ON THE PROLONGATION OF FIBER BUNDLES AND INFINITESIMAL STRUCTURES ( 1 )

Transcription

1 Du rolongement des esaces fibrés et des structures infinitésimales, Ann. Inst. Fourier, Grenoble 17, 1 (1967), ON THE PROLONGATION OF FIBER BUNDLES AND INFINITESIMAL STRUCTURES ( 1 ) By Ngô Van Que Translated by D. H. Delhenich Introduction. This aer has as its oint of dearture the fundamental aer of D. C. Sencer: Deformation of structures defined by transitive continuous seudogrous. D. C. Sencer has develoed a new method for differential geometry that he calls cohomological. This method, which was illustrated in the imortant treatise of I. M. Singer and S. Sternberg: On the infinite grous of Lie and Cartan, has, moreover, roved to an elegant and essential formalism for the study of differential oerators or linear differential systems in general (see the rofound thesis of D. G. Quillen at Harvard 1964), which unfortunately aeared while this aer was being conceived). Our goal is to mae more recise the general framewor in which the method of D. C. Sencer is alied and to aly it to the study of infinitesimal structures in articular. The various chaters of this aer are each receded by an exlanatory note that distills the essential results that are obtained; we thus refer the reader to them for a more detailed introduction. Professor B. Malgrange gave me the essence of the ideas that are contained in aragrah 4 of chater II. In articular, I owe the elegant form of theorem II-4-b in that aragrah to him. It would rofit me to resent him with my comlete acnowledgement here. Permit me to exress my rofound gratitude to Professor A. Lichnerowicz, whose benevolent advice and counsel have always guided my research. ( 1 ) The subject of this aer was a art of the thesis resented on 16 June 1966 to the Faculté des Sciences de Paris for obtaining the degree of Doctor of Sciences. The other art of this article, which develoed the ideas of D. C. Sencer on the theory of deformation will be resented in a later ublication.

2 TABLE OF CONTENTS CHAPTER I. LIE GROUPOIDS AND ASSOCIATED FIBER BUNDLES Lie grouoid Associated fiber bundle 6 3. Lie subgrouoid and grouoid extension. 8 CHAPTER II. PROLONGATION OF FIBER BUNDLES AND DIFFERENTIAL OPERATORS Prolongation of Lie grouoids Prolongation of vector bundles D oerator and Sencer exact sequence Differential oerator and its rolongation.. 20 CHAPTER III. CONNECTIONS OF HIGHER ORDER IN A VECTOR BUNDLE Connection in a Lie grouoid Connection in a vector bundle Covariant derivation and connections of order 1 31 CHAPTER IV. PROLONGATIONS OF THE TANGENT BUNDLE Structure of the sheaf of R-Lie algebras Notion of torsion Linear differential system associated to a regular infinitesimal structure 40 CHAPTER V. DIFFERENTIAL SYSTEM OF A G-STRUCTURE Tye and degree of the structure S-connection Integrables structures Structure tensor of D. Bernard Obstructions of higher order (to integrability). Case of structures of finite tye.. 59 BIBLIOGRAPHY. 60

3 CHAPTER ONE LIE GROUPOIDS AND ASSOCIATED FIBER BUNDLES In this aragrah, we recall the notion of Lie grouoid and associated fiber bundle. It seems necessary to us in what follows to introduce the notion of Lie grouoid, which is meanwhile equivalent to the well-nown notion of rincial fiber bundle with structure grou. 1. Lie grouoid. DEFINITION I.1.a A grouoid Φ on the set V (or, more recisely, with V as its set of units) is a set endowed with a ma: (a, b): Φ V V, z (a(z), b(z)) and a law of internal comosition that is associative and artial, and verifies the following axioms: 1) If z and z are two elements of Φ then the comosition z z is defined if and only if a(z) = b(z ), and one has: b(z z ) = b(z) and a(z z ) = a(z ). 2) x, x V, l x, which is an element of Φ such that: and: a(l x ) = b(l x ) = x, if z l x is defined then z l x = z, if l x z is defined then l x z = z. 3) z, z Φ, z 1, which is an element of Φ such that: z z 1 = l y, where y = b(z), z 1 z = l x, where x = a(z). The mas a and b are called the source and target mas of Φ, resectively. From axiom 2, the element l x that is associated with any element x of V is unique; it is called the unit of Φ at x. One verifies that the set of elements of Φ whose source and target coincide with the same element of V form a grou G x that is called the isotroy grou of Φ at x, and if z 0 is an element of Φ with source x and target y then: z 0 : G x G y, z z 0 z 1 z 0

4 Ngo Van Que On the rolongations of fiber bundles and infinitesimal structures. 4 is an isomorhism of the grou G x with G y. In the case where the ma (a, b) is surjective, we say that the grouoid Φ is transitive. DEFINITION I.1.b. Φ being a grouoid on V, Φ is a differentiable grouoid if there exists the structure of a differentiable manifold on Φ (of infinite class and aracomact) such that: 1) The ma (a, b) is (indefinitely) differentiable. 2) The ma: is differentiable. Φ Φ, z z 1, 3) For any differentiable manifold W that is endowed with two differentiable mas f and g of W into Φ that verify: a f = b g, the ma: f g: W Φ, z f(z) g(z) that is thus defined is differentiable. Following Matsushima, we say that a differentiable grouoid is a Lie grouoid if the ma (a, b) is a submersion (i.e., surjective and everywhere of maximal ran). A Lie grouoid is thus transitive. PROPOSITION I. 1. If Φ is a Lie grouoid on the manifold V then one has: 1) The isotroy grous of Φ are isomorhic Lie grous. 2) Uon setting: Φ x = {z, z Φ, such that a(z) = x}, Φ x is a differentiable rincial fiber bundle on V with the target rojection b, whose structure grou is the isotroy Lie grou G x. Indeed: a) Since the ma (a, b) is a submersion, the isotroy grous of Φ are closed submanifolds of Φ (Thom lemma), and conditions 2 and 3 of definition I.1.b entail that their algebraic structure is comatible with their differentiable structure. They are therefore Lie grous that are isomorhic, since Φ, being a Lie grouoid, is transitive. b) Liewise, since the ma a is also a submersion, Φ x is also a closed differentiable submanifold of Φ, and the ma b is a submersion of Φ onto V: Φ x is thus a differentiable fiber bundle over V. On the other hand, condition 3 of definition I.1.b entails that G x is a

