Abstract. We determine all natural transformations of the rth order cotangent bundle functor T

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POLONICI MATHEMATICI LVIII.1 (1993)

Natural transformations of higher order cotangent bundle functors

by Jan Kurek (Lublin)

Abstract. We determine all natural transformations of the rth order cotangent bundle functor T

^r∗

into T

^s∗

in the following cases: r = s, r < s, r > s. We deduce that all natural transformations of T

^r∗

into itself form an r-parameter family linearly generated by the pth power transformations with p = 1, . . . , r.

Using general methods developed in [2]–[5], we deduce that all natural transformations of the rth order cotangent bundle functor T

^r∗

into itself form an r-parameter family generated by the pth power transformations A

^r,r_p

with p = 1, . . . , r.

Then we deduce that all natural transformations of T

^r∗

into T

^(r+q)∗

form an r-parameter family generated by the generalized pth power transformations A

^r,r+q_p

with p = q + 1, . . . , q + r.

Moreover, we deduce that all natural transformations of T

^r∗

into T

^(r−q)∗

form an (r − q)-parameter family generated by the generalized pth power transformations A

^r,r−q_p

with p = 1, . . . , r − q.

The author is grateful to Professor I. Kol´ aˇ r for suggesting the problem and for valuable remarks and useful discussions.

1. Let M be a smooth n-dimensional manifold. Let T

^r∗

M = J

^r

(M, R)

0

be the space of all r-jets j

_x^r

f of smooth functions f : M → R with source at x ∈ M and target at 0 ∈ R. The fibre bundle π

M

: T

^r∗

M → M with source r-jet projection π

M

: j

_x^r

f 7→ x has a canonical structure of a vector bundle with

(1.1) j

_x^r

f + j

_x^r

g = j

^r_x

(f + g) , k · j

_x^r

f = j

_x^r

(k · f ) for x ∈ M and k ∈ R [1].

1991 Mathematics Subject Classification: Primary 58A20.

Key words and phrases: higher order cotangent bundle, jet, pth power transformation.

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The vector bundle π

M

: T

^r∗

M → M is called the r-th cotangent bundle over M .

Every local diffeomorphism ϕ : M → N is extended to a vector bundle morphism T

^r∗

ϕ : T

^r∗

M → T

^r∗

N , j

_x^r

f 7→ j

_ϕ(x)^r

(f ◦ϕ

⁻¹

), where ϕ

⁻¹

is locally defined. Hence, the rth cotangent bundle functor T

^r∗

is defined on the category Mf

n

of smooth n-dimensional manifolds with local diffeomorphisms as morphisms and has values in the category VB of vector bundles.

If (x

ⁱ

) are local coordinates on M , then we have the induced fibre coordinates (u

i

, u

i1i2

, . . . , u

i1...ir

) on T

^r∗

M (symmetric in all indices):

(1.2)

u

i

(j

_x^r

f ) = ∂f

∂x

ⁱ

(x)

, u

i1i2

(j

_x^r

f ) = ∂

²

f

∂x

ⁱ¹

∂x

ⁱ²

(x)

, . . . , u

i1...ir

(j

_x^r

f ) = ∂

^r

f

∂x

ⁱ¹

. . . ∂x

ⁱ^r

(x)

.

Since the functor T

^r∗

takes values in the category VB of vector bundles, we may define natural transformations A

^r,r_p

of T

^r∗

into itself for p = 1, . . . , r.

Definition 1. The natural transformation A

^r,r_p

of the rth cotangent bundle functor T

^r∗

into itself defined by

(1.3) A

^r,r_p

: j

_x^r

f 7→ j

_x^r

(f )

^p

,

where (f )

^p

denotes the pth power of f , is called the p-th power transformation.

Definition 2. The natural transformation A

^r,sp

of T

^r∗

into T

^s∗

defined by

(1.4) A

^r,s_p

: j

_x^r

f 7→ j

_x^s

(f )

^p

is called the generalized p-th power transformation.

We note that in the case s = r + q this definition is correct only for p = q + 1, . . . , q + r.

