� EXTRACTA MATHEMATICAE Volumen 33, Número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura EXTRACTA MATHEMATICAE Vol. 37, Num. 1 (2022), 111 – 138 doi:10.17398/2605-5686.37.1.111 Available online February 15, 2022 Prolongations of G-structures related to Weil bundles and some applications P.M. Kouotchop Wamba 1, G.F. Wankap Nono 2, A. Ntyam 1 1 Department of Mathematics, Higher Teacher Training college University of Yaoundé 1, PO.BOX 47 Yaoundé, Cameroon 2 Department of Mathematics and Computer Science, Faculty of Science University of Ngaoundéré, PO.BOX 454 Ngaoundéré, Cameroon wambapm@yahoo.fr , georgywan@yahoo.fr , achillentyam@yahoo.fr Received November 1, 2021 Presented by Manuel de León 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 TAM (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 ⊕NA, where NA 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 Rk into R with source 0 ∈ Rk and denoted by Jr0 (R k,R). The ideal of nilpotent elements is the finite vector space Jr0 (R k,R)0. Let A = R·1A⊕NA be a Weil algebra, we adopt the covariant approach of Weil functor described by I. Kolàr in [6]. We denote by NkA the ideal generated by the product of k elements of NA, there is one and only one natural number h such that N h A 6= 0 and Nh+1A = 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) https://doi.org/10.17398/2605-5686.37.1.111 mailto:wambapm@yahoo.fr mailto:georgywan@yahoo.fr mailto:achillentyam@yahoo.fr https://publicaciones.unex.es/index.php/EM https://creativecommons.org/licenses/by-nc/3.0/ 112 p.m.k. wamba, g.f. wankap nono, a. ntyam the vector space NA/N2A is said to the width of A. In this case, the Weil algebra A is called (k,h)-algebra. If %,%1 : J h 0 (R k,R) → A are two surjective algebra homomomorphisms, then there is an isomorphism σ : Jh0 (R k,R) → Jh0 (R k,R) such that: %1 ◦σ = %. We say that, two maps ϕ,ψ : Rk → M determine the same A-velocity if for every smooth map f : M → R % ( jh0 (f ◦ϕ) ) = % ( jh0 (f ◦ψ) ) . The equivalence class of the map ϕ : Rk → M is denoted by jAϕ and will called A-velocity at 0 (see [6], [7] or [8]). We denote by TAM the space of all A-velocities on M. More precisely, TAM = { jAϕ, ϕ : Rk → M } . TAM is a smooth manifold of dimension m × dim A. For a local chart( U,u1, . . . ,um ) of M, the local chart of TAM is ( TAU,ui0, . . . ,u i K ) such that:{ ui0 ( jAϕ ) = ui(ϕ(0)) uiα ( jAϕ ) = a∗α ( jA(ui ◦ϕ) ) 1 ≤ α ≤ K where (a0, . . . ,aK) is basis of A and (a ∗ 0, . . . ,a ∗ K) is a dual basis. We denote by πAM : T AM → M the natural projection such that πAM (j Aϕ) = ϕ(0), so( TAM,M,πAM ) is a fibered manifold. For every smooth map f : M → M, induces a smooth map TAf : TAM → TAM such that: for any jAϕ ∈ TAM, TAf(jAϕ) = jA(f ◦ϕ). In particular we have that ( f,TAf ) is a fibered morphism from ( TAM,M,πAM ) to ( TAM,M,πA M ) . This defines a bundle functor TA : Mf → FM called Weil functor induced by A. The bundle functor TA preserves product in the sense, that for any manifolds M and M, the map( TA(prM ),T A(prM ) ) : TA(M ×M) −→ TAM ×TAM where prM : M ×M → M and prM : M ×M → M are the projections, is an FM−isomorphism. Hence we can identify TA(M ×M) with TAM ×TAM. Let B be another (s,r) Weil algebra and µ : A → B be an algebra homo- morphism, %′ : Jr0 (R s,R) → B the surjective algebra homomorphism. Then prolongations of G-structures related to weil bundles 113 there is an algebra homomorphism µ̃ : Jh0 (R k,R) → Jr0 (R s,R) such that the following diagram Jh0 (R k,R) µ̃ −−−−→ Jr0 (R s,R) % y y %′ A −−−−→ µ B commutes. In particular, there is map fµ : Rs → Rk such that, µ̃(jh0 g) = jr0 (g◦fµ), where g ∈ C ∞(Rk). For any manifold M of dimension m ≥ 1, it is proved in [7] that there is smooth map µM : T AM → TBM defined by: µM (j Aϕ) = jB(ϕ◦fµ). More precisely, µM : T AM → TBM is a natural transformations and denoted by µ : TA → TB. 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 Fa : FR × FR → FR and Fλ : FR ×FR → FR is the vector addition and the algebra multiplication in the Weil algebra FR and F coincides with the Weil functor TFR. Every natural transformation µ : TA → TB are in bijection with the algebra homomorphism µR : A → B (see [8]). Since % : Jh0 (R k,R) → A is determined up to an isomorphism Jh0 (R k,R) → Jh0 (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 Rk into R with source 0 ∈ Rk denoted by Jr0 (R k,R), the corresponding Weil functor is the functor of k-dimensional velocities of order r and denoted by Trk . 