Nonlinear Conformal Electromagnetism

J.-F. Pommaret

CERMICS, Ecole des Ponts ParisTech, France.

DOI: 10.4236/jmp.2022.134031                

Abstract

In 1909 the brothers E. and F. Cosserat discovered a new nonlinear group theoretical approach to elasticity (EL), with the only experimental need to measure the EL constants. In a modern language, their idea has been to use the nonlinear Spencer sequence instead of the nonlinear Janet sequence for the Lie groupoid defining the group of rigid motions of space. Following H. Weyl, our purpose is to compute for the first time the nonlinear Spencer sequence for the Lie groupoid defining the conformal group of space-time in order to provide the mathematical foundations of electromagnetism (EM), with the only experimental need to measure the EM constant in vacuum. With a manifold of dimension n, the difficulty is to deal with the n nonlinear transformations that have been called “elations” by E. Cartan in 1922. Using the fact that dimension n=4 has very specific properties for the computation of the Spencer cohomology, we prove that there is thus no conceptual difference between the Cosserat EL field or induction equations and the Maxwell EM field or induction equations. As a byproduct, the well known field/matter couplings (piezzoelectricity, photoelasticity, streaming birefringence, …) can be described abstractly, with the only experimental need to measure the corresponding coupling constants. The main consequence of this paper is the need to revisit the mathematical foundations of gauge theory (GT) because we have proved that EM was depending on the conformal group and not on U(1), with a shift by one step to the left in the physical interpretation of the differential sequence involved.

Keywords

Nonlinear Differential Sequences, Linear Differential Sequences, Lie Groupoids, Lie Algebroids, Conformal Group, Spencer Cohomology, Maxwell Equations, Cosserat Equations

1. Introduction

Let us start this paper with a personal but meaningful story that has oriented my research during the last forty years or so, since the French “Grandes Ecoles” created their own research laboratories. Being a fresh permanent researcher of Ecole Nationale des Ponts et Chaussées in Paris, the author of this paper has been asked to become the scientific adviser of a young student in order to introduce him to research. As General Relativity was far too much difficult for somebody without any specific mathematical knowledge while remembering his own experience at the same age, he asked the student to collect about 50 books of Special Relativity and classify them along the way each writer was avoiding the use of the conformal group of space-time implied by the Michelson and Morley experiment, only caring about the Poincaré or Lorentz subgroups. After six months, the student (like any reader) arrived at the fact that most books were almost copying each other and could be nevertheless classified into three categories:

• 30 books, including the original 1905 paper ( [1] ) by Einstein, were at once, as a working assumption, deciding to restrict their study to a linear group reducing to the Galilée group when the speed of light was going to infinity. It must be noticed that people did believe that Einstein had not been influenced in 1905 by the Michelson and Morley experiment of 1887 till the discovery of hand written notes taken during lectures given by Einstein in Chicago (1921) and Kyoto (1922).

• 15 books were trying to “prove” that the conformal factor was indeed reduced to a constant equal to 1 when space-time was supposed to be homogeneous and isotropic.

• 5 books only were claiming that the conformal factor could eventually depend on the property of space-time, adding however that, if there was no surrounding electromagnetism or gravitation, the situation should be reduced to the preceding one but nothing was said otherwise.

The student was so disgusted by such a state of affair that he decided to give up on research and to become a normal civil engineer. As a byproduct, if group theory must be used, the underlying group of transformations of space-time must be related to the propagation of light by itself rather than by considering tricky signals between observers, thus must have to do with the biggest group of invariance of Maxwell Equations ( [2] [3] ). However, at the time we got the solution of this problem with the publication of ( [4] ) in 1988 (See [5] for recent results), a deep confusion was going on which is still not acknowledged though it can be explained in a few lines ( [6] ). Using standard notations of differential geometry, if the 2-form $F\in {\wedge }^2{T}^{\mathrm{<mo>\; *\; </mo>}}$ describing the EM field is satisfying the first set of Maxwell equations, it amounts to say that it is closed, that is killed by the exterior derivative $d\mathrm{<mo>\; :\; </mo>}{\wedge }^2{T}^{\mathrm{<mo>\; *\; </mo>}}\to {\wedge }^3{T}^{\mathrm{<mo>\; *\; </mo>}}$. The EM field can be thus (locally) parametrized by the EM potential 1-form $A\in T^{\mathrm{<mo>\; *\; </mo>}}$ with $dA=F$ where $d\mathrm{<mo>\; :\; </mo>}{T}^{\mathrm{<mo>\; *\; </mo>}}\to {\wedge }^2{T}^{\mathrm{<mo>\; *\; </mo>}}$ is again the exterior derivative, because $d^2=d\circ d=0$. Now, if E is a vector bundle over a manifold X of dimension n, then we may define its adjoint vector bundle $ad\left(E\right)={\wedge }^n{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes E^{\mathrm{<mo>\; *\; </mo>}}$ where $E^{\mathrm{<mo>\; *\; </mo>}}$ is obtained from E by inverting the transition rules, like $T^{\mathrm{<mo>\; *\; </mo>}}$ is obtained from $T=T\left(X\right)$ and such a construction can be extended to linear partial differential operators between (sections of) vector bundles. When $n=4$, it follows that the second set of Maxwell equations for the EM induction is just described by $ad\left(d\right)\mathrm{<mo>\; :\; </mo>}{\wedge }^4{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes {\wedge }^{2}T\to {\wedge }^4{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes T$, independently of any Minkowski constitutive relation between field and induction. Using Hodge duality with respect to the volume form $dx=d{x}^1\wedge \cdots \wedge d{x}^4$, this operator is isomorphic to $d\mathrm{<mo>\; :\; </mo>}{\wedge }^2{T}^{\mathrm{<mo>\; *\; </mo>}}\to {\wedge }^3{T}^{\mathrm{<mo>\; *\; </mo>}}$. It follows that both the first set and second set of Maxwell equations are invariant by any diffeomorphism and that the conformal group of space-time is the biggest group of transformations preserving the Minkowski constitutive relations in vacuum where the speed of light is truly c as a universal constant. It was thus natural to believe that the mathematical structure of electromagnetism and gravitation had only to do with such a group having:

$4translations+6rotations+1dilatation+4elations=15parameters$

the main difficulty being to deal with these later non-linear transformations. Of course, such a challenge could not be solved without the help of the non-linear theory of partial differential equations and Lie pseudogroups combined with homological algebra, that is before 1995 at least ( [7] ).

From a purely physical point of view, these new nonlinear methods have been introduced for the first time in 1909 by the brothers E. and F. Cosserat for studying the mathematical foundations of EL ( [8] - [14] ). We have presented their link with the nonlinear Spencer differential sequences existing in the formal theory of Lie pseudogroups at the end of our book “Differential Galois Theory” published in 1983 ( [15] ). Similarly, the conformal methods have been introduced by H. Weyl in 1918 for revisiting the mathematical foundations of EM ( [3] ). We have presented their link with the above approach through a unique differential sequence only depending on the structure of the conformal group in our book “Lie Pseudogroups and Mechanics” published in 1988 ( [4] ). However, the Cosserat brothers were only dealing with translations and rotations while Weyl was only dealing with dilatation and elations. Also, as an additional condition not fulfilled by the classical Einstein-Fokker-Nordström theory ( [16] ), if the conformal factor has to do with gravitation, it must be defined everywhere but at the central attractive mass as we already said.

From a purely mathematical point of view, the concept of a finite length differential sequence, now called Janet sequence, has been first described as a footnote by M. Janet in 1920 ( [17] ). Then, the work of D. C. Spencer in 1970 has been the first attempt to use the formal theory of systems of partial differential equations that he developed himself in order to study the formal theory of Lie pseudogroups ( [18] [19] [20] ). However, the nonlinear Spencer sequences for Lie pseudogroups, though never used in physics, largely supersede the “Cartan structure equations” introduced by E.Cartan in 1905 ( [21] [22] ) and are different from the “Vessiot structure equations” introduced by E. Vessiot in 1903 ( [23] ) or 1904 ( [24] ) for the same purpose but still not known today after more than a century because they have never been acknowledged by Cartan himself or even by his successors.

The purpose of the present difficult paper is to apply these new methods for studying the common nonlinear conformal origin of electromagnetism and gravitation, in a purely mathematical way, by constructing explicitly the corresponding nonlinear Spencer sequence. All the physical consequences will be presented in another paper.

2. Groupoids and Algebroids

Let us now turn to the clever way proposed by Vessiot in 1903 ( [23] ) and 1904 ( [24] ). Our purpose is only to sketch the main results that we have obtained in many books ( [4] [7] [13] [15], we do not know other references) and to illustrate them by a series of specific examples, asking the reader to imagine any link with what has been said. We break the study into 8 successive steps.

1) If $\mathcal{E}=X\times X$, we shall denote by ${\Pi }_q={\Pi }_q\left(X\mathrm{,}X\right)$ the open sub-fibered manifold of $J_q\left(X\times X\right)$ defined independently of the coordinate system by $det\left(y_i^k\right)\ne 0$ with source projection ${\alpha }_q\mathrm{<mo>\; :\; </mo>}{\Pi }_q\to X\mathrm{<mo>\; :\; </mo>}\left(x\mathrm{,}{y}_q\right)\to \left(x\right)$ and target projection ${\beta }_q\mathrm{<mo>\; :\; </mo>}{\Pi }_q\to X\mathrm{<mo>\; :\; </mo>}\left(x\mathrm{,}{y}_q\right)\to \left(y\right)$. 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. In order to construct another nonlinear sequence, we need a few basic definitions on Lie groupoids and Lie algebroids that will become substitutes for Lie groups and Lie algebras. The first idea is to use the chain rule for derivatives $j_q\left(g\circ f\right)=j_q\left(g\right)\circ j_q\left(f\right)$ whenever $f\mathrm{,}g\in aut\left(X\right)$ can be composed and to replace both $j_q\left(f\right)$ and $j_q\left(g\right)$ respectively by $f_q$ and $g_q$ in order to obtain the new section $g_q\circ f_q$. This kind of “composition” law can be written in a symbolic way by introducing another copy Z of X with local coordinates (z) as follows:

$\begin{array}{l}{\gamma }_q\mathrm{<mo>\; :\; </mo>}{\Pi }_q\left(Y\mathrm{,}Z\right){\times }_Y{\Pi }_q\left(X\mathrm{,}Y\right)\to {\Pi }_q\left(X\mathrm{,}Z\right)\mathrm{<mo>\; :\; </mo>}\\ \left(y\mathrm{,}z\mathrm{,}\frac{\partial z}{\partial y}\mathrm{,}\cdots \right)\mathrm{,}\left(x\mathrm{,}y\mathrm{,}\frac{\partial y}{\partial x}\mathrm{,}\cdots \right)\to \left(x\mathrm{,}z\mathrm{,}\frac{\partial z}{\partial y}\frac{\partial y}{\partial x}\mathrm{,}\cdots \right)\end{array}$

We may also define $j_q{\left(f\right)}^{-1}=j_q\left(f^{-1}\right)$ and obtain similarly an “inversion” law.

DEFINITION 2.1: A fibered submanifold ${\mathcal{R}}_q\subset {\Pi }_q$ is called a system of finite Lie equations or a Lie groupoid of order q if we have an induced source projection ${\alpha }_q\mathrm{<mo>\; :\; </mo>}{\mathcal{R}}_q\to X$, target projection ${\beta }_q\mathrm{<mo>\; :\; </mo>}{\mathcal{R}}_q\to X$, composition ${\gamma }_q\mathrm{<mo>\; :\; </mo>}{\mathcal{R}}_q{\times }_X{\mathcal{R}}_q\to {\mathcal{R}}_q$, inversion ${\iota }_q\mathrm{<mo>\; :\; </mo>}{\mathcal{R}}_q\to {\mathcal{R}}_q$ and identity $j_q\left(id\right)=i{d}_q\mathrm{<mo>\; :\; </mo>}X\to {\mathcal{R}}_q$. In the sequel we shall only consider transitive Lie groupoids such that the map $\left({\alpha }_q\mathrm{,}{\beta }_q\right)\mathrm{<mo>\; :\; </mo>}{\mathcal{R}}_q\to X\times X$ is an epimorphism and we shall denote by ${\mathcal{R}}_q^0=i{d}^{-1}\left({\mathcal{R}}_q\right)$ the isotropy Lie group bundle of ${\mathcal{R}}_q$. Also, one can prove that the new system ${\rho }_r\left({\mathcal{R}}_q\right)={\mathcal{R}}_{q+r}$ obtained by differentiating r times all the defining equations of ${\mathcal{R}}_q$ is a Lie groupoid of order $q+r$.

Let us start with a Lie pseudogroup $\Gamma \subset aut\left(X\right)$ defined by a system ${\mathcal{R}}_q\subset {\Pi }_q$ of order q. Roughly speaking, if $f\mathrm{,}g\in \Gamma \Rightarrow g\circ f\mathrm{,}{f}^{-1}\in \Gamma$ but such a definition is totally meaningless in actual practice as it cannot be checked most of the time. In all the sequel we shall suppose that the system is involutive ( [4] [7] [13] [15] [25] ) and that $\Gamma$ is transitive that is $\forall x,y\in X,\exists f\in \Gamma ,y=f\left(x\right)$ or, equivalently, the map $\left({\alpha }_q\mathrm{,}{\beta }_q\right)\mathrm{<mo>\; :\; </mo>}{\mathcal{R}}_q\to X\times X\mathrm{<mo>\; :\; </mo>}\left(x\mathrm{,}{y}_q\right)\to \left(x\mathrm{,}y\right)$ is surjective.

2) The Lie algebra $\Theta \subset T$ of infinitesimal transformations is then obtained by linearization, setting $y=x+t\xi \left(x\right)+\cdots$ and passing to the limit $t\to 0$ in order to obtain the linear involutive system $R_q=i{d}_q^{-1}\left(V\left({\mathcal{R}}_q\right)\right)\subset J_q\left(T\right)$ by reciprocal image with $\Theta =\left\{\xi \in T|j_q\left(\xi \right)\in R_q\right\}$. We define the isotropy Lie algebra bundle $R_q^0\subset J_q^0\left(T\right)$ by the short exact sequence $0\to R_q^0\to R_q\stackrel{{\pi }_0^q}{\to }T\to 0$.

3) Passing from source to target, we may prolong the vertical infinitesimal transformations $\eta ={\eta }^k\left(y\right)\frac{\partial }{\partial y^k}$ to the jet coordinates up to order q in order to obtain:

${\eta }^k\left(y\right)\frac{\partial }{\partial y^k}+\left(\frac{\partial {\eta }^k}{\partial y^r}{y}_i^r\right)\frac{\partial }{\partial y_i^k}+\left(\frac{{\partial }^2{\eta }^k}{\partial y^r\partial y^s}{y}_i^r{y}_j^s+\frac{\partial {\eta }^k}{\partial y^r}{y}_{ij}^r\right)\frac{\partial }{\partial y_{ij}^k}+\cdots$

where we have replaced $j_q\left(f\right)\left(x\right)$ by $y_q$, each component being the “formal” derivative of the previous one .

4) As $\left[\Theta \mathrm{,}\Theta \right]\subset \Theta$, we may use the Frobenius theorem in order to find a generating fundamental set of differential invariants $\left\{{\Phi }^{\tau }\left(y_q\right)\right\}$ up to order q which are such that ${\Phi }^{\tau }\left({\bar{y}}_q\right)={\Phi }^{\tau }\left(y_q\right)$ by using the chain rule for derivatives whenever $\bar{y}=g\left(y\right)\in \Gamma$ acting now on Y. Specializing the ${\Phi }^{\tau }$ at $i{d}_q\left(x\right)$ we obtain the Lie form ${\Phi }^{\tau }\left(y_q\right)={\omega }^{\tau }\left(x\right)$ of ${\mathcal{R}}_q$.

Of course, in actual practice one must use sections of $R_q$ instead of solutions and we now prove why the use of the Spencer operator becomes crucial for such a purpose. Indeed, using the algebraic bracket $\left\{j_{q+1}\left(\xi \right),j_{q+1}\left(\eta \right)\right\}=j_q\left(\left[\xi ,\eta \right]\right),\forall \xi ,\eta \in T$, we may obtain by bilinearity a differential bracket on $J_q\left(T\right)$ extending the bracket on T:

$\left[{\xi }_q\mathrm{,}{\eta }_q\right]=\left\{{\xi }_{q+1}\mathrm{,}{\eta }_{q+1}\right\}+i\left(\xi \right)D{\eta }_{q+1}-i\left(\eta \right)D{\xi }_{q+1}\mathrm{,}\forall {\xi }_q\mathrm{,}{\eta }_q\in J_q(\; T\; )$

which does not depend on the respective lifts ${\xi }_{q+1}$ and ${\eta }_{q+1}$ of ${\xi }_q$ and ${\eta }_q$ in $J_{q+1}\left(T\right)$. This bracket on sections satisfies the Jacobi identity and we set ( [4] [7] [13] [25] ):

DEFINITION 2.2: We say that a vector subbundle $R_q\subset J_q\left(T\right)$ is a system of infinitesimal Lie equations or a Lie algebroid if $\left[R_q\mathrm{,}{R}_q\right]\subset R_q$, that is to say ${\xi }_q\mathrm{,}{\eta }_q\in R_q\Rightarrow \left[{\xi }_q\mathrm{,}{\eta }_q\right]\in R_q$. Such a definition can be tested by means of computer algebra. We shall also say that $R_q$ is transitive if we have the short exact sequence $0\to R_q^0\to R_q\stackrel{{\pi }_0^q}{\to }T\to 0$. In that case, a splitting of this sequence, namely a map ${\chi }_q\mathrm{<mo>\; :\; </mo>}T\to R_q$ such that ${\pi }_0^q\circ {\chi }_q=i{d}_T$ or equivalently a section ${\chi }_q\in T^{\mathrm{<mo>\; *\; </mo>}}\otimes R_q$ over $i{d}_T\in T^{\mathrm{<mo>\; *\; </mo>}}\otimes T$, is called a $R_q$ -connection and its curvature ${\kappa }_q\in {\wedge }^2{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes R_q^0$ is defined by ${\kappa }_q\left(\xi ,\eta \right)=\left[{\chi }_q\left(\xi \right),{\chi }_q\left(\eta \right)\right]-{\chi }_q\left(\left[\xi ,\eta \right]\right),\forall \xi ,\eta \in T$.