5 Ngo Van Que On the rolongations of fiber bundles and infinitesimal structures. 5 Lie grou that oerates on Φ x on the right in a simly transitive fashion on the fibers. Then, by virtue of the theorem above, which generalizes a theorem of Gleason, Φ x is a differentiable fiber bundle over V, with the Lie grou G x for its structure grou (the rincial fiber bundle with structure grou being defined in sense of Steenrod, Toology of Fiber Bundles). THEOREM. Let E be a differentiable fiber bundle on a differentiable manifold V. If G is a Lie grou that oerates in the differentiable manifold E in a simly transitive fashion on the fiber then E is a differentiable rincial fiber bundle on V with structure grou G. Proof: LEMMA. Let E be a differentiable fiber bundle over V. Any oint of V admits a neighborhood U that is endowed with a differentiable ma s of U into E such that s = Id, being the rojection of E onto V. In other words, this lemma assures the existence of local differentiable sections in the neighborhood of any oint of V. This lemma is only an immediate consequence of roosition 2, age 80, of Theory of Lie Grous by C. Chevalley, which remains valid in the differentiable case. Therefore, let U be an oen subset of V that is endowed with a differentiable section s. Consider the ma: ϕ s : U G E U, (= 1 (U)) (x, g) s(x) g, where the dot on the right-hand side denotes the action of the element g of G in an element of E. The ma ϕ s is differentiable (lie the ma of the differentiable roduct manifold U G into the differentiable manifold E U ). It is also bijective, and one may easily see that is 1 tangent ma is a bijection at every oint. It is therefore a diffeomorhism: ϕ s exists and is differentiable. It remains to see that the coordinate change functions (see Steenrod) are differentiable with values in G. This amounts to seeing that if s denotes another differentiable section that is defined on U then the following ma: g: U G, x g(x) is such, that s (x) = s(x) g(x) is differentiable. Now, the ma g is nothing but the comosed ma: 1 f ϕ s, where f is the canonical rojection of U G onto G. Since each of the mas f, differentiable, the ma g is therefore differentiable. Q. E. D. s 1 ϕ s, s is

6 Ngo Van Que On the rolongations of fiber bundles and infinitesimal structures. 6 From roosition I-1, we easily see the corollary. COLROLLARY I.1. The Lie grouoid Φ is locally isomorhic to the trivial grouoid R n G R n, where n is the dimension of V and G is a Lie grou that is isomorhic to the isotroy grou of Φ. The trivial Lie grouoid R n G R n admits R n for its sace of units and the following comosition law: (z, g, y) (y, g, x) = (z, g g, x). In a more recise fashion, the corollary confirms that at every oint of V there exists an oen neighborhood U and a diffeomorhism: ϕ : (a, b) 1 (U U) R n G R n, z (y(z), g(z), x(z)), such that if a(z ) = b(z) then one has: and Examles. x(z ) = y(z) ϕ(z z) = ϕ(z ) ϕ(z). 1) V being a differentiable manifold, let Π (V) denote the set of invertible jets of order of V into V (see C. Ehresmann a ). Π (V) is a Lie grouoid on V with an isotroy grou that is isomorhic to the grou L. 2) If E is a (locally trivial) differentiable fiber bundle on V then the set Π(E) of linear isomorhisms of the fibers of E onto other fibers of E is a Lie grouoid on V. 3) If Φ and Φ are two Lie grouoids on V then let Φ Φ be the Whitney roduct of Φ and Φ, or the set of airs (z, z ) of elements of Φ and Φ that verify a(z) = a(z ) and b(z) = b(z ). The natural law of comosition: n ( z1, z 1) (z, z ) = (z 1 z, z 1 z ) determines a Lie grouoid structure on V in Φ Φ. 2. Associated fiber bundle. DEFINITION I.2. Let Φ be a Lie grouoid on V, and let E be a differentiable manifold that is fibered over V; i.e., endowed with a submersion onto V. We say that E

7 Ngo Van Que On the rolongations of fiber bundles and infinitesimal structures. 7 is a fiber bundle that is associated to Φ if and only if the following conditions are verified: 1) z, z Φ, with a(z) = x and b(z) = y, z determines a diffeomorhism of the fiber E x ( = 1 (x)) onto the fiber E y : and one has: zɶ : E x E y, e zɶ ( e), which is denoted z e, z z = zɶ z ɶ. 2) For any differentiable manifold W that is endowed with two differentiable mas f and g with values in Φ and E, resectively, such that: the ma f g, which is then defined: a f = g, is differentiable. f g : W E, x f(x) g(x), PROPOSITION I.2. If E is a fiber bundle that is associated with a Lie grouoid Φ then E is a locally trivial differentiable fiber bundle with fiber F and structure grou G, where F is a differentiable manifold that is diffeomorhic to any fiber of E, and G is a Lie grou that is isomorhic to the isotroy grou of Φ. Indeed, let F denote the fiber E x of E. It is easy to see that E is the fiber bundle that is obtained by modeling F on the rincial fiber Φ x, where the structure grou G x oerates on F, according to the definition I-2. We remar that if E is a differentiable fiber bundle that is obtained by modeling the manifold F on the rincial bundle Φ x then E is an associated fiber bundle, in the sense of the definition I-2 of the Lie grouoid Φ. Furthermore, when there exists an algebraic structure (grou, vector sace, algebra, etc.) on each fiber of E that is comatible with its differentiable structure and is such that z is an algebraic isomorhism for any element z of Φ, then E is a differentiable fiber bundle with algebraic structure (fibered into grous, vector saces, algebras, etc., res.). 1) Canonical grou fibration associated with a Lie grouoid. Let Φ be a Lie grouoid on V. Let G(Φ) denote the set (a, b) 1 ( ), where is the closed diagonal manifold in V V. Since the ma (a, b) is a submersion, from the Thom lemma, G(Φ) is a closed differentiable submanifold of Φ G(Φ) is obviously a differentiable fiber bundle on V under the ma a or b, and is such that each of its fibers is a Lie grou that is the isotroy grou of Φ G(Φ) is, on the other hand, canonically associated with Φ, in a manner that is comatible with the algebraic structure of its fibers: It is therefore a differentiable bundle on V that is fibered by grous.

