Lobachevskii Journal of Mathematics Vol. 12, 2003, 51 – 61 \begin{matrix}c \\ ◯ \end{matrix}Miroslav Kureš, Włodzimierz M. Mikulski

Miroslav Kureš, Włodzimierz M. Mikulski

LIFTINGS OF LINEAR VECTOR FIELDS TO PRODUCT PRESERVING GAUGE BUNDLE FUNCTORS ON VECTOR

BUNDLES

(submitted by B. N. Shapukov)

Abstract. All natural operators lifting linear vector fields to product preserving gauge bundle functors on vector bundles are classified. Some relevant properties of Weil modules are studied, too.

Introduction

Let F: VB → FM be a covariant functor from the category VB of all vector bundles and their vector bundle homomorphisms into the category FM of fibered manifolds and their fiber maps. Let B_{VB}: VB → Mf and B_{FM}: FM → Mf be the respective base functors. A gauge bundle functor on VB is a functor F satisfying

G I. (Base-preservation) B_{FM} ∘ F = B_{VB} G II. (Locality) for every inclusion of an open vector subbundle i_{E|U}: E|U →

E, F (E|U) is the restriction p_{E}^{−1}(U) of p_{E}: F E → B_{VB}(E) over U and

F i_{E|U} is the inclusion p_{E}^{−1}(U) → F E

Given two gauge bundle functors F_{1}, F_{2} on VB, by a natural transformation τ: F_{1} → F_{2} we mean a system of base preserving fibered maps τ_{E}: F_{1}E → F_{2}E for every vector bundle E satisfying F_{2}f ∘τ_{E} = τĒ ∘F_{1}f for every vector bundle homomorphism f: E → Ē. If proj_{1} and proj_{2} are projections in VB of E_{1} ×E_{2} to E_{1} and E_{2}, respectively, then we say that a gauge bundle functor F on VB is product preserving if F E_{1} ⟵^{F proj_{1}} F (E_{1} ×E_{2}) ⟶^{F proj_{2}} F E_{2} is a product diagram in FM.

The structure of the paper is the following. The properties of Weil modules are studied in the Section 1. We refer to [4], [5], where Weil modules are also used. We publicize a number of examples of Weil modules in 1.3 and generalize the classical result from the Lie group theory about the identification of algebra of derivations and Lie algebra of the group of automorphisms in 1.6. The canonical vector fields on vector bundles with m-dimensional base and ndimensional fibers are classified in Section 2. The Section 3 contains the main results of the paper. Namely, the theorem classifying all natural operators lifting linear vector fields to product preserving gauge bundle functors on vector bundles is expressed in 3.5.

2000 Mathematical Subject Classification. 58A32, 53A55. Secondary 22E60.

Key words and phrases. Lifting, product preserving functor, vector bundle, Weil module.

The first author was supported by GA CR, grant No. 201/02/0225..

All manifolds in the paper are assumed to be Hausdorff, finite-dimensional, without boundary and of class C^{∞}. All maps are assumed to be of class C^{∞}.

1. Weil modules

1.1. We denote by WA the category with Weil algebras as objects and algebra homomorphisms of Weil algebras as morphisms. Then the classical result of Kainz and Michor, Luciano and Eck reads as follows (see [2]). Theorem A. Product preserving bundle functors from the category Mf of manifolds into the category FM of fibered manifolds are in bijection with objects of WA and natural transformations between two such functors are in bijection with the morphisms of WA.

In particular, for a given Weil algebra A we can construct a product preserving bundle functor T^{A} by

T^{A}M:= ⋃_{z∈M}{ϕ | ϕ ∈ Hom(C_{z}^{∞}(M), A)},

where M is an object of Mf, C_{z}^{∞}(M) is the algebra of germs of smooth functions from M into ℝ in z ∈ M and Hom(C_{z}^{∞}(M), A) is the set of all algebra homomorphisms from C_{z}^{∞}(M) into A.