PROPOSITION 2.3: If $\left[R_q\mathrm{,}{R}_q\right]\subset R_q$, then $\left[R_{q+r}\mathrm{,}{R}_{q+r}\right]\subset R_{q+r}\mathrm{,}\forall r\ge 0$.

Proof: When $r=1$, we have ${\rho }_1\left(R_q\right)=R_{q+1}=\left\{{\xi }_{q+1}\in J_{q+1}\left(T\right)|{\xi }_q\in R_q,D{\xi }_{q+1}\in T^{*}\otimes R_q\right\}$ and we just need to use the following formulas showing how D acts on the various brackets (See [7] and [25] for more details):

$i\left(\zeta \right)D\left\{{\xi }_{q+1}\mathrm{,}{\eta }_{q+1}\right\}=\left\{i\left(\zeta \right)D{\xi }_{q+1}\mathrm{,}{\eta }_q\right\}+\left\{{\xi }_q\mathrm{,}i\left(\zeta \right)D{\eta }_{q+1}\right\}\mathrm{,}\forall \zeta \in T$

$\begin{array}{c}i\left(\zeta \right)D\left[{\xi }_{q+1}\mathrm{,}{\eta }_{q+1}\right]=\left[i\left(\zeta \right)D{\xi }_{q+1}\mathrm{,}{\eta }_q\right]+\left[{\xi }_q\mathrm{,}i\left(\zeta \right)D{\eta }_{q+1}\right]\\ +i\left(L\left({\eta }_1\right)\zeta \right)D{\xi }_{q+1}-i\left(L\left({\xi }_1\right)\zeta \right)D{\eta }_{q+1}\end{array}$

because the right member of the second formula is a section of $R_q$ whenever ${\xi }_{q+1}\mathrm{,}{\eta }_{q+1}\in R_{q+1}$. The first formula may be used when $R_q$ is formally integrable.

口

EXAMPLE 2.4: With $n=1,q=3,X=ℝ$ and evident notations, the components of $\left[{\xi }_3\mathrm{,}{\eta }_3\right]$ at order zero, one, two and three are defined by the totally unusual successive formulas:

$\left[\xi \mathrm{,}\eta \right]=\xi {\partial }_x\eta -\eta {\partial }_x\xi$

${\left(\left[{\xi }_1\mathrm{,}{\eta }_1\right]\right)}_x=\xi {\partial }_x{\eta }_x-\eta {\partial }_x{\xi }_x$

${\left(\left[{\xi }_2\mathrm{,}{\eta }_2\right]\right)}_{xx}={\xi }_x{\eta }_{xx}-{\eta }_x{\xi }_{xx}+\xi {\partial }_x{\eta }_{xx}-\eta {\partial }_x{\xi }_{xx}$

${\left(\left[{\xi }_3,{\eta }_3\right]\right)}_{xxx}=2{\xi }_x{\eta }_{xxx}-2{\eta }_x{\xi }_{xxx}+\xi {\partial }_x{\eta }_{xxx}-\eta {\partial }_x{\xi }_{xxx}$

For affine transformations, ${\xi }_{xx}=0,{\eta }_{xx}=0\Rightarrow {\left(\left[{\xi }_2,{\eta }_2\right]\right)}_{xx}=0$ and thus $\left[R_2\mathrm{,}{R}_2\right]\subset R_2$.

For projective transformations, ${\xi }_{xxx}=0,{\eta }_{xxx}=0\Rightarrow {\left(\left[{\xi }_3,{\eta }_3\right]\right)}_{xxx}=0$ and thus $\left[R_3\mathrm{,}{R}_3\right]\subset R_3$.

THEOREM 2.5: (prolongation/projection (PP) procedure) If an arbitrary system $R_q\subseteq J_q\left(E\right)$ is given, one can effectively find two integers $r\mathrm{,}s\ge 0$ such that the system $R_{q+r}^{\left(s\right)}$ is formally integrable or even involutive.

COROLLARY 2.6: The bracket is compatible with the PP procedure:

$\left[R_q\mathrm{,}{R}_q\right]\subset R_q\Rightarrow \left[R_{q+r}^{\left(s\right)}\mathrm{,}{R}_{q+r}^{\left(s\right)}\right]\subset R_{q+r}^{\left(s\right)}\mathrm{,}\forall r\mathrm{,}s\ge 0$

EXAMPLE 2.7: With $n=m=2$ and $q=1$, let us consider the Lie pseudodogroup $\Gamma \subset aut\left(X\right)$ of finite transformations $y=f\left(x\right)$ such that $y^{2}d{y}^1=x^{2}d{x}^1=\omega =\left(x^2,0\right)\in T^{*}$. Setting $y=x+t\xi \left(x\right)+\cdots$ and linearizing, we get the Lie operator $\mathcal{D}\xi =\mathcal{L}\left(\xi \right)\omega$ where $\mathcal{L}$ is the Lie derivative because it is well known that $\left[\mathcal{L}\left(\xi \right),\mathcal{L}\left(\eta \right)\right]=\mathcal{L}\left(\xi \right)\circ \mathcal{L}\left(\eta \right)-\mathcal{L}\left(\eta \right)\circ \mathcal{L}\left(\xi \right)=\mathcal{L}\left(\left[\xi ,\eta \right]\right)$ in the operator sense. The system $R_1\subset J_1\left(T\right)$ of linear infinitesimal Lie equations is:

$x^2{\partial }_1{\xi }^1+{\xi }^2=0,{\partial }_2{\xi }^1=0$

Replacing $j_1\left(\xi \right)$ by a section ${\xi }_1\in J_1\left(T\right)$, we have:

${\xi }_1^1=-\frac{1}{x^2}{\xi }^2,{\xi }_2^1=0$

Let us choose the two sections:

${\xi }_1=\left\{{\xi }^1=0,{\xi }^2=-x^2,{\xi }_1^1=1,{\xi }_2^1=0,{\xi }_1^2=0,{\xi }_2^2=0\right\}\in R_1$

${\eta }_1=\left\{{\eta }^1=x^2,{\eta }^2=0,{\eta }_1^1=0,{\eta }_2^1=-x^2,{\eta }_1^2=0,{\eta }_2^2=1\right\}\in R_1$

We let the reader check that $\left[{\xi }_1\mathrm{,}{\eta }_1\right]\in R_1$. However, we have the strict inclusion $R_1^{\left(1\right)}\subset R_1$ defined by the additional equation ${\xi }_1^1+{\xi }_2^2=0$ and thus ${\xi }_1\mathrm{,}{\eta }_1\notin R_1^{\left(1\right)}$ though we have indeed $\left[R_1^{\left(1\right)}\mathrm{,}{R}_1^{\left(1\right)}\right]\subset R_1^{\left(1\right)}$, a result not evident at all because the sections ${\xi }_1$ and ${\eta }_1$ have nothing to do with solutions. The reader may proceed in the same way with $x^{2}d{x}^1-x^{1}d{x}^2$ and compare.

5) The main discovery of Vessiot, as early as in 1903 and thus fifty years in advance, has been to notice that the prolongation at order q of any horizontal vector field $\xi ={\xi }^i\left(x\right)\frac{\partial }{\partial x^i}$ commutes with the prolongation at order q of any vertical vector field $\eta ={\eta }^k\left(y\right)\frac{\partial }{\partial y^k}$, 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 $y=f\left(x\right)$ can be lifted to a (local) transformation of the differential invariants between themselves of the form $u\to \lambda \left(u\mathrm{,}{j}_q\left(f\right)\left(x\right)\right)$ allowing to introduce a natural bundle $\mathcal{F}$ over X by patching changes of coordinates $\bar{x}=\phi \left(x\right),\bar{u}=\lambda \left(u,j_q\left(\phi \right)\left(x\right)\right)$. A section $\omega$ of $\mathcal{F}$ is called a geometric object or structure on X and transforms like $\bar{\omega }\left(f\left(x\right)\right)=\lambda \left(\omega \left(x\right),j_q\left(f\right)\left(x\right)\right)$ or simply $\bar{\omega }=j_q\left(f\right)\left(\omega \right)$. This is a way to generalize vectors and tensors ( $q=1$ ) or even connections ( $q=2$ ). As a byproduct we have $\Gamma =\left\{f\in aut\left(X\right)|{\Phi }_{\omega }\left(j_q\left(f\right)\right)=j_q{\left(f\right)}^{-1}\left(\omega \right)=\omega \right\}$ as a new way to write out the Lie form and we may say that $\Gamma$ preserves $\omega$. We also obtain ${\mathcal{R}}_q=\left\{f_q\in {\Pi }_q|f_q^{-1}\left(\omega \right)=\omega \right\}$. Coming back to the infinitesimal point of view and setting $f_t=exp\left(t\xi \right)\in aut\left(X\right)\mathrm{,}\forall \xi \in T$, we may define the ordinary Lie derivative with value in $F={\omega }^{-1}\left(V\left(\mathcal{F}\right)\right)$ by introducing the vertical bundle of $\mathcal{F}$ as a vector bundle over $\mathcal{F}$ and the formula:

$\mathcal{D}\xi =\mathcal{L}\left(\xi \right)\omega ={\frac{d}{dt}{j}_q{\left(f_t\right)}^{-1}\left(\omega \right)|}_{t=0}\Rightarrow \Theta =\left\{\xi \in T|\mathcal{L}\left(\xi \right)\omega =0\right\}$

while we have $x\to x+t\xi \left(x\right)+\cdots \Rightarrow u^{\tau }\to u^{\tau }+t{\partial }_{\mu }{\xi }^k{L}_k^{\tau \mu }\left(u\right)+\cdots$ where $\mu =\left({\mu }_1,\cdots ,{\mu }_n\right)$ is a multi-index as a way to write down the system $R_q\subset J_q\left(T\right)$ of infinitesimal Lie equations in the Medolaghi form:

${\Omega }^{\tau }\equiv {\left(\mathcal{L}\left(\xi \right)\omega \right)}^{\tau }\equiv -L_k^{\tau \mu }\left(\omega \left(x\right)\right){\partial }_{\mu }{\xi }^k+{\xi }^r{\partial }_r{\omega }^{\tau }\left(x\right)=0$

EXAMPLE 2.8: With $n=1$, let us consider the Lie group of projective transformations $y=\left(ax+b\right)/\left(cx+d\right)$ as a lie pseudogroup. Differentiating three times in order to eliminate the parameters, we obtain the third order Schwarzian OD equation and its linearization over $y=x$:

${\mathcal{R}}_3\subset {\Pi }_3,\Psi \equiv \frac{y_{xxx}}{y_x}-\frac{3}{2}{\left(\frac{y_{xx}}{y_x}\right)}^2=0$

$R_3\subset J_3\left(T\right),{\xi }_{xxx}=0$

Accordingly, the prolongation $\#\left({\eta }_3\right)$ of any ${\eta }_3\in J_3\left(T\left(Y\right)\right)$ over Y such that ${\eta }_{yyy}=0$ becomes:

$\begin{array}{l}\eta \left(y\right)\frac{\partial }{\partial y}+{\eta }_y\left(y\right)\left(y_x\frac{\partial }{\partial y_x}+y_{xx}\frac{\partial }{\partial y_{xx}}+y_{xxx}\frac{\partial }{\partial y_{xxx}}\right)\\ +{\eta }_{yy}\left(y\right)\left({\left(y_x\right)}^2\frac{\partial }{\partial y_{xx}}+3y_x{y}_{xx}\frac{\partial }{\partial y_{xxx}}\right)\end{array}$

It follows that $\Psi$ is a generating third order differential invariant and $R_3$ is in Lie form.

Now, we have:

$\begin{array}{l}\bar{x}=\phi \left(x\right)\Rightarrow y_x=y_{\bar{x}}{\partial }_x\phi ,y_{xx}=y_{\bar{x}\bar{x}}{\left({\partial }_x\phi \right)}^2+y_{\bar{x}}{\partial }_{xx}\phi ,\\ y_{xxx}=y_{\bar{x}\bar{x}\bar{x}}{\left({\partial }_x\phi \right)}^3+3y_{\bar{x}\bar{x}}{\partial }_x\phi {\partial }_{xx}\phi +y_{\bar{x}}{\partial }_{xxx}\phi \end{array}$

and the natural bundle $\mathcal{F}$ is thus defined by the transition rules:

$\bar{x}=\phi \left(x\right),u=\bar{u}{\left({\partial }_x\phi \right)}^2+\left(\frac{{\partial }_{xxx}\phi }{{\partial }_x\phi }-\frac{3}{2}{\left(\frac{{\partial }_{xx}\phi }{{\partial }_x\phi }\right)}^2\right)$

The general Lie form of ${\mathcal{R}}_3$ is:

$\frac{y_{xxx}}{y_x}-\frac{3}{2}{\left(\frac{y_{xx}}{y_x}\right)}^2+\omega \left(y\right){\left(y_x\right)}^2=\omega (\; x\; )$

We obtain $R_3\subset J_3\left(T\right)$ in Medolaghi form as follows:

$\Omega \equiv \mathcal{L}\left(\xi \right)\omega \equiv {\partial }_{xxx}\xi +2\omega \left(x\right){\partial }_x\xi +\xi {\partial }_x\omega \left(x\right)=0$

Using a section ${\xi }_3\in J_3\left(T\right)$, we finally get the formal Lie derivative:

$\Omega \equiv L\left({\xi }_3\right)\omega \equiv {\xi }_{xxx}+2\omega \left(x\right){\xi }_x+\xi {\partial }_x\omega \left(x\right)=0$

and let the reader ckeck directly that $\left[L\left({\xi }_3\right),L\left({\eta }_3\right)\right]=L\left(\left[{\xi }_3,{\eta }_3\right]\right)$, $\forall {\xi }_3,{\eta }_3\in J_3\left(T\right)$, a result absolutely not evident at first sight.

6) 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 $j_1\left(\omega \right)$, exactly like L.P. Eisenhart did for $\mathcal{F}=S_2{T}^{*}$ when finding the constant Riemann curvature condition for a metric $\omega$ with $det\left(\omega \right)\ne 0$ ( [7], Example 10, p 246 to 256, [26] ). Indeed, if any expression involving $\omega$ and its derivatives is a scalar object, it must reduce to a constant because $\Gamma$ is assumed to be transitive and thus cannot be defined by any zero order equation. Now one can prove that the CC for $\bar{\omega }$, thus for $\omega$ too, only depend on the $\Phi$ and take the quasi-linear symbolic form $v\equiv I\left(u_1\right)\equiv A\left(u\right)u_x+B\left(u\right)=0$, allowing to define an affine subfibered manifold ${\mathcal{B}}_1\subset J_1\left(\mathcal{F}\right)$ over $\mathcal{F}$. Now, if one has two sections $\omega$ and $\bar{\omega }$ of $\mathcal{F}$, the equivalence problem is to look for $f\in aut\left(X\right)$ such that $j_q{\left(f\right)}^{-1}\left(\omega \right)=\bar{\omega }$. 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 ( [25], p. 333).

7) Instead of the CC for the equivalence problem, let us look for the integrability conditions (IC) for 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 ( [23] with no proof, see [7], pp. 207-211) that such a property is held if and only if there is an equivariant section $c\mathrm{<mo>\; :\; </mo>}\mathcal{F}\to {\mathcal{F}}_1\mathrm{<mo>\; :\; </mo>}\left(x\mathrm{,}u\right)\to \left(x\mathrm{,}u\mathrm{,}v=c\left(u\right)\right)$ where ${\mathcal{F}}_1=J_1\left(\mathcal{F}\right)/{\mathcal{B}}_1$ is a natural vector bundle over $\mathcal{F}$ with local coordinates $\left(x\mathrm{,}u\mathrm{,}v\right)$. Moreover, any such equivariant section depends on a finite number of constants c called structure constants and the IC for the Vessiot structure equations $I\left(u_1\right)=c\left(u\right)$ are of a polynomial form $J\left(c\right)=0$.

