The first motivation for studying the methods used in this paper has been a 1000$ challenge proposed in 1970 by J. Wheeler in the physics department of Princeton University while the author of this paper was a student of D.C. Spencer in the closeby mathematics department:
Is it possible to express the generic solutions of Einstein equations in vacuum by means of the derivatives of a certain number of arbitrary functions like the potentials for Maxwell equations?
During the next 25 years and though surprising it may look like, no progress at all has been made towards any solution, either positive or negative. We now explain the way we found the (negative) solution of this challenge in 1995  .
Let us consider a manifold X of dimension n with local coordinates , tangent bundle T, cotangent bundle , vector bundle of q-symmetric covariant tensors and vector bundle of r-skew-symmetric covariant tensors or r-forms. The group of isometries of the non- degenerate metric with on X is defined by the nonlinear first order system in Lie form:
Linearizing at the identity transformation , we may introduce the corresponding Killing operator , which involves the Lie derivative and provides twice the so-called infinitesimal deformation tensor of continuum mechanics when is the Euclidean metric. We may consider the linear first order system of Medolaghi equations:
which is in fact a family of systems only depending on the geometric object and its derivatives. Introducing the Christoffel symbols , we may differentiate once and add the operator with the well known Levi-Civita isomorphism in order to obtain the linear second order system of Medolaghi equations:
Similarly, introducing the Jacobian determinant , the group of conformal transformations of the metric may be defined by the nonlinear first order system in Lie form:
while introducing the metric density as a new geometric object, rather than by eliminating a conformal factor as usual. The conformal Killing operator may be defined by linearization as above and we obtain:
We may introduce the trace with standard notations and obtain therefore because .
The reader may look at      for finding other examples of Lie groups or Lie pseudogroups of transformations along the approach initiated by E. Vessiot in 1903  .
In classical elasticity, the stress tensor density existing inside an elastic body is a symmetric 2-tensor density introduced by A. Cauchy in 1822.
Integrating by parts the implicit summation , we obtain the Cauchy
operator . When is the euclidean metric, the corresponding Cauchy stress equations can be written as where the right member describes the local density of forces applied to the body, for example gravitation. With zero second member, we study the possibility to “parametrize” the system of PD equations , namely to express its general solution by means of a certain number of arbitrary functions or potentials, called stress functions. Of course, the problem is to know about the number of such functions and the order of the parametrizing operator. For one may introduce the Euclidean metric while, for , one may consider the Minkowski metric. A few definitions used thereafter will be provided later on.
● When , the stress equations become , . Their second order parametrization , , has been provided by George Biddell Airy (1801-1892) in 1863  . It can be simply recovered in the following manner:
We get the linear second order system:
which is involutive with one equation of class 2, 2 equations of class 1 and it is easy to check that the 2 corresponding first order CC are just the stress equations.
When constructing a long prismatic dam with concrete as in   or in the Introduction of  , we may transform a problem of 3-dimensional elasticity into a problem of 2-dimensional elasticity by supposing that the axis of the dam is perpendicular to the river with and because of the rocky banks of the river. We may introduce the two Lamé constants and the Poisson coefficient in order to describe the usual constitutive relations of an homogeneous isotropic medium as follows, passing from the standard case to the restricted case just by setting:
even though . Let us consider the right square of the diagram below with locally exact rows:
Taking into account the formula (5.1.4) of  for the linearization of the only component of the Riemann tensor at when and substituting the Airy parametrization, we obtain:
where the linearized scalar curvature is allowing to define the Riemann operator in the previous diagram, namely the only compatibility condition (CC) of the Killing operator. It remains to exhibit an arbitrary homogeneous polynomial solution of degree 3 and to determine its 4 coefficients by the boundary pressure conditions on the upstream and downstream walls of the dam. Of course, the Airy potential has nothing to do with the perturbation of the metric and the Airy parametrization is nothing else but the formal adjoint operator of the Riemann operator, linearization of the Riemann tensor over , expressing the second order compatibility conditions (CC) of the inhomogeneous system .
● When , using now the left square of the following diagram with locally exact rows:
where the self-adjoint operator has ben introduced by E. Beltrami in 1892. We may substitute the 3-dimensional constitutive relations with Lamé constants in the Cauchy stress equations and get, when (gravity):
We discover at once that the origin of elastic waves is shifted by one step backwards, from the right square to the left square of the diagram. Indeed, using inertial forces for a medium with mass per unit volume in the right member of Cauchy stress equations because of Newton law and the vector identity , we discover the existence of two types of elastic waves with wave vector , period , pulsation with standard notations, namely the longitudinal and transversal waves with different speeds , which are really existing because that are responsible for earthquakes  :
It is this comment that pushed me to use the formal adjoint of an operator, knowing already that an operator and its (formal) adjoint have the same differential rank (See later on). In the case of the conformal Killing operator, the second order CC are generated by the Weyl operator, linearization of the Weyl tensor over when . The particular situation pour will be studied in the last section and its corresponding 5 third order CC are not known after one century  . Finally, the Bianchi operator describing the CC of the Riemann operator does not appear in this scheme.
Summarizing what we have just said, the study of elastic waves in continuum mechanics only depends on group theory because it has only to do with one differential sequence and its formal adjoint, combined together by means of constitutive relations. We have proved in many books   and in    that the situation is similar for Maxwell equations, a result leading therefore to revisit the mathematical foundations of both General Relativity (GR) and Gauge Theory (GT), thus also of Electromagnetism (EM).
