Product preserving functors and vector bundles ^{∗}

Jacek Gancarzewicz Włodzimierz Mikulski Zdzisław Pogoda

IMUJ PREPRINT 1997/24

Abstract

In the paper relantionships between a product preserving functor F and some operations on vector bundles are presented. Particularly, the following properties are verified: (1) F commutes with direct sum, it means there exists a natural isomorphism between F(E_{1} ⊕E_{2}) and F(E_{1})⊕F(E_{2}); (2) F(E_{1} ⊗_{R} E_{2}) is naturally isomorphic with F(E_{1})⊗_{A}F(E_{2}), where A = F(R) is the Weil algebra of F; (3) analogously F(⋀^{k(R)}E) and ⋀^{k(A)}F(E) are naturally isomorphic;

(4) there are natural vector bundle epimorphisms O_{E_{1},E_{2}} : F(E_{1})⊗_{R}F(E_{2}) ⟶ F(E_{1} ⊗_{R} E_{2}) and η̃_{E} : ⋀^{k(R)}F(E) ⟶ F(⋀^{k(R)}E); (5) there exists a natural isomorphism between F(E^{∗(R)}) and (F(E))^{∗(A)}; (6) for every linear function λ : A → R on the Weil algebra A there is a natural vector bundle homomorphism ω_{E}^{λ} : F(E^{∗(R)}) ⟶ (F(E))^{∗(R)}; (7) ω_{E}^{λ} is an isomorphism if and only if A × A ∋ (a, b) → λ(ab) ∈ R is a non-singular quadratic form on A. Obtained results are applied to study similar properties of bundles F(T^{∗(R)}M), T^{∗(A)}(FM) and T^{∗(R)}(FM).

0 Introduction

The product preserving functors defined on the category of differential manifolds and their differential mappings have many nice and important properties and generalize many situations in the differential geometry. The technique of product preserving functors gives the possibility to prove properties without using coordinates. These methods have been development in papers [5], [2] and others.

∗The paper was supported by Grant of KBN no 2 P 301 030 04

A product preserving functor F is uniquely determined by its so-called Weil algebra A = F(R). It is well known that if π : E ⟶ M is a vector bundle, then F(π) : F(E) ⟶ F(M) is an A-module bundle (see [2], [5], see also Section 1 of the presented paper).

Two product preserving functors have many interesting and important properties. Two product preserving functors F, G commute, it means, for every manifold M two bundles F(GM) and G(FM) are isomorphic as bundles over F(M). In this paper we will study the commutativity of a product preserving functor F with some operations on vector bundles and A-module bundles.

In the presented paper π : E ⟶ M is called an A-module bundle if:

(a) For each point x ∈ M the fiber E_{x} = π^{−1}(x) is a finite dimensional free Amodule, e.g, E_{x} is isomorphic with A^{p} for some integer p (we assume that dimensions of fibers are the same for each point of M).

(b) Each point of M has an open neighborhood U ⊂ M and a diffeomorphism f : π^{−1}(U) ⟶ U × A^{p} such that the diagram

[figure: π^{−1}(U); U × A^{p}; f; M; π; p_{1}]

commutes, where p_{1} is the projection on the first factor, and for each point x ∈ U the restriction f_{x} = f_{|E_{x}} : E_{x} ⟶ {x} × A^{p} = A^{p} is an A-linear isomorphism (the local triviality condition).

Firstly in Section 1 we recall basic properties of product preserving functors. In Sections 2 and 3 we prove that a product preserving functor F commutes with

direct sum, tensor product and exterior power of vector bundles. More precisely, we show that there are natural isomorphisms F(E_{1} ⊗_{R} E_{2}) ⟶ F(E_{1}) ⊗_{A} F(E_{2}) and F(⋀^{k(R)}E) ⟶ ⋀^{k(A)}F(E) of A-module bundles. The above isomorphisms define a natural vector bundle epimorphisms F(E_{1}) ⊗_{R} F(E_{2}) ⟶ F(E_{1} ⊗_{R} E_{2}) and ⋀^{k(R)}F(E_{1}) ⟶ F(⋀^{k(R)}E).

In Section 4 we prove that two A-module bundles F(E^{∗(R)}) and (FE)^{∗(A)} are naturally isomorphic. From this fact we deduce that any linear function λ : A = F(R) ⟶ R on the Weil algebra defines a natural homomorphism ω_{E}^{λ} : F(E^{∗(R)}) ⟶ (FE)^{∗(R)}. We show also that ω_{E}^{λ} is an isomorphism if and only if A × A ∋ (a, b) ⟶ λ(ab) ∈ R is a non-singular quadratic form.

In the next section for any linear function λ : A = F(R) ⟶ R on the Weil algebra such that A × A ∋ (a, b) ⟶ λ(ab) ∈ R is a non-singular quadratic form we construct natural monomorphisms μ_{E_{1},E_{2}}^{λ} : F(E_{1} ⊗_{R} E_{2}) ⟶ F(E_{1}) ⊗_{R} F(E_{2}) and μ_{E}^{λ} : F(⋀^{k(R)}E) ⟶ ⋀^{k(R)}F(E).

In the last section from results of Sections 2 — 5 we deduce similar properties for bundles F(T^{∗(R)}M), T^{∗(A)}(FM) and T^{∗(R)}(FM).

In the presented paper all manifolds and mappings are of class C^{∞}.

1 Product preserving functors

Let F be a covariant functor from the category of all manifolds and all mappings into the category of fibered manifolds and fibered mappings such that for every manifold M, F(M) is a fibered manifold over M, and for every mapping φ : M → N the induced mapping F(φ) : F(M) → F(N) covers φ, and if φ is an embedding of manifolds of the same dimension, then restrictions of F(φ) to fibers of F(M) are diffeomorphisms onto fibers of F(N).

F is called a product preserving functor if for all manifolds M_{1}, M_{2} the mapping

(F(π_{1}), F(π_{2})) : F(M_{1} × M_{2}) ⟶ F(M_{1}) × F(M_{2})

is a diffeomorphism, where π_{i} : M_{1} × M_{2} → M_{i} denotes the standard projection on the i-th factor (see [5], [2]).

It is possible to show (see [5]) that a product preserving functor F verifies also the regularity condition saying that if φ_{t} : M → N is a differentiable family of mappings, then F(φ_{t}) : F(M) → F(N) is a differentiable family of mappings.

The tangent bundle T M and the tangent bundle of p^{r}-velocities T_{p}^{r}M (see [5], [2]) are important examples of product preserving functors. The most general example of product preserving functor is so-called Weil functor (see [8], [5], [2]). In 1986 Eck [1], Kainz, Michor [4] and Luciano [6] have proved independently that any product preserving functor is equivalent to a Weil functor.

For a product preserving functor F we will always identify F(M_{1} × M_{2}) with F(M_{1}) × F(M_{2}) by the diffeomorphism from the definition. After this identification we have F(f_{1} × f_{2}) = F(f_{1}) × F(f_{2}) , F(f, g) = (F(f), F(g)) for all mappings f_{1} : M_{1} → N_{1}, f_{2} : M_{2} → N_{2}, f : M → N_{1} and g : M → N_{2} . Product preserving functors have many very interesting properties. We propose to consult [5], [2]. We recall only these properties which will be used in the paper.

For a product preserving functor F we construct a real associative, commutative and finite dimensional algebra structure on A = F(R). If +, · : R^{2} → R are the addition and the multiplication on R and m_{α} : R → R is the multiplication by α ∈ R, then F(+), F(·), F(m_{α}) are the operations in A, F(0) and F(1) are the zero and the unity in A (after identification of constant mappings with their values). The set N = F_{0}(R) is the ideal of nilpotent elements of A and we have A = R · 1 ⊕ N. The algebra A = F(R) is called Weil algebra of F.

This algebra is very important in the theory of product preserving functors. Two product preserving functors with isomorphic Weil algebras are equivalent.

For many structures on M we have a possibility to define their prolongations to F(M). In this paper we use prolongations of vector spaces and vector bundles.

If V is a finite dimensional vector space, then F(V) is a finite dimensional free A-module. If + : V × V → V is the addition mapping in V and d : R × V → V is the multiplication by scalars, then F(+) : F(V) × F(V) → F(V) is the sum mapping in F(V) and F(d) : A × F(V) → F(V) is the multiplication by scalars from A. If f : V → W is a linear mapping of two finite dimensional vector spaces, then F(f) : F(V) → F(W) is A-linear. We have a similar property for k-linear mappings (see [2]).