EXAMPLE 2.9: Comig back to Example 2.7 first considered by Vessiot as early as in 1903 ( [23] ), the geometric object $\omega =\left(\alpha \mathrm{,}\beta \right)\in T^{\mathrm{<mo>\; *\; </mo>}}{\otimes }_X{\wedge }^2{T}^{\mathrm{<mo>\; *\; </mo>}}$ must satisfy the Vessiot structure equation $d\alpha =c\beta$ with a single Vessiot structure constant $c=-1$ in the situation considered where $\alpha =x^{2}d{x}^1$ and $\beta =d{x}^1\wedge d{x}^2$ (See ( [27] ) for other examples and applications). As a byproduct, there is no conceptual difference between such a constant and the constant appearing in the constant Riemannian curvature condition of Eisenhart ( [26] ).

8) Finally, when Y is no longer a copy of X, a system ${\mathcal{A}}_q\subset J_q\left(X\times Y\right)$ is said to be an automorphic system for a Lie pseudogroup $\Gamma \subset aut\left(Y\right)$ if, whenever $y=f\left(x\right)$ and $\bar{y}=\bar{f}\left(x\right)$ are two solutions, then there exists one and only one transformation $\bar{y}=g\left(y\right)\in \Gamma$ such that $\bar{f}=g\circ f$. Explicit tests for checking such a property formally have been given in ( [15] ) and can be implemented on computer in the differential algebraic framework.

3. Nonlinear Sequences

Contrary to what happens in the study of Lie pseudogroups and in particular in the study of the algebraic ones that can be found in mathematical physics, nonlinear operators do not in general admit CC, unless they are defined by differential polynomials, as can be seen by considering the two following examples with $m=1,n=2,q=2$. With standard notations from differential algebra, if we are dealing with a ground differential field K, like $\mathbb{Q}$ in the next examples, we denote by $K\left\{y\right\}$ the ring (which is even an integral domain) of differential polynomials in y with coefficients in K and by $K\langle y\rangle =Q\left(K\left\{y\right\}\right)$ the corresponding quotient field of differential rational functions in y. Then, if $u\mathrm{,}v\in K\langle y\rangle$, we have the two towers $K\subset K\langle u\rangle \subset K\langle y\rangle$ and $K\subset K\langle v\rangle \subset K\langle y\rangle$ of extensions, thus the tower $K\subset K\langle u\mathrm{,}v\rangle \subset K\langle y\rangle$. Accordingly, the differential extension $K\langle u\mathrm{,}v\rangle /K$ is a finitely generated differential extension. If we consider u and v as new indeterminates, then $K\langle u\rangle$ and $K\langle v\rangle$ are both differential transcendental extensions of K and the kernel of the canonical differential morphism $K\left\{u\right\}{\otimes }_{K}K\left\{v\right\}\to K\langle y\rangle$ is a prime differential ideal in the differential integral domain $K\langle y\rangle {\otimes }_{K}K\langle v\rangle$, a way to describe by residue the smallest differential field containing $K\langle u\rangle$ and $K\langle v\rangle$ in $K\langle y\rangle$. Of course, the true difficulty is to find out such a prime differential ideal.

EXAMPLE 3.1: First of all, let us consider the following nonlinear system in y with second member $\left(u\mathrm{,}v\right)$:

$P\equiv y_{22}-\frac{1}{3}{\left(y_{11}\right)}^3=u,Q\equiv y_{12}-\frac{1}{2}{\left(y_{11}\right)}^2=v\Rightarrow y_{11}=\frac{u_1-v_2}{v_1}$

The differential ideal $\mathfrak{a}$ generated by P and Q in $\mathbb{Q}\left\{y\right\}$ is prime because $d_{2}Q+d_{1}P-y_{11}{d}_{1}Q=0$ and thus $\mathbb{Q}\left\{y\right\}/\left\{P\mathrm{,}Q\right\}\simeq \mathbb{Q}\left[y\mathrm{,}{y}_1\mathrm{,}{y}_2\mathrm{,}{y}_{11}\mathrm{,}{y}_{111}\mathrm{,}\cdots \right]$ is an integral domain.

We may consider the following nonlinear involutive system with two equations:

$\{\begin{array}{l}{y}_{22}-\frac{1}{3}{\left(y_{11}\right)}^3=0\hfill \\ y_{12}-\frac{1}{2}{\left(y_{11}\right)}^2=0\hfill \end{array}\boxed{\begin{array}{cc}1& 2\\ & \\ 1& •\end{array}}$

We have also the linear inhomogeneous finite type second order system with three equations:

$\{\begin{array}{l}{y}_{22}=u+\frac{1}{3}{\left(\frac{u_1-v_2}{v_1}\right)}^3\hfill \\ y_{12}=v+\frac{1}{2}{\left(\frac{u_1-v_2}{v_1}\right)}^2\hfill \\ y_{11}=\frac{u_1-v_2}{v_1}\hfill \end{array}\boxed{\begin{array}{cc}1& 2\\ & \\ 1& •\\ & \\ 1& •\end{array}}$

Though we have a priori two CC, we let the reader prove, as a delicate exercise, that there is only the single nonlinear second order CC obtained from the bottom dot:

$\boxed{d_2\left(\frac{u_1-v_2}{v_1}\right)-d_1\left(v+\frac{1}{2}{\left(\frac{u_1-v_2}{v_1}\right)}^2\right)=0}$

EXAMPLE 3.2: On the contrary, if we consider the following new nonlinear system:

$P\equiv y_{22}-\frac{1}{2}{\left(y_{11}\right)}^2=u,Q\equiv y_{12}-y_{11}=v\Rightarrow \left(y_{11}-1\right)y_{111}=v_2+v_1-u_1=w$

we obtain successively:

$d_{2}Q+d_{1}Q-d_{1}P\equiv \left(y_{11}-1\right)y_{111}$

$y_{111}\left(d_{12}Q+d_{11}Q-d_{11}P\right)-y_{1111}\left(d_{2}Q+d_{1}Q-d_{1}P\right)={\left(y_{111}\right)}^3$

The symbol at order 3 is thus not a vector bundle and no direct study as above can be used because the differential ideal generated by $\left(P\mathrm{,}Q\right)$ is not perfect as it contains ${\left(y_{111}\right)}^3$ without containing $y_{111}$ (See [15] and [28] for more details). The following nonlinear system is not involutive:

$\{\begin{array}{l}{y}_{22}-\frac{1}{2}{\left(y_{11}\right)}^2=0\hfill \\ y_{12}-y_{11}=0\hfill \end{array}\boxed{\begin{array}{cc}1& 2\\ & \\ 1& •\end{array}}$

We have the following four generic nonlinear additional finite type third order equations:

$\{\begin{array}{l}{y}_{222}-y_{11}\left(v_1+\frac{w}{y_{11}-1}\right)=u_2\hfill \\ y_{122}-y_{11}\frac{w}{y_{11}-1}=u_1\hfill \\ y_{112}-\frac{w}{y_{11}-1}=v_1\hfill \\ y_{111}-\frac{w}{y_{11}-1}=0\hfill \end{array}\boxed{\begin{array}{cc}1& 2\\ & \\ 1& •\\ & \\ 1& •\\ & \\ 1& •\end{array}}$

Though we have now a priori three CC and thus three additional equations because the system is not involutive, setting $y_{11}-1=z\Rightarrow y_{112}=z_2,y_{111}=z_1$, there is only the single additional nonlinear second order equation:

$\boxed{v_{11}{z}^2+\left(w_1-w_2\right)z+v_{1}w=0}$

Differentiating once and using the relation $z{z}_1=w$, we get:

$\boxed{v_{111}{z}^3+\left(w_{11}-w_{12}\right)z^2+\left(v_1{w}_1+3v_{11}w\right)z+\left(w_1-w_2\right)w=0}$

a result leading to a tricky resultant providing a third order differential polynomial in $\left(u\mathrm{,}v\right)$.

However, the kernel of a linear operator $\mathcal{D}\mathrm{<mo>\; :\; </mo>}E\to F$ is always taken with respect to the zero section of F, while it must be taken with respect to a prescribed section by a double arrow for a nonlinear operator. Keeping in mind the linear Janet sequence and the examples of Vessiot structure equations already presented, one obtains:

THEOREM 3.3: There exists a nonlinear Janet sequence associated with the Lie form of an involutive system of finite Lie equations:

$\begin{array}{ccccc}& \Phi \circ j_q& & I\circ j_1& \\ 0\to \Gamma \to aut\left(X\right)& \rightrightarrows & \mathcal{F}& \rightrightarrows & {\mathcal{F}}_1\\ & \omega \circ \alpha & & 0& \end{array}$

where the kernel of the first operator $f\to \Phi \circ j_q\left(f\right)=\Phi \left(j_q\left(f\right)\right)=j_q{\left(f\right)}^{-1}\left(\omega \right)$ is taken with respect to the section $\omega$ of $\mathcal{F}$ while the kernel of the second operator $\omega \to I\left(j_1\left(\omega \right)\right)\equiv A\left(\omega \right){\partial }_x\omega +B\left(\omega \right)$ is taken with respect to the zero section of the vector bundle ${\mathcal{F}}_1$ over $\mathcal{F}$.

COROLLARY 3.4: By linearization at the identity, one obtains the involutive Lie operator $\mathcal{D}\mathrm{<mo>\; :\; </mo>}T\to F_0\mathrm{<mo>\; :\; </mo>}\xi \to \mathcal{L}\left(\xi \right)\omega$ with kernel $\Theta =\left\{\xi \in T|\mathcal{L}\left(\xi \right)\omega =0\right\}\subset T$ satisfying $\left[\Theta \mathrm{,}\Theta \right]\subset \Theta$ and the corresponding linear Janet sequence:

$0\to \Theta \to T\stackrel{\mathcal{D}}{\to }{F}_0\stackrel{{\mathcal{D}}_1}{\to }{F}_1$

where $F_0=F={\omega }^{-1}\left(V\left(\mathcal{F}\right)\right)$ and $F_1={\omega }^{-1}\left({\mathcal{F}}_1\right)$.

Now we notice that T is a natural vector bundle of order 1 and $J_q\left(T\right)$ is thus a natural vector bundle of order $q+1$. Looking at the way a vector field and its derivatives are transformed under any $f\in aut\left(X\right)$ while replacing $j_q\left(f\right)$ by $f_q$, we obtain:

${\eta }^k\left(f\left(x\right)\right)=f_r^k\left(x\right){\xi }^r\left(x\right)\Rightarrow {\eta }_u^k\left(f\left(x\right)\right)f_i^u\left(x\right)=f_r^k\left(x\right){\xi }_i^r\left(x\right)+f_{ri}^k\left(x\right){\xi }^r(\; x\; )$

and so on, a result leading to:

LEMMA 3.5: $J_q\left(T\right)$ is associated with ${\Pi }_{q+1}={\Pi }_{q+1}\left(X\mathrm{,}X\right)$ that is we can obtain a new section ${\eta }_q=f_{q+1}\left({\xi }_q\right)$ from any section ${\xi }_q\in J_q\left(T\right)$ and any section $f_{q+1}\in {\Pi }_{q+1}$ by the formula:

$d_{\mu }{\eta }^k\equiv {\eta }_r^k{f}_{\mu }^r+\cdots \mathrm{<mo>\; =\; </mo>}{f}_r^k{\xi }_{\mu }^r+\cdots +f_{\mu +1_r}^k{\xi }^r\mathrm{,}\forall 0\le \left|\mu \right|\le q$

where the left member belongs to $V\left({\Pi }_q\right)$. Similarly $R_q\subset J_q\left(T\right)$ is associated with ${\mathcal{R}}_{q+1}\subset {\Pi }_{q+1}$.

More generally, looking now for transformations “close” to the identity, that is setting $y=x+t\xi \left(x\right)+\cdots$ when $t\ll 1$ is a small constant parameter and passing to the limit $t\to 0$, we may linearize any (nonlinear) system of finite Lie equations in order to obtain a (linear) system of infinitesimal Lie equations $R_q\subset J_q\left(T\right)$ for vector fields. Such a system has the property that, if $\xi \mathrm{,}\eta$ are two solutions, then $\left[\xi \mathrm{,}\eta \right]$ is also a solution. Accordingly, the set $\Theta \subset T$ of its solutions satisfies $\left[\Theta \mathrm{,}\Theta \right]\subset \Theta$ and can therefore be considered as the Lie algebra of $\Gamma$.

More generally, the next definition will extend the classical Lie derivative:

$\mathcal{L}\left(\xi \right)\omega =\left(i\left(\xi \right)d+di\left(\xi \right)\right)\omega ={\frac{d}{dt}{j}_q{\left(expt\xi \right)}^{-1}\left(\omega \right)|}_{t=0}.$

DEFINITION 3.6: We say that a vector bundle F is associated with $R_q$ if there exists a first order differential operator $L\left({\xi }_q\right)\mathrm{<mo>\; :\; </mo>}F\to F$ called formal Lie derivative and such that:

1) $L\left({\xi }_q+{\eta }_q\right)=L\left({\xi }_q\right)+L\left({\eta }_q\right)\forall {\xi }_q,{\eta }_q\in R_q$.

2) $L\left(f{\xi }_q\right)=fL\left({\xi }_q\right)\forall {\xi }_q\in R_q,\forall f\in C^{\infty }\left(X\right)$.

3) $\left[L\left({\xi }_q\right),L\left({\eta }_q\right)\right]=L\left({\xi }_q\right)\circ L\left({\eta }_q\right)-L\left({\eta }_q\right)\circ L\left({\xi }_q\right)=L\left(\left[{\xi }_q,{\eta }_q\right]\right)\forall {\xi }_q,{\eta }_q\in R_q$.

4) $L\left({\xi }_q\right)\left(f\eta \right)=fL\left({\xi }_q\right)\eta +\left(\xi f\right)\eta \forall {\xi }_q\in R_q,\forall f\in C^{\infty }\left(X\right),\forall \eta \in F$.

LEMMA 3.7: If E and F are associated with $R_q$, we may set on $E\otimes F$:

$L\left({\xi }_q\right)\left(\eta \otimes \zeta \right)=L\left({\xi }_q\right)\eta \otimes \zeta +\eta \otimes L\left({\xi }_q\right)\zeta \forall {\xi }_q\in R_q\mathrm{,}\forall \eta \in E\mathrm{,}\forall \zeta \in F$

If $\Theta \subset T$ denotes the solutions of $R_q$, then we may set $\mathcal{L}\left(\xi \right)=L\left(j_q\left(\xi \right)\right)\mathrm{,}\forall \xi \in \Theta$ but no explicit computation can be done when $\Theta$ is infinite dimensional. However, we have:

PROPOSITION 3.8: $J_q\left(T\right)$ is associated with $J_{q+1}\left(T\right)$ if we define:

$L\left({\xi }_{q+1}\right){\eta }_q=\left\{{\xi }_{q+1},{\eta }_{q+1}\right\}+i\left(\xi \right)D{\eta }_{q+1}=\left[{\xi }_q,{\eta }_q\right]+i\left(\eta \right)D{\xi }_{q+1}$

and thus $R_q$ is associated with $R_{q+1}$.

Proof: It is easy to check the properties 1, 2, 4 and it only remains to prove property 3 as follows.

$\begin{array}{l}\left[L\left({\xi }_{q+1}\right),L\left({\eta }_{q+1}\right)\right]{\zeta }_q\\ =L\left({\xi }_{q+1}\right)\left(\left\{{\eta }_{q+1},{\zeta }_{q+1}\right\}+i\left(\eta \right)D{\zeta }_{q+1}\right)-L\left({\eta }_{q+1}\right)\left(\left\{{\xi }_{q+1},{\zeta }_{q+1}\right\}+i\left(\xi \right)D{\zeta }_{q+1}\right)\\ =\left\{{\xi }_{q+1},\left\{{\eta }_{q+2},{\zeta }_{q+2}\right\}\right\}-\left\{{\eta }_{q+1},\left\{{\xi }_{q+2},{\zeta }_{q+2}\right\}\right\}+\left\{{\xi }_{q+1},i\left(\eta \right)D{\zeta }_{q+2}\right\}\\ -\left\{{\eta }_{q+1},i\left(\xi \right)D{\zeta }_{q+2}\right\}+i\left(\xi \right)D\left\{{\eta }_{q+2},{\zeta }_{q+2}\right\}-i\left(\eta \right)D\left\{{\xi }_{q+2},{\zeta }_{q+2}\right\}\\ +i\left(\xi \right)D\left(i\left(\eta \right)D{\zeta }_{q+2}\right)-i\left(\eta \right)D\left(i\left(\xi \right)D{\zeta }_{q+2}\right)\end{array}$

$\begin{array}{l}=\left\{\left\{{\xi }_{q+2},{\eta }_{q+2}\right\},{\zeta }_{q+1}\right\}+\left\{i\left(\xi \right)D{\eta }_{q+2},{\zeta }_{q+1}\right\}-\left\{i\left(\eta \right)D{\xi }_{q+2},{\zeta }_{q+1}\right\}+i\left(\left[\xi ,\eta \right]\right)D{\zeta }_{q+1}\\ =\left\{\left[{\xi }_{q+1},{\eta }_{q+1}\right],{\zeta }_{q+1}\right\}+i\left(\left[\xi ,\eta \right]\right)D{\zeta }_{q+1}\end{array}$

by using successively the Jacobi identity for the algebraic bracket and the last proposition.

