1, ξπ, 0, ξu) = F (p, ξ, π, 0, u) .

Further, set a

= −X/ξ

, a

= 1. Then by invariance F (p, ξ, π, X, u) = F (p, ξ, π + Xp/ξ, 0, u) = ψ(ξp, ξπ + Xp, ξu) = ψ(I, J, K). Hence we have proved that (15) holds on the dense subset ξ 6= 0, so by continuity it holds everywhere.

Now we complete the proof of our proposition. By (15) we have f (p, ξ, π, X)u = ψ(I, J, K) .

Differentiating this with respect to u we obtain f (p, ξ, π, X) = ∂ψ(I, J, K)

∂z · ξ

where z denotes the third variable of ψ(I, J, K). Setting u = 0 on the right side we get

f (p, ξ, π, X) = ϕ(I, J ) · ξ

where ϕ(x, y) = ∂ψ(x, y, 0)/∂z. This implies that for m = 1 the map f is of the form (10) as well. One finds easily that (11)–(14) are also true in this case.

2. Geometric interpretation. The canonical isomorphism ψ

:

T

T

M → T

T

M of Cantrijn et al. [1] corresponds to the constant values

F = 1, G = 1, H = 0, M = 0, N = 0 in (2). We first give another simple

geometric construction of this isomorphism. Gollek introduced a canonical

isomorphism κ : T

T

M → T

T

M which can be viewed as a generalization

of the canonical involution T T M → T T M [2]. Let q : T

M → M be the

bundle projection and let κ

be the above isomorphism T T

M → T

T M .

The map κ

has a simple geometric interpretation. Every C ∈ T T

M

is of the form C = (∂/∂t)|

j

γ(t, τ ), where γ is the map R × R → M , (t, τ ) 7→ γ(t, τ ), and j

means the partial jet with respect to the second variable. Then κ

(C) ∈ T

T M is defined by taking the partial jets in oppo- site order, i.e. κ

(C) = j

((∂/∂t)|

γ(t, τ )). Every A ∈ T

T

M is a 2-velocity of a curve α(t) = (x

(t), a

(t)) in T

M . Let v ∈ T

M be the point T

q(A).

If B ∈ T

T

M , then κ

(B) is a 2-velocity of a curve β(t) = (x

(t), b

(t)) in T M . Hence we can evaluate hα(t), β(t)i for every t and the expres- sion

d

dt

hα(t), β(t)i

depends only on A and B. Therefore it determines a linear map T

T

M → R, i.e. an element of T

T

M .

Now we present a geometric interpretation of the result (2). We shall proceed in four steps.

1. We can define the following multiplication by real numbers on the bundle T

N :

(16) k · (x

, y

, z

) = (x

, ky

, k

z

) .

There is a canonical inclusion T

N → T T N , (x

, y

, z

) 7→ (x

, y

, y

, z

), and the space T T N carries two vector bundle structures. Taking any (x

, y

, y

, z

) ∈ T T N , we can multiply it by k with respect to the first structure. We obtain

(17) (x

, y

, ky

, kz

) .

Further, multiplying (17) by k with respect to the second structure gives (x

, ky

, ky

, k

z

). This defines the multiplication (16), which we denote by A 7→ k · A. Another way of defining the multiplication (16) on T

N is to use the reparametrization x

(t) 7→ x

(kt).

Take any element A = (x

, p

, ξ

, X

, π

, P

) in T

T

M . Evaluating the result of multiplication of A by F , we get F · A = (x

, p

, F ξ

, F

X

, F π

, F

P

). Next, we can transform this into T

T

M by means of the canonical transformation ψ

. The coordinates of ψ

(F · A) are

y

= F ξ

, z

= F

X

, τ

= p

, %

= 2F π

, σ

= F

P

. Moreover, multiplying ψ

(F · A) by G with respect to the vector bundle structure of T

T

M we obtain an element

(18) Gψ

(F · A)

of T

T

M with coordinates

y

= F ξ

, z

= F

X

, τ

= Gp

, %

= 2F Gπ

, σ

= F

GP

.

2. The bundle projection q : T

M → M determines the projection r

: T

T

M → T

M , r

= T

q. Further, let r

: T

T

M → T T

M be the jet projection and let r

: T

T

M → T

M be the bundle projection.

