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Isomorphism of Intransitive Linear Lie Equations

We show that formal isomorphism of intransitive linear Lie equations along transversal to the orbits can be extended to neighborhoods of these transversal. In analytic cases, the word formal is dropped from theorems. Also, we associate an intransitive Lie algebra with each intransitive linear Lie eq...

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Опубліковано в: :Symmetry, Integrability and Geometry: Methods and Applications
Дата:2009
Автор: Veloso, Jose Miguel Martins
Формат: Стаття
Мова:Англійська
Опубліковано: Інститут математики НАН України 2009
Онлайн доступ:https://nasplib.isofts.kiev.ua/handle/123456789/149105
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Назва журналу:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:Isomorphism of Intransitive Linear Lie Equations / Jose Miguel Martins Veloso // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 27 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine
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author Veloso, Jose Miguel Martins
author_facet Veloso, Jose Miguel Martins
citation_txt Isomorphism of Intransitive Linear Lie Equations / Jose Miguel Martins Veloso // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 27 назв. — англ.
collection DSpace DC
container_title Symmetry, Integrability and Geometry: Methods and Applications
description We show that formal isomorphism of intransitive linear Lie equations along transversal to the orbits can be extended to neighborhoods of these transversal. In analytic cases, the word formal is dropped from theorems. Also, we associate an intransitive Lie algebra with each intransitive linear Lie equation, and from the intransitive Lie algebra we recover the linear Lie equation, unless of formal isomorphism. The intransitive Lie algebra gives the structure functions introduced by É. Cartan.
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fulltext Symmetry, Integrability and Geometry: Methods and Applications SIGMA 5 (2009), 103, 40 pages Isomorphism of Intransitive Linear Lie Equations? Jose Miguel Martins VELOSO Faculdade de Matematica, UFPA, Belem, PA, CEP 66075-110, Brasil E-mail: veloso@ufpa.br Received February 09, 2009, in final form November 11, 2009; Published online November 17, 2009 doi:10.3842/SIGMA.2009.103 Abstract. We show that formal isomorphism of intransitive linear Lie equations along transversal to the orbits can be extended to neighborhoods of these transversal. In analytic cases, the word formal is dropped from theorems. Also, we associate an intransitive Lie algebra with each intransitive linear Lie equation, and from the intransitive Lie algebra we recover the linear Lie equation, unless of formal isomorphism. The intransitive Lie algebra gives the structure functions introduced by É. Cartan. Key words: Lie equations; Lie groupoids; intransitive; isomorphism 2000 Mathematics Subject Classification: 58H05; 58H10 1 Introduction It is known [7] that the isomorphism of the fibers of transitive linear Lie equations at two points is sufficient to obtain the formal isomorphism of the Lie equations. This is proved by constructing a system of partial differential equations (SPDE) whose solutions would be these isomorphisms. This SPDE may be not integrable [2, 11], although it is formally integrable. If the data are analytic, the SPDE is integrable. In this paper we consider principally the extension of this theorem to intransitive linear Lie equations. Intransitive linear Lie equations generate a family of orbits on the manifold that we suppose locally to be a foliation. Given two intransitive Lie equations, consider the restriction of both Lie equations along two transversal to the orbits. If these restrictions are isomorphic in a certain sense, then we can construct a formally integrable SPDE whose solutions (if they exist) are isomorphisms of the two Lie equations. Therefore we prove, at least in the analytic case, that formal isomorphism of two linear Lie equations along transversal to the orbits can be extended locally to local isomorphism of the two linear Lie equations in a neighborhood of the two transversal. Specifically, consider M and M ′ manifolds of the same dimension, V and V ′ integrable distributions of the same dimension on M and M ′, respectively, N and N ′ submanifolds of M and M ′ such that each integral leaf of V and V ′ through points x ∈ M and x′ ∈ M ′ intersect N and N ′ (at least locally) at unique points ρx and ρ′x′, respectively. Let be φ : N → N ′ a local diffeomorphism, a ∈ N , a′ ∈ N ′, φ(a) = a′, Qk φ the manifold of k-jets of local diffeomorphisms f : M → M ′ such that φρ(x) = ρ′f(x), and Qk φ the sheaf of germs of invertible local sections of Qk φ. Furthermore, let be Rk ⊂ JkV , R′k ⊂ JkV ′ intransitive linear Lie equations such that R0 = J0V , R′0 = J0V ′. We say that Rk at point a is formally isomorphic to R′k at a point a′ if we can construct a formally integrable SPDE Sk ⊂ Qk φ such that any solution f of Sk satisfies (jk+1f)∗(Rk) = R′k (see Definition 5.3). We prove in Proposition 5.3 that this condition is ?This paper is a contribution to the Special Issue “Élie Cartan and Differential Geometry”. The full collection is available at http://www.emis.de/journals/SIGMA/Cartan.html mailto:veloso@ufpa.br http://dx.doi.org/10.3842/SIGMA.2009.103 http://www.emis.de/journals/SIGMA/Cartan.html 2 J.M.M. Veloso equivalent to the existence of F ∈ Qk+1 φ such that βF (a) = a′ and F∗(TM ⊕Rk) = TM ′ ⊕R′k. Then we prove: Theorem 5.1. Let be Φ : N → Qk+1 φ such that βΦ = φ and Φ∗ ( TN ⊕Rk|N ) = TN ′ ⊕R′k|N ′ . Then given a diffeomorphism f : M → M ′ such that f∗V = V ′ and f |N = φ there exists F ∈ Qk+1 φ satisfying F |N = Φ, βF = f and F∗ ( TM ⊕Rk ) = TM ′ ⊕R′k. This theorem for transitive linear Lie equations is in [7]. The next step is to define an intransitive Lie algebra representing TN ⊕Rk|N . Transitive Lie algebras were defined in [10] as the algebraic object necessary to study transitive infinitesimal Lie pseudogroups. This program was pursued in the papers [6, 7, 8, 9, 14, 16, 17, 20, 24] disclosing the fitness of transitive Lie algebras to study transitive linear Lie equations. At the same time, several tries were made to include intransitive linear Lie equations in this theory [12, 13, 18, 19]. Basically, they associated a family of transitive Lie algebras on a transversal to the orbits with an intransitive infinitesimal Lie pseudogroup, each transitive Lie algebra corresponding to the transitive infinitesimal Lie pseudogroup obtained by restriction to each orbit of the infinitesimal intransitive Lie pseudogroup. Another approach was to study the Lie algebra of infinite jets at one point of the infinitesimal intransitive Lie pseudogroup; this algebra is bigraded, and this bigraduation may give the power series of the structure functions introduced in [1]. In the first approach, it may happen to be impossible to relate transitive Lie algebras along the transversal and give a continuous structure to this family, as the following example in [27] shows: let be Θ the infinitesimal intransitive Lie pseudogroup acting in the plane R2 given by Θ = { θ(x, y) ∂ ∂y such that ∂θ ∂y = a(x) ∂θ ∂x with a 6= 0, a(0) = 0 } . Two of these infinitesimal intransitive Lie pseudogroups given by functions a(x) and ā(x) are iso- morphic if and only if there exists a C∞-function b(x), with b(0) 6= 0 such that ā(x) = b(x)a(x). The restriction of Θ to the orbits {(x0, y) : y ∈ R}, with a(x0) 6= 0 is the infinitesimal Lie pseu- dogroup of differentiable vector fields on R, and the restriction to the orbit where a(x0) = 0 are the infinitesimal translations on R. This example shows that, even if the intransitive linear Lie equation associated to Θ has all properties of regularity, by choosing an appropriate func- tion a(x), we obtain a highly discontinuous family of transitive Lie algebras along the transversal. If a(x) and ā(x) have the same power expansion series around 0, the bigraded Lie algebras of infinite jets of vector fields of Θ and Θ̄ at point (0, 0) are the same. So, the bigraded Lie algebras cannot distinguish between non isomorphic intransitive Lie pseudogroups. However, É. Cartan associated “structure functions” constant along the orbits with infinitesimal Lie pseudogroups. Therefore any definition of intransitive Lie algebras must contain a way of getting the germs of “structure functions” at the point considered. A way to define such algebra is the following: let be ON,a the ring of germs of real functions defined on N at the point a, Lm the ON,a-module of germs of sections of Rm|N at point a, and T Na the ON,a-module of germs of sections of TN at point a. The vector bundles Rm, with m > k, are the prolongations of Rk, which we suppose formally integrable. There is a R-bilinear map [[ , ]]m : (T Na ⊕ Lm)× (T Na ⊕ Lm) → (T Na ⊕ Lm−1), Isomorphism of Intransitive Linear Lie Equations 3 given by: the bracket of germs of vector fields in T Na; the germ of i(v)Dξ for v ∈ T Na, ξ ∈ Lm and D the linear Spencer operator; and the ON,a-bilinear map defined on holonomic sections by [[jkθ, jkη]]k = jk−1[θ, η]. Let be L = limprojmLm, L = T Na ⊕ L and [[ , ]]∞ = limprojm[[ , ]]m. Then (L, [[ , ]]∞) is the ON,a-intransitive R-Lie algebra associated with the formally integrable linear Lie equation Rk at the point a. From (L, [[ , ]]∞) we can obtain the germ of Rk|N at point a, and, by applying the Theorem of [25] and Theorem 5.1 of this paper, we get the germ of the linear Lie equation Rk at the point a, up to a formal isomorphism (cf. Theorems 6.1 and 6.2 below). We summarize the content of this paper. Section 2 presents basic facts on groupoids and algebroids of jets, the calculus on the diagonal introduced in [16, 17], and the construction of the first linear and non-linear Spencer complexes. We tried to be as complete as possible, and the presentation emphasizes the geometric relationship between the first linear and non-linear Spencer operators, and the left and right actions of a groupoid on itself. We hope this section will facilitate the reading of this paper, since several formulas given here are sometimes not easily identified in [16, 17, 14, 6, 7], due to the simultaneous use of the first, second and the sophisticated non-linear Spencer complex, and the identifications needed to introduce them. Section 3 contains the construction of partial connections on JkV . These partial connections are fundamental for Section 5. Section 4 introduces the basic facts on linear Lie equations and the associated groupoids. Section 5 presents the definition of formal isomorphism of linear Lie equations, and the proof of Theorem 5.1. Section 6 introduces the definition of intransitive Lie algebras, and the notion of isomorphism of these algebras. The last section discuss the classification of intransitive linear Lie equations of order one in the plane, with symbol g1 of dimension one. This classification contains the examples introduced above. 