8 Ngo Van Que On the rolongations of fiber bundles and infinitesimal structures. 8 2) A sace of infinitesimal rolongation of order of a differentiable manifold V is, by definition, a fiber bundle that is associated with the Lie grouoid Π (V). 3) If E and E are two fiber bundles over V that are associated with the Lie grouoids Φ and Φ, resectively, then their Whitney roduct on V is again a fiber bundle over V that is associated with the grouoid Φ Φ that is the Whitney roduct of Φ and Φ (see ex. 3, I-1). 3. Lie subgrouoid and grouoid extension. Let Φ and Φ be two grouoids on V. A functor of Φ to Φ is a ma f: such that: and f: Φ Φ, a f = a, b f = b f(z z ) = f(z) f(z ). When Φ and Φ are differentiable grouoids, a functor is always assumed to be differentiable. Lie subgrouoid. We say that Φ is a Lie subgrouoid of the Lie grouoid Φ if there exists an injective functor of Φ into Φ. As in the case of Lie grous, an injective functor is necessarily regular; i.e., everywhere of maximal ran: Φ is realized by a differentiable submanifold of Φ. DEFINITION I.3. Let E be a fiber bundle on V that is associated with the Lie grouoid Φ. A global (differentiable) section of V in E is called regular if and only if for every air of elements x and y in V there exists a z that is an element of Φ with source and target at x and y, resectively, and is such that: z s(x) = s(y). PROPOSITION I.3.a. Any regular section s of a fiber bundle E that is associated with a Lie grouoid Φ canonically defines a Lie subgrouoid Φ of Φ. Proof. Indeed, consider: Φ = {z, z Φ, such that z s(a(z)) = s(b(z))}. Φ is a subgrouoid of Φ that is transitive on V viz., the subgrouoid that leaves the section invariant.

9 Ngo Van Que On the rolongations of fiber bundles and infinitesimal structures. 9 Φ is a Lie subgrouoid if Φ is a differentiable submanifold of Φ. Now, in fact, it will suffice for us to consider things locally i.e., to suose that Φ is a trivial Lie grouoid R n G R n (see corollary I-1) and that E is a trivial fiber bundle R n F, where F is a differentiable manifold on which the Lie grou G oerates that is associated to F in the following manner: If z Φ, z = (x, g, y), and e E, with e = (y, f) then Moreover, let there be the regular section s: z e = (x, g f). s : R n R n F, x (x, s(x)). Consider the differentiable ma: S: R n G R n F F, (x, g, y) (s(x), g s(y)). It is clear that Φ is the set S 1 ( ), where is the diagonal submanifold of F F. However, since the section s is regular, one may suose that G oerates transitively on F, because otherwise one may tae the orbit submanifold G s(x) in lace of F. Furthermore, the ma S is transversal to, so from the Thom transversality theorem, S 1 ( ) is a closed differentiable submanifold of Φ. Q. E. D. Examles. An infinitesimal structure of order on the differentiable manifold V is the given of a (differentiable) section of a fiber bundle that is associated with the Lie grouoid Π (V). The infinitesimal structure is regular if that section is regular. It thus determines a Lie subgrouoid of Π (V), and that subgrouoid is what one calls a G-structure on V (G, a subgrou of L that is isomorhic to the isotroy subgrou of a subgrouoid). n 1) Let T * denote the cotangent bundle of V, so T * is associated with the Lie grouoid Π 1 (V). The symmetric roduct S 2 (T * ), in the Whitney sense, of T * with itself is again associated with Π 1 (V). Having said this, a seudo-riemannian structure on V is the given of a non-zero regular section of S 2 (T * ). 2) The exterior roduct Λ 2 T *, in the Whitney sense, of T * is liewise associated with Π 1 (V). A section of Λ 2 T * that is everywhere of the same ran is a regular section. In the case where the ran of the section (viz., the 2-form) is everywhere equal to the dimension of V, which must then be even, from a theorem of Leage, one has what one calls an almost-symlectic structure on V. An almost-symlectic structure is therefore regular.

10 Ngo Van Que On the rolongations of fiber bundles and infinitesimal structures. 10 Grouoid extension. Let Φ and Φ be two grouoids on V. Φ is called a grouoid extension of Φ if there exists a surjective functor ϕ of Φ onto Φ. Just as in the case of Lie grous, a surjective functor of a Lie grouoid Φ onto another Lie grouoid Φ is necessarily everywhere of maximal ran: i.e., Φ is fibered over Φ. Moreover, to any extension Φ of Φ, Φ and Φ being two Lie grouoids, there corresonds the following exact sequence of grou bundles on V: 1 N(ϕ) G(Φ) G(Φ ) 1. (1) If we call any functor ρ of Φ to Φ such that ϕ ρ = Id a reduction of Φ to Φ then to any reduction of Φ to Φ there corresonds a slitting of the exact sequence of grou bundles. Moreover, that slitting is regular, in the sense that for any air of elements x and y of V there exists a z, which is an element of Φ with source and target at x and y, resectively, such that: Conversely, we have the roosition: g G(Φ ), z ρ(g) z 1 = ρ(ϕ(z) g ϕ(z) 1 ). PROPOSITION I.3.b. If Φ is a Lie grouoid extension of the Lie grouoid Φ then any slitting of the exact sequence of grou bundles (1) determines a reduction of Φ to Φ when it has the roerty that for any oint x of V, G x (Φ ) oerates on N(ϕ) x by the adjoint oeration with no other fixed oint than the neutral element. Indeed, let ρ denote the lift of this slitting. One can immediately confirm that the set of elements z of Φ such that: z ρ(g) z 1 = ρ(ϕ(z) g ϕ(z) 1 ) for any g of G x (Φ ) with x = a(z) is a Lie subgrouoid of Φ that is isomorhic to Φ by the functor ϕ.

11 CHAPTER II PROLONGATIONS OF FIBER BUNDLES AND DIFFERENTIAL OPERATORS Suose is given (E,, V), i.e., a differentiable manifold E fibered over the differentiable manifold V by the submersion. Let J (E,, V), or, when there is no ris of confusion, simly J (E), the set of jets of order of (differentiable) sections of E. It is again a differentiable manifold that is fibered over V by the source ma that is called the rolongation of order of the fiber bundle E, and if s is a differentiable section of E then the ma: j s : V J (E,, V), x jx s is a differentiable section of V in J (E,, V). If E is a vector bundle then we can show that the rolongation of the bundle is also a vector bundle and by defining the Sencer oerator on the rolongation of the bundle we can mae a contribution to the study of differential oerators. 1. Prolongation of Lie grouoids. If Φ is a Lie grouoid on V then consider the set: Φ J (Φ, a, V), which is such that if X is a jet of order of a section of (Φ, a, V) then X Φ if and only if bx Π (V), bx denoting the comosition of jets. V. PROPOSITION II.1.a. The set Φ admits a canonical structure of a Lie grouoid on Indeed, consider the following mas to be the source and target mas of Φ onto V: a : Φ V, X α(x), where α(x) is the source of the jet X, and: b : Φ V, X b(β(x)), where β(x) the target of the jet X. If X and X are two elements of Φ that satisfy a (X) = b (X ) then one defines the comosition: X X = (X bx ) X, where X bx is the comosition of jets, and the dot in the right-hand side is comosed in the following manner: If Z = j x f and Z = j x g, f and g being two differentiable mas of

