In 1898, Jules Drach, a student of Ecole Normale Supérieure (ENS), published a thesis with a jury made by H. Poincaré, E. Picard and G. Darboux, the best French mathematicians at that time [1]. The thesis contains clever ideas on abstract differential fields and differential extensions, quite in advance at that time when people were more familiar with analysis. However, even if its extension to Lie pseudogroups, namely groups of transformations solutions of systems of OD or PD equations (called infinite groups at that time) was well understood, the definition of a kind of Galois group or Galois pseudogroup has been totally missed by Drach and the jury. Let us explain this problem by considering two elementary examples [2]-[5].

Let us consider the polynomial P ≡ y 3 − 3 y + 1 ∈ ℚ [ y ] which is generating a prime ideal p ⊂ ℚ [ y ] = K [ y ] and the field extension L / K with L = Q ( K [ y ] / p ) . A tedious substitution proves that if η is root of P = 0 , then both σ ( η ) = η 2 − 2 and σ 2 ( η ) = − η 2 + η + 2 are two other roots while σ 3 ( η ) = η . It follows that the splitting field of L / K is just L / K which is therefore a Galois extension. Another way is to introduce the discriminant δ = ( η 1 − η 2 ) ( η 1 − η 3 ) ( η 2 − η 3 ) as one can prove that δ 2 = 81 = 9 2 ≠ 0 and all the roots are different. We recall that:

while introducing the general polynomial P ( y ) = y 3 − ω 1 y 2 + ω 2 y − ω 3 ∈ K [ y ] .

Now, following Vessiot, we may introduce three copies ( y 1 , y 2 , y 3 ) of y and consider the general system made by P ( y 1 ) = 0 , P ( y 2 ) = 0 , P ( y 3 ) = 0 . Introducing the polynomial Δ = ( y 1 − y 2 ) ( y 1 − y 3 ) ( y 2 − y 3 ) ∈ K [ y 1 , y 2 , y 3 ] , we may consider the previous algebraic system as a linear system for ( ω 1 , ω 2 , ω 3 ) and the corresponding determinant is known to be the Vandermonde determinant ∆. Accordingly, if Δ ≠ 0 , the system is equivalent to:

This system, which is invariant by the permutation group S 3 with 6 elements, has been called by Vessiot an automorphic system because the search for all the solutions is equivalent to the search for a single one. Equivalently, the space of solutions is a principal homogeneous space (PHS) for the full group of permutation. However, for certain special ( ω 1 , ω 2 , ω 3 ) , the ideal generated in K [ y 1 , y 2 , y 3 ] is not prime but only perfect and thus an intersection of prime ideals. This is exactly the situation existing when ω 1 = 0 , ω 2 = − 3 , ω 3 = − 1 because Δ 2 − 81 = ( Δ − 9 ) ( Δ + 9 ) . It follows that, in this case, the system is not irreducible because one can add the compatible equation Δ − 9 = 0 or the compatible equation Δ + 9 = 0 as a way to reduce the group of invariance to the subgroup A 3 ⊲ S 3 having only 3 elements.

Following again Vessiot but in the differential case by using a ground ordinary differential field K , let us look for all the solutions of the OD equation y x x − ω 1 y x + ω 2 y = 0 when ( ω 1 , ω 2 ) ∈ K . Introducing two copies ( y 1 , y 2 ) of y and proceeding similarly, we may consider the OD system:

which is easily seen to be an automorphic system under the action of the linear group GL(2) with 4 elements made by 2 × 2 matrices with constant coefficients ( a , b , c , d ) that can be reduced to SL(2) with a d − b c = 1 by considering the new automorphic system with w ∈ K as Wronskian determinant such that ∂ x w = ω 1 w :

For the reader familiar with (differential) algebra, the concept of “irreducibility” is thus based on a deep confusion between maximal (differential) ideal m and prime (differential) ideal p because a maximal ideal is prime, but the converse may not be true. Surprisingly, as we shall now explain, the same confusion has been done by Drach and all his successors until... today.

Coming back to the historical framework, we may say for short that Lie has been caring about Lie groups of transformation around 1880 while providing his famous main three theorems, before discovering that they were only examples of Lie pseudogroups in 1890. Surprisingly, he had only to main French heirs who were in competition, namely E. Cartan [6] and E. Vessiot [7]. Cartan being an obscure provincial teacher before becoming famous for his relation with A. Einstein in 1930 [8] while Vessiot has been heading ENS during two periods of 10 years. Their common dream has been to introduce new mathematical methods in physics and mechanics, in particular exterior calculus for Cartan and geometric objects for Vessiot, both of them being fascinated by the work of E. Galois and trying to enlarge it in a kind of Differential Galois Theory(DGT) [9]. In 1909, the Italian Physicist Ugo Amaldi, in a prophetic comment, said that the two approaches were so different that only the future should show what is the best one. In fact, the structure equations of Cartan are using exterior calculus on jet spaces without any possibility to project them down on the base manifold, a possibility that has been done more than 60 years later by Spencer [10]-[12]. On the contrary, the structure equations of Vessiot are provided directly on the base manifold and can be considered as a kind of non-linear Janet sequence as we shall explain.