口

EXAMPLE 3.9: T and $T^{\mathrm{<mo>\; *\; </mo>}}$ both with any tensor bundle are associated with $J_1\left(T\right)$. For T we may define $L\left({\xi }_1\right)\eta =\left[\xi ,\eta \right]+i\left(\eta \right)D{\xi }_1=\left\{{\xi }_1,j_1\left(\eta \right)\right\}$. We have ${\xi }^r{\partial }_r{\eta }^k-{\eta }^s{\partial }_s{\xi }^k+{\eta }^s\left({\partial }_s{\xi }^k-{\xi }_s^k\right)=-{\eta }^s{\xi }_s^k+{\xi }^r{\partial }_r{\eta }^k$ and the four properties of the formal Lie derivative can be checked directly. Of course, we find back $\mathcal{L}\left(\xi \right)\eta =\left[\xi ,\eta \right],\forall \xi ,\eta \in T$. We let the reader treat similarly the case of $T^{\mathrm{<mo>\; *\; </mo>}}$.

PROPOSITION 3.10: There is a first nonlinear Spencer sequence:

$0\to aut\left(X\right)\stackrel{j_{q+1}}{\to }{\Pi }_{q+1}\left(X\mathrm{,}X\right)\stackrel{\bar{D}}{\to }{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes J_q\left(T\right)\stackrel{{\bar{D}}^{\prime }}{\to }{\wedge }^2{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes J_{q-1}(\; T\; )$

with $\bar{D}{f}_{q+1}\equiv f_{q+1}^{-1}\circ j_1\left(f_q\right)-i{d}_{q+1}={\chi }_q$ $\Rightarrow {\bar{D}}^{\prime }{\chi }_q\left(\xi ,\eta \right)\equiv D{\chi }_q\left(\xi ,\eta \right)-\left\{{\chi }_q\left(\xi \right),{\chi }_q\left(\eta \right)\right\}=0$. Moreover, setting ${\chi }_0=A-id\in T^{*}\otimes T$, this sequence is locally exact if $det\left(A\right)\ne 0$.

Proof: There is a canonical inclusion ${\Pi }_{q+1}\subset J_1\left({\Pi }_q\right)$ defined by $y_{\mu \mathrm{,}i}^k=y_{\mu +1_i}^k$ and the composition $f_{q+1}^{-1}\circ j_1\left(f_q\right)$ is a well defined section of $J_1\left({\Pi }_q\right)$ over the section $f_q^{-1}\circ f_q=i{d}_q$ of ${\Pi }_q$ like $i{d}_{q+1}$. The difference ${\chi }_q=f_{q+1}^{-1}\circ j_1\left(f_q\right)-i{d}_{q+1}$ is thus a section of $T^{\mathrm{<mo>\; *\; </mo>}}\otimes V\left({\Pi }_q\right)$ over $i{d}_q$ and we have already noticed that $i{d}_q^{-1}\left(V\left({\Pi }_q\right)\right)=J_q\left(T\right)$. For $q=1$ we get with $g_1=f_1^{-1}$:

${\chi }_{,i}^k=g_l^k{\partial }_i{f}^l-{\delta }_i^k=A_i^k-{\delta }_i^k,{\chi }_{j,i}^k=g_l^k\left({\partial }_i{f}_j^l-A_i^r{f}_{rj}^l\right)$

We also obtain from Lemma 3.5 the useful formula $f_r^k{\chi }_{\mu ,i}^r+\cdots +f_{\mu +1_r}^k{\chi }_{,i}^r={\partial }_i{f}_{\mu }^k-f_{\mu +1_i}^k$ allowing to determine ${\chi }_q$ inductively.

We refer to ( [7], p 215-216) for the inductive proof of the local exactness, providing the only formulas that will be used later on and can be checked directly by the reader:

${\partial }_i{\chi }_{,j}^k-{\partial }_j{\chi }_{,i}^k-{\chi }_{i,j}^k+{\chi }_{j,i}^k-\left({\chi }_{,i}^r{\chi }_{r,j}^k-{\chi }_{,j}^r{\chi }_{r,i}^k\right)=0$ (1)

${\partial }_i{\chi }_{l,j}^k-{\partial }_j{\chi }_{l,i}^k-{\chi }_{li,j}^k+{\chi }_{lj,i}^k-\left({\chi }_{,i}^r{\chi }_{lr,j}^k+{\chi }_{l,i}^r{\chi }_{r,j}^k-{\chi }_{l,j}^r{\chi }_{r,i}^k-{\chi }_{,j}^r{\chi }_{lr,i}^k\right)=0$ (2)

$\begin{array}{l}{\partial }_i{\chi }_{lr\mathrm{,}j}^k-{\partial }_j{\chi }_{lr\mathrm{,}i}^k-{\chi }_{lri\mathrm{,}j}^k+{\chi }_{lrj\mathrm{,}i}^k-({\chi }_{\mathrm{,}i}^s{\chi }_{lrs\mathrm{,}j}^k+{\chi }_{r\mathrm{,}i}^s{\chi }_{ls\mathrm{,}j}^k+{\chi }_{l\mathrm{,}i}^s{\chi }_{rs\mathrm{,}j}^k\\ +{\chi }_{lr\mathrm{,}i}^s{\chi }_{s\mathrm{,}j}^k-{\chi }_{\mathrm{,}j}^s{\chi }_{lrs\mathrm{,}i}^k-{\chi }_{r\mathrm{,}j}^s{\chi }_{ls\mathrm{,}i}^k-{\chi }_{l\mathrm{,}j}^s{\chi }_{rs\mathrm{,}i}^k-{\chi }_{lr\mathrm{,}j}^s{\chi }_{s\mathrm{,}i}^k)=0\end{array}$ (3)

There is no need for double-arrows in this framework as the kernels are taken with respect to the zero section of the vector bundles involved. We finally notice that the main difference with the gauge sequence is that all the indices range from 1 to n and that the condition $det\left(A\right)\ne 0$ amounts to $\Delta =det\left({\partial }_i{f}^k\right)\ne 0$ because $det\left(f_i^k\right)\ne 0$ by assumption.

口

COROLLARY 3.11: There is a restricted first nonlinear Spencer sequence:

$0\to \Gamma \stackrel{j_{q+1}}{\to }{\mathcal{R}}_{q+1}\stackrel{\bar{D}}{\to }{T}^{*}\otimes R_q\stackrel{{\bar{D}}^{\prime }}{\to }{\wedge }^2{T}^{*}\otimes J_{q-1}(\; T\; )$

DEFINITION 3.12: A splitting of the short exact sequence $0\to R_q^0\to R_q\stackrel{{\pi }_0^q}{\to }T\to 0$ is a map ${{\chi }^{\prime }}_q\mathrm{<mo>\; :\; </mo>}T\to R_q$ such that ${\pi }_0^q\circ {{\chi }^{\prime }}_q=i{d}_T$ or equivalently a section of $T^{\mathrm{<mo>\; *\; </mo>}}\otimes R_q$ over $i{d}_T\in T^{\mathrm{<mo>\; *\; </mo>}}\otimes T$ and is called a $R_q$ -connection. Its curvature ${{\kappa }^{\prime }}_q\in {\wedge }^2{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes R_q^0$ is defined by ${{\kappa }^{\prime }}_q\left(\xi ,\eta \right)=\left[{{\chi }^{\prime }}_q\left(\xi \right),{{\chi }^{\prime }}_q\left(\eta \right)\right]-{{\chi }^{\prime }}_q\left(\left[\xi ,\eta \right]\right)$. We notice that ${{\chi }^{\prime }}_q=-{\chi }_q$ is a connection with ${\bar{D}}^{\prime }{{\chi }^{\prime }}_q={{\kappa }^{\prime }}_q$ if and only if $A=0$. In particular $\left({\delta }_i^k\mathrm{,}-{\gamma }_{ij}^k\right)$ is the only existing symmetric connection for the Killing system.

REMARK 3.13: Rewriting the previous local formulas with A instead of ${\chi }_0$ we get:

${\partial }_i{A}_j^k-{\partial }_j{A}_i^k-A_i^r{\chi }_{r\mathrm{,}j}^k+A_j^r{\chi }_{r\mathrm{,}i}^k=0$ (1*)

${\partial }_i{\chi }_{l\mathrm{,}j}^k-{\partial }_j{\chi }_{l\mathrm{,}i}^k-A_i^r{\chi }_{lr\mathrm{,}j}^k+A_j^r{\chi }_{lr\mathrm{,}i}^k-{\chi }_{l\mathrm{,}i}^r{\chi }_{r\mathrm{,}j}^k+{\chi }_{l\mathrm{,}j}^r{\chi }_{r\mathrm{,}i}^k=0$ (2*)

$\begin{array}{l}{\partial }_i{\chi }_{lr\mathrm{,}j}^k-{\partial }_j{\chi }_{lr\mathrm{,}i}^k-A_i^s{\chi }_{lrs\mathrm{,}j}^k+A_j^s{\chi }_{lrs\mathrm{,}i}^k\\ -\left({\chi }_{r\mathrm{,}i}^s{\chi }_{ls\mathrm{,}j}^k+{\chi }_{l\mathrm{,}i}^s{\chi }_{rs\mathrm{,}j}^k+{\chi }_{lr\mathrm{,}i}^s{\chi }_{s\mathrm{,}j}^k-{\chi }_{r\mathrm{,}j}^s{\chi }_{ls\mathrm{,}i}^k-{\chi }_{l\mathrm{,}j}^s{\chi }_{rs\mathrm{,}i}^k-{\chi }_{lr\mathrm{,}j}^s{\chi }_{s\mathrm{,}i}^k\right)=0\end{array}$ (3*)

When $q=1,g_2=0$ and though surprising it may look like, we find back exactly all the formulas presented by E. and F. Cosserat in ( [10], p 123). Even more strikingly, in the case of a Riemann structure, the last two terms disappear but the quadratic terms are left while, in the case of screw and complex structures, the quadratic terms disappear but the last two terms are left. We finally notice that ${{\chi }^{\prime }}_q=-{\chi }_q$ is a $R_q$ -connection if and only if $A=0$, a result contradicting the use of connections in physics. However, when $A=0$, we have ${{\chi }^{\prime }}_0\left(\xi \right)=\xi$ and thus:

$\begin{array}{c}{\bar{D}}^{\prime }{\chi }_{q+1}=\left(D{\chi }_{q+1}\right)\left(\xi ,\eta \right)-\left(\left[{\chi }_q\left(\xi \right),{\chi }_q\left(\eta \right)\right]+i\left(\xi \right)D\left({\chi }_{q+1}\left(\eta \right)\right)-i\left(\eta \right)D\left({\chi }_{q+1}\left(\xi \right)\right)\right)\\ =-\left[{\chi }_q\left(\xi \right),{\chi }_q\left(\eta \right)\right]-{\chi }_q\left(\left[\xi ,\eta \right]\right)\\ =-\left(\left[{{\chi }^{\prime }}_q\left(\xi \right),{{\chi }^{\prime }}_q\left(\eta \right)\right]-{{\chi }^{\prime }}_q\left(\left[\xi ,\eta \right]\right)\right)\\ =-{{\kappa }^{\prime }}_q\left(\xi ,\eta \right)\end{array}$

does not depend on the lift of ${\chi }_q$.

COROLLARY 3.14: When $det\left(A\right)\ne 0$ there is a second nonlinear Spencer sequence stabilized at order q:

$0\to aut\left(X\right)\stackrel{j_q}{\to }{\Pi }_q\stackrel{{\bar{D}}_1}{\to }{C}_1\left(T\right)\stackrel{{\bar{D}}_2}{\to }{C}_2(\; T\; )$

where ${\bar{D}}_1$ and ${\bar{D}}_2$ are involutive and a restricted second nonlinear Spencer sequence:

$0\to \Gamma \stackrel{j_q}{\to }{\mathcal{R}}_q\stackrel{{\bar{D}}_1}{\to }{C}_1\stackrel{{\bar{D}}_2}{\to }{C}_2$

such that ${\bar{D}}_1$ and ${\bar{D}}_2$ are involutive whenever ${\mathcal{R}}_q$ is involutive.

Proof: With $\left|\mu \right|=q$ we have ${\chi }_{\mu ,i}^k=-g_l^k{A}_i^r{f}_{\mu +1_r}^l+terms\left(order\le q\right)$. Setting ${\chi }_{\mu ,i}^k=A_i^r{\tau }_{\mu ,r}^k$, we obtain ${\tau }_{\mu ,r}^k=-g_l^k{f}_{\mu +1_r}^l+terms\left(order\le q\right)$ and $\bar{D}\mathrm{<mo>\; :\; </mo>}{\Pi }_{q+1}\to T^{\mathrm{<mo>\; *\; </mo>}}\otimes J_q\left(T\right)$ restricts to ${\bar{D}}_1\mathrm{<mo>\; :\; </mo>}{\Pi }_q\to C_1\left(T\right)$.

Finally, setting $A^{-1}=B=id-{\tau }_0$, we obtain successively:

${\partial }_i{\chi }_{\mu ,j}^k-{\partial }_j{\chi }_{\mu ,i}^k+terms\left({\chi }_q\right)-\left(A_i^r{\chi }_{\mu +1_r,j}^k-A_j^r{\chi }_{\mu +1_r,i}^k\right)=0$

$B_r^i{B}_s^j\left({\partial }_i{\chi }_{\mu ,j}^k-{\partial }_j{\chi }_{\mu ,i}^k\right)+terms\left({\chi }_q\right)-\left({\tau }_{\mu +1_r,s}^k-{\tau }_{\mu +1_s,r}^k\right)=0$

We obtain therefore $D{\tau }_{q+1}+terms\left({\tau }_q\right)=0$ and ${\bar{D}}^{\prime }\mathrm{<mo>\; :\; </mo>}{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes J_q\left(T\right)\to {\wedge }^2{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes J_{q-1}\left(T\right)$ restricts to ${\bar{D}}_2\mathrm{<mo>\; :\; </mo>}{C}_1\left(T\right)\to C_2\left(T\right)$.

In the case of Lie groups of transformations, the symbol of the involutive system $R_q$ must be $g_q=0$ providing an isomorphism ${\mathcal{R}}_{q+1}\simeq {\mathcal{R}}_q\Rightarrow R_{q+1}\simeq R_q$ and we have therefore $C_r={\wedge }^r{T}^{*}\otimes R_q$ for $r=1,\cdots ,n$ like in the linear Spencer sequence.

口

REMARK 3.15: In the case of the (local) action of a Lie group G on X, we may consider the graph of this action, that is the morphism $X\times G\to X\times X\mathrm{<mo>\; :\; </mo>}\left(x\mathrm{,}a\right)\to \left(x\mathrm{,}y=f\left(x\mathrm{,}a\right)\right)$. If q is large enough, then there is an isomorphism $X\times G\to {\mathcal{R}}_q\subset {\Pi }_q\mathrm{<mo>\; :\; </mo>}\left(x\mathrm{,}a\right)\to j_q\left(f\right)\left(x\mathrm{,}a\right)$ obtained by eliminating the parameters and $C_r={\wedge }^r{T}^{*}\otimes R_q$. If $\left\{{\theta }_{\tau }\right\}$ with $1\le \tau \le dim\left(G\right)$ is a basis of infinitesimal generators of this action, there is a morphism of Lie algebroids over X, namely $X\times \mathcal{G}\to R_q\mathrm{<mo>\; :\; </mo>}{\lambda }^{\tau }\left(x\right)\to {\lambda }^{\tau }\left(x\right)j_q\left({\theta }_{\tau }\right)$ when q is large enough and the linear Spencer sequence $R_q\stackrel{D_1}{\to }{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes R_q\stackrel{D_2}{\to }{\wedge }^2{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes R_q\stackrel{D_3}{\to }\cdots$ is locally exact because it is locally isomoprphic to the tensor product by $\mathcal{G}$ of the Poincaré sequence ${\wedge }^0{T}^{\mathrm{<mo>\; *\; </mo>}}\stackrel{d}{\to }{\wedge }^1{T}^{\mathrm{<mo>\; *\; </mo>}}\stackrel{d}{\to }{\wedge }^2{T}^{\mathrm{<mo>\; *\; </mo>}}\stackrel{d}{\to }\cdots$ where d is the exterior derivative ( [7] ).

We may also consider similarly $dy=dax=da{a}^{-1}y$ and $dx=db{b}^{-1}dx=-a^{-1}dax$, depending on the choice of the independent variable among the source x or the target y.