12 Ngo Van Que On the rolongations of fiber bundles and infinitesimal structures. 12 W into Φ such that a f = b g then Z Z is the jet j x ( f g) of the ma f g (see definition I-1-b), a jet that deends uon only the jet of order of f and g at the oint x. When endowed with this law of internal and artial comosition, Φ is obviously a grouoid on V. To show that it is, in fact, a Lie grouoid on V it will suffice to consider things locally; i.e., to suose that Φ is a trivial Lie grouoid R n G R n, and this is left to the reader. Q. E. D. PROPOSITION II.1.b. If E is a differentiable fiber bundle that is associated with a Lie grouoid Φ then the rolongation of the bundle J (E) is canonically associated with the Lie grouoid Φ. Indeed, let Z Φ, with a (Z) = x, and let X J (E) with source x. Set: Z X = (Z (b Z) 1 ) (X (bz) 1 ), where the elements between arentheses are the comositions of jets and the dot in the right-hand side is defined as follows: If Y = j x f and Y = j x g, f and g being two mas of W into Φ and E, resectively, such that a f = g then one has Y Y = j ( f g), which is the jet of the ma f g of W into E (see definition II.2), a jet that deends uon only the jet of order of f and g at x. Z X is then a jet of a section of E with source y ( = b (Z)), and Φ thus oerates on J (E). For the condition of differentiability (axiom 2 of definition I.2), it is again obviously sufficient to regard matters locally; i.e., to suose that Φ is a trivial Lie grouoid R n G R n, and that E is the trivial bundle R n F, G being a Lie grouoid that oerates on the manifold F; this is left to the reader. x Q. E. D. Φ will be called the rolongation of order of the grouoid Φ. It is an extension grouoid of the roduct grouoid Φ Π (V) by the canonical functor: ρ: Φ Φ Π (V), Z (β(z), bz). Furthermore, let ρ r denote the canonical ma that associates any jet of order with the jet of lower order r. When alied to Φ, it is a surjective functor of Φ onto Φ r. When alied to J (E), it is a V-morhism of surjective bundles of J (E) onto J r (E), and one has: Z Φ, X J (E), ρ r (Z X) = ρ r (Z) ρ r (X).

13 Ngo Van Que On the rolongations of fiber bundles and infinitesimal structures Prolongation of vector bundles. In all of what follows, E will denote a (differentiable and locally trivial) vector bundle on a manifold V. Recall that Π(E) viz., the set of all linear isomorhisms of fibers of E to fibers of E is a Lie grouoid to which the fiber E is associated. PROPOSITION II.2.a. The bundle rolongation J (E) is a vector bundle. Indeed, if J (E) is a fiber bundle that is associated with the Lie grouoid Π (E) then it suffices to show that each fiber of J (E) is a vector bundle such that if Z is an element of Π (E) with source x and target y then Z is a linear isomorhism of J (E) x onto J (E) y. Therefore, let X = j x s and X = j s (s and s being two sections of E); set: x X + X = j ( s + s ), λ R, λx = j ( λ s). x When endowed with this law of comosition, J (E) x is obviously a vector bundle, and it is easy to confirm that the elements of Π (E) are linear isomorhisms of the fibers of J (E) to other fibers. Q. E. D. PROPOSITION II.2.b. For any integer, we have the following exact sequence of vector bundles over V: 0 E S (T * 1 ) J (E) ρ J 1 (E) 0, where E S (T * ) is the tensor roduct, in the Whitney sense, of E with S (T * ), which is the symmetric roduct, in the Whitney sense, of examles of the cotangent bundle T * of the base manifold V, and ρ 1 is the canonical morhism that associates any jet of a section of order with the jet of lower order 1. This roosition is an immediate consequence of the lemmas above. LEMMA 1. At any oint of V, we have the following exact sequence of vector saces: 0 E S (T * ) x J (E) ρ 1 x J 1 (E) x 0. Indeed, in order to rove the lemma, one may obviously suose that E is the trivial bundle R n F, n being the dimension of V, and F, a vector sace that is isomorhic to the fiber of E. Since J (E) is the trivial bundle R n T ( F ), denoting the set of jets of order of source 0 in R n into F by T ( F ) the lemma is nothing but the exact sequence of vector saces: n n x

14 Ngo Van Que On the rolongations of fiber bundles and infinitesimal structures E S (R n* 1 ) T ( F ) ρ n Tn 1 ( F) 0, an exact sequence that one establishes immediately by means of the olynomial reresentation of jets of T ( F ) by starting with the given of a basis for F and the canonical basis for R n. n LEMMA 2. For any Z that is an element of Π (E) with source x and target y, we have the following isomorhism of exact sequences: 0 E S (T * ) x J (E) x J 1 (E) x 0 ρ 0 (Z) Z ρ 1 (Z) 0 E S (T * ) y J (E) y J 1 (E) y 0 where ρ 0 is the canonical functor of Π (E) onto Π(E) Π 1 (V). In order to establish this isomorhism, it again suffices to mae a local study; i.e., by trivializing E in the neighborhood of x and y, which is left to the reader. Remar. Q. E. D. Recall that the vector bundles on V form an additive category. It is easy to establish that: J : E J (E) is an exact functor of that additive category into itself, and if h is a V-morhism of differentiable vector bundles: h : E E then one has: ρ 1 J (h) = J 1 (h) ρ 1, and, in articular, the restriction of J (h) to the sub-bundle E S (T * ) has its values in the sub-bundle E S (T * ), and that is nothing but the morhism h Id. 3. D oerator and the Sencer exact sequence. From the roosition II.2.b, we thus have, in articular: 0 J (E) T * J 1 [J (E)] ρ J (E) 0.

15 Ngo Van Que On the rolongations of fiber bundles and infinitesimal structures. 15 One confirms immediately that J +1 (E) is a differentiable sub-bundle of J 1 [J (E)], and that the restriction of the morhism ρ to the sub-bundle is nothing but the canonical morhism ρ. Consider a differentiable section s of J +1 (E); s and j 1 (ρ s) are two sections of J 1 [J (E)], which comose with the morhism ρ to give the same section of J (E). From the receding exact sequence, j 1 (ρ s) s is a section of J (E) T *. Therefore, we define an oerator D; i.e., an R-linear V-morhism of the sheaf of differentiable sections of J +1 (E) with values in the sheaf of differentiable sections of J (E) T * : D: J +1 (E) J (E) T *, s j 1 (ρ s) s, an oerator that we call the Sencer oerator (see Sencer ( 2 )). LEMMA 1. If s is a differentiable section of J +1 (E) and f is a differentiable function on the base manifold V then one has: where df is the exterior differential of f. D(fs) = f D(s) + (ρ s) df, This lemma is an immediate consequence of the following remar: If s is a differentiable section of the vector bundle E, and f is a differentiable function on the base manifold V then one has: j 1 (fs) = f j 1 (s) + s df, where s df is a differentiable section of E T*, which is a vector sub-bundle of J 1 (E), from roosition II.2.b Consider a V-morhism h of differentiable vector bundles: h: E E. From the remars made at the end of aragrah II.2, it is immediate that we have: LEMMA 2. D J +1 (h) = (J (h) Id) D. In articular, J +1 (E) is a differentiable sub-bundle of J r [J r+1 (E)], and the restriction of J r (ρ r ) to that subsace is nothing but the canonical morhism ρ 1. ( 2 ) Recall that the sheaf of (differentiable) sections of a vector bundle is a sheaf of D-modules, so it is, in articular R-linear, D being the sheaf of differentiable functions on the base manifold. We also oint out that in this aer any vector bundle and its sheaf of sections will be reresented by the same symbol, the context maing it recise in each case which interretation one must consider.