Knowing already M.P. Malliavin as I gave a seminar on the “Deformation Theory of Algebraic and Geometric Structures”   , I presented in 1995 a seminar at IHP in Paris, proving the impossibility to parametrize Einstein equations, a result I just found  . One of the participants called my attention on a recently published translation from japanese of the 1970 master thesis of M. Kashiwara that he just saw on display in the library of the Institute  . This has been the true starting of the story because I discovered that the duality involved in the preceding approach to physics was only a particular example of a much more sophisticated framework having to do with homological algebra       .
Let us explain this point of view by means of an elementary example. With for , we get for the CC . Then is defined by while is defined by but the CC of are generated by . Using operators, we have the two differential sequences:
where generates the CC of in the upper sequence but does not generate the CC of in the lower sequence, even though , contrary to what happened in the previous diagram. We shall see that this comment brings the need to introduce the first extension module of the differential module M determined by .
In a more intrinsic setting, using the same notation for a vector bundle and its set of (local) sections, we shall have:
In the meantime, following U. Oberst   , a few persons were trying to adapt these methods to control theory and, thanks to J.L. Lions, I have been able to advertise about this new approach in a european course, held with succes during 6 years  and continued for 5 other years in a slightly different form  . By chance I met A. Quadrat, a good PhD student interested by control and computer algebra and we have been staying alone because the specialists of Algebraic Analysis were pure mathematicians, not interested at all by applications. As a byproduct, it is rather strange to discover that the impossibility to parametrize Einstein equations, that we shall prove in Section 4, has never been acknowledged by physicists but can be found in a book on control because it is now known that a control system is controllable if and only if it is parametrizable   .
The following example of a double pendulum will prove that this result, still not acknowledged today by engineers, is not evident at all. For this, let us consider two pendula of respective length and attached at the ends of a rigid bar sliding horizontally with a reference position . If the pendula move with a respective (small) angle and with respect to the vertical, it is easy to prove from the Newton principle that the equations of the movements does not depend on the respective masses and of the pendula but only depend on the respective lengths and gravity along the two formulas:
where is the standard time derivative. It is experimentally visible and any reader can check it with a few dollars, that the system is controllable, that is the angles can reach any prescribed (small) values in a finite time when starting from equilibrium, if and only if and, in this case, we have the following (injective) 4th order parametrization:
of course, if , the system cannot be controllable because, setting , we obtain by substraction and thus , .
We end this Introduction explaining on a simple example why the second extension module must also be considered, especially in the study of Einstein equations, though surprising it may look like. To make a comparison, let us consider the following well known Poincaré sequence:
where is the exterior derivative. When , we have:
From their definition it follows that is parametrized by while is parametrized by . Also, in local coordinates, we have , , and the adjoint sequence is also the Poincaré sequence up to the sign. Let us nevertheless consider the new (minimal) parametrization of obtained by setting , namely   :
If we define the differential rank of an operator by the maximum number of differentially independent second member, this is clearly an involutive differential operator with differential rank equal to 2 because can be given arbitrarily and thus can be given arbitrarily or, equivalently, because the differential rank of is of course equal to 1 as has no CC. Now, the involutive system , , canot be parametrized by one arbitrary function because both and are autonomous in the sense that they both satisfy to at least one partial differential equation (PDE). Accordingly, we discover that can be parametrized by the through 3 arbitrary functions where may be given arbitrarily, the being itself parametrized by the , but can also be parametrized by another operator with less arbitrary functions or potentials which, in turn, cannot be parametrized again. Such a situation is similar to the one met in hunting rifles that may have one, two or more trigger mechanisms that can be used successively. It happens that the possibility to have one parametrization of is an intrinsic property described by the vanishing of where the differential module M is determined by while the property to have two successive parametrizations is an intrinsic property described by the vanishing of as we just said plus the vanishing of the second extension module , and so on, but such a result has no classical interpretation. It follows that certain parametrizations are “better” than others and no student should even imagine the minimal parametrization of that we have presented above. A similar procedure has been adopted by J.C. Maxwell  and G. Morera  when they modified the parametrization of the Cauchy stress equations obtained by E. Beltrami in 1892 (see  and  for more details and references or   and   for computer algebra calculations).
It is clear from the beginning of this Introduction that an isometry is a solution of a nonlinear system in Lie form    and that we have linearized this system over the identity transformation in order to study elastic waves. However, in general, no explicit solution may be known but most nonlinear systems of OD or PD equations of mathematical physics (constant riemannian curvature is a good example in  ) are defined by differential polynomials. This is particularly clear for riemannian, conformal, complex, contact, symplectic or unimodular structures on manifolds  . Hence, in Section 2 we shall provide the main results that exist in the formal theory of systems of nonlinear PD equations in order to construct a formal linearization. The proof of many results is quite difficult as it involves delicate chases in 3-dimensional diagrams    . In physics, the linear system obtained may have coefficients in a certain differential field and we shall need to revisit differential algebra in Section 3 because Spencer and Kolchin never clearly understood that their respective works could be combined. It will follow that the linear systems will have coefficients in a differential field K and we shall have to introduce the ring of differential operators with coefficients in K, which is even an integral domain. This fact will be particularly useful in order to revisit differential duality in Section 4 before applying it to physics in Section 5 and concluding in the last Section 6. This paper is an extended and improved version of a series of lectures given at the Albert Einstein Institute (Berlin/ Postdam), october 23-27, 2017, under the title: “General Relativity and Gauge Theory: Beyond the Mirror”.
These purely mathematical results question the origin and existence of gravitational waves.
