On The Equivalence Problem for Geometric Structures, II
aa r X i v : . [ m a t h . DG ] D ec ON THE EQUIVALENCE PROBLEM FORGEOMETRIC STRUCTURES, II
ANTONIO KUMPERA
Abstract.
This paper is a continuation of Part I where the gen-eral setup was developed. Here we discuss the general equivalenceproblem for geometric structures and provide criteria for the equiv-alence, local and global, of transitive structures. Cartan’s Flag Sys-tems illustrate the theory as a major example and, finally, someattention though little is given to non-transitive structures withregular orbits i.e. , intransitivity classes. Introduction
In the part I, we recalled the theory of Differential Invariants as con-ceived by Sophus Lie and placed it in a context best suited for the studyof the equivalence problem for geometric structures. It is interesting toadd some more remarks proper to what we shall discuss in the sequeland, firstly, let us recall what the Finiteness Theorem for the differentialinvariants, proved in Part 1, brings to the equivalence problem. A germof a structure S being fixed, we are interested in the formal equivalence,general or restricted, of S with the germs of neighboring structures orthose eventually located on other manifolds. We are as well interestedin global equivalences, these however offering several additional diffi-culties of quite a different nature since globalization is always a hardtopological task. Let us now place ourselves back into the neighbor-hoods W k as described in the statement of the Finiteness Theorem ofthe Part I. The constancy of the k − th order differential invariants pro-vides us with a necessary and sufficient condition for the k − th orderequivalence namely: The k − jet of a structure S’ is equivalent to the k − jet of S if and only if the k − th order differential invariants assumethe same values on both jets, these two jets belonging therefore to thesame orbit in W k . Similarly, two infinite jets of structures, elements of Date : June 2014.2010
Mathematics Subject Classification.
Primary 53C05; Secondary 53C15,53C17.
Key words and phrases. prolongation spaces · structures · equivalence · differen-tial invariants. lim proj W k , are equivalent whenever they are so at every finite order.Since from an integer µ onwards the higher order differential invariantsare all obtained by iterated formal admissible derivations of those oforder ≤ µ , we infer that the infinitely many equivalence conditions areall consequences of those of order ≤ µ and, moreover, just of a finitenumber of them, namely a finite fundamental system of such invariants.In the next section we shall examine the formal equivalence problem forthe formally transitive structures and, in later sections, more specifictypes of structures will also be discussed. As for the first mentionedstructures, we shall see that the formal equivalence takes place if andonly if the restrictions of the differential invariants of orders ≤ µ tothe flows of the same order associated to these structures assume thesame values and, further, the restrictions of the derived invariants tothe flows of higher orders vanish. We shall see as well that when theflows of orders ≤ µ associated to the germs of two structures are trans-verse to the trajectories - the opposite of transitivity - then a necessaryand sufficient condition for the formal equivalence is the existence of agerm of local diffeomorphism ϕ of P such that the restriction of everyinvariant of order ≤ µ to one of these jets of structure be equal tothe restriction to the other jet composed with ϕ , the same propertythereafter holding for all the invariants of higher order.It now only remains to determine criteria under which the hypotheses H and H , stated in Part I, section 9, are verified and it is precisely herethat the specific nature of the prolongation spaces E intervenes. Asfor the hypothesis H , it will be realised, for example, upon requiringthe local regularity, at a given order k , of the distributions ∆ k and ∆ k,k +1 together with the 2-acyclicity (or even involutivity) of ∆ k − ,k and arguing as in Quillen’s criterion (while assuming of course that1-acyclicity already holds on a sequence of open sets W k , this revivingnow the hypothesis H ). A more elaborate analysis (see [5]) shows thepossibility of linking the δ − complex associated to the kernels ∆ k − ,k with the corresponding δ − complex associated to the kernels of thespaces ˜ J k ( L ℓ ) and consequently, in view of the prolongation spacesstructure, with that constructed by means of the kernels of the spaces J ℓ + k L , the later introducing however a shifting, by one unit, in thecohomology groups. This matter will be discussed in more detail inlater sections. However, it is also worthwhile to mention that we canemploy, in this context, the method of characters by extending thereasoning of [6], §24, and relate the characters of the kernels of J ℓ + k L with those of ∆ k − ,k . N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, II 3
Finally concerning the hypothesis H , not much can be simplified in thespecific case of prolongation spaces. Apparently, the most efficaciouscriteria are provided, on the one hand, by the proposition 21.5 of [6]that involves the bracket of formal and holonomic derivations and, onthe other, by the proposition 25.4 that relies on the explicit formula(25.5) which, surprisingly, can already be found in Lie’s work. Thehypotheses underlying the last proposition are no other than conditionsof transversality, in the strict sense, of the k − th order flows of localsections S of the fibration E −→ P with the trajectories of ∆ k suchconditions being entirely antipodal to a formal transitivity hypothesis.We finally remark that it is possible to transcribe all the results statedabove in the case where ( E, π, P, p ) is no more than a prolongationspace relative to an infinitesimal Lie pseudo-algebra L ([2]).2. Subordinate prolongation spaces and formaltransitivity
Let us begin by stating a very naive lemma, consequence of thefunctoriality properties of the prolongation operation, that neverthelessis at the basis of everything that follows.
Lemma 1.
Let ( E, π, P, p ) be a finite prolongation space, ξ a localvector field defined on P and ϕ a local diffeomorphism such that α ( ξ ) = α ( ϕ ) . Under these conditions, ( pϕ ) ∗ pξ = p ( ϕ ∗ ξ ) . Let us next assume that the prolongation space has finite order ℓ . Inorder to render notations easier, we denote by Z Z h the projec-tion J k E → J h E and by Z → Z k − h the semi-holonomic inclusion J k − h ( J h E ) , the same notations applying as well to Π k P . We observethat the elements of J k − h ( J h E ) can be canonically identified with the ( k − h ) − th order holonomic contact elements of dimension n thatare transverse to the fibration J h E −→ P , the contact element Z k − h being issued at the point Y h . This being so, the groupoid Π ℓ + k +1 P operates to the left (or to the right if one prefers so) ona) J m E , m ≤ k + 1 , by means of the prolongation spaces law, in viewof the Lemma 2, section 2 in [9] or else by the left action (1) ( loc.cit. ,pg. 5, part I), and the projection Π ℓ + k +1 P −→ Π ℓ + k P .b) J m − h ( J h E ) , m ≤ k + 1 , by applying twice the above Lemma.c) J ℓ + k T P , by the standard action.
ANTONIO KUMPERA d) J k − h ( T J h E ) , thanks to the left action Λ k + h ( loc. cit. ) followed bythe standard action hence, in particular, on T J k E .These actions are compatible with all the fibrations and it is quite evi-dent that a) and b) are co-variant by means of the canonical inclusion.Furthermore, the preceding lemma implies that the infinitesimal pro-longation operation is compatible with these actions. In fact, taking Y ∈ Π ℓ + k +1 P , j ℓ + k ξ ( y ) ∈ J ℓ + k T P and Y ( j ℓ + k ξ ( y )) = j ℓ + k η ( y ) then, forall Z ∈ J h E composable with Y , the element transformed by Y via theaction d) of the jet j k − h ( p h ξ )( Z ) is equal to j k − h ( p h η )( Y ( Z )) . More-over, we can extend the action of Π ℓ + k +1 P to J ℓ + k T P × P J h E actingseparately on each factor. The previous considerations can now besummarized by the following statement ( cf . (7), [9], where we replace λ k by λ k + h , ℓ + k by ℓ + k + h and T J k E by J h ( T J k E ) ) Lemma 2.
The mapping λ h +( k − h ) : J ℓ + k T P × P J h E −→ J k − h ( T J h E ) is a differential co-variant with respect to the action by Π ℓ + k + h P or, inother words, this mapping commutes with both actions. In particular, we derive the following conclusions: Let Y ∈ Π ℓ + k +1 P , Z ∈ J k +1 E , Y ( j ℓ + k ξ ( y )) = j ℓ + k η ( y ′ ) and assume further that p h ξ istangent of order k − h to the contact element (of order k − h + 1 ) Z ( k +1) − h ∈ J k +1 − h ( J h E ) . Then the vector field p h η is tangent of order k − h to the contact element W ( k +1) − h , W = Y ( Z ) , since Y transforms Z ( k +1) − h onto W ( k +1) − h and j k − h p h ξ ( Z h ) onto j k − h p h η ( W h ) . Lemma 3.
The action of Π ℓ + k +1 P is compatible with the tangencyrelations between prolonged vector fields and contact elements. Let us now introduce the (eventually singular) vector bundles, withbase space J k E ,(1) R ℓ + k = { ( Y, Z ) | Y ∈ ( R ℓ + k ) Z , Z ∈ J k E } where ( R ℓ + k ) Z = { j ℓ + k ξ ( y ) ∈ J ℓ + k T P | ( p k ξ ) Z = 0 , y = α ( Z ) } and(2) g ℓ + k = ker ( R ℓ + k → R ℓ + k − ) , N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, II 5 the choice of the notations being justified by the fact that the linearisotropy as well as the symbol of orders ℓ + k of a given structure S ofspecies E can simply be obtained as the inverse image of these bundlesvia the flow j k S or, in other terms, by the relations(3) R ℓ + k ( S ) = ( j k S ) − R ℓ + k g ℓ + k ( S ) = ( j k S ) − g ℓ + k . The following sequence, where the third arrow is the morphism (7) in[9], is of course exact:(4) −→ R ℓ + k −→ J ℓ + k T P × P J k E λ k −→ ∆ k −→ The preceding lemmas show in particular that each term of this se-quence is invariant by the corresponding action of Π ℓ + k +1 P . Moreover,the action of Π ℓ + k +1 P induced on R ℓ + k factors to Π ℓ + k P since it isno other than the parts of the orders ≤ ℓ + k that operate on the to-tal isotropy J ℓ + k T P . Finally, we observe that the action of Π ℓ + k P on J ℓ + k T P × P J k E leaves invariant the sub-space g ℓ + k ⊂ ( S ℓ + k T ∗ P ⊗ T P ) × P J k E and the induced action on the tensorial term factors, via Π ℓ + k P −→ Π P ,since it is only the first order part that operates on the symbols. Lemma 4.
For all Z ∈ J k E , the symbol ( g ℓ + k ) Z is the k − th algebraicprolongation of ( g ℓ ) β ( Z ) and, consequently, g ℓ + k is entirely determinedby g ℓ via the relation g ℓ + k = g ( k ) ℓ ⊗ E J k E where g ( k ) ℓ is the k − th algebraicprolongation of g ℓ . In fact, it suffices to choose a section S of E such that Z = j k S ( y ) andapply the proposition 4 in [9]. The previous discussion also entails the Lemma 5.
Let Ω be an orbit of Π ℓ P in E. For every k, the restric-tion g ℓ | β − (Ω ) is a vector bundle of constant rank on the manifold β − (Ω ) . Further, taking any two points Z, Z ′ ∈ β − (Ω ) and assumingthat the jet Y ∈ Π ℓ P transforms β ( Z ) in β ( Z ′ ) , then ( g ℓ + k ) Z is trans-formed onto ( g ℓ + k ) Z ′ by the tensorial extension of the − jet Y , projec-tion of Y, considered as a linear transformation T α ( Z ) P −→ T α ( Z ′ ) P . ANTONIO KUMPERA
Corollary 1.