We genaralize these constructions to vector bundles. If π : E → M is a vector bundle, then F(π) : F(E) → F(M) is an A-module bundle. If V is the standard fibre of E and φ : E_{|U} → U × V is a trivialization over an open subset U ⊂ M, then the Amodule F(V) is the standard fibre of F(E) and F(φ) : F(E)_{|F(U)} → F(U) × F(V) is a trivialization over F(U) ⊂ F(M). If Ψ : E → E′ is a vector bundle homomorphism, then F(Ψ) : F(E) → F(E′) is an A-module bundle homomorphism. From this we deduce that if σ_{1} : U ⟶ E, …, σ_{p} : U ⟶ E is a local basis ^{1} of sections of E, then F(σ_{1}) : F(U) ⟶ F(E), …, F(σ_{p}) : F(U) ⟶ F(E) is a local basis over A of sections ^{2} of the A-module bundle F(E).

We also have a similar property for k-linear bundle homomorphisms (see [2]). For vector bundles E_{1} → M, …, E_{k} → M with the same base M and a vector bundle E′ → N, a mapping Ψ : E_{1} ×_{M} ⋯ ×_{M} E_{k} ⟶ E′ is called a k-linear bundle homomorphism if there exists a mapping ϕ : M ⟶ N such that the diagram

[figure: E_{1} ×_{M} ⋯ ×_{M} E_{k}; E′; M; N; Ψ; ψ; π; π]

commutes (we say that Ψ covers ψ) and restrictions of Ψ to fibers are k-linear. If Ψ : E_{1} ×_{M} ⋯ ×_{M} E_{k} ⟶ E′ is a k-linear bundle homomorphism, then F(Ψ) : F(E_{1})×_{F(M)}⋯×_{F(M)}F(E_{k}) ⟶ F(E′) is a k-linear (over A) bundle homomorphism.

There is a very interesting and nontrivial property saying that for two product preserving functors F_{1}, F_{2} there exists the natural isomorphism

F_{1}(F_{2}(M)) → F_{2}(F_{1}(M))

(see [5]). In particular case, if F_{1}(M) = T M is the tangent bundle the natural isomorphism η_{M} : F(T M) → T F(M) is a vector bundle isomorphism over the identity on F(M) (see [2]). Using the isomorphism η_{M} we introduce an A-module bundle structure on T F(M).

We go to study relantioships between a product preserving functor and some functors on vector bundles.

2 Direct sum and tensor product of vector bundles

In this section we will study relationships between a product preserving functor and direct sum and tensor product of vector bundles. Firstly we present the following proposition on direct sum.

1It means that σ_{j} are local sections such that for every point x of U elements σ_{1}(x), …, σ_{p}(x)

form a basis of E_{x}

2It means that for every point y of F(U) elements F(σ_{1})(y), …, F(σ_{p})(y) form an A-basis of F_{y}(E) (it is a free A-module).

Proposition 2.1 For vector bundles π_{1} : E_{1} ⟶ M and π_{2} : E_{2} ⟶ M there exists an A-module bundle isomorphism

ζ_{E_{1},E_{2}} : F(E_{1} ⊕ E_{2}) ⟶ F(E_{1}) ⊕ F(E_{2})

covering the identity on F(M) such that for two homomorphisms f_{1} : E_{1} ⟶ E′_{1} and f_{2} : E_{2} ⟶ E′_{2} of vector bundles over M and M′ respectively covering the same differential mapping φ : M ⟶ M′ the following diagram

[figure: F(E_{1} ⊕ E_{2}); F(E′_{1} ⊕ E′_{2}); F(f_{1} ⊕ f_{2}); F(E_{1}) ⊕ F(E_{2}); F(E′_{1}) ⊕ F(E′_{2}); ζ_{E_{1},E_{2}}; ζ_{E′_{1},E′_{2}}; F(f_{1}) ⊕ F(f_{2}); F(M); F(M′); F(φ); F(π); F(π′); Π; Π′]

commutes, where

\begin{aligned}π & : & E_{1} ⊕ E_{2} ⟶ M , & Π & : & F(E_{1}) ⊕ F(E_{2}) ⟶ F(M) , \\ π′ & : & E′_{1} ⊕ E′_{2} ⟶ M′ , & Π′ & : & F(E′_{1}) ⊕ F(E′_{2}) ⟶ F(M′) \end{aligned}

are the projections of corresponding bundles.

P r o o f. Let π_{1} : E_{1} ⊕ E_{2} ⟶ E_{1} and π_{2} : E_{1} ⊕ E_{2} ⟶ E_{2} be the standard projections on the first and the second factor. We define

ζ_{E_{1},E_{2}} = ( F(π_{1}), F(π_{2}) ) .

The standard verification show that the diagram from our proposition is commutative.

To prove that ζ_{E_{1},E_{2}} is an isomorphism, let σ_{11} : U ⟶ E_{1}, …, σ_{1p_{1}} : U ⟶ E_{1} be a local basis of sections of E_{1} and let σ_{21} : U ⟶ E_{2}, …, σ_{2p_{2}} : U ⟶ E_{2} be a local basis of sections of E_{2}.

Using the canonical inclusions

i_{1} : E_{1} ∋ y ⟶ (y, 0) ∈ E_{1} ⊕ E_{2} , i_{2} : E_{2} ∋ y ⟶ (0, y) ∈ E_{1} ⊕ E_{2}

we obtain that i_{1} ∘ σ_{11}, …, i_{1} ∘ σ_{1p_{1}}, i_{2} ∘ σ_{21}, …, i_{2} ∘ σ_{2p_{2}} is a local basis of sections of E_{1} ⊕ E_{2}.

From the local triviality of A-module bundles we deduce that

F(σ_{11}) : F(U) ⟶ F(E_{1}), …, F(σ_{1p_{1}}) : F(U) ⟶ F(E_{1}) ,
F(σ_{21}) : F(U) ⟶ F(E_{2}), …, F(σ_{2p_{2}}) : F(U) ⟶ F(E_{2}) ,
F(i_{1} ∘ σ_{11}), …, F(i_{1} ∘ σ_{1p_{1}}), F(i_{2} ∘ σ_{21}), …, F(i_{2} ∘ σ_{2p_{2}})

are local A-bases of sections of F(E_{1}), F(E_{2}) and F(E_{1} ⊕ E_{2}) respectively.

Applying the canonical inclusions

I_{1} : F(E_{1}) ⟶ F(E_{1}) ⊕ F(E_{2}) , I_{2} : F(E_{2}) ⟶ F(E_{1}) ⊕ F(E_{2})

we obtain that I_{1} ∘ F(σ_{11}), …, I_{1} ∘ F(σ_{1p_{1}}), I_{2} ∘ F(σ_{21}), …, I_{2} ∘ F(σ_{2p_{2}}) is a local A-basis of sections of F(E_{1}) ⊕ F(E_{2}). Now it is easy to see

ζ_{E_{1},E_{2}} ∘ F(i_{s} ∘ σ_{sj}) = I_{s} ∘ F(σ_{sj}) ,

for j = 1, …, p_{s} , s = 1, 2. It means that ζ_{E_{1},E_{2}} transform an A-basis onto an A-basis, i.e. ζ_{E_{1},E_{2}} is an isomorphism. The proof is finished. □

We prove also the following proposition on tensor product of vector bundles.

Proposition 2.2 For vector bundles π_{1} : E_{1} ⟶ M and π_{2} : E_{2} ⟶ M there exists an A-module bundle isomorphism

η_{E_{1},E_{2}} : F(E_{1}) ⊗_{A} F(E_{2}) ⟶ F(E_{1} ⊗_{R} E_{2})

covering the identity on F(M) such that for two homomorphisms f_{1} : E_{1} ⟶ E′_{1} and f_{2} : E_{2} ⟶ E′_{2} of vector bundles over M and M′ respectively covering the same differential mapping φ : M ⟶ M′ the following diagram

[figure: F(E_{1}) ⊗_{A} F(E_{2}); F(E′_{1}) ⊗_{A} F(E′_{2}); F(f_{1}) ⊗_{A} F(f_{2}); F(E_{1} ⊗_{R} E_{2}); F(E′_{1} ⊗_{R} E′_{2}); η_{E_{1},E_{2}}; η_{E′_{1},E′_{2}}; F(f_{1} ⊗_{R} f_{2}); F(M); F(M′); F(φ); Π; Π′; F(π); F(π′)]

commutes, where

\begin{aligned}π & : & E_{1} ⊗_{R} E_{2} ⟶ M , & Π & : & F(E_{1}) ⊗_{A} F(E_{2}) ⟶ F(M) , \\ π′ & : & E′_{1} ⊗_{R} E′_{2} ⟶ M′ , & Π′ & : & F(E′_{1}) ⊗_{A} F(E′_{2}) ⟶ F(M′) \end{aligned}

are the projections of corresponding bundles.

