DIFFERENTIAL GEOMETRY AND APPLICATIONS Satellite Conference of ICM in Berlin, Aug. 10 – 14, 1998, Brno Masaryk University in Brno (Czech Republic) 1999, 293–302

NATURAL T –FUNCTIONS ON THE COTANGENT BUNDLES

OF SOME WEIL BUNDLES

JIŘí TOMÁŠ

Abstract. For some Weil algebras A, we present a simple procedure giving all natural T -functions on the cotangent bundle of a Weil bundle T^{∗}T^{A}. For some other cases, we describe the additional natural functions. We outline the relations of this problem to the problem of lifting of vector fields to the cotangent bundle of a Weil bundle.

1. This paper is a contribution to the theory of Weil bundles. We follow some results by Kolář concerning the classification of all natural operators T → T T^{A} transforming vector fields on manifolds to vector fields on a Weil bundle T^{A}, [4],

[5] and some construction of his, transforming natural operators of this kind to natural T -functions on T^{∗}T^{A}, the cotangent bundle of T^{A}, [3].

For some cases of Weil algebras A, all natural T -functions in question can be obtained by this construction. We show, that in some other cases it is not so.

We start from the concept of a natural T -function. For a natural bundle F, a natural T -function f is a natural operator f_{M} transforming vector fields on a manifold M to functions on F M. The naturality condition reads as follows. For a local diffeomorphism φ : M → N between manifolds M, N and for vector fields X on M and Y on N satisfying T φ ∘ X = Y ∘ φ it holds f_{N} ∘ F φ = f_{M}. An absolute natural operator of this kind, i.e. independent of the vector field, is called a natural function on F.

Now we remind some basic results of Kolář. The first one, [4], [5], giving the classification of all natural operators T → T T^{A} reads as follows

All natural operators T → T T^{A} are of the form

L(c)T^{A} + λ_{D}

1991 Mathematics Subject Classification. 58A20, 53A55.

Key words and phrases. Natural bundle, natural operator, vector field, Weil bundle.

The author was supported by the grant No. 201/96/0079 of the GA ČR.

This paper is in final form and no version of it will be submitted for publication elsewhere.

where T^{A} denotes the flow prolongation on a natural bundle T^{A}, L(c) denotes the natural affinor on T^{A} induced by c ∈ A in the sense of Koszul, [4], [5] and λ_{D} denotes the absolute operator induced by a derivation D ∈ Der A.

We essentially use the following construction of Kolář, [3]. For a natural bundle F and a vector field Y on F M, Kolář defined an operation transforming Y into a function Ỹ : T^{∗}F M → ℝ as follows

Ỹ(ω) = ⟨Y (p(ω)), ω⟩

where p is the cotangent bundle projection and ω ∈ T^{∗}F M. One can immediately verify that this operation transforms a natural operator A : T → T F into a natural T -function Ã on T^{∗}F.

2. Let F = T^{A}, where A is considered as a factor algebra of the algebra of jets 𝔻_{k}^{r} = J_{0}^{r}(ℝ^{k}, ℝ) or as the factor of the algebra of polynomials ℝ[τ_{1}, …, τ_{k}].

Let D be a basis of Der A and C be a basis of A. Put I_{D} = λ̃_{D} and J_{c} = { \atop L(c)T^{A}}. Let M be a manifold satisfying dim M ≥ w(A) + 1, where w(A) denotes the width of A. We are going to prove, that for some cases of A, all natural T -functions f_{M} : T^{∗}T^{A}M → ℝ are of the form

(1)  h(J_{c,M}, I_{D,M}),  D ∈ D, c ∈ C

for an arbitrary smooth function h : ℝ^{dim A+dim Der A} → ℝ.

In what follows, we need the coordinate expressions of some natural operators. Let x_{α}^{i} denote the canonical coordinates on T_{k}^{r}M and p_{i}^{α} define the additional coordinates on T^{∗}T_{k}^{r}M. A map τ_{i} ↦ δ_{j}^{i}τ^{β} determines a derivation D_{j}^{β} ∈ Der 𝔻_{k}^{r}, [2], [8]. Denote by λ_{j}^{β} the associated natural operator λ_{D_{j}^{β}}. Its coordinate ex-

pression is of the form \frac{α_{j}}{α_{j}+β_{j}} \frac{(α+β)!}{α!}x_{α}^{i} \frac{∂}{∂x_{α+β−\{j\}}^{i}}. The following assertion yields all

natural T -functions on T^{∗}T_{k}^{r}, which are constructed by means of natural operators T → T T_{k}^{r}.

Proposition 1. For a manifold M satisfying dim M ≥ k + 1, all natural T - functions on T^{∗}T_{k}^{r}M are of the form

(2)  h(J_{c,M}, I_{D,M}),  D ∈ D, c ∈ C

where C is a basis of 𝔻_{k}^{r}, D is a basis of Der 𝔻_{k}^{r} and h is any smooth function ℝ^{dim 𝔻_{k}^{r}+dim Der 𝔻_{k}^{r}} → ℝ.

