Prolongations of G-structures related to Weil bundles and some applications

-

EXTRACTA MATHEMATICAE

Volumen 33, N´umero 2, 2018 instituto de investigaci´on de matem´aticas de la

universidad de extremadura

Vol. 37, Num. 1 (2022), 111 – 138 Available online February 15, 2022

Prolongations of G-structures related to Weil bundles and some applications

P.M. Kouotchop Wamba¹, G.F. Wankap Nono², A. Ntyam¹

1Department of Mathematics, Higher Teacher Training college University of Yaound´e 1, PO.BOX 47 Yaound´e, Cameroon

2Department of Mathematics and Computer Science, Faculty of Science University of Ngaound´er´e, PO.BOX 454 Ngaound´er´e, Cameroon [email protected] , [email protected] , [email protected]

Received November 1, 2021 Presented by Manuel de Le´on Accepted December 5, 2021

Abstract : Let M be a smooth manifold of dimension m ≥ 1 and P be a G-structure on M , where G is a Lie subgroup of linear group GL(m). In [8], it has been defined the prolongations of G-structures related to tangent functor of higher order and some properties have been established. The aim of this paper is to generalize these prolongations to a Weil bundles. More precisely, we study the prolongations of G-structures on a manifold M , to its Weil bundle T^AM (A is a Weil algebra) and we establish some properties. In particular, we characterize the canonical tensor fields induced by the A-prolongation of some classical G-structures.

Key words: G-structures, Weil-Frobenius algebras, Weil functors, gauge functors and natural trans- formations.

MSC (2020): 58A32, 53C15 secondary 58A20, 58A10.

Introduction

We recall that, a Weil algebra A is a real commutative algebra with unit which is of the form A = R · 1A⊕ N_A, where N_A is a finite dimensional ideal of nilpotent elements of A (see [4] or [8]). It exists several examples of Weil algebra, for instance the algebra generated by 1 and ε with ε² = 0 denoted by D (sometimes it is called the algebra of dual numbers, it is also the truncated polynomial algebra of degree 1). Another Weil algebra is given by the spaces of all r-jets of R^kinto R with source 0 ∈ R^kand denoted by J₀^r(R^k, R). The ideal of nilpotent elements is the finite vector space J₀^r(R^k, R)0. Let A = R·1A⊕N_A be a Weil algebra, we adopt the covariant approach of Weil functor described by I. Kol`ar in [6]. We denote by N_A^k the ideal generated by the product of k elements of N_A, there is one and only one natural number h such that N_A^h 6= 0 and N_A^h+1= 0. The integer h is called the order of A and the dimension k of

ISSN: 0213-8743 (print), 2605-5686 (online)

© The author(s) - Released under a Creative Commons Attribution License (CC BY-NC 3.0)

the vector space^NA/N_A² is said to the width of A. In this case, the Weil algebra A is called (k, h)-algebra. If %, %1 : J₀^h(R^k, R) → A are two surjective algebra homomomorphisms, then there is an isomorphism σ : J₀^h(R^k, R) → J0^h(R^k, R) such that: %₁◦ σ = %. We say that, two maps ϕ, ψ : R^k → M determine the same A-velocity if for every smooth map f : M → R

%

j₀^h(f ◦ ϕ)

= %

j₀^h(f ◦ ψ) .

The equivalence class of the map ϕ : R^k → M is denoted by j^Aϕ and will called A-velocity at 0 (see [6], [7] or [8]). We denote by T^AM the space of all A-velocities on M . More precisely,

T^AM =n

j^Aϕ, ϕ : R^k→ Mo .

T^AM is a smooth manifold of dimension m × dim A. For a local chart U, u¹, . . . , u^m
of M , the local chart of T^AM is T^AU, uⁱ₀, . . . , uⁱ_K
such that:

( uⁱ₀ j^Aϕ
= uⁱ(ϕ(0))

uⁱ_α j^Aϕ
= a^∗_α j^A(uⁱ◦ ϕ)
1 ≤ α ≤ K

where (a0, . . . , a_K) is basis of A and (a^∗₀, . . . , a^∗_K) is a dual basis. We denote by π_M^A : T^AM → M the natural projection such that π^A_M(j^Aϕ) = ϕ(0), so T^AM, M, π^A_M
is a fibered manifold. For every smooth map f : M → M , induces a smooth map T^Af : T^AM → T^AM such that: for any j^Aϕ ∈ T^AM ,

T^Af (j^Aϕ) = j^A(f ◦ ϕ).

In particular we have that f, T^Af
is a fibered morphism from T^AM, M, π^A_M
to 

T^AM , M , π^A

M



. This defines a bundle functor T^A : Mf → F M called Weil functor induced by A. The bundle functor T^A preserves product in the sense, that for any manifolds M and M , the map

T^A(pr_M), T^A(pr_M)
: T^A(M × M ) −→ T^AM × T^AM

where pr_M : M × M → M and pr_M : M × M → M are the projections, is an F M−isomorphism. Hence we can identify T^A(M × M ) with T^AM × T^AM .