Definition 3. The natural transformation P

^r,r−q

of T

^r∗

into T

^(r−q)∗

defined by

(1.5) P

^r,r−q

: j

_x^r

f 7→ j

_x^r−q

f is called a projection.

Note that

(1.6) A

^r,r−q_p

= A

^r−q,r−q_p

◦ P

^r,r−q

for p = 1, . . . , r − q .

2. In this part we determine, by induction on r, all natural transforma-

tions of T

^r∗

into itself.

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Theorem 1. All natural transformations A : T

^r∗

→ T

^r∗

form the r- parameter family

(2.1) A = k

1

A

^r,r₁

+ k

2

A

^r,r₂

+ . . . + k

r

A

^r,r_r

with any real parameters k

1

, k

2

, . . . , k

r

∈ R.

P r o o f. The functor T

^r∗

is defined on the category Mf

n

of n-dimensional smooth manifolds with local diffeomorphisms as morphisms and is of order r. Thus, its standard fibre S = (T

^r∗

R

ⁿ

)

0

is a G

^r_n

-space, where G

^r_n

is the group of all invertible r-jets from R

ⁿ

into R

ⁿ

with source and target at 0.

According to general standard methods [2]–[5], the natural transformations A : T

^r∗

→ T

^r∗

are in bijection with the G

^r_n

-equivariant maps f

^r,r

: (T

^r∗

R

ⁿ

)

0

→ (T

^r∗

R

ⁿ

)

0

of the standard fibres.

Let e a = a

⁻¹

denote the inverse element in G

^r_n

and let (2.2) t

(i1...ir)

= 1

r!

X

σ∈Sr

t

iσ(1)...iσ(r)

denote the symmetrization of a tensor with components t

i1...ir

.

By (1.2) the action of an element (a

ⁱ_j₁

, a

ⁱ_j₁_j₂

, . . . , a

ⁱ_j₁_...j_r

) ∈ G

^r_n

on (u

i1

, u

i1i2

, . . . , u

i1...ir

) ∈ (T

^r∗

R

ⁿ

)

0

is given by

(2.3)

u

i1

= u

j1

e a

^j_i₁¹

, u

i1i2

= u

j1j2

e a

^j_i¹₁

e a

^j_i₂²

+ u

j1

e a

^j_i₁¹_i₂

, . . . , u

i1...ir

= u

j1...jr

e a

^j_i₁¹

. . . e a

^j_i_r^r

+ u

j1...jr−1

r!

(r − 2)!2! e a

^j_(i¹

1

. . . e a

^j_i_r−2^r−2

e a

^j_i^r−1

r−1ir)

+ . . . + u

j1...jr−s

r!

(r − s − 1)!(s + 1)! e a

^j_(i¹

1

. . . e a

^j_i_r−s−1^r−s−1

e a

^jr−s_i

r−s...ir)

+ . . .

+ u

j1j2

r!

(r − 1)!1! e a

^j_(i¹

1

e a

^j_i²

2...ir)

+ . . .

+ u

j1

e a

^j_i₁¹_...i_r

.

I. Consider the case r = 2. The equivariance of a G

²_n

-equivariant map f

^2,2

= (f

i

, f

ij

) of (T

^2∗

R

ⁿ

)

0

into itself with respect to homotheties in G

²_n

: e a

ⁱ_j

= kδ

_jⁱ

, e a

ⁱ_j₁_j₂

= 0, gives the homogeneity conditions

(2.4) kf

i

(u

i

, u

ij

) = f

i

(ku

i

, k

²

u

ij

) , k

²

f

ij

(u

i

, u

ij

) = f

ij

(ku

i

, k

²

u

ij

) . By the homogeneous function theorem [2]–[5], we deduce that, first, the f

i

are linear in u

i

and independent of u

ij

, and secondly, the f

ij

are linear in u

ij

and quadratic in u

i

. Using the invariant tensor theorem for G

¹_n

[2]–[5], we obtain f

^2,2

in the form

(2.5) f

i

= k

1

u

i

, f

ij

= k

2

u

i

u

j

+ k

3

u

ij

with any real parameters k

1

, k

2

, k

3

∈ R.