NATURAL T –FUNCTIONS … 295

Proof. By the immersion theorem, there is an immersion element i ∈ T_{0}^{𝔻_{k}^{r}}ℝ^{m} satisfying x_{α}^{i} = 0 for |α| ≥ 2 and x_{j}^{i} = δ_{j}^{i}. By the general theory, [5], natural operators in question are of order r + 1 and they are determined by their values over j_{0}^{r+1}(\frac{∂}{∂x^{k+1}} ) on (T^{∗}T_{k}^{r})_{0}ℝ^{m}. By the action of the jet group G_{m}^{r+1} we can achieve on a dense subset of (J^{r+1}T)_{0}ℝ^{m} × (T^{∗}T_{k}^{r})_{0}ℝ^{m} the element j_{0}^{r+1}(\frac{∂}{∂x^{k+1}} ) and an element over i in (T^{∗}T_{k}^{r})_{0}ℝ^{m}. Applying a general lemma from [5], Chapter VI, Proposition 26.1 we obtain that every natural operator in question evaluated over i is of the form h(p_{j}^{α}), 0 ≤ |α| ≤ r, j ∈ {1, …, k + 1} for any smooth function h. It follows from the coordinate expressions of J_{τ^{α}} and λ̃_{j}^{β} that p_{k+1}^{α} coincide with J_{τ^{α}}, 0 ≤ |α| ≤ r over j_{0}^{r+1}(\frac{∂}{∂x^{k+1}} ) and i, while p_{j}^{β}, 1 ≤ |β| ≤ r, j ∈ \{1, …, k\} coincide with natural functions I_{j}^{β} = λ̃_{j}^{β} over i. The transformation laws of G_{m}^{r+1} on (T^{∗}T_{k}^{r})_{0}ℝ^{m} are of the form

(3)  ȳ_{α}^{i} = a_{l_{1}...l_{p}}^{i}y_{α_{1}}^{l_{1}}…y_{α_{p}}^{l_{p}},  p̄_{i}^{α} = ã_{l_{1}...l_{p}}^{j}ȳ_{γ_{1}}^{l_{1}}…ȳ_{γ_{s}}^{l_{s}}p_{j}^{αγ}

where a_{l_{1}...l_{p}}^{i} denote the canonical coordinates of G_{m}^{r+1} and ã_{l_{1}...l_{p}}^{i} denote its inverse.

It follows the existence of a subgroup H ⊆ G_{k+1}^{r+1} such that H ×id_{ℝ^{m−k}} stabilizes i and j_{0}^{r+1}(\frac{∂}{∂x^{k+1}} ). Put B_{m}^{s+1} = \{j_{0}^{s+1}φ ∈ B_{m}^{r+1}; j_{0}^{s}φ = j_{0}^{s} id_{ℝ^{m}}\}.

All p_{j}^{0}, j ∈ {1, …, k} are annihilated by H × id_{ℝ^{m−k}} ∩B_{m}^{r+1} namely by an element satisfying a_{j_{1}...j_{r+1}}^{i} = 0 except a_{j...j}^{j}, which completes the proof. □

3. In the following propositions, we investigate in more details the case A = 𝔻 ⊗ 𝔻_{k}^{r}, where 𝔻 = 𝔻_{1}^{1} denotes the algebra of dual numbers. Such a Weil algebra corresponds to the Weil bundle T T_{k}^{r}. The following lemma yields the description of all natural operators T → T T T^{A}.

Let E → M be a vector bundle and f : E → E be a vector bundle morphism over the identity on M. The vertical tangent bundle V E is canonicaly identified with E ×_{M} E. Let us denote by W f the map id_{E} ×f in this identification. Hence W f is a vertical vector field on E. The following result was deduced in [8].

Lemma 2. All natural operators T → T T T^{A} are of the form

T ∘ (L(a)T^{A} + λ_{D}) + V ∘ (L(b)T^{A} + λ_{E}) + W (L(c))

for a, b, c ∈ A, D, E ∈ Der A, T , T^{A} the flow operators associated to natural bundles T, T^{A} and the vertical lift V : T → T T.

Let x_{α}^{i} denote the canonical coordinates on T_{k}^{r}M and y_{α}^{i} = dx_{α}^{i} the additional ones on T T_{k}^{r}M. The additional coordinates on T^{∗}T_{k}^{r}M are defined by p_{i}^{α}dx_{α}^{i} + q_{i}^{α}dy_{α}^{i}.

We are going to prove that all natural T -functions on T^{∗}T T_{k}^{r} are obtained by the construction (1) only if r = 1 or k = 1 or r = 2 and k = 2.

Proposition 3. Let r = 1 or k = 1 or r = 2 and k = 2. Then for a manifold M satisfying dim M ≥ k + 2, all natural T -functions T^{∗}T T_{k}^{r}M → ℝ are of the form

h(T ∘ (L(a)T^{A} + λ_{D}), V ∘ (L(b)T^{A} + λ_{E}), W (L(c)))

where a, b, c are elements of the basis C of 𝔻_{k}^{r}, D as well as E are the elements of the basis D of 𝔻_{k}^{r} and h is any smooth function.