Shortly after the publication, Vessiot discovered that the main central result of the thesis on irreducibility was wrong. After he contacted Cartan, P. Painlevé and the jury, a “mathematical affair” started and became rather unpleasant because Drach refused to admit his mistake.

It is now time to explain how the author of this paper has been involved in this story. Indeed, I became a visiting student of D.C. Spencer in Princeton University for 6 months starting from the fall of 1969. I had in mind to use the new homological tools he had introduced in order to solve the very specific tool of diagram chasing for finding the number n 2 ( n 2 − 1 ) ( n − 2 ) / 24 of components of the Bianchi identities without referring to any combinatoric argument. During nights in the mathematical library, I discovered the work of Drach (1898), Vessiot (1903, 1904) and M. Janet (1920) [13]. I also discovered that Spencer was strictly unable to understand them and had no knowledge of general relativity though being a close friend of J. Wheeler with about the same age, and I decided to give it up by coming back to France in 1970. As I had dedicated my first Gordon and Breach book of 1978 to Janet while discovering that his tabular could be a key for future computer algebra (See the Introduction of [11]) and was still unable to understand the thesis of Drach, I imagined that the family of Janet could have kept some of his personal papers. Surprisingly, I discovered he was still alive in Paris, a few blocks away from my parents. Being a former close friend of Vessiot, he was so pleased by the dedication that he offered me a few private documents that I gave to the main library of ENS in Paris where they can be now consulted with gloves [5]. As a summary of what he told me at that time before dying in 1983 at the age of 95:

I lost three months trying vainly to understand the thesis but succeeded appreciating the work of Vessiot, in particular the Vessiot structure equations.He told me that the paper he wrote in 1920 had been used by Cartan without any reference to him, mainly because Cartan had not ben able to convince Einstein about the importance of exterior calculus.As a most important comment, he told me that the so-called differential groups introduced by J.F. Ritt in 1930 were useless for DGT because no non-trivial examples could be provided and that Ritt did miss Lie pseudogroups because he was mainly an analyst rather than a differential geometer [14][15]. However, he did lay down the foundations of differential algebra by adding the word “differential” to all concepts initially only concerned with pure algebra like the definition of prime and perfect differential ideals, a great achievement indeed.

Surprisingly, the same year, after loosing more than three months for nothing, the author lectured during a month at the Columbia University in New York along an invitation of E. Kolchin [16], who, even after discovering these facts, refused to provide a preface for the DGT book of 1983 [4]. It is only two years later that the author understood such a reaction when the second book of Kolchin on “Differential Algebraic Groups” did appear in 1985 [17], a fact needing no more comment on the conceptual side as we said. As a byproduct, it is easy for any reader to check, along the list of the differential algebra community made by about hundred persons including the former students of Kolchin, that not a single one did even quote once my DGT book of 1983 during forty years, despite the fact that Hopf algebras had been used for the first time in this book [3][4].

We now turn to the differential geometric background needed for the remaining of this paper.

If X is a manifold of dimension n with local coordinates x = ( x 1 , ⋯ , x n ) , we denote as usual by T = T ( X ) the tangent bundle of X , by T * = T * ( X ) the cotangent bundle, by ∧ r T * the bundle of r-forms and by S q T * the bundle of q-symmetric covariant tensors. More generally, let ℰ be a fibered manifold, that is a manifold with local coordinates ( x i , y k ) for i = 1 , ⋯ , n and k = 1 , ⋯ , m simply denoted by ( x , y ) , projection π : ℰ → X : ( x , y ) → ( x ) and changes of local coordinates x ¯ = φ ( x ) , y ¯ = ψ ( x , y ) . If ℰ and ℱ are two fibered manifolds over X with respective local coordinates ( x , y ) and ( x , z ) , we denote by ℰ × X ℱ the fibered product of ℰ and ℱ over X as the new fibered manifold over X with local coordinates ( x , y , z ) . We denote by f : X → ℰ : ( x ) → ( x , y = f ( x ) ) a global section of ℰ , that is a map such that π ∘ f = i d X but local sections over an open set U ⊂ X may also be considered when needed. Under a change of coordinates, a section transforms like f ¯ ( φ ( x ) ) = ψ ( x , f ( x ) ) and the derivatives transform like:

We may introduce new coordinates ( x i , y k , y i k ) transforming like:

in order to define the 1-jet bunble J 1 ( ℰ ) of ℰ . Differentiating q times and proceeding similarly, we shall denote by J q ( ℰ ) the q-jet bundle of ℰ with local coordinates ( x i , y k , y i k , y i j k , ⋯ ) = ( x , y q ) called jet coordinates and sections f q : ( x ) → ( x , f k ( x ) , f i k ( x ) , f i j k ( x ) , ⋯ ) = ( x , f q ( x ) ) transforming like the sections j q ( f ) : ( x ) → ( x , f k ( x ) , ∂ i f k ( x ) , ∂ i j f k ( x ) , ⋯ ) = ( x , j q ( f ) ( x ) ) where both f q and j q ( f ) are over the section f of ℰ . Of course J q ( ℰ ) is a fibered manifold over X with projection π q while J q + r ( ℰ ) is a fibered manifold over J q ( ℰ ) with projection π q q + r , ∀ r ≥ 0 .

DEFINITION 1.1: A (non-linear) system of order q on ℰ is a fibered submanifold ℛ q ⊂ J q ( ℰ ) and a solution of ℛ q is a section f of ℰ such that j q ( f ) is a section of ℛ q .

DEFINITION 1.2: When the changes of coordinates have the linear form x ¯ = φ ( x ) , y ¯ = A ( x ) y , we say that ℰ is a vector bundle over X and denote for simplicity a vector bundle and its set of sections by the same capital letter E . When the changes of coordinates have the form x ¯ = φ ( x ) , y ¯ = A ( x ) y + B ( x ) we say that ℰ is an affine bundle over X and we define the associated vector bundle E over X by the local coordinates ( x , v ) changing like x ¯ = φ ( x ) , v ¯ = A ( x ) v .

DEFINITION 1.3: If the tangent bundle T ( ℰ ) has local coordinates ( x , y , u , v ) changing like u ¯ j = ∂ i φ j ( x ) u i , v ¯ l = ∂ ψ l ∂ x i ( x , y ) u i + ∂ ψ l ∂ y k ( x , y ) v k , we may introduce the vertical bundle V ( ℰ ) ⊂ T ( ℰ ) as a vector bundle over ℰ with local coordinates ( x , y , v ) obtained by setting u = 0 and changes v ¯ l = ∂ ψ l ∂ y k ( x , y ) v k . Of course, when ℰ is an affine bundle over X with associated vector bundle E over X , we have V ( ℰ ) = ℰ × X E with notation E   ⋯   ℰ → π X .

For a later use, if ℰ is a fibered manifold over X and f is a section of ℰ , we denote by f − 1 ( V ( ℰ ) ) the reciprocal image of V ( ℰ ) by f as the vector bundle over X obtained when replacing ( x , y , v ) by ( x , f ( x ) , v ) in each chart. A similar construction may also be done for any affine bundle over ℰ .

We now recall a few basic geometric concepts that will be constantly used through this paper. First of all, if ξ , η ∈ T , we define their bracket [ ξ , η ] ∈ T by the local formula ( [ ξ , η ] ) i ( x ) = ξ r ( x ) ∂ r η i ( x ) − η s ( x ) ∂ s ξ i ( x ) leading to the Jacobi identity [ ξ , [ η , ζ ] ] + [ η , [ ζ , ξ ] ] + [ ζ , [ ξ , η ] ] = 0 , ∀ ξ , η , ζ ∈ T allowing to define a Lie algebra and to the useful formula [ T ( f ) ( ξ ) , T ( f ) ( η ) ] = T ( f ) ( [ ξ , η ] ) where T ( f ) : T ( X ) → T ( Y ) is the tangent mapping of a map f : X → Y .

When I = { i 1 < ⋯ < i r } is a multi-index, we may set d x I = d x i 1 ∧ ⋯ ∧ d x i r for describing ∧ r T * and introduce the exterior derivative d : ∧ r T * → ∧ r + 1 T * : ω = ω I d x I → d ω = ∂ i ω I d x i ∧ d x I with d 2 = d ∘ d = ∂ i j ω I d x i ∧ d x j ∧ d x I ≡ 0 in the Poincaré sequence:

The Lie derivative of an r -form with respect to a vector field ξ ∈ T is the linear first order operator ℒ ( ξ ) linearly depending on j 1 ( ξ ) and uniquely defined by the following three properties:

1) ℒ ( ξ ) f = ξ . f = ξ i ∂ i f , ∀ f ∈ ∧ 0 T * = C ∞ ( X ) .

2) ℒ ( ξ ) d = d ℒ ( ξ ) .