16 Ngo Van Que On the rolongations of fiber bundles and infinitesimal structures. 16 LEMMA 2. D ρ +1 = (ρ r Id) D. Finally, s being a differentiable section of J r+1 (E), we say that section is integrable if and only if s = j +1 σ, where σ is a section of E. LEMMA 3. If s is a differentiable section of J r+1 (E) then one has D(s) = 0 viz., the zero section of J r+1 (E) T * if and only if s is an integrable section. Indeed, it is clear from the definition itself of the oerator D that if s is an integrable section then D(s) = 0. We rove that if D(s) = 0 then s is an integrable section. Now, the roerty is obvious when = 0; it thus suffices to rove this by recurrence on the integer. Consider the section ρ s of J (E); from Lemma 2, one has: D(ρ s) = (ρ 1 Id) D(s) = 0. Since D(ρ s) is zero, by the recurrence hyothesis, one has: σ being a section of E. One then has: ρ s = j σ, D(s) = j 1 (ρ s) s = 0, s = j 1 (ρ s) = j 1 (j σ) = j +1 σ. THEOREM II.3.a. If E is a differentiable vector bundle on V then there exists one and only one oerator D of J +1 (E) into J (E) T * such that: 1) If s is a differentiable section of J +1 (E) then D(s) = 0 i.e., the zero section of J (E) T * if and only if s is an integrable section. 2) If f is a differentiable function on V then: in which df is the exterior differential of f. D(fs) = f D(s) + (ρ s) df From lemmas 1) and 3), the reviously-defined Sencer oerator verifies the two roerties of the theorem,. It remains for us to see that these roerties are indeed characteristic, or that if D is an oerator of J +1 (E) into J (E) T * that verifies these two roerties then one has D = D, the Sencer oerator. Now, these two oerators are identical if they coincide on the local sections of J +1 (E). Therefore, let s be a local section of J +1 (E): s = f i j +1 σ i (1 < i < q),

17 Ngo Van Que On the rolongations of fiber bundles and infinitesimal structures. 17 in which f i and σ i are differentiable functions on V and differentiable sections of E, resectively, and one has, from roerties 1) and 2): The roerty 2) ermits us to rolong in a natural fashion: D(s) = j σ i df i = D (s). Q. E. D. D: J +1 (E) Λ T * J (E) Λ +1 T *, s ω D(s) ^ω + (ρ s) dω, in which the notations have an obvious significance. Liewise, for the rolonged oerator we have: D (ρ Id) = (ρ Id) D. (1) Denoting the comosed oerator D D by D 2, we have: D 2 : J +1 (E) Λ T * D J (E) Λ +1 T * D J 1 (E) Λ +2 T *. LEMMA 1. D 2 = 0, the zero oerator, which associates any differentiable section of J +1 (E) Λ T * with the zero section of J 1 (E) Λ +2 T *. Indeed, since it is the comosition of two oerators, D 2 is again an oerator. It suffices for us to verify that for any local section s of J +1 (E) Λ T *, one has D 2 (s) = 0. Now, locally: s = j +1 σ i ω i (1 i q), where σ i are differentiable sections of E, and ω i are exterior -forms on V. Thus, one has: D(s) = j σ i dω i, D 2 (s) = j 1 σ i d 2 ω i = 0, because d 2 = 0. The restriction of the oerator D to the sub-sheaf: E S +1 (T * ) Λ T * Q. E. D. of the sheaf J +1 (E) Λ T * is, in fact, D-linear, and thus defines a V-morhism of differentiable vector bundles from: E S +r (T * ) Λ T * into J (E) Λ +r T * that is denoted by δ. Formula (1) above shows that this morhism taes its values in the vector sub-bundle E S (T * ) Λ +1 T *, and one sees that δ is nothing but the morhism:

18 Ngo Van Que On the rolongations of fiber bundles and infinitesimal structures. 18 δ: E S +r (T * ) Λ T * E S (T * ) Λ +r T *, e a +r ω ( + 1) e a (a ^ ω). We then have the following lemma, which was established by Koszul (Séminaire de Cartan, theorem, exosé 20, ). LEMMA 2. The following sequence of vector bundles is exact: 0 E S +1 (T * ) δ E S (T * ) T * δ E S 1 (T * ) Λ 2 T * δ THEOREM II.3.b. For any integer, we have the following exact sequence of R- linear sheaves: 0 E 1 j + J +1 (E) D J (E) T * D J 1 (E) Λ 2 T * D in which j +1 is the canonical oerator that associates any differentiable section of E with the differentiable section j +1 s of J +1 (E). (It is intended that in Lemma 2 and the theorem above that in everything we have adoted the following convention: and: S 0 (T*) = R V, the trivial fiber bundle over V, S (T*) = 0, if < 0, J 0 (E) = E, J (E) = 0 if < 0). Proof. 1) We indeed have the following exact sequence: 0 E 1 j + J +1 (E) D J +1 (E) T *, from the characteristic roerty 1) of the oerator D and the fact that the oerator j +1 is obviously injective. 2) The sequence that we defined is cohomological because D 2 = 0 (Lemma 1). It remains for us to rove the exactness of the sequence. Now, it is clear that the following sequence is exact: J 1 (E) Λ T * D E Λ +1 T * 0, since the restriction of the oerator D to the sub-sheaf E T * Λ +1 T * is already surjective, from Lemma 2.