Proof. Similarly as in the proof of Proposition 1, consider the immersion element i ∈ (T T_{k}^{r})_{0}ℝ^{m} which is of the form x_{α}^{i} = 0 for |α| ≥ 2, x_{j}^{i} = δ_{j}^{i} for j ∈ {1, …, k} and y_{0}^{i} = δ_{k+1}^{i}, y_{α}^{i} = 0 for |α| ≥ 1. On the dense subset of (J^{r+2}T)_{0}ℝ^{m} × (T^{∗}T T_{k}^{r})_{0}ℝ^{m} we can achieve by the action of the jet group G_{m}^{r+2} an element in ( T^{∗} T T_{k}^{r})_{0}ℝ^{m}

over i and j_{0}^{r+2}(\frac{∂}{∂x^{k+2}} ). In this case p_{k+2}^{α} coincide with T ∘ (L(τ^{α})T_{k}^{r}) and q_{k+2}^{α}

with V ∘ (L(τ^{α})T_{k}^{r}) for 0 ≤ |α| ≤ r. Moreover, p_{j}^{α} coincide with { \atop T ∘ λ_{j}^{β}} and q_{j}^{α}

with { \atop V ∘ λ_{j}^{β}} for 1 ≤ |α| ≤ r and j ∈ \{1, …, k\}. Finally, q_{k+1}^{α} coincide with natural

operators W (L(τ^{α}) for 0 ≤ |α| ≤ r.

Consider a subgroup H ⊆ G_{k+1}^{r+2} × id_{ℝ^{m−k−1}} stabilizing i and j_{0}^{r+2}(\frac{∂}{∂x^{k+2}} ). Its existence follows from the following transformation laws of the action of G_{m}^{r+2} on (J^{r+1}T)_{0}ℝ^{m} and on (T T_{k}^{r})_{0}ℝ^{m} given by the formulas

\begin{aligned}(4) & ȳ_{α}^{i} = a_{l_{1}...l_{p}}^{i}y_{α_{1}}^{l_{1}}…y_{α_{p}}^{l_{p}},  |α_{1}|, … |α_{p}| ≥ 1 \\ & z̄_{α}^{i} = a_{l_{1}...l_{s}}^{i}y_{α_{1}}^{l_{1}}…y_{α_{s−1}}^{l_{s−1}}z_{α_{s}}^{l_{s}}  |α_{1}|, … |α_{s−1}| ≥ 1, |α_{s}| ≥ 0 \end{aligned}

It follows that a_{j}^{i} = δ_{j}^{i} and for 2 ≤ |α| ≤ r, a_{α}^{i} = 0 and a_{k+1α}^{i} = 0 whenever α does not contain k + 1.

The following formulas yield the transformation laws of H × id_{ℝ^{m−k−1}} over i and j_{0}^{r+2}(\frac{∂}{∂x^{k+2}} ) for p_{k+1}^{α} and q_{i}^{0}

\begin{aligned}(5) & p̄_{k+1}^{α} = ã_{k+1k+1l_{1}...l_{s}}^{j}q_{j}^{αl_{1}...l_{s}} 0 ≤ s ≤ r − |α| \\ & q̄_{i}^{0} = q_{i}^{0} + ã_{il_{1}...l_{r}}^{j}q_{j}^{l_{1}...l_{r}} \end{aligned}

It follows that q_{i}^{0} can be annihilated by B_{m}^{r+1} ∩ (H × id_{ℝ^{m−k−1}}). All p_{k+1}^{α} can be annihilated by means of (H × id_{ℝ^{m−k−1}}) ∩ B_{m}^{r−|α|+2} even if |α| = r whenever k + 1 ≥ C(r + k − 1, k − 1) which is the number of multiindices of order r. It holds only for r = 1 or k = 1 or for r = 2 and k = 2. This completes the proof. □

4. We are searching all natural functions T^{∗}T T_{k}^{r}M → ℝ. In order to be simple, suppose M = ℝ^{k+1}, since for m > k +1 all our investigations can be easily modified. Consider a subgroup H stabilizing the immersion element i, which is of the form x_{j}^{i} = δ_{j}^{i}, j ∈ {1, …, k}, y_{0}^{i} = δ_{k+1}^{i} and x_{α}^{i} = 0 for |α| ≥ 2 and y_{α}^{i} = 0 for |α| ≥ 1. From the transformation laws for x_{α}^{i} and y_{α}^{i} we obtain a_{α}^{i} = 0 = a_{βk+1}^{i} for |α| ≥ 2 and |β| ≥ 1. Moreover, a_{j}^{i} = δ_{j}^{i} for j ∈ {1, …, k} and a_{k+1}^{i} = δ_{k+1}^{i}.