Let B be another (s, r) Weil algebra and µ : A → B be an algebra homo- morphism, %⁰ : J₀^r(R^s, R) → B the surjective algebra homomorphism. Then

there is an algebra homomorphism µ : Je ₀^h(R^k, R) → J₀^r(R^s, R) such that the following diagram

J₀^h(R^k, R) −−−−→ J^eµ ₀^r(R^s, R)

%



 y



 y ^%

0

A −−−−→

µ B

commutes. In particular, there is map fµ : R^s → R^k such that, µ(je ₀^hg) = j₀^r(g ◦ f_µ), where g ∈ C^∞(R^k). For any manifold M of dimension m ≥ 1, it is proved in [7] that there is smooth map µ_M : T^AM → T^BM defined by:

µ_M(j^Aϕ) = j^B(ϕ ◦ f_µ).

More precisely, µ_M : T^AM → T^BM is a natural transformations and denoted by µ : T^A → T^B. The fundamental result, which reads that every product preserving bundle functor on Mf is a Weil functor. More precisely, if F is a product preserving bundle functor on Mf , a : R × R → R and λ : R × R → R is the addition and the multiplication of reals, then F a : F R × F R → F R and F λ : F R × F R → F R is the vector addition and the algebra multiplication in the Weil algebra F R and F coincides with the Weil functor T^{F R}. Every natural transformation µ : T^A → T^B are in bijection with the algebra homomorphism µ_R : A → B (see [8]). Since % : J₀^h(R^k, R) → A is determined up to an isomorphism J₀^h(R^k, R) → J₀^h(R^k, R) it follows that this construction is independent of the choice of %. The Weil functor generalizes the tangent functor, more precisely, when A is the space of all r-jets of R^k into R with source 0 ∈ R^k denoted by J₀^r(R^k, R), the corresponding Weil functor is the functor of k-dimensional velocities of order r and denoted by T_k^r. For k = 1, it is called tangent functor of order r and denoted by T^r, this functor plays an essential role in the reduction of some hamiltonian systems of higher order. It has been clarified that, the theory of Weil functor represents a unified technique for studying a large class of geometric problems related with product preserving functor.

Let A = R · 1A⊕ N_A be a Weil algebra. For any multiindex 0 < |α| ≤ h, we put eα= j^A(x^α) is an element of NA. For any ϕ ∈ C^∞(R^k, R) , we have:

j^Aϕ = ϕ(0) · 1_A+ X

1≤|α|≤h 1

α!D_αϕ(0)e_α.

In particular the family {eα} generates the ideal N_A. We denote by B_A the set of all multiindex such that (e_α)_α∈B

A is a basis of N_A and B_A her

complementary with respect to the set of all multiindex γ ∈ Nⁿ such that 1 ≤| γ |≤ h. For β ∈ BA, we have eβ = λ^α_βeα. In particular,

e_α· e_β =

(e_α+β if α + β ∈ B_A, λ^γ_α+βeγ if α + β ∈ BA. It follows that, for any ϕ ∈ C^∞(R^k, R) , we have:

j^Aϕ = ϕ(0) · 1A+ X

α∈B_A 1

α!Dαϕ(0) + X

β∈BA

λ^α_β

β!Dβϕ(0)

! eα.

Let U, xⁱ
be a local coordinate system of M , a coordinate system induced by U, xⁱ
over the open T^AU of T^AM denoted by xⁱ, xⁱ_α
is given by

( xⁱ = xⁱ◦ π_M^A = xⁱ₀, xⁱ_α= xⁱ_α+P

β∈BAλ^α_βxⁱ_β,

where xⁱ_β(j^Ag) = _β!¹ · D_β(xⁱ◦ g)(0) and j^Ag ∈ T^AU . In the particular case where A = D, the local coordinate system of T M induced by U, xⁱ
is denoted by xⁱ, ˙xⁱ
.

Let M be a smooth manifold of dimension m ≥ 1, with (T M, M, πM) we denote its tangent bundle, and with (F (M ), M, p_M) we denote the frame bundles of M . Let G be a Lie subgroup of GL(m), a G-structure on a man- ifold M is a G-subbundle (P, M, π) of the frame bundle F (M ) of M . For the general theory of G-structures see, for instance [1]. The prolongations of G-structures from a manifold M to its tangent bundles of higher order T^rM has been studied by A. Morimoto in [12]. In particular, it proves that if a manifold M has an integrable structure (resp. almost complex structure, sym- plectic structure, pseudo-Riemannian structure), then T^rM has canonically the same kind of structure. Since the tangent functor of higher order T^r on the manifolds, considers all derivatives of higher order (up to order r), all the proofs are obtained by calculation in local coordinate. The situation should be much complicated for the Weil functor T^A. Thus, the aim of this paper is to define the prolongations of G-structures from a manifold M to its Weil bundle T^AM . In particular, we construct a canonical embedding j_A,E of T^A(F E) into F (T^AE), where F (E) denote the frame bundle of the vector bundle (E → M ).

Using the natural isomorphism κ_A,M : T^A(T M ) → T (T^AM ) (see [5]) and the embedding j_{A,T M}, we define this A-prolongation T^AP of a G-structure P of

a manifold M , to its Weil bundle T^AM . In particular, we prove that T^AP is integrable if and only if P is integrable. In the last section, we use the theory of lifting of tensor fields defined in [3] and [6], to characterized the canonical tensor fields induced by the A-prolongation of some classical G-structures.

In this paper, all manifolds and mappings are assumed to be differentiable of class C^∞. In the sequel A will be a Weil algebra of order h ≥ 2 and of width k ≥ 1.