2. Differential Geometry
If X is a manifold with local coordinates for , let be a fibered manifold over X with , that is a manifold with local coordinates for and simply denoted by , projection and changes of local coordinates . If and are two fibered manifolds over X with respective local coordinates and , we denote by the fibered product of and over X as the new fibered manifold over X with local coordinates . We denote by a global section of , that is a map such that but local sections over an open set may also be considered when needed. Under a change of coordinates, a section transforms like and the derivatives transform like:
We may introduce new coordinates transforming like:
We shall denote by the q-jet bundle of with local coordinates called jet coordinates and sections transforming like the sections where both and are over the section of . It will be useful to introduce a multi-index with length and to set . Finally, a jet coordinate is said to be of class i if . As the background will always be clear enough, we shall use the same notation for a vector bundle or a fibered manifold and their sets of sections   . We finally notice that is a fibered manifold over X with projection while is a fibered manifold over with projection     .
DEFINITION 2.1: A (nonlinear) system of order q on is a fibered submanifold and a global or local solution of is a section f of over X or such that is a section of over X or .
DEFINITION 2.2: When the changes of coordinates have the linear form , we say that is a vector bundle over X. Vector bundles will be denoted by capital letters and will have sections denoted by . In particular, we shall denote as usual by the tangent bundle of X, by the cotangent bundle, by the bundle of r-forms and by the bundle of q-symmetric covariant tensors. When the changes of coordinates have the form we say that is an affine bundle over X and we define the associated vector bundle over X by the local coordinates changing like .
DEFINITION 2.3: If the tangent bundle has local coordinates
changing like , we may introduce the vertical bundle as a vector bundle over with local coordinates obtained by setting and changes . Of course, when is an affine bundle over X with associated vector bundle E over X, we have . With a slight abuse of language, we shall set as a vector bundle over .
For a later use, if is a fibered manifold over X and f is a section of , we denote by the reciprocal image of by f as the vector bundle over X obtained when replacing by in each chart. A similar construction may also be done for any affine bundle over . Loking at the transition rules of , we deduce easily the following results:
PROPOSITION 2.4: is an affine bundle over modeled on but we shall not specify the tensor product in general.
PROPOSITION 2.5: There is a canonical isomorphism of vector bundles over given by setting at any order and a short exact sequence:
of vector bundles over allowing to establish a link with the formal theory of linear systems.
PROPOSITION 2.6: There is an exact sequence:
where is over with components is called the (nonlinear) Spencer operator. As , there is an induced exact sequence:
where is called the first Spencer operator.
DEFINITION 2.7: If is a system of order q on , then is called the first prolongation of and we may define the subsets . In actual practice, if the system is defined by PDE the first prolongation is defined by adding the PDE . accordingly, and as identities on X or at least over an open subset . Differentiating the first relation with respect to and substracting the second, we finally obtain:
and the Spencer operator restricts to . We set .
DEFINITION 2.8: The symbol of is the family of vector spaces over . The symbol of only depends on by a direct prolongation procedure. We may define the vector bundle over by the short exact sequence and we have the exact induced sequence .
Setting whenever and , we obtain:
In general, neither nor are vector bundles over .
On we may introduce the usual bases where we have set . In a purely algebraic setting, one has:
PROPOSITION 2.9: There exists a map which restricts to and .
Proof: Let us introduce the family of s-forms and set . We obtain at once and .
The kernel of each in the first case is equal to the image of the preceding but this may no longer be true in the restricted case and we set:
DEFINITION 2.10: Let and with be the coboundary space , cocycle space and cohomology space at of the restricted d-sequence which only depend on and may not be vector bundles. The symbol is said to be s- acyclic if , involutive if it is n-acyclic and finite type if becomes trivially involutive for r large enough. In particular, if is involutive and finite type, then . Finally, is involutive for any if we set .
Having in mind the example of with rank changing at , we have:
PROPOSITION 2.11: If is 2-acyclic and is a vector bundle over , then is a vector bundle over .
Proof: We may define the vector bundle over by the following ker/coker exact sequence where we denote by the image of the central map:
and we obtain by induction on r the following commutative and exact diagram of vector bundles over :
where all the maps have been given after Definition 2.9. The image of the central map of the top row is and a chase proves that is -acyclic whenever is s-acyclic by extending the diagram. The proposition finally follows by upper-semicontinuity from the relation:
LEMMA 2.12: If is involutive and is a vector bundle over , then is also a vector bundle over . In this case, changing linearly the local coordinates if necessary, we may look at the maximum number of equations that can be solved with respect to and the intrinsic number indicates the number of y that can be given arbitrarily.
Using the exactness of the top row in the preceding diagram and a delicate 3-dimensional chase, we have (See  and  , p. 336 for the details):
THEOREM 2.13: If is a system of order q on such that is a vector bundle over and is 2-acyclic, then there is an exact sequence:
where is called the r-curvature and is simply called the curvature of .
We notice that and in the following commutative diagram:
We also have because we have successively:
while chasing in the following commutative 3-dimensional diagram:
with a well defined map . We finally obtain the following crucial Theorem and its Corollary (Compare to  , p. 72-74 or  , p. 340 to  ):
THEOREM 2.14: Let be a system of order q on such that is a fibered submanifold of . If is 2-acyclic and is a vector bundle over , then we have for all .
DEFINITION 2.15: A system is said to be formally integrable if is an epimorphism of fibered manifolds for all and involutive if it is formally integrable with an involutive symbol . We have the following useful test     :
COROLLARY 2.16: Let be a system of order q on such that is a fibered submanifold of . If is 2-acyclic (involutive) and if the map is an epimorphism of fibered manifolds, then is formally integrable (involutive).
This is all what is needed in order to study systems of algebraic ordinary differential (OD) or partial differential (PD) equations.