2 Preliminaries In this section, we present some background material. The main references for this section are [16, 17, 14], and we will try to maintain the exposition as self-contained as possible, prin- cipally introducing geometrical proofs for actions of invertible sections of Qk+1M on sections of T ⊕ JkTM . 2.1 Groupoids and algebroids of jets Let be M a manifold and QkM = Qk the manifold of k-jets of local diffeomorphisms of M . This manifold has a natural structure of Lie groupoid given by composition of jets jk f(x)g.jk xf = jk x(gf), and inversion( jk xf )−1 = jk f(x)f −1, where f : U → V , g : V → W are local diffeomorphisms of M , and x ∈ U . The groupoid Qk has a natural submanifold of identities I = jkid, where id is the identity function of M . Then we have a natural identification of M with I, given by x 7→ I(x). Therefore we can think of M as a submanifold of Qk. There are two submersions α, β : Qk → M , the canonical projections source, α(jk xf) = x, and target, β(jk xf) = f(x). We also consider α and β with values in I, by the above identification of M with I. There are natural projections πk l = πl : Qk → Ql, for k ≥ l ≥ 0, defined by πl(jk xf) = jl xf . Observe that Q0 = M ×M and πk 0 = (α, β). The projections πk l commute with the operations of composition and inversion in Qk. 4 J.M.M. Veloso We denote by Qk(x) the α-fiber of Qk on x ∈ M , or Qk(x) = α−1(x); by Qk(·, y) the β-fiber of Qk on y ∈ M , in another way, Qk(·, y) = β−1(y) and Qk(x, y) = Qk(x) ∩ Qk(·, y). The set Qk(x, x) is a group, the so-called isotropy group of Qk at point x. If U, V are open sets of M , Qk(U) = ∪x∈UQk(x), Qk(·, V ) = ∪y∈V Qk(·, y), and Qk(U, V ) = Qk(U) ∩Qk(·, V ). A (differentiable) section F of Qk defined on an open set U of M is a differentiable map F : U → Qk such that α(F (x)) = x. If β(F (U)) = V and f = βF : U → V is a diffeomorphism, we say that the section F is invertible. We write U = α(F ) and V = β(F ). An invertible section F of Qk is said holonomic if there exists a diffeomorphism f : α(F ) → β(F ) such that F = jkf . In this case, βF = f . We denote byQk the set of invertible sections of Qk. NaturallyQk has a structure of groupoid. If F,H ∈ Qk with β(F ) = α(H), then HF (x) = H(f(x))F (x) and F−1(y) = F (f−1(y))−1, y ∈ β(F ). Similarly, we can introduce the groupoid of l-jets of invertible sections of Qk, and we denote this groupoid by QlQk. We have the inclusions λl : Qk+l → QlQk, jk+l x F 7→ jl xjkF. (2.1) An invertible section F , with α(F ) = U , β(F ) = V , defines a diffeomorphism F̃ : Qk(·, U) → Qk(·, V ), X 7→ F (β(X))X. The differential F̃∗ : TQk(·, U) → TQk(·, V ) depends, for each X ∈ Qk(·, U), only on j1 β(X)F . This defines an action j1 β(X)F : TXQk → TF (β(X))XQk, v 7→ j1 β(X)F · v = (F̃∗)X(v). (2.2) Then (2.2) defines a left action of Q1Qk on TQk Q1Qk × TQk → TQk,( j1 β(X)F, v ∈ TXQk ) 7→ j1 β(X)F · v ∈ TF (β(X))XQk. (2.3) If V k β ≡ ker β∗ ⊂ TQk denotes the subvector bundle of β∗ vertical vectors, then the action (2.2) depends only on F (β(X)): Qk × V k β → V k β , (F (β(X)), v ∈ (V k β )X) 7→ F (β(X)) · v ∈ (V k β )F (β(X))X . (2.4) In a similar way, F defines a right action which is a diffeomorphism F̄ : Qk(V ) → Qk(U), X 7→ XF (f−1(α(X))). (2.5) The differential F̄∗ of F̄ induces the right action TQk ×Q1Qk → TQk,( v ∈ TXQk, j1 f−1(α(X))F ) 7→ v · j1 f−1(α(X))F = (F̄∗)X(v). (2.6) Isomorphism of Intransitive Linear Lie Equations 5 As β(Y X) = β(Y ), it follows that β∗ ( v · j1 f−1(α(X))F ) = β∗(v), where β∗ : TQk → TM is the differential of β : Qk → M . We verify from (2.5) that the function F̄ restricted to the α-fiber Qk(y) depends only on the value of F in f−1(y). If V k α = ker α∗, then the right action (2.6) depends only on the value of F at each point, and the action (2.6) by restriction gives the action V k α ×Qk → V k α ,( v ∈ (V k α )X , Y ∈ Qk(·, α(X)) ) 7→ v · Y ∈ (V k α )XY . (2.7) A vector field ξ̄ on Qk with values in V k α is said right invariant if ξ̄(XY ) = ξ̄(X) · Y . The vector field ξ̄ is determined by its restriction ξ to I. Let be T = TM the tangent bundle of M , and T the sheaf of germs of local sections of T . We denote by JkT the vector bundle of k-jets of local sections of π : T → M . Then JkT is a vector bundle on M , and we also denote by π : JkT → M the map π(jk xθ) = x. If θ : U ⊂ M → TM is a local section, and ft is the 1-parameter group of local diffeomorphisms of M such that d dtft|t=0 = θ, then we get, for x ∈ U , d dt jk xft|t=0 = jk xθ. This means we have a natural identification V k α |I = JkT. Therefore, as TQk|I = TI ⊕ V k α |I , TQk|I ∼= T ⊕ JkT, (2.8) and if we denote by J̌kT = T ⊕ JkT, we have TQk|I ∼= J̌kT . Observe that J̌kT is a vector bundle on M . The restriction of β∗ : TQk → T to TQk|I , and the isomorphism TQk|I ∼= J̌kT defines the map β∗ : J̌kT → T, v + jk xθ ∈ (J̌kT )x 7→ v + θ(x), (2.9) which we denote again by β∗. For more details on this identification and the map β∗, see the Appendix of [14], in particular pages 260 and 274. If ξ is a section of JkT on U ⊂ M , let be ξ̄(X) = ξ(β(X)) ·X the right invariant vector field on Qk(·, U). Then ξ̄ has F̄t, −ε < t < ε, as the 1-parameter group of diffeomorphisms induced by invertible sections Ft of Qk such that d dt F̄t|t=0 = ξ̄. Therefore, F0 = I and d dt Ft(x)|t=0 = ξ(x). 6 J.M.M. Veloso Definition 2.1. The vector bundle JkT = V k α |I on M is the (differentiable) algebroid associated with the groupoid Qk. The Lie bracket [ , ]k on local sections of JkT is well defined, given by [ξ, η]k = [ ξ̄, η̄ ] |I , (2.10) where ξ, η are sections of πk : JkT → M defined on an open set U of M (or I). Proposition 2.1. If f is a real function on U , and ξ, η sections of JkT on U , then [fξ, η]k = f [ξ, η]k − (β∗η)(f)ξ. Proof. As fξ = (fβ)ξ̄, it follows [fξ, η]k = [ (fβ)ξ̄, η̄ ] |I = ( (fβ) [ ξ̄, η̄ ] − η̄(fβ)ξ̄ ) |I = f [ξ, η]k − (β∗η)(f)ξ. � If JkT denotes the sheaf of germs of local sections of JkT , then JkT is a Lie algebra sheaf, with the Lie bracket [ , ]k. Proposition 2.2. The bracket [ , ]k on JkT is determined by: (i) [jkξ, jkη]k = jk[ξ, η], ξ, η ∈ T , (ii) [ξk, fηk]k = f [ξk, ηk]k + (β∗ξk)(f)ηk, where ξk, ηk ∈ JkT and f is a real function on M . We denote by the same symbols as above the projections πk l = πl : JkT → J lT , l ≥ 0, defined by πl(jk xθ) = jl xθ. If ξk is a point or a section of JkT , let be ξl = πk l (ξk). The vector bundle J0T is isomorphic to T by the map β∗ : J0T → T , where β∗(j0 xθ) = θ(x), see (2.9). However, β∗ : JkT → T is not equal to πk 0 : JkT → J0T , but they are isomorphic maps. Again, we have the canonical inclusions, and we use the same notation as (2.1), λl : Jk+lT → J lJkT, jk+l x θ 7→ jl xjkθ. for θ ∈ T . Analogously to the definition of holonomic sections of Qk, a section ξk of JkT is holonomic if there exists ξ ∈ T such that ξk = jkξ. Therefore, if ξk is holonomic, we have ξk = jk(β∗ξk). If θ : U ⊂ M → JkT is a section, let be ξ = j1 xθ ∈ J1 xJkT , x ∈ U . Then ξ can be identified to the linear application ξ : Tx → Tθ(x)J kT, v 7→ θ∗(v). If η ∈ J1 xJkT is given by η = j1 xµ, with µ(x) = θ(x), then (π)∗(η − ξ)v = 0, and we remember that π : JkT → M is defined by π(jk xθ) = x. So η − ξ ∈ T ∗ x ⊗ Vπ1 0(ξ)J kT , where V JkT = kerπ∗. However JkT is a vector bundle, then Vπ1 0ξJ kT ∼= Jk xT , so η − ξ ∈ T ∗ x ⊗ Jk xT . The sequence 0 → T ∗ ⊗ JkT → J1JkT π1 0→ JkT → 0 (2.11) obtained in this way is exact, and we get an affine structure on J1JkT . Isomorphism of Intransitive Linear Lie Equations 7 The linear operator D defined by D : JkT → T ∗ ⊗ Jk−1T , ξk 7→ Dξk = j1ξk−1 − λ1(ξk), (2.12) is the linear Spencer operator. We remember that ξk−1 = πk k−1ξk and λ1 : JkT → J1Jk−1T, jk xξ 7→ j1 x(jk−1ξ). The difference in (2.12) is done in J1Jk−1T and is in T ∗ ⊗ Jk−1T , by (2.11). The operator D is null on a section ξk if and only if it is holonomic, i.e., Dξk = 0 if and only if there exists θ ∈ T such that ξk = jkθ. Proposition 2.3. The operator D is characterized by (i) Djk = 0, (ii) D(fξk) = df ⊗ ξk−1 + fDξk, with ξk ∈ JkT , ξk−1 = πk−1ξk and f is a real function on M . For a proof, see [14]. The operator D extends to D : ∧lT ∗ ⊗ JkT → ∧l+1T ∗ ⊗ Jk−1T , ω ⊗ ξk 7→ D(ω ⊗ ξk) = dω ⊗ ξk−1 + (−1)lω ∧Dξk. 2.2 The calculus on the diagonal Next, following [16, 17, 14], we will relate J̌kT to vector fields along the diagonal of M × M and actions of sections in Qk to diffeomorphisms of M ×M which leave the diagonal invariant. We denote the diagonal of M×M by ∆ = {(x, x) ∈ M×M |x ∈ M}, and by ρ1 : M×M → M and ρ2 : M × M → M , the first and second projections, respectively. The restrictions ρ1|∆ and ρ2|∆ are diffeomorphisms of ∆ on M . A sheaf on M will be identified to its inverse image by ρ1|∆. For example, if OM denotes the sheaf of germs of real functions on M , then we will write OM on ∆ instead of (ρ1|∆)−1OM . Therefore, a f ∈ OM will be considered in O∆ or in OM×M through the map f 7→ f ◦ ρ1. We denote by T (M ×M) the sheaf of germs of local sections of T (M ×M) → M ×M ; by R the subsheaf in Lie algebras of T (M × M), whose elements are vector fields ρ1-projectables; by HR the subsheaf in Lie algebras of R that projects on 0 by ρ2, i.e. HR = (ρ2)−1 ∗ (0)∩R; and by VR the subsheaf in Lie algebras defined by VR = (ρ1)−1 ∗ (0) ∩R. Clearly, R = HR ⊕ VR, and [HR,VR] ⊂ VR. Then (ρ1)∗ : HR−̃→T is an isomorphism, so we identify HR naturally with T by this isomorphism, and utilize both notations indistinctly. 