3) ℒ ( ξ ) ( α ∧ β ) = ( ℒ ( ξ ) α ) ∧ β + α ∧ ( ℒ ( ξ ) β ) , ∀ α , β ∈ ∧ T * .

It can be proved that ℒ ( ξ ) = i ( ξ ) d + d i ( ξ ) where i ( ξ ) is the interior multiplication ( i ( ξ ) ω ) i 1 ⋯ i r = ξ i ω i i 1 ⋯ i r and that [ ℒ ( ξ ) , ℒ ( η ) ] = ℒ ( ξ ) ∘ ℒ ( η ) − ℒ ( η ) ∘ ℒ ( ξ ) = ℒ ( [ ξ , η ] ) , ∀ ξ , η ∈ T .

Indeed, if α = α i ( x ) d x i ∈ T * , we have successively:

We now turn to group theory and start with two basic definitions:

Let G be a Lie group, that is another manifold with local coordinates a = ( a 1 , ⋯ , a p ) called parameters, a composition G × G → G : ( a , b ) → a b , an inverse G → G : a → a − 1 and an identity e ∈ G satisfying:

DEFINITION 1.4: G is said to act on X if there is a map X × G → X : ( x , a ) → y = a x = f ( x , a ) such that ( a b ) x = a ( b x ) = a b x , ∀ a , b ∈ G , ∀ x ∈ X and we shall say that we have a Lie group of transformations of X . In order to simplify the notations, we shall use global notations even if only local actions are existing. The set G x = { a ∈ G | a x = x } is called the isotropy subgroup of G at x ∈ X and the action is said to be effective if a x = x , ∀ x ∈ X ⇒ a = e .

DEFINITION 1.5: A Lie pseudogroup of transformations Γ ⊂ a u t ( X ) is a group of transformations solutions of a system of OD or PD equations such that, if y = f ( x ) and z = g ( y ) are two solutions, called finite transformations, that can be composed, then z = g ∘ f ( x ) = h ( x ) and x = f − 1 ( y ) = g ( y ) are also solutions while y = x is a solution and we shall set i d q = j q ( i d ) .

It is clear that Lie groups of transformations are particular examples of Lie pseudogroups of transformations as the system defining the finite transformations can be obtained by eliminating the parameters among the equations y q = j q ( f ) ( x , a ) when q is large enough. The underlying system may be non-linear and of high order. For example, y = a x b ⇒ ( y x x / y x ) − ( y x / y ) + 1 x = 0 and y = ( a x + b ) / ( c x + d ) ⇒ ( y x x x / y x ) − 3 2 ( y x x / y x ) 2 = 0 . Accordingly, we shall speak about an algebraic pseudogroup when the system is defined by differential polynomials, that is by polynomials in the derivatives. Looking for transformations “ close “ to the identity, that is setting y = x + t ξ ( x ) + ⋯ when t ≪ 1 is a small constant parameter, dividing by t and passing to the limit t → 0 , we may linearize the above non-linear system of finite lie equations in order to obtain a linear system of infinitesimal Lie equations of the same order for vector fields. Such a system has the property that, if ξ and η are two solutions, then [ ξ , η ] is also a solution. Accordingly, the set Θ ⊂ T of solutions of this new system satisfies [ Θ , Θ ] ⊂ Θ and can therefore be considered as the Lie algebra of Γ. In the two above examples, we obtain successively ξ x x − ( 1 / x ) ξ x + ( 1 / x 2 ) ξ = 0 and ξ x x x = 0 . However, such a definition has no meaning at all in actual practice because, in general, the explicit solutions cannot be known as we hall see on examples.

Let me end this rather historical introduction with a short explanation of the reasons that pushed me to study the results of Vessiot with the aim to use them in physics, particularly in general relativity (GR). Being interested by the mathematical foundations of GR, I obtained a PhD diploma in GR at the university of Paris with top marks before going to Princeton. It is exactly at this moment that I discovered a few papers on the deformation theory of finite dimensional Lie algebras leading to the Chavalley-Eilenberg purely algebraic cohomology obtained by introducing a parametrization c t = c + t C + ⋯ of the structure constants of a finite dimensional Lie algebra G when t ≪ 1 is a small dimensionless parameter (See the section 2 of [18] for more details). In the meantime, I discovered the work of Spencer on the deformation of a geometric structure on a manifold, obtained through the parametrization ω t = ω + t Ω + ⋯ in the case of a geometric object ω like a metric on a manifold and t is again a small parameter. Comparing to GR, it became clear that the idea of Einstein had been to “deform” the Minkowski metric by introducing the very small dimensionless parameter ϕ / c 2 ≪ 1 where ϕ = G m / r is the gravitational potential existing at the distance r of a central mass m and G is the gravitational constant while c is the speed of light in vacuum. When I discovered that the Vessiot structure equations obtained in the case of a Lie group G acting on a manifold X with d i m ( G ) = d i m ( X ) were just the Maurer-Cartan equations, such a result led me to use the Vessiot structure constants in a similar way but for an arbitrary Lie pseudogroup Γ ⊂ a u t ( X ) . The idea of studying the normalizer then came from the fact that the Poincaré group of space-time with 10 parameters is of codimension 1 in its normalizer which is the Weyl group of dimension 11 obtained by adding the only dilatation parameter. However, if the metric is not flat, that is has a non-vanishing scalar curvature, the corresponding group is self-normalizing as we shall see. It follows that all the structural results presented in this paper, namely the deformation cohomology, the deformation sequence and the study of the normalizer for a transitive Lie pseudogroup are new in the sense that no other reference can even be found elsewhere. In addition, all the motivating examples presented in section 5 are showing in an explicit way the role of the Vessiot structure constants.