1.2. Let A be a Weil algebra and V a A-module. If V is finite-dimensional as a real vector space, we name it the Weil A-module. More precisely, Weil A-module is a triple (A, V, ∗), where A is an object of WA, V is a finitedimensional real vector space, and ∗: A × V → V is a bilinear mapping endowing V with a structure of A-module; we write av instead ∗(a, v) (for a ∈ A, v ∈ V). If there is no confusion, we write (A, V) or only V instead (A, V, ∗) as well.

For a clear understanding, the roman font (Hom, End, Aut, …) is used for algebra homomorphisms and the italic font (Hom, End, Aut, …) for homomorphisms of vector spaces from now on. If we write no subscript, we think the homomorphisms over ℝ. If A, B be two Weil algebras, V a Weil A-module and W a Weil B-module, then we can consider a homomorphism ψ from V to W as a homomorphism of ℝ-modules (real vector spaces) and write ψ ∈ Hom(V, W). Apart from that, we can consider an algebra homomorphism ϕ from A to B, i.e. ϕ ∈ Hom(A, B), and a map ψ from V to W satisfying ψ(v + w) = ψ(v) + ψ(w), ψ(av) = ϕ(a)ψ(v) for all a ∈ A, v, w ∈ V. In this case we write ψ ∈ Hom_{A,B}(V, W). (If A = B, we write Hom_{A}(V, W) instead Hom_{A,A}(V, W).) Of course, ψ ∈ Hom_{A,B}(V, W) ⟹ ψ ∈ Hom(V, W). It is evident that, for a given ψ, the relation ψ(av) = ϕ(a)ψ(v) is not satisfied for an arbitrary ϕ ∈ Hom(A, B); if it is satisfied, we say that ψ is over ϕ. We write a pair of homomorphisms ϕ ∈ Hom(A, B), ψ ∈ Hom_{A,B}(V, W), where ψ is over ϕ, exclusively in brackets ⟨⟩, i.e. ⟨ϕ, ψ⟩.

We obtain the category WM, objects of which are Weil modules, and morphisms of which are pairs ⟨ϕ, ψ⟩ consisting of a homomorphism ϕ ∈ Hom(A, B), ψ ∈ Hom_{A,B}(V, W), where ψ is over ϕ.

1.3. We introduce some examples of Weil modules.

(i) An arbitrary ideal 𝔦 of a Weil algebra A is a Weil A-module. In particular, A is a Weil A-module.

(ii) If 𝔦 is an ideal of a Weil algebra A and V a Weil A-module, then 𝔦V is

a Weil A-module. (iii) If V and W are Weil A-modules of a Weil algebra A, the V ⊕ W is

a Weil A-module. Further, by a free Weil A-module we mean a Weil

A-module V in the form V = ⨁_{i=1}^{n} V_{i}, where every V_{i} is isomorphic with

the Weil A-module A. (iv) If A and B are Weil algebras and if A is a subalgebra of B, then B

is a Weil A-module, in which ∗ operates usually (not necessarily) as product in B.

(v) If ϕ ∈ Hom(B, A) is a homomorphism of Weil algebras and if V is a Weil A-module, then V is also B-module as we can define bv as ϕ(b)v (for b ∈ B and v ∈ V). In particular, for a Weil algebra with the order ord(A) = r and with the width w(A) = k there is an epimorphism π_{A}: 𝔻_{k}^{r} → A and that is why every A-module is a 𝔻_{k}^{r}-module as well.

(vi) If U is a vector space and if V is a Weil A-module, then Hom(U, V) is

a Weil A-module. (vii) If V and W are Weil A-modules, then Hom_{A}(V, W) is a Weil A-module. (viii) If A and B are Weil algebras and if V is a A-module and W is a B-

module, then V ⊗ W is a Weil (A ⊗ B)-module.