8 J.M.M. Veloso Proposition 2.4. The Lie bracket in R satisfies: [v + ξ, f(w + η)] = v(f)(w + η) + f [v + ξ, w + η], with f ∈ OM , v, w ∈ HR, ξ, η ∈ VR. In particular, the Lie bracket in VR is OM -bilinear. Proof. Let be f ∈ OM , ξ, η ∈ VR. Then [v + ξ, (f ◦ ρ1)(w + η)] = (v + ξ)(f ◦ ρ1)(w + η) + (f ◦ ρ1)[v + ξ, w + η]. As f◦ρ1 is constant on the submanifolds {x}×M and ξ is tangent to them, we obtain ξ(f◦ρ1) = 0, and the proposition is proved. � A vector field in VR is given by a family of sections of T parameterized by an open set of M . Therefore there exists a surjective morphism Υk : R→ T ⊕ JkT , v + ξ 7→ v + ξk, where v ∈ HR, ξ ∈ VR, and ξk(x) = jk (x,x)(ξ|{x}×M ). The kernel of morphism Υk is the subsheaf VRk+1 of VR constituted by vector fields that are null on ∆ at order k. Therefore R/VRk+1 is null outside ∆. It will be considered as a sheaf on ∆, and the sections in the quotient as sections on open sets of M . So the sheaf R/VRk+1 is isomorphic to the sheaf os germs of sections of the vector bundle T ⊕ JkT on M . So we have the isomorphism of sheaves on M , R/VRk+1 ≡ T ⊕ JkT . We usually denote by J̌kT = T ⊕ JkT and J̌kT = T ⊕ JkT . As [R,VRk+1] ⊂ VRk, the bracket on R induces a bilinear antisymmetric map, which we call the first bracket of order k, [[ , ]]k = (T ⊕ JkT )× (T ⊕ JkT ) → (T ⊕ Jk−1T ) (2.13) defined by [[v + ξk, w + ηk]]k = Υk−1([v + ξ, w + η]), where Υk(ξ) = ξk and Υk(η) = ηk. It follows from Proposition 2.4 that [[ , ]]k satisfies: [[v + ξk, f(w + ηk)]]k = v(f)(w + ηk−1) + f [[v + ξk, w + ηk]]k, (2.14) [[[[v + ξk, w + ηk]]k, z + θk−1]]k−1 + [[[[w + ηk, z + θk]]k, v + ξk−1]]k−1 + [[[[z + θk, v + ξk]]k, w + ηk−1]]k−1 = 0, for v, w, z ∈ T , ξk, ηk, θk ∈ JkT , f ∈ OM . In particular, the first bracket is OM -bilinear on JkT . Also, [[J0T , J0T ]]0 = 0. The following proposition relates [[ , ]]k to the bracket in T and the linear Spencer operator D in JkT . Isomorphism of Intransitive Linear Lie Equations 9 Proposition 2.5. Let be v, w, θ, µ ∈ T , ξk, ηk ∈ JkT and f ∈ OM . Then: (i) [[v, w]]k = [v, w], where the bracket at right is the bracket in T ; (ii) [[v, ξk]]k = i(v)Dξk; (iii) [[jkθ, jkµ]]k = jk−1[θ, µ], where the bracket at right is the bracket in T . Proof. (i) This follows from the identification of T with HR. (ii) First of all, if θ ∈ T , let be Θ ∈ VR defined by Θ(x, y) = θ(y). Then Υk(Θ) = jkθ. If v ∈ HR, then v and Θ are both ρ1 and ρ2 projectables, (ρ1)∗(Θ) = 0 and (ρ2)∗(v) = 0, so we get [v,Θ] = 0. Consequently [[v, jkθ]]k = Υk−1([v,Θ]) = 0. (2.15) Also by (2.14), we have [[v, fξk]]k = v(f)ξk−1 + f [[v, ξk]]k. (2.16) As (2.15) and (2.16) determine D (cf. Proposition 2.3), we get (ii). (iii) Given θ, µ ∈ T , we define Θ,H ∈ VR as in (ii), Θ(x, y) = θ(y) and H(x, y) = µ(y). Therefore [[jkθ, jkµ]]k = [[ΥkΘ,ΥkH]]k = Υk−1([Θ,H]) = jk−1[θ, η]. � Let be ṼR the subsheaf in Lie algebras of R such that ξ̃ ∈ ṼR if and only if ξ̃ is tangent to the diagonal ∆. If ξ̃ = ξH + ξ ∈ ṼR, ξH ∈ H, ξ ∈ V, then (ρ1)∗(ξH(x, x)) = (ρ2)∗(ξ(x, x)), where ρ1, ρ2 : M × M → ∆. Consequently, if ξk = Υk(ξ), then ξH = β∗(ξk), where we remember that β∗ : JkT → T is defined in (2.9). We can also write ξH = (ρ2)∗(ξk), since that (ρ2)∗VRk+1|∆ = 0. From now on, ξH denotes the horizontal component of ξ̃ ∈ ṼR, so ξ̃ = ξH +ξ, with ξH ∈ H and ξ ∈ V. We denote by JkT̃ the subsheaf of T ⊕ JkT = J̌kT , whose elements are ξ̃k = ξH + ξk, where ξH = β∗(ξk) or ξH = (ρ2)∗(ξk). Therefore JkT̃ identifies with ṼR/VRk+1, since that VRk+1 ⊂ ṼR. As [ṼR,VRk+1] ⊂ VRk+1, (2.17) since the vector fields in ṼR are tangents to ∆, it follows that the bracket in ṼR defines a bilinear antisymmetric map, called the second bracket, by [ , ]k : JkT̃ × JkT̃ → JkT̃ , (ξH + ξk, ηH + ηk) 7→ Υk([ξ̃, η̃]), (2.18) where Υk(ξ̃) = ξH + ξk and Υk(η̃) = ηH + ηk. Unlike the first bracket (2.13), we do not lose one order doing the bracket in JkT̃ . The second bracket [ , ]k is a Lie bracket on JkT̃ . The Proposition 2.6 below relates it to the bracket [ , ]k, defined in (2.10). The projection ν : HR ⊕ VR → VR, v + ξ 7→ ξ 10 J.M.M. Veloso quotients to νk : T ⊕ JkT → JkT , v + ξk 7→ ξk, and νk : JkT̃ → JkT is an isomorphism of vector bundles. Proposition 2.6. If ξ̃k, η̃k ∈ JkT̃ , then [ξk, ηk]k = νk([ξ̃k, η̃k]k), where ξk = νk(ξ̃k), ηk = νk(η̃k). Proof. We will verify properties (i) and (ii) of Proposition 2.2. If θ, µ ∈ T , let be Θ,H ∈ VR as in the proof of Proposition 2.5. Then: (i) νk([θ + jkθ, µ + jkµ]k) = νk(Υk([θ + Θ, µ + H])) = Υk(ν([θ, µ] + [Θ,H])) = jk([θ, µ]) = [jkθ, jkµ]k. (ii) νk([ξ̃k, f η̃k]k) = νk(f [ξ̃k, η̃k]k + ξH(f)η̃k) = fνk([ξ̃k, η̃k]k) + (β∗ξk)(f)ηk. � Corollary 2.1. If ξ̃k, η̃k ∈ JkT̃ , then νk([ξ̃k, η̃k]k) = [ξk, ηk]k = i(ξH)Dηk+1 − i(ηH)Dξk+1 + [[ξk+1, ηk+1]]k+1, where ξH = β∗ξk, ηH = β∗ηk ∈ T and ξk+1, ηk+1 ∈ Jk+1T projects on ξk, ηk, respectively. Proof. It follows from Propositions 2.5 and 2.6. � As a consequence of Proposition 2.6, we obtain that νk : JkT̃ → JkT , ξ̃k 7→ ξk is an isomorphism of Lie algebras sheaves, where the bracket in JkT̃ is the second bracket [ , ]k as defined in (2.18), and the bracket in JkT is the bracket [ , ]k as defined in (2.10). In a similar way, we obtain from (2.17) that we can define the third bracket as [ , ] k : Jk+1T̃ × J̌kT → J̌kT , (ξH + ξk+1, v + ηk) 7→ Υk([ξ̃, v + η]). where ξ̃ ∈ ṼR, v + η ∈ R, Proposition 2.7. The third bracket has the following properties: (i) [f ξ̃k+1, gη̌k] k = fξH(g)η̌k − v(f)gξ̃k + fg [ξ̃k+1, η̌k] k; (ii) [ξ̃k, [[η̌k, θ̌k]]k] k−1 = [[ [ξ̃k+1, η̌k] k, θ̌k]]k + [[η̌k, [ξ̃k+1, θ̌k] k ]] k ; (iii) [ξ̃k+1, η̌k] k = [[ξ̃k+1, η̌k+1]]k+1, where ξ̃k+1 = ξH + ξk+1 ∈ J̃k+1T , θ̌k ∈ J̌kT , η̌k+1 = v + ηk+1 ∈ J̌k+1T , ξ̃k = πk(ξ̃k+1), η̌k = πk(η̌k+1). Proof. The proof follows the same lines as the proof of Proposition 2.6. � Isomorphism of Intransitive Linear Lie Equations 11 Let’s now verify the relationship between the action of diffeomorphisms of M × M , which are ρ1-projectable and preserve ∆, on R, and actions (2.3) and (2.6) of Q1Qk on TQk. Let be σ a (local) diffeomorphism of M ×M that is ρ1-projectable. Then σ(x, y) = (f(x),Φ(x, y)), that is, σ is defined by f ∈ Diff M , and a function φ : M → Diff M, x 7→ φx, such that φx(y) = Φ(x, y). Particularly, when σ(∆) = ∆, then Φ(x, x) = f(x), for all x ∈ M , or φx(x) = f(x). As a special case, (φf−1(x)) −1(x) = f−1(x). Let’s denote by J the set of (local) diffeomorphisms of M ×M that are ρ1-projectable and preserve ∆. We naturally have the application J → Qk, σ 7→ σk, (2.19) where σk(x) = jk xφx, x ∈ M . If σ′ ∈ J , with σ′ = (f ′,Φ′), then (σ′ ◦ σ)(x, y) = σ′(f(x), φx(y)) = (f ′(f(x)), φ′f(x)(φx(y)) = ((f ′ ◦ f)(x), (φ′f(x) ◦ φx)(y)), and from this it follows (σ′ ◦ σ)k(x) = jk x(φ′f(x) ◦ φx) = jk f(x)φ ′ f(x).j k xφx = σ′k(f(x)).σk(x) = (σ′k ◦ σk)(x), for each x ∈ I. So (2.19) is a surjective morphism of groupoids. If φ ∈ Diff M , let be φ̃ ∈ J given by φ̃(x, y) = (φ(x), φ(y)). It is clear that (φ̃)k = jkφ. It follows from definitions of J and R that the action J ×R → R, (σ, v + ξ) 7→ σ∗(v + ξ), (2.20) is well defined. Then V and Ṽ are invariants by the action of J . Proposition 2.8. Let be σ ∈ J , v ∈ HR, ξ ∈ VR. We have: (i) (σ ∗ ξ)k = λ1σk+1.ξk.σ −1 k ; (ii) (σ ∗ ξ̃)k = f∗(ξH) + j1σk.ξk.σ −1 k ; (iii) (σ∗v)k = f∗(v) + (j1σk.v.λ1σ−1 k+1 − λ1σk+1.v.λ1σ−1 k+1). 12 J.M.M. Veloso Proof. If σ(x, y) = (f(x), φx(y)), then σ−1(x, y) = ( f−1(x), ( φ−1 ) x (y) ) , where( φ−1 ) x = ( φf−1(x) )−1 . (i) Let be ξ = d dt Vt ∣∣ t=0 , where Vt(x, y) = (x, ηt x(y)), with ηt x ∈ Diff M for each t, and gt(x) = ηt x(x). Then( σ ◦ Vt ◦ σ−1 ) (x, y) = ( x, ( φf−1(x) ◦ ηt f−1(x) ◦ (φf−1(x)) −1 ) (y) ) , and (σ∗ξ)(x, y) = d dt ( φf−1(x) ◦ ηt f−1(x) ◦ (φf−1(x)) −1 )∣∣ t=0 (y). Consequently Υk(σ∗ξ)(x) = jk x ( d dt ( φf−1(x) ◦ ηt f−1(x) ◦ (φf−1(x)) −1 ) |t=0 ) = d dt ( jk (gt◦f−1)(x)φf−1(x) ◦ jk f−1(x)η t f−1(x) ◦ jk x(φf−1(x)) −1 )∣∣ t=0 = d dt ( (φ̃f−1(x))k((gt ◦ f−1)(x)).jk f−1(x)η t f−1(x).((φ̃f−1(x)) −1)k(x) )∣∣ t=0 = d dt ( j1 f−1(x)(φ̃f−1(x))k.j k f−1(x)η t f−1(x).((φ̃f−1(x)) −1)k(x) ) |t=0 = λ1(σk+1(f−1(x))).ξk(f−1(x)).σ−1 k (x), since that j1 f−1(x)(φ̃f−1(x))k = j1 f−1(x)j kφf−1(x) = λ1(jk+1 f−1(x) φf−1(x)) = λ1(σk+1(f−1(x))). So we proved (σ ∗ ξ)k = λ1σk+1.ξk.σ −1 k . (ii) Let be, as in (i), ξ = d dtVt|t=0, where Vt(x, y) = (x, ηt x(y)), with ηt x ∈ Diff M for each t, and gt(x) = ηt x(x). Then ξ̃ = d dt Ṽt ∣∣ t=0 , where Ṽt(x, y) = (gt(x), ηt x(y)). Therefore( σ ◦ Ṽt ◦ σ−1 ) (x, y) = ( f ◦ gt ◦ f−1(x), ( φgt◦f−1(x) ◦ ηt f−1(x) ◦ (φf−1(x)) −1 ) (y) ) , Isomorphism of Intransitive Linear Lie Equations 13 and (σ∗ξ̃)(x, y) = f∗ξH(x) + d dt ( φgt◦f−1(x) ◦ ηt f−1(x) ◦ (φf−1(x)) −1 )∣∣ t=0 (y). By projecting, we obtain Υk(σ∗ξ̃)(x) = f∗ξH(x) + jk x ( d dt ( φgt◦f−1(x) ◦ ηt f−1(x) ◦ (φf−1(x)) −1 ) |t=0 ) = f∗ξH(x) + d dt ( jk (gt◦f−1)(x)φgt◦f−1(x) ◦ jk f−1(x)η t f−1(x) ◦ jk x(φf−1(x)) −1 )∣∣ t=0 = f∗ξH(x) + d dt ( σk((gt ◦ f−1)(x)).jk f−1(x)η t f−1(x).(σk)−1(x) )∣∣ t=0 = f∗ξH(x) + j1 f−1(x)σk.ξk(f−1(x)).σ−1 k (x), so (σ∗ξ̃)k = f∗ξH + j1σk.ξk.σ −1 k . Observe that this formula depends only on σk. (iii) By combining (i) and (ii), we obtain (σ ∗ ξH)k = (σ∗ξ̃)k − (σ ∗ ξ)k = (f∗ξH + j1σk.ξk.σ −1 k )− (λ1σk+1.ξk.σ −1 k ). As j1σk.ξk.σ −1 k = j1σk.ξk.λ 1σ−1 k+1 = j1σk.(ξk − ξH).λ1σ−1 k+1 + j1σk.ξH .λ1σ−1 k+1 = λ1σk+1.(ξk − ξH).λ1σ−1 k+1 + j1σk.ξH .λ1σ−1 k+1 = λ1σk+1.ξk.σ −1 k − λ1σk+1.ξH .λ1σ−1 k+1 + j1σk.ξH .λ1σ−1 k+1, where j1σk.(ξk − ξH) = λ1σk+1.(ξk − ξH) follows from β∗(ξk − ξH) = 0 by (2.4). By replacing this equality above we get (σ ∗ ξH)k = ( f∗ξH + ( λ1σk+1.ξk.σ −1 k − λ1σk+1.ξH .λ1σ−1 k+1 + j1σk.ξH .λ1σ−1 k+1 )) − ( λ1σk+1.ξk.σ −1 k ) = f∗ξH − λ1σk+1.ξH .λ1σ−1 k+1 + j1σk.ξH .λ1σ−1 k+1. � It follows from Proposition 2.8 that action (2.20) projects on an action ( )∗: Qk+1 × (T ⊕ JkT ) → T ⊕ JkT , (σk+1, v + ξk) 7→ (σk+1)∗(v + ξk), (2.21) where (σk+1)∗(v + ξk) = f∗v + ( j1σk.v.λ1σ−1 k+1 − λ1σk+1.v.λ1σ−1 k+1 ) + ( λ1σk+1.ξk.σ −1 k ) . This action verifies [[(σk+1)∗(v + ξk), (σk+1)∗(w + ηk)]]k = (σk)∗([[v + ξk, w + ηk]]k). (2.22) It follows from Proposition 2.8 and (2.21) that (σk+1)∗(ξk)(x) depends only on the value of σk+1(x) at the point x where ξ is defined, and (σk+1)∗(v)(x) depends on the value of σk+1 on a curve tangent to v(x). 14 J.M.M. Veloso Item (ii) of Proposition 2.8 says that restriction to JkT̃ of action (2.21) depends only on the section σk, so the action Qk × JkT̃ → JkT̃ , (σk, ξ̃k) 7→ (σk)∗(ξ̃k), is well defined, where (σk)∗(ξ̃k) = f∗ξH + j1σk.ξk.σ −1 k . In this case, we get [(σk)∗ξ̃k, (σk)∗η̃k]k = (σk)∗[ξ̃k, η̃k]k, and each νk(σk)∗ν−1 k acts as an automorphism of the Lie algebra sheaf JkT : [νk(σk)∗ν−1 k ξk, νk(σk)∗ν−1 k ηk]k = νk(σk)∗ν−1 k [ξk, ηk]k. If M and M ′ are two manifolds of the same dimension, we can define Qk(M,M ′) = {jk xf : f : U ⊂ M → U ′ ⊂ M ′ is a diffeomorphism, x ∈ U}. The groupoid Qk acts by the right on Qk(M,M ′), and Q ′k = Qk(M ′,M ′) acts by the left. Redoing the calculus of this subsection in this context, we obtain the analogous action of (2.21): Qk+1(M,M ′)× (T ⊕ JkT ) → T ′ ⊕ JkT ′, (σk+1, v + ξk) 7→ (σk+1)∗(v + ξk), (2.23) where Qk+1(M,M ′) denotes the set of invertible sections of Qk+1(M,M ′), (σk+1)∗(v + ξk) = f∗v + ( j1σk.v.