1. Preliminaries

1.1. Lifts of functions and vector fields. Let ` : A → R be a smooth function, for any smooth function f : M → R, we define the `-lift of f to T^AM by:

f<sup>(`)</sup> = ` ◦ T^A(f ) ; f^(`) is a smooth function on T^AM .

Remark 1. Let (eβ)_β∈B

A a basis of N_A, we denote by e⁰, e^β

β∈BA the dual basis of A. For ` = e<sup>α</sup>, the smooth function f<sup>(`)</sup> is denoted by f^(α). In particular, for any j^Aϕ ∈ T^AM ,

f^(α)(j^Aϕ) = _α!¹D_α(f ◦ ϕ)(z)

z=0+ X

β∈BA

λ^α_β

β!D_β(f ◦ ϕ)(z)   _z=0

and f⁽⁰⁾= f ◦ π_M^A. For a coordinate system U, x¹, . . . , x^m
in M , the induced coordinate system xⁱ₀, xⁱ_α on T^AM is such that, xⁱ_α= xⁱ
(α)

. Remark 2. For any smooth map ` : A → R, the map

C^∞(M ) −→ C^∞(T^AM ) f 7−→ f^(`)

is R-linear.

For all multiindex α such that |α| ≤ h, we denote by χ^(α) : T^A → T^A the natural transformation defined for any vector bundle (E → M ) and ϕ ∈ C^∞(R^k, E) by:

χ^(α)_E (j^Aϕ) = j^A(z^αϕ)

where z^αϕ is a smooth map defined for any z ∈ R^k by (z^αϕ)(z) = z^αϕ(z).

Proposition 1. Let A be a Weil algebra. There exists one and only one family κA,M : T^A(T M ) → T (T^AM ) of vector bundle isomorphisms such that π_TAM ◦ κ_A,M = T^A(π_M) and the following conditions hold:

1. For every smooth mapping f : M → N the following diagram T^A(T M ) ^T

A(T f )

−−−−−−→ T^A(T N )

κA,M



 y



 y^κA,N T (T^AM ) −−−−−−→

T (T^Af ) T (T^AN ) commutes.

2. For two manifolds M , N we have κ_{A,M ×N} = κ_A,M × κ_A,N. Proof. See [5].

Let X : M → T M be a vector field on a manifold M , then we put X^(α) = κ_A,M ◦ χ^(α)_{T M} ◦ T^A(X) .

It is a vector bundle field on T^A(M ) called α-lift of X to T^AM . In the particular case where α = 0, the vector field X⁽⁰⁾ is denoted by X^(c) and it is called complete lift of X to T^AM . We put X^(α)= 0, for |α| > h or α /∈ N^k.

Remark 3. For any |α| ≤ h, the map

X(M ) −→ X(T^AM ) X 7−→ X^(α)

is R-linear and for any smooth map ϕ : M → N and any ϕ-related vector fields X ∈ X(M ), Y ∈ X(N ), the vector fields X^(α) ∈ X(T^AM ), Y^(α) ∈ X(T^AN ) are T^A(ϕ) related.

Proposition 2. For X, Y ∈ X(M ), we have:

h

X^(α), Y^(β)i

= [X, Y ]^(α+β) for all 0 ≤ |α, β| ≤ h.

Proof. See [5].

Remark 4. The family of α-lift of vector fields is very important, because, if S and S⁰ are two tensor fields of type (1, p) or (0, p) on T^A(M ) such that, for all X1, . . . , Xp ∈ X (M ), and multiindex α₁, . . . , αp, the equality

S



X₁^(α1), . . . , Xp^(αp)



= S⁰



X₁^(α1), . . . , Xp^(αp)



holds, then S = S⁰ (see [2]).

1.2. Lifts of tensor fields of type (1, q). Let S be a tensor field of type (1, q), we interpret the tensor S as a q-linear mapping

S : T M ×_M· · · ×_M T M −→ T M

of the bundle product over M of q copies of the tangent bundle T M . For all 0 ≤ |α| ≤ h, we put:

S^(α) : T (T^AM ) ×_T^A_M· · · ×_TAM T (T^AM ) −→ T (T^AM ) with S^(α) = κA,M ◦ χ^(α)_{T M} ◦ T^A(S) ◦



κ⁻¹_A,M × · · · × κ⁻¹_A,M

. It is a tensor field of type (1, q) on T^A(M ) called α-prolongation of the tensor field S from M to T^A(M ). In the particular case where α = 0, it is denoted by S^(c) and is called complete lift of S from M to T^A(M ).

Proposition 3. The tensor S^(α) is the only tensor field of type (1, q) on T^A(M ) satisfying

S^(α)



X₁^(α1), . . . , Xq^(αq)



= S(X1, . . . , X_q)
(α+α1+···+αq)

for all X1, . . . , Xq ∈ X(M ) and multiindex α₁, . . . , αq. Proof. See [2].

For some properties of these lifts, see [2] and [3].

1.3. Lifts of tensor fields of type (0, s). We fix the linear map p : A → R, for any vector bundle (E, M, π), we consider the natural vector bundle morphism τ_A,E^p : T^AE^∗ → T^AE
∗

(see [10]) defined for any j^Aϕ ∈ T^AE^∗ and j^Aψ ∈ T^AE by:

τ_A,E^p (j^Aϕ)(j^Aψ) = p j^A(hψ, ϕi_E)

where hψ, ϕi_E : R^k → R, z 7→ hψ (z) , ϕ (z)i_E and h·, ·i_E the canonical pairing.