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The equivariance of f

^2,2

of the form (2.5) with respect to the kernel of the projection G

²_n

→ G

¹_n

: e a

ⁱ_j

= δ

_jⁱ

and e a

ⁱ_jk

arbitrary, gives

(2.6) k

3

= k

1

.

This proves our theorem for r = 2.

II. Suppose that the theorem holds for r − 1 and the G

^r−1_n

-equivariant maps f

^r−1,r−1

of (T

^(r−1)∗

R

ⁿ

)

0

into itself define the (r−1)–parameter family A = k

1

A

^r−1,r−1₁

+ . . . + k

r−1

A

^r−1,r−1_r−1

with any real parameters k

1

, . . . , k

r−1

∈ R.

Our aim is to obtain the general form of any G

^r_n

-equivariant map of (T

^r∗

R

ⁿ

)

0

into itself.

Let (u

1

, u

2

, . . . , u

r

) := (u

i1

, u

i1i2

, . . . , u

i1...ir

) denote the fibre coordinates on T

^r∗

M . We assume that a G

^r_n

-equivariant map f

^r,r

is of the general form f

^r,r

= (f

1

, . . . , f

r−1

, f

r

) and the given map f

^r−1,r−1

defines the first r − 1 components (f

1

, . . . , f

r−1

) of f

^r,r

.

Considering the equivariance of f

^r,r

with respect to the homotheties e a

ⁱ_j

= kδ

_jⁱ

, e a

ⁱ_j₁_j₂

= 0, . . . , e a

ⁱ_j₁_...j_r

= 0 in G

^r_n

, for the rth component f

r

we obtain the homogeneity condition

(2.7) k

^r

f

r

(u

1

, u

2

, . . . , u

r

) = f

r

(ku

1

, k

²

u

2

, . . . , k

^r

u

r

) .

By the homogeneous function theorem [2]–[5], f

r

is of the general form f

i1...ir

= h

r

u

i1

· . . . · u

_i_r

+ h

r−1

u

_(i₁

. . . u

ir−2

u

_i_r−1_i_r₎

(2.8)

+ . . . + h

2,1

u

(i1

u

i2...ir)

+ h

2,2

u

(i1i2

u

i3...ir)

+ . . . + h

1

u

i1...ir

with any real parameters h

1

, h

2,1

, h

2,2

, . . . , h

r−1

, h

r

∈ R. The equivariance of f

^r,r

with respect to the kernel of the projection G

^r_n

→ G

^r−1_n

: e a

ⁱ_j

= δ

_jⁱ

, e a

ⁱ_j₁_j₂

= 0, . . . , e a

ⁱ_j₁_...j_r−1

= 0 and e a

ⁱ_j₁_...j_r

arbitrary, gives

(2.9) h

1

= k

1

.

Thus, we obtain the 1st power transformation A

^r,r₁

.

Now, considering the equivariance of A−k

1

A

^r,r₁

with respect to the kernel of the projection G

^r−1_n

→ G

¹_n

: e a

ⁱ_j

= δ

_jⁱ

and e a

ⁱ_j₁_j₂

, . . . , e a

ⁱ_j₁_...j_r−1

arbitrary, we obtain

(2.10) h

2,1

= r!

1!(r − 1)! k

2

, h

2,2

= r!

2!(r − 2)! k

2

, . . . Thus, we obtain the 2nd power transformation A

^r,r₂

.

Then, considering the equivariance of A − k

1

A

^r,r₁

− k

2

A

^r,r₂

with respect

to the kernel of the projection G

^r−2_n

→ G

¹_n

, we obtain the general form

of the 3rd power transformation A

^r,r₃

. Continuing this procedure gives the

next power transformations A

^r,r₃

, . . . , A

^r,r_r−2

. The equivariance of A−k

1

A

^r,r₁

−

k

2

A

^r,r₂

−. . .−k

_r−2

A

^r,r_r−2

with respect to the kernel of the projection G

²_n

→ G

¹_n

:

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e a

ⁱ_j

= δ

_jⁱ

and e a

ⁱ_jk

arbitrary, leads to the next relation

(2.11) h

r−1

= r!

(r − 2)!2! k

r−1

.