P r o o f. Let s : E_{1}×_{M} E_{2} ⟶ E_{1}⊗_{R}E_{2} and S : F(E_{1})×_{F(M)}F(E_{2}) ⟶ F(E_{1})⊗_{A} F(E_{2}) be the standard mappings bilinear on each fibers (the first mapping — over R and the seconde one — over A) for tensor products of bundles. From Proposition 1.7 in [2] the induced mapping F(s) : F(E_{1}) ×_{F(M)} F(E_{2}) ⟶ F(E_{1} ⊗ E_{2}) is bilinear over A on each fiber over F(M). Now, the universal factorization property for tensor product implies that there exists one and only one A-module bundle homomorphism η_{E_{1},E_{2}} : F(E_{1}) ⊗_{A} F(E_{2}) ⟶ F(E_{1} ⊗_{R} E_{2}) which covers the identity mapping on F(M) and such that the diagram

[figure: F(E_{1}) ×_{F(M)} F(E_{2}); F(E_{1}) ⊗_{A} F(E_{2}); F(E_{1} ⊗_{R} E_{2}); η_{E_{1},E_{2}}; S; F(s)]

commutes. The commutativity of the diagram from our proposition is evident. We need to verify that η_{E_{1},E_{2}} is an isomorphism. Let σ_{11} : U ⟶ E_{1}, …, σ_{1p_{1}} : U ⟶ E_{1} be a local basis of sections of E_{1} and

let σ_{21} : U ⟶ E_{2}, …, σ_{2p_{2}} : U ⟶ E_{2} be a local basis of sections of E_{2}. Then F(σ_{11}) : F(U) ⟶ F(E_{1}), …, F(σ_{1p_{1}}) : F(U) ⟶ F(E_{1}) and F(σ_{21}) : F(U) ⟶ F(E_{2}), …, F(σ_{2p_{2}}) : F(U) ⟶ F(E_{2}) are local A-bases of sections of F(E_{1}) and F(E_{2}) respectively.

From the linear algebra we deduce that the family of sections σ_{1i} ⊗_{R} σ_{2j}, for i = 1, …, p_{1}, j = 1, …, p_{2} forms a local basis of sections of E_{1} ⊗_{R} E_{2} and the family F(σ_{1i}) ⊗_{A} F(σ_{2j}), for i = 1, …, p_{1}, j = 1, …, p_{2} is a local A-basis of sections of F(E_{1}) ⊗_{A} F(E_{2}).

Analogously we have

F(σ_{1i} ⊗_{R} σ_{2j}) , i = 1, …, p_{1}, j = 1, ….p_{2}

is an A-basis of sections of F(E_{1} ⊗_{R} E_{2}). Now we obtain

η_{E_{1},E_{2}} ∘ (F(σ_{1i}) ⊗_{A} F(σ_{2j})) = η_{E_{1},E_{2}} ∘ S ∘ (F(σ_{1i}), F(σ_{2j}))
= F(s) ∘ (F(σ_{1i}), F(σ_{2j}))
= F(s ∘ (σ_{1i}, σ_{2j}))
= F(σ_{1i} ⊗_{R} σ_{2j}) .

It means that that η_{E_{1},E_{2}} transforms an A-basis onto an A-basis, e.g. η_{E_{1},E_{2}} is an isomorphism. The proof is finished. □

Since any A-bilinear mappinig is also R-bilinear thus we have

Proposition 2.3 For vector bundles π_{1} : E_{1} ⟶ M and π_{2} : E_{2} ⟶ M there exists a vector bundle epimorphism

O_{E_{1},E_{2}} : F(E_{1}) ⊗_{R} F(E_{2}) ⟶ F(E_{1}) ⊗_{A} F(E_{2})

covering the identity on F(M) such that for two homomorphisms f_{1} : E_{1} ⟶ E′_{1} and f_{2} : E_{2} ⟶ E′_{2} of vector bundles over M and M′ respectively covering the same differential mapping φ : M ⟶ M′ the following diagram

[figure: F(E_{1}) ⊗_{R} F(E_{2}); F(E′_{1}) ⊗_{R} F(E′_{2}); F(f_{1}) ⊗_{R} F(f_{2}); F(E_{1}) ⊗_{A} F(E_{2}); F(E′_{1}) ⊗_{A} F(E′_{2}); O_{E_{1},E_{2}}; O_{E′_{1},E′_{2}}; F(f_{1}) ⊗_{A} F(f_{2}); F(M); F(M′); F(φ); Π_{R}; Π′_{R}; Π_{A}; Π′_{A}] (2.1)

commutes, where

Π_{A} : F(E_{1}) ⊗_{A} F(E_{2}) ⟶ F(M) ,  Π_{R} : F(E_{1}) ⊗_{R} F(E_{2}) ⟶ F(M) ,
Π′_{A} : F(E′_{1}) ⊗_{A} F(E′_{2}) ⟶ F(M′) ,  Π′_{R} : F(E′_{1}) ⊗_{R} F(E′_{2}) ⟶ F(M′)

are the projections of corresponding bundles.

P r o o f. Let

S_{R} : F(E_{1}) ×_{F(M)} F(E_{2}) ⟶ F(E_{1}) ⊗_{R} F(E_{2})
S_{A} : F(E_{1}) ×_{F(M)} F(E_{2}) ⟶ F(E_{1}) ⊗_{A} F(E_{2})

be the canonical mappings bilinear on all fibers (the first one — over R, and the second one — over A) for tensor pruducts over R and A respectively. From the universal factorization property for real tensor product we deduce that there exists an unique linear vector bundle homomorphism O_{E_{1},E_{2}} : F(E_{1}) ⊗_{R} F(E_{2}) ⟶ F(E_{1}) ⊗_{A} F(E_{2}) covering the identity on F(M) such that the diagram

[figure: F(E_{1}) ×_{F(M)} F(E_{2}); F(E_{1}) ⊗_{R} F(E_{2}); F(E_{1}) ⊗_{A} F(E_{2}); O_{E_{1},E_{2}}; S_{R}; S_{A}]

commutes. The commutativity of diagram (2.1) is evident.

Since images of restrictions of S_{A} to fibers generate (over A and over R) corresponding fibers of F(E_{1})⊗_{A} F(E_{2}), thus O_{E_{1},E_{2}} is a vector bundle epimorphism. The proof is finished. □

From Propositions 2.2 and 2.3 we obtain

Proposition 2.4 For vector bundles π_{1} : E_{1} ⟶ M and π_{2} : E_{2} ⟶ M there exists a vector bundle epimorphism

η̃_{E_{1},E_{2}} : F(E_{1}) ⊗_{R} F(E_{2}) ⟶ F(E_{1} ⊗_{R} E_{2})

covering the identity on F(M) such that for two homomorphisms f_{1} : E_{1} ⟶ E′_{1} and f_{2} : E_{2} ⟶ E′_{2} of vector bundles over M and M′ respectively covering the same differential mapping φ : M ⟶ M′ the following diagram

[figure: F(E_{1}) ⊗_{R} F(E_{2}); F(E′_{1}) ⊗_{R} F(E′_{2}); F(f_{1}) ⊗_{R} F(f_{2}); F(E_{1} ⊗_{R} E_{2}); F(E′_{1} ⊗_{R} E′_{2}); η̃_{E_{1},E_{2}}; η̃_{E′_{1},E′_{2}}; F(f_{1} ⊗_{R} f_{2}); F(M); F(M′); F(φ); Π; Π′; F(π); F(π′)]

commutes, where

\begin{aligned}π & : & E_{1} ⊗_{R} E_{2} ⟶ M , & Π & : & F(E_{1}) ⊗_{R} F(E_{2}) ⟶ F(M) , \\ π′ & : & E′_{1} ⊗_{R} E′_{2} ⟶ M′ , & Π′ & : & F(E′_{1}) ⊗_{R} F(E′_{2}) ⟶ F(M′) \end{aligned}

are the projections of corresponding vector bundles.

P r o o f. We set η̃_{E_{1},E_{2}} = η_{E_{1},E_{2}} ∘ O_{E_{1},E_{2}}.□

Let us remark that in similar way we can formulate and prove analogical propositions for tensor products of any number of vector bundles, for symmetric tensor products and for exterior products. In the next paragraph we will explain it for exterior products of vector bundles.