For any manifold M of dimension m, we consider the vector bundle mor- phism

ε^p_A,M =h

κ⁻¹_A,Mi∗

◦ τ_{A,T M}^p : T^AT^∗M −→ T^∗T^AM.

It is clear that the family of maps

 ε^p_A,M



defines a natural transformation be- tween the functors T^A◦T^∗and T^∗◦T^Aon the category Mfmof m-dimensional manifolds and local diffeomorphisms, denoted by

ε^p_A,∗: T^A◦ T^∗−→ T^∗◦ T^A.

When (A, p) is a Weil-Frobenius algebra (see [4]), the mapping ε^p_A,M is an isomorphism of vector bundles over id_T^A_M. Beingx¹, . . . , x^m a local coor- dinate system of M , we introduce the coordinates xⁱ, ˙xⁱ
in T M , xⁱ, π_i
in T^∗M , (xⁱ, ˙xⁱ, xⁱ_β, ˙xⁱ_β) in T^AT M , (xⁱ, πj, xⁱ_β, π^β_j) in T^AT^∗M , (xⁱ, xⁱ_β, ˙xⁱ, ˙xⁱ_β) in T T^AM and (xⁱ, xⁱ_β, ξ_j, ξ^β_j) in T^∗T^AM . We have

ε^p_A,M



xⁱ, πj, xⁱ_β, π^β_j



=



xⁱ, xⁱ_β, ξ_j, ξ^β_j



with







ξ_j = π_jp₀+ P

µ∈BA

π^µ_jp_µ, ξ^β_j = P

µ∈BA

π^µ−β_j p_µ, and pα= p (eα).

Let G be a tensor fields of type (0, s) on a manifold M . It induces the vector bundle morphism G^] : T M ×_M · · · ×_M T M → T^∗M of the bundle product over M of s − 1 copies of T M . We define,

G^(p): T (T^AM ) ×_T^A_M · · · ×_TAMT (T^AM ) −→ T^∗(T^AM ) as G^(p) = ε^p_A,M ◦ T^A(G^]) ◦



κ⁻¹_A,M × · · · × κ⁻¹_A,M

. It is a T^AM -morphism of vector bundles, so G^(p) is tensor field of type (0, s) on T^AM called p- prolongation of G from M to T^AM .

Example 1. In a particular case, where s = 2 and locally G = Gijdxⁱ⊗ dx^j then

G^(p)= Gijp0dxⁱ⊗ dx^j + X

α∈BA

pα

X

β∈BA

G^(α−β)_ij

!

dxⁱ⊗ dx^j_β

+ X

µ,β∈BA

X

α∈B_A

pαG^{(α−β−µ)}_ij

!

dxⁱ_µ⊗ dx^j_β.

In the particular case where A = J₀^r(R^k, R) and p(j₀^rϕ) = _α!¹ D_α(ϕ(z))|_z=0, then G^(p) coincides with the α-prolongation of G from M to T_k^rM defined in [13].

Example 2. If ΩM is a Liouville 2-form on T^∗M defined in local coordi- nate system xⁱ, ξj
by:

ΩM = dxⁱ∧ dξ_i, then we have:

Ω^(p)_M = p0dxⁱ∧ dξ_i+ X

α∈BA

pαdxⁱ∧ dξ^α_i + X

α,β∈BA

pαdxⁱ_β∧ dξ^α−β_i .

Proposition 4. The tensor field G^(p)is the only tensor field of type (0, s) on T^A(M ) satisfying, for all X₁, . . . , X_s∈ X(M ) and multiindex α₁, . . . , α_s

G^(p)

X₁^(α1), . . . , X_s^(αs)

= (G(X₁, . . . , X_s))(^{p◦lα1+···+αs}) where la: A → A is given by la(x) = ax.

Proof. See [5].

2. The natural transformations jA,E : T^A(F E) → F (T^AE) Let V be a real vector space of dimension n, we denote by GL(V ) the Lie group of automorphisms of V .

2.1. The embedding jA,V : T^A(GL(V )) → GL(T^AV ). Let G be a Lie group and M be a m−dimensional manifold, m ≥ 1. We consider the differential action ρ : G × M → M , then the Lie group T^AG acts to T^AM by the differential action T^Aρ : T^AG × T^AM → T^AM .

Lemma 1. If the Lie group G operates on M effectively, then T^AG oper- ates on T^AM effectively by the differential action T^A(ρ).

Proof. See [5].

Let ρ_V : GL(V ) × V → V be the canonical action of GL(V ), then the Lie group T^A(GL(V )) operates effectively on the vector space T^AV by the induced action

T^A(ρV) : T^A(GL(V )) × T^AV −→ T^AV j^Aϕ, j^Au

7−→ j^A(ϕ ∗ u)

where ϕ ∗ u : R^k→ V is defined for any z ∈ R^k by:

ϕ ∗ u(z) = ϕ(z)(u(z)).

We deduce an injective map j_A,V : T^A(GL(V )) → GL(T^AV ) such that, j_A,V(j^Ag) : T^AV −→ T^AV

j^Aξ 7−→ j^A(g ∗ ξ) .

Proposition 5. The map jA,V : T^A(GL(V )) → GL(T^AV ) is an embed- ding of Lie groups.

Proof. By calculation, it is clear that jA,V is a homomorphism of Lie groups.