1.4. The second author have generalized Theorem A as follows (see [3]). Theorem B. Product preserving gauge bundle functors from the category VB of vector bundles into the category FM of fibered manifolds are in bijection with objects of WM and natural transformations between two such functors are in bijection with the morphisms of WM.

In particular, for a given Weil module V we can construct a product preserving bundle functor T^{A,V} by

T^{A,V} E:= ⋃_{z∈M}{⟨ϕ, ψ⟩|ϕ ∈ Hom(C_{z}^{∞}(M), A), ψ ∈ Hom_{C_{z}^{∞}(M),A}(C_{z}^{∞, f.l.}(E), V)},

where E is an object of VB, C_{z}^{∞}(M) is the algebra of germs of smooth functions from M into ℝ in z ∈ M, C_{z}^{∞, f.l.}(E) is the C_{z}^{∞}(M)-module of germs of smooth fiber linear functions from E into ℝ in z ∈ M, Hom(C_{z}^{∞}(M), A) is the set of all algebra homomorphisms from C_{z}^{∞}(M) into A and Hom_{C_{z}^{∞}(M),A}(C_{z}^{∞, f.l.}(E), V) is the set of all module homomorphisms from C_{z}^{∞, f.l.}(E) into V.

1.5. Let A be a Weil algebra. The group Aut(A) of automorphisms of A is a closed subgroup of the Lie group GL(A), so therefore Aut(A) is a Lie group. The Lie algebra of GL(A) is 𝔤𝔩(A) = End(A) and the Lie algebra of Aut(A) is

𝔞𝔲𝔱(A) = {D ∈ End(A) | exp(tD) ∈ Aut(A) ∀t ∈ ℝ}.

The algebra of derivations of A is defined as

Der(A) = {D ∈ End(A) | D(ab) = D(a)b + aD(b) ∀a, b ∈ A}.

In the classical Lie group theory, the identification Der(A) = 𝔞𝔲𝔱(A) is proved.

1.6. Let A be a Weil algebra and V a Weil A-module. The group Aut(A, V) consists of pairs ⟨ϕ, ψ⟩, where ϕ ∈ Aut(A) and ψ ∈ Aut_{A}(V); Aut(A, V) is a closed subgroup of the Lie group GL(A) × GL(V), so therefore Aut(A, V) is a Lie group. The Lie algebra of Aut(A, V) is

𝔞𝔲𝔱(A, V) = {(D, Δ) ∈ End(A) × End(V) | ⟨exp(tD), exp(tΔ)⟩ ∈ Aut(A) × Aut_{A}(V)
∀t ∈ ℝ}.

Further, the algebra of derivations of (A, V) is

Der(A, V) = {(D, Δ) ∈ End(A) × End(V) | D(ab) = D(a)b + aD(b) ∀a, b ∈ A,

Δ(av) = D(a)v + aΔ(v) ∀a ∈ A, v ∈ V }.

We improve the classical result from 1.5 by the following assertion. Another useful generalization is e.g. in [6].

Proposition 1. Der(A, V) = 𝔞𝔲𝔱(A, V).

Proof. Step 1. Let us consider (D, Δ) ∈ 𝔞𝔲𝔱(A, V). The condition exp(tD) ∈ Aut(A) means

exp(tD)(ab) = exp(tD)(a) exp(tD)(b).

The differentiating with respect to t gives

exp(tD)D(ab) = exp(tD)D(a) exp(tD)(b) + exp(tD)(a) exp(tD)D(b),

or when evaluated at t = 0,

D(ab) = D(a)b + aD(b).

Similarly, exp(tΔ) is over exp(tD) means

exp(tΔ)(av) = exp(tD)(a) exp(tΔ)(v).

The differentiating with respect to t gives

exp(tΔ)Δ(av) = exp(tD)D(a) exp(tΔ)(v) + exp(tD)(a) exp(tΔ)Δ(v),

or when evaluated at t = 0,

Δ(av) = D(a)v + aΔ(v).

We have proved (D, Δ) ∈ 𝔞𝔲𝔱(A, V) ⟹ (D, Δ) ∈ Der(A, V).