For all k, the Spencer δ − cohomology complexes asso-ciated to the symbols ( g ℓ + k ) Z and ( g ℓ + k ) Z ′ taken at any two arbitrarypoints of β − (Ω ) are isomorphic by means of the restriction of anisomorphism, of the total complexes, produced by the linear part ofan element belonging to Π ℓ P that sends T α ( Z ) P into T α ( Z ′ ) P . Con-sequently, the symbols have the same homological properties along thesub-manifold β − (Ω ) . In particular, when S is a section of E takingits values in β − (Ω ) then the symbol g ℓ + k ( S ) is a vector bundle ofconstant rank and each fibre has the same homological properties. Lemma 6.
Let Ω be an orbit of Π ℓ P in E. The for every k, the vectorbundle g ( k ) ℓ | Ω is of constant rank and is invariant by the action ofthe groupoid Π ℓ P on ( S ℓ + k T ∗ P ⊗ T P ) × P Ω , the action on thefirst term factoring by Π ℓ P −→ Π P . Furthermore, g ℓ + k | β − (Ω ) = g ( k ) ℓ × Ω β − (Ω ) . Moreover, if S is a section of E taking its values in Ω , then the groupoid R ( S ) is transitive on α ( S ) and the linear actionof ρ ( R ( S )) leaves invariant the vector bundle g ℓ + k ( S ) . Lemma 7.
Let F k be an orbit, in J k E , of the finite or infinitesimal ac-tion (i.e., an orbit Ω k or ω k ) of Π ℓ + k P and let us write F h = ρ F k , ≤ h ≤ k . Under these conditions:i) F h is a finite or infinitesimal orbit of J h E according to the natureof the corresponding k − orbit.ii) The action of Π ℓ + k +1 P on J ℓ + h T P × P F h , ≤ h ≤ k , leavesinvariant the sub-bundles R ℓ + h | F h and g ℓ + k | F h , the induced actionthen factoring to Π ℓ + k P , and operating transitively in the base space F h . The isotropies ( R ℓ + h ) Z and ( R ℓ + h ) Z ′ as well as the symbols ( g ℓ + h ) Z and ( g ℓ + h ) Z ′ , at any two points Z, Z ′ ∈ F h , are therefore isomorphicby means of the action by elements of Π ℓ + k P , resp. Π P .iii) The action of Π ℓ + k +1 P on T J h E | F h , ≤ h ≤ k , leaves invari-ant the sub-bundle ∆ h | F h , the fibres at any two arbitrary points beingisomorphic via the above action and, moreover, ∆ h | F h = T F h .iv) The restrictions R ℓ + h | F h , g ℓ + k | F h and ∆ h | F h = T F h arevector bundles of constant rank. By all that has already been said, the proof is obvious. Let us mentionhowever that, in view of the exactness of the sequence (4), the vector
N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, II 7 bundles R ℓ + h and ∆ h are simultaneously regular i.e. , when one is reg-ular then the other is also necessarily regular. We next state (withoutproof!) a rather long lemma that outlines all the basic techniques tobe used from here onwards. Lemma 8.
The hypotheses being those of the preceding lemma, for eachsolution S of F k namely, a section S of the prolongation space E suchthat j k S takes its values in F k , the equations R ℓ + h ( S ) and R ℓ + h ( S ) , ≤ h ≤ k , are transitive in α ( S ) and the bundles R ℓ + h ( S ) , R ℓ + h ( S ) and g ℓ + h ( S ) are all of constant rank. Further, R ℓ + h ( S ) is a closed andlocally trivial Lie (Differentiable) sub-groupoid, its differentiable struc-ture being regularly embedded in Π ℓ + h α ( S ) . The non-linear isotropy R ℓ + h ( S ) is a closed and locally trivial sub-bundle of Π ℓ + h α ( S ) , eachfibre being a closed Lie sub-group of the total group. The projection ρ ℓ + h − R ℓ + h ( S ) is a locally trivial Lie sub-groupoid of Π ℓ + h − α ( S ) andthe non-linear symbol g ℓ + h ( S ) = ker ( R ℓ + h ( S ) −→ ρ ℓ + h − R ℓ + h ( S )) is a locally trivial affine sub-bundle (unless when ℓ = 1 and h = 0 ,in which case the fibre is a linear group) of the total symbol above ρ ℓ h − R ℓ + h ( S ) . Finally, the groupoid R ℓ + h ( S ) leaves invariant the sub-bundles R ℓ + h ( S ) and g ℓ + h ( S ) via the standard action on J ℓ + h T P and S ℓ + h T ∗ P ⊗ T P respectively. Inasmuch, R ℓ + h ( S ) leaves invariant theisotropy R ℓ + h ( S ) via the action on Π ℓ + h P defined by conjugation, aswell as the non-linear symbol g ℓ + h ( S ) via the standard action on thetotal symbol defined by the translations to the left or to the right. Wededuce that any two fibres of R ℓ + h ( S ) and g ℓ + h ( S ) | Id = α ( S ) arealways isomorphic as Lie groups and that those of g ℓ + h ( S ) are alsoisomorphic as homogeneous spaces. Lemma 9.
The data as well as the hypotheses being those of the pre-ceding Lemma, we also have the following equalities, for ≤ h ≤ k − , R ℓ + h +1 ( S ) = p R ℓ + h ( S ) and R ℓ + h +1 ( S ) = p R ℓ + h ( S ) where p is thestandard prolongation operator for partial differential equations. We finally introduce the (eventually singular) vector bundle with basis J k +1 E (5) R ℓ + k = { ( Y, Z ) | Y ∈ ( R ℓ + k ) Z , Z ∈ J k +1 E } ANTONIO KUMPERA where ( R ℓ + k ) Z = { j ℓ + k ξ ( z ) ∈ J ℓ + k T P | ( p k ξ ) Z = Z ( k +1) − } . Observe that Z ( k +1) − ∈ J ( J k E ) is a first order contact element namely,a transverse vector sub-space of T Z k J k E . The choice of these notationsis justified by the relation(6) R ℓ + k ( S ) = ( j k +1 S ) − R ℓ + k . Lemma 10.
The fibre bundle R ℓ + k is invariant under the action of Π ℓ + k +1 P on J ℓ + k T P × P J k +1 E . If F k +1 is a finite or infinitesimalorbit in J k +1 E , then the restriction R ℓ + k | F k +1 is of constant rankand any two fibres are isomorphic by the action of some element in Π ℓ + k +1 P . Let us recall ( cf. the Theorem 2 in [9]) that each trajectory Ω k is a locally trivial sub-bundle of J k E isomorphic to the quotientspace (Π ℓ + k P ) y / H ( Y ) . Inasmuch, an infinitesimal trajectory ω k isa locally trivial sub-bundle since it is isomorphic to the quotient of theconnected component of the unit y in (Π ℓ + k P ) y by the isotropy (weshall be careful to assume P connected otherwise ω k is just a bundlewhose basis is the connected component of P containing the point y ).We can thus consider any trajectory F k , finite or infinitesimal, as a non-linear partial differential equation (a non-linear differential system) oforder k on the fibration (and prolongation space) π : E −→ P thatwe shall call a fundamental equation of order k . Lemma 11.
Let F k +1 be a fundamental equation, F k its projection at k − th order and assume that F k +1 ∩ p F k = φ . Under these conditions, F k +1 ⊂ p F k and therefore p F k is a union of trajectories. Lemma 12.
A fundamental equation F k is − integrable namely, theproperty ρ k ( p F k ) = F k holds, if and only if p F k = φ (for every fibre).This condition being verified, the prolongation p F k is a locally trivialaffine sub-bundle of J k +1 E −→ F k where E is assumed to be a finiteprolongation space. Let us denote by p h F k the prolongation of order k of F k . Arguingas previously, we can also show the N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, II 9
Corollary 2.
A fundamental equation F k verifies the extended equality ρ k ( p h F k ) = F k if and only if p h F k = 0 and, whenever this condition isverified, the following properties also hold:i) For every µ ≤ h , p µ F k is a locally trivial (and non void) sub-bundle of J k + µ E −→ F k saturated by the orbits of order k + µ , finiteor infinitesimal according to the nature of F k .ii) p µ ( p η F k ) = p µ + η F k , µ + η ≤ h . We observe, however, that the property p h F k = 0 does not imply, ingeneral, the h − integrability of F k namely, the property ρ µ ( p µ +1 F k ) = p µ F k , µ ≤ h. The above property only implies the − integrability.Let us next examine the tangent symbol of a fundamental equation.On the one hand, it is clear that this symbol at the point Z ∈ J k E issimply the contact element (∆ k − , k ) Z = ker [(∆ k ) Z −→ (∆ k − ) ρ k − Z ] , (∆ k − , k ) Z ⊂ [( S k T ∗ P ⊗ V E ) × E J k E ] Z = [ S k T ∗ αZ P ⊗ V βZ E ] × { Z } , and, on the other, we see immediately that the diagram below is com-mutative and exact ( cf. , the second diagram at the outset of section3 in [9]). Curiously enough, this diagram cannot be completed ev-erywhere by surjectivities, as is so common in commutative diagrams,since structures do not behave always as nicely as one would like to andthe problem resides in the lack of h-integrability or, stated equivalently,in the difficulty of approximating what we really look for namely, theintegrability . Lemma 13.
The first vertical sequence in (7) is surjective at the endif and only if the first horizontal sequence verifies the same property. (7) ↓ ↓ ↓ −→ g ℓ + k −→ ( S ℓ + k T ∗ P ⊗ T P ) × P J k E ℓ k −→ ∆ k − , k ↓↓ ↓ | ↓↓ −→ R ℓ + k −→ J ℓ + k T P × P J k E λ k −→ | ∆ k −→ ↓↓ ρ ℓ + k − ↓ | ↓↓ −−−−−−←−−−−−−−−←−−−−−−−−←−−−−−−−−←−−−− → R ℓ + k − × P J k E −→ J ℓ + k − T P × P J k E λ k − −−−→ ∆ k − × J k − E J k E −→ ↓ ↓ In fact, the above snake arrow tells us that the following sequence isexact: → ( S ℓ + k T ∗ P ⊗ T P ) × P J k E ℓ k −→ ∆ k − , k → coker ρ ℓ + k − → The Lemma 1 can now be extended to the following assertion:
Lemma 14.
Let ( E, π, P, p ) be a finite prolongation space of order ℓ, ζ a local vector field defined on E, ϕ a local diffeomorphism of P and weassume that α ( pϕ ) = α ( ζ ) . Then, [ p k ( pϕ )] ∗ p k ζ = p k [( pϕ ) ∗ ζ ] . We infer the following corollaries by taking also into account the The-orem 13.1 as well as the Proposition 14.2 in [6].
N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, II 11
Lemma 15.