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 ⊕NA be a Weil algebra. For any multiindex 0 < |α| ≤ h, we put eα = j A(xα) is an element of NA. For any ϕ ∈ C∞(Rk,R) , we have: jAϕ = ϕ(0) · 1A + ∑ 1≤|α|≤h 1 α! Dαϕ(0)eα. In particular the family {eα} generates the ideal NA. We denote by BA the set of all multiindex such that (eα)α∈BA is a basis of NA and BA her 114 p.m.k. wamba, g.f. wankap nono, a. ntyam complementary with respect to the set of all multiindex γ ∈ Nn such that 1 ≤| γ |≤ h. For β ∈ BA, we have eβ = λαβeα. In particular, eα ·eβ = { eα+β if α + β ∈ BA , λ γ α+βeγ if α + β ∈ BA . It follows that, for any ϕ ∈ C∞(Rk,R) , we have: jAϕ = ϕ(0) · 1A + ∑ α∈BA ( 1 α! Dαϕ(0) + ∑ β∈BA λα 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 TAU of TAM denoted by ( xi,xiα ) is given by{ xi = xi ◦πAM = x i 0 , xiα = x i α + ∑ β∈BA λ α βx i β , where xiβ(j Ag) = 1 β! · Dβ(xi ◦ g)(0) and jAg ∈ TAU. In the particular case where A = D, the local coordinate system of TM induced by ( U,xi ) is denoted by ( xi, ẋi ) . Let M be a smooth manifold of dimension m ≥ 1, with (TM,M,πM ) we denote its tangent bundle, and with (F(M),M,pM ) 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 TrM 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 TrM has canonically the same kind of structure. Since the tangent functor of higher order Tr 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 TA. Thus, the aim of this paper is to define the prolongations of G-structures from a manifold M to its Weil bundle TAM. In particular, we construct a canonical embedding jA,E of T A(FE) into F(TAE), where F(E) denote the frame bundle of the vector bundle (E → M). Using the natural isomorphism κA,M : T A(TM) → T(TAM) (see [5]) and the embedding jA,TM , we define this A-prolongation T AP of a G-structure P of prolongations of G-structures related to weil bundles 115 a manifold M, to its Weil bundle TAM. 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 TAM by: f(`) = `◦TA(f) ; f(`) is a smooth function on TAM. Remark 1. Let (eβ)β∈BA a basis of NA, we denote by ( e0,eβ ) β∈BA the dual basis of A. For ` = eα, the smooth function f(`) is denoted by f(α). In particular, for any jAϕ ∈ TAM, f(α)(jAϕ) = 1 α! Dα(f ◦ϕ)(z) ∣∣ z=0 + ∑ β∈BA λα β β! Dβ(f ◦ϕ)(z) ∣∣∣ z=0 and f(0) = f◦πAM . For a coordinate system ( U,x1, . . . ,xm ) in M, the induced coordinate system { xi0,x i α } on TAM is such that, xiα = ( xi )(α) . Remark 2. For any smooth map ` : A → R, the map C∞(M) −→ C∞(TAM) f 7−→ f(`) is R-linear. For all multiindex α such that |α| ≤ h, we denote by χ(α) : TA → TA the natural transformation defined for any vector bundle (E → M) and ϕ ∈ C∞(Rk,E) by: χ (α) E (j Aϕ) = jA(zαϕ) where zαϕ is a smooth map defined for any z ∈ Rk by (zαϕ)(z) = zαϕ(z). 116 p.m.k. wamba, g.f. wankap nono, a. ntyam Proposition 1. Let A be a Weil algebra. There exists one and only one family κA,M : T A(TM) → T(TAM) of vector bundle isomorphisms such that πTAM ◦κA,M = TA(πM ) and the following conditions hold: 1. For every smooth mapping f : M → N the following diagram TA(TM) TA(Tf) −−−−−−→ TA(TN) κA,M y yκA,N T(TAM) −−−−−−→ T(TAf) T(TAN) commutes. 2. For two manifolds M, N we have κA,M×N = κA,M ×κA,N . Proof. See [5]. Let X : M → TM be a vector field on a manifold M, then we put X(α) = κA,M ◦χ (α) TM ◦T A(X) . It is a vector bundle field on TA(M) called α-lift of X to TAM. 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 TAM. We put X(α) = 0, for |α| > h or α /∈ Nk. Remark 3. For any |α| ≤ h, the map X(M) −→ X(TAM) 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(TAM), Y (α) ∈ X(TAN) are TA(ϕ) related. Proposition 2. For X,Y ∈ X(M), we have:[ X(α),Y (β) ] = [X,Y ] (α+β) for all 0 ≤ |α,β| ≤ h. Proof. See [5]. prolongations of G-structures related to weil bundles 117 Remark 4. The family of α-lift of vector fields is very important, because, if S and S′ are two tensor fields of type (1,p) or (0,p) on TA (M) such that, for all X1, . . . ,Xp ∈ X (M), and multiindex α1, . . . ,αp, the equality S ( X (α1) 1 , . . . ,X (αp) p ) = S′ ( X (α1) 1 , . . . ,X (αp) p ) holds, then S = S′ (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 : TM ×M · · ·×M TM −→ TM of the bundle product over M of q copies of the tangent bundle TM. For all 0 ≤ |α| ≤ h, we put: S(α) : T(TAM) ×TAM · · ·×TAM T(T AM) −→ T(TAM) with S(α) = κA,M ◦ χ (α) TM ◦ T A(S) ◦ ( κ−1A,M ×···×κ −1 A,M ) . It is a tensor field of type (1,q) on TA(M) called α-prolongation of the tensor field S from M to TA(M). In the particular case where α = 0, it is denoted by S(c) and is called complete lift of S from M to TA (M). Proposition 3. The tensor S(α) is the only tensor field of type (1,q) on TA(M) satisfying S(α) ( X (α1) 1 , . . . ,X (αq) q ) = ( S(X1, . . . ,Xq) )(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 p A,E : T AE∗ → ( TAE )∗ (see [10]) defined for any jAϕ ∈ TAE∗ and jAψ ∈ TAE by: τ p A,E(j Aϕ)(jAψ) = p ( jA(〈ψ,ϕ〉E) ) 118 p.m.k. wamba, g.f. wankap nono, a. ntyam where 〈ψ,ϕ〉E : R k → R, z 7→ 〈ψ (z) ,ϕ (z)〉E and 〈·, ·〉E the canonical pairing. For any manifold M of dimension m, we consider the vector bundle mor- phism ε p A,M = [ κ−1A,M ]∗ ◦ τpA,TM : T AT∗M −→ T∗TAM. It is clear that the family of maps ( ε p A,M ) defines a natural transformation be- tween the functors TA◦T∗ and T∗◦TA on the category Mfm of m-dimensional manifolds and local diffeomorphisms, denoted