19 Ngo Van Que On the rolongations of fiber bundles and infinitesimal structures. 19 We then rove by recurrence on the integer that we have the following exact sequence of R-linear sheaves: J +1 (E) Λ 1 T * D J (E) Λ T * D J 1 (E) Λ +1 T *. Therefore, let s be a differentiable section of J (E) Λ T * such that D( s ) = 0. The section ρ 1 s of J (E) Λ T * is also such that D(ρ 1 s ) = (ρ 1 Id) D( s ) = 0. Then, by the recurrence hyothesis, there exists a section that: 1 D( σ ) = ρ 1 s. Since the morhism ρ is surjective, we may find Λ 1 T * such that: 1 ρ s = σ. 1 s σ of J (E) Λ 1 T * such, which is a section of J +1(E) 1 Consider the section s D( s + 1 ) of J (E) Λ T *. It is, in fact, a section of the subbundle E S (T * ) Λ T *, because: ρ ( s D( s )) = ρ s D( ρ s )) = It is, moreover, annulled by the morhism δ: δ ( s D( s )) = D( s D( s + 1 )) = D( s ) = 0. From lemma 2, there thus exists a section n + of: 1 1 such that: Hence: E s 1 (T * ) Λ 1 T *, δ ( ) 1 n + 1 = s = 1 s D( s + 1 ). D( n + s ), when n + s is a section J +1 (E) Λ 1 T * Q. E. D. If we let ρ denote the canonical morhism of J 1 [J (E)] onto J (E), and again let D be the Sencer oerator of J 1 [J (E)] into J (E) T * then the restriction of this morhism and that oerator to the differentiable sub-bundle J +1 (E) of J 1 [J (E)] are the mas that were considered above that are denoted by the same letters.

20 Ngo Van Que On the rolongations of fiber bundles and infinitesimal structures. 20 COROLLARY. In order for a differentiable section s of J 1 [J (E)] to be a section with values in the sub-bundle J +1 (E), it is necessary and sufficient that: 1) D(ρ s) = ρ 1 D(s), 2) D 2 (s) = 0, the zero section of J 1 (E) Λ 2 T *. The conditions are obviously necessary. They are sufficient; indeed, let s be a section of J 1 [J (E)] such that conditions 1) and 2) are verified. D(s) is then a section of J (E) T * such that: D D(s) = D 2 (s) = 0, so, from the last theorem, there exists a section σ of J +1 (E) such that: D(σ) = D(s). However, s σ is then a section of J 1 [J (E)] such that: D(s σ) = D(s) D(σ) = 0. From the first characteristic roerty of the oerator D, s σ = j 1 χ, with χ a differentiable section of J (E), and one has: D(χ) = D(ρ s ρ s) = (ρ 1 Id) D(s σ) = 0, χ = j η, with η a section of E and: s = j 1 χ + σ = j 1 (j η) + σ, = j +1 η + σ, a section of J +1 (E). Q. E. D. We conclude this aragrah by maing the remar that the Sencer oerator D slits the sheaf J 1 (E) into a direct sum of two R-linear sheaves E and E T *, or more recisely, two sections s and s of J (E) are identical if and only if: and: ρ 1 s = ρ 1 s, D(s) = D(s ). 4. A differential oerator and its rolongation. Let E and F be two vector bundles over the same base V. Recall that we call any R- linear V-morhism of the sheaf E into the sheaf F an oerator.

21 Ngo Van Que On the rolongations of fiber bundles and infinitesimal structures. 21 DEFINITION II.4. An oerator of E into F will be called a differential oerator of order if, s being a differentiable section of E, jx s = 0 entails that the (s) in F is zero at the oint x. Any V-morhism of differential fiber bundles is obviously equivalent to a differential oerator of order 0, which is nothing but a D-linear oerator (D, the sheaf of differentiable functions on the base manifold V). The Sencer oerator D is, from its second characteristic roerty, a differential oerator of order 1 of J (E) into J 1 (E) T *. If is a differential oerator of order of E into F, and is a differential oerator of order r of F into G then the comosition is obviously a differential oerator of order r + of E into G. Consider the oerator: j : E J (E). It is a differential oerator of order, and one easily establishes the following theorem (see R. Palais, cha. IV, Seminar on the Atiyah-Singer Index Theorem). THEOREM II.4. To any differential oerator of order of E into F there corresonds one and only one V-morhism of the vector bundle h( ) of J (E) into F such that: = h( ) j. Let be a differential oerator of order of E into F. We call the oerator j that mas E into J r (F) the rolongation of order of. We have, in an obvious fashion: h(j ) = J r (h( )), or more exactly, the restriction of the V-morhism J r (h( )) to the sub-bundle J +1 (E) of J r [J (E)]. For any integer r, let: S r = er(h(j r )) be the sub-sheaf of -modules of the sheaf J r (E) that is formed from the sections that are annulled by the oerator h(j r ), and agree that: and that: S r = J r (E) for r <, S r = S r Λ T * is the tensor roduct, in the Whitney sense, of the two sheaves of D-modules. From the definition of h(j r ), we have: 1) ρ r Id: S r+ 1 S r,

22 Ngo Van Que On the rolongations of fiber bundles and infinitesimal structures. 22 2) D : S r+ 1 S. + 1 r Hence, one has the Sencer cohomology sequence relative to the differential oerator : 0 Θ r j S r D S r 1 D 2 S r 2 D, where Θ denotes the sheaf of solutions of i.e., of sections of E that are annulled by the oerator. For r, the start of the sequence: r j D 0 Θ S r S r 1 is obviously exact. Uon denoting the sub-sheaf of D-modules of S r+ 1, we have the cohomology sequence of sheaves of D-modules: N N N, δ q δ q 1 δ δ δ r+ q r + q 1 r whose corresonding cohomology sheaves will be denoted by H r. Suose that the oerator ρ r 1 : S r S r 1 is surjective, so liewise for any, the oerator ρ r 1 Id: Sr S r 1 is also surjective. Let λ denote an arbitrary lift i.e. a V- morhism of sheaves: λ: S r 1 S r, (ρ r 1 Id) λ = Id. The morhism: 2 D λ D: S r S r 1 is such that: (ρ r 1 Id) D λ D = D (ρ r 1 Id) λ D = D 2 = 0. 2 It thus mas S r into N r 1 and taes values that are annulled by the oerators D or δ; it thus asses to the quotient to define a morhism: m = D λ D: S r H r 1. One immediately verifies that this morhism is a D-linear oerator ( H r 1 being obviously a sheaf of D-modules that is defined indeendently of the choice of the lift λ). THEOREM II.4.b If the oerator: ρ r 1 : S r S r 1 is surjective then this canonically defines a D-linear oerator:

23 Ngo Van Que On the rolongations of fiber bundles and infinitesimal structures. 23 m : S r 2 H r 1, such that the following sequence of sheaves of D-modules is exact: Proof. S S H. ρr m 2 r + 1 r r 1 Indeed, if the oerator m is defined as before then one confirms immediately that: m ρ r = 0. It remains for us to rove the exactness of the sequence. The reasoning consists of two stes, which we mae recise in the form of a Lemma. LEMMA 1. If s is a section of S r then there exists a section σ of J 1 [J r (E)] that verifies the following conditions: 1) J 1 [h(j r )](σ) = 0, 2) ρ(σ) = s, 3) (ρ r 1 Id) D(σ) = D(s). Indeed, tae a lift λ of 1 S r 1 to 1 S r. Let η denote the section λ D(s), which we consider to be a section of J r (E) T *, which is a sub-bundle of J 1 [J r (E)]. The section η is therefore such that: 1) J 1 [h(j r )](η) = (h(j r ) Id) λ D(s) = 0, 2) ρ(η) = 0, 3) D(η) = λ D(s). The section j 1 s is a section of J 1 [J r (E)] such that: 1) J 1 [h(j r )](j 1 s) = j 1 [h(j r ) (s) = 0, 2) ρ(j 1 s) = s, 3) D(j 1 s) = 0. The section σ = j 1 s + η is therefore the section that resonds to the conditions of the lemma. LEMMA 2. If s is a section of S r such that m(s) = 0 then there exists a section χ of J 1 [J r (E)] that verifies the three conditions of lemma 1, and the following fourth condition: 4) D D(χ) = 0. Indeed, always letting λ be a lift of S r 1 into S r, we say that m(s) = 0, i.e., that:

24 Ngo Van Que On the rolongations of fiber bundles and infinitesimal structures. 24 D λ D = δ(η), 1 with η, a section of N r. We may consider η to be a section of J r (E) T *, which is a sub-bundle of J 1 [J r (E)] such that: 1) J 1 [h(j r )](η) = 0, 2) ρ(η) = 0, 3) (ρ r 1 Id) D(η) = (ρ r 1 Id)(η) = 0. If σ is a section as in the Lemma then the section: χ = σ + η is a section of J 1 [J r (E)] that verifies the conditions of Lemma 2. Now, from the corollary to the theorem II.3.b, the section χ of Lemma 2, in fact, has its values in the sub-bundle J r+2 (E) of J 1 [J r (E)]. It is therefore a section of S r+1 such that ρ r (χ) = s. Remar. Q. E. D. 1) One of the imortant roblems of analysis is to now whether the Sencer cohomology sequence relative to a given differential oerator is exact or not ( 3 ). 2) being a differential oerator of order, for r, the exactness of the sequence: 0 Θ r j S r D S r 1, signifies that the integrable sections j r s of S r, are nothing but the rolongations of the sections of E that are solutions of, i.e.: (s) = 0. We say that the differential oerator of order is comletely integrable to order r ( ) if and only if S r is locally generated by the integrable sections, i.e.: If s is a section of S r then one has locally, in the neighborhood of any oint of V: s = f i j r σ i, 1 i, where f i are differentiable functions on the base manifold V, and j r σ i are integrable sections of S r. If the differential oerator is comletely integrable of order r then the morhism: ( 3 ) Indeed, D. G. Quillen has roved that showing the exactness of the Sencer sequence is equivalent to finding the necessary and sufficient conditions on the section f of F for there to exist a section s of E such that (s) = f.

25 Ngo Van Que On the rolongations of fiber bundles and infinitesimal structures. 25 is obviously surjective. The oerator m: ρ r : S r+1 S r m: S r+1 2 H r 1 if it defined is then zero. The oerator m thus resents us with an obstruction, in an obvious sense, to the comlete integrability of the differential oerator of order r. 3) Let E r denote the set: {e, e J r (E) such that h(j r )(e) = 0}. The set E r is what one calls the linear differential system that is associated with the differential oerator j r of order r. The sets E r are not necessarily vector subsaces of the fibers of J r (E), but fibered by fibers over V, so each fiber of E r is a vector subsace of the fiber of J r (E); we may thus form the tensor roduct, in the Whitney sense, of E r with Λ T *, and we obviously have: S r = E r Λ T *, in which the right-hand side denote the differentiable sections of J r (E) Λ T * with values in the subset of E r Λ T *. We say that the differential system E is homogeneously linear if E is a vector subbundle of J r (E). Any homogeneous, linear, differential system can be considered as the associated system to the differential oerator of order : j : E j J (E) J (E) / E, where J (E) / E denotes the quotient bundle of J (E) by the sub-bundle E, and is the canonical rojection morhism. 4) Theorem II.4.b is a revised form of a nown roosition of Quillen (Singer and Sternberg, On the infinite grous of Lie and Cartan. ) In the analytic case, it is equivalent (E and F being analytic vector bundles, and the differential oerator of order transorting any analytic section of E to an analytic section of F) to the celebrated Cartan-Kaehler theorem on involutive linear differential systems (C. Buttin, Existence of local solutions for analytic systems of equations. ).

26 CHAPTER III CONNECTIONS OF HIGHER ORDER IN A VECTOR BUNDLE In this chater, we shall introduce the theory of connections, whose role one nows of from differential geometry. Our essential result is that if E is a vector bundle that is associated with a Lie grouoid Φ then a connection of order, in the sense of C. Ehresmann, in the Lie grouoid Φ canonically determines a slitting of the exact sequence of vector bundles: 0 J 0 ( E ) J (E) ρ E 0. Conversely, any slitting of this sequence is determined by a connection of order in the Lie grouoid Π(E), the grouoid of all linear isomorhisms from fibers of E to fibers. 1. Connection is a Lie grouoid. Let Φ be a Lie grouoid on the differentiable base manifold V. From geometric considerations, which we shall not recall, C. Ehresmann was led to mae the following definition: DEFINITION III.1.a A connection element of order in Φ is a jet of a section X: X J (Φ, b, V), such that: 1) β(x) = l α(x), α(x) and β(x) are the source and target of the jet X, resectively. j α 2) ax = ( X ), the jet of order of the constant ma that mas V to the oint α(x). PROPOSITION III.1a. The sace Q (Φ) of all connection elements of order of Φ is a differentiable fiber bundle over V under the source rojection a that is associated with the rolonged Lie grouoid Φ. Proof. Indeed, let X be a connection element of source x, and let Z be an element of Φ with source x and target y. Let Y denote: Y = bz, Y Π (V) (see cha. II-1). One has a new connection element X of source y: X = (Z Y 1 ) (X Y 1 ) β(z) 1,