Surprisingly, in the case of Lie pseudogroups or Lie groupoids, the situation is quite different. We recall the way to introduce a groupoid structure on ${\Pi }_{q\mathrm{,1}}\subset J_1\left({\Pi }_q\right)$ from the groupoid structure on ${\Pi }_q$ when $\Delta =det\left({\partial }_i{f}^k\left(x\right)\right)\ne 0$, that is how to define $j_1\left(h_q\right)=j_1\left(g_q\circ f_q\right)=j_1\left(g_q\right)\circ j_1\left(f_q\right)$. We get successively with $y=f\left(x\right)$:

$h\left(x\right)=\left(g\circ f\right)\left(x\right)=g\left(f\left(x\right)\right)\Rightarrow \frac{\partial h^r}{\partial x^i}=\frac{\partial g^r}{\partial y^k}\frac{\partial f^k}{\partial x^i}\Rightarrow h_i^r\left(x\right)=g_k^r\left(f\left(x\right)\right)f_i^k(\; x\; )$

$\begin{array}{l}\frac{\partial h_i^r}{\partial x^j}=\frac{\partial g_k^r}{\partial y^l}{f}_i^k\frac{\partial f^l}{\partial x^j}+g_k^r\frac{\partial f_i^k}{\partial x^j}\\ \Rightarrow h_{ij}^r\left(x\right)=g_{kl}^r\left(f\left(x\right)\right)f_i^k\left(x\right)f_j^l\left(x\right)+g_k^r\left(f\left(x\right)\right)f_{ij}^k(\; x\; )\end{array}$

$\frac{\partial h_{ij}^r}{\partial x^s}=\frac{\partial g_{ki}^r}{\partial y^u}{f}_i^k{f}_j^l\frac{\partial f^u}{\partial x^s}+g_{kl}^r\left(\frac{\partial f_i^k}{\partial x^s}{f}_j^l+f_i^k\frac{\partial f_j^l}{\partial x^s}\right)+\frac{\partial g_k^r}{\partial y^u}\frac{\partial f^u}{\partial x^s}{f}_{ij}^k+g_k^r\frac{\partial f_{ij}^k}{\partial x^s}$

$\Rightarrow h_{ijs}^r=g_{klu}^r{f}_i^k{f}_j^l{f}_s^u+g_{kl}^r\left(f_{is}^k{f}_j^l+f_i^k{f}_{js}^l\right)+g_{ku}^r{f}_s^u{f}_{ij}^k+g_k^r{f}_{ijs}^k$

and so on with more and more involved formulas.

Now, if we want to obtain objects over the source x according to the non-linear Spencer sequence, we have only two possibilities in actual practice, namely:

$\begin{array}{l}{\chi }_q=f_{q+1}^{-1}\circ j_1\left(f_q\right)-i{d}_{q+1}\in T^{*}\otimes J_q\left(T\right)\\ \leftrightarrow {\bar{\chi }}_q=j_1{\left(f_q\right)}^{-1}\circ f_{q+1}-i{d}_{q+1}\in T^{*}\otimes J_q(\; T\; )\end{array}$

As we have already considered the first, we have now only to study the second. In $J_1\left({\Pi }_q\right)$, we have:

$\begin{array}{l}{\chi }_q+i{d}_{q+1}=\left(A_r^k,{\chi }_{i,r}^k,{\chi }_{ij,r}^k,\cdots \right)\text{and}{\bar{\chi }}_q+i{d}_{q+1}=\left({\bar{A}}_r^k,{\bar{\chi }}_{i,r}^k,{\bar{\chi }}_{ij,r}^k,\cdots \right)\\ \text{over}\left(x,x,\delta ,0,\cdots \right)\end{array}$

LEMMA 3.16: ${\bar{\chi }}_q$ is a quasi-linear rational function of ${\chi }_q$, $\forall q\ge 0$. With more details, when $q=0$, we have ${\bar{\chi }}_0=\bar{A}-id$ and ${\chi }_0=A-id$ with $\bar{A}=A^{-1}=B$ and when $q\ge 1$, we have ${\bar{\chi }}_q\circ A=-{\chi }_q$, that is to say ${\bar{\chi }}_q=-{\tau }_q$.

Proof: In the groupoid framework, we have:

$\left({\bar{\chi }}_q+i{d}_{q+1}\right)\circ \left({\chi }_q+i{d}_{q+1}\right)=i{d}_{q+1}\in J_1(\; \Pi \; q\; )$

Doing the substitutions:

$\frac{\partial g^r}{\partial y^k}\to {\bar{A}}_k^r\mathrm{,}\frac{\partial g_k^r}{\partial y^l}\to {\bar{\chi }}_{k\mathrm{,}l}^r\mathrm{,}\frac{\partial g_{kl}^r}{\partial y^u}\to {\bar{\chi }}_{kl\mathrm{,}u}^r$

$\frac{\partial f^k}{\partial x^i}\to A_i^k\mathrm{,}\frac{\partial f_i^k}{\partial x^j}\to {\chi }_{i\mathrm{,}j}^k\mathrm{,}\frac{\partial f_{ij}^k}{\partial x^s}\to {\chi }_{ij\mathrm{,}s}^k$

while using the fact that $f_i^k={\delta }_i^k,f_{ij}^k=0,\cdots$ and $g_k^r={\delta }_k^r,g_{kl}^r=0,\cdots$, we obtain at once:

${\bar{A}}_k^r{A}_i^k={\delta }_i^r,{\bar{\chi }}_{k,l}^r{A}_j^l+{\chi }_{i,j}^k=0,{\bar{\chi }}_{ij,u}^r{A}_s^u+{\chi }_{ij,s}^r=0,\cdots$

Proceeding by induction, we finally obtain:

${\bar{\chi }}_{\mu ,r}^k{A}_s^r+{\chi }_{\mu ,i}^k=0$

that is to say ${\bar{\chi }}_{\mu ,i}^k+{\tau }_{\mu ,i}^k=0$ because $\Delta \ne 0\Rightarrow det\left(A\right)\ne 0$, thus ${\bar{\chi }}_q\circ A=-{\chi }_q$ or, equivalently, ${\bar{\chi }}_q=-{\tau }_q$.

口

REMARK 3.17: The passage from ${\chi }_q$ to ${\tau }_q$ is exactly the one done by E. and F. Cosserat in ( [10], p 190), even though it is based on a subtle misunderstanding that we shall correct later on.

REMARK 3.18: According to the previous results, the “field” must be a section of the natural bundle $\mathcal{F}$ of geometric objects if we use the nonlinear Janet sequence or a section of the first Spencer bundle $C_1$ if we use the nonlinear Spencer sequence. The aim of this paper is to prove that the second choice is by far more convenient for mathematical physics.

4. Variational Calculus

It remains to graft a variational procedure adapted to the previous results. Contrary to what happens in analytical mechanics or elasticity for example, the main idea is to vary sections but not points. Hence, we may introduce the variation $\delta f^k\left(x\right)={\eta }^k\left(f\left(x\right)\right)$ and set ${\eta }^k\left(f\left(x\right)\right)={\xi }^i{\partial }_i{f}^k\left(x\right)\left(x\right)$ along the “vertical machinery” but notations like $\delta x^i={\xi }^i$ or $\delta y^k={\eta }^k$ have no meaning at all.

As a major result first discovered in specific cases by the brothers Cosserat in 1909 and by Weyl in 1916, we shall prove and apply the following key result:

THE PROCEDURE ONLY DEPENDS ON THE LINEAR SPENCER OPERATOR AND ITS FORMAL ADJOINT.

In order to prove this result, if $f_{q+1}\mathrm{,}{g}_{q+1}\mathrm{,}{h}_{q+1}\in {\Pi }_{q+1}$ can be composed in such a way that ${g^{\prime }}_{q+1}=g_{q+1}\circ f_{q+1}=f_{q+1}\circ h_{q+1}$, we get:

$\begin{array}{c}\bar{D}{g^{\prime }}_{q+1}=f_{q+1}^{-1}\circ g_{q+1}^{-1}\circ j_1(g_q)\circ j_1\left(f_q\right)-i{d}_{q+1}=f_{q+1}^{-1}\circ \bar{D}{g}_{q+1}\circ j_1\left(f_q\right)+\bar{D}{f}_{q+1}\\ =h_{q+1}^{-1}\circ f_{q+1}^{-1}\circ j_1\left(f_q\right)\circ j_1\left(h_q\right)-i{d}_{q+1}=h_{q+1}^{-1}\circ \bar{D}{f}_{q+1}\circ j_1\left(h_q\right)+\bar{D}{h}_{q+1}\end{array}$

Using the local exactness of the first nonlinear Spencer sequence or ( [25], p 219), we may state:

LEMMA 4.1: For any section $f_{q+1}\in {\mathcal{R}}_{q+1}$, the finite gauge transformation:

${\chi }_q\in T^{*}\otimes R_q\to {{\chi }^{\prime }}_q=f_{q+1}^{-1}\circ {\chi }_q\circ j_1\left(f_q\right)+\bar{D}{f}_{q+1}\in T^{*}\otimes R_q$

exchanges the solutions of the field equations ${\bar{D}}^{\prime }{\chi }_q=0$.

Introducing the formal Lie derivative on $J_q\left(T\right)$ by the formulas:

$L\left({\xi }_{q+1}\right){\eta }_q=\left\{{\xi }_{q+1},{\eta }_{q+1}\right\}+i\left(\xi \right)D{\eta }_{q+1}=\left[{\xi }_q,{\eta }_q\right]+i\left(\eta \right)D{\xi }_{q+1}$

$\left(L\left(j_1\left({\xi }_{q+1}\right)\right){\chi }_q\right)\left(\zeta \right)=L\left({\xi }_{q+1}\right)\left({\chi }_q\left(\zeta \right)\right)-{\chi }_q\left(\left[\xi \mathrm{,}\zeta \right]\right)$

LEMMA 4.2: Passing to the limit over the source with $h_{q+1}=i{d}_{q+1}+t{\xi }_{q+1}+\cdots$ for $t\to 0$, we get an infinitesimal gauge transformation leading to the infinitesimal variation:

$\delta {\chi }_q=D{\xi }_{q+1}+L\left(j_1\left({\xi }_{q+1}\right)\right){\chi }_q$ (3)

which does not depend on the parametrization of ${\chi }_q$. Setting ${\bar{\xi }}_{q+1}={\xi }_{q+1}+{\chi }_{q+1}\left(\xi \right)$, we get:

$\delta {\chi }_q=D{\bar{\xi }}_{q+1}-\left\{{\chi }_{q+1},{\bar{\xi }}_{q+1}\right\}$ (3*)

LEMMA 4.3: Passing to the limit over the target with ${\chi }_q=\bar{D}{f}_{q+1}$ and $g_{q+1}=i{d}_{q+1}+t{\eta }_{q+1}+\cdots$, we get the other infinitesimal variation where $D{\eta }_{q+1}$ is over the target:

$\delta {\chi }_q=f_{q+1}^{-1}\circ D{\eta }_{q+1}\circ j_1\left(f_q\right)$ (4)

which depends on the parametrization of ${\chi }_q$.

EXAMPLE 4.4: We obtain for $q=1$:

$\begin{array}{c}\delta {\chi }_{,i}^k=\left({\partial }_i{\xi }^k-{\xi }_i^k\right)+\left({\xi }^r{\partial }_r{\chi }_{,i}^k+{\chi }_{,r}^k{\partial }_i{\xi }^r-{\chi }_{,i}^r{\xi }_r^k\right)\\ =\left({\partial }_i{\bar{\xi }}^k-{\bar{\xi }}_i^k\right)+\left({\chi }_{r,i}^k{\bar{\xi }}^r-{\chi }_{,i}^r{\bar{\xi }}_r^k\right)\end{array}$

$\begin{array}{c}\delta {\chi }_{j,i}^k=\left({\partial }_i{\xi }_j^k-{\xi }_{ij}^k\right)+\left({\xi }^r{\partial }_r{\chi }_{j,i}^k+{\chi }_{j,r}^k{\partial }_i{\xi }^r+{\chi }_{r,i}^k{\xi }_j^r-{\chi }_{j,i}^r{\xi }_r^k-{\chi }_{,i}^r{\xi }_{jr}^k\right)\\ =\left({\partial }_i{\bar{\xi }}_j^k-{\bar{\xi }}_{ij}^k\right)+\left({\chi }_{rj,i}^k{\bar{\xi }}^r+{\chi }_{r,i}^k{\bar{\xi }}_j^r-{\chi }_{j,i}^r{\bar{\xi }}_r^k-{\chi }_{,i}^r{\bar{\xi }}_{jr}^k\right)\end{array}$

Introducing the inverse matrix $B=A^{-1}$, we obtain therefore equivalently:

$\delta A_i^k={\xi }^r{\partial }_r{A}_i^k+A_r^k{\partial }_i{\xi }^r-A_i^r{\xi }_r^k\iff \delta B_k^i={\xi }^r{\partial }_r{B}_k^i-B_k^r{\partial }_r{\xi }^i+B_r^i{\xi }_k^r$

both with:

$\delta {\chi }_{j\mathrm{,}i}^k=\left({\partial }_i{\xi }_j^k-A_i^r{\xi }_{jr}^k\right)+\left({\xi }^r{\partial }_r{\chi }_{j\mathrm{,}i}^k+{\chi }_{j\mathrm{,}r}^k{\partial }_i{\xi }^r+{\chi }_{r\mathrm{,}i}^k{\xi }_j^r-{\chi }_{j\mathrm{,}i}^r{\xi }_r^k\right)$

For the Killing system $R_1\subset J_1\left(T\right)$ with $g_2=0$, these variations are exactly the ones that can be found in ( [10], (50) + (49), p 124 with a printing mistake corrected on p 128) when replacing a 3 × 3 skew-symmetric matrix by the corresponding vector. The last unavoidable Proposition is thus essential in order to bring back the nonlinear framework of finite elasticity to the linear framework of infinitesimal elasticity that only depends on the linear Spencer operator.

For the conformal Killing system ${\hat{R}}_1\subset J_1\left(T\right)$ (see next section) we obtain:

$\delta {\chi }_{r\mathrm{,}i}^r=\left({\partial }_i{\xi }_r^r-{\xi }_{ri}^r\right)+\left({\xi }^r{\partial }_r{\chi }_{s\mathrm{,}i}^s+{\chi }_{s\mathrm{,}r}^s{\partial }_i{\xi }^r-{\chi }_{\mathrm{,}i}^s{\xi }_{rs}^r\right)$

but ${\chi }_{r\mathrm{,}i}^r\left(x\right)d{x}^i$ is far from being a 1-form. However, $\left({\chi }_{j\mathrm{,}i}^k+{\gamma }_{js}^k{\chi }_{\mathrm{,}i}^s\right)\in T^{\mathrm{<mo>\; *\; </mo>}}\otimes T^{\mathrm{<mo>\; *\; </mo>}}\otimes T$ and thus $\left({\alpha }_i={\chi }_{r,i}^r+{\gamma }_{rs}^r{\chi }_{,i}^s\right)\in T^{*}$ is a pure 1-form if we replace $\left({\chi }_{r\mathrm{,}i}^r\mathrm{,}{\chi }_{\mathrm{,}i}^r\right)$ by $\left({\alpha }_i\mathrm{,0}\right)$. Hence, $\alpha \left(\zeta \right)$ is a scalar for any $\zeta \in T$ and we have $L\left({\xi }_1\right)\left(\alpha \left(\zeta \right)\right)-\alpha \left(\left[\xi \mathrm{,}\zeta \right]\right)=\left({\alpha }_r{\partial }_i{\xi }^r+{\xi }^r{\partial }_r{\alpha }_i\right){\zeta }^i$. As we shall see in section V.A, we have ${\left(L\left({\xi }_2\right)\gamma \right)}_{ij}^k={\xi }_{ij}^k$ for any section ${\xi }_2\in J_2\left(T\right)$ and we obtain therefore successively:

$\delta {\alpha }_i=\left({\partial }_i{\xi }_r^r-{\xi }_{ri}^r\right)+\left({\alpha }_r{\partial }_i{\xi }^r+{\xi }^r{\partial }_r{\alpha }_i\right)$

${\phi }_{ij}={\partial }_i{\alpha }_j-{\partial }_j{\alpha }_i\Rightarrow \delta {\phi }_{ij}=\left({\partial }_j{\xi }_{ri}^r-{\partial }_i{\xi }_{rj}^r\right)+\left({\phi }_{rj}{\partial }_i{\xi }^r+{\phi }_{ir}{\partial }_j{\xi }^r+{\xi }^r{\partial }_r{\phi }_{ij}\right)$

These are exactly the variations obtained by Weyl ( [3], (76), p. 289) who was assuming implicitly $A=0$ when setting ${\bar{\xi }}_r^r=0\iff {\xi }_r^r=-{\alpha }_i{\xi }^i$ by introducing a connection. Accordingly, ${\xi }_{ri}^r$ is the variation of the EM potential itself, that is the $\delta A_i$ of engineers used in order to exhibit the Maxwell equations from a variational principle ( [3], p. 26) but the introduction of the Spencer operator is new in this framework.

The explicit general formulas of the two lemma cannot be found somewhere else (The reader may compare them to the ones obtained in [19] by means of the so-called “diagonal” method that cannot be applied to the study of explicit examples). The following unusual difficult proposition generalizes well known variational techniques used in continuum mechanics and will be crucially used for applications:

PROPOSITION 4.5: The same variation is obtained whenever ${\eta }_q=f_{q+1}\left({\xi }_q+{\chi }_q\left(\xi \right)\right)$ with ${\chi }_q=\bar{D}{f}_{q+1}$, a transformation only depending on $j_1\left(f_q\right)$ and invertible if and only if $det\left(A\right)\ne 0$.