In a quite more general homological framework, which is out of the frame of this paper, the work of Vessiot is fitting with the Janet sequence while the work of Cartan is fitting with the Spencer sequence (See the fundamental diagram I of section 3 and refer to [3][10]-[12] for more details).

We finally insist on the fact that the confusion done by Eisenhart between IC and CC which is explained in example 5.4 by the corresponding technical lemma is still unknown and leads to revisiting the foundation of classical and conformal Riemannian geometry. Accordingly, and though striking it may be today, the mathematical foundations of GR must be revisited along these new structural results that question the origin and existence of gravitational waves and black holes.

We now turn to the clever theory proposed by Vessiot in 1903 [7] and sketch in a few successive steps the main results that we have presented through many books [4][10][11][18][19]. We invite the reader to follow all the definitions and results on the examples (5.2 + 5.3) provided by Vessiot himself in 1903 (!) that we revisit, by the example 5.4 which is by far the most striking one and by example 5.8 still not known though it is quite useful in analytical mechanics [3].

A) FINITE LIE EQUATIONS: If ℰ = X × X , we shall denote by Π q = Π q ( X , X ) the open subfibered manifold of J q ( X × X ) defined independently of the coordinate system by d e t ( y i k ) ≠ 0 with source projection α q : Π q → X : ( x , y q ) → ( x ) and target projection β q : Π q → X : ( x , y q ) → ( y ) . We shall sometimes introduce a copy Y of X with local coordinates ( y ) in order to avoid any confusion between the source and the target manifolds. Let us start with a Lie pseudogroup Γ ⊂ a u t ( X ) defined by a system ℛ q ⊂ Π q of order q . In all the sequel we shall suppose that the system is involutive (see next section) and that Γ is transitive that is ∀ x , y ∈ X , ∃ f ∈ Γ , y = f ( x ) or, equivalently, the map ( α q , β q ) : ℛ q → X × X : ( x , y q ) → ( x , y ) is surjective.

B) INFINITESIMAL LIE EQUATIONS: The Lie algebra Θ ⊂ T of infinitesimal transformations is then obtained by linearization, that is to say setting y = x + t ξ ( x ) + ⋯ , dividing by t and passing to the limit t → 0 as we already said, in order to obtain the linear involutive system R q = i d q − 1 ( V ( ℛ q ) ) ⊂ J q ( T ) by reciprocal image with Θ = { ξ ∈ T | j q ( ξ ) ∈ R q } while taking into account the fact that T = i d − 1 ( V ( X × X ) ) . From now on we shall suppose that R q is transitive, that is to say the canonical projection π 0 q : J q ( T ) → T induces an epimorphism π 0 q : R q → T with kernel R q 0 ⊂ R q and we have the useful short exact sequence 0 → R q 0 → R q → π 0 q T → 0 .

C) DIFFERENTIAL INVARIANTS: Passing from source to target, we may prolong the vertical infinitesimal transformations η = η k ( y ) ∂ ∂ y k to the jet coordinates up to order q in order to obtain for any η ∈ T ( Y ) :

where we have replaced j q ( f ) ( x ) by y q , each component being the “formal” derivative of the previous one obtained by introducing d i = ∂ i + y μ + 1 i k ∂ ∂ y μ k . Replacing the derivatives of η with respect to the target y by sections of R q ( Y ) over Y , we get for example:

It is possible to prove that [ ♯ ( R q ( Y ) ) , ♯ ( R q ( Y ) ) ] = ♯ ( [ R q ( Y ) , R q ( Y ) ] ) where the differential bracket involved over the target will be defined independently over the source in Definition 4.2.

