Natural affinors on time-dependent higher order cotangent bundles

U N I V E R S I T A T I S M A R I A E C U R I E - S K Ł O D O W S K A L U B L I N – P O L O N I A

VOL. LVIII, 2004 SECTIO A 47–58

ST ˇˇ EP ´AN DOPITA

Natural affinors on time-dependent higher order cotangent bundles

Abstract. We study natural affinors on time-dependent natural bundles.

Then we determine all natural affinors on the time-dependent higher order cotangent bundle T^r∗M × R.

1. Introduction. Recently, it has been pointed out that natural tensor fields of type (1, 1) (in other words affinors) play an important role in differ- ential geometry. In particular, I. Kol´aˇr and M. Modugno have used natural affinors to introduce the general concept of the torsion of a connection, [6].

Using such a point of view, it is useful to classify all natural affinors on some natural bundles. Such an approach has been used e.g. in [3], [4] and [6].

Further, non-autonomous Lagrangian dynamics can be considered as an extension of autonomous Lagrangian dynamics by introducing the addi- tional time coordinate. For example, M. de León and R. P. Rodrigues have introduced the concept of time-dependent (or dynamical) connection, [10].

Quite analogously, one can define dynamical vector fields, affinors, sprays and other structures. M. Doupovec and I. Kol´aˇr have classified all na- tural affinors on time-dependent Weil bundles, [2]. It is well known that Weil algebras and Weil functors generalize many geometric structures and constructions. In particular, there is a complete description of all product

2000 Mathematics Subject Classification. 53A55, 58A20.

Key words and phrases. Time-dependent bundle, natural affinor, higher order cotan- gent bundle.

48 S. Dopita

preserving functors on the category of all smooth manifolds and all smooth maps in terms of Weil functors, [7].

The aim of this paper is twofold. First, we study natural affinors on time- dependent natural bundles from a general point of view. In Example 2 we introduce the new natural affinor on a time-dependent natural bundle, which was not included in [2]. Second, we classify all natural affinors on time- dependent higher order cotangent bundles. We remark that such bundles are used e.g. in higher order mechanics, [14]. In this paper we essentially use the results [8] and [9] of J. Kurek.

All manifolds and maps are assumed to be infinitely differentiable.

2. Natural affinors on time-dependent bundles. In general, an affi- nor on a manifold M is a tensor field of type (1, 1) on M , which can be interpreted as a linear morphism T M → T M over the identity of M . By the Fr¨olicher–Nijenhuis theory, affinors are exactly tangent-valued one-forms on M, i.e. sections from C^∞(T M ⊗T^∗M ). Given a fibered manifold p : Y → M , an affinor Q on Y is called vertical, if Q has values in the vertical bundle VY, i.e. Q ∈ C^∞(V Y ⊗ T^∗Y ).

Further, let T^∗M ⊂ T^∗Y be the canonical inclusion of cotangent bun- dles. By [11], vertical affinors of the form Q ∈ C^∞(V Y ⊗ T^∗M ) are called soldering forms. Let F be a natural bundle F on the category Mf_m of all m-dimensional manifolds and their local diffeomorphisms. We recall that a natural affinor on a natural bundle F is a system of affinors Q_M : T F M → T F M for every m-manifold M satisfying T F f ◦ Q_M = Q_N◦ T F f for every local diffeomorphism f : M → N . An example of a natural affinor is the classical almost tangent structure on T M .

Definition 1. The time-dependent natural bundle F_R corresponding to the natural bundle F is defined by F_RM = F M × R for every m-dimensional manifold M and by F_Rf = F f × Id_R: F_RM → F_RN for every local diffeo- morphism f : M → N .

Clearly, the time-dependent natural bundle F_Rgeneralizes the well known time-dependent tangent bundle T M × R and also the time-dependent Weil bundle T^A

R from [2], if we restrict T^A

R to the category Mf_m.

In what follows we introduce some examples of natural affinors on time- -dependent bundles.