3 Exterior power of vector bundles

We denote by ⋀^{p(R)} the exterior p-power of real vector spaces and real vector bundles and by ⋀^{p(A)} we denote the exterior p-power of free finite dimensional A-modules and A-module bundles. We have the following proposition:

Proposition 3.1 For a vector bundle π : E ⟶ M there exists an A-module bundle isomorphism

η_{E} : ⋀^{p(A)}F(E) ⟶ F(⋀^{p(R)}E)

covering the identity on F(M) such that for a homomorphism f : E ⟶ E′ of vector bundles over M and M′ respectively covering a differential mapping φ : M ⟶ M′ the following diagram

[figure: ⋀^{p(A)}F(E); ⋀^{p(A)}F(E′); ⋀^{p(A)}F(f); F(⋀^{p(R)}E); F(⋀^{p(R)}E′); η_{E}; η_{E′}; F(⋀^{p(R)}f); F(M); F(M′); F(φ); Π; Π′; F(π); F(π′)]

commutes, where

\begin{aligned}π & : & ⋀^{p(R)}E ⟶ M , & Π & : & ⋀^{p(A)}F(E) ⟶ F(M) \\ π′ & : & ⋀^{p(R)}E′ ⟶ M′ , & Π′ & : & ⋀^{p(A)}F(E′) ⟶ F(M′) \end{aligned}

are the projections of corresponding bundles.

P r o o f. Let

s : E ×_{M} ⋯ ×_{M} E ⟶ ⋀^{p(R)}E
S : F(E) ×_{F(M)} ⋯ ×_{F(M)} F(E) ⟶ ⋀^{p(A)}F(E)

be the standard mappings for exterior products of bundles which are p-linear skewsymmetric on all fibers (the first mapping — over R and the second one — over A). From Propopsition 1.7 in [2] the induced mapping F(s) : F(E) ×_{F(M)} ⋯ ×_{F(M)} F(E) ⟶ F(⋀^{p(R)}) is p-linear over A on all fibers. Now, the universal factorization property for exterior power implies that there exists one and only one A-module bundle homomorphism η_{E} : ⋀^{p(A)}F(E) ⟶ F(⋀^{p(R)}E) of bundles which covers the identity mapping on F(M) and such that the diagram

[figure: F(E) ×_{F(M)} ⋯ ×_{F(M)} F(E); ⋀^{p(A)}F(E); F(⋀^{p(R)}E); η_{E}; S; F(s)]

commutes. The commutativity of the diagram from our proposition is evident.

In similar way as in Proposition 2.2 we verify that η_{E} is an isomorphism. The proof is finished. □

Since any A-multilinear mappinig is also R-multilinear thus we have

Proposition 3.2 For a vector bundle π : E ⟶ M there exists a vector bundle epimorphism

O_{E} : ⋀^{p(R)}F(E) ⟶ ⋀^{p(A)}F(E)

covering the identity on F(M) such that for a homomorphism f : E ⟶ E′ of vector bundles over M and M′ respectively covering a differential mapping φ : M ⟶ M′ the following diagram

[figure: ⋀^{p(R)}F(E); ⋀^{p(R)}F(E′); ⋀^{p(R)}F(f); ⋀^{p(A)}F(E); ⋀^{p(A)}F(E′); O_{E}; O_{E′}; ⋀^{p(A)}F(f); F(M); F(M′); F(φ); Π_{R}; Π′_{R}; Π_{A}; Π′_{A}]

commutes, where

Π_{A} : ⋀^{p(A)}F(E) ⟶ F(M) ,  Π_{R} : ⋀^{p(R)}F(E) ⟶ F(M) ,
Π′_{A} : ⋀^{p(A)}F(E′) ⟶ F(M′) ,  Π′_{R} : ⋀^{p(R)}F(E′) ⟶ F(M′)

are the projections of corresponding vector bundles.

P r o o f. We verify the proposition in a similar way as Proposition 2.3. □

From Propositions 3.1 and 3.2 we obtain

Proposition 3.3 For a vector bundle π : E ⟶ M there exists a vector bundle epimorphism

η̃_{E} : ⋀^{p(R)}F(E) ⟶ F(⋀^{p(R)}E)

covering the identity on F(M) such that for a homomorphism f : E ⟶ E′ of vector bundles covering a smooth mapping φ : M ⟶ M′ the following diagram

[figure: ⋀^{p(R)}F(E); ⋀^{p(R)}F(E′); ⋀^{p(R)}F(f); F(⋀^{p(R)}E); F(⋀^{p(R)}E′); η̃_{E}; η̃_{E′}; F(⋀^{p(R)}f); F(M); F(M′); F(φ); Π; Π′; F(π); F(π′)]

commutes, where

\begin{aligned}π & : & ⋀^{p(R)}(E) ⟶ M , & Π & : & ⋀^{p(R)}F(E) ⟶ F(M) , \\ π′ & : & ⋀^{p(R)}(E′) ⟶ M′ , & Π′ & : & ⋀^{p(R)}F(E′) ⟶ F(M′) \end{aligned}

are the projections of corresponding vector bundles.

P r o o f. We set η̃_{E} = η_{E} ∘ O_{E}. □

4 Dual bundles

We begin this section by some algebraic introduction.

Let V, W be two finite dimensional vector spaces. A bilinear mapping φ : V × W ⟶ R is called non-singular if for any vector v ∈ V the following implication

∀ w ∈ W : φ(v, w) = 0 ⟹ v = 0

holds. For v ∈ V we denote by i_{v}φ : W ⟶ R a linear mapping given by i_{v}φ(w) = φ(v, w). A bilinear mapping φ is non-singular if and only if

i_{φ} : V ∋ v ⟶ i_{v}φ ∈ W^{∗(R)}

is a monomorphism, where W^{∗(R)} denotes the dual vector space. If dim V = dim W, then φ is non-singular if and only if i_{φ} is an isomorphism.

Let v_{1}, …, v_{n} and w_{1}, …, w_{k} be bases of V and W respectively. A matrix K with coefficients a_{ij} given by a_{ij} = φ(v_{i}, w_{j}) is called a matrix of φ with respect to bases v_{1}, …, v_{n} and w_{1}, …, w_{k}. It is clear, that K is also a matrix of linear mapping i_{φ} with respect to bases v_{1}, …, v_{n} and w_{1}^{∗}, …, w_{k}^{∗}, where w_{1}^{∗}, …, w_{k}^{∗} is the dual basis to w_{1}, …, w_{k}.

Now, let us observe that φ is non-singular if and only if n ≤ k and K is of maximal rang n.

Analogously, if V, W are two finite dimensional free A-modules, then an A-bilinear mapping φ : V × W ⟶ A is called non-singular if for any vector v ∈ V the following implication

∀ w ∈ W : φ(v, w) = 0 ⟹ v = 0

holds. For v ∈ V we denote by i_{v}φ : W ⟶ A an A-linear mapping given by i_{v}φ(w) = φ(v, w). An A-bilinear mapping φ is non-singular if and only if

i_{φ} : V ∋ v ⟶ i_{v}φ ∈ W^{∗(A)}

is a monomorphism, where W^{∗(A)} denotes A-dual module, it means W^{∗(A)} is the A-module of all A-linear mappings W ⟶ A. If dim_{A} V = dim_{A} W, then φ is nonsingular if and only if i_{φ} is an isomorphism.

Also we can define a matrix of φ. Namely, we consider two A-bases v_{1}, …, v_{n} and w_{1}, …, w_{k} of A-modules V and W respectively. A matrix K with coefficients a_{ij} ∈ A given by a_{ij} = φ(v_{i}, w_{j}) is called a matrix of φ with respect to A-bases v_{1}, …, v_{n} and w_{1}, …, w_{k}. It is clear, that K is also a matrix of A-linear mapping i_{φ} with respect to A-basis v_{1}, …, v_{n} and w_{1}^{∗(A)}, …, w_{k}^{∗(A)}, where w_{1}^{∗(A)}, …, w_{k}^{∗(A)} is the dual basis to w_{1}, …, w_{k} (it means that w_{i}^{∗(A)} : W ⟶ A is an A-linear mapping such that w_{i}^{∗(A)}(w_{j}) = δ_{ij}). Now, φ is non-singular if and only if n ≤ k and K is of maximal rang n.

For a vector bundle π : E ⟶ M we consider the dual bundle E^{∗(R)} and we denote by the same symbol π : E^{∗(R)} ⟶ M its projection. If f : E ⟶ E′ is a vector bundles homomorphism which is also an imbedding, where vector bundles π : E ⟶ M and π′ : E′ ⟶ M′ are such that dim M = dim M′ and dim E = dim E′, then we define the induced mapping D^{R}(f) : E^{∗(R)} ⟶ E^{′∗(R)} by

D^{R}(f)_{|E_{x}^{∗(R)}} = (f_{x}^{∗(R)})^{−1} ,

where f_{x} = f_{|E_{x}} : E_{x} ⟶ E′_{B(f)(x)} and f_{x}^{∗(R)} : E_{B(f)(x)}^{′∗(R)} ⟶ E_{x}^{∗(R)} is the transposed mapping. Our assumptions on f and on the equality dimensions imply that f_{x} is an isomorphism.

We want to emphasize that the induced mapping D^{R}(f) is well-defined only if dimM = dim M′, dim E = dim E′ and a vector bundle homomorphism f : E ⟶ E′ is an imbedding. A such f : E ⟶ E′ will be called a vector bundle imbedding.