by ε p A,∗ : T A ◦T∗ −→ T∗ ◦TA. When (A,p) is a Weil-Frobenius algebra (see [4]), the mapping ε p A,M is an isomorphism of vector bundles over idTAM . Being { x1, . . . ,xm } a local coor- dinate system of M, we introduce the coordinates ( xi, ẋi ) in TM, ( xi,πi ) in T∗M, (xi, ẋi,xiβ, ẋ i β) in T ATM, (xi,πj,x i β,π β j ) in T AT∗M, (xi,xiβ, ẋ i, ẋ i β) in TTAM and (xi,xiβ,ξj,ξ β j ) in T ∗TAM. We have ε p A,M ( xi,πj,x i β,π β j ) = ( xi,xiβ,ξj,ξ β j ) with   ξj = πjp0 + ∑ µ∈BA π µ j pµ , ξ β j = ∑ µ∈BA π µ−β j pµ , 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] : TM ×M · · · ×M TM → T∗M of the bundle product over M of s− 1 copies of TM. We define, G(p) : T(TAM) ×TAM · · ·×TAM T(T AM) −→ T∗(TAM) as G(p) = ε p A,M ◦ T A(G]) ◦ ( κ−1A,M ×···×κ −1 A,M ) . It is a TAM-morphism of vector bundles, so G(p) is tensor field of type (0,s) on TAM called p- prolongation of G from M to TAM. Example 1. In a particular case, where s = 2 and locally G = Gijdx i ⊗ dxj then G(p) = Gijp0dx i ⊗dxj + ∑ α∈BA pα ( ∑ β∈BA G (α−β) ij ) dxi ⊗dxjβ + ∑ µ,β∈BA ( ∑ α∈BA pαG (α−β−µ) ij ) dxiµ ⊗dx j β. prolongations of G-structures related to weil bundles 119 In the particular case where A = Jr0 (R k,R) and p(jr0ϕ) = 1 α! Dα (ϕ(z))|z=0, then G(p) coincides with the α-prolongation of G from M to TrkM 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 = dx i ∧dξi, then we have: Ω (p) M = p0dx i ∧dξi + ∑ α∈BA pαdx i ∧dξαi + ∑ α,β∈BA pαdx i β ∧dξ α−β i . Proposition 4. The tensor field G(p) is the only tensor field of type (0,s) on TA(M) satisfying, for all X1, . . . ,Xs ∈ X(M) and multiindex α1, . . . ,αs G(p) ( X (α1) 1 , . . . ,X (αs) s ) = (G(X1, . . . ,Xs)) (p◦lα1+···+αs) where la : A → A is given by la(x) = ax. Proof. See [5]. 2. The natural transformations jA,E : T A(FE) → F(TAE) 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(TAV ). 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 TAG acts to TAM by the differential action TAρ : TAG×TAM → TAM. Lemma 1. If the Lie group G operates on M effectively, then TAG oper- ates on TAM effectively by the differential action TA(ρ). Proof. See [5]. Let ρV : GL(V ) × V → V be the canonical action of GL(V ), then the Lie group TA(GL(V )) operates effectively on the vector space TAV by the induced action TA(ρV ) : T A(GL(V )) ×TAV −→ TAV( jAϕ,jAu ) 7−→ jA(ϕ∗u) 120 p.m.k. wamba, g.f. wankap nono, a. ntyam where ϕ∗u : Rk → V is defined for any z ∈ Rk by: ϕ∗u(z) = ϕ(z)(u(z)). We deduce an injective map jA,V : T A(GL(V )) → GL(TAV ) such that, jA,V (j Ag) : TAV −→ TAV jAξ 7−→ jA(g ∗ ξ) . Proposition 5. The map jA,V : T A(GL(V )) → GL(TAV ) 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, . . . ,en} be a basis of V (dim V = n), we consider the global coordinate system of V , ( e1, . . . ,en ) , we denote by ( yij ) the global coordinate of GL(V ), for any f ∈ GL(V ), yij(f) = 〈 ei,f(ej) 〉 where 〈·, ·〉 is the duality bracket V ∗×V → R. We deduce that, the coordinate system of TA(GL(V )) is denoted by ( yij,y i j,α ) α∈BA . On the other hand, the global coordinate system of TAV is ( ei,eiα ) , such that:  ei ( jAu ) = ei(u(0)) , eiα ( jAu ) = 1 α! Dα(e i ◦u)(z) ∣∣ z=0 + ∑ β∈BA λα β β! Dβ(e i ◦u)(z) ∣∣ z=0 , j Au ∈ TAV, the global coordinate of GL(TAV ) denoted ( zij,z i,β j,α ) α,β∈BA is such that:  zij(ξ) = 〈 ei,πA,V (ξ)(ej) 〉 , z i,β j,α(ξ) = 〈 eiβ,ξ ( eαj )〉 , ξ ∈ GL(TAV ) , we deduce that the local coordinate of the map jA,V is given by: jA,V ( yij,y i j,α ) =   yij 0 · · · · · · 0 ... . . . . . . ... ... . . . . . . ... ... . . . 0 · · · · yij,α · · · y i j   prolongations of G-structures related to weil bundles 121 In fact, z i,β j,α ( jA,V ( jAg )) = 〈 eiβ,jA,V ( jAg )( eαj )〉 = 1 β! Dβ ( tα 〈 ei,g(t)(ej) 〉)∣∣ t=0 + ∑ µ∈BA λ µ β µ! Dµ ( tα 〈 ei,g(t)(ej) 〉)∣∣ t=0 for any jAg ∈ TA(GL(V )). 2.2. Frame gauge functor on the vector bundles. We denote by VBm the category of vector bundles with m-dimensional base together with local isomorphism. Let BVBm : VBm → Mf and BFM : FM → Mf be the respective base functors. Definition 1. (See [11]) A gauge bundle functor on VBm is a covariant functor F : VBm →FM satisfying: 1. (Base preservation) BFM ◦F = BVBm; 2. (Locality) for any inclusion of an open vector bundle ıE|U : E|U → E, F (E|U ) is the restriction p−1E (U) of pE : E → VBm(E) over U and F ( ıE|U ) is the inclusion p−1E (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 θαβ : Uα ∩ Uβ → G which satisfies the cocycle condition (θαβ(x) · θβγ(x) = θαγ(x) for x ∈ Uα ∩Uβ ∩Uγ and θαα (x) = e) and for each x ∈ Uα ∩Uβ , for each g ∈ G, ϕα ◦ϕ−1β (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 Ex and we set FE = ⋃ x∈MFxE, it is clear that FE is an open set of the manifold hom(M ×V, E). We denote by pE : FE → M the canonical projection. Let (Uα,ψα)α∈Λ the fiber bundle atlas of (E,M,p), so for all x ∈ Uα∩Uβ and v ∈ V , ψα◦ψ−1β (x,v) = (x,θαβ(x)(v)), where θαβ : Uα ∩ Uβ → GL(V ) satisfies the cocycle condition. We