Example 1. For any natural bundle F we have three simple constructions of natural affinors on F_R. First, every natural affinor Q on F induces a natural affinor eQ on F_Rby means of the product structure F M × R. Quite analogously, the identity Id_{T R}of T R determines another affinor fId_{T R}on F_R. The third type of natural affinors on F_R can be defined by tensor products X ⊗ dt of absolute vector fields on FM with the canonical one-form dt on R.

We recall that an absolute vector field can be interpreted as an absolute natural operator transforming vector fields on M into vector fields on FM, [7]. Clearly, absolute vector fields are natural in the following sense.

Definition 2. A natural vector field X on natural bundle F is a system of vector fields X_M : F M → T F M for every m-manifold M satisfying T F f ◦ X_M = X_N ◦ F f for all local diffeomorphisms f : M → N .

If F is a natural vector bundle, then the classical Liouville vector field L_{F M} on FM is natural. Clearly, L_{F M} is generated by the one-parameter family of homotheties. More generally, let Φ(t) be a smooth one-parameter family of natural transformations F → F , where smoothness means that the map Φ(t)_M : F M × R → F M is smooth for every manifold M . Then the formula X_M = _dt^d

0Φ(t)_M defines a natural vector field X_M : F M → T F M . By [7], every natural vector field X on F is vertical. This yields that natural affinors X ⊗ dt on F_R from Example 1 are soldering forms.

Example 2. Let F be a natural vector bundle and let f be a natural function on TF. We recall that this is a system of functions f_M : T F M → R for every m-dimensional manifold M satisfying f_M = f_N◦ T F ϕ for all local diffeomorphisms ϕ : M → N . Denote by π_M : F M → M the bundle projection and by p_M : T M → M the tangent bundle projection. For any X ∈ T F_RM = T F M × T R we have pF_RM(X) ∈ F_RM , pr1(pF_RM(X)) ∈ F M and x := π_M(pr1(p_FRM(X))) ∈ M . Let s : M → F M be a zero section.

Then the cartesian product of s(x) with f_M(pr₁(X)) defines an element R(X) := s(x) × f_M(pr₁(X)) ∈ F M × R = FRM.

As FM is a vector bundle, F_RM is a vector bundle too. For X ∈ T F_RM we have

P (X) := (p_FRM(X), R(X)) ∈ (F_RM ⊕ F_RM ) ∼= V F_RM ⊂ T F_RM.

This defines a natural affinor P on F_RM .

We remark that natural affinors from Example 2 did not appear in the description of all natural affinors on time-dependent Weil bundles, [2]. We also point out that the classical Liouville one-form of the cotangent bundle T^∗M is the simplest example of a natural function on T T^∗.

It is well known that natural affinors play a significant role in the theory of torsions of connections. In particular, if we interpret a general connection Γ : F M → J¹F M as its horizontal projection (denoted by the same symbol) Γ : T F M → T F M , we obtain an affinor on F . Further, I. Kol´aˇr and M.

Modugno introduced the generalized torsion of Γ as the Fr¨olicher–Nijenhuis bracket [Γ, Q] of Γ with some natural affinor Q on F , [6]. Such an approach has been used e.g. in [3], [4] and [6]. There are also many papers which

50 S. Dopita

classify all natural affinors on some natural bundles, see [5], [8], [12] and [13].

Denote by T^Athe Weil functor corresponding to a Weil algebra A, [2]. By the general theory, every product preserving functor F on the category Mf of all smooth manifolds and all smooth maps is the Weil functor F = T^A, where A = F R. M. Doupovec and I. Kol´aˇr have determined all natural affinors on the time-dependent Weil bundle T^A

RM , [2]. It is interesting to point out that all natural affinors on T^A

RM are generated only by affinors from Example 1. Using this result, M. Doupovec has described torsions of dynamical connections on time-dependent Weil bundles, [1].

Further, natural affinors on time-dependent higher order tangent bundles were determined by I. Kol´aˇr and J. Gancarzewicz, [5]. Such affinors are also generated only by three affinors from Example 1.