The induced mapping D^{R}(f) : E^{∗(R)} ⟶ E^{′∗(R)} is a vector bundle imbedding which covers the same mapping of bases as f.

Analogously, for an A-module bundle π : E ⟶ M we define A-dual A-module bundle E^{∗(A)}. Its projection we denote by the same symbol as a projection of a given A-module bundle. Namely, we denote by E_{x}^{∗(A)} the set of all A-linear mappings E_{x} ⟶ A and E^{∗(A)} = ⋃_{x∈M} E_{x}^{∗(A)}. Now π : E^{∗(A)} ⟶ M is an A-module bundle called A-dual bundle.

Let π : E ⟶ M and π′ : E′ ⟶ M′ be two A-module bundles such that dim M = dim M′ and dim E = dim E′. An A-module bundle homomorphism f : E ⟶ E′ is called an A-module bundle imbedding if f is an imbbeding. If x is a point of M, then f_{x} = f_{|E_{x}} : E_{x} ⟶ E′_{B(f)(x)} is an A-linear isomorphism, and in cosequence, the transposed mapping f_{x}^{∗(A)} : E_{B(f)(x)}^{′∗(A)} ⟶ E_{x}^{∗(A)} is also an A-linear isomorphism. Now we can define the induced mapping

D^{A}(f) : E^{∗(A)} ⟶ (E′)^{∗(A)} , D^{A}(f)_{|E_{x}} = (f_{x}^{∗(A)})^{−1} .

We have the following proposition:

Proposition 4.1 For any vector bundle π : E ⟶ M there exists an isomorphism

Λ_{E} : F(E^{∗(R)}) ⟶ (F(E))^{∗(A)}

of A-module bundles covering the identity on F(M) satisfying the following naturality property:

for two vector bundles π : E ⟶ M, π′ : E′ ⟶ M′ such that dim M = dim M′, dim E = dim E′ and any vector bundle imbbeding f : E ⟶ E′ the diagram

(4.1)^{[figure: F(E^{∗(R)}); F(E^{′∗(R)}); F(D^{R}(f)); D^{A}(F(f)); (F(E))^{∗(A)}; (F(E′))^{∗(A)}; Λ_{E}; Λ_{E′}; F(M); F(M′); F(B(f))]} commutes, where B(f) : M ⟶ M′ is a mapping covered by f and arrows ↘,

↙, ↗, ↖ are the coresponding bundle projections.

P r o o f. We consider a mapping ( , )_{E} : E^{∗(R)} ×_{M} E ⟶ R defined by

(α, v)_{E} = α(v) , v ∈ E , α ∈ E^{∗(R)} , π(v) = π(α) , (4.2)

which is bilinear on all fibers. Now F(E) and F(E^{∗(R)}) are A-module bundles as well as the induced mapping

( , )_{F(E)} = F(( , )_{E}) : F(E^{∗(R)}) ×_{F(M)} F(E) ⟶ F(R) = A

is bilinear over A on all fibers (see [2]). We will verify that it is non-singular on all fibers.

Let σ_{1}, …, σ_{p} be a local basis of sections of E, and let σ_{1}^{∗(R)}, …, σ_{p}^{∗(R)} denote the dual basis of sections of the dual vector bundle E^{∗(R)}. Now F(σ_{1}), …, F(σ_{p}) and F(σ_{1}^{∗(R)}), …, F(σ_{p}^{∗(R)}) are local A-bases of sections of A-module bundles F(E) and F(E^{∗(R)}) respectively.

From (σ_{i}^{∗(R)}, σ_{j})_{E} = δ_{ij} we deduce

(F(σ_{i}^{∗(R)}), F(σ_{j}))_{F(E)} = F(δ_{ij}) = \{ \begin{matrix}F(1) = 1 , & if i = j \\ F(0) = 0 , & if i ≠ j \end{matrix} (4.3)

It means that matrix of ( , )_{F(E)} is non-sigular, and in consequense, ( , )_{F(E)} is non-singular on all fibers.

It implies that there exists an A-module bundle monomorphism

Λ_{E} : F(E^{∗(R)}) ⟶ (F(E))^{∗(A)}

such that

(α̃, ṽ)_{F(E)} = Λ_{E}(α̃)(ṽ) . (4.4)

From formula (4.4) we deduce immediately that Λ_{E} covers the identity on F(M), and restrictions of Λ_{E} to fibers are A-linear isomorphisms because fibers of F(E^{∗(R)}) and (F(E))^{∗(A)} have the same dimensions. Thus, Λ_{E} is an A-module bundle isomorphism.

We need to verify that for arbitrary vector bundles π : E ⟶ M, π′ : E′ ⟶ M′ such that dim M = dim M′, dim E = dim E′ and any vector bundle imbedding f : E ⟶ E′ the diagram (4.1) commutes.

Let us observe that from the definition of D^{R}(f) we deduce

( , )_{E′} ∘ (D^{R}(f) × f) = ( , )_{E} .

Applying F we obtain

( , )_{F(E′)} ∘ (F(D^{R}(f)) × F(f)) = ( , )_{F(E)} . (4.5)

Let y ∈ F(M) and α̃ ∈ F_{y}(E^{∗(R)}). To verify the commutativity of diagram (4.1) we need to show the equality

(Λ_{E′} ∘ F(D^{R}(f))(α̃) = (D^{A}(F(f)) ∘ Λ_{E})(α̃) .

According to definition of D^{A}(F(f)), it is equivalent to

Λ_{E′}(F(D^{R}(f))(α̃)) = Λ_{E}(α̃) ∘ (F_{y}(f))^{−1} . (4.6)

The left side and the right side of equality (4.6) are elements of

(F_{y′}(E′))^{∗(A)} = Hom_{A}(F_{y′}(E′), A) ,

where y′ = F(B(f))(y) ∈ F(M′). So, to verify formula (4.6) we need to show

(Λ_{E′}(F(D^{R}(f))(α̃)))(ṽ′) = (Λ_{E}(α̃) ∘ (F_{y}(f))^{−1})(ṽ′)

for all ṽ′ ∈ F_{y′}(E′). According to (4.4), this formula is equivalent to the following one

(F(D^{R}(f))(α̃), ṽ′)_{F(E′)} = (α̃ , (F_{y}(f))^{−1}(ṽ′))_{F(E)} . (4.7)

Since F_{y}(f) : F_{y}(E) ⟶ F_{y′}(E′) is an isomorphism, thus there exists ṽ ∈ F_{y}(E) such that ṽ′ = F(f)(ṽ). Using formula (4.5) we obtain

(F(D^{R}(f))(α̃), ṽ′)_{F(E′)} = (F(D^{R}(f))(α̃), F(f)(ṽ))_{F(E′)}
= (α̃, ṽ)_{F(E)}
= (α̃, (F_{y}(f))^{−1}(ṽ′))_{F(E)} .

Formula (4.7) is verified and the proof of the proposition is ended. □

Next we prove

Proposition 4.2 Let λ : A ⟶ R be an R-linear mapping. For an A-module bundle π : E ⟶ M the mapping

ξ_{E}^{λ} : E^{∗(A)} ∋ α ⟶ λ ∘ α ∈ E^{∗(R)} (4.8)

is a vector bundle homomorphism covering the identity on M and satisfying the following naturality property:

if π : E ⟶ M and π′ : E′ ⟶ M′ are two A-module bundles such that dim M = dim M′, dim E = dim E′ and f : E ⟶ E′ is an A-module bundle imbedding, then the diagram

(4.9)^{[figure: E^{∗(A)}; E^{′∗(A)}; D^{A}(f); D^{R}(f); E^{∗(R)}; E^{′∗(R)}; ξ_{E}^{λ}; ξ_{E′}^{λ}; M; M′; f]}

commutes, where arrows ↘, ↙, ↗, ↖ are the corresponding bundle projections.

P r o o f. From formula (4.8) we deduce immediately that ξ_{E}^{λ} covers identity on M. Since for an element α ∈ E^{∗(A)} we have

[ξ_{E′}^{λ} ∘ D^{A}(f)](α) = λ ∘ α ∘ f_{x}^{−1} = [D^{R}(f) ∘ ξ_{E}^{λ}](α) ,

thus diagram (4.9) commutes. The proof is finished. □

An interesting problem is to study equivalence of the dual bundle functors. We prove the following proposition:

Proposition 4.3 Assume dim E > dim M. Then ξ_{E}^{λ} is a vector bundle isomorphism if and only if the R-bilinear symmetric form

A × A ∋ (a, b) ⟶ λ(ab) ∈ R

associated with λ is non-singular.