Step 2. Let us consider (D, Δ) ∈ Der(A, V). Take P (t):= exp(tD)(ab), S(t):= exp(tD)(a) exp(tD)(b). Then

\frac{d}{dt}P (t) = \frac{d}{dt} exp(tD)(ab) = exp(tD)D(ab) = D exp(tD)(ab) = DP (t)

and

P (0) = ab.

Also,

\frac{d}{dt}S(t) = \frac{d}{dt}(exp(tD)(a) exp(tD)(b)) = exp(tD)D(a) exp(tD)(b) + exp(tD)(a) exp(tD)D(b)
= D exp(tD)(a) exp(tD)(b) + exp(tD)(a)D exp(tD)(b)D exp(tD)(a) exp(tD)(b)
= DS(t)

and

S(0) = ab.

We have

\frac{d}{dt}P (t) = DP (t) and P (0) = ab,
\frac{d}{dt}S(t) = DS(t) and S(0) = ab.

It means P (t), S(t) satisfy the same first order differential equations and initial conditions. Hence P (t) = S(t).

Now, take Π(t):= exp(tΔ)(av), Σ(t):= exp(tD)(a) exp(tΔ)(v). Then

\frac{d}{dt}Π(t) = \frac{d}{dt} exp(tΔ)(av) = exp(tΔ)Δ(av)Δ exp(tΔ)(av) = ΔΠ(t)

and

Π(0) = av.

Also,

\frac{d}{dt}Σ(t) = \frac{d}{dt}(exp(tD)(a) exp(tΔ)(v)) = exp(tD)D(a) exp(tΔ)(v) + exp(tD)(a) exp(tΔ)Δ(v)
= D exp(tD)(a) exp(tΔ)(v) + exp(tD)(a)Δ exp(tΔ)(v)
= Δ exp(tD)(a) exp(tΔ)(v) = ΔΣ(t)

and

Σ(0) = av.

We have

\frac{d}{dt}Π(t) = ΔΠ(t) and Π(0) = av,
\frac{d}{dt}Σ(t) = ΔΣ(t) and Σ(0) = av.

It means Π(t), Σ(t) satisfy the same first order differential equations and initial conditions. Hence Π(t) = Σ(t). We have proved (D, Δ) ∈ Der(A, V) ⟹ (D, Δ) ∈ 𝔞𝔲𝔱(A, V), too. □

1.7. Now, we derive the result about the important case V = A. For it, we define the algebra of fissions of A as

Fis(A) = {F ∈ End(A) | F (ab) = aF (b) ∀a, b ∈ A}.

Then the following assertion holds.

Proposition 2. Der(A, A) = {(D, D + F) | D ∈ Der(A), F ∈ Fis(A)}.

Proof. By the definition,

Der(A, A) = {(D, Δ) ∈ End(A) × End(A) | D(ab) = D(a)b + aD(b),
Δ(ab) = D(a)b + aΔ(b) ∀a, b ∈ A}.

It follows

D(ab) − aD(b) = Δ(ab) − aΔ(b).

Let F:= Δ−D. Then F ∈ End(A) and relation above reads as F (ab) = aF (b). Conversely, if we take D ∈ Der(A) and F ∈ Fis(A), then for Δ:= D + F

Δ(ab) = D(ab) + F (ab) = D(a)b + aD(b) + aF (b) = D(a)b + aD(b) + a(Δ(b) − D(b))
= D(a)b + aΔ(b)

is satisfied. □

1.8. We illustrate the previous result on the Weil algebra A = 𝔻 = ℝ[t]/⟨t^{2}⟩. The elements A can be expressed as

k_{1} + k_{2}t, where k_{1}, k_{2} ∈ ℝ.

It is not difficult derive that every D ∈ Der(A) has a form

k_{1} + k_{2}t ↦ qk_{2}t, q ∈ ℝ,

and every F ∈ Fis(A) has a form

k_{1} + k_{2}t ↦ qk_{1} + (rk_{1} + qk_{2})t, q, r ∈ ℝ.