λ1σ−1 k+1 − λ1σk+1.v.λ1σ−1 k+1 ) + ( λ1σk+1.ξk.σ −1 k ) , and f = βσk+1 : M → M ′. This action also verifies (2.22). 2.3 The Lie algebra sheaf ∧(J̌∞T )∗ ⊗ (J̌∞T ) In this subsection, we continue to follow the presentation of [16, 17]. We denote by J∞T the projective limit of JkT , say, J∞T = lim proj JkT , and J̌∞T = T ⊕ J∞T = lim proj J̌kT . As T⊕JkT ∼= TQk|I , we have the identification of J̌∞T with lim proj Γ(TQk|I), where Γ(TQk|I) denotes the sheaf of germs of local sections of the vector bundle TQk|I → I. From the fact that T ⊕ JkT is a OM -module, we get J̌∞T is a OM -module. In the following, we use the notation ξ̌ = v + lim proj ξk, η̌ = w + lim proj ηk ∈ T ⊕ J∞T . We define the first bracket in J̌∞T as: [[ξ̌, η̌]]∞ = lim proj [[v + ξk, w + ηk]]k. (2.24) Isomorphism of Intransitive Linear Lie Equations 15 With the bracket defined by (2.24), J̌∞T is a Lie algebra sheaf. Furthermore, [[ξ̌, f η̌]]∞ = v(f)η̌ + f [[ξ̌, η̌]]∞. We extend now, as in [16, 17, 14], the bracket on J̌∞T to a Nijenhuis bracket (see [3]) on ∧(J̌∞T )∗ ⊗ (J̌∞T ), where (J̌∞T )∗ = lim ind (J̌kT )∗. We introduce the exterior differential d on ∧(J̌∞T )∗, by: (i) if f ∈ OM , then df ∈ (J̌∞T )∗ is defined by 〈df, ξ̌〉 = v(f). (ii) if ω ∈ (J̌∞T )∗, then dω ∈ ∧2(J̌∞T )∗ is defined by 〈dω, ξ̌ ∧ η̌〉 = θ(ξ̌)〈ω, η̌〉 − θ(η̌)〈ω, ξ̌〉 − 〈ω, [[ξ̌, η̌]]∞〉, where θ(ξ̌)f = 〈df, ξ̌〉. We extend this operator to forms of any degree as a derivation of degree +1 d : ∧r(J̌∞T )∗ → ∧r+1(J̌∞T )∗. The exterior differential d is linear, d(ω ∧ τ) = dω ∧ τ + (−1)rω ∧ dτ, for ω ∈ ∧r(J̌∞T )∗, and d2 = 0. Remember that (ρ1)∗ : T ⊕ J∞T → T is the projection given by the decomposition in direct sum of J̌∞T = T ⊕ J∞T . (We could use, instead of (ρ1)∗, the natural map α∗ : T ⊕ JkT → T , given by α∗ : TQk|I → T , and the identification (2.8)). Then (ρ1)∗ : T ∗ → (J̌∞T )∗, and this map extends to (ρ1)∗ : ∧T ∗ → ∧(J̌∞T )∗. If ω ∈ ∧r(J̌∞T )∗, then 〈(ρ1)∗ω, ξ̌1 ∧ · · · ∧ ξ̌r〉 = 〈ω, v1 ∧ · · · ∧ vr〉, where ξ̌j = vj + ξj , j = 1, . . . , r. It follows that d((ρ1)∗ω) = (ρ1)∗(dω). We identify ∧T ∗ with its image in ∧(J̌∞T )∗ by (ρ1)∗, and we write simply ω instead of (ρ1)∗ω. Let be u = ω ⊗ ξ̌ ∈ ∧(J̌∞T )∗ ⊗ (J̌∞T ), τ ∈ ∧(J̌∞T )∗, with deg ω = r and deg τ = s. We also define deg u = r. Then we define the derivation of degree (r − 1) i(u) : ∧s(J̌∞T )∗ → ∧s+r−1(J̌∞T )∗ by i(u)τ = i(ω ⊗ ξ̌)τ = ω ∧ i(ξ̌)τ (2.25) and the Lie derivative θ(u) : ∧r(J̌∞T )∗ → ∧r+s(J̌∞T )∗ by θ(u)τ = i(u)dτ + (−1)rd(i(u)τ), (2.26) which is a derivation of degree r. If v = τ ⊗ η̌, we define [u,v] = [ω ⊗ ξ̌, τ ⊗ η̌] = ω ∧ τ ⊗ [[ξ̌, η̌]]∞ + θ(ω ⊗ ξ̌)τ ⊗ η̌ − (−1)rsθ(τ ⊗ η̌)ω ⊗ ξ̌. (2.27) 16 J.M.M. Veloso A straightforward calculation shows that: [u, τ ⊗ η̌] = θ(u)τ ⊗ η̌ + (−1)rsτ ∧ [u, η̌]− (−1)rs+sdτ ∧ i(η̌)u, (2.28) where i(η̌)u = i(η̌)(ω ⊗ ξ̌) = i(η̌)ω ⊗ ξ̌. We verify that [u,v] = −(−1)rs[v,u] and [u, [v,w]] = [[u,v],w] + (−1)rs[v, [u,w]], (2.29) where deg u = r, deg v = s. With this bracket, ∧(J̌∞T )∗ ⊗ (J̌∞T ) is a Lie algebra sheaf. Furthermore, if [θ(u), θ(v)] = θ(u)θ(v)− (−1)rsθ(v)θ(u) (2.30) then [θ(u), θ(v)] = θ([u,v]). (2.31) In particular, we have the following formulas: Proposition 2.9. If u,v ∈ ∧1(J̌∞T )∗ ⊗ (J̌∞T ), ω ∈ ∧1(J̌∞T )∗, ξ̌, η̌ ∈ J̌∞T , then: (i) 〈θ(u)ω, ξ̌ ∧ η̌〉 = θ(i(ξ̌)u)〈ω, η̌〉 − θ(i(η̌)u)〈ω, ξ̌〉 − 〈ω, [[i(ξ̌)u, η̌]]∞ + [[ξ̌, i(η̌)u]]∞ − i([[ξ̌, η̌]]∞)u〉, (ii) i(ξ̌)[u, η̌] = [[i(ξ̌)u, η̌]]∞ − i([[ξ̌, η̌]]∞)u, (iii) 〈[u,v], ξ̌ ∧ η̌〉 = [[i(ξ̌)u, i(η̌)v]]∞ − [[i(η̌)u, i(ξ̌)v]]∞ − i([[i(ξ̌)u, η̌]]∞ − [[i(η̌)u, ξ̌]]∞ − i([[ξ̌, η̌]]∞)u)v − i([[i(ξ̌)v, η̌]]∞ − [[i(η̌)v, ξ̌]]∞ − i([[ξ̌, η̌]]∞)v)u. Proof. It is a straightforward calculus applying the definitions. � If we define the groupoid Q∞ = lim proj Qk, then for σ = lim proj σk ∈ Q∞, we obtain, from (2.21), σ∗ : J̌∞T → J̌∞T , ξ = v + lim proj ξk 7→ σ∗ξ = lim proj (σk+1)∗(v + ξk), so the action Q∞ × J̌∞T → J̌∞T , (σ, ξ) 7→ σ∗ξ, is well defined. It follows from (2.22) that σ∗ : J̌∞T → J̌∞T is an automorphism of Lie algebra sheaf. Given σ ∈ Q∞, σ acts on ∧(J̌T )∗: σ∗ : ∧(J̌T )∗ → ∧(J̌T )∗, ω 7→ σ∗ω, Isomorphism of Intransitive Linear Lie Equations 17 where, if ω is a r-form, 〈σ∗ω, ξ̌1 ∧ · · · ∧ ξ̌r〉 = 〈ω, σ−1 ∗ (ξ̌1) ∧ · · · ∧ σ−1 ∗ (ξ̌r)〉. (2.32) Consequently, Q∞ acts on ∧(J̌∞T )∗ ⊗ (J̌T ): Q∞ × (∧(J̌∞T )∗ ⊗ (J̌∞T )) → ∧(J̌∞T )∗ ⊗ (J̌∞T ), (σ,u) 7→ σ∗u, where σ∗u = σ∗(ω ⊗ ξ̌) = σ∗(ω)⊗ σ∗(ξ̌). (2.33) The action of σ∗ is an automorphism of the Lie algebra sheaf ∧(J̌∞T )∗ ⊗ (J̌∞T ), i.e., [σ∗u, σ∗v] = σ∗[u,v]. 2.4 The first non-linear Spencer complex In this subsection we will study the subsheaf ∧T ∗ ⊗ J∞T and introduce linear and non-linear Spencer complexes. Principal references are [16, 17, 14]. Proposition 2.10. The sheaf ∧T ∗⊗J∞T is a Lie algebra subsheaf of ∧(J̌∞T )∗⊗ (J̌∞T ), and [ω ⊗ ξ, τ ⊗ η] = ω ∧ τ ⊗ [[ξ, η]]∞, where ω, τ ∈ ∧T ∗, ξ, η ∈ J∞T . Proof. Let be u = ω⊗ξ ∈ ∧T ∗⊗J∞T . For any τ ∈ ∧T ∗, i(ξ)τ = 0, then, by applying (2.25) we obtain i(u)τ = 0, and by (2.26), θ(u)τ = 0. So (2.27) implies [ω⊗ξ, τ ⊗η] = ω∧τ ⊗ [[ξ, η]]∞. � Let be the fundamental form χ ∈ (J̌∞T )∗ ⊗ (J̌∞T ) defined by i(ξ̌)χ = (ρ1)∗(ξ̌) = v, where ξ̌ = v + ξ ∈ T ⊕ J∞T . In another words, χ is the projection of J̌∞T on T , parallel to J∞T . If u = limuk, we define Du = lim Duk. Proposition 2.11. If ω ∈ ∧T ∗, and u ∈ ∧T ∗ ⊗ J∞T , then: (i) θ(χ)ω = dω; (ii) [χ, χ] = 0; (iii) [χ,u] = Du. Proof. Let be ξ̌ = v + ξ, η̌ = w + η ∈ T ⊕ J∞T . (i) As θ(χ) is a derivation of degree 1, it is enough to prove (i) for 0-forms f and 1-forms ω ∈ (J̌∞T )∗. From (2.26) we have θ(χ)f = i(χ)df = df . It follows from Proposition 2.9(i) that 〈θ(χ)ω, ξ̌ ∧ η̌〉 = θ(v)〈ω, η̌〉 − θ(w)〈ω, ξ̌〉 − 〈ω, [[v, η̌]]∞ + [[ξ̌, w]]∞ − χ([[ξ̌, η̌]]∞)〉 = θ(v)〈ω, w〉 − θ(w)〈ω, v〉 − 〈ω, [v, w] + [v, w]− [v, w]〉 = 〈dω, ξ̌ ∧ η̌〉. 18 J.M.M. Veloso (ii) By applying Proposition 2.9(iii), we obtain 〈1 2 [χ, χ], ξ̌ ∧ η̌〉 = [v, w]− i([[v, η̌]]∞ − [[w, ξ̌]]∞ − ρ1[[ξ̌, η̌]]∞)χ = [v, w]− ([v, w]− [w, v]− [v, w]) = 0. (iii) It follows from (2.28) that, for u = ω ⊗ ξ, [χ,u] = θ(χ)ω ⊗ ξ + (−1)rω ∧ [χ, ξ]− (−1)2rdω ∧ i(ξ)χ = dω ⊗ ξ + (−1)rω ∧ [χ, ξ]. As D is characterized by Proposition 2.3, it is enough to prove [χ, ξ] = Dξ. It follows from Propositions 2.5(ii) and 2.9(ii) that i(η̌)[χ, ξ] = [[i(η̌)χ, ξ]]∞ − i([[η̌, ξ]]∞)χ = [[w, ξ]]∞ = i(η̌)Dξ. � If u,v ∈ ∧T ∗ ⊗ J∞T , with deg u = r, deg v = s, then we get from (2.29) and Proposi- tion 2.11(iii) that D[u,v] = [Du,v] + (−1)r[u, Dv], and [χ, [χ,u]] = [[χ, χ],u]− [χ, [χ,u]] = −[χ, [χ,u]]. Therefore, D2u = 0, or D2 = 0. Then the first linear Spencer complex, 0 → T j∞→ J∞T D→ T ∗ ⊗ J∞T D→ ∧2T ∗ ⊗ J∞T D→ · · · D→ ∧mT ∗ ⊗ J∞T → 0, where dim T = m, is well defined. This complex projects on 0 → T jk → JkT D→ T ∗ ⊗ Jk−1T D→ ∧2T ∗ ⊗ Jk−2T D→ · · · D→ ∧mT ∗ ⊗ Jk−mT → 0, and is exact (see [16, 17, 14]). Let be γk the kernel of πk : JkT → Jk−1T . Denote by δ the restriction of D to γk. It follows from Proposition 2.3(ii) that δ is OM -linear and δ : γk → T ∗ ⊗ γk−1. This map is injective, in fact, if ξ ∈ γk, then by (2.11), δξ = −λ1(ξ) is injective. As i(v)D(i(w)Dξ)− i(w)D(i(v)Dξ)− i([v, w])Dπk−1ξ = 0, for v, w ∈ T , ξ ∈ γk ⊂ JkT , we obtain that δ is symmetric, i(v)δ(i(w)δξ) = i(w)δ(i(v)δξ). Observe that we get the map ι : γk → S2T ∗ ⊗ γk−2 defined by i(v, w)ι(ξ) = i(w)δ(i(v)δξ), and, if we go on, we obtain the isomorphism γk ∼= SkT ∗ ⊗ J0T, where, given a basis e1, . . . , em ∈ T with the dual basis e1, . . . , em ∈ T ∗, we obtain the basis fk1,k2,...,km l = 1 k1!k2! · · · km! (e1)k1(e2)k2 · · · (em)km ⊗ j0el (2.34) Isomorphism of Intransitive Linear Lie Equations 19 of SkT ∗ ⊗ J0T , where k1 + k2 + · · ·+ km = k, k1, . . . , km ≥ 0 and l = 1, . . . ,m. In this basis δ ( fk1,k2,...,km l ) = − m∑ i=1 ei ⊗ f k1,...,ki−1,ki−1,ki+1,...,km l . From the linear Spencer complex, we obtain the exact sequence of morphisms of vector bundles 0 → γk δ→ T ∗ ⊗ γk−1 δ→ ∧2T ∗ ⊗ γk−2 δ→ · · · δ→ ∧mT ∗ ⊗ γk−m → 0. (2.35) Let’s now introduce the first non-linear Spencer operator D. The “finite” form D of the linear Spencer operator D is defined by Dσ = χ− σ−1 ∗ (χ), (2.36) where σ ∈ Q∞. Proposition 2.12. The operator D take values in T ∗ ⊗ J∞T , so D : Q∞ → T ∗ ⊗ J∞T , and i(v)(Dσ)k = λ1σ−1 k+1.j 1σk.v − v, (2.37) where σ = lim proj σk ∈ Q∞. Proof. By applying (2.32) and (2.33), it follows that for ξ ∈ J∞T , i(ξ)Dσ = i(ξ)χ− σ−1 ∗ (i(σ∗(ξ))χ) = 0, and for v ∈ T , i(v)Dσ = i(v)χ− σ−1 ∗ (i(σ∗(v))χ) = v − σ−1 ∗ (f∗v), where f = β ◦ σ. By Proposition 2.8(iii), i(v)(Dσ)k = v − ( f−1 ∗ (f∗v) + j1σ−1 k .f∗v.λ1σk+1 − λ1σ−1 k+1.f∗v.λ1σk+1 ) . (2.38) By posing v = d dtxt|t=0, we obtain j1σk.v.j1σ−1 k = d dt ( σk(xt).σ−1 k (f(xt)) )∣∣ t=0 = d dt f(xt) ∣∣ t=0 = f∗v, (2.39) and by replacing (2.39) in (2.38), we get i(v)(Dσ)k = −j1σ−1 k . ( j1σk.v.j1σ−1 k ) .λ1σk+1 + λ1σ−1 k+1. ( j1σk.v.j1σ−1 k ) .λ1σk+1 = ( λ1σ−1 k+1.j 1σk.v − v ) .j1σ−1 k .λ1σk+1 = ( λ1σ−1 k+1.j 1σk.v − v ) .σ−1 k .σk = λ1σ−1 k+1.j 1σk.v − v, since that λ1σ−1 k+1.j 1σk.v − v is α-vertical (cf. (2.7)). � Corollary 2.2. We have Dσ = 0 if and only if σ = j∞(βσ), where β : Q∞ → Diff M . 20 J.M.M. Veloso Corollary 2.3. If σk+1 ∈ Qk+1, then (σk+1)∗(v) = f∗v + (σk+1)∗(i(v)Dσk+1), for v ∈ T . Proof. It follows from (2.21) and Proposition 2.12 that (σk+1)∗(i(v)Dσk+1) = λ1σk+1.(i(v)Dσk+1).λ1σ−1 k+1 = λ1σk+1. ( λ1σ−1 k+1.j 1σk.v − v ) .λ1σ−1 k+1 = j1σk.v.λ1σ−1 k+1 − λ1σk+1.v.λ1σ−1 k+1 = (σk+1)∗(v)− f∗v. � Proposition 2.13. The operator D has the following properties: (i) If σ, σ′ ∈ Q∞, D(σ′ ◦ σ) = Dσ + σ−1 ∗ (Dσ′). In particular, Dσ−1 = −σ∗(Dσ). (ii) If σ ∈ Q∞, u ∈ ∧T ∗ ⊗ J∞T , D(σ−1 ∗ u) = σ−1 ∗ (Du) + [ Dσ, σ−1 ∗ u ] . (iii) If ξ = d dtσt|t=0, with ξ ∈ J∞T , and σt ∈ Q∞ is the 1-parameter group associated to ξ, then Dξ = d dt Dσt ∣∣ t=0 . Proof. (i) D(σ′ ◦ σ) = χ− σ−1 ∗ (χ) + σ−1 ∗ (χ− (σ′)−1 ∗ (χ)) = Dσ + σ−1 ∗ (Dσ′). (ii) D(σ−1 ∗ u) = σ−1 ∗ [σ∗χ,u] = σ−1 ∗ [χ−Dσ−1,u] = σ−1 ∗ (Du)− [σ−1 ∗ (Dσ−1), σ−1 ∗ u] = σ−1 ∗ (Du) + [Dσ, σ−1 ∗ u]. (iii) d dt Dσt ∣∣ t=0 = − d dt (σ−1 t )∗(χ) = −[ξ, χ] = Dξ. � Proposition 2.12 says that D is projectable: D : Qk+1 → T ∗ ⊗ JkT , σk+1 7→ Dσk+1, where i(v)Dσk+1 = λ1σ−1 k+1.j 1σk.v − v. It follows from [χ, χ] = 0 that 0 = σ−1 ∗ ([χ, χ]) = [σ−1 ∗ (χ), σ−1 ∗ (χ)] = [χ−Dσ, χ−Dσ] = [Dσ,Dσ]− 2D(Dσ), therefore D(Dσ)− 1 2 [Dσ,Dσ] = 0. (2.40) Isomorphism of Intransitive Linear Lie Equations 21 If we define the non-linear operator D1 : T ∗ ⊗ J∞T → ∧2T ∗ ⊗ J∞T , u 7→ Du− 1 2 [u,u], then we can write (2.40) as D1D = 0. The operator D1 projects in order k to D1 : T ∗ ⊗ JkT → ∧2T ∗ ⊗ Jk−1T , where D1u = Du− 1 2 [[u,u]]k. Here [[u,u]]k denotes the analogous of formulas (2.27) and (2.28) projected in the order k, so that the extension of first bracket makes sense. We will leave the details to the reader. We define the first non-linear Spencer complex by 1 → Diff M jk+1 → Qk+1 D→ T ∗ ⊗ JkT D1→ ∧2T ∗ ⊗ Jk−1T , which is exact in Qk+1. It is possible to define the first nonlinear Spencer complex D for invertible sections of Q∞(M,M ′) by: Dσ = χ− σ−1 ∗ (χ′), (2.41) where σ ∈ Q∞(M,M ′) and χ′ ∈ (J̌∞T ′)∗ ⊗ (J̌∞T ′) is the fundamental form. The operator D take values in T ∗ ⊗ J∞T , so D : Q∞(M,M ′) → T ∗ ⊗ J∞T , and the same formula of Proposition 2.12 holds: i(v)(Dσ)k = λ1σ−1 k+1.j 1σk.v − v, where σ = lim proj σk ∈ Q∞(M,M ′). Other properties can easily be generalised. 