P r o o f. Let σ^{i} : U ⟶ E, i = 1, …, p, be a local A-basis of sections of E and let a_{1}, …, a_{K} be a basis (over R) of Weil algebra A. We denote by (σ^{i})^{∗(A)} the A-dual basis of sections of E^{∗(A)}. Clearly, we have

a_{ν} · σ^{i} , i = 1, …, p , ν = 1, …, K

is a basis of E over R a_{ν} · (σ^{i})^{∗(A)} , i = 1, …, p , ν = 1, …, K is a basis of E^{∗(A)} over R (4.10)

We want to evaluate a rang (over R) of ξ_{E}^{λ}, so we need to find a rang of matrix of ξ_{E}^{λ} which is composed with elements

A_{(i,ν),(j,ρ)} = [ξ_{E}^{λ}(a_{ν} · (σ^{i})^{∗(A)})](a_{ρ} · σ^{j}) . (4.11)

Applying formulas (4.10), (4.11) and definitions of all used mappings we obtain

A_{(i,ν),(j,ρ)} = [ξ_{E}^{λ}(a_{ν} · (σ^{i})^{∗(A)})](a_{ρ} · σ^{j}) = [λ ∘ (a_{ν} · (σ^{i})^{∗(A)})](a_{ρ} · σ^{j})
= [λ ∘ (a_{ν}a_{ρ} · (σ^{i})^{∗(A)})](σ^{j})
= δ_{ij}λ(a_{ν}a_{ρ}) .

From this equality we deduce that the matrix with elements A_{(i,ν),(j,ρ)} given by (4.11) is of rank pr, where r is a rank of R-bilinear symmetric form associated to λ.

The proof of Proposition 4.3 is finished. □

Composing the natural transformations from Propositions 4.1 and 4.2 we obtain

Proposition 4.4 Let λ : A ⟶ R be an R-linear mapping. If π : E ⟶ M is a vector bundle, then the composition

ω_{E}^{λ} = ξ_{F(E)}^{λ} ∘ Λ_{E} : F(E^{∗(R)}) ⟶ (F(E))^{∗(A)} ⟶ (F(E))^{∗(R)}

is a homomorphism of vector bundles covering the identity on F(M).

The familly of mappings {ω_{E}^{λ}} is natural in the sense, that for two vector bundles π : E ⟶ M and π′ : E′ ⟶ M′ such that dim M = dim M′ and dim E = dim E′, and for any vector bundle imbedding f : E ⟶ E′ the diagram

[figure: F(E^{∗(R)}); F(E^{′∗(R)}); F(D^{R}(f)); (F(E))^{∗(R)}; (F(E′))^{∗(R)}; ω_{E}^{λ}; ω_{E′}^{λ}; D^{R}(F(f)); F(M); F(M′); F(B(f)); F(π); F(π′); Π; Π′] (4.12)

commutes, where π, π′, Π, Π′ are the projections of corresponding vector bundles.

If dim E > dim M, then ω_{E}^{λ} is a vector bundle isomorphism if and only if the symmetric 2-form A × A ∋ (a, b) ⟶ λ(ab) ∈ R associated with λ is non-singular.

P r o o f. The first part of the proposition is now an immediate consequence of Propositions 4.1 and 4.2, and the second one — of Propositions 4.1 and 4.3. □

We have the following classification of natural transformations of bundles F(E^{∗(R)}) and (F(E))^{∗(R)} (see [3])

Proposition 4.5 Let E ⟶ M be a vector bundle such that dim E > dim M > dim A. If ω_{E} : F(E^{∗(R)}) ⟶ (F(E))^{∗(R)} is a natural homomorphism of vector bundles over F(M) (it means that ω_{E} covers the identity on F(M) and diagram (4.12) commutes), then there exists an A-linear function λ : A ⟶ R such that ω = ω^{λ}.

From Propositions 4.4 and 4.5 we deduce the following sufficient and necessary condition for the commutativity of real dual bundle and a product preserving functor.

Proposition 4.6 Let dim E > dim M > dim A. There exists a natural isomorphism between two vector bundles F(E^{∗(R)}) and (F(E))^{∗(R)} over F(M) if and only if there exists a linear function λ : A ⟶ R such that the bilinear mapping A×A ∋ (a, b) ⟶ λ(ab) ∈ R is non-singular.

5 Natural monomorphisms

In Sections 2 and 3 we have found the natural epimorphism F(E_{1}) ⊗_{R} F(E_{2}) ⟶ F(E_{1} ⊗_{R} E_{2}) and ⋀^{k(R)}F(E) ⟶ F(⋀^{k(R)}E). In this section we shall look for a condition for F under which there exist natural monomorphisms F(E_{1} ⊗_{R} E_{2}) ⟶ F(E_{1}) ⊗_{R} F(E_{2}) and F(⋀^{k(R)}E) ⟶ ⋀^{k(R)}F(E). We hope that these monomorphisms will be very usefull to study liftings of geometric structures from M to F(M). In these constructions we will apply results of Section 4. We begin uppon some general remarks.

Let E_{1} → M_{1} and E_{2} → M_{2} be two vector bundles and let f : E_{1} ⟶ E_{2} be a vector bundle homomorphism covering φ : M_{1} ⟶ M_{2}. For each point x of M we consider the double-transposed mapping f_{x}^{∗∗(R)} : (E_{1})_{x}^{∗∗(R)} ⟶ (E_{2})_{φ(x)}^{∗∗(R)} . We can glue this family {f_{x}^{∗∗(R)}} to obtain a vector bundle homomorphism

f^{∗∗(R)} : E_{1}^{∗∗(R)} ⟶ E_{2}^{∗∗(R)}

defined by f^{∗∗(R)} | (E_{1})_{x}^{∗∗(R)} = f_{x}^{∗∗(R)} . Of course, f^{∗∗(R)} is a vector bundle homomorphism covering φ.

Let us observe that if f : E_{1} ⟶ E_{2} is a vector bundle imbedding and dim E_{1} = dim E_{2}, dim M_{1} = dim M_{2}, then

f^{∗∗(R)} = D^{R}(D^{R}(f)) .

The natural isomorphism between a vector space V and its bidual space V^{∗∗} implies a vector bundle isomorphism

k_{E} : E ⟶ E^{∗∗(R)}

covering the idendity. This family of mappings is natural in the sense that for any vector bundle homomorphism f : E ⟶ E′ covering φ : M ⟶ M′ the following diagram

(5.1)^{[figure: E; E′; f; E^{∗∗(R)}; E^{′∗∗(R)}; k_{E}; k_{E′}; f^{∗∗(R)}; M; M′; φ]}

commutes.

The double-transposed mapping f^{∗∗(R)} is defined for any vector bundle homomorphism f : E_{1} ⟶ E_{2}. A transposed mapping is defined only for vector bundle homomorphisms satisfying certain supplementary condition. We will use transposed mapping for vector bundle homomorphisms covering the identity.

Let E_{1} → M and E_{2} → M be two vector bundles over the same base M and let f : E_{1} ⟶ E_{2} be a vector bundle homomorphism covering the identity on M. Now for each point x of M we consider the transposed mapping f_{x}^{∗(R)} : (E_{2})_{x}^{∗(R)} ⟶ (E_{1})_{x}^{∗(R)} . We can glue this family {f_{x}^{∗(R)}} to obtain a vector bundle homomorphism

f^{∗(R)} : E_{2}^{∗(R)} ⟶ E_{1}^{∗(R)}

defined by f^{∗(R)} | (E_{2})_{x}^{∗(R)} = f_{x}^{∗(R)} . Of course, f^{∗(R)} is a vector bundle homomorphism covering the identity on M.

Let us observe that for a vector bundle homomorphism f : E_{1} ⟶ E_{2} covering the identity on M we have

f^{∗∗(R)} = (f^{∗(R)})^{∗(R)}

and for a vector bundle isomorphism f : E_{1} ⟶ E_{2} covering the identity on M we have

D^{R}(f) = (f^{∗(R)})^{−1} .

From the linear algebra we obtain immediately

Lemma 5.1 If f : E_{1} ⟶ E_{2} is a vector bundle epimorphism covering the identity, then f^{∗(R)} : E_{2}^{∗(R)} ⟶ E_{1}^{∗(R)} is a vector bundle monomorphism covering the identity.