Remark 5. Let {e1, . . . , e_n} be a basis of V (dim V = n), we consider the global coordinate system of V , e¹, . . . , eⁿ
, we denote by 

yⁱ_j



the global coordinate of GL(V ), for any f ∈ GL(V ),

yⁱ_j(f ) =eⁱ, f (ej)

where h·, ·i is the duality bracket V^∗×V → R. We deduce that, the coordinate system of T^A(GL(V )) is denoted by

 y_jⁱ, y_j,αⁱ



α∈BA

. On the other hand, the global coordinate system of T^AV is eⁱ, eⁱ_α
, such that:







eⁱ j^Au

= eⁱ(u(0)) ,

eⁱ_α j^Au
= _α!¹ Dα(eⁱ◦ u)(z)

z=0+ P

β∈BA

λ^α_β

β! D_β(eⁱ◦ u)(z)

z=0, j^Au ∈ T^AV, the global coordinate of GL(T^AV ) denoted 

z_jⁱ, z_j,α^i,β

α,β∈B_A is such that:







z_jⁱ(ξ) =eⁱ, π_A,V(ξ)(e_j) , z_j,α^i,β(ξ) =D

eⁱ_β, ξ e^α_jE

, ξ ∈ GL(T^AV ) , we deduce that the local coordinate of the map j_A,V is given by:

j_A,V y_jⁱ, y_j,αⁱ
=



















yⁱ_j 0 · · · 0 ... . .. ... ... ... . .. ... ...

... . .. 0

· · · · y_j,αⁱ · · · y_jⁱ



















In fact,

z_j,α^i,β j_A,V j^Ag

= eⁱ_β, j_A,V j^Ag
e^α_j


= 1

β! Dβ t^αeⁱ, g(t)(ej)
 t=0

+ X

µ∈BA

λ^µ_β

µ! D_µ t^αeⁱ, g(t)(e_j)
 t=0

for any j^Ag ∈ T^A(GL(V )).

2.2. Frame gauge functor on the vector bundles. We denote by VB_m the category of vector bundles with m-dimensional base together with local isomorphism. Let BVBm : VB_m → Mf and B_{F M} : F M → Mf be the respective base functors.

Definition 1. (See [11]) A gauge bundle functor on VB_m is a covariant functor F : VBm→ F M satisfying:

1. (Base preservation) BF M◦ F = BVBm;

2. (Locality) for any inclusion of an open vector bundle ı_E|U : E|U → E, F (E|U) is the restriction p⁻¹_E (U ) of p_E : E → VB_m(E) over U and F ıE|U
is the inclusion p⁻¹_E (U ) → FE.

Definition 2. Let G be a Lie group. A principal fiber bundle is a fiber bundle (P, M, π) of standard fiber G such that: there is a fiber bundle atlas

Uα, ϕα : π⁻¹(Uα) → Uα× G

α∈A, the family of smooth maps θ_αβ : Uα ∩ U_β → G which satisfies the cocycle condition (θ_αβ(x) · θ_βγ(x) = θ_αγ(x) for x ∈ Uα∩ U_β∩ U_γ and θαα(x) = e) and

for each x ∈ Uα∩ U_β, for each g ∈ G , ϕα◦ ϕ⁻¹_β (x, g) = (x, θαβ(x) · g) . Example 3. Let (E, M, π) be a vector bundle of standard fiber the real vector space V of dimension n ≥ 1. For any x ∈ M , we denote by FxE the set of all linear isomorphisms of V on E_x and we set F E =S

x∈MF_xE, it is clear that F E is an open set of the manifold hom(M × V, E). We denote by pE : F E → M the canonical projection. Let (Uα, ψα)α∈Λthe fiber bundle atlas of (E, M, p), so for all x ∈ Uα∩ U_β and v ∈ V , ψα◦ ψ⁻¹_β (x, v) = (x, θ_αβ(x)(v)), where θ_αβ : U_α∩ U_β → GL(V ) satisfies the cocycle condition. We consider

the smooth map ϕ_α : p⁻¹_E (U_α) → U_α× GL(V ) such that, for any x ∈ U_α and fx∈ p⁻¹_M(Uα),

ϕα(fx) = (x, ψα|_Ex◦ f_x) .

It is clear that, (Uα, ϕα)_α∈Λ is the fiber bundle atlas of (F E, M, pE). As ϕ_β ◦ ϕ⁻¹_α (x, f ) = (x, θ_αβ(x) ◦ f ), it follows that (F E, M, p_E) is a principal bundle of standard fiber, the linear Lie group GL(V ). It is called the frame bundle of the vector bundle (E, M, π).

Remark 6. Let U, xⁱ
be a local coordinate system of M , we denote by

 xⁱ, xⁱ_j

the local coordinate of F M induced by U, xⁱ
, it is such that:

( xⁱ(ξ) = xⁱ(p_E(ξ)) , xⁱ_j(ξ) = dxⁱ, (ξ(ej)) , for ξ ∈ F M and (e₁, . . . , e_n) is a basis of V .

Definition 3. Φ : (P, M, p, G) → (P⁰, M⁰, p⁰, G⁰) is a homomorphism of principal bundles over the homomorphism of Lie groups φ : G → G⁰ if Φ : P → P⁰ is smooth and satisfies

for each u ∈ P , for each g ∈ G , Φ(u · g) = Φ(u) · φ(g) .