3. Natural affinors on time-dependent higher order cotangent bun- dles. Let M be a smooth m-dimensional manifold and denote by T^r∗M = J^r(M, R)0 the space of all r-jets from M into R with target 0. Every local diffeomorphism f : M → N can be extended to a vector bundle morphism T^r∗f : T^r∗M → T^r∗N by j_x^rϕ 7→ j_{f (x)}^r (ϕ ◦ f⁻¹), where f⁻¹ is constructed locally. Then π_M : T^r∗M → M is a natural vector bundle which is called the r-th order cotangent bundle. Clearly, T^1∗M = T^∗M is the classical cotangent bundle.

Denote by

q_M : T^r∗M → T^∗M

the bundle projection defined by q_M(j_x^rf ) = j_x¹f . If X ∈ T T^r∗M , then T π_M(X) ∈ T M and q_M(p_T^r∗_M(X)) ∈ T^∗M . So we can define a map

λ_M : T T^r∗M → R, λM(X) = hq_M(p_T^r∗_M(X)), T π_M(X)i, which is called the generalized Liouville form on T^r∗M .

Further, let A^r_s : T^r∗M → T^r∗M be the s-th power natural transforma- tion defined by A^r_s(j^r_xf ) = j_x^r(f )^s, where (f )^s denotes s-th power of f. Since πM : T^r∗M → M is a vector bundle, the vertical bundle V T^r∗M can be identified with the Whitney sum T^r∗M ⊕ T^r∗M . Using this identification we can define natural affinors Q^s_M : T T^r∗M → V T^r∗M by

Q^s_M(X) = (p_T^r∗_M(X), λ_M(X)A^r_s(p_T^r∗_M(X))).

In what follows we will use the following results, which were proved by J. Kurek.

Lemma 1 ([8]). All natural affinors on the r-th order cotangent bundle T^r∗M are of the form

k0IdT^r∗M + k1Q_M¹ + · · · + krQ^r_M, ki∈ R.

Lemma 2 ([9]). All natural transformations T^r∗M → T^r∗M are of the form

k₁A^r₁+ · · · + k_rA^r_r, k_i∈ R.

Multiplying the s-th power transformation A^r_s by a real number t, we obtain a smooth one-parameter family of natural transformations (tA^r_s) : T^r∗M → T^r∗M . This generates a vector field Ls: T^r∗M → V T^r∗M by

Ls(u) = d dt

   0

(u + tA^r_s(u)) .

Clearly, L₁ is the classical Liouville vector field on T^r∗ and L_s can be also defined by L_s(u) = (u, A^r_s(u)).

Using Example 1 and Example 2, we have four types of natural affinors on the time-dependent bundle T^r∗

R M = T^r∗M × R:

I) Each natural affinor on T^r∗M from Lemma 1 induces a natural affinor on T^r∗

R M by means of the product structure. In this way we obtain natural affinors eQ¹_M, . . . , eQ^r_M and fId_T^r∗_M.

II) The identity of T R induces a natural affinor fId_R on T^r∗

R M .

III) Natural vector fields L_s : T^r∗M → T T^r∗M induce natural affinors (Ls⊗ dt) on T^r∗

R M .

IV) Clearly, the generalized Liouville form λ_M : T T^r∗M → R is a natural function on T T^r∗M . By Example 2, this natural function determines a natural affinor P on T^r∗

R M .

In the rest of this paper we prove that natural affinors from I–IV generate all natural affinors on T^r∗

R M . We first introduce the coordinate form of affinors from I–IV.

The canonical coordinates (xⁱ) on M induce the additional fiber coordi- nates (u_i, u_ij, . . . , u_i1...ir) on T^r∗M , which are symmetric in all indices, [4].

Denoting by t the coordinate on R, the coordinates on T T_R^r∗M are of the form

(xⁱ, t, ui, . . . , ui1···ir, Xⁱ= dxⁱ, T = dt, Ui = dui, . . . , Ui1···ir = dui1···ir).