We apply this construction to the natural epimorphism η̃_{E_{1},E_{2}} from Proposition 2.4. From Proposition 2.4 and Lemma 5.1 we deduce that the family

η̃_{E_{1},E_{2}}^{∗(R)} = (η̃_{E_{1},E_{2}})^{∗(R)} : (F(E_{1} ⊗_{R} E_{2}))^{∗(R)} ⟶ (F(E_{1}) ⊗_{R} F(E_{2}))^{∗(R)}

of vector bundle monomorphisms over the identity on F(M) is natural in the following sense:

If E_{1}, E_{2}, E′_{1}, E′_{2} are vector bundles over M and M′ such that dim M = dim M′, dim E_{1} = dim E′_{1} and dim E_{2} = dim E′_{2}, then for two vector bundle imbedings f_{1} : E_{1} ⟶ E′_{1} and f_{2} : E_{2} ⟶ E′_{2} covering the same imbedding φ : M ⟶ M′ the following diagram

[figure: (F(E_{1}) ⊗_{R} F(E_{2}))^{∗(R)}; (F(E′_{1}) ⊗_{R} F(E′_{2}))^{∗(R)}; D^{R}(F(f_{1}) ⊗_{R} F(f_{2})); (F(E_{1} ⊗_{R} E_{2}))^{∗(R)}; (F(E′_{1} ⊗_{R} E′_{2}))^{∗(R)}; η̃_{E_{1},E_{2}}^{∗(R)}; η̃_{E′_{1},E′_{2}}^{∗(R)}; D^{R}(F(f_{1} ⊗_{R} f_{2})); F(M); F(M′); F(φ)] (5.2)

commutes, where the arrows ↘, ↙, ↗, ↖ are the projections of corresponding vector bundles.

We will be used also the natural isomorphism

I_{E_{1},E_{2}} : (E_{1} ⊗_{R} E_{2})^{∗(R)} ⟶ E_{1}^{∗(R)} ⊗_{R} E_{2}^{∗(R)}

induced by the standard isomorphism of vector spaces (V_{1} ⊗ V_{2})^{∗} and V_{1}^{∗} ⊗ V_{2}^{∗}. The family of mappings {I_{E_{1},E_{2}}} is natural in the sense that for vector bundles E_{1} → M, E_{2} → M, E′_{1} → M′, E′_{2} → M′ such that dim M = dim M′, dim E_{1} = dim E′_{1}, dim E_{2} = dim E′_{2} and for two vector bundle monomorphisms f_{1} : E_{1} ⟶ E′_{1} and f_{2} : E_{2} ⟶ E′_{2} covering the same imbedding φ : M ⟶ M′ the diagram

[figure: (E_{1} ⊗_{R} E_{2})^{∗(R)}; (E′_{1} ⊗_{R} E′_{2})^{∗(R)}; D^{R}(f_{1} ⊗ f_{2}); E_{1}^{∗(R)} ⊗_{R} E_{2}^{∗(R)}; E^{′∗(R)} ⊗_{R} E_{2}^{′∗(R)}; I_{E_{1},E_{2}}; I_{E′_{1},E′_{2}}; D^{R}(f_{1}) ⊗ D^{R}(f_{2}); M; M′; φ] (5.3)

commutes.

After this introduction we formulate our proposition.

Proposition 5.2 Let λ : A ⟶ R be an R-linear mapping such that the symmetric form

A × A ∋ (a, b) ⟶ λ(ab) ∈ R

is non-singular. Then for vector bundles E_{1} → M and E_{2} → M there exists a vector bundle monomorphism

μ_{E_{1},E_{2}}^{λ} : F(E_{1} ⊗_{R} E_{2}) ⟶ F(E_{1}) ⊗_{R} F(E_{2})

covering the identity on F(M). The family of mappings {μ_{E_{1},E_{2}}^{λ}} satisfies the following naturality condition:

For vector bundles E_{1} → M, E_{2} → M, E′_{1} → M′, E′_{2} → M′ such that dim M = dim M′, dim E_{1} = dim E′_{1}, dim E_{2} = dim E′_{2} and for two vector bundle monomorphisms f_{1} : E_{1} ⟶ E′_{1} and f_{2} : E_{2} ⟶ E′_{2} covering the same imbedding φ : M ⟶ M′ the diagram

(5.4)^{[figure: F(E_{1} ⊗_{R} E_{2}); F(E′_{1} ⊗_{R} E′_{2}); F(f_{1} ⊗ f_{2}); F(E_{1}) ⊗_{R} F(E_{2}); F(E′_{1}) ⊗_{R} F(E′_{2}); μ_{E_{1},E_{2}}^{λ}; μ_{E′_{1},E′_{2}}^{λ}; F(f_{1}) ⊗ F(f_{2}); F(M); F(M′); F(φ)]}

commutes.

P r o o f. To simplify the notations in the proof of this proposition we will write ⊗ instead of ⊗_{R} , ( )^{∗} instead of ( )^{∗(R)} and D instead of D^{R}. We define μ_{E_{1},E_{2}}^{λ} as the composition

\begin{matrix}F(E_{1} ⊗_{R} E_{2}) & F((E_{1} ⊗ E_{2})^{∗∗}) & \overline{} & F(k_{E_{1}⊗E_{2}}) & & ω_{(E_{1}⊗E_{2})^{∗}}^{λ} & (F((E_{1} ⊗ E_{2})^{∗}))^{∗} & \overline{} & D(F(I_{E_{1},E_{2}})) & (F(E_{1}^{∗} ⊗ E_{2}^{∗}))^{∗} & & (F(E_{1}^{∗}) ⊗ F(E_{2}^{∗}))^{∗} & η̃_{E_{1}^{∗},E_{2}^{∗}}^{∗} & \overline{} & I_{F(E_{1}^{∗}),F(E_{2}^{∗})} & (F(E_{1}^{∗}))^{∗} ⊗ (F(E_{2}^{∗}))^{∗} & & D(ω_{E_{1}^{∗}}^{λ}) ⊗ D(ω_{E_{2}^{∗}}^{λ}) & F(E_{1})^{∗∗} ⊗ F(E_{2})^{∗∗} & F(E_{1}) ⊗_{R} F(E_{2}) & \overline{} & k_{F(E_{1})}^{−1} ⊗ k_{F(E_{2})}^{−1} \\ (5.5) \end{matrix}

Let us observe that μ_{E_{1},E_{2}}^{λ} is well-defined under the condition that 2-form A×A ∋ (a, b) → λ(ab) ∈ R is no-singular because only under this condition the mapping D(ω_{E_{1}}^{λ}) ⊗ D(ω_{E_{2}}^{λ}) is defined.

All arrows in diagram (5.5) are vector bundle homomorphisms covering the identity on F(M). According to Proposition 2.4 and Lemma 5.1 η̃_{E_{1}^{∗},E_{2}^{∗}}^{∗} is a vector bundle monomorphism. All other arrows in diagram (5.5) are vector bundle isomorphisms. Namely, F(k_{E_{1}⊗E_{2}}) and k_{F(E_{1})}^{−1} ⊗ k_{F(E_{2})}^{−1} are isomorphisms because k_{E_{1}⊗E_{2}}, k_{F(E_{1})}^{−1} and k_{F(E_{2})}^{−1} are isomorphisms, ω_{(E_{1}⊗E_{2})^{∗}}^{λ} and D(ω_{E_{1}^{∗}}^{λ}) ⊗ D(ω_{E_{2}^{∗}}^{λ}) are isomorphisms according to Proposition 4.4.

The commutativity of diagram (5.4) is an immediate consequence of Proposition 4.4 and the commutativity of diagrams (5.1), (5.2) and (5.3). □

In similar way using Proposition 3.3 instead of Proposition 2.4 we can verify

Proposition 5.3 Let λ : A ⟶ R be an R-linear mapping such that the symmetric form

A × A ∋ (a, b) ⟶ λ(ab) ∈ R

is non-singular. Then for a vector bundle E → M there exists a vector bundle monomorphism

μ_{E}^{λ} : F(⋀^{k(R)}E) ⟶ ⋀^{k(R)}F(E)

covering the identity on F(M). The family of mappings {μ_{E}^{λ}} satisfies the following naturality condition:

For vector bundles E_{1} → M and E′ → M′ such that dim M = dim M′, dim E_{1} = dim E′_{1} and for a vector bundle monomorphism f : E ⟶ E′ covering an imbedding φ : M ⟶ M′ the diagram

[figure: F(⋀^{k(R)}E); F(⋀^{k(R)}E′); F(⋀^{k(R)}f); ⋀^{k(R)}F(E); ⋀^{k(R)}F(E′); μ_{E}^{λ}; μ_{E′}^{λ}; ⋀^{k(R)}F(f); F(M); F(M′); F(φ)]

commutes.

P r o o f. We define μ_{E}^{λ} as a similar composition used to define μ_{E_{1},E_{2}}^{λ} in Proposition 5.2. □

We can generalise Proposition 5.2 for a finite sequence of vector bundles and we can formulate a similar proposition as Proposition 5.3 for symmetric tensor product.

6 Product preserving functors, the cotangent bundle and exterior power of the tangent bundle

In this section we will study bundles F(T^{∗(R)}M) and T^{∗(R)}(FM) and we apply results obtained in Section 4.