The groupoid Π ℓ + k +1 P operates differentiably on the spaces ˜ J k T E , T J k E and ( S k T ∗ P ⊗ V E ) × E J k E . The Lie sub-fibration ˜ J k V E as well as the vector sub-bundle
V J k E are invariant by this ac-tion and the following exact sequences −→ J k T P × P J k E Σ k −→ ˜ J k T E p k −→ T J k −→ and −→ ( S k T ∗ P ⊗ V E ) × E J k E ǫ k −→ T J k E T ρ k − ,k −−−−→ T J k − E × J k − E J k E −→ are co-variant. Furthermore, the action on the term S k T ∗ P ⊗ V E factors via ρ ℓ +1 : Π ℓ + k +1 P −→ Π ℓ +1 P and that on the term J k T P via ρ k : Π ℓ + k +1 P −→ Π k P . Lemma 16.
Let F k be a fundamental equation and F k − its projection.By means of the canonical identification, ∆ k − , k | F k = ker [ T F k −→ T F k − ] is a locally trivial vector sub-bundle of S k T ∗ P ⊗ V E ) × E F k invariantunder the action of Π ℓ + k P . When k ≥ and when Z, Z ′ ∈ F k and Y ∈ Π ℓ + k P are such that Y ( Z ) = Z ′ , then Y transforms (∆ k − , k ) Z in (∆ k − , k ) Z ′ by means of the tensorial extension of the ( ℓ + 1) − jet Y ℓ +1 ,projection of Y, considered as a linear map T β ( Z ) E −→ T β ( Z ′ ) E . Inthe case where k = 0 , ∆ − , | F = ker [ T π : T F −→ T P ] ⊂ V E | F , Y ∈ Π ℓ P, and we can take any jet Y ℓ +1 ∈ Π ℓ +1 P projecting upon Y. When F k isan infinitesimal trajectory, we shall take care to restrict the action justto the open subset of Π ℓ + k P that leaves invariant F k . We next consider a pair F k +1 and F k of fundamental equations, k ≥ , and verifying ρ k ( F k +1 ) = F k . We shall study the geomet-rical properties of the fibration F k +1 −→ F k inasmuch as a sub-fibration of the affine bundle J k +1 E −→ F k above the base space F k (more precisely, it would be convenient to replace the term J k +1 E ) by ρ − k ( F k ) ⊂ J k +1 E where ρ k : J k +1 E −→ J k E ). Since F k +1 is an orbit,this fibration is obviously a locally trivial sub-bundle, any two fibresbeing isomorphic by the action of the elements belonging to Π ℓ + k +1 P .We next recall ([6], §19) that J k +1 E −→ J k E is an affine bundle, theunderlying vector bundle being equal to ( S k +1 T ∗ P ⊗ V E ) × E J k E ,the affine action being defined as follows: The point Z ∈ J k E beingfixed, a vector v ∈ S k +1 T ∗ y P ⊗ V z E , y = α ( Z ) , z = β ( Z ) , defines bymeans of the canonical identification ([6], Proposition 14.2) a vectorfield ν ( v ) along the fiber A Z of J k +1 E above the point Z . This vectorfield admits a global 1-parameter group ( ϕ t ) of affine transformationsand the affine action of v on W ∈ A Z is defined by W + v = ϕ ( W ) .We thus see that the set { ν ( v ) | v ∈ S k +1 T ∗ y P ⊗ V z E } is the abelianLie algebra of all the infinitesimal translations of the affine space A Z .If we examine again the diagram (14.1) in [6], we shall observe thatthe pair ( v, W ) identifies to an element in ˜ J k +1 V E that projects onto (0 , Z ) by the morphism ˜ ρ k, k +1 (see the first line of the diagram (14.1)).This means however, in setting W = j k +1 σ ( y ) and Z = ρ k ( W ) , that ( v, W ) identifies to a jet j k +1 ( ζ ◦ σ )( y ) where ζ is a vertical vector fieldon E null to order k (i.e., it vanishes up to order k ) along the imageof the section σ namely, j k ( ζ ◦ σ )( y ) = 0 . Equivalently, ζ is a verticalvector field, on E, tangent to order k along the image of σ and at thepoint σ ( y ) . This being so, we see readily, and according to the diagram (14 . , that ν ( v ) W = ( p k +1 ζ ) W .Let us now return to the fibration ρ k : F k +1 −→ F k . Since F k +1 is an orbit under the action of Π ℓ + k +1 P on J k +1 E , the fibre ( F k +1 ) W , W ∈ F k , is the orbit of an arbitrary point W ′ ∈ ( F k +1 ) W under the action of the sub-group K ( W ) of (Π ℓ + k +1 P ) y , inverse im-age of the isotropy H ( Z ) by the projection ρ k, k +1 . Consequently, thefibre ( F k +1 ) W is isomorphic to the homogeneous space K ( W ) /H ( Z ) that, in general, is not connected. However, we can easily deter-mine its connected components since they are simply the orbits of theconnected component of the unity in K ( W ) or, equivalently, the or-bits of the infinitesimal action of the Lie algebra k ( W ) of the group K ( W ) . We shall therefore study the algebra q ( W ) of the vector fieldsof A Z , images by the infinitesimal action of k ( W ) . We first remarkthat T Z ( F k +1 ) W = (∆ k, k +1 ) Z and that consequently, every vector w ∈ T Z ( F k +1 ) W is obtained as follows: We take a local vector field ξ of P defined in a neighborhood of y and consider its prolongation p k +1 ξ = p k +1 ( pξ ) . Under these conditions, ( p k +1 ξ ) W ∈ (∆ k, k +1 ) W ifand only if T ρ k ( p k +1 ξ ) Z = ( p k ξ ) Z = 0 , Z = ρ k ( W ) which means pre-cisely that j ℓ + k +1 ξ ( y ) ∈ k ( W ) . Moreover, the condition ( p k ξ ) Z = 0 N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, II 13 also means, in writing Z = j k σ ( y ) , z = σ ( y ) , that ( c.f. , [6], p.317, theremark just following the Corollary)a) pξ ( z ) = 0 andb) pξ is tangent to order k along im σ at the point z .Since we have nullity at the point z , we infer that the k − th ordertangency condition along the section σ and at the point z only dependson the k − th order jet of σ at the point y or, in other terms, pξ will alsobe tangent up to order k along another section τ and at the point z assoon as j k τ ( y ) = j k σ ( y ) . The vector field ξ determines, by k − th orderprolongation, a vector field p k +1 ξ tangent to A Z and we shall write ˜ ν ( ξ ) = p k +1 ξ | A Z = p k +1 ( pξ ) | A Z . Since the prolongation operation preserves the brackets i.e. , [ p k +1 ( pξ ) , p k +1 ζ ] = p k +1 ([ pξ, ζ ]) , where ζ is an arbitrary local vector field on E , then1) [ p k +1 ξ , p k +1 ζ ] | A Z = [˜ ν ( ξ ) , ν ( v )] = p k +1 ([ pξ, ζ ]) | A Z ,2) [ pξ, ζ ] is a vertical vector field on E ,3) [ pξ, ζ ] is null up to order k along the section σ at the point σ ( y ) since ζ verifies the same property and pξ is tangent to order k along σ and vanishes at the point σ ( y ) .We then infer that [˜ ν ( ξ ) , ν ( v )] = ν ( v ′ ) where v ′ ∈ S k +1 T ∗ y P ⊗ V z E is the element that, for all W ∈ A Z , determines the identification ( v ′ , W ) ≡ ( j k +1 ([ pξ, ζ ] ◦ τ )( z ) ∈ J k +1 V E , W = j k +1 τ ( z )) . The above relation shows that any vector field ˜ ν ( ξ ) belongs to thenormalizer, in W ( A Z ) , of the algebra of infinitesimal transformationsof the affine space A Z .Consequently, q ( W ) is a finite dimensional Liealgebra, sub-algebra of the affine infinitesimal transformations of A Z .Since the above argumentation only puts in evidence the infinitesimal Nullité.
Vice qui ôte à un acte toute sa valeur . Larousse transformations, we see readily that the preceding results remain validwhen F k +1 and F k are replaced just by infinitesimal orbits. Theorem 1.
Let ( F k +1 , F k ) be a pair of fundamental equations verify-ing ρ k F k +1 = F k . The connected components of the fibre ( F k +1 ) Z , Z ∈ F k , are the orbits of the finite action, on A Z , of the connected affineLie group whose Lie algebra is equal to q ( Z ) . If Y ∈ Π ℓ + k +1 P andif ρ k Y ( Z ) = W , then Y [( F k +1 ) Z ] = ( F k +1 ) W and the affine transfor-mation ( J k +1 E ) Z −→ ( J k +1 E ) W , induced by Y, transforms q ( Z ) in q ( W ) conjugating the corresponding affine groups. Let us next assume that the first line of the diagram (7) is surjectiveat the end at a point Y ∈ ( F k +1 ) Z . Since F k +1 is an orbit, theLemma 1 implies that the same property will hold in any other point Y ′ ∈ ( F k +1 ) Z . Furthermore, the surjectivity of ( ℓ k +1 ) Y shows that anyvector in T Y ( F k +1 ) Z = (∆ k, k +1 ) Y can be obtained by prolongation of avector field ξ satisfying the property j ℓ + k +1 ξ ( y ) ∈ S ℓ + k +1 T ∗ y P ⊗ T y P .On account of the two commutative diagrams in the beginning of thesection 3 in Part I ([9]), we finally infer that p k +1 ξ | A Z is an infinitesimaltranslation . Corollary 3.
When ℓ k +1 is surjective at a point W ∈ F k +1 then it isalso surjective at any other point of F k +1 and the Lie algebra q ( Z ) , Z ∈ F k , contains a sub-algebra t ( Z ) of infinitesimal translations whose orbitissued from a point of ( F k +1 ) Z is equal to that of q ( Z ) . Each connectedcomponent of ( F k +1 ) Z is an affine sub-space of A Z . Corollary 4.
When ℓ k +1 is surjective at a point in F k +1 and if fur-ther the fibre ( F k +1 ) Z is connected, then ρ k : F k +1 −→ F k is a locallytrivial affine sub-bundle of J k +1 E −→ F k , any two fibres being linearlyisomorphic under the action of Π ℓ + k +1 P . Lemma 17.
Let ( F k ) k ≥ µ be a family of fundamental equations verifyingthe following properties: ρ k F k +1 = F k and F k +1 ∩ p F k = φ . Under theseconditions, F k +1 ⊂ p F k for all k and there exists an integer µ suchthat F k +1 is an open sub-bundle of p F k , k ≥ µ . The integer µ is theorder of stability (1-acyclicity) of the Spencer δ − complex associated tothe tangent symbols of the equations F k (and their prolongations). The proof is always essentially the same and so will be omitted.
N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, II 15
We now recall that a fundamental equation does not have necessarilyany solution at all. Fortunately or not, there are non-analytic though involutive differential systems that do not possess any solution! As forthe fundamental equations, if ever they possess a germ of a solutionpassing by one of their points then the same will hold for all the otherpoints and the equation will be completely (or everywhere) integrable .It is worthwhile to state the following Corollary, consequence of thepreceding Lemma, though containing some repetitions.
Corollary 5. If ( F k ) k ≥ µ is a family of integrable fundamental equationsand if further the following property holds: ρ k F k +1 = F k , then F k +1 ⊂ p F k for all k and F k +1 is an open sub-bundle of p F k for k ≥ µ .We can even assume that µ = 0 since the projection of an integrablefundamental equation is also integrable. Let us now return to the study of symbols . We know that g ℓ + k +1 isalways the algebraic prolongation of g ℓ + k but this is not the case, ingeneral, for the kernels ∆ k − ,k . Nevertheless, Lemma 18.