The collection of principal bundles and their homomorphisms form a cat- egory, it is called the category of principal bundles and denoted by PB. In particular, it is subcategory of the category F M.

Example 4. Let f : E₁ → E₂ an isomorphism of vector bundles over the diffeomorphism f : M₁ → M₂. The smooth map F (f ) : F E₁ → F E₂ defined for any ϕx∈ F_xE1 by:

F (f )(ϕ_x) = f_x◦ ϕ_x∈ F_{f (x)}E₁

is such that f , F (f )
: (F E₁, M1, pE1) → (F E2, M2, pE2) is an isomorphism of principal bundles. We obtain in particular a functor F : VB_n → PB, it is a covariant functor.

Proposition 6. The functor F : VBn→ F M is a gauge bundle functor on VBnwhich do not preserves the fiber product. It is called the frame gauge functor on VB_n.

Proof. The properties of gauge functor F : VB_n→ F M are easily verified by calculation. Since do not exists an isomorphism between the Lie groups GL(V1) × GL(V2) and GL(V1 ⊕ V₂), it follows that the gauge functor F do not preserves the fiber product.

Remark 7. Let (P, M, π) be a principal fiber bundle with total space P , base space M , projection π and structure group G. If {Uα}_α∈Λ is an open covering of M , for each α ∈ Λ, P giving a trivial bundle over U_α, and if gαβ : Uα∩ U_β → G are the transition functions of P , we express this fiber bundle by P = {Uα, g_αβ}. When G is a Lie subgroup of a Lie group G⁰ and j : G → G⁰ is the injection map, then there is a fiber bundle P⁰ = {U_α, j ◦ g_αβ} and an injection j : P → P⁰ which is a bundle homomorphism i.e. j(p · a) = j(p) · a, for any p ∈ P and a ∈ G.

2.3. The natural embedding j_A,E : T^A(F E) → F (T^AE). We de- note with (E, M, π) a vector bundle of standard fiber the real vector space V of dimension n ≥ 1. Then, T^AE, T^AM, T^Aπ
is a real vector bundle of standard fiber T^AV , in particular the frame bundle of this vector bundle is a GL(T^AV )-principal F (T^AE), T^AM, p_TAE
. On the other hand, (F E, M, p_E) is a GL(V )-principal bundle, so T^A(F E), T^AM, T^A(pE)
is a T^A(GL(V ))- principal bundle. Let (Uα, ψα)_α∈Λ a fiber bundle atlas of (E, M, π), so that

T^AU_α, T^Aψ_α

α∈Λ is a fiber bundle atlas of T^AE, T^AM, T^Aπ
. The bundle atlas of the principal bundle (F E, M, p_E) is denoted by (U_α, ϕ_α)_α∈Λ where

ϕα: p⁻¹_E (Uα) −→ U_α× GL(V ) g 7−→ 

pE(g), (ψα)_p

E(g)◦ g , we deduce that T^AU_α, T^A(ϕ_α)

α∈Λ is the following fiber bundle atlas of T^A(F E), T^AM, T^A(p_E)
,

T^A(ϕα) : T^ApE

−1

T^AUα

−→ T^AUα× T^A(GL(V )) j^Ag 7−→ T^Ap_E(j^Ag), j^A(ψ_α· g)
, where (ψα· g)(z) = (ψ_α)_p

E(g(z))◦ g(z) : V → V is a linear isomorphism, for all z ∈ R^k.

As T^AU_α, T^Aψ_α

α∈Λ is a fiber bundle atlas of T^AE, T^AM, T^Aπ
, it fol- lows that the fiber bundle atlas of the principal bundle F (T^AE), T^AM, p_TAE

is denoted by T^AU_α, ϕ_α,A

α∈Λ where

ϕ_α,Ap⁻¹_TAE(T^AU_α) −→ T^AU_α× GL(T^AV ) ξ 7−→ 

p_TAE(ξ), T^A(ψα)

p_{T AE}(ξ)◦ ξ and ϕ⁻¹_α,A

 x, ee ξ



= T^Aψα

−1

(ex, ·) ◦ eξ, for any

 x, ee ξ



∈ T^AUα× GL(T^AV ).

For any α ∈ Λ, we put

j_A,Uα= ϕ⁻¹_α,A◦ (id_TAUα, j_A,V) ◦ T^A(ϕα) : T^Ap_E
−1

(T^AUα) −→ p⁻¹_TAE(T^AUα) and for any j^Ag ∈ T^Ap_E
−1

(T^AU_α), we have:

j_A,Uα(j^Ag) = ϕ⁻¹_α,A j^A(p_E ◦ g), j_A,V j^A(ψα· g)

= T^Aψ_α
⁻¹

j^A(p_E ◦ g), ·
◦ j_A,V j^A(ψ_α· g)
. For β ∈ Λ such that U_α ∩ U_β 6= ∅, we have j_A,Uα

(T^ApE)⁻¹(^TAUα∩TAUβ) = j_A,Uβ

(T^ApE)⁻¹(^TAUα∩TAUβ), it follows that, it exists one and only one principal fiber bundle homomorphism j_A,E : T^A(F E) → F (T^AE) such that, for any α ∈ A, jA,E

(T^ApE)⁻¹(T^AUα) = jA,Uα. In particular, for any eξ ∈ T^A(F E) and eu ∈ T^A(GL(V )),

j_A,E ξ ·e ue

= j_A,E ξe

· j_A,V (u) .e

Theorem 1. The map jA,E : T^A(F E) → F (T^AE) is a principal fiber bun- dle homomorphism over the homomorphism of Lie groups jA,V : T^A(GL(V ))

→ GL(T^AV ). In particular, j_A,E is an embedding.