As [ Θ , Θ ] ⊂ Θ and thus [ R q , R q ] ⊂ R q , we may use the Frobenius theorem in order to find a generating fundamental set of differential invariants { Φ τ ( y q ) } up to order q which are such that Φ τ ( y ¯ q ) = Φ τ ( y q ) by using the chain rule for derivatives whenever y ¯ = g ( y ) ∈ Γ acting now on Y . Of course, in actual practice one must use sections of R q instead of solutions but it is only in section 4 through Definition 4.2 that we shall see why the use of the Spencer operator will be crucial for this purpose. Specializing the Φ τ at i d q ( x ) we obtain the Lieform Φ τ ( y q ) = ω τ ( x ) of ℛ q . Finally, if Φ τ is any differential invariant at the order q , then d i Φ τ is a differential invariant at order q + 1 , ∀ i = 1 , ⋯ , n and so on.

D) GEOMETRIC OBJECTS: It has been the clever discovery of Vessiot in 1903 [7] to notice that a natural bundle ℱ over X could be associated with any Lie pseudogroup Γ ⊂ a u t ( X ) , both with a section ω of ℱ called geometric object or structure on X , transforming the same way. Of course, as he also pointed out, different sections can provide the same Lie pseudogroup and this will be the starting point of deformation theory. Indeed, the prolongation ♭ ( j q ( ξ ) ) at order q of any horizontal vector field ξ = ξ i ( x ) ∂ ∂ x i commutes with the prolongation at order q of any vertical vector field η = η k ( y ) ∂ ∂ y k , exchanging therefore the differential invariants. Keeping in mind the well-known property of the Jacobian determinant while passing to the finite point of view, any (local) transformation x ¯ = φ ( x ) can be lifted to a (local) transformation of the differential invariants between themselves of the form u → λ ( u , j q ( φ ) ( x ) ) allowing to introduce a natural bundle ℱ over X by patching changes of coordinates x ¯ = φ ( x ) , u ¯ = λ ( u , j q ( φ ) ( x ) ) . A section ω of ℱ is called a geometric object or structure on X and transforms like ω ¯ ( f ( x ) ) = λ ( ω ( x ) , j q ( f ) ( x ) ) or simply ω ¯ = j q ( f ) ( ω ) for any (local) transformation y = f ( x ) . This is a way to generalize vectors and tensors ( q = 1 ) or even connections ( q = 2 ). As a byproduct we have Γ = { f ∈ a u t ( X ) | Φ ω ( j q ( f ) ) ≡ j q ( f ) − 1 ( ω ) = ω } as a new way to write out the Lie form and we may say that Γpreserves ω . Replacing j q ( f ) by f q , we also obtain ℛ q = { f q ∈ Π q | f q − 1 ( ω ) = ω } . Coming back to the infinitesimal point of view and setting f t = exp ( t ξ ) ∈ a u t ( X ) , ∀ ξ ∈ T , we may define the ordinary Lie derivative with value in F = ω − 1 ( V ( ℱ ) ) by the formula:

along with the commutative diagram:

We have x → x ¯ = x + t ξ ( x ) + ⋯ ⇒ u τ → u ¯ τ = u τ + t ∂ μ ξ k L k τ μ ( u ) + ⋯ where μ = ( μ 1 , ⋯ , μ n ) is a multi-index and we may write down the system of infinitesimal Lie equations in the Medolaghi form (care to signs):

as a way to state the invariance of the section ω of ℱ , that is u τ − ω τ ( x ) = 0 ⇒ u ¯ τ − ω ¯ τ ( x ¯ ) = 0 .

Finally, replacing j q ( ξ ) by a section ξ q ∈ J q ( T ) over ξ ∈ T , we may define R q ⊂ J q ( T ) on sections by the linear (non-differential) equations:

and obtain the first prolongation R q + 1 ⊂ J q + 1 ( T ) by adding:

When ξ q + 1 is replaced by j q + 1 ( ξ ) , we obtain the PD Medolaghi equations of order q + 1 .

E) COMPATIBILITY CONDITIONS(CC): By analogy with “special” and “general” relativity, we shall call the given section special and any other arbitrary section general. The problem is now to study the formal properties of the linear system just obtained with coefficients only depending on j 1 ( ω ) , exactly like we shall proceed in the motivating examples later on. In particular, if any expression involving ω and its derivatives is a scalar object, it must reduce to a constant because Γ is assumed to be transitive and thus cannot be defined by any zero order equation. Now let us prove that the CC for ω ¯ , thus for ω too, only depend on the Φ and take the quasi-linear symbolic form v ≡ I ( u 1 ) ≡ A ( u ) u x + B ( u ) = 0 with u 1 = ( u , u x ) , allowing to define an affine subfibered manifold ℬ 1 ⊂ J 1 ( ℱ ) over ℱ . Indeed, if ∑ A ( y q ) d x Φ is a minimum sum of formal derivatives of differential invariants of order q not containing any jet coordinate of strict order q + 1 , we may suppose by division that the first A in the sum is equal to 1. Applying the prolonged distribution of vector fields introduced in step 3 at order q + 1 , we obtain a new sum with less terms and a contradiction unless all the A are again differential invariants at order q and thus functions of the Φ because of the Frobenius theorem. A similar comment can be done for the B . Now, if one has two sections ω and ω ¯ of ℱ , the equivalence problem is to look for f ∈ a u t ( X ) such that j q ( f ) − 1 ( ω ) = ω ¯ . When the two sections satisfy the same CC, the problem is sometimes locally possible (Lie groups of transformations, Darboux problem in analytical mechanics, ...) but sometimes not ([10], p 333).