Let ( Z k ) be an element of J ∞ E and let us assume, for k ≥ µ , that the morphism ℓ k in the first horizontal sequence of (7) issurjective at the point Z k . Under these conditions,1) ∆ k,k +1 ⊂ p ∆ k − ,k for k ≥ µ and2) The Spencer δ − complex corresponding to the family (∆ k − ,k ) k ≥ µ is p − acyclic at order µ p (i.e., for k ≥ µ p ) if and only if the complexconstructed with the aid of the family ( g ℓ + k ) k ≥ is ( p + 1) − acyclic atorder ℓ + µ p − As for the proof, it will suffice to recall ([9], sect.3) that the family ( ℓ k ) is a natural transformation of the corresponding δ − cohomologycomplexes compatible with the algebraic prolongation operations per-formed on the principal parts and proceed with the diagram chasing below by confronting the term ∧ q +1 T ∗ P ⊗ g ℓ + k − with ∧ q T ∗ P ⊗ ∆ k − , k .As for the notations, we suggest to have a glance at [12], p.83.An entirely analogous argument, where we replace ∆ k − , k by h k , showsas well the following lemma that transcribes what we can expect whenthe surjectivity only occurs at the level µ . AN T O N I OK U M PE R A → ∧ q − T ∗ P ⊗ g ℓ + k +1 → ∧ q − T ∗ P ⊗ ( S ℓ + k +1 T ∗ P ⊗ T P ) Id ⊗ ℓ k +1 −−−−−→ ∧ q − T ∗ P ⊗ (∆ k, k +1 → ↓ δ ↓ δ ↓ δ → ∧ q T ∗ P ⊗ g ℓ + k −→ ∧ q T ∗ P ⊗ ( S ℓ + k T ∗ P ⊗ T P ) Id ⊗ ℓ k −−−→ ∧ q T ∗ P ⊗ ∆ k − , k −→ (8) ↓ δ ↓ δ ↓ δ → ∧ q +1 T ∗ P ⊗ g ℓ + k − → ∧ q +1 T ∗ P ⊗ ( S ℓ + k − T ∗ P ⊗ T P ) Id ⊗ ℓ k − −−−−−→ ∧ q +1 T ∗ P ⊗ ∆ k − , k − → ↓ δ ↓ δ → ∧ q +2 T ∗ P ⊗ g ℓ + k − → ∧ q +2 T ∗ P ⊗ ( S ℓ + k − T ∗ P ⊗ T P ) N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, II 17
Lemma 19.
Let Z ∈ J µ E be an element such that the first horizontalsequence in (7) is surjective. Setting h µ = (∆ µ − ,µ ) and indicating by h k , k ≥ µ the ( k − µ ) − th algebraic prolongation of h µ , then h k is p − acyclic if and only if g ℓ + k − is ( p + 1) − acyclic. Theorem 2.
The hypotheses being those of the Lemma , let us fur-ther assume that there exists a formal solution ( Z k ) k ≥ µ of the funda-mental system F = ( F k ) k ≥ µ (i.e., Z k ∈ F k and ρ k Z k +1 = Z k ) such thatthe first sequence of (7) is surjective at the end along ( Z k ) . This beingso, we infer that1) the same property of surjectivity also holds for any other formalsolution W = ( W k ) of F ,2) F k +1 = p F k for k ≥ µ and3) µ = aup { µ, µ } , where µ is the integer such that the symbol g ℓ + k , k ≥ µ − , becomes − acyclic .We recall that the symbol g ℓ + k ( Z k ) , Z k ∈ F k , is entirely determined by g ℓ ( Z ) and that the homological properties of this symbol are uniformalong a trajectory F k in E (cf., the Proposition 4 in [9] , sect. 6 and theCorollary 1). Remark . Under the hypotheses of the preceding Theorem, we haveassumed the surjectivity of the morphisms ℓ k , for k ≥ µ , in view ofbeing able to guarantee the stability of the symbols tangent to thefundamental equations F k by means of the − acyclicity of the algebraicsymbols g ℓ + k . If we let down this property, very useful in applications,it will suffice to assume the surjectivity of the morphism ℓ k for k ≥ µ + 1 . Corollary 6.
The hypotheses being those of the last Theorem, the so-lutions of F µ coincide with the simultaneous solutions of the (infinitelymany) fundamental equations F = ( F k ) k ≥ µ i.e., with the local sectionsS of E satisfying the conditions im j k S ⊂ F k , k ≥ µ . Corollary 7.
Let F = ( F k ) k ≥ be a family of integrable fundamentalequations verifying the property ρ k F k +1 = F k and let us assume furtherthat ℓ k , k ≥ µ + 1 , is surjective along a formal solution of F , µ beingthe integer that stabilizes the symbols tangent to the equations F k . Under these conditions, we claim that F k +1 = p F k for all k ≥ µ andthat the solutions of F µ coincide with the simultaneous solutions of thesystem F . We also conclude thereafter that any particular solution ofa given equation F k is as well a simultaneous solution. The sorites being terminated (17 pages), let us return to geometryand try to do some adequate work. We thus begin by considering afinite prolongation space ( E, π, P, p ) of order ℓ and take a local or globalsection S of E namely, a structure of species E . We shall say that S is homogeneous or transitive in the base space if the germs of S at anytwo points y, y ′ ∈ α ( S ) (the source of S ) are always equivalent. Inother terms, there exists a germ of local diffeomorphism φ of P withsource y and target y’ such that φ ( S y ) def = pφ ◦ S y ◦ φ − = S y ′ where S y denotes the germ of S at the point y . Inasmuch, we shallsay that S is homogeneous of order k when the k − jets of S at any twoarbitrary points y, y ′ ∈ α ( S ) are k − th order equivalent which meansthat there exists an invertible jet Y ∈ Π ℓ + k P such that Y ( j k S ( y )) = j k S ( y ′ ) , y = α ( Y ) , y ′ = β ( Y ) , and where, by definition, Y ( j k S ( y )) = Y ( k ) · j k S ( y ) · Y − , Y ( k ) being the invertible k − jet on E that corresponds,by prolongation, to the ( ℓ + k ) − jet Y on P . We shall say, finally, that S is formally homogeneous when the above k − th order condition isverified for all integer k .We now assume that S is homogeneous of order k which means thatthe set { j k S ( y ) | y ∈ α ( S ) } is contained in a single trajectory, denotedby Ω k ( S ) , under the action of the pseudo-group Γ ell + k on J k E , naturalprolongation to k − th order of the general pseudo-group Γ( P ) of alllocal diffeomorphisms of the manifold P . Furthermore, when α ( S ) iaconnected, the above set Ω k ( S ) is also contained in a single trajectory ω k ( S ) of the infinitesimal action L ℓ + k since these last trajectories arethe connected components of the finite trajectories of Γ ell + k . Each Ω k ( S ) is a finite fundamental equation of order k and each ω k ( S ) isan infinitesimal fundamental equation of the same order. When S isformally homogeneous, we shall denote by Ω( S ) = (Ω k ( S )) the family Sorite.
Du grecque "sorkites". Argument composé d’une suite de propositionsliées entre elles de manière que l’attribut de chacune d’elles devienne le sujet dela suivante, et ainsi de suite, jusqu’à la conclusion, qui a pour sujet le sujet de lapremière et pour attribut l’attribut de l’avant dernière.
Larousse, quoque turbatio.
N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, II 19 of all the respective fundamental equations. If, further, j k D takes all itsvalues in a given infinitesimal trajectory, we denote by ω ( S ) = ( ω k ( S )) the corresponding family. Proposition 1.
Let S be a homogeneous structure of order k and Ω k ( S ) its finite fundamental equation of the same order. Then Ω h +1 ( S ) ⊂ p Ω h ( S ) for every integer h < k . If furthermore S is formally homo-geneous, then there exists an integer µ such that Ω h +1 ( S ) is an opensub-bundle of p Ω h ( S ) for any h ≥ µ . The integer µ is the order ofstability ( − acyclicity) of the Spencer δ − complex associated to the sym-bols tangent to the equations Ω k ( S ) . If, moreover, the symbol (i.e., thefibre) of Ω h +1 ( S ) above Ω h ( S ) is, along j ∞ S ( y ) and for h ≥ µ , an affinesub-space of the total symbol (more generally, if each connected com-ponent of the symbol is an affine sub-space), then Ω h +1 ( S ) = p Ω h ( S ) for h ≥ h = suo { µ , µ } . This being so, the solutions of Ω h ( S ) co-incide with the simultaneous solutions of Ω( S ) . The same propertiesalso remain valid for ω h +1 ( S ) and ω ( S ) , the integers µ and h beingthe same as those for Ω( S ) . It is clear that the solutions S ′ of Ω k ( S ) are precisely the structures ofspecies E k − th order equivalent to the model S , the k − th order equiv-alence taking place for all couples of points y ∈ α ( S ) and y ′ ∈ α ( S ′ ) .The structures S’ are of course homogeneous of order k and any struc-ture of this type whose jet at a single point is k − th order equivalent toa k − jet of S is a structure k − th order equivalent to S . Likewise, thesimultaneous solutions of the family Ω( S ) are the structure formallyequivalent to the model S , such structures being all formally homoge-neous. We shall say that a structure is connected whenever α ( S ) isconnected, a connected component of an arbitrary structure S being,by definition, the restriction of S to a connected component of α ( S ) .Its image is a connected component of im S . Proposition 2.
Let S be a formally homogeneous structure. Then,the local or global structures of species E and formally equivalent tothe model S are the solutions of Ω( S ) . Moreover, when the hypothesesof the last proposition concerning the affine nature of the symbols of Ω − h ( S ) are verified, the formal equivalence is then a consequence ofthe equivalence at order h . Finally, when ω ( S ) is defined (which oc-curs especially when S is connected), the connected structures formallyequivalent to the model S are the solutions of ω ( S ) . The system Ω( S ) is, according to the terminology of [13], the fundamen-tal differential system for the structures of species E that are formallyequivalent to S . In his article, the author considers the model structuredefined on a manifold P eventually distinct from P . In ou case, wesimply identify P to an open set of P since it is always possible to transfer the model given on P onto a model defined on an open set of P without, for that matter, modifying the equivalence relation.We could hereafter contemplate in defining the reduced fundamentalsystem and in transcribing some of the results of [13]. However, weleave such transcriptions, not all together evident, to the care of thereader since, at present, we are inclined to examine other importantaspects of the theory.Let us first observe that the different notions of homogeneity introducedabove refer essentially to the pseudo-group Γ( S ) of all automorphismsof S as well as to the groupoids R ℓ + k ( S ) . In fact, S is homogeneous when Γ( S ) is transitive, homogeneous of order k when the groupoid R ℓ + k ( S ) is transitive and finally formally homogeneous when the tran-sitivity of R ℓ + k ( S ) occurs for all k . We can therefore introduce the cor-responding infinitesimal notions namely, we can say that S is infinitesi-mally homogeneous when L ( S ) is transitive, k − th order infinitesimallyhomogeneous when R ℓ + k ( S ) is transitive i.e., β ( R ℓ + k ( S )) y = T y P , forall y ∈ P , and formally infinitesimally homogeneous when the last con-dition is verified for all k . When S is only defined on an open set U ,we shall simply replace, in the above definitions, the manifold P by theopen set U .Let S be a structure of species E and let us inquire for conditionsrendering the target map β : ( R ℓ + k S ) y −→ T y P surjective. For this,we set Y = j ℓ + k S ( y ) and recall that ( R ℓ + k S ) y = { j ℓ + k ξ ( y ) ∈ J ℓ + k T P | ( p k ξ ) Y ∈ T Y ( im j k S ) } . However, since ξ y = β ( j ℓ + k ξ ( y )) = α ∗ ( p k ξ ) Y , α : J k E −→ P , we seepromptly that the surjectivity condition is given by T Y ( im j k S ) ⊂ (∆ k ) Y . Consequently, a structure S is infinitesimally homogeneous oforder k if and only if im j k S is an integral sub-manifold of the distri-bution (differential system) ∆ k . Taking into account the integrability(involutivity) of ∆ k ([9], Theorem 2) and the fact that the integralleaves of ∆ k are the connected components of the trajectories Ω k of Γ ℓ + k , we obtain the following result: N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, II 21
Proposition 3.