Proof. It is clear that, j_A,E : T^A(F E) → F (T^AE) is a principal fiber bundle homomorphism over jA,V, because for any eξ ∈ T^A(F E) and u ∈e T^A(GL(V )),

jA,E

 ξ ·e ue



= jA,E

 ξe



· j_A,V (eu) . On the other hand, for any α ∈ A, j_A,E

(T^ApE)⁻¹(T^AUα) = j_A,Uα, it follows that jA,E is an embedding.

Remark 8. Let π⁻¹(U ), xⁱ, y^j
be a fiber chart of E, then the local coor- dinate of F E and T^AE are



p⁻¹_E (Ui), xⁱ, y^j_k

 and



T^Aπ
−1

(T^AU ), xⁱ_α, yα^j

 .

We deduce that, the local coordinate of T^A(F E) and F (T^AE) are given by



T^A(p⁻¹_E (Ui)), xⁱ_α, y_k^j, y_k,α^j

 and



p⁻¹_TAE T^AU
, xⁱ_α, y^j,α_k,β



, so the local expres- sion of j_A,E is given by:

j_A,E

(T^ApE)⁻¹(T^AU )



xⁱ_α, y^j_k, y^j_k,α

=











 xⁱ_α,













y_k^j 0 · · · 0 ... . .. ... ...

... . .. 0

· · · y_k,α^j · · · y_k^j























 .

Proposition 7. Let f : E → E⁰ is an isomorphism of vector bundles over the diffeomorphism f : M → M⁰. The following diagram

T^A(F E) ^T

A(F f ))

−−−−−−−→ T^A(F E⁰)

jA,E



 y



 y^jA,E0 F (T^AE) −−−−−−→

F (T^Af )

F (T^AE⁰) commutes.

Proof. Let (Uα, ψα)_α∈Λ and (U_α⁰, ψ_α⁰)_α∈Λ the bundle atlas of (E, M, π) and (E⁰, M⁰, π⁰) such that f (U_α) = U_α⁰, α ∈ Λ. As f : E → E⁰ is an isomorphism of vector bundles over f , it follows that it exists a smooth map fα: Uα× V → V such that ψ_α⁰ ◦ f

π⁻¹(Uα)◦ ψ_α⁻¹(x, v) = f (x), fα(x, v)
, for any (x, v) ∈ U_α× V and f_α(x, ·) is a linear isomorphism. It follows that, the diagram

p⁻¹_E (U_α)

F f p−1

E (Uα)

−−−−−−−−−→ p⁻¹_E0(U_α⁰)

ϕα



 y



 y^ϕ

0 α

Uα× GL(V ) −−−−−−−→

ffα

U_α⁰ × GL(V )

commutes, and ffα(x, g) = f (x), fα(x, ·) ◦ g
, for each (x, g) ∈ U_α× GL(V ).

It is clear that the following diagram T^Ap_E
−1

T^AU_α

T^A(F f )

(T ApE)⁻¹(T AUα)

−−−−−−−−−−−−−−−−−−→ T^Ap_E⁰
−1

T^AU_α⁰

T^Aϕα



 y



 y^T

Aϕ⁰_α

T^AUα× T^A(GL(V )) −−−−−−−−−−−−−−−→

T^A ffα

 T^AU_α⁰ × T^A(GL(V ))

commutes. On the other hand, as the diagram following commutes

T^AU_α× T^A(GL(V ))

T^A ffα



−−−−−−−→ T^AU_α⁰ × T^A(GL(V )) (^idUα,jA,V)



 y



 y

 id_{U 0}

α,jA,V



T^AU_α× GL T^AV

−−−−−−→

fgα,A

T^AU_α⁰ × GL T^AV

with gf_α,A

 ex, eξ



=



T^Af (x) , fe _α,A(x, ·) ◦ ee ξ

 where

T^A(ψ_α⁰) ◦ T^Af

(T^Aπ)⁻¹(T^AUα)◦ T^Aψ_α
⁻¹

(x, v) = Te ^Af (x) , fe _α,A(x, ·)
,e it follows that

id_Uα⁰, jA,V
◦ T^A ffα



j^Au, j^Aξ

= id_Uα⁰, j_A,V


T^Af j^Au
, j^A

f (u, ·) ◦ ξe 

=



T^Af (j^Au), jA,V

 j^A



f (u, ·) ◦ ξe



.