Consider the group projection π_{H} : G_{k+1}^{r+1} → H. Then G_{k+1}^{r+1} = H ⊕ Ker π_{H} and every j_{0}^{r+1}φ can be decomposed as π_{H}(j_{0}^{r+1}φ) ∘ (π_{H}(j_{0}^{r+1}φ^{−1}) ∘ j_{0}^{r+1}φ). In coordinates, every element of Ker π_{H} must satisfy a_{k+1k+1α}^{i} = 0 for any multiindex α including the empty one.

Weil algebra 𝔻 ⊗ 𝔻_{k}^{r} can be considered as the factor of the algebra of 𝔻_{k+1}^{r+1} by the ideal < τ_{k+1}^{2} >. Consider an inclusion ι : T T_{k}^{r} → T_{k+1}^{r+1} defined by x̄_{α}^{i} = x_{α}^{i} and x̄_{k+1α}^{i} = y_{α}^{i} if α is any multiindex not containing k + 1, 0 ≤ |α| ≤ r and other x̄_{α}^{i} put zeros. Further, let p : T_{k+1}^{r+1} → T T_{k}^{r} be the canonical projection induced by the homomorphism p̃ : 𝔻_{k+1}^{r+1} → 𝔻 ⊗ 𝔻_{k}^{r} of Weil algebras. Denote by Λ̄_{k+1}^{β} the absolute natural operator on T_{k+1}^{r+1} determined by the derivation D_{k+1}^{β} ∈ Der(𝔻_{k+1}^{r+1}), D_{k+1}^{β}(τ^{i}) = δ_{k+1}^{i}τ^{β}. Put A_{k+1}^{β} = T p ∘ Λ̄_{k+1}^{β} ∘ ι.

The operators A_{k+1}^{β} are not natural but the following lemma presents some properties of theirs, used later in the construction of the additional natural T - functions on T^{∗}T T_{k}^{r}.

Lemma 4. The functions { \atop A_{k+1}^{β}} are natural with respect to the action of Ker π_{H}.

Proof. The coordinate expression of { \atop A_{k+1}^{β}} is of the form y_{α}^{i}p_{i}^{α+β} if we distinguish the order in multiindices. Let p̄_{k+1}^{β} denote the value of p_{k+1}^{β} over the immersion element i, which can be achieved by the action of Ker π_{H}.

It follows from the transformation laws that p̄_{k+1}^{β} = b̃_{k+1}^{j}p_{j}^{αβ}, where (b_{α}^{i}) denote the coordinates of the inverse element to (a_{α}^{i}) transforming i onto (x_{α}^{i}, y_{α}^{i}). Thus we have p̄_{k+1}^{β} = y_{α}^{i}p_{i}^{αβ}, which completes the proof. □

In what follows, we shall need one notation for absolute natural operators T → T T T_{k}^{r} of all three kinds and for operators A_{k+1}^{β} defined before Lemma 4.

Let β not contain k + 1 and j ∈ {1, …, k}. We have four sets of operators T → T T T_{k}^{r}, the first three of them being natural.

(i) A_{j}^{β} = T p ∘ Λ̄_{j}^{β} ∘ ι = T ∘ λ_{j}^{β} for 1 ≤ |β| ≤ r, where Λ̄_{j}^{β} are absolute natural operators T → T T_{k+1}^{r+1} corresponding to derivations D ∈ Der(𝔻_{k+1}^{r+1}), which are of the form D : τ_{i} → δ_{j}^{i}τ^{β}.

(ii)A_{j}^{k+1β} = T p ∘ Λ̄_{j}^{k+1β} ∘ ι = V ∘ λ_{j}^{β} for 1 ≤ |β| ≤ r, where Λ̄_{j}^{k+1β} are absolute operators T → T T_{k+1}^{r+1} corresponding to derivations D ∈ Der(𝔻_{k+1}^{r+1}) of the form D : τ_{i} → δ_{j}^{i}τ^{k+1}τ^{β}.

(iii)A_{k+1}^{k+1β} = T p ∘ Λ̄_{k+1}^{k+1β} ∘ ι = W (L(τβ)), where Λ̄_{k+1}^{k+1β} are natural operators T → T T_{k+1}^{r+1} corresponding to derivations on Der(𝔻_{k+1}^{r+1}), which are of the form D : τ_{i} → δ_{k+1}^{i}τ^{k+1}τ^{β}.

(iv)A_{k+1}^{β} = T p∘Λ̄_{k+1}^{β}∘ι considered in Lemma 4 and corresponding to derivations on Der 𝔻_{k+1}^{r+1} which are of the form τ^{i} = δ_{k+1}^{i}τ^{β}.

Now, the value of every natural function in question over i is of the form h(p_{i}^{α}, q_{i}^{α}). Clearly, p_{j}^{α} and q_{j}^{α} for j ∈ {1, …, k} and 1 ≤ |α| ≤ r as well as q_{j}^{α} for 0 ≤ |α| ≤ r correspond to natural functions constructed from natural operators T → T T T_{k}^{r} by the operations . Further, q_{j}^{0} for j ∈ {1, …, k} and p_{i}^{0} can be easily annihilated by H. It follows from Lemma 4, that p_{k+1}^{α}, 1 ≤ |α| ≤ r correspond to A_{k+1}^{α} invariant with respect to the action of Ker π_{H}.