Proof: First of all, setting ${\bar{\xi }}_q={\xi }_q+{\chi }_q\left(\xi \right)$, we get $\bar{\xi }=A\left(\xi \right)$ for $q=0$, a transformation which is invertible if and only if $det\left(A\right)\ne 0$. In the nonlinear framework, we have to keep in mind that there is no need to vary the object $\omega$ which is given but only the need to vary the section $f_{q+1}$ as we already saw, using ${\eta }_q\in R_q\left(Y\right)$ over the target or ${\xi }_q\in R_q$ over the source. With ${\eta }_q=f_{q+1}\left({\xi }_q\right)$, we obtain for example:

$\begin{array}{l}\delta f^k={\eta }^k=f_r^k{\xi }^r\\ \delta f_i^k={\eta }_u^k{f}_i^u=f_r^k{\xi }_i^r+f_{ri}^k{\xi }^r\\ \delta f_{ij}^k={\eta }_{uv}^k{f}_i^u{f}_j^v+{\eta }_u^k{f}_{ij}^u=f_r^k{\xi }_{ij}^r+f_{ri}^k{\xi }_j^r+f_{rj}^k{\xi }_i^r+f_{rij}^k{\xi }^r\end{array}$

and so on. Introducing the formal derivatives $d_i$ for $i=1,\cdots ,n$, we have:

$\delta f_{\mu }^k={\zeta }_{\mu }^k\left(f_q,{\eta }_q\right)=d_{\mu }{\eta }^k={\eta }_u^k{f}_{\mu }^u+\cdots =f_r^k{\xi }_{\mu }^r+\cdots +f_{\mu +1_r}^k{\xi }^r$

We shall denote by $\#\left({\eta }_q\right)={\zeta }_{\mu }^k\left(y_q\mathrm{,}{\eta }_q\right)\frac{\partial }{\partial y_{\mu }^k}\in V\left({\mathcal{R}}_q\right)$ with ${\zeta }^k={\eta }^k$ the corresponding vertical vector field, namely:

$\begin{array}{c}\#\left({\eta }_q\right)=0\frac{\partial }{\partial x^i}+{\eta }^k\left(y\right)\frac{\partial }{\partial y^k}+\left({\eta }_u^k\left(y\right)y_i^u\right)\frac{\partial }{\partial y_i^k}\\ +\left({\eta }_{uv}^k\left(y\right)y_i^u{y}_j^v+{\eta }_u^k\left(y\right)y_{ij}^u\right)\frac{\partial }{\partial y_{ij}^k}+\cdots \end{array}$

However, the standard prolongation of an infinitesimal change of source coordinates described by the horizontal vector field $\xi$, obtained by replacing all the derivatives of $\xi$ by a section ${\xi }_q\in R_q$ over $\xi \in T$, is the vector field:

$\begin{array}{c}♭\left({\xi }_q\right)={\xi }^i\left(x\right)\frac{\partial }{\partial x^i}+0\frac{\partial }{\partial y^k}-\left(y_r^k{\xi }_i^r\left(x\right)\right)\frac{\partial }{\partial y_i^k}\\ -\left(y_r^k{\xi }_{ij}^r\left(x\right)+y_{rj}^k{\xi }_i^r\left(x\right)+y_{ri}^k{\xi }_j^r\left(x\right)\right)\frac{\partial }{\partial y_{ij}^k}+\cdots \end{array}$

It can be proved that $\left[♭\left({\xi }_q\right),♭\left({{\xi }^{\prime }}_q\right)\right]=♭\left(\left[{\xi }_q,{{\xi }^{\prime }}_q\right]\right),\forall {\xi }_q,{{\xi }^{\prime }}_q\in R_q$ over the source, with a similar property for $\#\left(\mathrm{.}\right)$ over the target ( [25] ). However, $♭\left({\xi }_q\right)$ is not a vertical vector field and cannot therefore be compared to $\#\left({\eta }_q\right)$. The solution of this problem explains a strange comment made by Weyl in ( [3], p 289 + (78), p 290) and which became a founding stone of classical gauge theory. Indeed, ${\xi }_r^r$ is not a scalar because ${\xi }_i^k$ is not a 2-tensor. However, when $A=0$, then $-{\chi }_q$ is a $R_q$ -connection and ${\bar{\xi }}_r^r={\xi }_r^r+{\chi }_{r,i}^r{\xi }^i$ is a true scalar that may be set equal to zero in order to obtain ${\xi }_r^r=-{\chi }_{r,i}^r{\xi }^i$, a fact explaining why the EM-potential is considered as a connection in quantum mechanics instead of using the second order jets ${\xi }_{ri}^r$ of the conformal system, with a shift by one step in the physical interpretation of the Spencer sequence (See [4] for more historical details).

The main idea is to consider the vertical vector field $T\left(f_q\right)\left(\xi \right)-♭\left({\xi }_q\right)\in V\left({\mathcal{R}}_q\right)$ whenever $y_q=f_q\left(x\right)$. Passing to the limit $t\to 0$ in the formula $g_q\circ f_q=f_q\circ h_q$, we first get $g\circ f=f\circ h\Rightarrow f\left(x\right)+t\eta \left(f\left(x\right)\right)+\cdots =f\left(x+t\xi \left(x\right)+\cdots \right)$. Using the chain rule for derivatives and substituting jets, we get successively:

$\begin{array}{l}\delta f^k\left(x\right)={\xi }^r{\partial }_r{f}^k,\delta f_i^k={\xi }^r{\partial }_r{f}_i^k+f_r^k{\xi }_i^r,\\ \delta f_{ij}^k={\xi }^r{\partial }_r{f}_{ij}^k+f_{rj}^k{\xi }_i^r+f_{ri}^k{\xi }_j^r+f_r^k{\xi }_{ij}^r\end{array}$

and so on, replacing ${\xi }^r{f}_{\mu +1_r}^k$ by ${\xi }^r{\partial }_r{f}_{\mu }^k$ in ${\eta }_q=f_{q+1}\left({\xi }_q\right)$ in order to obtain:

$\delta f_{\mu }^k={\eta }_r^k{f}_{\mu }^r+\cdots ={\xi }^i\left({\partial }_i{f}_{\mu }^k-f_{\mu +1_i}^k\right)+f_{\mu +1_r}^k{\xi }^r+\cdots +f_r^k{\xi }_{\mu }^r$

where the right member only depends on $j_1\left(f_q\right)$ when $\left|\mu \right|=q$.

Finally, we may write the symbolic formula $f_{q+1}\left({\chi }_q\right)=j_1\left(f_q\right)-f_{q+1}=D{f}_{q+1}\in T^{*}\otimes V\left({\mathcal{R}}_q\right)$ in the explicit form:

$f_r^k{\chi }_{\mu ,i}^r+\cdots +f_{\mu +1_r}^k{\chi }_{,i}^r={\partial }_i{f}_{\mu }^k-f_{\mu +1_i}^k$

Substituting in the previous formula provides ${\eta }_q=f_{q+1}\left({\xi }_q+{\chi }_q\left(\xi \right)\right)$ and we just need to replace q by $q+1$ in order to achieve the proof.

Checking directly the proposition is not evident even when $q=0$ as we have:

$\left(\frac{\partial {\eta }^k}{\partial y^u}-{\eta }_u^k\right){\partial }_i{f}^u=f_r^k\left[\left({\partial }_i{\bar{\xi }}^r-{\bar{\xi }}_i^r\right)-\left({\chi }_{\mathrm{,}i}^s{\bar{\xi }}_s^r-{\chi }_{s\mathrm{,}i}^r{\bar{\xi }}^s\right)\right]$

but cannot be done by hand when $q\ge 1$.

口

For an arbitrary vector bundle E and involutive system $R_q\subseteq J_q\left(E\right)$, we may define the r-prolongations ${\rho }_r\left(R_q\right)=R_{q+r}=J_r\left(R_q\right)\cap J_{q+r}\left(E\right)\subset J_r\left(J_q\left(E\right)\right)$ and their respective symbols $g_{q+r}={\rho }_r\left(g_q\right)$ defined from $g_q\subseteq S_q{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes E$ where $S_q{T}^{\mathrm{<mo>\; *\; </mo>}}$ is the vector bundle of q-symmetric covariant tensors. Using the Spencer δ-map, we now recall the definition of the Spencer bundles:

$\begin{array}{l}{C}_r={\wedge }^r{T}^{*}\otimes R_q/\delta \left({\wedge }^{r-1}{T}^{*}\otimes g_{q+1}\right)\\ \subseteq {\wedge }^r{T}^{*}\otimes J_q\left(E\right)/\delta \left({\wedge }^{r-1}{T}^{*}\otimes S_{q+1}\right)T^{*}\otimes E=C_r(\; E\; )\end{array}$

and of the Janet bundles:

$F_r={\wedge }^r{T}^{*}\otimes J_q\left(E\right)/\left({\wedge }^r{T}^{*}\otimes R_q+\delta {\wedge }^{r-1}{T}^{*}\otimes S_{q+1}{T}^{*}\otimes E\right)$

When $D=\Phi \circ j_q$, we may obtain by induction on r the following fundamental diagram I relating the second linear Spencer sequence to the linear Janet sequence with epimorphisms $\Phi ={\Phi }_0,\cdots ,{\Phi }_n$:

$\boxed{\begin{array}{ccccccccccccc}& & & & & 0& & 0& & 0& & 0& \\ & & & & & \downarrow & & \downarrow & & \downarrow & & \downarrow & \\ & 0& \to & \Theta & \stackrel{j_q}{\to }& C_0& \stackrel{D_1}{\to }& C_1& \stackrel{D_2}{\to }& C_2& \stackrel{D_3}{\to }\cdots \stackrel{D_n}{\to }& C_n& \to 0\\ & & & & & \downarrow & & \downarrow & & \downarrow & & \downarrow & \\ & 0& \to & E& \stackrel{j_q}{\to }& C_0\left(E\right)& \stackrel{D_1}{\to }& C_1\left(E\right)& \stackrel{D_2}{\to }& C_2\left(E\right)& \stackrel{D_3}{\to }\cdots \stackrel{D_n}{\to }& C_n\left(E\right)& \to 0\\ & & & \parallel & & \downarrow {\Phi }_0& & \downarrow {\Phi }_1& & \downarrow {\Phi }_2& & \downarrow {\Phi }_n& \\ 0\to & \Theta & \to & E& \stackrel{\mathcal{D}}{\to }& F_0& \stackrel{{\mathcal{D}}_1}{\to }& F_1& \stackrel{{\mathcal{D}}_2}{\to }& F_2& \stackrel{{\mathcal{D}}_3}{\to }\cdots \stackrel{{\mathcal{D}}_n}{\to }& F_n& \to 0\\ & & & & & \downarrow & & \downarrow & & \downarrow & & \downarrow & \\ & & & & & 0& & 0& & 0& & 0& \end{array}}$

Chasing in the above diagram, the Spencer sequence is locally exact at $C_1$ if and only if the Janet sequence is locally exact at $F_0$ because the central sequence is locally exact (See [7] [13] [25] for more details). In the present situation, we shall always have $E=T$. The situation is much more complicate in the nonlinear framework and we provide details for a later use.

Let $\bar{\omega }$ be a section of $\mathcal{F}$ satisfying the same CC as $\omega$, namely $I\left(j_1\left(\omega \right)\right)=0$. As $\mathcal{F}$ is a quotient of ${\Pi }_q$, we may find a section $f_q\in {\Pi }_q$ such that:

$\begin{array}{l}\Phi \circ f_q\equiv f_q^{-1}\left(\omega \right)=\bar{\omega }\\ \Rightarrow {\rho }_1\left(\Phi \right)\circ j_1\left(f_q\right)\equiv j_1\left(f_q^{-1}\right)\left(j_1\left(\omega \right)\right)=j_1\left(f_q^{-1}\left(\omega \right)\right)=j_1(\; \omega \; ¯\; )\end{array}$

Similarly, as $\mathcal{F}$ is a natural bundle of order q, then $J_1\left(\mathcal{F}\right)$ is a natural bundle of order $q+1$ and we can find a section $f_{q+1}\in {\Pi }_{q+1}$ such that:

${\rho }_1\left(\Phi \right)\circ f_{q+1}\equiv f_{q+1}^{-1}\left(j_1\left(\omega \right)\right)=j_1(\; \omega \; ¯\; )$

and we are facing two possible but quite different situations:

• Eliminating $\bar{\omega }$, we obtain:

$\begin{array}{l}{j}_1\left(f_q^{-1}\right)\left(j_1\left(\omega \right)\right)=f_{q+1}^{-1}\left(j_1\left(\omega \right)\right)\\ \Rightarrow {\left(f_{q+1}\circ j_1\left(f_q^{-1}\right)\right)}^{-1}\left(j_1\left(\omega \right)\right)-j_1\left(\omega \right)=L\left({\sigma }_q\right)\omega =0\end{array}$

and thus ${\sigma }_q=\bar{D}{f}_{q+1}^{-1}\in T^{*}\otimes R_q=-f_{q+1}\circ {\chi }_q\circ j_1{\left(f\right)}^{-1}$ over the target if we set ${\chi }_q=\bar{D}{f}_{q+1}=f_{q+1}^{-1}\circ j_1\left(f_q\right)-i{d}_{q+1}$ over the source, even if $f_{q+1}$ may not be a section of ${\mathcal{R}}_{q+1}$. As ${\sigma }_q$ is killed by ${\bar{D}}^{\prime }$, we have related cocycles at $\mathcal{F}$ in the Janet sequence over the source with cocycles at $T^{\mathrm{<mo>\; *\; </mo>}}\otimes R_q$ or $C_1$ over the target.

• Eliminating $\omega$, we obtain successively:

$\begin{array}{l}\left(f_{q+1}^{-1}\circ j_1\left(f_q\right)\right)\left(j_1\left(\bar{\omega }\right)\right)-j_1\left(\bar{\omega }\right)\\ =-\left(f_{q+1}^{-1}\circ j_1\left(f_q\right)\right)\left[{\left(f_{q+1}^{-1}\circ j_1\left(f_q\right)\right)}^{-1}\left(j_1\left(\bar{\omega }\right)\right)-j_1\left(\bar{\omega }\right)\right]\\ =-\left(f_{q+1}^{-1}\circ j_1\left(f_q\right)\right)L\left({\chi }_q\right)\bar{\omega }\end{array}$

where we have over the source:

$L\left({\chi }_q\right)\bar{\omega }=\left({\bar{\Omega }}_i^{\tau }\equiv -L_k^{\tau \mu }\left(\bar{\omega }\left(x\right)\right){\chi }_{\mu ,i}^k+{\chi }_{,i}^r{\partial }_r{\bar{\omega }}^{\tau }\left(x\right)\right)\in T^{*}\otimes F_0$

However, we know that $F_0$ is associated with $R_q$ and is thus not affected by $f_{q+1}^{-1}\circ j_1\left(f_q\right)$ which projects onto $f_q^{-1}\circ f_q=i{d}_q$. Hence, only $T^{\mathrm{<mo>\; *\; </mo>}}$ is affected by $f_1^{-1}\circ j_1\left(f\right)=A$ in a covariant way and we obtain therefore over the source:

$\left(f_{q+1}^{-1}\circ j_1\left(f_q\right)\right)\left(j_1\left(\bar{\omega }\right)\right)-j_1\left(\bar{\omega }\right)=-BL\left({\chi }_q\right)\bar{\omega }=-L\left({\tau }_q\right)\bar{\omega }=0$

where $B=A^{-1}$. It follows that ${\chi }_q\in T^{\mathrm{<mo>\; *\; </mo>}}\otimes R_q\left(\bar{\omega }\right)$ with ${\bar{D}}^{\prime }{\chi }_q=0$ in the first non-linear Spencer sequence for $R_q\left(\bar{\omega }\right)\subset J_q\left(T\right)$.

We invite the reader to follow all the formulas involved in these technical results on the next examples. Of course, whenever ${\mathcal{R}}_q$ is formally integrable and $f_{q+1}\in {\mathcal{R}}_{q+1}$ is a lift of $f_q\in {\mathcal{R}}_q$, then we have $\bar{\omega }=\omega$ and ${\xi }_q\in T^{\mathrm{<mo>\; *\; </mo>}}\otimes R_q$ because $R_q\left(\omega \right)=R_q$.

EXAMPLE 4.6: In the case of Riemannian structures, we have $\mathcal{F}\in S_2{T}^{\mathrm{<mo>\; *\; </mo>}}$ because we deal with a non-degenerate metric $\omega =\left({\omega }_{ij}\right)\in S_2{T}^{\mathrm{<mo>\; *\; </mo>}}$ with $det\left(\omega \right)\ne 0$ and may introduce ${\omega }^{-1}=\left({\omega }^{ij}\right)\in S_{2}T$. We have by definition ${\omega }_{kl}\left(f\left(x\right)\right)f_i^k\left(x\right)f_j^l\left(x\right)={\bar{\omega }}_{ij}\left(x\right)$ that we shall simply write ${\omega }_{kl}\left(f\right)f_i^k{f}_j^l={\bar{\omega }}_{ij}\left(x\right)$ and obtain therefore:

${\omega }_{kl}\left(f\right)f_j^l{\partial }_r{f}_i^k+{\omega }_{kl}\left(f\right)f_i^k{\partial }_r{f}_j^l+\frac{\partial {\omega }_{kl}}{\partial y^u}\left(f\right)f_i^k{f}_j^l{\partial }_r{f}^u-{\partial }_r{\bar{\omega }}_{ij}\left(x\right)=0$

Our purpose is now to compute the expression:

${\omega }_{kl}\left(f\right)f_j^l{f}_{ir}^k+{\omega }_{kl}\left(f\right)f_i^k{f}_{jr}^l+\frac{\partial {\omega }_{kl}}{\partial y^u}\left(f\right)f_i^k{f}_j^l{f}_r^u-{\partial }_r{\bar{\omega }}_{ij}\left(x\right)\ne 0$

In order to eliminate the derivatives of $\omega$ over te target we may multiply the first equation by B and substract from the second while using the fact that ${\omega }_{kl}\left(f\right)={\bar{\omega }}_{ij}\left(x\right)g_k^i{g}_l^j$ with ${\chi }_0=A-i{d}_T\Rightarrow {\tau }_0=B{\chi }_0=i{d}_T-B$ in order to get:

$-\left({\bar{\omega }}_{sj}{\tau }_{i\mathrm{,}r}^s+{\bar{\omega }}_{is}{\tau }_{j\mathrm{,}r}^s+{\tau }_{\mathrm{,}r}^s{\partial }_s{\bar{\omega }}_{ij}\right)=-{\left(L\left({\tau }_1\right)\bar{\omega }\right)}_{ij\mathrm{,}r}$

These results can be extended at once to any tensorial geometric object but the conformal case needs more work and we let the reader treat it as an exercise. He will discover that the standard elimination of a conformal factor is not the best way to use in order to understand the conformal structure which has to do with a tensor density and no longer with a tensor.