The following statements are equivalent:1. S is infinitesimally homogeneous of order k.2. Each connected component of im j k S is contained in an infini-tesimal trajectory of L ℓ + k .3. Each connected component of im j k S is contained in a finitetrajectory of Γ ℓ + k .4. Each connected component of im S is homogeneous of order k. Corollary 8.
The following statements hold:1. Every connected structure that is infinitesimally homogeneous oforder k is also homogeneous of order k and every homogeneous structureof order k is also infinitesimally homogeneous of the same order.2. Let S be a connected and infinitesimally homogeneous model struc-ture of order k. Then the class of all the connected structures S’ equiv-alent to order k to the model S is given by the connected solutions ofthe fundamental equation ω k ( S ) . Each such equivalent structure S’ isinfinitesimally homogeneous of order k.3. Let S be a connected and infinitesimally formally homogeneousmodel structure. Then the class of all the connected structures S’ for-mally equivalent to the model S is given by the connected solutions ofthe infinitesimal fundamental system ω ( S ) or, when the hypotheses ofthe Proposition 1 are verified, by the connected solutions of ω h ( S ) inwhich case the formal equivalence is a consequence of the equivalenceat order h . Any solution S’ is infinitesimally formally homogeneous. We next discuss the notion of transitivity . The structure S is said to be transitive of order k or k − th order transitive when it is homogeneous oforder k and if further the projections R ℓ + k ( S ) −→ R ℓ + h ( S ) , ≤ h ≤ k , are all surjective (in fact, surmersions on account of the Lemma 8).The structure S is said formally transitive when the above conditionsare verified for all k and, finally, just transitive when it is formallytransitive and, moreover, when R ℓ + k ( S ) = J ℓ + k Γ( S ) . In other terms,this means that any equivalence of finite order among the jets of S is actually realised (achieved) by the jet of a local automorphism of S . In essentially the same way, we can introduce the notions of k − th order infinitesimal transitivity, formal infinitesimal transitivity and, forshort, transitivity by simply replacing Γ( S ) by L ( S ) and R ℓ + k ( S ) by R ℓ + k ( S ) .We observe that the notions of k − th order and formal finite or infinites-imal transitivity, are invariant by k − th order and formal equivalencesrespectively. Thus, if we start with a formally transitive resp., k − th or-der model, every solution of the fundamental system Ω( S ) resp, Ω k ( S ) is a formally transitive resp., k − th order structure. The same remarksapply of course for the infinitesimal context taking, however, the carein taking for S a formally homogeneous and infinitesimally formallytransitive resp., k − th order infinitesimally transitive structure.Let us now make abstracrion of the model. It is clear that a structure S is homogeneous of order k if and only if this structure is a solution of afundamental equation Ω k . Moreover, each connected component of S isa solution of an infinitesimal equation ω k contained in Ω k . Inasmuch, S is infinitesimally homogeneous of order k if and only if each con-nected component is a solution of a finite or infinitesimal fundamentalequation of order k , though these equations may vary along with theconnected component. We have analogous conclusions in the formalcase, the number of equations being infinite unless the hypotheses ofthe Proposition 1 be verified.We next inquire on the equations bearing on the finite jets of a structure S and displaying their k − th order or their formal transitivity and startexamining the infinitesimal aspect that will curiously place in evidencea new element namely, the Medolaghi-Vessiot equations .Let us be given a structure S of species E , let us fix an integer k andlet us pose ourselves the following problem. What are the conditionsto which are bound the finite jets j k +1 S ( y ) , y ∈ α ( S ) ⊂ P in such away that a ) the projection R ℓ + k +1 ( S ) −→ R ℓ + k ( S ) be surjective ( − integrableat order ℓ + k ) and b ) R ℓ + k ( S ) −→ T P be surjective (infinitesimal homogeneity oforder k ).We observe immediately that the second condition can be replace, inview of (a), by the surjectivity of R ℓ + k +1 ( S ) −→ T P . Moreover, wecan replace the above problem by the equivalent problem a ) R ℓ + k +1 ( S ) −→ R ℓ + k ( S ) be surjective and b ) R ℓ + k +1 ( S ) −→ T P be surjective,
N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, II 23 since the small diagram hereafter is commutative and exact. −→ R ℓ + k +1 ( S ) −→ R ℓ + k +1 ( S ) −→ T P −→ ↓ ↓ || −→ R ℓ + k ( S ) −→ R ℓ + k ( S ) −→ T P −→ ... ... We first examine the problem a . The relations (3) lead us to introducethe equation Θ k +1 of order k + 1 defined on the fibration E −→ P Θ k +1 = { Z ∈ J k +1 E | ( R ℓ + k +1 ) Z −→ ( R ℓ + k ) ρ ( Z −→ is exact } . The following Lemma is evident on account of the Lemma 1.
Lemma 20.
The equation Θ k +1 is invariant under the finite actionof Γ ℓ + k +1 as well as under the infinitesimal action of L ℓ + k +1 or, inother terms, Θ k +1 is a union of trajectories. Inasmuch, this equation isinvariant under the left action of the groupoid Π ℓ + k +1 P on J k E and theinfinitesimal action of the algebroid J ℓ + k +1 T E . The solutions of Θ k +1 are the local or global structures of species E verifying the condition a . Since the condition b is no other than the ( k + 1) − order infinitesimalhomogeneity condition, we derive the following result: Proposition 4.
A structure of species E verifies the conditions ( a ) and ( b ) if and only if each of its connected components is a solution of afinite or infinitesimal fundamental equation contained in the equation Θ k +1 . The connected structures satisfying ( a ) and ( b ) are the connectedsolutions of the infinitesimal dundamental equations contained in Θ k +1 . We thus see that there exists a whole family of differential equationsof order k+1 on the fibration E −→ P solving the proposed problem.Whereas the fundamental equations are well behaved with respect toprolongations, this does not happen with the equations Θ k +1 . In fact,there does not exist, a priori , any plausible relation between Θ k +1 and p Θ k . Inasmuch, we can even note that there does not exist any ap-parent relation between ρ k (Θ k +1 ) and Θ k , all depending on the specificstructure of the prolongation space E . We shall therefore search for thedesired results with the help of the Theorem of Quillen. With this in mind, let us first recall that this theorem strongly relieson the − acyclicity of the symbols and the Corollary 1 tells us thatthe homological properties of the symbols g ℓ + k are uniform along theinverse image of an orbit of degree zero hence, in particular, along afundamental equation F k . On the other hand, if S is a solution of F k , S takes its values in the orbit of order zero F = β ( F k ) . Consequently,when F k is integrable then F k ⊂ J k F and this leads us naturally toonly consider the prolongation space ( F , π, P, p ) of order ℓ that will becalled subordinate to ( E, π, P, p ) and whose prolongation structure issimply obtained by restricting the structure of E ( cf. [9], Theorem 2).When F is a finite trajectory, the prolongation laws of E admit naturalrestrictions. Quite to the contrary, when the trajectory is infinitesimal,the infinitesimal prolongation law restricts without any problem whilethe finite law will have to be restricted, at each order k , to the opensub-groupoid of Π ℓ + k P , composed of those finite jets that preserve the k − th order transverse contact elements of dimension n tangent to F but also restricted to the solutions of this sub-groupoid. In partic-ular, the α − connected component (Π ℓ + k P ) is contained in this sub-groupoid. It is clear that J k F is invariant in J k E (always respectingthe above restrictions concerning the infinitesimal orbits) and we shalldenote by Θ k +1 ( F ) and g ℓ + k ( F the equation Θ k +1 and the symbol g ℓ + k respectively, when restricted to J k +1 F and J k F . The restrictedsymbol has the same homological properties everywhere. Lastly, weshall call F − admissible any fundamental equation contained in Θ k +1 thus finding the so claimed Medolaghi-Vessiot equations ([15], p.436,eqs.(58) and (59)). Theorem 3.
Let ( E, π, P, p ) be a finite prolongation space of order ℓ , ( F , π, P, p ) the prolongation space subordinate to an orbit F and η = η ( F ) the integer from whereon the symbol g ℓ + k ( F ) , k ≥ η ,becomes − acyclic.The linear equation R ℓ + k ( S ) associated to any so-lution of an F − admissible fundamental equation of order k + 1 ≥ η + 1 is then transitive and formally integrable i.e., formally tran-sitive. Moreover, when η ∞ = η ∞ ( F ) is the integer where from therestricted symbol g ℓ + k ( F ) , k ≥ η ∞ , becomes involutive, then theequation R ℓ + k ( S ) , k ≥ η ∞ , is involutive. The argument is as follows. S being a solution of a fundamental equa-tion of order k + 1 , we know ( cf. the Lemma 8) that R ℓ + k ( S ) as wellas R ℓ + k +1 ( S ) are regular equations and that R ℓ + k +1 ( S ) = p R ℓ + k ( S ) .Furthermore, the symbol g ℓ + k ( S ) = ( j k S ) − g ℓ + k ( F ) is − acyclic and N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, II 25 the morphism R ℓ + k +1 ( S ) −→ R ℓ + k ( S ) is surjective since S is a so-lution of the Medolaghi-Vessiot equations of order k + 1 . We thusre-encounter the hypotheses of Quillen’s Theorem ([12]) and conse-quently R ℓ + k ( S ) is formally integrable. We also observe, en passant ,that R ℓ + k +2 ( S ) = p R ℓ + k +1 ( S ) since the equation of order ℓ + k + 1 isregular and the − acyclicity implies that that R ℓ + k +2 ( S ) is also reg-ular. An inductive argument will then show that all the equations R ℓ + k + h ( S ) are regular and that R ℓ + k + h +1 ( S ) = p R ℓ + k + h ( S ) , which bythe way is already stated in the Theorem of Quillen. We finally observethat this Theorem does not require the regularity of the first two equa-tions since it suffices in fact that R ℓ + k ( S ) , hence also its prolongation,be defined by a differential operator which is always the case for theequations R ℓ + k ( S ) . The regularity is just a Black Friday extra bonus. Corollary 9.