As jA,V

 j^A



f (u, ·) ◦ ξe



j^Av
= j^A

f (u, ·) ◦ ξe



· v and



f (u, ·) ◦ ξe



· v (z) = ef (u(z), ξ(z)(v(z))) ,

for any z ∈ R^k, thus,

F (T^Af ) ◦ jA,Uα j^Au, j^Aξ
= F (T^Af ) j^Au, jA,V j^Aξ

=



T^Af j^Au
, T^Af je ^Au, ◦
◦ j_A,V j^Aξ
 . For any j^Av ∈ T^AV , as j_A,V j^Aξ

j^Av

= j^A(ξ ∗ v) with ξ ∗ v(z) = ξ(z)(v(z)), for all z ∈ R^k, we deduce that

T^Af je ^Au, ◦
◦ j_A,V j^Aξ

j^Av
= T^Af je ^Au, j^A(ξ ∗ v)

= j^A



f (u, ξ ∗ v)e

 ,

so T^Af je ^Au, ◦
◦ j_A,V j^Aξ
j^Av

= jA,V

 j^A



f (u, ·) ◦ ξe



j^Av

for any j^Av ∈ T^AV . More precisely, jA,U_α⁰ ◦ T^A

ffα



= gfα,A◦ j_A,Uα, j_A,E⁰

(T^Ap_E0)⁻¹(T^AU_α⁰)◦ T^A(F f ) = ϕ⁰_α,A⁻¹ ◦ j_A,U⁰

α◦ T^A ϕ⁰_α
◦ T^A(F f )

= ϕ⁰_α,A⁻¹ ◦ j_A,U⁰

α ◦ T^A ϕ⁰_α◦ F f ◦ ϕ⁻¹_α
◦ T^A(ϕ⁻¹_α )

= ϕ⁰_α,A⁻¹ ◦ j_A,U⁰

α ◦ T^A ff_α

◦ T^A(ϕ⁻¹_α )

= ϕ⁰_α,A⁻¹ ◦ gfα,A◦ j_A,Uα◦ T^A(ϕ⁻¹_α )

=



ϕ⁰_α,A⁻¹ ◦ gfα,A◦ ϕ_α,A

◦ ϕ⁻¹_α,A◦ j_A,Uα◦ T^A(ϕ⁻¹_α )

= F (T^Af ) ◦ j_A,E

(T^ApE)⁻¹(T^AUα), thus, j_A,E⁰ ◦ T^A(F f ) = F (T^Af ) ◦ jA,E.

Let (E, M, π) be a vector bundle of standard fiber V , for any t ∈ R, we consider the linear automorphism of E, gt : E → E defined by: gt(u) = exp(t)u, for any u ∈ E. We consider the principal bundle isomorphism over id_M, ϕ_t+ F (gt) : F E → F E such that, for any x ∈ M ,

ϕ_t

FxE : F_xE −→ F_xE hx 7−→ h_x◦ g_t.

In particular, we deduce a smooth map ϕ : R × F E → F E, (t, ξ) 7→ ϕt(ξ).

For any multi index α, we consider the smooth map ϕα,E : T^A(F E) −→ T^A(F E)

ξ 7−→ T^Aϕ(eα, ξ).

Then T^A(p_E) ◦ ϕ_α,E = T^A(p_E). In particular, it is a homomorphism of prin- cipal bundle of T^A(F E) in to T^A(F E).

Proposition 8. Let f : E → E⁰ be an isomorphism of vector bundles over the diffeomorphism f : M → M⁰. Then the following diagram

T^A(F E) ^T

A(F f )

−−−−−−−→ T^A(F E⁰)

ϕα,E



 y



 y

ϕ_α,E0

T^A(F E) −−−−−−−→

T^A(F f )

T^A(F E⁰) commutes.

Proof. Let j^Aξ ∈ T^A(F E), we have:

ϕ_α,E⁰◦ T^A(F f ) j^Aξ
= ϕ_α,E⁰ j^A(F (f ) ◦ ξ)

= T^Aϕ j^A(t^α), j^A(F (f ) ◦ ξ)

= j^A(ϕ(t^α, F (f ) ◦ ξ))

= j^A(F (f ) ◦ ϕ(t^α, ξ))

= T^A(F (f )) ◦ ϕ_α,E j^Aξ
. Therefore, ϕα,E⁰◦ T^A(F f ) = T^A(F (f )) ◦ ϕα,E.

3. Prolongations of G-structures to Weil bundles

3.1. The natural embedding j_A,M : T^A(F M ) → F (T^AM ). Let M be a smooth manifold of dimension n ≥ 1, we denote by GL(n) the Lie group GL(Rⁿ) and (F (M ), M, pM) the frame bundle of the tangent vector bundle (T M, M, π_M), so that T^A(F M ), T^AM, T^A(p_M)
is a principal fiber bundle over the Lie group T^A(GL(n)). By the same way F (T^AM ), T^AM, p_TAM
is a frame bundle of the vector bundle T (T^AM ), T^AM, π_T^A_M
. If f : M → N is a local diffeomorphism, we denote with F (f ) the principal bundle homo- morphism F (T f ) : F M → F N .

Let M be a smooth n-dimensional manifold,

F (κ_A,M) : F (T^AT M ) −→ F (T^AM )

is an isomorphism of principal bundles over id_TAM and p_TAM ◦ F (κ_A,M) = p_TAT M, where κ_A,M : T^A(T M ) → T (T^AM ) is the canonical isomorphism defined in [7]. We put

jA,M = F (κA,M) ◦ jA,T M : T^A(F M ) −→ F (T^AM )

such that p_TAM◦ j_A,M = T^A(p_M) and j_A,M(˜x · g) = j_A,M(ex) · j_A,Rⁿ(g). In par- ticular j_A,M is a homomorphism of principal bundles over j_A,Rⁿ. We identify T^ARⁿ with the euclidian vector space R^{n×dim A}, it follows that T^A(GL(n)) is a Lie subgroup of GL(n × dim A).

Proposition 9. Let M and N be two manifolds and f : M → N be a diffeomorphism between them. Then the following diagram