3. Differential Algebra
We now present in an independent manner two OD examples and two PD examples showing the difficulties met when studying differential ideals and ask the reader to revisit them later on while reading the main Theorems. As only a few results will be proved, the interested reader may look at    for more details and compare to    .
EXAMPLE 3.1: If , y is a differential indeterminate and is a formal derivation, we may set and so on in order to introduce the differential ring . We consider the (proper) differential ideal generated by the differential polynomial . We have and cannot be a prime differential ideal. Hence, looking for the “solutions” of , we must have either or and thus where c should be a “constant” with no clear meaning. However, we have successively:
and thus is neither prime nor perfect, that is equal to its radical, but is perfect as it is the intersection of the prime differential ideal generated by y with the prime differential ideal generated by and , both containing .
EXAMPLE 3.2: With the notations of the previous Example, let us consider the (proper) differential ideal generated by the differential polynomial . We have and cannot be prime differential ideal. Hence, looking for the “solutions” of , we must have either or and . However, we have:
and thus is neither prime or perfect as before but is the prime differential ideal generated by and .
EXAMPLE 3.3: If as before, y is a differential indeterminate and are two formal derivations, let us consider the differential ideal generated by and in . Using crossed derivatives, we get successively:
and thus is neither prime nor perfect but is a perfect differential ideal and even a prime differential ideal because we obtain easily from the last section that the resisual differential ring is a differential integral domain. Its quotient field is thus the differential field with the rules:
as a way to avoid looking for solutions. The formal linearization is the linear system obtained in the last section where it was defined over , but not over K, by the two linear second order PDE:
changing slightly the notations for using the letter v only when looking at the symbols. It is at this point that the problem starts because is indeed a fibered manifold with arbitrary parametric jets but is no longer a fibered manifold because the dimension of its symbol changes when . We understand therefore that there should be a close link existing between formal integrability and the search for prime differential ideals or differential fields. The solution of this problem has been provided as early as in 1983 for studying the “Differential Galois Theory” but has never been acknowledged and is thus not known today (   ). The idea is to add the third order PDE and thus the linearized PDE obtaining therefore a third order involutive system well defined over K with symbol . We invite the reader to treat similarly the two previous examples and to compare.
EXAMPLE 3.4: If as before, y is a differential indeterminate and are two formal derivations, let us consider the differential ideal
generated by and in . Using crossed derivatives, we get successively:
and thus . As the symbol is involutive, there is an infinite number of parametric jets and thus is a differential integral domain with
, , . It follows that is a prime differential ideal with . The second order linearized system is:
is now well defined over the differential field and is involutive.
DEFINITION 3.5: A differential ring is a ring A with a finite number of commuting derivations such that , that can be extended to derivations of the ring of quotients by setting . We shall suppose from now on that A is even an integral domain and use the differential field . For example, if are indeterminates over , then is a differential ring for the standard with quotient field .
If K is a differential field as above and are indeterminates over K, we transform the polynomial ring into a differential ring by introducing as usual the formal derivations and we shall set .
DEFINITION 3.6: We say that is a differential ideal if it is stable by the , that is if . We shall also introduce the radical and say that is a perfect (or radical) differential ideal if . If S is any subset of A, we shall denote by the differential ideal generated by S and introduce the (non-differential) ideal in A.
LEMMA 3.7: If is a differential ideal, then is a differential ideal containing .
Proof: If d is one of the derivations, we have and thus:
LEMMA 3.8: If , we set with and . We have in general and the problem will be to know when we may have equality.
We shall say that a differential extension is a finitely generated differential extension of K and we may define the evaluation epimorphism with kernel by calling or the residue of y modulo . If we study such a differential extension , by analogy with Section 2, we shall say that or is a vector bundle over if one can find a certain number of maximum rank determinant that cannot be all zero at a generic solution of defined by differential polynomials , that is to say, according to the Hilbert Theorem of Zeros, we may find polynomials such that:
The following Lemma will be used in the next important Theorem:
LEMMA 3.9: If is a prime differential ideal of , then, for q sufficiently large, there is a polynomial such that and:
THEOREM 3.10: (Primality test) Let and be prime ideals such that and . If the symbol of the algebraic variety defined by is 2-acyclic and if its first prolongation is a vector bundle over , then is a prime differential ideal with .
COROLLARY 3.11: Every perfect differential ideal of can be expressed in a unique way as the non-redundant intersection of a finite number of prime differential ideals.
COROLLARY 3.12: (Differential basis) If is a perfect differential ideal of , then we have for q sufficiently large.
EXAMPLE 3.13: As is a polynomial ring with an infinite number of variables it is not noetherian and an ideal may not have a finite basis. With and , then is a prime differential ideal.
PROPOSITION 3.14: If is differentially algebraic over and is differentially algebraic over K, then is differentially algebraic over K. Setting , it follows that, if is a differential extension and are both differentially algebraic over K, then , and are differentially algebraic over K.
If , and are such that and , we have the two towers and of differential extensions and we may therefore define the new tower . However, if only and are known and we look for such an N containing both L and M, we may use the universal property of tensor products an deduce the existence of a differential morphism by setting whenever . The construction of an abstract composite differential field amounts therefore to look for a prime differential ideal in which is a direct sum of integral domains  .
DEFINITION 3.15: A differential extension L of a differential field K is said to be differentially algebraic over K if every element of L is differentially algebraic over K. The set of such elements is an intermediate differential field , called the differential algebraic closure of K in L. If is a differential extension, one can always find a maximal subset S of elements of L that are differentially transcendental over K and such that L is differentially algebraic over . Such a set is called a differential transcedence basis and the number of elements of S is called the differential transcendence degree of .