When g ℓ ( F ) is − acyclic then every solution S of an F − admissible fundamental equation of order is an infinitesimallyformally transitive structure of species E (more so, of species F ). If,further, g ℓ ( F ) is involutive, the same will be true for the structure Si.e., for the equation R ℓ ( S ) . We can now consider the section j k S as being a structure of species J k E , prolongation of order k of the structure S and the last Theoremcan be paraphrased by the Corollary 10.
Let ( F , π, P, p ) be the prolongation space subordinedto an orbit F of ( E, π, P, p ) . The k − th order prolongation of any F − admissible fundamental equation of order k + 1 ≥ η + 1 is astructure of species J k E (viz. of species J k ) that moreover is infinites-imally formally transitive. Inasmuch, the prolongation of order ≥ η ∞ is an involutive structure We need not play, for the prolonged structures, the Ehresmannian semi-or non-holonomic game i.e. , by considering spaces like J h ( J ℓ + k E ) , sincewe can see promptly that R ( ℓ + k )+ h ( j k S ) = R ℓ + k + h ( S ) as soon as wedefine these equations by the contact order of the prolonged vectorfields with the sections defining respectively the structure. Corollary 11.
Let F k +1 be an integrable fundamental equation con-tained in Θ k +1 and assume that g ℓ + k ( F ) , F = ρ F k +1 , is − acyclic.Then, for any h ≥ , the h − th prolongation p h F k +1 is an F − admissible fundamental equation of order k + h + 1 namely, the unique integrablefundamental equation projecting onto F k +1 . a) The structure S is formally homogeneous hence im j k + h S ⊂ F k + h , i.e., F k + h is integrable and consequently F k + h ⊂ p F k + h ⊂ p h F k +1 . b) The morphism ( R ℓ + k + h ) Y k + h −→ ( R ℓ + k + h − ) Y k + h − is surjective hence F k + h is F − admissible and, on account of the Lemma12, the mapping ( ℓ k + h ) Y k + h is also surjective. The − acyclicity of g ℓ + k ( F ) implies (Lemma 15) that µ = k +1 is equal to the integer thatstabilizes the symbols tangent to the equations F k and consequently(Corollary 10), F k + h +1 = p F k + h or, in other terms, F k + h +1 = p h F k +1 .The uniqueness of the equation F k + h +1 is obvious since any other inte-grable fundamental equation of order k + h + 1 that projects onto F k +1 is necessarily contained hence equal to p h F k +1 .This Corollary shows that given an F − admissible and integrable fun-damental equation F k +1 , k + 1 ≥ η + 1 ( cf. the Theorem 3), therewill be no place in pushing any further the calculations that is to say,search for solutions S of F − admissible fundamental equations F k +1+ h projecting upon F k +1 since we shall find no other than the solutions of F k +1 , no additional restriction being therefore possible.We thus see that the local or global structure of species E admittingan infinitesimally formally transitive prolongation can be searched foramong the solutions of rhe F − admissible and integrable fundamentalequations of orders k + 1 ≥ η ( F ) + 1 . Moreover, this method exhauststhe connected structures. In fact, if j k S is an infinitesimally formallytransitive structure, the equation R ℓ + k ( S ) is formally integrable andtransitive. Taking, if necessary, an integer k ′ > k , this equation ac-quires a − acyclic symbol. Since, by definition, this equation verifiesthe conditions ( a ) and ( b ) , the section S is the solution of a fundamen-tal equation F k +1 contained in Θ k +1 (Proposition 3) and thereafter is F − admissible, where F = ρ F k +1 . Since g ℓ + k ( F ) is − acyclic, wecan find the structure S by the method of the Theorem 3. When thestructure is not connected, each of its components can be determinedby the above method. N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, II 27
The preceding results involve integrable fundamental equations henceit is high time for us to examine the formal integrability of the lessrestrictive F − admissible fundamental equations. The next Lemmaas well as the Theorem show that, subject to reasonable hypotheses,the Medolaghi-Vessiot equations have good properties with respect toprolongations and formal integrability. Lemma 21.
Let F k +1 be an F − admissible fundamental equation ver-ifying the two properties: p F k +1 = φ and g ℓ + k ( F ) is − acyclic. Then p F k +1 is also an F − admissible fundamental equation. In order to prove this rather remarkable result, we first observe thatthe condition p F k +1 = φ entails (Corollary 2) that p F k +1 is a regularequation of order k + 2 namely, a locally trivial affine sub-bundle of J k +2 E −→ F k +1 and, in particular, that its tangent symbol ( γ k +2 ) Z ,at a point Z ∈ p F k +1 is the algebraic prolongation of (∆ k,k +1 ) Y , Y = ρ k +1 Z , the latter being the symbol tangent to F k +1 at the point Y .We also know (Lemma 11) that p F k +1 is a union of trajectories i.e., the trajectory F k +2 that contains the point Z is entirely contained in p F k +1 . We infer that the tangent symbol (∆ k +1 ,k +2 ) Z of F k +2 at thepoint Z is contained in ( γ k +2 ) Z . Let us now re-examine the diagram (8) at its stage (1 , k + 1) where we shall replace the kernel ∆ k − ,k by im ℓ k . The fact that F k +1 be F − admissible implies (Lemma 12) that im ( ℓ k +1 ) Y = (∆ k,k +1 ) Y and the proper definition of ℓ k ( cf. , (7) k ) showsthat im ( ℓ k +2 ) Z ⊂ (∆ k +1 ,k +2 ) Z . Last but not least, the − acyclicityof g ℓ + k ( F ) implies that the third vertical sequence in (8) i.e. , thesequence im ( ℓ k +2 ) Z δ −→ T ∗ y P ⊗ (∆ k, k +1 ) Y δ −→ ∧ T ∗ y ⊗ im ( ℓ k ) Y Y = ρ k Z , is exact. But this means precisely that im ( ℓ k +2 ) Z is the al-gebraic prolongation of (∆ k,k +1 ) Y or, in other terms, that im ( ℓ k +2 ) Z =( γ k +2 ) Z ⊃ ∆ k +1 ,k +2 ) Z and consequently that im ( ℓ k +2 ) Z = ∆ k +1 ,k +2 ) Z .The trajectory F k +2 is F − admissible (Lemma 12) and F k +2 = p F k +1 since ℓ k +2 ) Z is surjective (Corollary 6) and the symbols tangent to bothof these equations coincide ( cf. , the proof of the Lemma 14). Theorem 4.
Let F k +1 be an F − admissible fundamental equation andlet us further assume that the symbol g ℓ + k ( F ) is − acyclic. Then F k +1 is formally integrable if and only if p F k +1 = φ (on each fibre!) and, thisbeing the case, each p h F k +1 is an F − admissible fundamental equation.If, moreover, g ℓ + k ( F ) is involutive then F k +1 is also involutive. The proof is almost obvious. We know, according to the Corollary 2,that p F k +1 = φ if and only if F k +1 is − integrable and, whenever thiscondition is fulfilled, that p F k +1 is also a regular equation. More pre-cisely, p F k +1 is a locally trivial affine sub-bunbdle of J k +2 E −→ F k +1 .The − acyclicity hypothesis on g ℓ + k ( F ) entails, having in mind theLemma 16, that the tangent symbol ∆ k,k +1 of F k +1 is everywhere − acyclic. We can then complete the proof using a non-linear ver-sion of the Theorem of Quillen and showing the formal integrability of F k +1 by induction on the integer h so as to propagate the result of thepreceding Lemma to all the prolongations ( see, for a hint [12] and [7]). Scholium 1.
When ( E, π, P, p ) is an analytic prolongation space andwhen F k +1 is an F − admissible − integrable fundamental equationwhose symbol g ℓ + k ( F ) is − acyclic, then F k +1 is integrable, everyoneof its element being the ( k + 1) − jet of an analytic solution S. Further-more, the solutions of any such equation are analytic structures whoseprolongations of order k + 1 are infinitesimally transitive. Prolonging if necessary the equation F k +1 , we find an equation with in-volutive symbol hence the result as a consequence of the Cartan-Kähleror the Cauchy-Kovalevskaya (or even perhaps the Monge-Ampère?) Theorems. The last assertion follows from the above mentioned theo-rems and an eventual prolongation procedure of the formally integrableequation R ℓ + k ( S ) so as to attain involutivity.The condition p F k +1 = φ is a point wise imposition since, for this to beso, it is necessary and sufficient that, at a point Y ∈ F k +1 , the tangentspace T Y F k +1 contain a contact element of order 1 transverse (to thefibration J k +1 E −→ P ) and holonomic . This condition correspoonds,when k = 0 , to what the author of [14] calls The generalized Jacobirelations , that seem rather out of place.Let us next show that the restrictions Θ k +1 ( F ) of Θ k +1 to the sub-ordinate prolongation spaces are closed sub-sets of J k +1 F . We shallprove simultaneously that the restrictions Θ k +1 ∩ β − k +1 ( F ) are closedsub-spaces in β − k +1 ( F ) where β k +1 : J k +1 E −→ J k E . The argumentwill be based on a local coordinates calculation which, in its essence,can already be found in [1], Vol.2, Chap.V, §9 ( Cenni sulle ricerche diEngel-Medolaghi-Vessiot ).Let ( x i ) be a system of local coordinates defined on an open set U of P , ( x i , y λ ) an adapted system on the open set U of E and ( x i , y λ , y λα ) | α |≤ k N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, II 29 the corresponding system on β − k ( U ) ⊂ J k E . Let us further denoteby ( z i ) the same coordinates ( x i ) on U and then start our coordinatejuggling game that will be written in French so as to spare those readersthat suffer intolerance with coordinates.En chaque point fixé z = ( z i ) de U , le champ de vecteurs ξ βi,z =(1 /β !)