Thus we fix i and consider the action of H on p_{k+1}^{α}. It follows from the transformation laws for the action of H on p_{i}^{α} over i that

(5)  p̄_{k+1}^{α} = p_{k+1}^{α} − a_{k+1k+1γ}^{j}q_{j}^{αγ}

Let 0 ≤ s ≤ r. Then there are N_{s} = C(s + k − 1, k − 1) multiindices α satisfying |α| = s. The number of a_{k+1k+1γ}^{j} is (k + 1)N_{r−s} and all p_{k+1}^{α} can be annihilated by B_{k+1}^{r−s+2} ∩ H whenever (k + 1)N_{r−s} ≥ N_{s}.

Let (k + 1)N_{r−s} < N_{s} and M be a minimal number of this property. Then (5) can be considered in the matrix form p̄_{k+1} = p_{k+1} − Qa, which will be in more details disscussed later. This way we obtain that almost every G_{k+1}^{r+1} orbit in the sense of density forms a (k + 1) Σ_{s=M}^{r} N_{r−s}-dimensional affine subspace in the Σ_{s=M}^{r} N_{s}-dimensional affine space parametrized by p_{i}^{α} and q_{i}^{α} corresponding to natural functions constructed from natural operators T → T T T_{k}^{r}. Thus we obtain Σ_{s=M}^{r} N_{s} − (k + 1) Σ_{s=M}^{r} N_{r−s} base H-invariants and consequentely we obtain Σ_{s=M}^{r} N_{s} − (k + 1) Σ_{s=M}^{r} N_{r−s} base additional natural functions, while every natural function in question is obtained as a smooth combination of those additional ones and those constructed from natural operators T → T T T_{k}^{r}.

Now we describe the matrix Q. It can be considered as a block-matrix Q_{t}^{s} of r − M + 1 block-lines and block-columns, where s denote indices of block-lines and t indices of block-columns for s, t ∈ {0, 1, …, r − M + 1}. A block Q_{t}^{s} = 0 if t > s. Let s ≥ t. Then Q_{t}^{s} is a block-column matrix of N_{t} block-columns. Every block Q_{t,l}^{s} for l ∈ {1, …, N_{t}} contains of N_{s} lines and k + 1 columns.

We are going to construct the additional natural functions. The equation (5) in the matrix form p̄_{k+1} = p_{k+1}−Qa can be considered as systems of linear equations in a_{k+1k+1γ}^{j}q_{j}^{αγ} cosidered by block-lines if we put for some multindices the lefthand sides zeros. Then a_{k+1k+1γ}^{j} are uniquely determined and the values of other p̄_{k+1}^{α} in the block-line determine the H-invariants. In the next block-line Q_{s} we use the already expressed a_{k+1k+1δ}^{j} for |δ| ≤ s and by them we express further H-invariants. We use the identifications

(6)  p_{k+1}^{α} ≅ { \atop A_{k+1}^{α}}  and  q_{j}^{α} ≅ { \atop A_{j}^{αk+1}}

and this way obtain the additional natural functions on T^{∗}T T_{k}^{r}, which follows from Lemma 4.

Now we construct the base of the additional natural functions. Let Mult_{s} denote the set of all multiindices of order s.

In the first step we costruct natural functions from p_{k+1}^{α}, where α are multiindices of order r. Let I_{r} denote a selection of multiindices α_{1}, …, α_{k+1} for which the vectors q^{α_{1}}, …, q^{α_{k+1}} are linearly independent, which is possible on a dense subset of ℝ^{(k+1)N_{r}}. In the first block-line of (5) considered in the matrix form put p̄_{k+1}^{α_{1}} = ⋯ = p̄_{k+1}^{α_{k+1}} = 0, which uniquely determines all a_{k+1k+1}^{j}. Their coordinate expressions are of the form

(7)  a_{k+1k+1}^{j} = \frac{{ \atop A_{k+1}^{α_{s}}} · Ort^{j}({ \atop A_{i}^{k+1α_{s}}})}{{ \atop A_{j}^{k+1α_{s}}} · Ort^{j}({ \atop A_{i}^{k+1α_{s}}})}  α_{s} ∈ I_{r}, i, j ∈ \{1, …, k + 1\}

where Ort^{j}({ \atop A_{i}^{k+1α_{s}}}) denotes the orthogonal complement of column vectors form-

ing the matrix ({ \atop A_{i}^{k+1α_{s}}}) of order k + 1 and · denotes the scalar product. By means of a_{k+1k+1}^{j} we obtain natural functions F_{I_{r}}^{l} corresponding to the selection of multiindices I_{r}