Denote by s the isomorphism T T

M → T

T M of Modugno and Ste- fani [7]. We recall the coordinate expression of s. Having the canonical coordinates x

, ζ

= dx

on T R

, the expression α

dx

+ β

dζ

determines the additional coordinates α

, β

on T

T R

. Further, let x

, p

, ξ

= dx

, π

= dp

be the canonical coordinates on T T

R

. Then the equations of the isomorphism s are [5]

ζ

= ξ

, α

= π

, β

= p

.

Moreover, there is an inclusion i : T

M ×

T

T M → T

T

M ,

(x

, y

, z

, α

, β

) 7→ (x

= x

, y

= y

, z

= z

, σ

= α

, %

= β

, τ

= 0) . Having an arbitrary element A = (x

, p

, ξ

, X

, π

, P

) in T

T

M , we can evaluate i(r

F · A, s(r

F · A)). Next, multiplying this by the function M on the vector bundle T

T

M we obtain an element

(19) M i(r

F · A, s(r

F · A)) of T

T

M with coordinates

y

= F ξ

, z

= F

X

, τ

= 0 , %

= M p

, σ

= F M π

. 3. Denote by j the inclusion T

M ×

T

M → T

T

M ,

(x

, y

, z

, α

) 7→ (x

= x

, y

= y

, z

= z

, σ

= α

, %

= 0, τ

= 0) . Applying a similar procedure to step 2, we associate to any A ∈ T

T

M an element

(20) N j(r

F · A, r

F · A) of T

T

M . The coordinate form of (20) is

y

= F ξ

, z

= F

X

, τ

= 0 , %

= 0 , σ

= N p

.

4. It is well known that T

M → T M is an affine bundle associated to the pullback p

T M of T M → M over p

: T M → M . In particular, T

T

M → T T

M is an affine bundle whose associated vector bundle is the pullback of T T

M → T

M over p

: T T

M → T

M . Hence we have defined the addition of vectors in T T

M to points in T

T

M :

(x

, ξ

, X

, p

, π

, P

) + (x

, p

, v

, u

) = (x

, ξ

, X

+ v

, p

, π

, P

+ u

) . Using the canonical isomorphism ψ

: T

T

M → T

T

M we can transform this addition in the affine bundle T

T

M to an addition ⊕ in the bundle T

T

M :

(x

, y

, z

, τ

, %

, σ

) ⊕ (x

, v

, τ

, u

) = (x

, y

, z

+ v

, τ

, %

, σ

+ u

) .

Now we can complete the geometric interpretation of (2). Given an arbi- trary A = (x

, p

, ξ

, X

, π

, P

) ∈ T

T

M we have constructed geometrically three elements (18), (19) and (20) in T

T

M . Then their sum

B = Gψ

(F · A) + M i(r

F · A, s(r

F · A)) + N j(r

F · A, r

F · A) with respect to the vector bundle structure of T

T

M has coordinates

x

= x

, y

= F ξ

, z

= F

X

, τ

= Gp

,

%

= 2F Gπ

+ M p

, σ

= F

GP

+ F M π

+ N p

.

Taking further the vector (x

, ξ

, p

, π

) ∈ T T

M and multiplying by G in the vector bundle structure T T

M → T M we get (x

, ξ

, Gp

, Gπ

). Moreover, multiplying this by H in T T

M → T

M we obtain C = (x

, Hξ

, Gp

, HGπ

). Finally, the sum B ⊕ C gives (x

, F ξ

, F

X

+ Hξ

, Gp

, 2F Gπ

+ M p

, F

Gp

+ F M π

+ N p

+ HGπ

). This corresponds to (2).

3. A geometric characterization of the isomorphism ψ

. The natural equivalence s : T T

M → T

T M of Modugno and Stefani can be distinguished among all natural transformations by an explicit geometric construction [5]. We show that a similar result is true for the natural equiv- alence ψ

: T

T

M → T

T

M of Cantrijn et al.

Every vector field ξ on the manifold M induces the flow prolongation T

ξ = ∂

∂t    

(T

exp tξ)

on T

M . Further, if ω : M → T

M is any 1-form on M , then hω, ξi : M → R and we can construct T

hω, ξi : T

M → T

R. Let δ

hω, ξi or δ

hω, ξi be the second and third component of the map T

hω, ξi, respectively. We have T

ω : T

M → T

T

M , so that ψ

T

ω : T

M → T

T

M is a 1-form on T

M . Hence we can evaluate hψ

T

ω, T

ξi : T

M → R.