THEOREM 3.16: The number of elements in a differential basis of does not depent on the generators of and his value is . Moreover, if are differential fields, then .
THEOREM 3.17: If is a finitely generated differential extension, then any intermediate differential field between K and L is also finitely generated over K.
EXAMPLE 3.18: With , let us introduce the manifolds X with local coordinate x and Y with local coordinates . We may consider the algebraic Lie pseudogroup of (local, invertible) transformations of Y preserving the 1-form , that is to say made up by transformations solutions of the Pfaffian system . Equivalently, we have to look for the invertible solutions of the algebraic first order involutive system defined over by the first order involutive system of algebraic PD equations in Lie form:
By chance one can obtain the generic solution , where g is an arbitrary function of one variable. Now, if we introduce a function and consider the corresponding transformations of the jets , we obtain the only generating differential invariant . Hence, setting and , we have the tower of differential extensions . As any intermediate differential field is finitely generated, let us consider . Then:
allows to define a Lie subpseudogroup with generating differential invariants in such a way that, if we set , we have the strict inclusions and it does not seem possible to obtain a differential Galois correspondence between algebraic subpseudogroups and intermediate differential fields, similar to the classical one. We have explained in  how to overcome this problem but this is out of the scope of this paper. It is finally important to notice that the fundamental differential isomorphism    :
is the Hopf dual of the projective limit of the action graph isomorphisms between fibered manifolds:
of fibered dimension . The corresponding automorphic system in Lie form where is a geometric object as in the Introduction and its prolongations has been introduced as early as in 1903 by E. Vessiot   as a way to study principal homogeneous spaces (PHS) for Lie pseudogroups, namely if is a solution and is another solution, then there exists one and only one transformation of such that .
This is all what is needed in order to study systems of infinitesimal Lie equations defined, like the classical and conformal Killing systems, over where is a geometric object solution of a system of algebraic Vessiot structure equations (constant Riemann curvature, zero Weyl tensor).
4. Differential Duality
Let A be a unitary ring, that is and even an integral domain ( or ) with field of fractions . However, we shall not always assume that A is commutative, that is may be different from in general for . We say that is a left module over A if or a right module over B if the operation of B on M is . If M is a left module over A and a right module over B with , then we shall say that is a bimodule. Of course, is a bimodule over itself. We define the torsion submodule and M is a torsion module if or a torsion-free module if . We denote by the set of morphisms such that . We finally recall that a sequence of modules and maps is exact if the kernel of any map is equal to the image of the map preceding it.
When A is commutative, is again an A-module for the law as we have . In the non-commutative case, things are more complicate and, given and , then becomes a right module over B for the law .
DEFINITION 4.1: A module F is said to be free if it is isomorphic to a (finite) power of A called the rank of F over A and denoted by while the rank of a module M is the rank of a maximum free submodule . It follows from this definition that is a torsion module. In the sequel we shall only consider finitely presented modules, namely finitely generated modules defined by exact sequences of the type where and are free modules of finite ranks and often denoted by m and p in examples. A module P is called projective if there exists a free module F and another (projective) module Q such that .
PROPOSITION 4.2: For any short exact sequence , we have the relation , even in the non-commutative case. As a byproduct, if M admits a finite length free resolution , we may introduce the Euler-Poincaré characteristic (See  , p. 469).
The following proposition will be used many times in Section 5, in particular for exhibiting the Weyl tensor from the Riemann tensor (  , p. 73) (  , p. 33):
PROPOSITION 4.3: We shall say that the following short exact sequence splits if one of the following equivalent three conditions holds:
● There exists a monomorphism called lift of g and such that .
● There exists an epimorphism called lift of and such that .
● There exist isomorphisms and that are inverse to each other and provide an isomorphism with and thus .
These conditions are automatically satisfied if is free or projective.
Using the notation , for any morphism , we shall denote by the morphism which is defined by and satisfies (See  , Corollary 5.3, p. 179). We may take out in
order to obtain the deleted sequence and apply in order to get the sequence .
PROPOSITION 4.4: The extension modules and do not depend on the resolution chosen and are torsion modules for .
Let A be a differential ring, that is a commutative ring with n commuting derivations , that is while and . We shall use thereafter a differential integral domain A with unit whenever we shall need a differential field of coefficients, that is a field ( ) with , in order to exhibit solved forms for systems of partial differential equations as in the preceding section. Using an implicit summation on multi-indices, we may introduce the (noncommutative) ring of differential operators with elements such that and . The highest value of with is called the order of the operator P and the ring D with multiplication is filtred by the order q of the operators with the filtration . Moreover, it is clear that D, as an algebra, is generated by and with if we identify an element with the vector field of differential geometry, but with now. It follows that is a bimodule over itself, being at the same time a left D-module by the composition and a right D-module by the composition with in any case.
If we introduce differential indeterminates , we may extend to for . Therefore, setting and calling the differential module of equations, we obtain by residue the differential module or D- module , introducing the canonical projection and denoting the residue of by when there can be a confusion. Introducing the two free differential modules , we obtain equivalently the free presentation of order q when . It follows that M can be endowed with a quotient filtration obtained from that of which is defined by the order of the jet coordinates in . We shall suppose that the system is formally integrable. We have therefore the inductive limit with which is the dual of the projective limit if we set with and , the main reason for using a differential field K. We have in general with . Also, R is a left D-module with
More generally, introducing the successive CC as in the preceding Section while changing slightly the numbering of the respective operators, we may finally obtain the free resolution of M, namely the exact sequence where p is the canonical projection. Also, with a slight abuse of language, when is involutive, that is to say when is involutive, one should say that M has an involutive presentation of order q or that is involutive.