Thus, we obtain the (r − 1)th power transformation A

^r,r_r−1

. Finally, the G

^r_n

-equivariant map

(2.12) A − k

1

A

^r,r₁

− k

₂

A

^r,r₂

− . . . − k

_r−1

A

^r,r_r−1

= h

r

A

^r,r_r

is defined by the rth power transformation with any real parameter h

r

∈ R.

If we put h

r

= k

r

, this proves our theorem.

3. In this part we determine all natural transformations T

^r∗

→ T

^s∗

in two cases: r < s and r > s.

Theorem 2. All natural transformations A : T

^r∗

→ T

^(r+q)∗

form the r-parameter family

(3.1) A = k

q+1

A

^r,r+q_q+1

+ k

q+2

A

^r,r+q_q+2

+ . . . + k

q+r

A

^r,r+q_q+r

with any real parameters k

q+1

, k

q+2

, . . . , k

q+r

∈ R.

P r o o f. We apply induction on q.

I. Consider the case q = 1. According to general standard methods [2]–

[5], the natural transformations A : T

^r∗

→ T

^(r+1)∗

are in bijection with the G

^r+1_n

-equivariant maps of the standard fibres f

^r,r+1

: (T

^r∗

R

ⁿ

)

0

→ (T

^(r+1)∗

R

ⁿ

)

0

.

Considering the equivariance of f

^r,r+1

= (f

1

, . . . , f

r

, f

r+1

) with respect to homotheties: e a

ⁱ_j

= kδ

_jⁱ

, e a

ⁱ_j₁_j₂

= 0, . . . , e a

ⁱ_j₁_...j_r+1

= 0, we obtain the homogeneity conditions

(3.2)

kf

1

(u

1

, u

2

, . . . , u

r

) = f

1

(ku

1

, k

²

u

2

, . . . , k

^r

u

r

), . . . , k

^r

f

r

(u

1

, u

2

, . . . , u

r

) = f

r

(ku

1

, k

²

u

2

, . . . , k

^r

u

r

), k

^r+1

f

r+1

(u

1

, u

2

, . . . , u

r

) = f

r+1

(ku

1

, k

²

u

2

, . . . , k

^r

u

r

).

Additionally, using the equivariance of f

^r,r

= (f

1

, . . . , f

r

) with respect to the kernel of the projection G

^r_n

→ G

¹_n

, we obtain, by Theorem 1, the r- parameter family of the form (2.1): A = k

1

A

^r,r₁

+ k

2

A

^r,r₂

+ . . . + k

r

A

^r,r_r

with any real parameters k

1

, k

2

, . . . , k

r

∈ R.

Moreover, by the homogeneous function theorem and the invariant tensor theorem [2]–[5], we deduce that the (r+1)th component f

r+1

is of the general form

f

i1...ir+1

= l

r+1

u

i1

u

i2

. . . u

ir+1

+ l

r

u

(i1

. . . u

ir−1

u

irir+1)

(3.3)

+ . . . + l

2,1

u

_(i₁

u

_i₂_...i_r+1₎

+ l

2,2

u

_(i₁_i₂

u

_i₃_...i_r+1₎

+ . . .

with any real parameters l

2,1

, l

2,2

, . . . , l

r

, l

r+1

∈ R.

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The equivariance of f

^r,r+1

with respect to the kernel of the projections G

^r+1_n

→ G

^r_n

and G

^r+1_n

→ G

¹_n

gives the relations

(3.4) k

1

= 0 ,

(3.5) l

2,1

= (r + 1)!

r!1! k

2

, l

2,2

= (r + 1)!

(r − 1)!2! k

2

, . . . , l

r

= (r + 1)!

(r − 1)!2! k

r

. If we put l

r+1

= k

r+1

, this gives the r-parameter family f

^r,r+1

= (f

^r,r

, f

r+1

) of the form

(3.6) A = k

2

A

^r,r+1₂

+ . . . + k

r+1

A

^r,r+1_r+1

with any real parameters k

2

, . . . , k

r+1

∈ R.