2. Absolute operators

2.1. We denote by VB_{m,n} the category of vector bundles with m-dimensional base and n-dimensional fibers together with local vector bundle isomorphisms.

Proposition 3. Let m, n ∈ ℕ and let F: VB → FM be a product preserving gauge bundle functor. Every natural transformation τ^{⋄}: F |_{VB_{m,n}} → F |_{VB_{m,n}} can be extended uniquely to a natural transformation τ: F → F.

Proof. Step 1. Let x^{1}, …, x^{m}, y^{1}, …, y^{n} be the usual coordinates on the trivial vector bundle ℝ^{m} × ℝ^{n} → ℝ^{m}, which is an object of VB_{m,n}. Let us consider a natural transformation τ^{⋄}: F |_{VB_{m,n}} → F |_{VB_{m,n}}. As we have F (ℝ^{m} × ℝ^{n}) ≅ A^{m} × V^{n}, the transformation τ^{⋄} corresponds with the map

T: A^{m} × V^{n} → A^{m} × V^{n},
T: (a^{1}, …, a^{m}, v^{1}, …, v^{n}) ↦ (ϕ^{1}(a^{1}, …, a^{m}, v^{1}, …, v^{n}), …, ϕ^{m}(a^{1}, …, a^{m}, v^{1}, …, v^{n}),

ψ^{1}(a^{1}, …, a^{m}, v^{1}, …, v^{n}), …, ψ^{n}(a^{1}, …, a^{m}, v^{1}, …, v^{n})),

where a^{1}, …, a^{m} ∈ A, v^{1}, …, v^{n} ∈ V. Thus, by the invariance of τ^{⋄} with respect to the homotheties

(x^{1}, …, x^{m}, y^{1}, …, y^{n}) ↦ (K_{1}x^{1}, …, K_{m}x^{m}, L_{1}y^{1}, …, L_{n}y^{n}),

(K_{1}, …, K_{m}, L_{1}, …, L_{n} ∈ ℝ^{+}) which represent morphisms of VB_{m,n}, we have obtained the homogeneity conditions

K_{1}ϕ^{1}(a^{1}, …, a^{m}, v^{1}, …, v^{n}) = ϕ^{1}(K_{1}a^{1}, …, K_{m}a^{m}, L_{1}v^{1}, …, L_{n}v^{n})
…
K_{m}ϕ^{m}(a^{1}, …, a^{m}, v^{1}, …, v^{n}) = ϕ^{m}(K_{1}a^{1}, …, K_{m}a^{m}, L_{1}v^{1}, …, L_{n}v^{n})
L_{1}ψ^{1}(a^{1}, …, a^{m}, v^{1}, …, v^{n}) = ψ^{1}(K_{1}a^{1}, …, K_{m}a^{m}, L_{1}v^{1}, …, L_{n}v^{n})
…
L_{n}ψ^{n}(a^{1}, …, a^{m}, v^{1}, …, v^{n}) = ψ^{n}(K_{1}a^{1}, …, K_{m}a^{m}, L_{1}v^{1}, …, L_{n}v^{n}).

Inasmuch the homogeneous function theorem (see e.g. [2]) holds, ϕ_{i} depends linearly on a_{i} (i = 1, …, m), ψ_{j} depends linearly on v_{j} (j = 1, …, n) and they do not depend on the remaining a’s and v’s. As to permissible permutations in A^{m} × V^{n}, we have that ϕ_{1} = ⋯ = ϕ_{m} =: ϕ belongs to End(A) and ψ_{1} = ⋯ = ψ_{m} =: ψ belongs to End(V).