3 Partial connections In this section we will develop the concept of partial connections or partial covariant derivatives associated with the vector bundle H ⊕ JkV in the directions of the distribution V ⊂ T . We thank the referee for pointing out that this concept is already in [15, p. 24]. The construction of connections for JkT , the transitive case, is in [7]. Let be V an involutive subvector bundle of T , a ∈ M , and N a (local) submanifold of M such that TaN ⊕ Va = TaM . Then there exists a coordinate system (x, y) in a neighborhood of a such that a = (0, 0), the submanifolds given by points with coordinates x constant are integral submanifolds of V , and N is given by the submanifold y = 0. At least locally, we can suppose that the integral manifolds of V are the fibers of a submersion ρ : M → N . In the coordinates (x, y), we get ρ(x, y) = (x, 0). If we denote by H the subvector bundle of T given by 22 J.M.M. Veloso vectors tangent to the submanifolds defined by y constant, then H is involutive and T = H⊕V . Also, T ∗ = H∗ ⊕ V ∗. We denote by H and V the sheaves of germs of H and V , respectively. We denote by Qk V the subgroupoid of Qk whose elements are the k-jets of local diffeomor- phisms h of M which are ρ-projectable on the identity of N . In the coordinates (x, y), h(x, y) = (x, h2(x, y)). The sheaf of germs of invertible local α-sections of Qk V will be denoted by Qk V . The algebroid associated with Qk V is JkV , and we denote by JkV the sheaf of germs of local sections of JkV . Then the first non-linear Spencer operator D can be restricted to Qk+1 V , D : Qk+1 V → T ∗ ⊗ JkV, (3.1) and the linear Spencer operator D can be restricted to Jk+1V, D : Jk+1V → T ∗ ⊗ JkV. A vector u ∈ T decomposes in u = uH +uV , uH ∈ H, uV ∈ V . If d is the exterior differential, we get the decomposition d = dH + dV . The fundamental form χ decomposes in χ = χH + χV , where χH(u) = χ(uH) and χV (u) = χ(uV ). The linear Spencer operator D also decomposes in D = DH + DV , and it follows from Proposition 2.11(iii) that DHξ = [χH , ξ] and DV ξ = [χV , ξ], for ξ ∈ JkT . Proposition 3.1. If ω ∈ ∧V∗, and u ∈ ∧V∗ ⊗ J∞T , then: (i) θ(χV )ω = dV ω; (ii) [χV , χV ] = 0; (iii) [χV ,u] = DV u. Proof. Let be ξ̌ = v + ξ, η̌ = w + η ∈ T ⊕ J∞T , v = vH + vV , w = wH + wV ∈ H ⊕ V. (i) As θ(χV ) is a derivation of degree 1, it is enough to prove (i) for 0-forms f and 1-forms ω ∈ V∗. From (2.26) we have θ(χV )f = i(χV )df = dV f. It follows from Proposition 2.9(i) that 〈θ(χV )ω, ξ̌ ∧ η̌〉 = θ(vV )〈ω, η̌〉 − θ(wV )〈ω, ξ̌〉 − 〈ω, [[vV , η̌]]∞ + [[ξ̌, wV ]]∞ − χV ([[ξ̌, η̌]]∞)〉 = θ(vV )〈ω, wV 〉 − θ(wV )〈ω, vV 〉 − 〈ω, [vV , w]V + [v, wV ]V − [v, w]V 〉 = 〈dω, vV ∧ wV 〉 = 〈dV ω, ξ̌ ∧ η̌〉, since that [vV , w]V + [v, wV ]V − [v, w]V = [vV , wV ] + [vV , wH ]V + [vH , wV ]V + [vV , wV ]− [vH , wV ]V − [vV , wH ]V − [vV , wV ] = [vV , wV ]. (ii) By applying Proposition 2.9(iii), we obtain 〈1 2 [χV , χV ], ξ̌ ∧ η̌〉 = [vV , wV ]− i([[vV , η̌]]∞ − [[wV , ξ̌]]∞ − (ρ1)∗[[ξ̌, η̌]]∞)χV = [vV , wV ]− ([vV , w]− [wV , v]− [v, w])V = 0. Isomorphism of Intransitive Linear Lie Equations 23 (iii) It follows from (2.28) that [χV ,u] = θ(χV )ω ⊗ ξ + (−1)rω ∧ [χV , ξ]− (−1)2rdω ∧ i(ξ)χV = dV ω ⊗ ξ + (−1)rω ∧ [χV , ξ]. It is enough to prove [χV , ξ] = DV ξ. It follows from Propositions 2.5(ii) and 2.9(ii) that i(η̌)[χV , ξ] = [[i(η̌)χV , ξ]]∞ − i([[η̌, ξ]]∞)χV = [[wV , ξ]]∞ = i(η̌)DV ξ. � From item (ii) of this proposition, (2.30) and (2.31) we obtain d2 V = 0. (3.2) The first non-linear Spencer operator D also decomposes naturally in D = DH + DV , where i(u)DH = i(uH)D and i(u)DV = i(uV )D. We obtain: Proposition 3.2. If F ∈ Qk+1 V is such that f = βF satisfies f∗H = H, then DHF = χH − F−1 ∗ (χH), and DV F = χV − F−1 ∗ (χV ). Proof. From hypothesis, we get f∗(uH) = (f∗u)H and f∗(uV ) = (f∗u)V , and from equa- tion (2.36), we get i(u)DHF = i(uH)DF = i(uH)χ− F−1 ∗ (i(f∗(uH))χ) = i(uH)χ− F−1 ∗ (i((f∗u)H)χ) = i(u)χH − F−1 ∗ (i(f∗u)χH) = i(u)(χH − F−1 ∗ (χH)), so DHF = χH − F−1 ∗ (χH). The proof of the second formula is analogous. � If we apply Proposition 3.2 to GF , we get DV (GF ) = χV − F−1 ∗ G−1 ∗ (χV ) = χV + F−1 ∗ (DV G− χV ) = DV F + F−1 ∗ (DV G), and if we pose F = G−1, we get DV G−1 = −G∗(DV G). Definition 3.1. A partial connection ∇ on H ⊕ JkV is a R-linear map ∇ : H⊕ JkV → V∗ ⊗ (H⊕ JkV) such that ∇(f ξ̌) = (dV f)⊗ ξ̌ + f∇ξ, for f ∈ OM , ξ̌ ∈ H ⊕ JkV. We extend ∇ to ∧V ∗ ⊗ (H ⊕ JkV ) by ∇(α⊗ ξ̌) = dV α⊗ ξ̌ + (−1)|α|α ∧∇ξ̌, where α ∈ ∧V∗ and |α| is the degree of α. 24 J.M.M. Veloso It follows from (3.2) that ∇2(α⊗ ξ̌) = α ∧∇2ξ̌, so ∇2 : H⊕ JkV → ∧2V∗ ⊗ (H⊕ JkV) is a tensor, called the curvature tensor of the partial connection ∇. If ∇2 = 0 we say that ∇ is flat. Let be ω ∈ V∗ ⊗ Jk+1V such that β∗(i(w)ω) = w, for w ∈ V, and ω̃ = χV + ω, so ω̃ ∈ V∗ ⊗ Jk+1Ṽ, where Jk+1Ṽ = Jk+1T̃ ∩ (T ⊕ Jk+1V). In the sequel, for ξ̌ ∈ H ⊕ JkV, we denote by [ω̃, ξ̌] k and [ω̃, ω̃]k+1 the analogous of formu- las (2.27) and (2.28) projected in the order k and k+1, respectively, so that the same construction of third and second bracket make sense. We will use this convention in the present section when it makes sense, and leave the details to the reader. Proposition 3.3. The operator ∇ defined by ∇ξ̌ = [ω̃, ξ̌] k, for ω̃ ∈ V∗ ⊗ Jk+1Ṽ, ξ̌ = u + ξ ∈ H ⊕ JkV is a partial connection on H ⊕ JkV with curvature ∇2 = 1 2 [ω̃, ω̃]k+1. Proof. If we apply formula (ii) of Proposition 2.9, we obtain i(η̌) [ω̃, ξ̌] k = [i(η̌)ω̃, ξ̌] k − i([[η̌k+1, ξ̌k+1]]k+1)πkω̃, where η̌ = v+η ∈ T ⊕JkT , and η̌k+1, ξ̌k+1 ∈ T ⊕Jk+1T projects on η̌, ξ̌, respectively. If v ∈ H, the right side is 0, and if v ∈ V i(η̌) [ω̃, ξ̌] k = [v, u]H + [i(v)ω̃, ξ] k − i([v, u])πkω − i(u)D(i(v)ω) ∈ H ⊕ JkV. Then ∇ξ̌ ∈ V∗ ⊗ JkV. Also, i(η̌) [ω̃, f ξ̌] k = [i(η̌)ω̃, f ξ̌] k − i([[η̌k+1, f ξ̌k+1]]k+1)πkω̃ = vV (f)ξ̌ + f [i(η̌)ω̃, ξ̌] k − i(v(f)ξ̌ + f [[η̌k+1, ξ̌k+1]]k+1)πkω̃ = vV (f)ξ̌ + f [i(η̌)ω̃, ξ̌] k − i([[η̌k+1, ξ̌k+1]]k+1)(fπkω̃) = vV (f)ξ̌ + i(η̌)(f [ω̃, ξ̌] k) = i(η̌)(dV f)ξ̌ + i(η̌)(f [ω̃, ξ̌] k), so ∇(f ξ̌) = dV f ⊗ ξ̌ + f∇ξ̌. If α⊗ ξ̌ ∈ ∧V∗ ⊗ (H⊕ JkV), we know from (2.28) that [ω̃, α⊗ ξ̌] k = θ(ω̃)α⊗ ξ̌ + (−1)|α|α ∧ [ω̃, ξ̌] k − (−1)2|α|dα ∧ i(ξ̌)ω̃, and from (2.26) that θ(ω̃)α = i(ω̃)dα− d(i(ω̃)α) = i(χV )dα− d(i(χV )α) = θ(χV )α = dV α, and as i(ξ̌)ω̃ = 0, then [ω̃, α⊗ ξ̌] k = dV α⊗ ξ̌ + (−1)|α|α ∧ [ω̃, ξ̌] k = dV α⊗ ξ̌ + (−1)|α|α ∧∇ξ̌, Isomorphism of Intransitive Linear Lie Equations 25 so ∇(α⊗ ξ̌) = [ω̃, α⊗ ξ̌] k. Therefore, ∇2ξ̌ = [ω̃, [ω̃, ξ̌] k] k = [[ω̃, ω̃]k+1, ξ̌] k −∇2ξ̌, and from this it follows that ∇2ξ̌ = 1 2 [[ω̃, ω̃]k+1, ξ̌] k . � Let be σy : N → Qk+1 V a family of differentiable sections such that σy(x, 0) ∈ Qk+1 V ((x, 0), (x, y)), (3.3) with σ0(x, 0) = jk+1 (x,0)id. Then σ : M → Qk+1 V , given by σ(x, y) = σy(x, 0), is a differentiable β-section of Qk+1 V |N = {X ∈ Qk+1 V : α(X) ∈ N}. If v(x, y) = d dt(x, y(t))|t=0, define ω ∈ V∗ ⊗ Jk+1V by i(v(x, y))ω = d dt ( σy(t)(x, 0)σy(x, 0)−1 )∣∣ t=0 ∈ Jk+1 (x,y)V. (3.4) Proposition 3.4. The partial connection ∇ defined by ω̃ = χV + ω, where ω is defined as in (3.4), is flat. Proof. First of all, β∗(i(v(x, y))ω) = d dt β ( σy(t)(x, 0)σy(x, 0)−1 )∣∣ t=0 = d dt (x, y(t))|t=0 = v(x, y), so ω satisfies the condition to define a partial connection. We will show that ω̃ satisfies [ω̃, ω̃]k+1 = 0, i.e., the partial connection ∇ defined by ω̃ is flat. Let be, for v ∈ V, i(v)ω the right invariant vector field defined by i(v)ω(X) = i(v)ω(β(X)).X, where X ∈ Qk+1 V . We prove that i(v)ω is tangent to σ(M), which follows from i(v)ω(σ(x, y)) = i(v(x, y))ω.σ(x, y) = d dt [ σy(t)(x, 0)σy(x, 0)−1 ] t=0 .σ(x, y) = d dt [ σy(t)(x, 0)σy(x, 0)−1σ(x, y) ] t=0 = d dt [ σy(t)(x, 0) ] t=0 ∈ T (σ(M)). To finish, we know that, for v, w ∈ V, i(v)ω, i(w)ω and i([v, w])ω are tangent to the submani- fold σ(M), and, as β∗([i(v)ω, i(w)ω] = [v, w], it follows that [i(v)ω, i(w)ω] = i([v, w])ω, so from Proposition 2.6, [i(v)ω̃, i(w)ω̃]k+1 = i([v, w])ω̃. From Proposition 2.9(iii) we obtain i(v ∧ w) (1 2 [ω̃, ω̃]k+1 ) = [i(v)ω̃, i(w)ω̃]k+1 − i ( [[i(v)ω̃k+2, w]]k+2 − [[i(w)ω̃k+2, v]]k+2 − i([v, w])ω̃ ) ω̃ = [i(v)ω̃, i(w)ω̃]k+1 − i ([v, w]− [w, v]− [v, w]) ω̃ = 0, where ωk+2 is a section in Jk+2V that projects on ω. The proposition is proved. � 26 J.M.M. Veloso Therefore, given a section u + ξ : N → (H ⊕ JkV )|N , there exists only one section (U + Ξ) ∈ H⊕JkV such that (U +Ξ)|N = u+ξ and ∇(U +Ξ) = 0. The following proposition characterizes these sections: Proposition 3.5. Let be (U + Ξ) ∈ H ⊕ JkV such that (U + Ξ)|N = u + ξ : N → (H ⊕ JkV )|N , and ∇(U + Ξ) = 0. Then (U + Ξ)(x, y) = σy ∗((u + ξ)(x, 0)). (3.5) Proof. Choose v ∈ V such that v is H-projectable, i.e., the 1-parameter group ft of v is given in coordinates by ft(x, y) = (x, ht(y)). If we define Ft(x, y) = σht(y)(x, 0)σy(x, 0)−1, then Ft ∈ Qk+1 V . Furthermore, βFt = ft, and (FsFt)(x, y) = ( σhs(ht(y))(x, 0)σht(y)(x, 0)−1 )( σht(y)(x, 0)σy(x, 0)−1 ) = σhs+t(y)(x, 0)σy(x, 0)−1 = Fs+t(x, y). So, Ft is the 1-parameter group such that d dtFt|t=0 = ω̃(v). If U + Ξ is defined by (3.5), we get ((Ft)∗(U + Ξ))(x, y) = (Ft)∗((U + Ξ)(x, h−t(y))) = (Ft)∗(σ h−t(y) ∗ ((u + ξ)(x, 0))) = ( σht(h−t(y)) ( σh−t(y) )−1) ∗ ( σh−t(y) ) ∗((u + ξ)(x, 0)) = σy ∗((u + ξ)(x, 0)) = (U + Ξ)(x, y), so [i(v)ω̃, U + Ξ] k = 0. Also, (U + Ξ)(x, 0) = σ0 ∗((u + ξ)(x, 0)) = (jk+1 (x,0)id)∗((u + ξ)(x, 0)) = (u + ξ)(x, 0). Let ū ∈ H be the vector field ρ-projectable such that ū|N = u. Then [ū, v] = 0, and from (3.5) and Corollary 2.3 we get U(x, y) = ū(x, y) + σy ∗ (i(u(x, 0))Dσy) ∈ H ⊕ JkV, so i(v)∇(U + Ξ) = [i(v)ω̃, U + Ξ] k − i([v, ū])πkω̃ = 0. � We will now verify how a partial connection defined by ω̃ changes. Let M ′, V ′, a′, N ′, (x′, y′), H ′, ρ′, be as above, with the same properties and dimensions. Denote by T ′ = TM ′. Let be φ : N → N ′ a (local) diffeomorphism, with φ(a) = a′, and denote by Qk φ the submanifold of Qk(M,M ′) of k-jets of local diffeomorphisms τ : M → M ′ such that ρ′τ = φρ. This means τ(x, y) = (φ(x), b(x, y)). Let be Qk φ the sheaf of germs of invertible local sections of Qk φ. Then, by restriction of action (2.23), there exists an action of Qk+1 φ (similar to (2.21)) on T ⊕ JkV : Qk+1 φ × (T ⊕ JkV) → T ′ ⊕ JkV ′, (σk+1, v + ξk) 7→ (σk+1)∗(v + ξk). The operator D : Qk+1(M,M ′) → T ∗ ⊗ JkT defined in (2.41) restricts, as (3.1), to D : Qk+1 φ → T ∗ ⊗ JkV, and as above D decomposes in D = DH +DV . The analogous of Proposition 3.2 holds: Isomorphism of Intransitive Linear Lie Equations 27 Proposition 3.6. If F ∈ Qk+1 φ is such that f = βF satisfies f∗H = H ′, then DHF = χH − F−1 ∗ (χH′), and DV F = χV − F−1 ∗ (χV ′). Denote by Qk+1 φ−1 = { Φ ∈ Qk+1(M ′,M) : Φ−1 ∈ Qk+1 φ } . If F ∈ Qk+1 φ satisfy for f = βF , f∗(H) = H ′, then DV ′F −1 = χV ′ − F∗(χV ) (3.6) and if G ∈ Qk+1 φ−1 satisfy α(G) = β(F ), and for g = βG we have g∗(H ′) = H, then by applying Propositions 3.2 and 3.6 to FG we get DV ′(FG) = χV ′ −G−1 ∗ F−1 ∗ (χV ′) = χV ′ + G−1 ∗ (DV F − χV ) = DV ′G + G−1 ∗ (DV F ). By posing G = F−1 we get DV ′F −1 = −F∗(DV F ). (3.7) Choose Φ ∈ Qk+1 φ with ϕ = π0Φ satisfying ϕ(N) = N ′ and ϕ∗(H) = H ′. Then ϕ|N = φ and (x′, y′) = ϕ(x, y) = (φ(x), b(y)). Define σ′y ′ as σ′y ′ (x′, 0) = Φ(x, y)σy(x, 0)Φ(x, 0)−1, (3.8) and let be ω′ and ∇′ as in (3.4) and Proposition 3.3, respectively. Following the proof of Proposition 3.5, take v ∈ V such that v is H-projectable, i.e., the 1-parameter group ft of v is given in coordinates by ft(x, y) = (x, ht(y)). Define Ft ∈ Qk+1 V by Ft(x, y) = σht(y)(x, 0)σy(x, 0)−1. If v′ = ϕ∗v, then v′ ∈ V ′ is H ′-projectable, f ′t = ϕftϕ −1 is the associated 1-parameter group of v′, and the 1-parameter group associated with ω′(v′) satisfies F ′ t′(x ′, y′) = σ′ht′ (y ′)(x′, 0)σy′(x′, 0)−1 = ( Φ(x, ht(y))σht(y)(x, 0)Φ(x, 0)−1 )( Φ(x, y)σy(x, 0)Φ(x, 0)−1 )−1 = Φ(x, ht(y))Ft(x, y)Φ(x, y)−1 = (ΦFtΦ−1)(x′, y′), i.e., F ′ t = ΦFtΦ−1. From this, we get i(v′)ω̃′ = Φ∗(i(v)ω̃), and i(v′)(Φ∗ω̃) = Φ∗(i(Φ−1 ∗ (v′))ω̃) = Φ∗(i(f−1 ∗ (v′))ω̃) = Φ∗(i(v)ω̃) = i(v′)ω̃′, so Φ∗ω̃ = ω̃′. (3.9) Then ∇′(Φ∗(U + Ξ)) = [ω̃′,Φ∗(U + Ξ)] k = Φ∗ [ω̃, U + Ξ] k, 28 J.M.M. Veloso i.e., ∇′Φ∗ = Φ∗∇, (3.10) which shows that Φ∗(U + Ξ) is parallel with respect to ∇′ if and only if U + Ξ is parallel with respect to ∇. Taking account of (3.6), the equation (3.9) projected in order k is equivalent to, πk(χV ′ + ω′) = πk(Φ∗(χV + ω)) = Φ∗(χV ) + Φ∗(πkω) = ( χV ′ −DV ′Φ−1 ) + Φ∗(πkω), or, considering (3.7), χV ′ + πkω ′ = χV ′ + Φ∗(DV Φ) + Φ∗(πkω), so πkω ′ = Φ∗(DV Φ + πkω). (3.11) 4 Linear Lie equations Definition 4.1. Let be Rk a subvector bundle of JkT . We define the prolongation Rk+1 of Rk by Rk+1 = (λ1)−1(J1Rk ∩ λ1(Jk+1T )) ⊂ Jk+1T, where the intersection is done in J1JkT . We denote the prolongation (Rk+1)+1 of Rk+1 by Rk+2 and so on, and by Rk+l the sheaf of germs of local sections of Rk+l, for l ≥ 0. Proposition 4.1. A section ξ ∈ Jk+1T is in Rk+1 if and only if πkξ ∈ Rk and Dξ ∈ T ∗⊗Rk. Proof. From Definition 4.1 and (2.12), since j1πkξ ∈ J1Rk. � Definition 4.2. A subvector bundle Rk of JkT is a linear Lie equation if the prolongation Rk+1 of Rk is a subvector bundle of Jk+1T such that (i) πk(Rk+1) = Rk; (ii) [[Rk+1,Rk+1]]k+1 ⊂ Rk. It follows from Proposition 4.1 and Definition 4.2 that [[T ⊕Rk+1, T ⊕Rk+1]]k+1 ⊂ T ⊕Rk, and from this, [R̃k, R̃k]k ⊂ R̃k, (4.1) and [R̃k+1,Rk] k ⊂ Rk. (4.2) Proposition 4.2. If Rk ⊂ JkT is a linear Lie equation, then [[T ⊕Rk+l, T ⊕Rk+l]]k+l ⊂ T ⊕Rk+l−1, for l ≥ 2. Isomorphism of Intransitive Linear Lie Equations 29 Proof. Let’s prove this for l = 2. The other proofs for l > 2 are equal. Suppose ξk+2, ηk+2 ∈ Rk+2. Then, by Proposition 4.1, ξk+1 = πk+1ξk+2, ηk+1 = πk+1ηk+2 ∈ Rk+1, and Dξk+2, Dηk+2 ∈ T ⊗Rk+1. So, πk[[ξk+2, ηk+2]]k+2 = [[ξk+1, ηk+1]]k+1 ∈ R k, and D[[ξk+2, ηk+2]]k+2 = [[Dξk+2, ηk+1]]k+1 + [[ξk+1, Dηk+2]]k+1 ∈ T ∗ ⊗Rk. Therefore, by the same Proposition 4.1, [[ξk+2, ηk+2]]k+2 ∈ Rk+1, and the proposition follows. � It does not follow from this proposition that Rk+l is a vector bundle, and that πk+l : Rk+l → Rk+l−1 is onto, for l ≥ 2. To obtain this, we need an additional condition. Definition 4.3. We say that the linear Lie equation Rk is formally integrable if (i) Rk+l is a subvector bundle of Jk+lT , (ii) πk+l : Rk+l+1 → Rk+l is onto, for l ≥ 1. The symbol gk of Rk is the kernel of πk−1 : Rk → Jk−1T . Also, gk+l is the kernel of πk+l−1 : Rk+l → Rk+l−1, for l ≥ 1. It follows from Proposition 4.1 and from (2.35) that we have the subcomplex 0 → gk+l δ→ T ∗ ⊗ gk+l−1 δ→ ∧2T ∗ ⊗ gk+l−2 δ→ ∧3T ∗ ⊗ γk+l−3 (4.3) for l ≥ 2. Definition 4.4. We say that the symbol gk is 2-acyclic if the subcomplex (4.3) is exact for l ≥ 2. The following proposition is in [4, 5]. For an alternative proof, see [21, 22, 23, 26]. Proposition 4.3. If Rk ⊂ JkT is such that (i) Rk+1 is a subvector bundle of Jk+1T , (ii) πk : Rk+1 → Rk is onto, (iii) gk is 2-acyclic, then πk+l−1 : Rk+l → Rk+l−1 is onto for l ≥ 2. A consequence of this proposition is: Corollary 4.1. If Rk ⊂ JkT is a linear Lie equation and gk is 2-acyclic, then Rk is formally integrable. Given a linear Lie equation Rk, let be the distribution B ⊂ TQk defined by BX = Rk β(X).X, for X ∈ Qk. It follows from (2.10) and (4.1) that the distribution B is involutive. Let be P k(x) the integral leaf of B that contains the point I(x), and P k = ∪x∈MP k(x). Then P k is a groupoid, and a differentiable submanifold at a neighborhood of I. As our problem is local, we will suppose that P k is a differentiable groupoid, the differentiable groupoid associated with the linear Lie equation Rk. Then the linear Lie equation Rk is the Lie algebroid associated with P k. As before, we denote by Pk the groupoid of invertible sections of P k. We define the prolongation P k+1 of P k by P k+1 = (λ1)−1 ( Q1P k ∩ λ1Qk+1 ) , where λ1 : Qk+1 → Q1Qk and Q1P k is the groupoid of 1-jets of invertible sections of P k. The following is Proposition 6.9(ii) of [17]: 30 J.M.M. Veloso Proposition 4.4. Let be F ∈ Qk+1 such that (i) πkF ∈ Pk, (ii) DF ∈ T ∗ ⊗Rk. Then F ∈ Pk+1. Proof. It follows from (2.37) i(v)DF = λ1F−1.j1πkF.v − v, where v ∈ T , so λ1F.(i(v)DF ) = j1πkF.v − λ1F.v. As i(v)DF ∈ Rk, we get λ1F.(i(v)DF ) ∈ T Pk. Also, we get from πkF ∈ Pk that j1πkF.v ∈ T Pk. Therefore, λ1F.v = j1πkF.v − λ1F.(i(v)DF ) ∈ T Pk, so F ∈ Pk+1. � If the linear Lie equation Rk is formally integrable, and P k is the differentiable groupoid associated with Rk, it is true (cf. Proposition 6.1, [17]) that the prolongation P k+l of P k is the groupoid associated with the linear Lie equation Rk+l. Therefore, πk+l : P k+l+1 → P k+l are submersions, for l ≥ 0. 5 Formal isomorphism of intransitive linear Lie equations In the following sections, we consider intransitive linear Lie equations. Definition 5.1. We say that a linear Lie equation Rk ⊂ JkT is intransitive if there exists an integrable distribution V ⊂ T such that Rk ⊂ JkV and π0(Rk) = J0V . In reality, considering (4.1), we need only to verify that π0(Rk) is a subvector bundle of J0T . Our basic problem in this section is to determine the conditions for two intransitive linear Lie equations to be isomorphic. This means that there exists a diffeomorphism that sends one equation onto the other. In the sequel, we give a brief description of the system of partial differential equations that we should solve to obtain a class of diffeomorphisms f : M → M ′ such that (jk+1f)∗(Rk) = R′k. We utilize the same notation of Section 3. Consider Rk ⊂ JkV and R′k ⊂ JkV ′ intransitive linear Lie equations, and P k ⊂ Qk V and P ′k ⊂ Qk V ′ the associated groupoids. Definition 5.2. We say that a submanifold Sk ⊂ Qk φ is automorphic by P k if α : Sk → M , β : Sk → M ′ are submersions, and for every X ∈ Sk(a, b), where a ∈ M and b ∈ M ′, Sk(·, b) = X ◦ P k(·, a). We denote by Sk the set of invertible sections of Sk. Proposition 5.1. Let Sk+1 be the prolongation of Sk. Then an invertible section F ∈ Qk+1 φ is such that F (x) ∈ Sk+1(x) for every x ∈ α(F ) if and only if πkF ∈ Sk and DF ∈ T ∗ ⊗Rk. Proof. The same proof of Proposition 4.4 applies. � Isomorphism of Intransitive Linear Lie Equations 31 We define the symbol gk S = {v ∈ TSk : (πk−1)∗v = 0}. The symbol gk S of Sk is isomorphic to the symbol gk of Rk, and we get an complex analogous to (4.3), and we define that gk S is 2-acyclic in the same way. From the formal integrability theorem (see [5]) we obtain: Proposition 5.2. Let be Sk ⊂ Qk φ automorphic by P k such that (i) Sk+1 is a submanifold of Qk+1 φ , (ii) πk : Sk+1 → Sk is onto, (iii) gk S is 2-acyclic. Then Sk is formally integrable, and each prolongation Sk+r is automorphic by P k+r, for r ≥ 1. Definition 5.3. We say that the intransitive linear Lie equation Rk ⊂ JkV is formally iso- morphic to the intransitive linear Lie equation R′k ⊂ JkV ′ at points a and a′, respectively, if there exists a diffeomorphism φ : N → N ′, and a submanifold Sk ⊂ Qk φ automorphic by P k and formally integrable, such that: (i) S′k = {X−1 : X ∈ Sk} ⊂ Qk φ−1 is automorphic by P ′k; (ii) Sk(a, a′) 6= ∅. If there exists a solution f : M → M ′ of Sk, i.e., a diffeomorphism f such that jkf is a section of Sk, and f(a) = a′, then Rk at point a is said isomorphic to R′k at point a′. This definition is essentially local. A most useful way to verify the formal isomorphism is given by proposition below, analogous of Proposition 5.1: Proposition 5.3. Suppose that Rk ⊂ JkV , R′k ⊂ JkV ′ are intransitive linear Lie equations, N and N ′ submanifolds of M and M ′ transversal to integral submanifolds of V and V ′, re- spectively, and φ : N → N ′ a diffeomorphism, a ∈ N , a′ ∈ N ′, and φ(a) = a′. Suppose furthermore that the symbol gk of Rk is 2-acyclic. If there exists F ∈ Qk+1 φ such that βF |N = φ, F∗(Rk) = R′k, and DF ∈ T ∗ ⊗Rk, then Rk at a is formally isomorphic to R′k at a′. Proof. Define Sk+1 = { Y F (x)X : X ∈ P k+1(·, x), Y ∈ P ′k+1(f(x), ·), x ∈ α(F )}, where f = βF : α(F ) ⊂ M → β(F ) ⊂ M ′. Let be U = ρ−1(ρ(α(F ))) ⊂ M and U ′ = ρ′−1(ρ′(β(F ))) ⊂ M ′. Observe that Sk+1 ⊂ Qk+1 φ and α × β : Sk+1 → U × U ′ is onto (at least locally). If Sk = πkS k+1, then it is a straightforward verification that Sk is automorphic by P k and S′k is automorphic by P ′k. Given an invertible section G ∈ Sk+1, then in the neighborhood of each point of α(G), there are invertible sections G1 ∈ Pk+1 and G2 ∈ P ′k+1 such that G = G2FG1. In fact, given a point x ∈ α(G), there is an open set Vx ⊂ α(G), with x ∈ Vx, and an invertible section G1 of P k+1 defined on Vx, such that β(G1) ⊂ α(F ). Let be G2 = GG−1 1 F−1, defined on f(β(G1)). Then, G2 is an invertible section of P ′k+1, and G|Vx = G2FG1. It follows from Proposition 2.13(i) and Proposition 5.1 that DG ∈ T ∗ ⊗ Rk on the open set Vx. As the Vx’s cover α(G), we get this property on all α(G). Therefore, Sk is formally integrable, and conditions of Definition 5.3 are satisfied. � 32 J.M.M. Veloso Corollary 5.1. Suppose that Rk ⊂ JkV , R′k ⊂ JkV ′ are intransitive linear Lie equations, N and N ′ submanifolds of M and M ′, transversal to integral submanifolds of V and V ′, re- spectively. Let be φ : N → N ′ a diffeomorphism, a ∈ N , a′ ∈ N ′, and φ(a) = a′. Suppose furthermore that the symbol gk of Rk is 2-acyclic. If there exists F ∈ Qk+1 φ such that βF |N = φ, and F∗(T ⊕Rk) = T ′ ⊕R′k, then Rk at a is formally isomorphic to R′k at a′. Let’s now show the existence of a flat partial connection that leaves Rk invariant. Proposition 5.4. Let be Rk ⊂ JkV an intransitive linear Lie equation. Then there exists a flat partial connection ∇ : H⊕ JkV → V∗ ⊗ (H⊕ JkV), such that, restricted to H⊕Rk, it satisfies ∇ : H⊕Rk → V∗ ⊗ (H⊕Rk). Furthermore, if U +Ξ is a parallel section of H⊕JkV and (U +Ξ)|N is a section of (H⊕Rk)|N , then U + Ξ ∈ H ⊕Rk. Proof. Choose a family of differentiable sections σy introduced in (3.3) satisfying σy(N) ⊂ P k+1. As P k+1 is the groupoid associated with Rk+1, the form ω defined by (3.4) belongs to V∗ ⊗Rk+1, and the partial connection ∇ : H⊕ JkV → V∗ ⊗ (H⊕ JkV) defined by ω̃, restricted to H ⊕ Rk sends H ⊕Rk to V∗ ⊗ (H ⊕Rk), as a consequence of (4.2). The proof now follows from Propositions 3.4 and 3.5. � Now we prove the fundamental theorem for formal isomorphism of linear Lie equations: Theorem 5.1. Suppose that Rk ⊂ JkV and R′k ⊂ JkV ′ are intransitive linear Lie equations, N and N ′ submanifolds of M and M ′ transversal to integral submanifolds of V and V ′, respec- tively, and φ : N → N ′ a diffeomorphism. Suppose furthermore that there exists Φ : N → Qk+1 φ such that βΦ = φ, and Φ∗ ( TN ⊕Rk|N ) = TN ′ ⊕R′k|N ′ . Then given a diffeomorphism f : M → M ′ such that f∗V = V ′, f |N = φ, there exists F ∈ Qk+1 φ satisfying F |N = Φ, βF = f , and F∗ ( T ⊕Rk ) = T ′ ⊕R′k. Proof. Let be families σy : N → P k+1, σ′y ′ : N ′ → P ′k+1, as in the proof of Proposition 5.4, and defining flat partial connections ∇ : H⊕ JkV → V∗ ⊗ ( H⊕ JkV ) , and ∇′ : H′ ⊕ JkV ′ → V ′∗ ⊗ ( H′ ⊕ JkV ′ ) such that ∇ ( H⊕Rk ) ⊂ V∗ ⊗ ( H⊕Rk ) and ∇′(H′ ⊕R′k) ⊂ V ′∗ ⊗ ( H′ ⊕R′k). Isomorphism of Intransitive Linear Lie Equations 33 Observe that by Proposition 5.4, ω ∈ V∗ ⊗Rk+1, and ω′ ∈ V ′∗ ⊗R′k+1. Redefine H ′ = f∗H, if necessary, to obtain (x′, y′) = f(x, y) = (a(x), b(y)), and define F ∈ Qk+1 φ by F (x, y) = σ′y ′ (x′, 0)Φ(x, 0)σy(x, 0)−1. Then, we get from (3.8) F (x, y)σy(x, 0)F (x, 0)−1 = σ′y ′ (x′, 0), and from (3.10) we get ∇′F∗ = F∗∇. By hypothesis F∗ ( TN ⊕Rk|N ) = TN ′ ⊕R′k|N ′ , then by Proposition 5.4 we obtain F∗ ( H ⊕Rk ) = H ′ ⊕R′k. (5.1) From this and Corollary 2.3 we obtain DHF ∈ H∗ ⊗Rk. (5.2) It follows from (3.11) that πkω ′ = F∗(DV F + πkω). So, from πkω ∈ Rk, πkω ′ ∈ R′k and (5.1) we get DV F ∈ V∗ ⊗Rk. Combining this with (5.1) and (5.2), we get DF ∈ T ∗ ⊗Rk and F∗(Rk) = R′k, and by Proposi- tion 5.3 the theorem follows. � Corollary 5.2. Suppose that Rk ⊂ JkV and R′k ⊂ JkV ′ are intransitive linear Lie equations, that N and N ′ are submanifolds of M and M ′ transversal to integral submanifolds of V and V ′, respectively. Let be φ : N → N ′ a diffeomorphism such that φ(a) = a′, where a ∈ N and a′ ∈ N ′. Suppose furthermore that the symbol gk of Rk is 2-acyclic. If there exists Φ : N → Qk+1 φ such that βΦ = φ, and Φ∗(TN ⊕ Rk|N ) = TN ′ ⊕ R′k|N ′, then Rk at point a is formally isomorphic to R′k at point a′. Proof. The corollary follows from Theorem 5.1 and Corollary 5.1. � 6 Intransitive Lie algebras In this section, we associate an intransitive Lie algebra with a germ of an intransitive linear Lie equation. This definition must generalize the definition of transitive Lie algebra, and incorpo- rate the fact that we can reconstruct an intransitive linear Lie equation from its restriction to a transversal to the orbits, unless of formal isomorphism, as the Theorem of [25] and Theorem 5.1 above shows. We continue, in this section, to suppose that Rk ⊂ JkV is an intransitive linear Lie equation and gk is 2-acyclic. We remember that it follows from these hypotheses, see Corollary 4.1, that the prolongations Rk+l of Rk, l ≥ 1, satisfy: (i) Rk+l is a subvector bundle of Jk+lV ; (ii) πk+l : Rk+l+1 → Rk+l is onto; 34 J.M.M. Veloso (iii) [[T ⊕Rk+l, T ⊕Rk+l]]k+l ⊂ T ⊕Rk+l−1. We also assume that Rl = πlR k is a subvector bundle of J lV for every 0 ≤ l ≤ k − 1, in particular, R0 = J0V . We denote by ON,a the R-algebra of germs at point a ∈ N of local C∞ real functions on N . The ON,a-module (T N)a of germs at point a of local sections of TN is isomorphic to Der ON,a, the ON,a-module of derivations of ON,a. We denote by Lj the ON,a-module of germs at point a of local C∞-sections of Rj |N , considered as a vector bundle on N by the map π|N : Rk|N → N . Let be the ON,a-modules L = lim proj Lj and L = Der ON,a ⊕ L. The bilinear antisymmetric map [[ , ]]k+l : (T ⊕Rk+l)× (T ⊕Rk+l) → T ⊕Rk+l−1 induces a well defined R-bilinear antisymmetric map [[ , ]]j : (Der ON,a ⊕ Lj)× (Der ON,a ⊕ Lj) → Der ON,a ⊕ Lj−1. As we saw in (2.24), the projective limit of [[ , ]]j induces a R-Lie bracket [[ , ]]∞ : L × L → L, so that L is a R-Lie algebra. The structure of the Lie algebra L is the semi-direct product of the Lie algebras Der ON,a and L, where the action of Der ON,a on Lm is given by the restriction to N of DH : Rj → H∗ ⊗Rj−1. If v ∈ Der ON,a and ξ is a section of Rj |N defined in a neighborhood of a ∈ N , we get (see Proposition 2.5) [[v, ξ]]j = i(v)Dξ. The map (ρ1)∗ : L → Der ON,a is the canonical projection given by the direct sum, and [[ξ, fη]]∞ = ((ρ1)∗ξ)(f)η + f [[ξ, η]]∞, where f ∈ ON,a, ξ, η ∈ L. The restriction of [[ , ]]∞ to L is ON,a-bilinear, so L is a ON,a-Lie algebra. Each Lj is a free ON,a-module finitely generated. Definition 6.1. We call L the ON,a-intransitive R-Lie algebra associated with the formally integrable linear Lie equation Rk at the point a (and transversal N). In particular, we denote by D(V ) the ON,a-intransitive Lie algebra associated with the linear Lie equation J0V , and call it the ON,a-intransitive R-Lie algebra associated with the involutive distribution V at point a ∈ M . Clearly L ⊂ D(V ). If Lj = Der ON,a⊕Lj , then (Lj , [[ , ]]j) is called the truncated ON,a-intransitive R-Lie algebra of order j associated with Rk at point a (and transversal N). Then we can state the Theorem of [25] as: Theorem 6.1. Let be Lk+2 ⊂ Dk+2(V ) a truncated ON,a-intransitive R-Lie algebra. Then there exists a vector sub-bundle R′k+1 ⊂ Jk+1V such that: Isomorphism of Intransitive Linear Lie Equations 35 (i) R′k = πkR ′k+1 is a vector sub-bundle of JkV ; (ii) [[T ⊕R′k+1, T ⊕R′k+1]]k+1 ⊂ T ⊕R′k; (iii) the truncated ON,a-intransitive R-Lie algebra associated with R′k+1 is Lk+1 = πk+1Lk+2. Furthermore, if hk = {ξ ∈ Lk|πk−1ξ = 0} is 2-acyclic, then R′k is formally integrable. The definitions of L and D(V ) depend on the choice of the transversal N . Let’s now introduce a notion of isomorphism inspired in Theorem 5.1 such that the intransitive Lie algebras obtained at point a taking different transversal submanifolds are isomorphic. We maintain the notation of Section 5. Suppose that Rk ⊂ JkV , R′k ⊂ JkV ′ are formally integrable intransitive linear Lie equa- tions, N and N ′ submanifolds of M and M ′ transversal to integral submanifolds of V and V ′, respectively, and φ : N → N ′ a diffeomorphism, a ∈ N , a′ ∈ N ′, and φ(a) = a′. We denote also by φ the isomorphism of R-algebras φ : ON,a → ON ′,a′ , defined by φ(f) = fφ−1. Let be Φj+1 : N → Qj+1 φ α-sections such that φ = βΦj+1 and πjΦj+1 = Φj for j ≥ 0. Put Φ = lim proj Φj . We get maps (Φj+1)∗ : Dj(V ) → Dj(V ′) and Φ∗ : D(V ) → D(V ′). The map Φ∗ is R-linear, commutes with [[ , ]]∞, and, if f ∈ ON,a and ξ ∈ D(V ), then Φ∗(fξ) = φ(f)Φ∗(ξ). Definition 6.2. We say that Φ∗ is an isomomorphism from intransitive Lie algebra L ⊂ D(V ) onto intransitive Lie algebra L′ ⊂ D(V ′) if Φ∗L = L′. If Φ∗ is an isomorphism, then L is said isomorphic to L′. Proposition 6.1. Suppose that Rk ⊂ JkV is a formally integrable intransitive linear Lie equa- tion, a, b points of M , N , N1 transversal to the orbits of Rk through the points a, b, and L, L1 the intransitive Lie algebras associated with Rk at the points a, b (and transversal N , N1), respectively. Let be ρ : M → N the fibration (at least locally) defined by the leaves of V . If ρ(a) = ρ(b), then the ON,a-intransitive Lie algebra L is isomorphic to the ON1,b-intransitive Lie algebra L1. Proof. If x, y ∈ M , ρ(x) = ρ(y), there exists X ∈ Pm with α(X) = x, β(X) = y. Therefore, we can choose Φj : N → P j such that βΦj(N) = N1. Then (Φj)∗ ( TN ⊕Rj−1|N ) = TN1 ⊕Rj−1|N1 . It follows from the formal integrability of P k that we can choose the family {Φj : j ≥ 1} such that Φj+1 projects on Φj , for j ≥ 1. If Φ = lim proj Φj , then Φ∗L = L1. � With these definitions, we can state Corollary 5.2 as: Theorem 6.2. Suppose that Rk ⊂ JkV is a linear Lie equation with symbol gk 2-acyclic, and R′k ⊂ JkV ′ another linear Lie equation. Let be a ∈ M , a′ ∈ M ′, N and N ′ transversal to the orbits of Rk and R′k through the points a and a′, Lk and L′k the truncated intransitive Lie algebras associated with Rk and R′k, at points a, a′ and tranversal N and N ′, respectively. If there exists Φ : N → Qk+1 φ such that βΦ = φ : N → N ′, φ(a) = a′, and Φ∗Lk = L′k, then Rk at point a is formally isomorphic to R′k at point a′. 