(8) F_{I_{r}}^{l} = { \atop A_{k+1}^{α_{l}}} − \frac{{ \atop A_{k+1}^{α_{s}}} · Ort^{j}({ \atop A_{i}^{k+1α_{s}}})}{{ \atop A_{j}^{k+1α_{s}}} · Ort^{j}({ \atop A_{i}^{k+1α_{s}}})}{ \atop A_{j}^{k+1α_{s}}}  α_{s} ∈ I_{r}, l ∈ Mult_{r} \I_{r}

In the points, where the denominator is equal zero, F_{I_{r}}^{l} can be multiplied in the neighbourhood and F_{I_{r}}^{l} can be replaced by G_{I_{r}}^{l}, which are of the form

(9) G_{I_{r}}^{l} = ({ \atop A_{j}^{k+1α_{s}}} · Ort^{j}({ \atop A_{i}^{k+1α_{s}}})){ \atop A_{k+1}^{α_{l}}}−({ \atop A_{k+1}^{α_{s}}}· Ort^{j}({ \atop A_{i}^{k+1α_{s}}})){ \atop A_{j}^{k+1α_{s}}}

Clearly, G_{I_{r}}^{l} takes the zero value in the point, where the denominator is zero and it is a natural T^{∗}T T_{k}^{r}-function.

In the next steps, we are searching the additional natural T^{∗}T T_{k}^{r}-functions corresponding to multiindices of lower orders by a reccurent procedure. Fix an element ω = (p_{i}^{α}, p_{i}^{α}) over i. Let r−s ≥ M and suppose the selections I_{r}, …, I_{r−s+1} of multiindices for which the corresponding G_{I_{r}}^{l}, …, G_{I_{r},...,I_{r−s+1}}^{l} take a non-zero value. It is satisfied on the dense subset.

Select a set of multiindices of order r − s denoted by I_{r−s} so that the vectors q^{α_{1}}, …, q^{α_{(k+1)N_{r−s}}} be linearly independent. Similarly as in the first step we consider the formula (5) in the matrix form and put p̄_{k+1}^{α_{1}} = ⋯ = p̄_{k+1}^{α_{(k+1)N_{r−s}}} = 0 on the left hand side. Using the identifications (6) we have

(10)  F_{I_{r},...,I_{r−s}}^{l} = { \atop A_{k+1}^{α_{l}}} − a_{k+1k+1γ}^{j} { \atop A_{j}^{α_{l}γk+1}},  0 ≤ |γ| ≤ s

where all a_{k+1k+1γ}^{j} for |γ| ≤ s − 1 are determined in the previous steps. By the following formula we can reccurently compute a_{k+1k+1γ}^{j} for |γ| = s

(11)  a_{k+1k+1γ}^{j} = \frac{({ \atop A_{k+1}^{α_{t}}} − a_{k+1k+1δ}^{h}{ \atop A_{h}^{k+1α_{t}δ}}) · Ort_{γ}^{j}({ \atop A_{i}^{α_{t}εk+1}})}{{ \atop A_{j}^{α_{t}γk+1}} · Ort_{γ}^{j}({ \atop A_{i}^{α_{t}εk+1}})}

where δ, ε are multiindices satisfying |ε| = |γ| and 0 ≤ |δ| < |γ|. If the denominator in (11) is zero, we can consider the natural T^{∗}T T_{k}^{r}- functions G_{I_{r},...,I_{r−s}}^{l} instead F_{I_{r},...,I_{r−s}}^{l}, where

(12)  G_{I_{r},...,I_{r−s}}^{l} = ({ \atop A_{j}^{α_{t}γk+1}} · Ort_{γ}^{j}({ \atop A_{i}^{α_{t}εk+1}}))F_{I_{r},...,I_{r−s}}^{l}

Then G_{I_{r},...,I_{r−s}}^{l} take zero value, in the points where the denominator in (10) is zero.

Thus we proved the following theorem, giving all natural T -functions on T^{∗}T T_{k}^{r}, since the step including the non-absolute natural T -functions is the same as in Proposition 1 or in Proposition 3.

Theorem 5. Let M be a manifold satisfying dim M ≥ k + 2. Then all natural T -functions T^{∗}T T_{k}^{r}M → ℝ are of the form

(13)  h(T ∘ ( L( a) T_{k}^{r}), V ∘ ( L( b) T^{A}), W (L(c))), G_{I_{r},...,I_{r−s}}^{l})

for any smooth function h, 0 ≤ s ≤ M − r and l ∈ {(k + 1)N_{r−s} + 1, …, N_{s}}.

Moreover, in the s-th step we can choose only N_{r−s} − (k + 1)N_{s} base additional natural functions G_{I_{r},...,I_{r−s}}^{l}.

5. We present a corollary of Theorem 5, giving the classification of all natural operators T → T T^{∗}T_{k}^{r}, which is the result published in [8] . This result is obtained in the different form and we present it little modified.