REMARK 4.5: In actual practice, one must never forget that acts on the left on column vectors in the operator case and on the right on row vectors in the module case. For this reason, when E is a (finite dimensional) vector bundle over X, we may apply the correspondence with and between jet bundles and left differential modules in order to be able to use the double dual isomorphism in both cases. We shall say that is the the left differential module induced by . Hence, starting from a differential operator , we may obtain a finite presentation and conversely, keeping the same operator matrix if we act on the right of row vectors. This comment becomes particularly useful when dealing with the Poincaré sequence in electromagnetism ( ) or even as we already saw in the Introduction ( ).
Roughly speaking, homological algebra has been created in order to find intrinsic properties of modules not depending on their presentations or even on their resolutions and we now exhibit another approach by defining the formal adjoint of an operator P and an operator matrix :
DEFINITION 4.6: Setting , we have and . Such a definition can be extended to any matrix of operators by using the transposed matrix of adjoint operators and we get:
from integration by part, where is a row vector of test functions and the usual contraction. We quote the useful formulas as in (  or  , p. 339-341).
The following technical Lemma is crucially used in the next proposition:
LEMMA 4.7: If is a local diffeomorphisms on X, we may set and we have the identity:
PROPOSITION 4.8: If we have an operator , we may obtain by duality an operator .
Now, with operational notations, let us consider the two differential sequences:
where generates all the CC of . Then but may not generate all the CC of as we already saw in the Introduction. Passing to the module framework, we just recognize the definition of when M is determined by .
As is a bimodule, then is a right D-module according to Lemma 3.1 and we may thus define a right module by the ker/coker long exact sequence but we have     :
THEOREM 4.9: We have the side changing procedures and with and .
Now, exactly like we defined the differential module M from , we may define the differential module N from . For any other presentation of M with an accent, we have   :
THEOREM 4.10: The modules and are projectively equivalent, that is one can find two projective modules and such that and we obtain therefore .
THEOREM 4.11: The operator is simply parametrizable if and doubly parametrizable if and . Moreover, we have the ker/coker long exact sequence:
where whenever and we have .
Proof: We prove first that . Indeed, if , then one may find such that and thus because is an integral domain and thus .
Let us now start with a free presentation of :
Applying , we may define and exhibit the following free resolution of N by right D-modules:
where . The deleted sequence is:
Applying again and using the canonical isomorphism for any free module F of finite rank, we get the sequence of left D-modules:
Denoting as usual a coboundary space by B, a cocycle space by Z and the corresponding cohomology by , we get the commutative and exact diagram:
An easy chase provides at once . It follows that is a torsion module and, as we already know that , we finally obtain . Also, as and , we obtain . Accordingly, a torsion-free ( injective)/reflexive ( bijective) module is described by an operator that admits respectively a single/double step parametrization.
We now turn to the operator framework;
DEFINITION 4.12: If a differential operator is given, a direct problem is to find generating compatibility conditions (CC) as an operator such that . Conversely, given , the inverse problem will be to look for such that generates the CC of and we shall say that is parametrized by if such an operator is existing. We finally notice that any operator is the adjoint of a certain operator because and we get:
THEOREM 4.13: (reflexivity test) In order to check whether M is reflexive or not, that is to find out a parametrization if which can be again parametrized, the test has 5 steps which are drawn in the following diagram where generates the CC of and generates the CC of while generates the CC of and generates the CC of :
COROLLARY 4.14: In the differential module framework, if is a finite free presentation of with , then we may obtain an exact sequence of free differential modules where is the parametrizing operator. However, there may exist other parametrizations called minimal parametrizations such that is a torsion module and we have thus .
REMARK 4.15: The following chains of inclusions and short exact sequences allow to compare the main procedures used in the respective study of differential extensions and differential modules:
where F is a maximum free submodule of M, is a torsion-module and is a torsion-free module. The next examples open the way towards a new domain of research.
EXAMPLE 4.16: With , let us consider the first order nonlinear involutive system:
This system defines a prime differential ideal and the differential extension is differentially algebraic over with parametric jets .
The linearized system over L is:
Multiplying by test functions and integrating by part, we get in the form:
Using only the parametric jets for y and in the PD equations provided, we get:
and the only CC over L:
Multiplying by a test function and integrating by part, we get over L in the form:
admitting the CC of course but also the additional zero order CC:
which provides a torsion element satisfying . Setting as the standard variational notation used by engineers, we obtain easily and cannot therefore admit an integrating factor, a result showing that K is its own differential algebraic closure in L.
EXAMPLE 4.17: If , the linear system obtained over by eliminating the factor in the linear system admits the injective parametrization , , . It defines therefore a free differential module which is thus reflexive and even projective. Any resolution of this module splits, like the short exact sequence , and the corresponding differential sequence of operators is locally exact like the Poincaré sequence (  , p. 684-691).
We start this section with a general (difficult) result on the actions of Lie groups, covering at the same time the study of the classical and conformal Killing systems. For this, we notice that the involutive first Spencer operator of order one is induced by the Spencer operator . Introducing the Spencer bundles , the first order involutive ( )-Spencer operator is induced by . We obtain therefore the canonical linear Spencer sequence (  , p. 150 or  ) (See     for other applications):
PROPOSITION 5.1: The Spencer sequence for the Lie operator describing the infinitesimal action of a Lie group G is (locally) isomorphic to the tensor product of the Poincaré sequence by the Lie algebra where is the identity element. It follows that generates the CC of generates the CC of , a result not evident at all.