Step 2. We need to prove that ϕ ∈ End(A) and ψ is over ϕ. As T is invariant with respect to the morphism of VB_{m,n}

(x^{1}, …, x^{m}, y^{1}, …, y^{n}) ↦ (x^{1} + (x^{1})^{2}, x^{2}, …, x^{m}, y^{1}, …, y^{n}),

we have

ϕ(a + a^{2}) = ϕ(a) + (ϕ(a))^{2}

and the linearity of ϕ yields

ϕ(a^{2}) = (ϕ(a))^{2}.

It gives for a = b + c

ϕ(b^{2} + 2bc + c^{2}) = ϕ((b + c)^{2}) = (ϕ(b + c))^{2} = (ϕ(b) + ϕ(c))^{2},

whence

ϕ(bc) = ϕ(b)ϕ(c)

is satisfied for all b, c ∈ A and that is why ϕ ∈ End(A).

Further, as T is invariant with respect to the morphism of VB_{m,n}

(x^{1}, …, x^{m}, y^{1}, …, y^{n}) ↦ (x^{1}, …, x^{m}, y^{1}, y^{1} + x^{1}y^{1}, y^{2}, …, y^{n}),

we have

ψ(v + av) = ψ(v) + ϕ(a)ψ(v)

and the linearity of ψ yields

ψ(av) = ϕ(a)ψ(v)

and that is why ψ is over ϕ.

Thus, ⟨ϕ, ψ⟩ is a morphism of the category WM.

Step 3. We denote by τ: F → F the natural transformation which correspond with ⟨ϕ, ψ⟩. Clearly, τ^{⋄} is the restriction of τ. If τ̄: F → F is another natural transformation, restriction of which is τ^{⋄}, then τ̄ = τ, because τ and τ̄ determine the same morphism ⟨ϕ, ψ⟩. □

In particular, the corollary of [3] and the previous proposition is, that natural transformations of F |_{VB_{m,n}} into itself correspond exactly to Aut(A, V).

2.2. Let us consider (D, Δ) ∈ Der(A, V) = 𝔞𝔲𝔱(A, V). Then ⟨exp(tD), exp(tΔ)⟩ is a 1-parameter subgroup of Aut(A, V), i.e. it can be considered as a 1parameter subgroup of natural transformations F |_{VB_{m,n}} → F |_{VB_{m,n}}. If Y is an object of VB_{m,n}, then ⟨exp(tD), exp(tΔ)⟩ defines the flow on F Y of a vector field on F M, and accordingly, this vector field. We denote it by op(D, Δ). So, the vector field op(D, Δ) on F Y is the canonical VB_{m,n}-invariant.

Proposition 4. Let m, n ∈ ℕ and let F: VB → FM be a product preserving gauge bundle functor with the corresponding Weil module (A,V). Then every canonical vector field (or absolute operator) on F have a form op(D, Δ) for some (D, Δ) ∈ Der(A, V).

Proof. Let us consider such a vector field Ξ; the flow of Ξ on F Y for an object Y of VB_{m,n} is VB_{m,n}-invariant. We denote the flow by Fl_{τ}^{Ξ}. Using homotheties, we can easily show that there exists u ∈ F Y such that F Y is the orbit of a neighborhood U ⊂ F Y of u. This yields Ξ is complete, i.e. Fl_{τ}^{Ξ} is global. Hence Fl_{τ}^{Ξ} corresponds to some 1-parameter subgroup in Aut(A, V). Thus, the corresponding pair ⟨D, Δ⟩ ∈ Der(A, V) is determined. Clearly, Ξ = op(D, Δ).

□

3. Linear vector fields

3.1. We recall the following noted Kolář’s result. (see [1]). Theorem C. Let F = T^{A} be a Weil bundle. All natural operators transforming vector field on a manifold M into a vector field on F M (i.e. natural operators T ⇝ T T^{A}) are of the form

F^{(a)} + op(D),

where F^{(a)} := af(a) ∘ F (af(a) is a natural affinor determined by a ∈ A) and D ∈ Der(A).