Let x_{α}^{i} denote the canonical coordinates on T_{k}^{r}M and q_{i}^{α}dx_{α}^{i} define the additional coordinates on T^{∗}T_{k}^{r}M. Let

(14) A_{M}X(x_{α}^{i}, q_{i}^{α}) = Y_{α}^{i}(j_{0}^{r+1}X, x_{α}^{i}, q_{i}^{α})\frac{∂}{∂x_{α}^{i}} + Q_{i}^{α}(j_{0}^{r+1}X, x_{α}^{i}, q_{i}^{α})\frac{∂}{∂q_{i}^{α}}

be the expression of a natural operator A_{M} in coordinates, X being a vector field on M. Let us define by π_{i}^{α}dx_{α}^{i} + ρ_{α}^{i}dq_{i}^{α} the additional coordinates on T^{∗}T^{∗}T_{k}^{r}M.

Since natural operators T M → T T^{∗}T_{k}^{r}M are in the canonical bijection with natural T -functions g_{M} : T^{∗}T^{∗}T_{k}^{r}M → ℝ linear on fibers of T^{∗}T^{∗}T_{k}^{r}M → T^{∗}T_{k}^{r}M,

(14) can be identified with

(15)  g_{M} = Y_{α}^{i}(j_{0}^{r+1}X, x_{α}^{i}, q_{i}^{α})π_{i}^{α} + Q_{i}^{α}(j_{0}^{r+1}X, x_{α}^{i}, q_{i}^{α})ρ_{α}^{i}.

Natural T -functions g_{M} are in the canonical bijection with natural T -functions f_{M} : T^{∗}T T_{k}^{r}M → ℝ, given by f_{M} = g_{M} ∘ t_{T_{k}^{r}M} ∘ s_{T_{k}^{r}M}^{−1}, where s_{M} : T T^{∗}M → T^{∗}T M is the natural equivalence by Modugno-Stefani, [7] and t_{M} : T T^{∗}M → T^{∗}T^{∗}M is the natural equivalence by Kolář-Radziszewski, [6].

Let y_{α}^{i} = dx_{α}^{i} define the additional coordinates on T T_{k}^{r}M and p_{i}^{α}dx_{α}^{i} + s_{i}^{α}dy_{α}^{i} on T^{∗}T T_{k}^{r}M. The coordinate expression of t_{T_{k}^{r}M} ∘ s_{T_{k}^{r}M}^{−1} yields

(16)  y_{α}^{i} = −ρ_{α}^{i}  p_{i}^{α} = π_{i}^{α}  s_{i}^{α} = q_{i}^{α}

Thus natural operators A_{M} : T M → T T^{∗}T_{k}^{r}M are identified with natural T - functions f_{M} : T^{∗}T T_{k}^{r}M → ℝ which are of the form Y_{α}^{i}(j_{0}^{r+1}X, x_{α}^{i}, q_{i}^{α})p_{i}^{α} − Q_{i}^{α}(j_{0}^{r+1}X, x_{α}^{i}, q_{i}^{α})y_{α}^{i}. Having a natural operator A_{M} : T M → T T_{k}^{r}M such that the corresponding natural T -function Ã_{M} on T^{∗}T T_{k}^{r}M is linear in (y_{α}^{i}, p_{i}^{α}), we

obtain a natural operator ˜ Ã_{M} : T M → T T^{∗}T_{k}^{r}M. The base of natural opeators A_{M} : T M → T T_{k}^{r}M satisfying these properties of linearity is formed by T ∘L(c)T_{k}^{r}, T ∘ λ_{j}^{β} and W (L(c)).

Proposition 6. For a manifold M satisfying dim M ≥ k + 2 all natural operators T M → T T^{∗}T_{k}^{r}M are of the form

h(V ∘ ( L( d) T_{k}^{r}), V ∘ λ_{t}^{μ}) T ∘ ( L(c) T_{k}^{r}) + h_{β}^{j}(V ∘ ( L( d) T_{k}^{r}), V ∘ λ_{t}^{μ}) T ∘ λ_{j}^{β}+

h_{c}(V ∘ ( L( d) T_{k}^{r}), V ∘ λ_{t}^{μ}) W (L(c))

Proof. Among all natural T -functions f_{M} : T^{∗}T T_{k}^{r}M → ℝ we are searching for those, which are linear in (y_{α}^{i}, p_{i}^{α}). All natural T -functions f_{M} are of the form

h(T ∘( L( a) T_{k}^{r}), V ∘( L( b) T^{A}), W (L(c))), G_{I_{r},...,I_{r−s}}^{l}). The coordinate expression of G_{I_{r},...,I_{r−s}}^{l} is a function of products y_{α}^{i}p_{i}^{β} and natural T -functions constructed from natural operators T → T T T_{k}^{r}. It holds Y_{α}^{i} = \frac{∂f}{∂p_{i}^{α}} = \frac{∂f}{∂p_{i}^{α}} _{|y_{β}^{i}=0=p_{i}^{γ}} =_{\frac{∂f}{∂G_{Ir ,...,I_{r−s}}^{l}}} ·

0+f_{1}(j_{0}^{r+1}X, x_{α}^{i}, q_{i}^{α}). Thus Y_{α}^{i} does not depend on G_{I_{r},...,I_{r−s}}^{l} and analogously we prove that neither does Q_{i}^{α}. This completes the proof. □

We are going to give a more geometrical form of this proposition. We need the following lemma.