27 Ngo Van Que On the rolongations of fiber bundles and infinitesimal structures. 27 where the elements between arentheses are the comositions of jets, β(z) 1 must be considered as the jet of order of the constant ma of V onto β(z) 1, and the dots have the same significance as they did in the roof of roosition II.1. Set X = Z X; it is then clear that Φ oerates on Q (Φ). It remains for us to verify the differentiability conditions, which we may do locally. Now, if Φ is trivial Lie grouoid: Φ = R n G R n then one has Q (Φ) = T, ( G ) R n, where T, ( G ) is, we recall, the set of jets of order n e n e of R n into G with source 0 and the neutral element e of G for target. Recall that Φ is then a trivial grouoid Φ = R n G R n, where G is the semi-direct roduct of T, ( G ) with the roduct grou G L n. This being the case, the reviously-determined oerator Φ on Q (Φ) is defined by the oeration of G on T, ( G ) in the following manner: Z G, Z = (g, Y) X, with (g, Y) G n e L and X T, ( G ), Z: T, ( G) T, ( G ), X (g, Y) X X (g, Y) 1, n e n e which are roducts of elements of G that belong to the subgrou T, ( G ). This indeed roves that Q (Φ) is a differentiable fiber bundle that is associated with the Lie grouoid Φ. Q. E. D. DEFINITION III.1b. A connection of order in Φ is the given of a differentiable section of the bundle Q (Φ). Since the fiber of Q (Φ) is isomorhic to T, ( G ), it is therefore contractible. We remar that since the differentiable manifold V is assumed to be aracomact, there always exists a connection of order in the Lie grouoid Φ. From the receding, we remar that the isotroy grou G of Φ oerates transitively on the fiber Q (Φ) x. Any section of Q (Φ) is therefore regular; it canonically determines a Lie subgrouoid of Φ, namely, the subgrou that leaves it invariant. It is easy to rove that this subgrouoid is isomorhic to the grouoid Φ Π (V) by the surjective functor of Φ onto Φ Π (V). We thus have the roosition: PROPOSITION III.1b. Any connection of order in Φ canonically defines a reduction of Φ Π (V) in Φ. Liewise, one immediately establishes that: n e x n n e n e n e

28 Ngo Van Que On the rolongations of fiber bundles and infinitesimal structures. 28 PROPOSITION III.1c. If ϕ is a reduction of Φ Π (V) in Φ such that at a certain oint x of V the isotroy grou G of ϕ(φ Π (V)) leaves invariant a connection element of order at x of Φ then there exists a connection of order of Φ that defines that reduction. 2. Connections in a vector bundle. Suose one is given (E,, V); i.e., a differentiable manifold that is fibered by a submersion on V. Let F (E,, V) or simly F (E), when there is no ris of confusion denote the set of jets of order of V into E, such that: X F (E), X = j x is the jet of order of the constant ma of V onto the oint x = α(x), which is the source of X. F (E) is obviously a differentiable manifold that is fibered by the source ma α onto V. In a more recise fashion, we have the roosition: PROPOSITION III.2. If E is a fiber bundle that is associated with the Lie grouoid Φ then F (E) is a fiber bundle that is associated with the roduct Lie grouoid Φ Π (V). The roof is identical to that of roosition II.1b, if we remar that the grouoid Φ Π (V) oerates on F (E) in the following fashion: If (z, Y) Φ Π (V), with source x and target y, X F (E), with source x, and: X = j x g, where g is a ma of V into the fiber E x, Y = jx f, where f is a local diffeomorhism of V, then one has: (z, Y) X = j y (z (g f 1 )), in which z (g f 1 ) is, by definition of the oeration (denoted by a dot) of Φ on E, a differentiable ma that is defined in a neighborhood of y from V into the fiber E y. THEOREM III.2a. If E is a fiber bundle that is associated with the Lie grouoid Φ then any connection of order in Φ determines a V-isomorhism of differentiable bundles of F (E) into J (E). Indeed, let C x be a connection element of order at x of Φ: C x = jx f, where f is a differentiable ma that is defined in the neighborhood of x from V into Φ such that: a f = ˆx, the constant ma of V to x, b f = Id.

29 Ngo Van Que On the rolongations of fiber bundles and infinitesimal structures. 29 C x determines a diffeomorhism of F (E) x with J (E) x in the following manner: C x : F (E) x J (E) x, X = j g X = j ( f g), x x where g is a ma of V into the fiber E x and f g is a section of E that is defined in a neighborhood of x. Therefore, a connection C of order in Φ determines a bijective ma of F (E) into J (E) that diffeomorhically transforms the fiber F (E) x onto the fiber J (E) x at any oint x of V. In order for this ma to be a differentiable isomorhism of fibers, it remains for us to rove that if s is a differentiable section of F (E) then C s is a differentiable section of J (E); one may do this by (locally) trivializing E. Obviously, we have the following injective V-morhism: i: E F (E), e j eˆ, x Q. E. D. where x = (e), and ê is the constant ma of V to the oint e. When it is comosed with the morhism C that is defined by a connection, we have an injective V-morhism: such that: C i: E J (E), ρ C i : J (E) E (where r is the canonical morhism): ρ C i = Id. Indeed, let e be in E such that (e) = x, and let: so we have: Thus: C x = jx f, C i(e) = j x (f ê ). ρ C i(e) = f(x) e = e, because, from the definition of a connection element at x, one has: f(x) = l x, the unit at x of Φ. In the case where E is a differentiable vector bundle, F (E) is also a vector bundle and the reviously-defined morhisms i and C are morhisms of vector bundles. We thus have:

30 Ngo Van Que On the rolongations of fiber bundles and infinitesimal structures. 30 COROLLARY If E is a vector bundle that is associated with a Lie grouoid Φ then any connection of order in Φ canonically determines a slitting of the following exact sequence of differentiable vector bundles: 0 J 0 ( E ) J (E) ρ E 0. In the case of vector bundles, we also have the following theorem: THEOREM III.2b. If E is a differentiable vector bundle over V then any slitting of the exact sequence: 0 0 ρ J ( E ) J (E) E 0 is determined by a connection of order in the Lie grouoid Π(E) of all linear isomorhisms of fibers of E onto fibers that is associated with the bundle E. Proof. Let λ be a lift of the slitting, and let (e 1, e 2,, e q ) be a basis system for the vector sace E x, which is the fiber of E at x. Set: λ(e i ) = j s, x i where one may obviously tae differentiable sections s i that are defined in the same neighborhood of x in V, and are such that in this neighborhood the s i form a basis for the sheaf E of locally free D-modules. On this same neighborhood of x, consider the differentiable section f of (Π(E), b, V) such that: 1) a f = ˆx, ˆx : V x, 2) f(y) e i = s i (y). One must also have: because for any i: f(x) = l x, the unit of Π(E) at x, f(y) e i = s i (y) = e i. The jet X = jx f is then a connection element of order at x of Π(E), and this connection element is obviously defined in a manner that is indeendent of the choice of basis (e 1,, e q ) of E, and the lift λ at the oint x recisely. To any oint x of V, we thus associate a connection element or order in Π(E); in other words, we have a section that we denote by C of V in the connection sace Q [Π(E)]; it is a differentiable section. Indeed, let Φ be the subgrouoid of Π (E) leaves this section invariant. It is a transitive subgrouoid on V, because Π (E) oerates in a transitive fashion on the sace Q [Π(E)]. The subgrouoid Φ is also the subgrouoid of Π (E) that leaves the lift invariant, considered as a differentiable section of J (E) E *