Clearly, we have

fId_T^r∗_M(dxⁱ, dt, du_i, . . . , du_i1···ir) =(dxⁱ, 0, du_i, . . . , du_i1···ir) fId_R(dxⁱ, dt, dui, . . . , dui1···ir) =(0, dt, 0, . . . , 0).

Obviously, the generalized Liouville form λ_M has the coordinate expres- sion u_idxⁱ. Then

P (dxⁱ, dt, dui, . . . , dui1···i_r) = (0, ujdx^j, 0, . . . , 0).

52 S. Dopita

J. Kurek has computed the coordinate form of affinors Q¹_M, . . . , Q^r_M on T^r∗M . Using [8], we have

Qe¹_M(dxⁱ, dt, du_i, . . . , du_i1···ir) = (0, 0, u_iu_jdx^j, . . . , u_i1···iru_jdx^j) ...

Qe^s_M(dxⁱ, dt, dui, . . . , dui1···i_r) = 0, 0, 0, . . . , ui1···i_sujdx^j , (s + 1)!

(s − 1)!2! u_(i1· · · u_is−1u_isis+1)ujdx^j, . . . , r!

(s − 1)!(r − s + 1)! u_(i1· · · u_is−1u_is···ir)u_jdx^j
...

Qe^r_M(dxⁱ, dt, du_i, . . . , du_i1···ir) = (0, 0, 0, . . . , 0, u_i1· · · u_iru_jdx^j), where (i_i· · · i_r) denotes the symmetrization.

Finally, the natural vector field L_s is of the form L_s= u_i1· · · u_is ∂

∂u_i1···i_s

+· · ·+ r!

(s − 1)!(r − s + 1)!u_(i1· · · u_is−1u_is···ir) ∂

∂u_i1···i_r

, see [4]. So we have

(L1⊗ dt)(dxⁱ, dt, dui, . . . , dui1···ir) = (0, 0, uidt, . . . , ui1···irdt) ...

(Lr⊗ dt)(dxⁱ, dt, dui, . . . , dui1···ir) = (0, 0, 0, . . . , 0, ui1ui2· · · u_irdt).

Proposition 1. All natural affinors F^r : T T_R^r∗M → T T_R^r∗M are of the form

F^r = a(t)fId_T^r∗_M + b(t)fId_R+ a₁(t) eQ¹_M + · · · + a_r(t) eQ^r_M +b₁(t)L₁⊗ dt + · · · + b_r(t)L_r⊗ dt + c(t)P, where a(t), . . . , c(t) are arbitrary smooth functions of R.

Proof. Denote by G^r_m the group of all invertible r-jets of R^m into R^m with the source and the target zero. By the general theory, [7], it suffices to find all G^r+1_m -equivariant linear maps T (T^r∗

R R^m)0 → T (T^r∗

R R^m)0of standard fibers.

Let (aⁱ_j, aⁱ_jk, . . . , aⁱ_j1j2...jr) be the coordinates on G^r_m and denote by a tilde the inverse element. By standard evaluations we find the action of G^r+1_m on the standard fibre T (T^r∗

R R^m)0

ui=ea^j_iuj

(1)

u_i1i2 =ea^j_i¹

1ea^j_i²

2u_j1j2 +ea^j_i¹

1i2u_j1 (2)

...

ui1...ir =ea^j_i1¹. . .ea^j_ir^ruj1...jr

+ r!

(r − 2)!2!ea^j_(i¹

1. . .ea^j_i^r−2

r−2ea^j_i^r−1

r−1ir)u_j1...jr−1 + · · · +

 r!

(r − 1)!1!ea^j_(i¹

1ea^j_i²

2...ir)+ · · ·



u_j1j2+ea^j_i

1...iru_j (3)

(4) Xⁱ= aⁱ_jX^j

(5) T = T

U_i=ea^j_iU_j+ea^j_ika^k_lX^lu_j (6)

Ui1i2 =ea^j_i¹

1ea^j_i²

2Uj1j2 +ea^j_i¹

1i2Uj1

+

 ea^j_i1¹ea^j_i²

2ka^k_lX^l+ea^j_i²₂ea^j_i¹

1ka^k_lX^l



uj1j2 +ea^j_i¹

1i2ka^k_lX^luj1

(7)

...