( x − z ) β ∂/∂x i , ( x − z ) β = ( x − z ) β . . . ( x n − z n ) β n , ≤ | β | ≤ k ,s’annule au point z et par conséquent ǫ βi ( z ) = j k ξ βi,z ( z ) ∈ ( J k T P ) z . Onremarque en plus que l’ensemble { ǫ βi ( z ) } est une base de cette algèbrede Lie, la base de Engel suivant la terminologie de [14] car, lorsque ξ = P ξ i ∂/∂x i , alors j k ξ ( z ) = P ( ∂ | β | ξ i /∂x β )( z ) ǫ βi ( z ) .Les sections ǫ βi : U −→ J k T P sont différentiables et constituent unabase locale du fibré J k T P | U . Plaçons nous maintenant dans le cadred’un espace de prolongement E d’ordre ℓ et reprenons les notations ci-dessus avec k remplacé par ℓ + k . À chaque section ǫ βi , ≤ | β | ≤ ℓ + k ,de J k T P correspond un champ de vecteurs prolongé p k ǫ βi de J k E définisur l’ouvert α − k ( U ) et prenant ses valeurs dans le fibré vectoriel V J k E .L’ensemble de ces champs prolongés n’est pas en général linéairementindépendant en tous les points mais pourtant il engendre la distributionverticale ∆ k ∩ V J k E . Or, pour tout Z ∈ α − k ( U ) , l’isotropie ( R ℓ + k ) Z est l’ensemble des combinaisons linéaires de la base de Engel d’ordre ℓ + k au point z dont lea coefficients sont ceux qui établissent unerelation de dépendance linéaire entre les vecteurs prolongéa p Z ǫ βi . Pourmieux décrire l’espace vectoriel de ces relations linéaires revenons, avecles modifications évidentes, à la suite exacte (7) de [9] −→ R ℓ + k −→ J ℓ + k T P × P J k E λ k −→ V J k E et posons λ k = X | α |≤ k a ( λ,α ) ∂/∂y λα . Les coefficients a ( λ,α ) sont des formes linéaires définies sur le fibré vec-toriel J ℓ + k T P × P J k E et il est évident que W ∈ R ℓ + k si et seule-ment si a ( λ,α ) ( W ) = 0 pour tout ( λ, α ) . La base locale { ǫ βi } de J ℓ + k T P | U se remonte, par image réciproque, en une base locale dufibré J ℓ + k T P × P α − k ( U ) . Écrivons a ( λ,α ) = X a ( λ,α )( β,i ) ( ǫ βi ) ∗ où { ( ǫ βi ) ∗ } est la base duale de { ǫ βi } . La matrice de λ k par rapport auxbases locales { ǫ βi } de J ℓ + k T P × P β − k ( U ) et { ∂/∂y λα } de V β − k ( U ) ⊂ V J k E n’est autre que a ( λ,α )( β,i ) , matrice dont les coefficients sont desfonctions définies sur l’ouvert β − k ( U ) . En plus, la dimension de ( R ℓ + k ) Z est égale au co-rang de cette matrice car l’isotropie ci-dessus est lenoyau simultané des formes a ( λ,α ) . Prenons maintenant un point W ∈ β − k +1 ( U ) ⊂ J k +1 E se projetant en Z et indiquons par (cid:16) b ( λ,α )( β,i ) (cid:17) , α ≤| α | ≤ k + 1 , ≤ | β | ≤ k + 1 , la matrice de λ k +1 . Le diagrammecommutatif J ℓ + k +1 T P × P J k +1 E λ k +1 −−−→ V J k +1 E ↓ ↓ T ρJ ℓ + k T P × P J k E λ k −→ V J k E montre que T ρ ◦ λ k +1 se factorise à J ℓ + k T P × P J k E en le morphisme λ k et par conséquent la matrice de λ k +1 se transcrit par (cid:18) B CA (cid:19) où A est la matrice de λ k et les blocs ( B C ) sont formés par les com-posantes des formes b ( λ,α ) , | α | = k + 1 , le bloc C étant formé par les b ( λ,α )( β,j ) avec | β | = ℓ + k + 1 . Les coordonnées étant fixées, nous pouvonsidentifier ( J ℓ + k T P ) z à un facteur direct de ( J ℓ + k +1 T P ) z , les formes a ( λ,α ) Z devenant des formes linéaires sur le deuxième espace. Dans cesconditions, la surjectivité de ( R ℓ + k +1 ) W −→ ( R ℓ + k ) Z veut dire toutsimplement que le sous-espace quotient [ b ( λ,α ) W ] / [ a ( λ,α ) Z ] se projette in-jectivement dans l’espace [ ǫ βi ( z ) ∗ ] | β | = k +1 où [ ] indique le sous-espaceengendré. Or, le rang de cette projection est donné par le rang dubloc C W et par conséquent, la surjectivité se traduit par la conditionsuivante: Le rang, au point W , de la matrice ci-dessus est égal au rangde (cid:18) CA (cid:19) Puisque le rang de C peut varier, cette condition ne peut pas êtretraduite localement par l’annulation d’un certain nombre de détermi-nants. Remarquons pourtant que ( g ℓ + k +1 ) W est le noyau simultanédes restrictions, au symbole total, des formes b ( λ,α ) , | α | = k + 1 . Ilen résulte que le rang de C W , égal à la codimension de ( g ℓ + k +1 ) W , not to be confounded with the indices λ N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, II 31 est constant au dessus d’une trajectoire F de E et par conséquent, onpeut, dans l’image réciproque β − k +1 ( F ) , déterminer localement l’égalitédes rangs de ces deux matrices par la nullité d’un certain nombre dedéterminants. L’équation Θ k +1 est donc fermée dans β − k +1 ( F ) . Enparticulier, Θ k +1 ( F ) est fermée dans J k +1 F .Les raisonnements ci-dessus reviennent de fait à un long procéssusd’élimination. En effet, si l’on écrit explicitement les équations linéaires,portant sur les composantes ξ iβ d’un jet j ℓ + k +1 ξ ( z ) ∈ J ℓ + k +1 T P , quidéfinissent le sous-espace ( R ℓ + k +1 ) W , alors la projection de cette isotro-pie dans ( R ℓ + k ) Z aura, outre les équations définissant ( R ℓ + k ) Z , toutescelles qui peuvent s’obtenir à partir des équtions b ( λ,α ) = 0 , | α | = k +1 ,par élimination d’inconnues ξ iβ , | β | = ℓ + k + 1 . Or, cette éliminationconsiste à prendre les relations linéaires et linéairement indépendantesentre les lignes de C c’est-à-dire les combinaisons linéaires qui annulentles lignes de C et retranscrire ces mêmes relations à l’aide des lignes de B . On obtiendra ainsi les équations supplémentaires qui définissent laprojection de ( R ℓ + k +1 ) W . La condition d’égalité des rangs de ces deuxmatrices veut dire précisément que ces équations supplémentaires sontdes combinaisons linéaires des équations définissant ( R ℓ + k ) Z , c’est-à-dire les équations dont les coefficients sont donnés par les lignes de A .On retrouve ainsi (vraisemblablement) les arguments de [14], p.297,qui sont faits à l’ordre 1 et pour les presque-structures modelées sur unpseudogroupe de Lie transitif. Proposition 5.
When ( F , π, P, p ) is the prolongation space subor-dinate to an orbit F contained in E then the equation Θ k +1 ( F ) in J k +1 F is a closed sub-set locally defined by the vanishing of a cer-tain number of determinants namely, those that establish the equalityof ranks of the previously indicated matrices. Moreover, the rank of thefirst matrix is equal to the fibre co-dimension of R ℓ + k +1 | J k +1 F and therank of the sub-matrix C is equal to the co-dimension of g ℓ + k +1 ( F ) . A prolongation space ( E, π, P, p ) of order ℓ is said to be homogeneous when Γ ℓ or, equivalently, Π ℓ P operate transitively on E . Examplesof such spaces are the total spaces of ℓ − th order G − structures aswell as those of almost-structures modeled on a transitive Lie pseudo-group and all the previous results transcribe replacing F by E (= F ) . Moreover, the only proper subordinate prolongation spaces arethe infinitesimal orbits in E . We terminate this section by resumingall the data concernig formal transitivity. Theorem 5.
Let ( E, π, P, p ) be a finite prolongation space of order ℓ and ( F , π, P, p ) the prolongation space subordinate to an orbit F contained in E . Let us denote by η = η ( F ) the integer where afterthe symbol g ℓ + k ( F ) , k ≥ η becomes − acyclic and by η ∞ the integerwhere after this symbol becomes involutive. Under these conditions:i) The non-linear equations R ℓ + k + h ( S ) , h ≥ , associated to ev-ery solution S of an F − admissible fundamental equation of order k +1 ≥ η + 1 , are transitive (in ( α ( S ) ) Lie sub-groupoids that fur-ther are closed, locally trivial and regularly embedded in Π ℓ + k + h α ( S ) , R ℓ + k + h +1 ( S ) = p R ℓ + k + h ( S ) and the morphism R ℓ + k + h +1 ( S ) −→ R ℓ + k + h ( S ) is a submersion. If, moreover, the isotropy groups ( R ℓ + k + h S ) y at apoint y ∈ α ( S ) are all connected (or else project one upon the other)then the above submersions become surjective and the equation R ℓ + k ( S ) will be formally integrable. This being the case, the k − th order pro-longation of the structure S is also formally transitive above the openset α ( S ) .ii) The α − connected components R ℓ + k + h ( S ) also verify the sameregularity properties, the transitivity taking place in each connectedcomponent of α ( S ) hence everywhere when S is connected. Finally,the prolongation-projections properties stated above also transcribe forthe connected components namely, R ℓ + k + h +1 ( S ) = p R ℓ + k + h ( S ) and R ℓ + k + h +1 ( S ) −→ R ℓ + k + h ( S ) is a submersion or, in other terms, the equation R ℓ + k ( S ) is formallyintegrable and becomes involutive when k ≥ η ∞ . The proof is essentially provided by all the preceding sorites .Let us next observe that the fibres of the fibration R ℓ + k + h +1 ( S ) −→ R ℓ + k + h ( S ) are affine sub-spaces. Consequently, if for some integer h the isotropy R ℓ + k + h ( S ) y is connected, the same will occur for all higher order isotropiesat that point and the transitivity of R ℓ + k + h ( S ) will further imply that R ℓ + k + h ( S ) y ′ is also connected at any other point y ′ ∈ α ( S ) . N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, II 33
Corollary 12.