consider 122 p.m.k. wamba, g.f. wankap nono, a. ntyam the smooth map ϕα : p −1 E (Uα) → Uα ×GL(V ) such that, for any x ∈ Uα and fx ∈ p−1M (Uα), ϕα(fx) = (x,ψα|Ex ◦fx) . It is clear that, (Uα,ϕα)α∈Λ is the fiber bundle atlas of (FE,M,pE). As ϕβ ◦ ϕ−1α (x,f) = (x,θαβ(x) ◦f), it follows that (FE,M,pE) 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,xij ) the local coordinate of FM induced by ( U,xi ) , it is such that: { xi (ξ) = xi(pE(ξ)) , xij (ξ) = 〈 dxi, (ξ(ej)) 〉 , for ξ ∈ FM and (e1, . . . ,en) is a basis of V . Definition 3. Φ : (P,M,p,G) → (P ′,M ′,p′,G′) is a homomorphism of principal bundles over the homomorphism of Lie groups φ : G → G′ if Φ : P → P ′ 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 FM. Example 4. Let f : E1 → E2 an isomorphism of vector bundles over the diffeomorphism f : M1 → M2. The smooth map F(f) : FE1 → FE2 defined for any ϕx ∈ FxE1 by: F(f)(ϕx) = fx ◦ϕx ∈ Ff(x)E1 is such that ( f,F(f) ) : (FE1,M1,pE1 ) → (FE2,M2,pE2 ) is an isomorphism of principal bundles. We obtain in particular a functor F : VBn → PB, it is a covariant functor. Proposition 6. The functor F : VBn → FM is a gauge bundle functor on VBn which do not preserves the fiber product. It is called the frame gauge functor on VBn. prolongations of G-structures related to weil bundles 123 Proof. The properties of gauge functor F : VBn →FM 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 Uα, and if gαβ : Uα ∩Uβ → G are the transition functions of P , we express this fiber bundle by P = {Uα,gαβ}. When G is a Lie subgroup of a Lie group G′ and j : G → G′ is the injection map, then there is a fiber bundle P ′ = {Uα,j ◦gαβ} and an injection j : P → P ′ 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 jA,E : T A(FE) → F(TAE). We de- note with (E,M,π) a vector bundle of standard fiber the real vector space V of dimension n ≥ 1. Then, ( TAE,TAM,TAπ ) is a real vector bundle of standard fiber TAV , in particular the frame bundle of this vector bundle is a GL(TAV )-principal ( F(TAE),TAM,pTAE ) . On the other hand, (FE,M,pE) is a GL(V )-principal bundle, so ( TA(FE),TAM,TA(pE) ) is a TA(GL(V ))- principal bundle. Let (Uα,ψα)α∈Λ a fiber bundle atlas of (E,M,π), so that( TAUα,T Aψα ) α∈Λ is a fiber bundle atlas of ( TAE,TAM,TAπ ) . The bundle atlas of the principal bundle (FE,M,pE) is denoted by (Uα,ϕα)α∈Λ where ϕα : p −1 E (Uα) −→ Uα ×GL(V ) g 7−→ ( pE(g), (ψα)pE(g) ◦g ) , we deduce that ( TAUα,T A(ϕα) ) α∈Λ is the following fiber bundle atlas of( TA(FE),TAM,TA(pE) ) , TA(ϕα) : ( TApE )−1 ( TAUα ) −→ TAUα ×TA(GL(V )) jAg 7−→ ( TApE(j Ag),jA(ψα ·g) ) , where (ψα · g)(z) = (ψα)pE(g(z)) ◦ g(z) : V → V is a linear isomorphism, for all z ∈ Rk. As ( TAUα,T Aψα ) α∈Λ is a fiber bundle atlas of ( TAE,TAM,TAπ ) , it fol- lows that the fiber bundle atlas of the principal bundle ( F(TAE),TAM,pTAE ) 124 p.m.k. wamba, g.f. wankap nono, a. ntyam is denoted by ( TAUα,ϕα,A ) α∈Λ where ϕα,Ap −1 TAE (TAUα) −→ TAUα ×GL(TAV ) ξ 7−→ ( pTAE(ξ), ( TA(ψα) ) p TAE (ξ) ◦ ξ ) and ϕ−1α,A ( x̃, ξ̃ ) = ( TAψα )−1 (x̃, ·) ◦ ξ̃, for any ( x̃, ξ̃ ) ∈ TAUα × GL(TAV ). For any α ∈ Λ, we put jA,Uα = ϕ −1 α,A ◦ (idTAUα,jA,V ) ◦T A(ϕα) : ( TApE )−1 (TAUα) −→ p−1TAE(T AUα) and for any jAg ∈ ( TApE )−1 (TAUα), we have: jA,Uα(j Ag) = ϕ−1α,A ( jA(pE ◦g),jA,V ( jA(ψα ·g) )) = ( TAψα )−1 ( jA(pE ◦g), · ) ◦ jA,V ( jA(ψα ·g) ) . For β ∈ Λ such that Uα ∩ Uβ 6= ∅, we have jA,Uα ∣∣ (TApE) −1(TAUα∩TAUβ) = jA,Uβ ∣∣ (TApE) −1(TAUα∩TAUβ) , it follows that, it exists one and only one principal fiber bundle homomorphism jA,E : T A(FE) → F(TAE) such that, for any α ∈ A, jA,E ∣∣ (TApE) −1 (TAUα) = jA,Uα. In particular, for any ξ̃ ∈ TA(FE) and ũ ∈ TA(GL(V )), jA,E ( ξ̃ · ũ ) = jA,E ( ξ̃ ) · jA,V (ũ) . Theorem 1. The map jA,E : T A(FE) → F(TAE) is a principal fiber bun- dle homomorphism over the homomorphism of Lie groups jA,V : T A(GL(V )) → GL(TAV ). In particular, jA,E is an embedding. Proof. It is clear that, jA,E : T A(FE) → F(TAE) is a principal fiber bundle homomorphism over jA,V , because for any ξ̃ ∈ TA(FE) and ũ ∈ TA(GL(V )), jA,E ( ξ̃ · ũ ) = jA,E ( ξ̃ ) · jA,V (ũ) . On the other hand, for any α ∈ A, jA,E ∣∣ (TApE) −1 (TAUα) = jA,Uα, it follows that jA,E is an embedding. Remark 8. Let ( π−1(U),xi,yj ) be a fiber chart of E, then the local coor- dinate of FE and TAE are ( p−1E (Ui),x i,y j k ) and (( TAπ )−1 (TAU),xiα,y j α ) . prolongations of G-structures related to weil bundles 125 We deduce that, the local coordinate of TA(FE) and F(TAE) are given by( TA(p−1E (Ui)),x i α,y j k, y j k,α ) and ( p−1 TAE ( TAU ) ,xiα,y j,α k,β ) , so the local expres- sion of jA,E is given by: jA,E ∣∣ (TApE) −1 (TAU) ( xiα,y j k,y j k,α ) =  xiα,   y j k 0 · · · 0 ... . . . . . . ... ... . . . 