Ui1···ir =ea^j_i1¹· · ·ea^j_ir^rUi1···ir + r!

(r − 2!)2!ea^j_(i¹

1· · ·ea^j_ir−2^r−2ea^j_i^r−1

r−1ir)Uj1···jr−1

+ · · · +

 r!

(r − 1)!1!ea^j_(i¹

1ea^j_i²

2···i_r)+ · · ·

 U_j1j2 +ea^j_i1¹_···irUj1+

h ea^j_i¹

1k· · ·ea^j_ir^ra^k_lX^l+ · · · i

uj1···jr

+

 r!

(r − 2)!2!ea^j_(i¹

1k· · ·ea^j_ir−2^r−2ea^j_i^r−1

r−1ir)a^k_lX^l+ · · ·



uj1···j_r−1+ · · · +

 r!

(r − 1)!1!ea^j_(i¹

1kea^j_i¹

2···i_r)a^k_lX^l+ · · ·



u_j1j2 +ea^j_i¹

1···irka^k_lX^lu_j1. (8)

Write u = (u_i, u_ij, . . . , u_i1···ir). Any linear map of the standard fibre into itself has the form

(9) T = αj(t, u)X^j+ β(t, u)T + A^j(t, u)Uj+ · · · + A^j1···jr(t, u)Uj1···jr

(10) Xⁱ = γ_jⁱ(t, u)X^j + δⁱ(t, u)T + B^ij(t, u)U_j+ · · · + B^ij1···jr(t, u)U_j1···jr

54 S. Dopita

Ui = ηij(t, u)X^j+ζi(t, u)T +C_i^j(t, u)Uj+· · ·+C_i^j1···jr(t, u)Uj1···j_r

(11)

...

Ui1···ir = ηi1···irj(t, u)X^j + ζi1···ir(t, u)T + C_i^j

1···ir(t, u)Uj

(12)

+ · · · + C_i^j₁¹_···i^···j_r^r(t, u)Uj1···jr .

Considering equivariancy of (9) with respect to the homotheties aⁱ_j = kδ_jⁱ we obtain

1

k αj(t, ui, . . . , ui1···ir) = αj

 t, 1

kui, . . . , 1 k^rui1···ir



β(t, ui, . . . , ui1···i_r) = β

 t, 1

kui, . . . , 1 k^rui1···i_r



A^j(t, u_i, . . . , u_i1···ir) = 1 kA^j

 t,1

ku_i, . . . , 1 k^ru_i1···ir

 ...

A^j1···jr(t, u_i, . . . , u_i1···i_r) = 1 k^rA^j1···jr

 t,1

ku_i, . . . , 1 k^ru_i1···i_r

 .

By the homogenous function theorem from [7] we compute α_j(t, u) = α(t)u_j, β(t, u) = β(t), A^j(t, u) = 0, A^j1···jr(t, u) = 0. Thus (9) can be written in the form

(13) T = α(t)u_jX^j+ β(t)T.

Quite analogously we prove

(14) Xⁱ= γ(t)Xⁱ.

Further, equivariancy of (11) implies 1

k² η_ij(t, u_i, . . . , u_i1···i_r) = η_ij

 t,1

ku_i, . . . , 1 k^ru_i1···i_r



1

kζ_i(t, u_i, . . . , u_i1···ir) = ζ_i

 t, 1

ku_i, . . . , 1 k^ru_i1···ir



C_i^j(t, u_i, . . . , u_i1···i_r) = C_i^j

 t, 1

ku_i, . . . , 1 k^ru_i1···i_r



C_i^j1j2(t, u_i, . . . , u_i1···ir) = 1 kC_i^j1j2

 t,1

ku_i, . . . , 1 k^ru_i1···ir

 ...