3.2. Let F = T^{A,V} be a product preserving gauge bundle functor on vector bundles. Let Y → M be an object of VB_{m,n}. Then T Y → T M is a vector bundle. A vector field X: Y → T Y is called linear, if it represents a homomorphism of vector bundles Y → M and T Y → T M. Equivalently, the flow Fl_{t}^{X} is a VB_{m,n}-morphism. In the remaining text, we shall study natural operators transforming linear vector fields on a vector bundle Y with m-dimensional base and n-dimensional fibers into a vector fields on F Y. Respecting the notation of [2], we write such natural operators as T_{lin|VB_{m,n}} ⇝ T T^{A,V}.

3.3. Now, we construct for F = T^{A,V} the flow operator F: T_{lin|VB_{m,n}} ⇝ T T^{A,V}. Let Y → M be an object of VB_{m,n}. Let us consider a linear vector field X on Y. Applying F to the flow Fl_{t}^{X}, we obtain the flow F (Fl_{t}^{X}) of a vector field on F Y. This corresponding vector field on F Y determines the operator F by

FX:= \frac{∂}{∂t}|_{0}F (Fl_{t}^{X}).

The natural operator F is called the flow operator of F |_{VB_{m,n}}.

3.4. The construction of the flow operator of F |_{VB_{m,n}} can be generalized by the following way. The linear vector field X on Y can be assumed to be complete thanks to locality. The flow Fl^{X} of X can be considered as the morphism of vector bundles:

\xymatrix{ℝ × Y \ar[r]_{_{Fl_{X}}} \ar[d]_{id_{ℝ} ×p} & Y \ar[d]_{p} \\ ℝ × M \ar[r] & M}

Applying F, we obtain

F (Fl^{X}): A × F Y → F Y.

The flow condition

Fl^{X}(t_{1} + t_{2}, y) = Fl^{X}(t_{1}, Fl^{X}(t_{2}, y))

(t_{1}, t_{2} ∈ ℝ, y ∈ Y) transforms to

F (Fl^{X})(a_{1} + a_{2}, w) = F (Fl^{X})(a_{1}, Fl^{X}(a_{2}, w))

(a_{1}, a_{2} ∈ A, w ∈ F Y). Hence

Φ_{t}^{X,a} := F (Fl^{X})(at, )

is a flow on F Y. The corresponding vector field on F Y determines the natural operator F^{(a)} : T_{lin|VB_{m,n}} ⇝ T F.

Clearly, F^{(1)} = F and the map

a ↦ F^{(a)}

belongs to Mon(A, S), where S is the vector space of all linear natural operators T_{lin|VB_{m,n}} ⇝ T F.

3.5. Let F = T^{A,V} be a product preserving gauge bundle functor on vector bundles. We present the following assertion analogous to Theorem C.

Theorem 1. All natural operators transforming linear vector field on a vector bundle Y with m-dimensional base and n-dimensional fibers into a vector field on F Y (i.e. natural operators T_{lin|VB_{m,n}} ⇝ T T^{A,V}) are of the form

L = F^{(a)} + op(D, Δ),

where (D, Δ) ∈ Der(A, V).

Proof. Let x^{1}, …, x^{m}, y^{1}, …, y^{n} be the usual coordinates on the trivial vector bundle ℝ^{m} × ℝ^{n} → ℝ^{m}, which is the standard object of VB_{m,n}. Let x̃^{1}, …, x̃^{m}: F (ℝ^{m} ×ℝ^{n}) → A, ỹ^{1}, …, ỹ^{n}: F (ℝ^{m} ×ℝ^{n}) → V be the induced coordinates (cf. Example 1 in [3]). Let λ_{α}: A → ℝ, α = 1, …, dim_{A}, μ_{β}: V → ℝ, β = 1, …, dim_{V} be a basis of the dual vector spaces A^{∗} and V^{∗}. Then

x^{i,α}:= λ_{α} ∘ x̃^{i}

together with

y^{j,β}:= μ_{β} ∘ ỹ^{j}

(i = 1, …, m, j = 1, …, n) form a coordinate system on F (ℝ^{m} × ℝ^{n}). Let \frac{∂}{∂x^{1}} be the standard linear vector field on ℝ^{m} × ℝ^{n}. Since any non-vanishing linear vector field can be expressed locally as \frac{∂}{∂x^{1}} in a suitable vector bundle coordinates, then L is uniquely determined by L(\frac{∂}{∂x^{1}} ).