Proposition 2. ψ

is the only natural transformation T

T

→ T

T

over the identity transformation of T

satisfying

(21) hψ

T

ω, T

ξi = δ

hω, ξi for every vector field ξ and every 1-form ω.

P r o o f. Let x

= x

, p

= a

(x) be the coordinate expression of ω. Then the coordinate expression of T

ω is

x

= x

, p

= a

(x) , ξ

= ξ

, X

= X

, π

= ∂a

∂x

ξ

, P

= ∂

a

∂x

∂x

ξ

ξ

+ ∂a

∂x

X

.

Applying transformation (2) with F = 1, H = 0 we get x

= x

, y

= ξ

, z

= X

, τ

= Ga

, %

= 2G ∂a

∂x

ξ

+ M a

, σ

= G ∂

a

∂x

∂x

ξ

ξ

+ G ∂a

∂x

X

+ M ∂a

∂x

ξ

+ N a

.

The fact that F = 1, H = 0 follows from the assumption that our natural transformation is over the identity of T

. Further, the coordinate expression of the flow prolongation T

ξ is

dx

= b

(x) , dy

= ∂b

∂x

ξ

, dz

= ∂

b

∂x

∂x

ξ

ξ

+ ∂b

∂x

X

, provided the b

(x) are the coordinates of a vector field ξ. We can write

δ

hω, ξi =  ∂a

∂x

b

+ a

∂b

∂x

 ξ

. Hence (21) reads

G

 ∂

a

∂x

∂x

ξ

ξ

+ ∂a

∂x

X



b

+ M ∂a

∂x

ξ

b

+ N a

b

+ M a

∂b

∂x

ξ

+ 2G ∂a

∂x

ξ

∂b

∂x

ξ

+ Ga

∂

b

∂x

∂x

ξ

ξ

+ Ga

∂b

∂x

X

= ∂

a

∂x

∂x

b

ξ

ξ

+ 2 ∂a

∂x

∂b

∂x

ξ

ξ

+ a

∂

b

∂x

∂x

ξ

ξ

+ ∂a

∂x

b

X

+ a

∂b

∂x

X

. This implies G = 1, M = 0, N = 0.

Acknowledgement. The author thanks Prof. I. Kol´ aˇ r for his help during the work on this paper.

References

[1] F. C a n t r i j n, M. C r a m p i n, W. S a r l e t and D. S a u n d e r s, The canonical isomor- phism between T^kT^∗M and T^∗T^kM , C. R. Acad. Sci. Paris 309 (1989), 1509–1514.

[2] H. G o l l e k, Anwendungen der Jet-Theorie auf Faserb¨undel und Liesche Transforma- tionsgruppen, Math. Nachr. 53 (1972), 161–180.

[3] J. J a n yˇs k a, Geometrical properties of prolongation functors, ˇCas. Pˇest. Mat. 110 (1985), 77–86.

[4] P. K o b a k, Natural liftings of vector fields to tangent bundles of 1-forms, ibid., to appear.

[5] I. K o l ´aˇr and Z. R a d z i s z e w s k i, Natural transformations of second tangent and cotangent functors, Czechoslovak Math. J. 38 (113) (1988), 274–279.

[6] I. K o l ´aˇr, P. W. M i c h o r and J. S l o v ´a k, Natural Operations in Differential Geo- metry, to appear.

[7] M. M o d u g n o and G. S t e f a n i, Some results on second tangent and cotangent spaces, Quaderni dell’Instituto di Matematica dell’Universit`a di Lecce, Q. 16, 1978.

[8] A. N i j e n h u i s, Natural bundles and their general properties, in: Differential Geo- metry in honor of Yano, Kinokuniya, Tokyo 1972, 317–334.

DEPARTMENT OF MATHEMATICS TECHNICAL UNIVERSITY OF BRNO TECHNICK ´A 2

616 69 BRNO CZECHOSLOVAKIA

Re¸cu par la R´edaction le 28.5.1990 R´evis´e le 1.8.1990