0 · · · yjk,α · · · y j k     . Proposition 7. Let f : E → E′ is an isomorphism of vector bundles over the diffeomorphism f : M → M ′. The following diagram TA(FE) TA(Ff)) −−−−−−−→ TA(FE′) jA,E y yjA,E′ F(TAE) −−−−−−→ F(TAf) F(TAE′) commutes. Proof. Let (Uα,ψα)α∈Λ and (U ′ α,ψ ′ α)α∈Λ the bundle atlas of (E,M,π) and (E′,M ′,π′) such that f(Uα) = U ′ α, α ∈ Λ. As f : E → E′ is an isomorphism of vector bundles over f, it follows that it exists a smooth map fα : Uα×V → V such that ψ′α◦f ∣∣ π−1(Uα) ◦ψ−1α (x,v) = ( f(x), fα(x,v) ) , for any (x,v) ∈ Uα×V and fα(x, ·) is a linear isomorphism. It follows that, the diagram p−1E (Uα) Ff ∣∣ p −1 E (Uα) −−−−−−−−−→ p−1E′ (U ′ α) ϕα y yϕ′α Uα ×GL(V ) −−−−−−−→ f̃α U ′α ×GL(V ) commutes, and f̃α(x,g) = ( f(x), fα(x, ·) ◦g ) , for each (x,g) ∈ Uα ×GL(V ). It is clear that the following diagram ( TApE )−1 ( TAUα ) TA(Ff)∣∣(TApE)−1(TAUα)−−−−−−−−−−−−−−−−−−→ (TApE′)−1 (TAU ′α) TAϕα y yTAϕ′α TAUα ×TA(GL(V )) −−−−−−−−−−−−−−−→ TA ( f̃α ) TAU ′α ×TA(GL(V )) 126 p.m.k. wamba, g.f. wankap nono, a. ntyam commutes. On the other hand, as the diagram following commutes TAUα ×TA(GL(V )) TA ( f̃α ) −−−−−−−→ TAU ′α ×TA(GL(V )) (idUα,jA,V ) y y(idU′α,jA,V ) TAUα ×GL ( TAV ) −−−−−−→ f̃α,A TAU ′α ×GL ( TAV ) with f̃α,A ( x̃, ξ̃ ) = ( TAf (x̃) , fα,A(x̃, ·) ◦ ξ̃ ) where TA(ψ′α) ◦T Af ∣∣ (TAπ) −1 (TAUα) ◦ ( TAψα )−1 (x̃,v) = ( TAf (x̃) , fα,A(x̃, ·) ) , it follows that( idU′α,jA,V ) ◦TA ( f̃α )( jAu,jAξ ) = ( idU′α,jA,V )( TAf ( jAu ) ,jA ( f̃ (u, ·) ◦ ξ )) = ( TAf(jAu), jA,V ( jA ( f̃ (u, ·) ◦ ξ ))) . As jA,V ( jA ( f̃ (u, ·) ◦ ξ ))( jAv ) = jA (( f̃ (u, ·) ◦ ξ ) ·v ) and ( f̃(u, ·) ◦ ξ ) ·v (z) = f̃ (u(z),ξ(z)(v(z))) , for any z ∈ Rk, thus, F(TAf) ◦ jA,Uα ( jAu,jAξ ) = F(TAf) ( jAu, jA,V ( jAξ )) = ( TAf ( jAu ) ,TAf̃ ( jAu,◦ ) ◦ jA,V ( jAξ )) . For any jAv ∈ TAV , as jA,V ( jAξ )( jAv ) = jA (ξ ∗v) with ξ ∗ v(z) = ξ(z)(v(z)), for all z ∈ Rk, we deduce that TAf̃ ( jAu,◦ ) ◦ jA,V ( jAξ )( jAv ) = TAf̃ ( jAu,jA (ξ ∗v) ) = jA ( f̃ (u,ξ ∗v) ) , prolongations of G-structures related to weil bundles 127 so TAf̃ ( jAu,◦ ) ◦ jA,V ( jAξ )( jAv ) = jA,V ( jA ( f̃ (u, ·) ◦ ξ ))( jAv ) for any jAv ∈ TAV . More precisely, jA,U′α ◦T A ( f̃α ) = f̃α,A ◦ jA,Uα, jA,E′ ∣∣ (TApE′) −1 (TAU′α) ◦TA(Ff) = ϕ ′−1 α,A ◦ jA,U′α ◦T A ( ϕ′α ) ◦TA(Ff) = ϕ ′−1 α,A ◦ jA,U′α ◦T A ( ϕ′α ◦Ff ◦ϕ −1 α ) ◦TA(ϕ−1α ) = ϕ ′−1 α,A ◦ jA,U′α ◦T A ( f̃α ) ◦TA(ϕ−1α ) = ϕ ′−1 α,A ◦ f̃α,A ◦ jA,Uα ◦T A(ϕ−1α ) = ( ϕ ′−1 α,A ◦ f̃α,A ◦ϕα,A ) ◦ϕ−1α,A ◦ jA,Uα ◦T A(ϕ−1α ) = F(TAf) ◦ jA,E ∣∣ (TApE) −1 (TAUα) , thus, jA,E′ ◦TA(Ff) = F(TAf) ◦ 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 idM , ϕt + F(gt) : FE → FE such that, for any x ∈ M, ϕt ∣∣ FxE : FxE −→ FxE hx 7−→ hx ◦gt . In particular, we deduce a smooth map ϕ : R × FE → FE, (t,ξ) 7→ ϕt(ξ). For any multi index α, we consider the smooth map ϕα,E : T A(FE) −→ TA(FE) ξ 7−→ TAϕ(eα,ξ). Then TA(pE) ◦ϕα,E = TA(pE). In particular, it is a homomorphism of prin- cipal bundle of TA(FE) in to TA(FE). Proposition 8. Let f : E → E′ be an isomorphism of vector bundles over the diffeomorphism f : M → M ′. Then the following diagram TA(FE) TA(Ff) −−−−−−−→ TA(FE′) ϕα,E y yϕα,E′ TA(FE) −−−−−−−→ TA(Ff) TA(FE′) commutes. 128 p.m.k. wamba, g.f. wankap nono, a. ntyam Proof. Let jAξ ∈ TA(FE), we have: ϕα,E′ ◦TA(Ff) ( jAξ ) = ϕα,E′ ( jA(F(f) ◦ ξ) ) = TAϕ ( jA(tα),jA(F(f) ◦ ξ) ) = jA (ϕ(tα,F(f) ◦ ξ)) = jA (F(f) ◦ϕ(tα,ξ)) = TA(F(f)) ◦ϕα,E ( jAξ ) . Therefore, ϕα,E′ ◦TA(Ff) = TA(F(f)) ◦ϕα,E. 3. Prolongations of G-structures to Weil bundles 3.1. The natural embedding jA,M : T A(FM) → F(TAM). 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 (TM,M,πM ), so that ( TA(FM),TAM,TA(pM ) ) is a principal fiber bundle over the Lie group TA(GL(n)). By the same way ( F(TAM),TAM,pTAM ) is a frame bundle of the vector bundle ( T(TAM),TAM,πTAM ) . If f : M → N is a local diffeomorphism, we denote with F(f) the principal bundle homo- morphism F(Tf) : FM → FN. Let M be a smooth n-dimensional manifold, F(κA,M ) : F(T ATM) −→ F(TAM) is an isomorphism of principal bundles over idTAM and pTAM ◦ F(κA,M ) = pTATM , where κA,M : T A(TM) → T(TAM) is the canonical isomorphism defined in [7]. We put jA,M = F(κA,M ) ◦ jA,TM : TA(FM) −→ F(TAM) such that pTAM ◦jA,M = TA(pM ) and jA,M (x̃·g) = jA,M (x̃)·jA,Rn(g). In par- ticular jA,M is a homomorphism of principal bundles over jA,Rn. We identify TARn with the euclidian vector space Rn×dim A, it follows that TA(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 prolongations of G-structures related to weil bundles 129 TA(FM) TA(Ff) −−−−−−−→ TA(FN) jA,M y yjA,N F(TAM) −−−−−−−→ F(TAf) F(TAN) commutes. Proof. Let f : M → N a diffeomorphism, jA,N ◦TA(Ff) = F(κA,N ) ◦ jA,TN ◦TA(Ff) = F(κA,N ) ◦F(TATf) ◦ jA,TM = F ( κA,N ◦TA(Tf) ) ◦ jA,TM = F(T(TAf) ◦κA,M ) ◦ jA,TM = F(T(TAf)) ◦F(κA,M ) ◦ jA,TM = F(TAf) ◦F(κA,M ) ◦ jA,TM. We deduce that jA,N ◦TA(Ff) = F ( TAf ) ◦ jA,M . Remark 9. Let ( U,xi ) be a local coordinate on a manifold M, the local coordinate of FM is denoted by ( p−1M (U),x i,xij ) , ( TAU,xi,xiα ) the local co- ordinate of TAM , (( TApM )−1 ( TAU ) ,xi,xij,x i α,x i j,α ) the local coordinate of TA(FM) and ( p−1 TAM ( TAU ) ,xi,xiα,x i j,x i,β j,α ) local coordinate of F(TAM). The formula jA,M (x i,xij,x i α,x i j,α) =  xi, xiα,   xij 0 · · · 0 ... . . . . . . ... ... . . . 