F) INTEGRABILITY CONDITIONS (IC): Instead of the CC for the equivalence problem, let us look for the integrability conditions (IC) of the system of infinitesimal Lie equations and suppose that, for the given section, all the equations of order q + r are obtained by differentiating r times only the equations of order q , then it was claimed by Vessiot in [7] but with no proof that such a property is held if and only if there is an equivariant section c : ℱ → ℱ 1 : ( x , u ) → ( x , u , v = c ( u ) ) where ℱ 1 = J 1 ( ℱ ) / ℬ 1 is a natural vector bundle over ℱ with local coordinates ( x , u , v ) . Moreover, any such equivariant section depends linearly on a finite number of constants c called structure constants and the IC for the Vessiot structure equations I ( u 1 ) = c ( u ) are of a polynomial form J ( c ) = 0 . It is important to notice that, according to its construction, the form of the Vessiot structure equations is invariant under any change of coordinate system. In actual practice, this study can be divided into two parts according to the following commutative and exact diagram where R q + 1 is the first prolongation of R q and the symbol g q + 1 of R q + 1 is the kernel of the (not necessarily surjective) map π q q + 1 : R q + 1 → R q induced by the short exact sequence 0 → S q + 1 T * ⊗ T → J q + 1 ( T ) → π q q + 1 J q ( T ) → 0 :

Indeed, chasing in this diagram, we discover that π q q + 1 : R q + 1 → R q is an epimorphism if and only if π q q + 1 : R q + 1 0 → R q 0 is an epimorphism and π 0 q + 1 : R q + 1 → T is also an epimorphism. Looking at the form of the corresponding Medolaghi equations L ( ξ q ) ω = 0 and L ( ξ q + 1 ) j 1 ( ω ) = 0 , these two conditions respectively bring the Vessiot structure equations of first kind I * ( u 1 ) = 0 not invoving any structure constant and the Vessiot stucture equations of second kind I * * ( u 1 ) = c involving the same number of different structure constants. Such a study, only depending now on linear algebraic techniques, can be achieved by means of computer algebra, in particular when dealing with algebraic pseudogroups defined by differential polynomials.

Finally, looking at the formal integrability of the system ℬ 1 ⊂ J 1 ( ℱ ) defined by the equations A ( u ) u x + B ( u ) = 0 and their first prolongation A ( u ) u x x + ∂ A ( u ) ∂ u u x u x + ∂ B ( u ) ∂ u u x = 0 , the symbols only depend on A ( u ) and we may obtain equations of the form a ( u ) u x u x + b ( u ) u x = 0 by eliminating the jets of order 2. Using local coordinates ( x , u , v = A ( u ) u x + B ( u ) ) for ℱ 1 , and substituting ( u , v ) in place of ( u , u x ) , we obtain equations of the form α ( u ) v v + β ( u ) v + γ ( u ) = 0 . As we may suppose that c = 0 for the special section, we finally get equations of the form α ( u ) I I + β ( u ) I = 0 and it only remains to set I ( u 1 ) = c ( u ) in order to get polynomial Jacobi conditions of degree ≤ 2 which may not depend on u anymore because these equations are invariant in form under any change of coordinates. Though Vessiot claimed that these results had been found by Lie, we have never seen them despite a close examination of his collected works.

When q = 1 , a close examination of the Medolaghi equations and their first prolongation show at once that we can choose v = A ( u ) u x and we get homogeneous Jacobi conditions of degree 2 but systems of order q = 2 may even provide linear Jacobi relations as in [19]. Other examples will be provided in Section 5. The reader interested may find more details and explicit examples in [4][10][11]. Finally, we invite again the reader to look carefully at the example of Riemann structure in order to understand the deep confusion that has been done by L.P. Eisenhart and all successors between integrability conditions and compatibility conditions in the study of Killing equations that are only specific first order Medolaghi equations that may not provide an involutive system.