The given data and the hypotheses being those of thetheorem, let us assume further that R ℓ + k ( S ) y be connected at a point y ∈ α ( S ) . Then R ℓ + k ( S ) is formally integrable and the k − th orderprolongation of any solution S of F k +1 is a formally transitive structure. We thus see that the infinitesimal formal transitivity properties implythe same properties for the finite formal transitivity as long as werestrict the finite order equivalences to the α − connected componentsof the non-linear Lie equations associated to the structure S . Wheneverone of the isotropy groups is connected, all the other goups are alsoconnected and the formal transitivity is assured without any restriction.We can equally define the ( k + 1) − st order equations ˜Θ k +1 by consid-ering the set of all Z ∈ J k +1 E such that the morphism ( R ℓ + k +1 ) Z −→ ( R ℓ + k ) Z is surjective. These equations are obviously invariant by the action of Γ ℓ + k +1 and Π ℓ + k +1 P and ˜Θ k +1 ⊂ Θ k +1 . Moreover, every solution ofa finite fundamental equation of order k + 1 contained in ˜Θ k +1 verifiesthe following two properties:a) R ℓ + k +1 ( S ) −→ R ℓ + k ( S ) is surjective andb) R ℓ + k ( S ) is homogeneous of order k .The fact that S is a solution of a fundamental equation implies (Lemma8.) that both equations are "good" Lie sub-groupoids. Taking a finitetrajectory F in E as well as the corresponding subordinate prolon-gation space, we shall obtain all the structures admitting a formallytransitive prolongation - at least each connected component of suchstructures - as the solutions of F − admissible fundamental equations( i.e. , contained in ˜Θ k +1 ∩ J k +1 F , in applying the non-linear version ofQuillen’s Theorem at the level of − acyclicity. Finally, if we ascend tothe level of involutivity, we shall obtain involutive equations and con-sequently transitive structures, for short, in the analytic case. Besides,it is clear that the solutions of these fundamental equations are, upto a prolongation, infinitesimally formally transitive and that the pre-ceding results concerning the properties of F − admissible fundamentalequations can be entirely transcribed when the admissibility is takenwith respect to ˜Θ k +1 since the conditions (a) and (b) for the non-linearequations imply the corresponding conditions for the linear equations. Unfortunately, The equations ˜Θ k +1 are not susceptible of a good ana-lytical description since the surjectivity of the isotropy groups is morelikely a global topological problem involving connected componentsonce this surjectivity is verified at the Lie algebra level i.e. , when thelinear sequence ( R ℓ + k +1 ) Z −→ ( R ℓ + k ) ρ k Z −→ is exact.We finally remark that all the preceding definitions and results arebased on the general pseudo-group Γ( P ) of P and the groupoids Π ℓ + k P .We can however and without any additional difficulties by simply adapt-ing the definitions, re-write the entire section in the restricted contextthough it seems unavoidable, due to the nature of the various concepts,to consider no other but the transitive pseudo-groups. In fact, when Γ is a transitive Lie pseudo-group of order k operating on the manifold P and L the corresponding infinitesimal pseudo-algebra, we can replacethe general equivalence problem by the restricted one, the fundamentalequations then becoming the finite or infinitesimal trajectories of theprolonged pseudo-groups or pseudo-algebras Γ ℓ + k and L ℓ + k . In muchthe same way, we shall replace, in the notions of homogeneity and tran-sitivity, the general equivalence by the restricted one limiting ourselvesto the elements of J ℓ + k Γ and J ℓ + k L .3. Formally transitive structures
Let ( E, π, P, p ) be a finite prolongation space of order ℓ that weassume, from now on, homogeneous or else we restrict our attention toa subordinate prolongation space relative to an orbit of order zero.We next consider a formally homogeneous structure S of species E andwill say that another structure S’ is formally equivalent to S when, forall integers k ≥ and for any pair of points y ∈ α ( S ) and y ′ ∈ α ( S ′ ) ,there exists Y ∈ Π ℓ + k such that Y ( j k S ( y )) = j k S ′ ( y ′ ) or, in otherterms, when the two jets belong to the same k − th order finite orbitnamely, the orbit containing im j k S . The jet Y will be named a k − th order finite equivalence. Further, we shall say that the formal equiva-lence is dominated when there exists an integer ν such that every finiteequivalence Y of order k ≥ ν is "dominated" by a finite equivalence ˜ Y of order k + 1 where ρ k ˜ Y = Y . This being so, every finite equivalenceof order k ≥ ν is induced, at order k , by an infinite order equivalencenamely, by an infinite jet ( Y h ) h ≥ where ( Y h ) is a finite equivalence oforder h and ρ h Y h +1 = Y h . Such an infinite jet is, in view of the Theo-rem of Borel , the infinite jet j ∞ ϕ ( y ) of a local diffeomorphism ϕ of the N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, II 35 manifold P . The notion of dominated formal equivalence is actually thenotion of equivalence most often adopted in the study of G − structures.The recurrent construction of principal bundles whose elements are the k − th order equivalences of the flat model with the G − structure, thisbeing achieved by requiring the nullity of the successive structure ten-sors or else by equivalent conditions imposed on the fundamental formswhere after culminating in the desired dominated formal equivalenceWe shall say that two germs of structures of species E are formallyequivalent when they admit formally equivalent representatives, thedominated equivalence being defined similarly with the help of repre-sentatives. The study of the formal equivalence of formally infinitesi-mally transitive structures is a consequence of the Corollary 15. Theorem (of formal equivalence) 6.
Let S be a connected infinites-imally formally transitive structure of type E and η = η ( S ) the integerwhere after the symbol g ℓ + k ( S ) , k ≥ η , is − acyclic. In order thata structure S’ be formally equivalent to S it is necessary and sufficientthat S’ be a solution of the fundamental equation F η +1 that contains im j η +1 S . Such a structure S’ is infinitesimally formally transitiveand its symbol g ℓ + k ( S ′ ) has, for all k ≥ , the same homological prop-erties of the initial one, the formal equivalence being a consequence ofthe finite equivalence of order η + 1 . Moreover, if S or one of its pro-longations is formally transitive, that will notably take place when thenon-linear isotropy groups ( R ℓ + k S ) y at a point y project, for k ≥ k ,one upon the other or else when one of these groups is connected, thenthe same will hold for S’ and the formal equivalence will be of dominatedtype. The structure S being connected, it is also homogeneous at any orderand consequently im j η +1 S is contained in a fundamental equation F η +1 that is admissible due to the ( η + 1) − st order infinitesimaltransitivity of S . The uniformity of the homological properties of thesymbols implies that g ℓ + k = g ℓ + k ( S ) is − acyclic for k ≥ η and theCorollary 15. then shows that S and S’ are, for all k ≥ , solu-tions of p k F η +1 which is the admissible fundamental equation, orbitof im j η + k +1 S . The property relative to the symbols g ℓ + k ( S ′ ) is animmediate consequence of the Lemma 5. Finally, the part concerningthe formal transitivity of S’ as well as the dominated equivalence resultby an easy argument on the isotropy groups. We shall remark, for thatmatter, that the set of all the k − th order equivalences with a givensource y and a given target y’ is a homogeneous space of the isotropygroups ( R ℓ + k S ) y and ( R ℓ + k S ) y ′ . Remark.
The connectedness hypothesis of the structure S can be re-placed by the formal homogeneity or even by the homogeneiy of order η + 1 , in such a way as to assure that im j η +1 S be contained withinan orbit. Corollary 13.
Let S be a connected structure or a formally homoge-neous one whose µ − th prolongation is infinitesimally transitive an let η be the integer from where on all the symbols g ℓ + k ( S ) , k ≥ η , become − acyclic. In order that S’ be formally equivalent to S, it is necessaryand sufficient that S’ be a solution of the fundamental equation F k +1 , k = max { µ, η } , that contains im j k +1 S . The µ − th prolongationof such a structure S’ is infinitesimally formally transitive. If, further,one of the prolongations of S is formally transitive, the same will occurwith S’ and the formal equivalence will be dominated. The formal problem being resolved, we can proceed with our inquiryand look at what really matters namely, the local equivalence of thetwo structures. But here there is no way out other than to integratethe fundamental equations.As for non-transitive structures, we should point out that much canbe said in the case where the pseudo-group of local automorphisms orthe pseudo-algebra of infinitesimal automorphisms have regular orbitsand, more precisely, these orbits being distributed within a regularfoliation. We can even assume that their prolongations to E provideas well regular foliations. In this case we can define, at least locally,quotient prolongation spaces and quotient structures and, addaptingaccordingly the definitions, we shall be able to rewrite most of whatwas stated in this and the previous sections and consequently studythe local as well as the global equivalence problems. We should alsomention that Pradine’s holonomy groupoid is certainly the main toolthat will hopefully determine the equivalences.We finally say a few words on a rather outstanding example of a tran-sitive and homogeneous structure, one of the many Chef − d ′ Oeuvres given to us by Élie Cartan ([3], [4]). We are referring here to the
Systèmes de Pfaff en Drapeau or, in celtic,
Flag Systems .These are very special non-integrable Pfaffian systems enjoying theproperty of "slowly increasing their manifestation" of non-integrabilityor, in other terms, the successive derived systems decrease their di-mensions just by one unit and, of course, terminate by the null system.The most remarkable about these systems is the fact that they provide
N THE EQUIVALENCE PROBLEM FOR GEOMETRIC STRUCTURES, II 37 a vivid example of one of Cartan’s merihedric (mériédrique) prolonga-tion and equivalence processes. Since all this can already be learnt in[11], [10] and [8], we shall only tell "a nice story" about them. Firstof all, and this is the main fact, the automorphism groups and al-gebras of arbitrary Flag Systems are all canonically isomorphic and,more specifically, they are all canonically isomorphic to the
Darbouxautomorphisms emanating from the equation dx − x dx = . Wecan however study and classify these structures with the help of theisotropy groups and algebras of arbitrary order. These are intrinsi-cally defined objects and the classification comes out upon looking atthe differences in the co-ranks of these (Fréchet infinite dimensional)groups and algebras at least for those flags that the author in [9] calls elementary . When the systems are non-elementary, their classifica-tion becomes much more delicate and a new phenomenon does occurnamely, the appearance of continuous or even differentiable deforma-tions (variations) of a given flag via non-equivalent flags. However, wecan also cope with such phenomena by considering again the isotropyalgebras as discrete (finite number of variables) moduli. Most authorsstill employ the rather clumsy and outdated pseudo-normal forms in-troduced, way back in 1980, by the present author though the aboveterminology is due to others. At present, we can do no better than apol-ogize for having considered these silly and uncouth objects and whereseveral authors come forth with fathom deep proofs that nobody willever understand except, and with all the blessing and mercy of the AllMighty, those authors that wrote them.And what about the totally intransitive or rigid structures. For this,we shall be forced to " admirer le beau et fragile papillon qui ouvre sesailes à Aix-en-Provence et provoque non le dépliement des ailes d’unautre papillon mais en fait une tsuname à Fukushima ". This being arather long subject we shall not even try to say a word about it hopingthat other authors give us much to read and enjoy. References [1] U. Amaldi.
Introduzione alla teoria dei gruppi continui infiniti di trasfor-mazioni . Libreria dell’Università, Roma, 1942.[2] E. Cartan. Sur la structure des groupes infinis de transformation.
Annales Sci.École Norm. Sup. , 21:153–206, 1904.[3] E. Cartan. Les systèmes de Pfaff à cinq variables et les équations aux dérivéespartielles du second ordre.
Annales Sci. École Norm. Sup. , 27:109–192, 1910.[4] E. Cartan. Sur l’équivalence absolue de certains systèmes d’équations différen-tielles et sur certaines familles de courbes.
Bull. Soc. Math. de France , 42:12–48,1914. [5] A. Kumpera. Étude des invariants différentiels attachés à un pseudogroupe deLie.
Thèse, Paris , 1967.[6] A. Kumpera. Invariants différentiels d’un pseudogroupe de Lie, I, II.
J. Differ.Geom. , 10:279–335, 347–417, 1975.[7] A. Kumpera. On the Lie and Cartan theory of invariant differential systems.
J. Math. Sci. Univ. Tokyo , 6:229–314, 1999.[8] A. Kumpera. Automorphisms of Flag Systems. arXiv , 2014.[9] A. Kumpera. On the equivalence problem for geometric structures, I. pre-print ,2014.[10] A. Kumpera and J. Rubin. Multi-Flag Systems and Ordinary DifferentialEquations.
Nagoya Math. J. , 166:1–27, 2002.[11] A. Kumpera and C. Ruiz. Sur l’équivalence locale des systèmes de Pfaff endrapeau.
F. Gherardelli (Ed.), Monge-Ampère Equations and Related Topics,Firenze 1980, Proceedings, Istit. Naz. Alta Mat. "Francesco Severi", Roma ,pages 201–248, 1982.[12] A. Kumpera and D. Spencer.
Lie Equations, Vol.1: General Theory . PrincetonUniversity Press, 1972.[13] P. Molino. Sur quelques propriétés des G-structures.
J. Differ. Geom. , 6:489–518, 1972.[14] J. Pommaret. Théorie des déformations de structures.
Ann. Inst. HenriPoincaré , 18:285–352, 1973.[15] E. Vessiot. Théorie des groupes continus.
Ann. Sci. École Norm. Sup. , 20:411–451, 1903.(Antonio Kumpera)
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