0 · · · xij,α · · · x i j     is a local expression of the natural embedding jA,M . 3.2. Prolongations of G-structures. Let G be a Lie subgroup of GL(n), we denote by GA,n the image of T AG by the homomorphism jA,Rn, i.e. GA,n = jA,Rn(T AG). Clearly GA,n is a Lie subgroup of GL(n × dim A). 130 p.m.k. wamba, g.f. wankap nono, a. ntyam Let (P,M,π) be a G-structure on M, we denote by πA the restriction of the projection pTAM : F(T AM) → TAM to the subbundle T AP = jA,M (TAP). Then we obtain a GA,n-structure ( T AP,TAM,πA ) on the Weil bundle TAM of M related to A. It is called the A-prolongation of the G-structure P to the Weil bundle TAM to M. Proposition 10. Let P (resp. P ′) be a G-structure on M (resp. M ′) and f : M → M ′ be a diffeomorphism. Then f is an isomorphism of P on P ′ if and only if TAf : TAM → TAM ′ is an isomorphism of T AP on T AP ′. Proof. The diffeomorphism f : M → M ′ is an isomorphism of P on P ′, if and only if F(f)(P) = P ′. By the equality jA,M′ ◦TA(Ff) = F ( TAf ) ◦jA,M it follows that, if f is an isomorphism of P on P ′, then T AP ′ = jA,M′ ( TAP ′ ) = jA,M′ ◦TA(Ff) ( TAP ) = F ( TAf ) ◦ jA,M ( TAP ) = F ( TAf )( T AP ) . Inversely, if TAf : TAM → TAM ′ is an isomorphism of T AP on T AP ′, then jA,M′(T AP ′) = F(TAf)(TAP) = F(TAf) ◦ jA,M (TAP) = jA,M′ ◦TA(Ff)(TAP). Therefore, TAP ′ = TA(Ff)(TAP). In particular, P ′ = πA,P ′(T AP ′) = πA,P ′◦ TA(Ff)(TAP) = F(f)◦πA,P (TAP) = F(f)(P). So f is an isomorphism of P on P ′. Corollary 1. Let f be a diffeomorphism of M into itself, and P be a G-structure on M. Then f is an automorphism of P if and only if TAf is an automorphism of the A-prolongation T AP . Let φ : M → FM be a smooth section, the we define φ̃A = jA,M ◦TA(φ), where jA,M : T A(FM) → F(TAM) is the natural embedding from Subsection 3.1. It is a smooth section of the frame bundle F(TAM) called complete lift of φ to F(TAM). Remark 10. Let ( U,x1, . . . ,xn ) be a local coordinate of M, we introduce the coordinate ( TAU,xiα ) of TAM. Let φ : M → FM be a smooth section such that φ ∣∣ U = φij ( ∂ ∂xi ) ⊗ej, prolongations of G-structures related to weil bundles 131 then φ̃A ∣∣ TAU = ( φij )(α−β) ( ∂ ∂xiα ) ⊗ejβ , where { ei } i=1,...,n and { ei,eiα } (i,α)∈{1,...,n}×BA are the dual basis of the canon- ical basis of Rn and TA(Rn). Definition 4. Let (P,M,π) be a G-structure on M. The G-structure P is called integrable (or flat) if for each point x ∈ M, there is a coordinate neighborhood U with local coordinate system ( x1, . . . ,xn ) such that the frame(( ∂ ∂x1 ) y , . . . , ( ∂ ∂xn ) y ) ∈ Py for any y ∈ U. Proposition 11. Let P be a G-structure on a manifold M. Then, P is integrable if and only if the A-prolongation T AP of P is integrable. Proof. We suppose that P is integrable, then there is a cross section φ : U → P of P over U ⊂ M of FM such that φ = n∑ i=1 ( ∂ ∂xi ) ⊗ei. Then φ̃A = jA,M ◦TA(φ) is a cross section of T AP over TAU and, φ̃A = ∑ α∈BA ( ∂ ∂xiα ) ⊗eiα so, the A-prolongation T AP of P is integrable. Inversely, taking (a1, . . . ,aK) be a basis of NA over R. We consider the basis B = (1A,a1, . . . ,aK) as a linear isomorphism A → RK+1 and let παB : A → R be the composition of B with the projection RK+1 → R on α-factor, α = 1, . . . ,K + 1. For a coordinate system ( U,xi ) in M we define the induced coordinate system { xi0,x i α } on TAM by:{ xi0 = x i ◦πAM , xiα = ( xi )(παB) , α = 1, . . . ,K . Using these arguments, the proof is similar as for the case of tangent bundle of higher order establish in [12]. 132 p.m.k. wamba, g.f. wankap nono, a. ntyam 4. Prolongations of some classical G-structures 4.1. Complex structures. We take J0 : R2n → R2n a linear auto- morphism such that J0 ◦ J0 = −idRn and denote by G(n,J0) the group of all a ∈ GL(2n) such that a ◦ J0 = J0 ◦ a. We consider {1A,eα, α ∈ BA} be a basis of A over R. We consider this basis as a linear isomorphism TA(R2n) → R2n dim A. The map TA(J0) is a linear automorphism of TA(R2n) such that TA(J0) ◦TA(J0) = −idTA(Rn). We put, G̃ = jA,R2n ( TA(G(n,J0)) ) . Proposition 12. The Lie group G̃ is a Lie subgroup of G ( n × dim A, TA(J0) ) . Proof. Let ã ∈ G̃, then there is an element X ∈ TA(G(n,J0)), such that ã = jA,Rn(X). We put X = j Aϕ, with ϕ : Rk → G(n,J0) smooth map. For any jAξ ∈ TARn, we have: TA(J0) ◦ ã ( jAξ ) = TA(J0) ( jA(ϕ∗ ξ) ) = jA(J0 ◦ (ϕ∗ ξ)). As, for any z ∈ Rk, J0 ◦ (ϕ∗ ξ)(z) = J0 ◦ϕ(z)(ξ(z)) = ϕ(z) ◦J0(ξ(z)) = ϕ∗ (J0 ◦ ξ)(z) , we deduce that TA(J0) ◦ ã ( jAξ ) = jA(ϕ∗ (J0 ◦ ξ)) = jA,Rn ( jAϕ )( jA(J0 ◦ ξ) ) = jA,Rn(X) ◦TA(J0) ( jAξ ) . So, TA(J0) ◦ ã ( jAξ ) = ã◦TA(J0) ( jAξ ) , for all jAξ ∈ TARn. Remark 11. Let M be a smooth manifold of dimension 2n, M has an almost complex structure if and only if M has a G (n,J0)-structure P . Ap- plying Subsection 2.2, we see that TAM has canonically a