We now provide a survey of the work done by D.C. Spencer and coworkers in [20][21] with the purpose to prove that the work of Cartan has been superseded by the work of Spencer while the work of Vessiot has been superseded by the work of Janet, namely that curvature+torsion is in the Spencer sequence while curvature alone is in the Janet sequence, a big gap indeed for physics, particularly for Einstein general relativity [22][23].

Let μ = ( μ 1 , ⋯ , μ n ) be a multi-index with length | μ | = μ 1 + ⋯ + μ n . We shall say that μ is of class i if μ 1 = ⋯ = μ i − 1 = 0 , μ i ≠ 0 and set μ + 1 i = ( μ 1 , ⋯ , μ i − 1 , μ i + 1 , μ i + 1 , ⋯ , μ n ) . We set y q = { y μ k | 1 ≤ k ≤ m , 0 ≤ | μ | ≤ q } with y μ k = y k when | μ | = 0 . If E is a vector bundle over X with local coordinates ( x i , y k ) for i = 1 , ⋯ , n and k = 1 , ⋯ , m , we denote by J q ( E ) the q -jet bundle of E with local coordinates simply denoted by ( x , y q ) and sections f q : ( x ) → ( x , f k ( x ) , f i k ( x ) , f i j k ( x ) , ⋯ ) transforming like the section j q ( f ) : ( x ) → ( x , f k ( x ) , ∂ i f k ( x ) , ∂ i j f k ( x ) , ⋯ ) when f is an arbitrary section of E . Then both f q ∈ J q ( E ) and j q ( f ) ∈ J q ( E ) are over f ∈ E and the Spencer operator just allows to distinguish them by introducing a kind of “difference” through the operator d : J q + 1 ( E ) → T * ⊗ J q ( E ) : f q + 1 → j 1 ( f q ) − f q + 1 with local components ( ∂ i f k ( x ) − f i k ( x ) , ∂ i f j k ( x ) − f i j k ( x ) , ⋯ ) and more generally ( d f q + 1 ) μ , i k ( x ) = ∂ i f μ k ( x ) − f μ + 1 i k ( x ) . Indeed, the composite map J q + 1 ( E ) → J q ( E ) → j 1   J 1 ( J q ( E ) ) and the canonical map J q + 1 ( E ) → J 1 ( J q ( E ) ) identifying y μ , i with y μ + 1 i when | μ | = q are both over the identity map J q ( E ) = J q ( E ) and their difference has therefore an image in T * ⊗ J q ( E ) because of the short exact sequence:

In a symbolic way, when changes of coordinates are not involved, it is sometimes useful to write down the components of d in the form d i = ∂ i − δ i and the restriction of d to the kernel S q + 1 T * ⊗ E of the canonical projection π q q + 1 : J q + 1 ( E ) → J q ( E ) is minus the Spencer map δ = d x i ∧ δ i : S q + 1 T * ⊗ E → T * ⊗ S q T * ⊗ E which can be extended to:

with δ ∘ δ ≡ 0 if we set ω = ( ω μ , I k d x I ) and define ( δ ω ) μ k = d x i ∧ ω μ + 1 i k when | μ | = q .

The kernel of d is made by sections such that f q + 1 = j 1 ( f q ) = j 2 ( f q − 1 ) = ⋯ = j q + 1 ( f ) . Finally, if R q ⊂ J q ( E ) is a system of order q on E locally defined by linear equations Φ τ ( x , y q ) ≡ a k τ μ ( x ) y μ k = 0 and local coordinates ( x , z ) for the parametric jets up to order q , the r -prolongation R q + r = ρ r ( R q ) = J r ( R q ) ∩ J q + r ( E ) ⊂ J r ( J q ( E ) ) is locally defined when r = 1 by the linear equations Φ τ ( x , y q ) = 0 ⇒ d i Φ τ ( x , y q + 1 ) ≡ a k τ μ ( x ) y μ + 1 i k + ∂ i a k τ μ ( x ) y μ k = 0 and has symbol g q + r = R q + r ∩ S q + r T * ⊗ E ⊂ J q + r ( E ) if one looks at the top order terms. If f q + 1 ∈ R q + 1 is over f q ∈ R q , differentiating the identity a k τ μ ( x ) f μ k ( x ) ≡ 0 with respect to x i and subtracting the identity a k τ μ ( x ) f μ + 1 i k ( x ) + ∂ i a k τ μ ( x ) f μ k ( x ) ≡ 0 , we obtain the identity a k τ μ ( x ) ( ∂ i f μ k ( x ) − f μ + 1 i k ( x ) ) ≡ 0 and thus the restriction d : R q + 1 → T * ⊗ R q . This first order operator induces, up to sign, the purely algebraic monomorphism 0 → g q + 1 → δ T * ⊗ g q on the symbol level [10]-[12].