We can write

L(t\frac{∂}{∂x^{1}} ) ∑_{i,α} f_{i,α}(t, a, v)\frac{∂}{∂x^{i,α}} (a, v) + ∑_{j,β} g_{j,β}(t, a, v)\frac{∂}{∂y^{j,β}} (a, v).

Using the invariance of L with respect to the homotheties K id_{ℝ^{m}×ℝ^{n}}, we obtain the homogeneity conditions

f_{i,α}(Kt, Ka, Kv) = Kf_{i,α}(t, a, v)
g_{j,β}(Kt, Ka, Kv) = Kg_{j,β}(t, a, v).

Inasmuch the homogeneous function theorem holds, each f_{i,α}, g_{j,β} depend linearly on t, a, v. Further, L(0) corresponds to an absolute vector. So, L(0) = op(D, Δ) for some (D, Δ) ∈ Der(A, V). Replacing L by L − L(0), we can assume that L(0) = 0, i.e.

f_{i,α}(0, a, v) = 0
g_{j,β}(0, a, v) = 0.

Thus,

f_{i,α}(t, a, v) = A_{i,α}t
g_{j,β}(t, a, v) = B_{j,β}t

for some A_{i,α}, B_{j,β} ∈ ℝ. Further, using the invariance of L with respect to the homotheties

(x^{1}, …, x^{m}, y^{1}, …, y^{n}) ↦ (x^{1}, Kx^{2}, …, Kx^{m}, Ky^{1}, …, Ky^{n})

(K ≠ 0), we obtain the conditions

A_{i,α} = 0  for i = 2, …, m
B_{j,β} = 0  for j = 1, …, n.

Hence L is uniquely determined by A_{1,α}, α = 1, …, dim A. It implies that the vector space S of all linear natural operators T_{lin|VB_{m,n}} ⇝ T F with L(0) = 0 has the dimension less or equal dim A.

Of course, the vector space {F^{(a)}}_{a∈A} has the dimension exactly dim A. It follows S identifies with {F^{(a)}}_{a∈A} and then L = F^{(a)} for some a ∈ A. □

LIFTINGS OF LINEAR VECTOR FIELDS 61

References

1. Kolář, I., On the natural operators on vector fields, Ann. Global Anal. Geom., 6, No. 2, 109–117 (1988)

2. Kolář, I., Michor, P.W. and Slovák, J., Natural Operations in Differential Geometry, Springer Verlag 1993

3. Mikulski, W.M., Product preserving gauge bundle functors on vector bundles, Colloq. Math., 90, 277–285 (2001)

4. Shurygin, V.V., The strucuture of smooth mappings over Weil algebras and the category of manifolds over algebras, Lobachevskii Journal of Math., 5, 29–55 (1999)

5. Shurygin, V.V., Smolyakova, L.B., An analog of the Vaisman-Molino cohomology for manifolds modelled on some types of modules over Weil algebras and its application, Lobachevskii Journal of Math., 9, 55–75 (2001)

6. Tomáš, J., Natural operators transforming projectable vector fields to product preserving bundles, Supplem. ai Rendic. del Circ. Matem. di Palermo, 59, 181–187 (1999)

Miroslav Kureš, Department of Mathematics, Brno University of Technology, Technická 2, 61669 Brno, Czech Republic E-mail address: kures@mat.fme.vutbr.cz Włodzimierz M. Mikulski, Institute of Mathematics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland E-mail address: mikulski@im.uj.edu.pl

Received January 28, 2003