Lemma 7. Let Y : T M → T T M be a linear vector field, ([5]). Then there is a

unique vector field ˜ Ỹ : T^{∗}M → T T^{∗}M satisfying ˜ ˜ Ỹ ∘ t_{M} ∘ s_{M}^{−1} = Ỹ. Moreover, ˜ Ỹ is dual to Y.

Proof. Let Y = ξ^{i}(x)\frac{∂}{∂x^{i}} + η_{j}^{i}(x)y^{j} \frac{∂}{∂y^{i}}. Further, let p_{i}dx^{i} + q_{i}dx^{i} define the additional coordinates on T^{∗}T M. Then Ỹ = ξ^{i}(x)p_{i} + η_{j}^{i}(x)y^{j}q_{i}. Define the additional coordinates on T^{∗}M by α_{i}dx^{i} and on T^{∗}T^{∗}M by ρ_{i}dx^{i} + σ^{i}dα_{i}. If

˜Ỹ = U^{i} \frac{∂}{∂x^{i}} + V_{i} \frac{∂}{∂α_{i}}, then ˜ ˜ Ỹ = U^{i}ρ_{i} + V_{i}σ^{i}. It follows from the definitions of the natural equivalences by Modugno-Stefani, ([7]), Kolář-Radziszewski, ( [6] ) and their coordinate descriptions that ρ_{i} = p_{i} and σ^{i} = −y^{i} and consquentely we have

˜˜Ỹ = p_{i}U^{i} − y^{i}V_{i}. Thus we have V_{i} = −η_{i}^{j}(x)q_{j}, which completes the proof. □

Lemma 8. Let Y = T X be the flow prolongation of a vector field X. Then

˜Ỹ = T^{∗}X.

Proof. Using the coordinates from Lemma 7, we have

T X=X^{i}(x)\frac{∂}{∂x^{i}} + X_{j}^{i}(x)y^{j}\frac{∂}{∂y^{i}} .

One can easily verify that

T^{∗}X = X^{i}(x)\frac{∂}{∂x^{i}} − X_{i}^{j}(x)\frac{∂}{∂q_{i}} .

Thus we have T ∘ (L(c)T_{k}^{r}) = T^{∗} ∘ (L(c)T_{k}^{r}) and ; { \atop T ∘ λ_{j}^{β}} = T^{∗} ∘ λ_{j}^{β}. Moreover,

we have ; { \atop W (L(c))} = W^{∗}(L(c)) = −W (L^{∗}(c)), where L^{∗}(c) denotes the dual map to L(c). Now we can present Proposition 6 in the following form.

Proposition 9. All natural operators T → T T^{∗}T_{k}^{r} are of the form

h_{c}({ \atop L(d)T_{k}^{r}}, λ̃_{t}^{μ})T^{∗} ∘ (L(c)T_{k}^{r}) + h_{β}^{j}({ \atop L(d)T_{k}^{r}}, λ̃_{t}^{μ})T^{∗} ∘ λ_{j}^{β} + h_{c}({ \atop L(d)T_{k}^{r}}, λ̃_{t}^{μ})W (L^{∗}(c))

for any basis of 𝔻_{k}^{r} formed by c, d, 1≤ |β|,|μ|≤r and for any smooth functions h_{c}, h_{β}^{j}.

Proof. It remains to investigate the arguments of functions, h_{c}, h_{β}^{j}. This is quite similar to [8].

I thank prof. I. Kolář for his useful help and suggestions.

References

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[2] Kolář, I., Covariant Approach to Natural Transformations of Weil Bundles, Comment. Math. Univ. Carolinae (1986).

[3] Kolář, I., On Cotagent Bundles of Some Natural Bundles, Rendiconti del Circolo Matematico di Palermo Serie II-Numero 37 (1994), 115–120.

[4] Kolář, I., On the Natural Operators on Vector Fields, Ann. Global Anal. Geometry 6 (1988), 109–117.

[5] Kolář, I., Michor, P. W., Slovák, J., Natural Operations in Differential Geometry, Springer – Verlag, 1993.

[6] Kolář, I., Radziszewski, Z., Natural Transformations of Second Tangent and Cotangent Bundles, Czechoslovak Math. (1988), 274–279.

[7] Modugno, M., Stefani, G., Some results on second tangent and cotangent spaces, Quaderni dell’Università di Lecce (1978).

[8] Tomáš, J., Natural operators on vector fields on the cotangent bundles of the bundles of (k, r)-velocities, Rendiconti del Circolo Matematico di Palermo Serie II Numero 54 (1998), 113–124.

[9] Tomáš, J., Some natural operators on vector fields, Archivum Mathematicum 31 (1995), 239–249.

Department of Mathematics, Faculty of Civil Engineering Technical University Brno Žižkova 17, 602 00 Brno, CZECH REPUBLIC E-mail: mdtom@fce.vutbr.cz