36 J.M.M. Veloso 7 Application As an application of this theory, we could utilize the definition of intransitive Lie algebras to obtain the intransitive linear Lie equations in the plane obtained by É. Cartan in [1]. We will be limited to classifying the first order intransitive linear Lie equations, with dim g1 = 1. This will include the example we presented in the introduction, which was not presented by Cartan in his table, suppressed by a nullity hypothesis. Let be V a 1-dimensional distribution on R2, which we can suppose is generated by the vector field ∂ ∂y . We will use the coordinate system pj,l, j, l ≥ 0, 0 ≤ j + l ≤ k, in JkV , defined by pj,l(jk (a,b)Θ) = ∂j+lθ ∂xj∂yl (a, b), where Θ(x, y) = θ(x, y) ∂ ∂y . Let’s consider (0, 0) as point base and the transversal N = R× {0}. Then Der ON,(0,0) is generated, as ON,(0,0)-module, by ∂ ∂x |N . Let be gk V the symbol of JkV . This symbol is generated by f j,l ⊗ j0 ∂ ∂y , where f j,l = 1 j!l! (dx)j(dy)l, with j, l ≥ 0 and j + l = k (cf. (2.34)). Then[[ ∂ ∂x , f j,l ⊗ j0 ∂ ∂y ]] k = −f j−1,l ⊗ j0 ∂ ∂y ∈ gk−1 V , and if Y ∈ JkV is such that π0(Y ) = j0 ∂ ∂y , then[[ Y, f j,l ⊗ j0 ∂ ∂y ]] k = f j,l−1 ⊗ j0 ∂ ∂y ∈ gk−1 V . Let be R1 ⊂ J1V a formally integrable linear Lie equation, with R0 = J0V . Suppose g1 is the symbol of R1, with dim g1 = 1. Then g1 is generated by an element X1 = (Af1,0+Bf0,1)⊗j0 ∂ ∂y , with A(x, y)2 + B(x, y)2 6= 0. Lemma 7.1. If R2 is the prolongation of R1, then the symbol g2 of R2 is generated by( Af1,0 + Bf0,1 )2 ⊗ j0 ∂ ∂y . Proof. Consider Y0, Y1 ∈ R2 such that π0Y0 = j0 ∂ ∂y and π1Y1 = (Af1,0 + Bf0,1) ⊗ j0 ∂ ∂y is a section of g1. Let be Y = (a20f 2,0 + a11f 1,1 + a02f 0,2)⊗ j0 ∂ ∂y a section of g2. Then[[ ∂ ∂x , Y ]] 2 = − ( a20f 1,0 + a11f 0,1 ) ∈ g1, so (a20f 1,0 + a11f 0,1) = λ(Af1,0 + Bf0,1), for a real function λ. In a similar way, [[Y0, Y ]]2 = ( a11f 1,0 + a02f 0,1 ) ∈ g1, and (a11f 1,0+a02f 0,1) = µ(Af1,0+Bf0,1), for some real function µ. Therefore, we get a20 = λA, a11 = λB = µA and a02 = µB. So, there exists r such that λ = rA and µ = rB. From this, a20 = rA2, a11 = rAB and a02 = rB2. Then Y = r ( Af1,0 + Bf0,1 )2 ⊗ j0 ∂ ∂y . � Isomorphism of Intransitive Linear Lie Equations 37 A similar argument shows that gk is one dimensional and is generated by (Af1,0 + Bf0,1)k ⊗ j0 ∂ ∂y . Consider now the complex (4.3), 0 → gl+1 δ→ T ∗ ⊗ gl δ→ ∧2T ∗ ⊗ gl−1→0 for l ≥ 2. As dim gk = 1 and dim T = 2, this complex is clearly exact, so g1 is 2-acyclic. Let’s now verify conditions on the truncated intransitive Lie algebra L2. The ON,(0,0)-module T N(0,0) is generated by Y−1 = ∂ ∂x ∣∣∣ N , and the generators of L2 are Y0, Y1, defined in the proof of lemma and Y2 = 1 2(af1,0 + bf0,1)2⊗ j0 ∂ ∂y , restricted to N , again denoted by the same letters. Here, a(x) = A(x, 0) and b(x) = B(x, 0). We have [[Y−1, Y0]]2 = b00π1(Y0) + b01π1(Y1), [[Y0, Y1]]2 = bπ1(Y0) + a01π1(Y1), [[Y−1, Y1]]2 = −aπ1(Y0) + b11π1(Y1), [[Y0, Y2]]2 = bπ1(Y1), [[Y−1, Y2]]2 = −aπ1(Y1), [[Y1, Y2]]2 = 0, for a, b, b00, b01, b11, a01 in ON,(0,0). It follows from [[Y−1, [[Y0, Y1]]2]]1 = [[[[Y−1, Y0]]2, π1(Y1)]]1 + [[π1(Y0), [[Y−1, Y1]]2]]1 that ( ∂b ∂x − aa01 − bb11)π0(Y0) = 0, so ∂b ∂x − aa01 − bb11 = 0. As a(0) or b(0) is not null, we can solve this equation for a01 or b11. Then we can find a truncated intransitive Lie algebra L2 that projects on L1. Now by Theorem 6.2 we classify the isomorphism class of L1. We must examine two cases: Case 1. We suppose b(0) 6= 0. By dividing by b, we can suppose b = 1. Then the algebra L1 is generated by Y−1, π0Y0, π1Y1, and they satisfy, from above, [[Y−1, π1Y0]]1 = b00π0(Y0), [[Y−1, π1Y1]]1 = −aπ0(Y0), [[π1Y0, π1Y1]]1 = π0(Y0). Let be L′1 a truncated intransitive Lie algebra generated by X−1, X0 and X1, such that [[X−1, X0]]1 = 0, [[X−1, X1]]1 = 0, [[X0, X1]]1 = π0(X0), and f1 : L′1 → L1 defined by f1(X−1) = Y−1 + aπ1(Y0) + b00π1(Y1), f1(X0) = π1(Y0), f1(X1) = π1(Y1). Let’s verify that f1 is an isomorphism of truncated intransitive Lie algebras of order 1. In fact, [[f1(X−1), f1(X0)]]1 = [[Y−1 + aπ1(Y0) + b00π1(Y1), π1(Y0)]]1 = 0 = f0([[X−1, X0]]1), [[f1(X−1), f1(X1)]]1 = [[Y−1 + aπ1(Y0) + b00π1(Y1), π1(Y1)]]1 = 0 = f0([[X−1, X1]]1), and [[f1(X0), f1(X1)]]1 = [[π1(Y0), π1(Y1)]]1 = π0(Y0) = f0(π0(X0)) = f0([[X0, X1]]1). 38 J.M.M. Veloso The truncated Lie algebra L′1 generated asON,(0,0)-module by X−1, X0, X1 can be represented by X−1 = ∂ ∂x ∣∣∣ N , X0 = j1 ∂ ∂y , X1 = f0,1 ⊗ j0 ∂ ∂y . The linear Lie equation associated with L′1 is R′1 = { (p0,0, p1,0, p0,1) ∈ J1V : p1,0 = 0 } , so R1 is formally isomorphic to R′1. The infinitesimal pseudogroup of solutions of R′1 is{ Θ(x, y) = θ(y) ∂ ∂y } . Case 2. We suppose b(0) = 0. In this case, a(0) 6= 0, and, dividing it by a, we can suppose a = 1. Then the truncated intransitive Lie algebra L1 is generated by Y−1, π0Y0, π1Y1, and they satisfy, from above, [[Y−1, π1Y0]]1 = b00π0(Y0), [[Y−1, π1Y1]]1 = −π0(Y0), [[π1Y0, π1Y1]]1 = bπ0(Y0). Replacing π1Y0 by π1Y0+b00π1Y1, we obtain [[Y−1, π1Y0 + b00π1Y1]]1 = 0, and the other products remain unchanged. Without loss of generality, we can suppose b00 = 0. Let be L′1 a truncated intransitive Lie algebra generated by X−1, X0 and X1, such that [[X−1, X0]]1 = 0, [[X−1, X1]]1 = −π0(X0), [[X0, X1]]1 = βπ0(X0), and f1 : L′1 → L1 defined by f1(X−1) = c−1Y−1, f1(X0) = π1(Y0), f1(X1) = cπ1(Y1), with c(0) 6= 0. Then [[f1(X−1), f1(X0)]]1 = [[c−1Y−1, π1(Y0)]]1 = 0 = f0([[X−1, X0]]1), [[f1(X−1), f1(X1)]]1 = [[c−1Y−1, cπ1(Y1)]]1 = −π0(Y0) = f0([[X−1, X1]]1), and [[f1(X0), f1(X1)]]1 = [[π1(Y0), cπ1(Y1)]]1 = cbπ0(Y0) = f0(βπ0(X0)) = f0([[X0, X1]]1), if cb = β. Then f1 is an isomorphism of truncated intransitive Lie algebras of order 1 if and only if there exists c ∈ ON,(0,0) with c(0) 6= 0 such that cb = β. The class of truncated intransitive Lie algebras is the class of equivalence of b, b(0) = 0, and b ≡ β if and only if there exists c, c(0) 6= 0, with bc = β. The truncated Lie algebra L′1 generated as OR-module by X−1, X0, X1 can be represented by X−1 = ∂ ∂x , X0 = j1 ∂ ∂y , X1 = ( f1,0 + βf0,1 ) ⊗ j0 ∂ ∂y , and the linear Lie equation associated with L′1 is R′1 = { (p0,0, p1,0, p0,1) ∈ J1V : p0,1 = βp1,0 } , Isomorphism of Intransitive Linear Lie Equations 39 so R1 is formally isomorphic to R′1. The infinitesimal pseudogroup of solutions of R′1 is{ Θ(x, y) = θ(x, y) ∂ ∂y : ∂θ ∂y = β ∂θ ∂x } . If β(x) = x, we obtain{ Θ(x, y) = θ(xey) ∂ ∂y } ; if β(x) = xk, k ≥ 2, we obtain{ Θ(x, y) = θ ( xk−1 (k − 1)yxk−1 − 1 ) ∂ ∂y } ; if β(x) = 0, we obtain{ Θ(x, y) = θ(x) ∂ ∂y } . In the classification of [1], case 2 is represented only by β = 0. 8 Conclusion The results of this paper show that the intransitive Lie algebra here introduced to represent a linear Lie equation at a point is sufficient to guarantee the existence and formal isomorphism of intransitive linear Lie equations. This brings a new way to pursue the study of intransitive Lie groups and the applications envisaged by Sophus Lie on the integrability of partial differential equations with a pseudogroup of invariants. It is clear that several problems can still exist, as the relationship between subalgebras of transitive algebras and intransitive algebras, and the notion of equivalence of intransitive algebras. 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[18] Morimoto T., On the intransitive Lie algebras whose transitive parts are infinite and primitive, J. Math. Soc. of Japan 29 (1977), 35–65. [19] Ngô Van Quê V.V.T., Définition des pseudogroupes infinitesimaux de Lie intransitifs. Théorème fondamental de réalization en dimension deux, Nagoya Math. J. 86 (1982), 211–228. [20] Petitjean A., Rodrigues A.M., Correspondance entre algébres de Lie abstraites et pseudo-groupes de Lie transitifs, Ann. of Math. (2) 101 (1975), 268–279. [21] Ruiz C., Prolongement formel des systèmes différentiels extérieur d’ordre supérieur, C. R. Acad. Sci. Paris Sér. A-B 285 (1977), A1077–A1080. [22] Ruiz C., Complexe de Koszul du symbole d’un système différentiel extérieur, C. R. Acad. Sci. Paris Sér. A-B 286 (1978), A55–A58. [23] Ruiz C., Propriétès de dualité du prolongement formel des systèmes différentiels extérieurs, C. R. Acad. Sci. Paris Sér. A-B 286 (1978), A99–A101. [24] Singer I.M., Sternberg S., The infinite groups of Lie and Cartan. I. The transitive groups, J. Analyse Math. 15 (1965), 1–114. [25] Veloso J.M.M., Lie’s third theorem for intransitive Lie equations, J. Differential Geom. 32 (1990), 185–198. [26] Veloso J.M., Prolongation projection commutativity theorem, in CR-Geometry and Overdetermined Systems (Osaka, 1994), Adv. Stud. Pure Math., Vol. 25, Math. Soc. Japan, Tokyo, 1997, 386–405. [27] Veloso J.M.M., New classes of intransitive simple Lie pseudogroups, Bull. Soc. Sci. Lett. Lodz 36 (1986), no. 19, 7 pages. 1 Introduction 2 Preliminaries 2.1 Groupoids and algebroids of jets 2.2 The calculus on the diagonal 2.3 The Lie algebra sheaf ( T)*( T) 2.4 The first non-linear Spencer complex 3 Partial connections 4 Linear Lie equations 5 Formal isomorphism of intransitive linear Lie equations 6 Intransitive Lie algebras 7 Application 8 Conclusion References
id nasplib_isofts_kiev_ua-123456789-149105
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn 1815-0659
language English
last_indexed 2025-12-07T18:17:24Z
publishDate 2009
publisher Інститут математики НАН України
record_format dspace
spelling Veloso, Jose Miguel Martins
2019-02-19T17:25:25Z
2019-02-19T17:25:25Z
2009
Isomorphism of Intransitive Linear Lie Equations / Jose Miguel Martins Veloso // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 27 назв. — англ.
1815-0659
2000 Mathematics Subject Classification: 58H05; 58H10
https://nasplib.isofts.kiev.ua/handle/123456789/149105
We show that formal isomorphism of intransitive linear Lie equations along transversal to the orbits can be extended to neighborhoods of these transversal. In analytic cases, the word formal is dropped from theorems. Also, we associate an intransitive Lie algebra with each intransitive linear Lie equation, and from the intransitive Lie algebra we recover the linear Lie equation, unless of formal isomorphism. The intransitive Lie algebra gives the structure functions introduced by É. Cartan.
This paper is a contribution to the Special Issue “Elie Cartan and Differential Geometry”. I would like to thank the referees for the several suggestions to improve this paper.
en
Інститут математики НАН України
Symmetry, Integrability and Geometry: Methods and Applications
Isomorphism of Intransitive Linear Lie Equations
Article
published earlier
spellingShingle Isomorphism of Intransitive Linear Lie Equations
Veloso, Jose Miguel Martins
title Isomorphism of Intransitive Linear Lie Equations
title_full Isomorphism of Intransitive Linear Lie Equations
title_fullStr Isomorphism of Intransitive Linear Lie Equations
title_full_unstemmed Isomorphism of Intransitive Linear Lie Equations
title_short Isomorphism of Intransitive Linear Lie Equations
title_sort isomorphism of intransitive linear lie equations
url https://nasplib.isofts.kiev.ua/handle/123456789/149105
work_keys_str_mv AT velosojosemiguelmartins isomorphismofintransitivelinearlieequations

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