II. Suppose that the theorem holds for q − 1 and the G

^r+q−1_n

-equivariant maps f

^r,r+q−1

: (T

^r∗

R

ⁿ

)

0

→ (T

^(r+q−1)∗

R

ⁿ

)

0

define the r-parameter family (3.7) A = k

q

A

^r,r+q−1_q

+ k

q+1

A

^r,r+q−1_q+1

+ . . . + k

q+r−1

A

^r,r+q−1_q+r−1

with any real parameters k

q

, k

q+1

, . . . , k

q+r−1

∈ R.

Consider a G

^r+q_n

-equivariant map f

^r,r+q

: (T

^r∗

R

ⁿ

)

0

→ (T

^(r+q)∗

R

ⁿ

)

0

of the form f

^r,r+q

= (f

^r,r+q−1

, f

r+q

).

The equivariance of f

^r,r+q

with respect to the homotheties in G

^r+q_n

: e a

ⁱ_j

= kδ

_jⁱ

, e a

ⁱ_j₁_j₂

= 0, . . . , e a

ⁱ_j₁_...j_r+q

= 0, gives for the (r + q)th component f

r+q

the homogeneity condition

(3.8) k

^r+q

f

r+q

(u

1

, u

2

, . . . , u

r

) = f

r+q

(ku

1

, k

²

u

2

, . . . , k

^r

u

r

) .

By the homogeneous function theorem and the invariant tensor theorem [2]–[5], f

r+q

is of the form

f

i1...ir+q

= l

r+q

u

i1

. . . u

ir+q

+ l

r+q−1

u

_(i₁

. . . u

ir+q−2

u

_i_r+q−1_i_r+q₎

(3.9)

+ . . . + l

q+1

u

(i1

. . . u

iq

u

iq+1...iq+r)

with any real parameters l

q+1

, . . . , l

q+r−1

, l

q+r

∈ R.

The equivariance of f

^r,r+q

with respect to the kernel of the projections G

^r+q_n

→ G

^r_n

and G

^r+q_n

→ G

¹_n

gives the relations

(3.10) k

q

= 0, . . . ,

(3.11) l

q+1

= (q + r)!

r!q! k

q+1

, . . . , l

q+r−1

= (q + r)!

(q + r − 2)!2! k

q+r−1

. If we put l

q+r

= k

q+r

, this proves our theorem.

Theorem 3. All natural transformations A : T

^r∗

→ T

^(r−q)∗

form the (r − q)-parameter family

(3.12) A = k

1

A

^r,r−q₁

+ k

2

A

^r,r−q₂

+ . . . + k

r−q

A

^r,r−q_r−q

with any real parameters k

1

, k

2

, . . . , k

r−q

∈ R.

P r o o f. Applying the same general procedure, we conclude that each A :

T

^r∗

→ T

^(r−q)∗

is the composition of the projection P

^r,r−q

: T

^r∗

→ T

^(r−q)∗

(7)

and a transformation A of T

^(r−q)∗

into itself: A = A ◦ P

^r,r−q

. This proves our theorem.

References

[1] J. G a n c a r z e w i c z, Differential Geometry , PWN, Warszawa 1987 (in Polish).

[2] I. K o l ´ aˇ r, Some natural operators in differential geometry , in: Proc. Conf. Diff. Geom.

and its Applications, Brno 1986, Kluwer, Dordrecht 1987, 91–110.

[3] I. K o l ´ aˇ r and G. V o s m a n s k a, Natural transformations of higher order tangent bundles and jet spaces, ˇ Casopis Pˇ est. Mat. 114 (2) (1989), 181–186.

[4] I. K o l ´ aˇ r, P. M i c h o r and J. S l o v a k, Natural Operations in Differential Geometry , to appear.

[5] J. K u r e k, On natural operators on sectorform fields, ˇ Casopis Pˇ est. Mat. 115 (3) (1990), 232–239.

INSTITUTE OF MATHEMATICS

MARIA CURIE-SK LODOWSKA UNIVERSITY PL. M. CURIE-SK LODOWSKIEJ 1

20-031 LUBLIN, POLAND

Re¸ cu par la R´ edaction le 20.5.1991

R´ evis´ e le 10.2.1992 et 29.6.1992