Proof: We may introduce a basis of infinitesimal generators of the action with and the commutation relations discovered by S. Lie giving the structure constants c of (See  and  for more details). Any element can be written . “Gauging” such an element, that is to say replacing the constants by functions or, equivalently, introducing a map , we may obtain locally a map or, equivalently, vector fields of the form , keeping the index i for 1-forms. More generally, we can introduce a map:
that we can lift to the element . It follows from the definitions that by introducing any element of through its representative . We obtain therefore the crucial formula:
allowing to identify locally the Spencer sequence with a tensor product of the Poincaré sequence, because . When the action is effective, the map is injective. We obtain therefore an isomorphism when q is large enough allowing to exhibit an isomorphism between the canonical Spencer sequence and the tensor product of the Poincaré sequence by when q is large enough in such a way that is involutive with and .
We now study what happens when because the case has already been provided, proving that conformal geometry must be entirely revisited.
● : Using the euclidean metric , we have 6 components of with in the case of the classical Killing system/operator and obtain easily the components of the second order Riemann operator, linearization of the Riemann tensor at . We have first order Bianchi identities (  , p. 625). Introducing the respective adjoint operators while taking into account the last Proposition and the fact that the extension modules do not depend on the resolution used (a difficult result indeed !), we get the following diagram where we have set for historical reasons  and each operator generates the CC of the next one:
As in the Introduction where , the Beltrami operator is now parametrizing the 3 Cauchy stress equations  but it is rather striking to discover that the central second order operator is self-adjoint and can be given as follows:
The study of the conformal case is much more delicate. As can be described by trace-free symmetric tensors, we have and it remains to discover the operator that will replace the Riemann operator. Having in mind the diagram of Proposition 2.11 and the fact that while , we have successively:
● NO CC order 1: .
● NO CC order 2: .
● OK CC order 3: .
Once again, the central third order operator is self-adjoint as can be easily seen by proving that the last operator, obtained in  by means of computer algebra, can be chosen to be the transpose of the first conformal Killing operator, just by changing columns.
This result can also be obtained by using the fact that, when an operator/a system is formally integrable, the order of the generating CC is equal to the number of prolongations needed to get a 2-acyclic symbol plus 1 (  , p. 120,  ). In the present case, neither nor are 2-acyclic while is trivially involutive, so that .
● : In the classical case, we may proceed as before for exibiting the 20 components of the second order Riemann operator and the 20 components of the first order Bianchi operator.
The study of the conformal case is much more delicate and still unknown. Indeed, the symbol is 2-acyclic when and 3-acyclic when . Accordingly, the Weyl operator, namely the CC for the conformal Killing operator, is second order like the Riemann operator. However, when only (care), the symbol of the Weyl system is not 2-acyclic while its first prolongation becomes 2-acyclic. It follows that the CC for the Weyl operator are second order, ... and so on. For example, we have the long exact sequence:
and deduce that , a result that can be ckecked by computer algebra in a few milliseconds but is still unknown.
We shall finally prove below that the Einstein parametrization of the stress equations is neither canonical nor minimal in the following diagrams:
obtained by using the fact that the Einstein operator is self-adjoint, where by Einstein operator we mean the linearization of the Einstein equations at the Minkowski metric, the 6 terms being exchanged between themselves   .
Indeed, setting with , it is essential to notice that the Ricci operator is not self-adjoint because we have for example:
and provides a term appearing in but not in because we have, as in (5.1.4) of  :
The upper induced by Bianchi has nothing to do with the lower Cauchy stress equations, contrary to what is still believed today while the 10 on the right of the lower diagram has nothing to do with the perturbation of a metric which is the 10 on the left in the upper diagram. It also follows that the Einstein equations in vacuum cannot be parametrized as we have the following diagram of operators recapitulating the five steps of the parametrizability criterion (See   for more details or   for a computer algebra exhibition of this result):
We are facing only two possibilities, both leading to a contradiction:
1) If we use the operator in the geometrical setting, the on the left has indeed someting to do with the perturbation of the metric but the on the right has nothing to do with the stress.
2) If we use the adjoint operator in the physical setting, then on the left has of course something to do with the stress but the on the right has nothing to do with the perturbation of a metric.
These purely mathematical results question the origin and existence of gravitational waves.
We may summarize these results, which do not seem to be known, by the following differential sequences where the order of an operator is written under its arrow:
THEOREM 5.2: Recalling that we have and thus:
we have the following commutative and exact “fundamental diagram II”:
The following theorem will provide all the classical formulas of both Riemannian and conformal geometry in one piece but in a totally unusual framework not depending on any conformal factor:
THEOREM 5.3: All the short exact sequences of the preceding diagram split in a canonical way, that is in a way compatible with the underlying tensorial properties of the vector bundles involved.
Proof: First of all, we recall that:
Now, if , then we have:
and we may set with and such a formula does not depend on any conformal factor  . We have:
● The splitting of the lower row is obtained by setting in such a way that .
Similarly, and .
● The most important result is to split the right column. For this, we first need to describe the monomorphism which is in fact produced by a diagonal north-east snake type chase. Let us choose . Then, we may find by deciding that in and apply in order to get such that and thus . We obtain:
Contracting in k and i while setting simply , we get:
Substituting, we finally obtain and thus the tricky formula:
Contracting in k and i, we check that indeed, obtaining therefore the desired canonical lift . Finally, using Proposition 4.3, the epimorphism is just described by the formula:
which is just the way to define the Weyl tensor. We notice that and by using indices or a circular chase showing that . This purely algebraic result only depends on the metric and does not depend on any conformal factor. In actual practice, the lift is described by but it is not evident at all that the lift is described by the strict inclusion providing a short exact sequence as in Proposition 4.3 because by composition.