G̃-structure T AP . By Proposition 9, T AP induces canonically a G(n dim A,TA(J0))-structure P̃A. Which means that TAM has a canonical almost complex structure. Theorem 2. The canonical almost complex structure J̃A on TAM in- duced by a G(n dim A,TA(J0))-structure P̃ A is just the complete lift J(c) of the associated almost complex structure J with P . prolongations of G-structures related to weil bundles 133 Proof. Let φ : M → P be a smooth section, then J(x) = φ(x)◦J0◦φ(x)−1, for any x ∈ M. Consider the vector ei,α = jA(zαei), with α ∈ BA and i ∈ {1, . . . , 2n}. The family (ei,α) is a basis of the real vector space TA(Rn). If φ ∣∣ U = φ j i ( ∂ ∂xj ) ⊗ei then φ̃A ∣∣ TAU = ( φ j i )(α−β) ( ∂ ∂x j α ) ⊗eiβ. In particular φ̃A(ei,α) = ( φ j i )(α−β) ( ∂ ∂x j β ) = (φ(ei)) (α) , so J̃A ◦ φ̃A(ei,α) = J̃A ( (φ(ei)) (α) ) . For any jAξ ∈ TAM, we have φ̃A ◦TA(J0)(ei,α) ( jAξ ) = κA,M ◦ jA,TM ( TA(φ) ◦TA(J0) ( jA(zαei) ))( jAξ ) = κA,M ◦ jA,TM ( jA(φ◦ ξ) )( jA(zαJ0(ei)) ) = κA,M ( jA ((φ◦ ξ) ∗ (zαJ0(ei))) ) . For any z ∈ Rk, (φ◦ ξ) ∗ (zαJ0(ei))(z) = φ(ξ(z))(zαJ0(ei)) = zαφ(ξ(z)) ◦J0(ei) = zαJ(ξ(z)) ◦φ(ξ(z))(ei) = J(ξ(z)) ◦φ(zαei)(ξ(z)), we deduce that φ̃A ◦TA(J0)(ei,α) ( jAξ ) = κA,M ◦TAJ ( χ (α) TM ◦T A(φ(ei)) )( jAξ ) = ( κA,M ◦TA(J) ◦k−1A,M ) ◦ ( κA,M ◦χ (α) TM ◦T A(φ(ei)) )( jAξ ) = J(c) ( (φ(ei)) (α) )( jAξ ) . As φ̃A ◦ TA(J0)(ei,α) = J̃A ◦ φ̃A(ei,α), we deduce that, J̃A ( (φ(ei)) (α) ) = J(c) ( (φ(ei)) (α) ) , for any α ∈ BA. So J̃A is the complete lift of J. 4.2. Almost symplectic structure. Let f : R2n × R2n → R be a skew-symmetric non degenerate bilinear form on R2n. We denote by G(f) the group of all a ∈ GL(2n) such that f(a(x),a(y)) = f(x,y), for all x,y ∈ R2n. We consider the basis of A over R, {1A,eα, α ∈ BA} as a linear isomorphism TA(R2n) → R2n dim A. We suppose that, A is a Weil-Frobenius algebra, so 134 p.m.k. wamba, g.f. wankap nono, a. ntyam there is a linear form p : A → R such that the bilinear form q : A×A → R, (a,b) 7→ p (ab) is non degenerate. The map p◦TA(f) : TA(R2n)×TA(R2n) → R is a skew-symmetric non degenerate bilinear form on TA ( R2n ) . We put, f(A) = p◦TA(f) and G̃ = jA,R2n ( TA(G(f)) ) . Proposition 13. The Lie group G̃ is a Lie subgroup of G(f(A)). Proof. Let u = jAξ ∈ TA(G(f)), then jA,R2n(u) = ũ is the linear automor- phism of TA(R2n) defined for any jAϕ ∈ TA(R2n) by: ũ ( jAϕ ) = jA(ξ ∗ϕ) where (ξ ∗ϕ)(z) = ξ(z)(ϕ(z)), for any z ∈ Rk. For any jAϕ,jAψ ∈ TA(R2n), we have: f(A) ( ũ ( jAϕ ) , ũ ( jAψ )) = f(A) ( jA(ξ ∗ϕ),jA(ξ ∗ψ) ) = p◦TA(f) ( jA(ξ ∗ϕ),jA(ξ ∗ψ) ) = p ( jA(f(ξ ∗ϕ,ξ ∗ψ)) ) . On the other hand, for any z ∈ Rk, f(ξ ∗ϕ,ξ ∗ψ)(z) = f(ξ(z)(ϕ(z)),ξ(z)(ψ(z))) = f(ϕ(z),ψ(z)). Therefore, f(A) ( ũ ( jAϕ ) , ũ ( jAψ )) = p◦TA(f) ( jAϕ,jAψ ) = f(A) ( jAϕ,jAψ ) . Theorem 3. The almost symplectic form on TAM associated with the A-prolongation of a G(f) structure P on a manifold M is the p-prolongation of the almost symplectic form associated with the G-structure P . Proof. Let φ : M → P be a smooth section, consider the vector ei,α = jA(zαei), with α ∈ BA and i ∈ {1, . . . , 2n}. The family (ei,ei,α) is a basis of the real vector space TA(Rn). If φ ∣∣ U = φ j i ( ∂ ∂xj ) ⊗ ei then φ̃A ∣∣ TAU =( φ j i )(α−β) ( ∂ ∂x j α ) ⊗eiβ. In particular, φ̃A(ei,α) = ( φ j i )(α−β) ( ∂ ∂x j β ) = (φ(ei)) (α). prolongations of G-structures related to weil bundles 135 We denote by ω the almost symplectic form induced by P and ωA the almost symplectic form induced by T AP . For all i,j ∈ {1, . . . , 2n} and α,β ∈ BA, we have: ωA ( (φ(ei)) (α), (φ(ej)) (β) ) = f(A) (( φ̃A )−1 ( (φ(ei)) (α) ) , ( φ̃A )−1 ( (φ(ej)) (β) )) = p◦TA(f)(ei,α,ej,β) = p◦TA(f) ( jA(zαei),j A(zβej) ) = p ( jA(f(zαei,z βej)) ) = p ( jA(zα+βf(ei,ej)) ) = (ω(φ(ei),φ(ej))) (α+β) = ω(p) ( (φ(ei)) (α), (φ(ej)) (β) ) . It follows that, ωA = ω (p), where ω(p) is the complete p-lift of ω described in [9] and [10]. Remark 12. When f : Rn × Rn → R is a bilinear symmetric non de- generate form and G the Lie subgroup generated by all elements of linear group invariant with respect to f, then, the pseudo riemannian metric on TAM associated with the A-prolongation of a G-structure P on a manifold M is the p-prolongation of the pseudo riemannian metric associated with the structure P . 4.3. Regular foliations induced by A-prolongations of G(V )- structures. Let V be a vector subspace of Rn (dim V = p). We denote by G(V ) the group of all a ∈ GL(n) such that a(V ) = V . We consider the basis {1A,eα, α ∈ BA} of A over R and the linear isomorphism induced TA(Rn) → Rn dim A. Therefore GL(TA(R2n)) is identified to GL(n dim A). Proposition 14. The Lie group G̃ = jA,Rn(T A(G(V ))) is a Lie subgroup of G(TA(V )). Proof. Let X = jA,Rn ( jAϕ ) where ϕ : Rk → G(V ) is a smooth map. So that, X : TA(Rn) → TA(Rn) is a linear isomorphism and for any jAξ ∈ TA(Rn), X ( jAξ ) = jA(ϕ∗ ξ). For any jAξ ∈ TA(V ), we have X ( jAξ ) = jA(ϕ ∗ ξ), as for any z ∈ Rk, (ϕ ∗ ξ)(z) = ϕ(z)(ξ(z)) ∈ V , it follows that X ( jAξ ) ∈ TA(V ). Thus, X ( TA(V ) ) ⊂ TA(V ). 