C_i^j1···jr

1···ir(t, u_i, . . . , u_i1···ir) = 1

k^r−1C_i^j1···jr

1···ir

 t,1

ku_i, . . . , 1 k^ru_i1···ir

 .

Using the homogenous function theorem we obtain

(15) Ui= (1ηij(t)uij+ 2ηij(t)uiuj) X^j+ ζ(t)uiT + C(t) Ui . Finally, the equivariancy of (12) leads to following relations:

1

k^r+1ηi1···i_rj(t, ui, . . . , ui1···i_r) = ηi1···i_rj

 t, 1

kui, . . . , 1 k^rui1···i_r

 1

k^rζ_i1···ir(t, u_i, . . . , u_i1···ir) = ζ_i1···ir

 t, 1

ku_i, . . . , 1 k^ru_i1···ir

 1

k^r−1 C_i^j

1···ir(t, ui, . . . , ui1···ir) = C_i^j

1···ir

 t,1

kui, . . . , 1 k^rui1···ir

 1

k^r−2 C_i^j₁¹_···i^j2_r(t, ui, . . . , ui1···i_r) = C_i^j₁¹_···i^j2_r

 t,1

kui, . . . , 1 k^rui1···i_r

 ...

C_i^j₁¹_···i^···j_r^r(t, ui, . . . , ui1···ir) = C_i^j₁¹_···i^···j_r^r

 t,1

kui, . . . , 1 k^rui1···ir

 . By the homogenous function theorem, the function η_i1···i_rj is a sum of the polynomials of degree a_s in u_i1···is satisfying the relation

r + 1 = a1+ 2a2+ · · · + rar. This has the following solutions:

a1= r + 1, a2 = · · · = ar= 0

a1= r − 1, a2 = 1, a3 = · · · = ar= 0 a₁= r − 2, a₃ = 1, a₂ = · · · = a_r= 0

...

a₁= 1, a_r= 1, a₂ = · · · = a_r−1= 0

56 S. Dopita

so that η_i1···irj can be written in the form

ηi1···i_rj(t, u) = rηi1···i_rj(t)ui1ui2· · · u_iruj

+ r−1,1ηi1···irj(t)u₍i1i2· · · u_ir−2u_ir−1ir)uj

+ r−1,2ηi1···irj(t)u₍i1i2· · · u_ir−2uir−1u_ir)j + · · · + ₁η_i1···irj(t)u_i1···irj .

By similar computations we find the expression of ζ_i1···i_r, C_i^j_1···ir, . . . , C_i^j₁¹_···i^···j_r^r and we obtain

(16)

U_i1···ir = [_rη_i1···irj(t)u_i1u_i2· · · u_iru_j

+ _r−1,1η_i1···irj(t)u₍i₁i₂· · · u_ir−2u_ir−1ir)u_j + r−1,2ηi1···i_rj(t)u₍i1i2· · · u_ir−2uir−1u_ir)jk + · · · + ₁η_i1···irj(t)u_i1···irj] X^j

+

rζ_i1···ir(t)u_i1· · · u_ir+ _r−1ζ_i1···ir(t)u_(i1· · · u_ir−2u_ir−1ir) + · · · + ₁ζ_i1···ir(t)u_i1···ir] T

+ h

r−1C_i^j

1···ir(t)δ^j_(i¹

1ui2· · · u_ir) + · · · + 1C_i^j_1···ir(t)δ_(i^j1

1u_i2···i_r)

i Uj1

+ · · · + C_i^j₁¹_···i^···j_r^r(t)δ^j_(i¹

1· · · δ^j_i^r

r)Ui1···i_r .