COROLLARY 5.4: When , each component of the Weyl tensor is a torsion element killed by the Dalembert operator whenever the Einstein equations in vacuum are satisfied by the metric. Hence, there exists a second order operator such that we have an identity:
Proof: According to Proposition 4.4, each extension module is a torsion module, . It follows that each additional CC in which is not already in is a torsion element as it belongs to this module. One may also notice that:
The differential ranks of the Einstein and Riemann operators are thus equal, but this is a pure coincidence because has only to do with the operator induced by contracting the Bianchi identities, while has only to do with the classical Killing operator and the fact that the corresponding differential module is a torsion module because we have
a Lie group of transformations having parameters
(translations + rotations). Hence, as the Riemann operator is a direct sum of the Weyl operator and the Einstein or Ricci operator according to the previous theorem, each component of the Weyl operator must be killed by a certain operator whenever the Einstein or Ricci equations in vacuum are satisfied. A direct tricky computation can be found in (  , p. 206) and (  , exercise 7.7).
REMARK 5.5: In a similar manner, the EM wave equations are easily obtained when the second set of Maxwell equations in vacuum is satisfied, avoiding therefore the Lorenz (no “t”) gauge condition for the EM potential  . Indeed, let us start with the Minkowski constitutive law with electric constant and magnetic constant such that in vacuum:
where , is the EM field and the induction is thus a contravariant skewsymmetric 2-tensor density. From the Maxwell equations we have:
REMARK 5.6: Using Proposition 4.3 and the splittings of Theorem 5.3 for the second column, we obtain the following commutative and exact diagram:
It follows that the 10 components of the Weyl tensor must satisfy a first order linear system with 16 equations, having 6 generating first order CC. The differential rank of the corresponding operator is thus equal to and such an operator defines a torsion module in which we have to look separately for each component of the Weyl tensor in order to obtain Corollary 5.4. The situation is similar to that of the Cauchy-Riemann equations when . Indeed, any complex transformation must be solution of the (linear) first order system of finite Lie equations though we obtain , that is and are separately killed by the second order Laplace operator .
Collecting the above results, we obtain the striking theorem:
THEOREM 5.7: The Cauchy operator can be parametrized by the operator (with only 4 terms) and there is thus no need to introduce the Einstein operator (with 6 terms) in GR.
Proof: Linearizing the Ricci tensor over the Minkowski metric, we obtain the Ricci operator :
The Einstein operator is defined by setting that we shall write where is a symmetric matrix only depending on , which is invertible whenever . We may also introduce the linear transformation and the unknown composite operator in such a way that where is defined by (See  , 5.1.5 p. 134):
Now, introducing the test functions , we get:
Integrating by parts, we obtain:
Moreover, suppressing the “bar” for simplicity, we have:
As Einstein is a self-adjoint operator (contrary to the Ricci operator), we have the identities:
because C is a symmetric matrix, we have and we know that . Accordingly, the operator parametrizes the Cauchy equations, without any reference to the Einstein operator which has no mathematical origin, in the sense that it cannot be obtained by any diagram chasing. The three terms after the Dalembert operator factorize through the divergence operator . We may thus add the differential constraints without any reference to a gauge transformation in order to obtain a (minimum) relative parametrization (see  and  for details and explicit examples). When we finally obtain the adjoint sequences:
without any reference to the Bianchi operator or the induced operator.
This last result even strengthens the doubts we already had about the origin and existence of gravitational waves.
Whenever is an involutive system of order q on E, we may define the Janet bundles for by the short exact sequences:
We may pick up a section of , lift it up to a section of that we may lift up to a section of and apply D in order to get a section of that we may project onto a section of in order to construct an operator generating the CC of in the canonical linear Janet sequence (  , p. 145):
If we have two involutive systems , the Janet sequence for projects onto the Janet sequence for and we may define inductively canonical epimorphisms for by comparing the previous sequences for and .
A similar procedure can also be obtained if we define the Spencer bundles for by the short exact sequences:
We may pick up a section of , lift it to a section of , lift it up to a section of and apply D in order to construct a section of that we may project to in order to construct an operator generating the CC of in the canonical linear Spencer sequence which is another completely different resolution of the set of (formal) solutions of :
However, if we have two systems as above, the Spencer sequence for is now contained into the Spencer sequence for and we may construct inductively canonical monomorphisms for by comparing the previous sequences for and .
When dealing with applications, we have set and considered systems of finite type Lie equations determined by Lie groups of transformations and generates the CC of while generates the CC of . We have obtained in particular when comparing the classical and conformal Killing systems, but these bundles have never been used in physics. Therefore, instead of the classical Killing system defined by and or the conformal Killing system defined by and , we may introduce the intermediate differential system defined by with and , for the Weyl group obtained by adding the only dilatation with infinitesimal generator to the Poincaré group. We have but the strict inclusions and we discover exactly the group scheme used through this paper, both with the need to shift by one step to the left the physical interpretation of the various differential sequences used. Indeed, as , the first Spencer operator is induced by the usual Spencer operator and thus projects by cokernel onto the induced operator . Composing with , it projects therefore onto as in EM and so on by using the fact that and d are both involutive or the composite epimorphisms . The main result we have obtained is thus to be able to increase the order and dimension of the underlying jet bundles and groups, proving therefore that any 1-form with value in the second order jets (elations) of the conformal Killing system (conformal group) can be decomposed uniquely into the direct sum where R is a section of the Ricci bundle and the EM field F is a section of as conjectured by H. Weyl in 1918    .
The mathematical structures of electromagnetism and gravitation only depend on the second order jets.