The bundles F(T^{∗(R)}M), T^{∗(A)}(FM) and T^{∗(R)}(FM) are obtained by composition of three functors. The first bundle is obtained by composition: the tangent bundle functor, the dual bundle functor D^{R} and a product preserving functor F, the second one is a composition of: a product preserving functor F, the tangent bundle and the dual bundle functor D^{A} and the last bundle is a composition of: a product preserving functor F, the tangent bundle and the dual bundle functor D^{R}. It means that F(T^{∗(R)}M) = F((T M)^{∗(R)}), T^{∗(A)}(FM) = (T (FM))^{∗(A)} and T^{∗(R)}(FM) = (T (FM))^{∗(R)}.

Thus from Proposition 4.1 we have the following proposition.

Proposition 6.1 For each manifold M there exists an isomorphism of A-module bundles

Ω_{M} : F(T^{∗(R)}M) ⟶ T^{∗(A)}(FM)

covering the identity on F(M).

The family of mappings {Ω_{M}} is natural in the sense that if M, M′ are two manifolds of the same dimension and f : M ⟶ M′ is an imbedding, then the diagram

(6.1)^{[figure: F(T^{∗(R)}M); F(T^{∗(R)}M′); F(D^{R}(T f)); D^{A}(T F(f)); T^{∗(A)}(FM); T^{∗(A)}(FM′); Ω_{M}; Ω_{M′}; F(M); F(M′); F(f)]}

commutes.

P r o o f. It is sufficient to define

Ω_{M} = D^{A}(η_{M}) ∘ Λ_{T M} ,

where Λ_{T M} is from Proposition 4.1, η_{M} : F(T M) ⟶ T (FM) is the natural isomorphism between A-module bundles F(T M) and T (FM) (see [5], [2]). □

From Propositions 6.1 and 4.2 we have immediately

Proposition 6.2 For every R-linear function λ : A ⟶ R the family of mappings

Ω_{M}^{λ} = ξ_{T (FM)}^{λ} ∘ Ω_{M} : F(T^{∗(R)}M) ⟶ T^{∗(R)}(FM)

is a natural transformation of A-module bundles covering the identity on F(M). It means that the family of mappings {Ω_{M}^{λ}} satisfies the condition:

if f : M ⟶ M′ is an imbedding of two manifolds M and M′ of the same dimension, then the diagram

[figure: F(T^{∗(R)}M); F(T^{∗(R)}M′); F(D^{R}(T f)); D^{R}(T F(f)); T^{∗(R)}(FM); T^{∗(R)}(FM′); Ω_{M}^{λ}; Ω_{M′}^{λ}; F(M); F(M′); F(f)] (6.2)

commutes.

An interesting problem is to study the commutativity of the cotangent bundle functor with a product preserving functor. From Propositions 4.4 and 6.2 we deduce:

Proposition 6.3 The transformation Ω_{M}^{λ} is a natural isomorphism of vector bundles F(T^{∗(R)}M) and T^{∗(R)}(FM) if and only if the R-bilinear symmetric form

A × A ∋ (a, b) ⟶ λ(ab) ∈ R

associated with λ is non-singular.

From Proposition 6.3 we obtain

Proposition 6.4 There exists a natural isomorphism T^{r}(T^{∗(R)}M) ⟶ T^{∗(R)}(T^{r}M), where T^{r}M = J_{0}^{r}(R, M) is the tangent bundle of order r.

P r o o f. In the case F(M) = T^{r}M the Weil algebra is

A = J_{0}^{r}(R, R) = { j_{0}^{r}f : f : R ⟶ R } .

If a = j_{o}^{r}(id), then a^{0} = 1, a, a^{2}, …, a^{r−1} is a basis of A. We consider then R-linear function λ : A ⟶ R given by

λ(a^{i}) = \{ \begin{matrix}1 , & if i = r − 1 \\ 0 , & in other cases \end{matrix}

Now the associated form A × A ∋ (x, y) ⟶ λ(xy) ∈ R is non-singular because its matrix is

[ \begin{matrix}0 & 0 & … & 0 & 1 \\ 0 & 0 & … & 1 & 0 \\ ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 1 & … & 0 & 0 \\ 1 & 0 & … & 0 & 0 \end{matrix} ]

Now, by Proposition 6.3 there is a natural isomorphism T^{r}(T^{∗}M) ⟶ T^{∗}(T^{r}M). □

For r = 1 we have

Proposition 6.5 (Modugno, Stefani [7]) The tangent bundle and the cotangent bundle functors commute, e.g., there exists a natural isomorphism T (T^{∗}M) ⟶ T^{∗}(T M).

We can also apply Propositions 5.2 and 5.3 to the tangant bundle T M. We obtain

Proposition 6.6 Let λ : A ⟶ R be an R-linear mapping such that the symmetric form

A × A ∋ (a, b) ⟶ λ(ab) ∈ R

is non-singular. Then for a manifold M there exists a vector bundle monomorphism

μ̃_{M}^{λ} : F(⨂^{k(R)}T M) ⟶ ⨂^{k(R)}T (FM)

covering the identity on F(M). The family of mappings {μ̃_{M}^{λ}} satisfies the following naturality condition:

For two manifolds M, M′ such that dim M = dim M′ and for its imbedding φ : M ⟶ M′ the diagram

[figure: F(⨂^{k(R)}T M); F(⨂^{k(R)}T M′); F(⨂^{k(R)}T φ); ⨂^{k(R)}T (FM); ⨂^{k(R)}T (FM′); μ̃_{M}^{λ}; μ̃_{M′}^{λ}; ⨂^{k(R)}T F(φ); F(M); F(M′); F(φ)]

commutes.

P r o o f. We set μ̃_{M}^{λ} = ⨂^{k(R)}η_{M} ∘ μ_{T M,...,T M}^{λ}. □

Proposition 6.7 Let λ : A ⟶ R be an R-linear mapping such that the symmetric form

A × A ∋ (a, b) ⟶ λ(ab) ∈ R

is non-singular. Then for a manifold M there exists a vector bundle monomorphism

μ̃_{M}^{λ} : F(⋀^{k(R)}T M) ⟶ ⋀^{k(R)}T (FM)

covering the identity on F(M). The family of mappings {μ̃_{M}^{λ}} satisfies the following naturality condition:

For two manifolds M, M′ such that dim M = dim M′ and for an imbedding φ : M ⟶ M′ the diagram

[figure: F(⋀^{k(R)}T M); F(⋀^{k(R)}T M′); F(⋀^{k(R)}T φ); ⋀^{k(R)}T (FM); ⋀^{k(R)}T (FM′); μ̃_{M}^{λ}; μ̃_{M′}^{λ}; ⋀^{k(R)}T F(φ); F(M); F(M′); F(φ)]

commutes.

P r o o f. We set μ̃_{M}^{λ} = ⋀^{k(R)}η_{M} ∘ μ_{T M}^{λ}. □

We can use these monomorphisms to lift tensor fields of type (k, 0) (multivectors).

Let F be a product preserving functor which admits an R-linear function λ : A ⟶ R on its Weil algebra such that the associated symmetric form A × A ∋ (a, b) ⟶ λ(ab) ∈ R is non-singular.

If t is a k-multivector (a tensor field of type (k, 0) ) on M, then we define its λ-lift to F(M) by

t^{λ} = μ̃_{M}^{λ} ∘ F(t) .

It is a k-multivector on F(M). If t is a skewsymmetric, then so is t^{λ}.

References

[1] D. J. Eck, Product-precerving functors on smooth manifolds J. Pure Appl. Algebra 42 (1986), 133 — 140

[2] J. Gancarzewicz, W. Mikulski, Z. Pogoda, Lifts of some tensor fields and linear connections to product preserving functors, Nagoya Math. J., 135(1994), 1 — 41

[3] J. Gancarzewicz, W. Mikulski, Z. Pogoda, Product preserving functors and dual bundles, (in press)

[4] G. Kainz, P. Michor, Natural transformations in differential geometry, Czech. Math. J. 37(112),1987, 584 — 607

[5] I. Kolář, P. W. Michor, J. Slovák, Natural operations in differential geometry, ed. Spriger Verlag, Berlin, 1993.

[6] O. O. Luciano, Categories of multiplicative functors and Morimoto’s conjecture, Preprint 46, Inst. Fourier, Laboratoire des Mathématiques, Grenoble, 1986

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

[8] A. Morimoto, Prolongations of connections to bundles of infinitely near points, J. Diff. Geom. 11(1976), 476 — 498

Authors’ address:

Instytut Matematyki UJ, ul. Reymonta 4, 30-059 Kraków, POLAND e.-mail address:

gancarze@im.uj.edu.pl

mikulski@im.uj.edu.pl

pogoda@im.uj.edu.pl