We first prove our assertion for r = 2. Formulas (13)–(16) for r = 2 are of the form

(17) T = α(t)u_kX^k+ β(t)T

(18) Xⁱ= γ(t)Xⁱ

(19) Ui = (1ηik(t)uik+ 2ηik(t)uiuk) X^k+ ζ(t)uiT + C(t) Ui

(20)

Uij = 2ηijk(t)uiujuk+ 1,1ηijk(t)uijuk+ 1,2ηijk(t)u_(iu_j)k +1ηijk(t)uijk) X^k+ (2ζij(t)uiuj+ 1ζij(t)uij) T + Cⁱ(t)δ_(i^ku_j)Uk+ C^ij(t)Uij .

The equivariancy of (19) with respect to the kernel of the jet projection G²_m→ G¹_m given by aⁱ_j = δ_jⁱ and aⁱ_jk arbitrary leads to relations

C(t) = γ(t), 1ηik = 0

so that (19) is of the form

(21) U_i= η_i(t)u_iu_kX^k+ ζ_i(t)u_iT + γ(t) U_i .

Finally, the equivariancy of (20) with respect to the kernel of the jet pro- jection G³_m→ G¹_m given by aⁱ_j = δⁱ_j and aⁱ_jk, aⁱ_jklarbitrary leads to relations

Cⁱ = 0, C^ij(t) = γ(t),₁ζ_ij(t) = ζ_i(t),

1,1η_ijk(t) = η_i(t),₁η_ijk = 0,_1,2η_ijk = 0, so that (20) is of the form

(22) Uij = ηij(t)uiuju_kX^k+ ηi(t)uiju_kX^k+ ζij(t)uiujT + ζ(t)u_ijT + γ(t)U_ij .

Hence we have proved

(23) F²= a(t)fId_{T T}2∗M + b(t)fId_{T R}+ a₁(t) eQ¹_M + a₂(t) eQ²_M + b₁(t)(L₁⊗ dt) + b₂(t)(L₂⊗ dt) + c(t)P, where

a(t) = γ(t), b(t) = β(t), a1(t) = ηi(t), a2(t) = ηij(t) b₁(t) = ζ_i(t), b₂(t) = ζ_ij(t), c(t) = α(t).

This proves our proposition for r = 2. To finish the proof, we will use the induction with respect to r. Suppose now, that our proposition is true for r − 1, i.e.

(24) F^r−1= a(t)fId_{T T}(r−1)∗M + b(t)fId_{T R}+ a₁(t) eQ¹_M + · · · a_r−1(t) eQ^r−1_M + b₁(t)(L₁⊗ dt) + · · · + b_r−1(t)(L_r−1⊗ dt) + c(t)P.

Using the homogenous function theorem we deduce easily that the com- ponents of F^r at T, Xⁱ, Ui, . . . , Ui1···ir are exactly the corresponding com- ponents of F^r−1. That is why it suffices to determine the last (r + 2)-th component of F^r, which is given by (16). The equivariancy with respect to the kernel of the projection G^r+1_m → G¹_m leads to the relations

C_i^j1···jr

1···i_r(t) = a(t), C_i^j

1···i_r(t) = · · · = C_i^{j1···jr−1}

1···i_r (t) = 0,

1η_i1···irj(t) = a₁(t), _s−1,1η_i1···irj(t) = a_s−1(t) where s = 2, . . . , r,

s−1,2η_i1···i_rj(t) = 0 where s = 2, . . . , r,

rηi1···i_rj(t) = ar(t) is a new function,

1ζi1···ir(t) = b1(t), s−1ζi1···ir(t) = bs−1(t) where s = 2, . . . , r,

rζ_i1···ir(t) = b_r(t) is a new function.

58 S. Dopita

This completes the proof. 

Corollary 1. All natural affinors on the time-dependent cotangent bundle T^∗

RM are of the form

Xⁱ = a(t)Xⁱ (25)

Ui = a1(t)uiu_kX^k+ b1(t)uiT + a(t) Ui

(26)

T = c(t)u_kX^k+ b(t)T.

(27)

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Stˇˇ ep´an Dopita

Institute of Mathematics

Faculty of Mechanical Engineering Brno University of Technology Technick´a 2, 616 69 Brno Czech Republic

e-mail: ste.dopita@centrum.cz

Received July 16, 2004