136 p.m.k. wamba, g.f. wankap nono, a. ntyam Let D be a smooth regular distribution on M of rank p, we denote by XD the set of all local vector fields X such that: for all x ∈ M, X(x) ∈ Dx. Let us notice that for a completely integrable distribution D, the family XD is a Lie subalgebra of the Lie algebra of vector fields on M. We denote by D(A) the distribution generated by the family { X(α), 0 ≤ α ≤ h } . As [ X(α),X(β) ] = [X,Y ] (α+β) and by the Frobenius theorem, it follows that D(A) is a smooth reg- ular and completely integrable distribution on TAM. It is called A-complete lift of D from M to TAM. In particular D(A) = κA,M ( TA(D) ) ⊂ T ( TAM ) . Proposition 15. If S ⊂ M is a leaf of regular completely integrable distribution D, then TAS is a leaf of the regular distribution D(A). Proof. As S is connected, then TAS is also connected. In fact, let ξ1,ξ2 ∈ TAS, we put πAS (ξi) = si, i = 1, 2. We consider X0 : M → T AM the smooth section defined by for any x ∈ M by: X0(x) = j A(z 7→ x). In particular πAS ◦ X0(si) = si, for i = 1, 2. There is a continuous curve α1 : [0, 1] → TAs1M such that α1(0) = ξ1 and α1(1) = X0(s1). By the same way, there is a continuous curve α2 : [0, 1] → TAs2M such that α2(0) = X0(s2) and α2(1) = ξ2. Let α0 : [0, 1] → S be a continuous curve such that α0(0) = s1 and α0(1) = s2. Consider the following curve α : [0, 1] → TAS defined by: α(t) =   α1(3t) if 0 ≤ t ≤ 13 X0 ◦α0(3t− 1) if 13 ≤ t ≤ 2 3 , α2(3t− 2) if 23 ≤ t ≤ 1 . The curve α is continuous and α(0) = ξ1, α(1) = ξ2. So, T AS is connected. For any ξ ∈ TAS, we have, Tξ(T AS) = Tξ (( πAM )−1 (S) ) = ( Tξπ A M )−1 ( TπA M (ξ)S ) = ( Tξπ A M )−1 ( DπA M (ξ) ) = D (A) ξ . Thus, TAS is a leaf of D(A). prolongations of G-structures related to weil bundles 137 Theorem 4. The regular foliation on TAM associated with the A- prolongation of a G(V )-structure P on a manifold M is the A-complete lift of the regular foliation associated with the structure P . Proof. Let φ : M → P be a smooth section. If locally φ ∣∣ U = φ j i ( ∂ ∂xj ) ⊗ei then φ̃A ∣∣ TAU = ( φ j i )(α−β) ( ∂ ∂x j α ) ⊗eiβ. In particular, φ̃A(ei,α) = ( φ j i )(α−β) ( ∂ ∂x j β ) = (φ(ei)) (α). Let D the regular smooth distribution induced by the G(V )-structure P and D̃ the smooth distribution induced by T AP , for any ξ ∈ TAM, D̃ξ = φ̃A(ξ) ( TAV ) = 〈 φ̃A(ξ)(ei,α), i ∈{1, . . . ,p} , 0 ≤ |α| ≤ h 〉 = 〈 (φ(ei)) (α)(ξ), i ∈{1, . . . ,p} , 0 ≤ |α| ≤ h 〉 . It follows that, D̃ξ = D (A) ξ . References [1] D. Bernard, Sur la géométrie différentielle des G-structures, Ann. Inst. Fourier (Grenoble) 10 (1960), 151 – 270. [2] A. Cabras, I. Kolář, Prolongation of tangent valued forms to Weil bun- dles, Arch. Math. (Brno) 31 (2) (1995), 139 – 145. [3] J. Debecki, Linear natural operators lifting p-vectors to tensors of type (q, 0) on Weil bundles, Czechoslovak Math. J. 66 (2) (2016), 511 – 525. [4] M. Doupovec, M. Kureš, Some geometric constructions on Frobenius Weil bundles, Differential Geom. Appl. 35 (2014), 143 – 149. [5] J. Gancarzewicz, W. Mikulski, Z. Pogoda, Lifts of some tensor fields and connections to product preserving functors, Nagoya Math. J. 135 (1994), 1 – 41. [6] I. Kolář, On the geometry of Weil bundles, Differential Geom. Appl. 35 (2014), 136 – 142. [7] I. Kolář, Covariant approach to natural transformations of Weil functors, Comment. Math. Univ. Carolin. 27 (4) (1986), 723 – 729. [8] I. Kolář, P. Michor, J. Slovak, “ Natural Operations in Differential Geometry ”, Springer-Verlag, Berlin, 1993. [9] P.M. Kouotchop Wamba, A. Ntyam, Prolongations of Dirac structures related to Weil bundles, Lobachevskii J. Math. 35 (2014), 106 – 121. 138 p.m.k. wamba, g.f. wankap nono, a. ntyam [10] P.M. Kouotchop Wamba, A. Mba, Characterization of some natural transformations between the bundle functors TA◦T∗ and T∗◦TA on Mfm, IMHOTEP J. Afr. Math. Pures Appl. 3 (2018), 21 – 32. [11] M. Kures, W. Mikulski, Lifting of linear vector fields to product pre- serving gauge bundle functors on vector bundles, Lobachevskii J. Math. 12 (2003), 51 – 61. [12] A. Morimoto, Prolongations of G-structure to tangent bundles of higher order, Nagoya Math. J. 38 (1970), 153 – 179. [13] A. Morimoto, Lifting of some types of tensor fields and connections to tangent bundles of pr-velocities, Nagoya Math. J. 40 (1970), 13 – 31. Preliminaries Lifts of functions and vector fields. Lifts of tensor fields of type (1,q). Lifts of tensor fields of type (0,s). The natural transformations jA,E: TA(FE) F(TAE) The embedding jA,V: TA(GL(V))GL(TAV). Frame gauge functor on the vector bundles. The natural embedding jA,E: TA(FE)F(TAE). Prolongations of G-structures to Weil bundles The natural embedding jA,M: TA(FM)F(TAM). Prolongations of G-structures. Prolongations of some classical G-structures Complex structures. Almost symplectic structure. Regular foliations induced by A-prolongations of G(V)-structures.