In the non-linear case, the non-linear CC of the system ${\mathcal{R}}_q$ defined by $\Phi \left(y_q\right)=\bar{\omega }\left(x\right)$ only depend on the differential invariants and are exactly the ones satisfied by $\omega$ in the sense that they have the same Vessiot structure constants whenever ${\mathcal{R}}_q$ is formally integrable, in particular involutive as shown in Example 2.7. Accordingly, we can always find $f_{q+1}$ over $f_q$. In the linear case, the procedure is similar but slightly simpler. Indeed, if $\mathcal{D}\mathrm{<mo>\; :\; </mo>}T\to F_0$ is an involutive Lie operator, we may consider only the initial part of the fundamental diagram I:

$\begin{array}{cccccccccc}& & & & & 0& & 0& & 0\\ & & & & & \downarrow & & \downarrow & & \downarrow \\ & 0& \to & \Theta & \stackrel{j_q}{\to }& C_0& \stackrel{D_1}{\to }& C_1& \stackrel{D_2}{\to }& C_2\\ & & & & & \downarrow & & \downarrow & & \downarrow \\ & 0& \to & T& \stackrel{j_q}{\to }& C_0\left(T\right)& \stackrel{D_1}{\to }& C_1\left(T\right)& \stackrel{D_2}{\to }& C_2\left(T\right)\\ & & & \parallel & & \downarrow {\Phi }_0& & \downarrow {\Phi }_1& & \\ 0\to & \Theta & \to & T& \stackrel{\mathcal{D}}{\to }& F_0& \stackrel{{\mathcal{D}}_1}{\to }& F_1& & \\ & & & & & \downarrow & & \downarrow & & \\ & & & & & 0& & 0& & \end{array}$

$\begin{array}{ccccccccc}& & & & 0& & 0& & \\ & & & & \downarrow & & \downarrow & & \\ & & 0& \to & g_{q+1}& \stackrel{-\delta }{\to }& \delta \left(g_{q+1}\right)& \to & 0\\ & & & & \downarrow & & \downarrow & & \\ 0& \to & \Theta & \stackrel{j_{q+1}}{\to }& R_{q+1}& \stackrel{D}{\to }& T^{*}\otimes R_q& & \\ & & \parallel & & \downarrow & & \downarrow & & \\ 0& \to & \Theta & \stackrel{j_q}{\to }& R_q& \stackrel{D_1}{\to }& C_1& & \\ & & & & \downarrow & & \downarrow & & \\ & & & & 0& & 0& & \end{array}$

and study the linear inhomogeneous involutive system $\mathcal{D}\xi =\Omega$ with $\Omega \in F_0$ and ${\mathcal{D}}_1\Omega =0$. If we pick up any lift ${\xi }_q\in C_0\left(T\right)=J_q\left(T\right)$ of $\Omega$ and chase, we notice that $X_1=D_1{\xi }_q\in C_1\subset C_1\left(T\right)$ is such that $D_2{X}_1=0$.

EXAMPLE 4.7: In the Example 2.7, using the involutive system ${R^{\prime }}_1=R_1^{\left(1\right)}\subset R_1\subset J_1\left(T\right)$, we have $m=n=2,q=1$ and the fundamental diagram I:

$\boxed{\begin{array}{cccccccccccc}& & & & & 0& & 0& & 0& & \\ & & & & & \downarrow & & \downarrow & & \downarrow & & \\ & 0& \to & \Theta & \stackrel{j_1}{\to }& 3& \stackrel{D_1}{\to }& 5& \stackrel{D_2}{\to }& 2& \to & 0\\ & & & & & \downarrow & & \downarrow & & \parallel & & \\ & 0& \to & 2& \stackrel{j_1}{\to }& 6& \stackrel{D_1}{\to }& 6& \stackrel{D_2}{\to }& 2& \to & 0\\ & & & \parallel & & \downarrow {\Phi }_0& & \downarrow {\Phi }_1& & \downarrow & & \\ 0\to & \Theta & \to & 2& \stackrel{\mathcal{D}}{\to }& 3& \stackrel{{\mathcal{D}}_1}{\to }& 1& \to & 0& & \\ & & & & & \downarrow & & \downarrow & & & & \\ & & & & & 0& & 0& & & & \end{array}}$

with fiber dimensions:

$\begin{array}{ccccccccc}& & & & 0& & 0& & \\ & & & & \downarrow & & \downarrow & & \\ & & 0& \to & 1& \stackrel{-\delta }{\to }& 1& \to & 0\\ & & & & \downarrow & & \downarrow & & \\ 0& \to & \Theta & \stackrel{j_2}{\to }& 4& \stackrel{D}{\to }& 6& & \\ & & \parallel & & \downarrow & & \downarrow & & \\ 0& \to & \Theta & \stackrel{j_1}{\to }& 3& \stackrel{D_1}{\to }& 5& & \\ & & & & \downarrow & & \downarrow & & \\ & & & & 0& & 0& & \end{array}$

It is important to point out the importance of formal integrability and involution in this case. For this, let us start with a 1-form $\alpha =\left({\alpha }_1\mathrm{,}{\alpha }_2\right)$, denote its variation by $A=\left(A_1\mathrm{,}{A}_2\right)$ and consider only the linear inhomogeneous system $\mathcal{D}\xi =\mathcal{L}\left(\xi \right)\alpha =A$ with no CC for A. If the ground differential field is $K=\mathbb{Q}\left(x^1\mathrm{,}{x}^2\right)$ with commuting derivations $\left(d_1\mathrm{,}{d}_2\right)$, let us choose $\alpha =x^{2}d{x}^1=\left(x^2,0\right)$, $A=\left(x^2,x^1\right)$. As a lift ${\xi }_1\in J_1\left(T\right)$ of A, we let the reader check that we may choose in K:

${\xi }^1=0,{\xi }^2=0,{\xi }_1^1=1,{\xi }_2^1=\frac{x^1}{x^2},{\xi }_1^2=0,{\xi }_2^2=0$

Using one prolongation, we have:

$\begin{array}{l}{d}_1{A}_1\equiv x^2{\xi }_{11}^1+{\xi }_1^2=0,d_2{A}_1\equiv x^2{\xi }_{12}^1+{\xi }_1^1+{\xi }_2^2=1,\\ d_1{A}_2\equiv x^2{\xi }_{12}^1=1,d_2{A}_2\equiv x^2{\xi }_{22}^1+{\xi }_2^1=0\end{array}$

If $\beta =-d\alpha =d{x}^1\wedge d{x}^2$, we may denote its variation by B and get at once $B=d_2{A}_1-d_1{A}_2\equiv {\xi }_1^1+{\xi }_2^2=0$. Such a result is contradicting our initial choice $1+0=1$ and we cannot therefore find a lift ${\xi }_2$ of $j_1\left(A\right)$. Hence, we have to introduce the new geometric object $\omega =\left(\alpha \mathrm{,}\beta \right)$ with $\Omega =\left(A,B\right)$ and CC $d\alpha +\beta =0$ leading to $d_1{A}_2-d_2{A}_1+B=0$ while using the previous diagrams. We can therefore lift $\Omega =\left(A\mathrm{,}B\right)$ to ${\xi }_1\in J_1\left(T\right)$ by choosing in K:

${\xi }^1=0,{\xi }^2=0,{\xi }_1^1=1,{\xi }_2^1=\frac{x^1}{x^2},{\xi }_1^2=0,{\xi }_2^2=-1$

However, we have now to add:

$d_{1}B\equiv {\xi }_{11}^1+{\xi }_{12}^2=0,d_{2}B\equiv {\xi }_{12}^1+{\xi }_{22}^2=0$

and lift $j_1\left(\Omega \right)$ to ${\xi }_2\in J_2\left(T\right)$ over ${\xi }_1\in J_1\left(T\right)$ by choosing in K:

${\xi }_{11}^1=0,{\xi }_{12}^1=\frac{1}{x^2},{\xi }_{22}^1=-\frac{x^1}{{\left(x^2\right)}^2},{\xi }_{11}^2=0,{\xi }_{12}^2=0,{\xi }_{22}^2=-\frac{1}{x^2}$

The image of the Spencer operator is $X_1=D{\xi }_2=j_1\left({\xi }_1\right)-{\xi }_2$ that is to say:

$X_{,1}^1=-1,X_{,2}^1=-\frac{x^1}{x^2},X_{,1}^2=0,X_{,2}^2=1,$

$X_{1,1}^1=0,X_{2,1}^1=0,X_{1,2}^1=-\frac{1}{x^2},X_{2,2}^1=0,X_{1,1}^2=0,X_{2,1}^2=0,X_{1,2}^2=0,X_{2,2}^2=\frac{1}{x^2}$

and we check that $X_1\in T^{\mathrm{<mo>\; *\; </mo>}}\otimes R_1$, namely:

$x^2{X}_{1,i}^1+X_{,i}^2=0,X_{2,i}^1=0,X_{1,i}^1+X_{2,i}^2=0,\forall i=1,2$

a result which is not evident at first sight and has no meaning in any classical approach because we use sections and not solutions.

Now, if $f_{q+1}\mathrm{,}{f^{\prime }}_{q+1}\in {\Pi }_{q+1}$ are such that $f_{q+1}^{-1}\left(j_1\left(\omega \right)\right)={f^{\prime }}_{q+1}^{-1}\left(j_1\left(\omega \right)\right)=j_1\left(\bar{\omega }\right)$, it follows that $\left({f^{\prime }}_{q+1}\circ f_{q+1}^{-1}\right)\left(j_1\left(\omega \right)\right)=j_1\left(\omega \right)\Rightarrow \exists g_{q+1}\in {\mathcal{R}}_{q+1}$ such that ${f^{\prime }}_{q+1}=g_{q+1}\circ f_{q+1}$ and the new ${{\sigma }^{\prime }}_q=\bar{D}{f^{\prime }}_{q+1}^{-1}$ differs from the initial ${\sigma }_q=\bar{D}{f}_{q+1}^{-1}$ by a gauge transformation.

Conversely, let $f_{q+1}\mathrm{,}{f^{\prime }}_{q+1}\in {\Pi }_{q+1}$ be such that ${\sigma }_q=\bar{D}{f}_{q+1}^{-1}=\bar{D}{f^{\prime }}_{q+1}^{-1}={{\sigma }^{\prime }}_q$. It follows that $\bar{D}\left(f_{q+1}^{-1}\circ {f^{\prime }}_{q+1}\right)=0$ and one can find $g\in aut\left(X\right)$ such that ${f^{\prime }}_{q+1}=f_{q+1}\circ j_{q+1}\left(g\right)$ providing ${\bar{\omega }}^{\prime }={f^{\prime }}_q^{-1}\left(\omega \right)={\left(f_q\circ j_q\left(g\right)\right)}^{-1}\left(\omega \right)=j_q{\left(g\right)}^{-1}\left(f_q^{-1}\left(\omega \right)\right)=j_q{\left(g\right)}^{-1}\left(\bar{\omega }\right)$.

PROPOSITION 4.8: Natural transformations of $\mathcal{F}$ over the source in the nonlinear Janet sequence correspond to gauge transformations of $T^{\mathrm{<mo>\; *\; </mo>}}\otimes R_q$ or $C_1$ over the target in the nonlinear Spencer sequence. Similarly, the Lie derivative $\mathcal{D}\xi =\mathcal{L}\left(\xi \right)\omega \in F_0$ in the linear Janet sequence corresponds to the Spencer operator $D{\xi }_{q+1}\in T^{\mathrm{<mo>\; *\; </mo>}}\otimes R_q$ or $D_1{\xi }_q\in C_1$ in the linear Spencer sequence.

With a slight abuse of language $\delta f=\eta \circ f\iff \delta f\circ f^{-1}=\eta \iff f^{-1}\circ \delta f=\xi$ when $\eta =T\left(f\right)\left(\xi \right)$ and we get $j_q{\left(f\right)}^{-1}\left(\omega \right)=\bar{\omega }\Rightarrow j_q{\left(f+\delta f\right)}^{-1}\left(\omega \right)=\bar{\omega }+\delta \bar{\omega }$ that is $j_q{\left(f^{-1}\circ \left(f+\delta f\right)\right)}^{-1}\left(\bar{\omega }\right)=\bar{\omega }+\delta \bar{\omega }\Rightarrow \delta \bar{\omega }=\mathcal{L}\left(\xi \right)\bar{\omega }$ and $j_q{\left(\left(f+\delta f\right)\circ f^{-1}\circ f\right)}^{-1}\left(\omega \right)=j_q{\left(f\right)}^{-1}\left(j_q{\left(\left(f+\delta f\right)\circ f^{-1}\right)}^{-1}\left(\omega \right)\right)$ $\Rightarrow \delta \bar{\omega }=j_q{\left(f\right)}^{-1}\left(\mathcal{L}\left(\eta \right)\omega \right)$.

Passing to the infinitesimal point of view, we obtain the following generalization of Remark 3.12 which is important for applications.

COROLLARY 4.9: $\bar{\Omega }=\delta \bar{\omega }=L\left({\xi }_q\right)\bar{\omega }=f_q^{-1}\left(L\left({\eta }_q\right)\omega \right)\Rightarrow \delta \bar{\omega }=\mathcal{L}\left(\xi \right)\bar{\omega }=j_q{\left(f\right)}^{-1}\left(\mathcal{L}\left(\eta \right)\omega \right)$.

Recapitulating the results so far obtained concerning the links existing between the source and the target points of view, we may set in a symbolic way:

$\delta f_q\stackrel{\left(f_q\right)}{\leftrightarrow }{\eta }_q\stackrel{\left(f_{q+1}\right)}{\leftrightarrow }{\bar{\xi }}_q\stackrel{\left({\chi }_q\right)}{\leftrightarrow }{\xi }_q$

In order to help the reader maturing the corresponding nontrivial formulas, we compute explicitly the case $n=1,q=1,2$ and let the case n arbitrary left to the reader as a difficult exercise that cannot be achieved by hand when $q\ge 3$:

EXAMPLE 4.10: Using the previous formulas, we have $\delta f\left(x\right)=\eta \left(f\left(x\right)\right)$, $\delta f_x\left(x\right)={\eta }_y\left(f\left(x\right)\right)f_x\left(x\right)$ and:

$\begin{array}{l}{\eta }_1=f_2\left({\bar{\xi }}_1\right)\Rightarrow (\eta \left(f\left(x\right)\right)=f_x\left(x\right)\bar{\xi }\left(x\right),\\ {\eta }_y\left(f\left(x\right)\right)f_x\left(x\right)=f_x\left(x\right){\bar{\xi }}_x\left(x\right)+f_{xx}\left(x\right)\bar{\xi }(\; x\; ))\end{array}$

The delicate point is that we have successively:

${\chi }_{,x}=\frac{{\partial }_{x}f}{f_x}-1=A-1,{\chi }_{x,x}=\frac{1}{f_x}\left({\partial }_x{f}_x-\frac{{\partial }_{x}f}{f_x}{f}_{xx}\right)$

$\begin{array}{l}\bar{\xi }=\xi +{\chi }_{,x}\left(\xi \right)=\frac{{\partial }_{x}f}{f_x}\xi =A\xi ,{\bar{\xi }}_x={\xi }_x+{\chi }_{x,x}\xi \\ \Rightarrow \eta ={\partial }_{x}f\xi ,{\eta }_y={\xi }_x+\frac{{\partial }_x{f}_x}{f_x}\xi \end{array}$

$f_x{\eta }_{yy}={\xi }_{xx}+\frac{f_{xx}}{f_x}{\xi }_x+\left(\frac{{\partial }_x{f}_{xx}}{f_x}-\frac{f_{xx}}{{\left(f_x\right)}^2}{\partial }_x{f}_x\right)\xi$

When $z=g\left(y\right),y=f\left(x\right)\Rightarrow z=\left(g\circ f\right)\left(x\right)=h\left(x\right)$, we obtain therefore the simple groupoid composition formulas $h_x\left(x\right)=g_y\left(f\left(x\right)\right)f_x\left(x\right)$ and thus:

$\begin{array}{l}\zeta ={\partial }_{x}h\xi ={\partial }_{y}g\eta ={\partial }_{y}g{\partial }_{x}f\xi ,\\ {\zeta }_z={\eta }_y+\frac{{\partial }_y{g}_y}{g_y}\eta ={\xi }_x+\left(\frac{{\partial }_y{g}_y}{g_y}{\partial }_{x}f+\frac{{\partial }_x{f}_x}{f_x}\right)\xi ={\xi }_x+\frac{{\partial }_x{h}_x}{h_x}\xi \end{array}$

Using indices in arbitrary dimension, we get successively:

$\begin{array}{l}{\eta }^k=f_r^k{\bar{\xi }}^r,{\eta }_u^k{f}_i^u=f_r^k{\bar{\xi }}_i^r+f_{ri}^k{\bar{\xi }}^r{\eta }^k\\ \Rightarrow {\eta }^k={\xi }^r{\partial }_r{f}^k,{\eta }_u^k{f}_i^u=f_s^k\left({\xi }_i^s+g_u^s\left({\partial }_r{f}_i^u-A_r^t{f}_{ti}^u\right){\xi }^r\right)+f_{ti}^k{A}_r^t{\xi }^r\end{array}$

${\eta }_u^k=g_u^i{f}_s^k{\xi }_i^s+{\xi }^r{g}_u^i{\partial }_r{f}_i^k\Rightarrow {\eta }_k^k={\xi }_r^r+{\xi }^r{g}_u^i{\partial }_r{f}_i^u$

As a very useful application, we obtain successively:

$\Delta \left(x\right)=det\left({\partial }_i{f}^k\left(x\right)\right)\Rightarrow \delta \Delta \mathrm{<mo>\; =\; </mo>}\Delta \frac{\partial {\eta }^k}{\partial y^k}=\Delta {\partial }_r{\xi }^r+{\xi }^r{\partial }_r\Delta ={\partial }_r\left({\xi }^r\Delta \right)$

$\delta det\left(A\right)=det\left(A\right)\left(\frac{\partial {\eta }^k}{\partial y^k}-{\eta }_k^k\right)=det\left(A\right)\left({\partial }_r{\xi }^r-{\xi }_r^r\right)+{\xi }^r{\partial }_{r}det(\; A\; )$

where sections of jet bundles are used in an essential way, and the important lemma:

LEMMA 4.11: When the transformation $y=f\left(x\right)$ is invertible with inverse $x=g\left(y\right)$, we have the fundamental identity over the source or over the target:

$\begin{array}{l}\frac{\partial }{\partial x^i}\left(\Delta \left(x\right)\frac{\partial g^i}{\partial y^k}\left(f\left(x\right)\right)\right)\equiv \mathrm{0,}\forall x\in X\\ \iff \frac{\partial }{\partial y^k}\left(\frac{1}{\Delta \left(g\left(y\right)\right)}\frac{\partial f^k}{\partial x^i}\left(g\left(y\right)\right)\right)\equiv \mathrm{0,}\forall y\in Y\end{array}$

EXAMPLE 4.12: We proceed the same way for studying the links existing between ${\chi }_q=\bar{D}{f}_{q+1}$ over the source, ${\chi }_q^{-1}={\sigma }_q=\bar{D}{f}_{q+1}^{-1}$ over the target and the nonlinear Spencer operator. First of all, we notice that:

$\begin{array}{l}{\sigma }_q=f_{q+1}\circ j_1\left(f_q^{-1}\right)-i{d}_{q+1}=f_{q+1}\circ \left(i{d}_{q+1}-f_{q+1}^{-1}{j}_1\left(f_q\right)\right)\circ j_1{\left(f_q\right)}^{-1}\\ =-f_{q+1}\circ {\chi }_q\circ j_1{\left(f_q\right)}^{-1}\end{array}$

and the components of ${\sigma }_q$ thus factor through linear combinations of the components of ${\chi }_q$. After tedious computations, we get successively when $m=n=1$:

${\chi }_{,x}=\frac{{\partial }_{x}f}{f_x}-1=A-1=\frac{1}{f_x}\left({\partial }_{x}f-f_x\right)$

${\chi }_{x,x}=\frac{1}{f_x}\left({\partial }_x{f}_x-\frac{{\partial }_{x}f}{f_x}{f}_{xx}\right)=\frac{1}{f_x}\left({\partial }_x{f}_x-f_{xx}\right)-\frac{f_{xx}}{{\left(f_x\right)}^2}\left({\partial }_{x}f-f_x\right)$

$\begin{array}{l}{\chi }_{xx\mathrm{,}x}=\frac{1}{f_x}\left({\partial }_x{f}_{xx}-\frac{{\partial }_{x}f}{f_x}{f}_{xxx}\right)-2\frac{f_{xx}}{{\left(f_x\right)}^2}\left({\partial }_x{f}_x-\frac{{\partial }_{x}f}{f_x}{f}_{xx}\right)\\ =\frac{1}{f_x}\left({\partial }_x{f}_{xx}-f_{xxx}\right)-2\frac{f_{xx}}{{\left(f_x\right)}^2}\left({\partial }_x{f}_x-f_{xx}\right)+\left(2\frac{{\left(f_{xx}\right)}^2}{{\left(f_x\right)}^3}-\frac{f_{xxx}}{{\left(f_x\right)}^2}\right)\left({\partial }_{x}f-f_x\right)\end{array}$

These formulas agree with the successive constructive/inductive identities:

$\{\begin{array}{l}{\chi }_{\mathrm{,}x}{f}_x={\partial }_{x}f-f_x\\ {\chi }_{x\mathrm{,}x}{f}_x+{\chi }_{\mathrm{,}x}{f}_{xx}={\partial }_x{f}_x-f_{xx}\\ {\chi }_{xx\mathrm{,}x}{f}_x+2{\chi }_{x\mathrm{,}x}{f}_{xx}+{\chi }_{\mathrm{,}x}{f}_{xxx}={\partial }_x{f}_{xx}-f_{xxx}\end{array}$

showing that ${\chi }_q$ is linearly depending on $D{f}_{q+1}$ and we finally get:

$\{\begin{array}{l}{\sigma }_{,y}=-\left({\partial }_{x}f-f_x\right)\frac{1}{{\partial }_{x}f}=\frac{f_x}{{\partial }_{x}f}-1=-f_x{\chi }_{,x}\frac{1}{{\partial }_{x}f}\\ {\sigma }_{y,y}=-\frac{1}{f_x}\left({\partial }_x{f}_x-f_{xx}\right)\frac{1}{{\partial }_{x}f}=-\left({\chi }_{x,x}+\frac{f_{xx}}{f_x}{\chi }_{,x}\right)\frac{1}{{\partial }_{x}f}\\ {\sigma }_{yy,y}=-\left(\frac{1}{{\left(f_x\right)}^2}\left({\partial }_x{f}_{xx}-f_{xxx}\right)-\frac{f_{xx}}{{\left(f_x\right)}^3}\left({\partial }_x{f}_x-f_{xx}\right)\right)\frac{1}{{\partial }_{x}f}\\ =-\left(\frac{1}{f_x}{\chi }_{xx,x}+\frac{f_{xx}}{{\left(f_x\right)}^2}{\chi }_{x,x}+\left(\frac{f_{xxx}}{{\left(f_x\right)}^2}-\frac{{\left(f_{xx}\right)}^2}{{\left(f_x\right)}^3}\right){\chi }_{,x}\right)\frac{1}{{\partial }_{x}f}\end{array}$

while using successively the relations $g_y{f}_x=1$, ${\partial }_{y}g{\partial }_{x}f=1$, $g_{yy}{\left(f_x\right)}^2+g_y{f}_{xx}=0$ and so on when $x=g\left(y\right)$ is the inverse of $y=f\left(x\right)$, in a coherent way with the action of $f_3$ on $J_2\left(T\right)$ which is described as follows:

$\{\begin{array}{l}\eta =f_x\xi \\ {\eta }_y={\xi }_x+\frac{f_{xx}}{f_x}\xi \\ {\eta }_{yy}=\frac{1}{f_x}{\xi }_{xx}+\frac{f_{xx}}{{\left(f_x\right)}^2}{\xi }_x+\left(\frac{f_{xxx}}{{\left(f_x\right)}^2}-\frac{{\left(f_{xx}\right)}^2}{{\left(f_x\right)}^3}\right)\xi \end{array}$

Restricting these formulas to the affine case defined by $y_{xx}=0\Rightarrow {\xi }_{xx}=0$, we have thus $y_{xx}=0,y_{xxx}=0\Rightarrow f_{xx}=0,f_{xxx}=0$. It follows that $\eta =f_x\xi ,{\eta }_y={\xi }_x,{\eta }_{yy}=\frac{1}{f_x}{\xi }_{xx}=0$ on one side and ${\chi }_{xx,x}=0\iff {\sigma }_{yy,y}=0$ in a coherent way. It is finally important to notice that these results are not evident, even when $m=n=1$, as soon as second order jets are involved.

We shall use all the preceding formulas in the next example showing that, contrary to what happens in elasticity theory where the source is usually identified with the Lagrange variables, in both the Vessiot/Janet and the Cartan/Spencer approaches, the source must be identified with the Euler variables without any possible doubt.

EXAMPLE 4.13: With $n=1,q=1,\mathcal{F}=T^{*}$ and the finite OD Lie equation $\omega \left(y\right)y_x=\omega \left(x\right)$ with $\omega \in T^{\mathrm{<mo>\; *\; </mo>}}$ and corresponding Lie operator $\mathcal{D}\xi \equiv \mathcal{L}\left(\xi \right)\omega =\omega \left(x\right){\partial }_x\xi +\xi {\partial }_x\omega \left(x\right)$ over the source, we have:

$\omega \left(f\left(x\right)\right)f_x\left(x\right)=\bar{\omega }\left(x\right)\mathrm{,}\omega \left(f\left(x\right)\right)f_{xx}\left(x\right)+{\partial }_y\omega \left(f\left(x\right)\right)f_x^2\left(x\right)={\partial }_x\bar{\omega }(\; x\; )$

Differentiating once the first equation and substracting the second, we obtain therefore:

$\omega {\sigma }_{y,y}+{\sigma }_{,y}{\partial }_y\omega \equiv -\omega \left(1/f_x\right)\left({\partial }_x{f}_x-f_{xx}\right)\left(1/{\partial }_{x}f\right)+\left(\left(f_x/{\partial }_{x}f\right)-1\right){\partial }_y\omega =0$

whenever $y=f\left(x\right)$. Finally, setting $\omega \left(f\left(x\right)\right){\partial }_{x}f\left(x\right)=\bar{\omega }\left(x\right)$, we get over the target:

$\delta \bar{\omega }=\omega \left(f\left(x\right)\right)\frac{\partial \eta }{\partial y}{\partial }_{x}f\left(x\right)+{\partial }_{x}f\left(x\right)\frac{\partial \omega }{\partial y}\left(f\left(x\right)\right)\eta ={\partial }_{x}f\mathcal{L}\left(\eta \right)\omega$

Differentiating $\eta =\xi {\partial }_{x}f$ in order to obtain $\frac{\partial \eta }{\partial y}={\partial }_x\xi +\xi \left({\partial }_{xx}f/{\partial }_{x}f\right)$, we get over the source:

$\delta \bar{\omega }=\bar{\omega }{\partial }_x\xi +\xi {\partial }_x\bar{\omega }=\mathcal{L}\left(\xi \right)\bar{\omega }$

We may summarize these results as follows:

$\delta \bar{\omega }=\mathcal{L}\left(\xi \right)\bar{\omega }\stackrel{\left(j_1\left(f\right)\right)}{\to }\delta \bar{\omega }={\partial }_{x}f\mathcal{L}\left(\eta \right)\omega$

We invite the reader to extend this result to an arbitrary dimension $n\ge 2$.

EXAMPLE 4.14: The case of an affine stucture needs more work with $n=m=1,q=2$. Indeed, let us consider the action of the affine Lie group of transformations $\bar{y}=ay+b$ with $a,b=cst$ acting on the target $y\in Y$ considered as a copy of the real line X. We obtain the prolongations up to order 2 of the 2 infinitesimal generators of the action:

$a\to y\frac{\partial }{\partial y}+y_x\frac{\partial }{\partial y_x}+y_{xx}\frac{\partial }{\partial y_{xx}}\mathrm{,}b\to \frac{\partial }{\partial y}+0\frac{\partial }{\partial y_x}+0\frac{\partial }{\partial y_{xx}}$

There cannot be any differential invariant of order 1 and the only generating one of order 2 can be $\Phi \equiv y_{xx}/y_x$. When $\bar{x}=\phi \left(x\right)$ we get successively $y_x=y_{\bar{x}}{\partial }_x\phi$, $y_{xx}=y_{\bar{x}\bar{x}}{\left({\partial }_x\phi \right)}^2+y_{\bar{x}}{\partial }_{xx}\phi$ and $\Phi$ transforms like $u={\partial }_x\phi \bar{u}+\frac{{\partial }_{xx}\phi }{{\partial }_x\phi }$ a result providing the bundle of geometric objects $\mathcal{F}$ with local coordinates $\left(x\mathrm{,}u\right)$ and corresponding transition rules. For any section $\gamma$, we get the Vessiot general system ${\mathcal{R}}_2\subset {\Pi }_2$ of second order finite Lie equations $\frac{y_{xx}}{y_x}+\gamma \left(y\right)y_x=\gamma \left(x\right)$ which is already in Lie form and relates the jet coordinates $\left(x\mathrm{,}y\mathrm{,}{y}_x\mathrm{,}{y}_{xx}\right)$ of order 2. The special section is $\gamma =0$ and we may consider the automorphic system $\Phi \equiv \frac{y_{xx}}{y_x}=\bar{\gamma }\left(x\right)$ obtained by introducing any second order section $f_2\left(x\right)=\left(f\left(x\right),f_x\left(x\right),f_{xx}\left(x\right)\right)$, for example $f_2=j_2\left(f\right)$ providing $\left(f\left(x\right)\mathrm{,}{\partial }_{x}f\left(x\right)\mathrm{,}{\partial }_{xx}f\left(x\right)\right)$. It is not at all evident, even on such an elementary example, to compute the variation $\bar{\Gamma }=\delta \bar{\gamma }$ induced by the previous formulas and to prove that, like any field quantity, it only depends on $\bar{\gamma }$ on the condition to use only source quantities. For this, setting $\frac{f_{xx}\left(x\right)}{f_x\left(x\right)}=\bar{\gamma }\left(x\right)$, varying and substituting, we obtain:

$\bar{\Gamma }=\delta \bar{\gamma }=\frac{\delta f_{xx}}{f_x}-\frac{f_{xx}}{{\left(f_x\right)}^2}\delta f_x=f_x{\eta }_{yy}={\xi }_{xx}+\bar{\gamma }{\xi }_x+\xi {\partial }_x\bar{\gamma }$

Now, linearizing the preceding Lie equation over the identity transformation $y=x$, we get the Medolaghi equation:

$L\left({\xi }_2\right)\gamma \equiv {\xi }_{xx}+\gamma \left(x\right){\xi }_x+\xi {\partial }_x\gamma \left(x\right)=0,\forall {\xi }_2\in R_2\subset J_2(\; T\; )$

and the striking formula $\bar{\Gamma }=\delta \bar{\gamma }=L\left({\xi }_2\right)\bar{\gamma }$ over the source for an arbitrary ${\xi }_2\in J_2\left(T\right)$. We finally point out the fact that, as we have just shown above and contrary to what the brothers Cosserat had in mind, the first order operators involved in the nonlinear Spencer sequence have strictly nothing to do with the operators involved in the nonlinear Janet sequence whenever $q\ge 2$. For example, in the present situation, ${\chi }_{,x}=\frac{{\partial }_{x}f}{f_x}-1$ has nothing to do with $\Phi \equiv \frac{f_{xx}}{f_x}$. Similarly, using the comment before example 4.7 in the linear framework, we have the first order Spencer operator $D_1\mathrm{<mo>\; :\; </mo>}\left(\xi \mathrm{,}{\xi }_x\right)\to \left({\partial }_x\xi -{\xi }_x\mathrm{,}{\partial }_x{\xi }_x\right)$ on one side and the second order Lie operator $\mathcal{D}\mathrm{<mo>\; :\; </mo>}\xi \to {\partial }_{xx}\xi$ on the other side.

The next delicate example proves nevertheless that target quantities may also be used.

EXAMPLE 4.15: In the last example, depending on the way we use $\bar{\gamma }\left(x\right)$ on the source or $\gamma \left(y\right)$ on the target, we may consider the two (very different) Medolaghi equations:

${\xi }_{xx}+\bar{\gamma }\left(x\right){\xi }_x+\xi {\partial }_x\bar{\gamma }\left(x\right)=0\leftrightarrow {\eta }_{yy}+\gamma \left(y\right){\eta }_y+\eta {\partial }_y\gamma \left(y\right)=0$

Now, starting from the single OD equation $\frac{f_{xx}}{f_x}=\bar{\gamma }\left(x\right)$ in sectional notations, we may successively differentiate and prolongate once in order to get:

$\frac{{\partial }_x{f}_{xx}}{f_x}-\frac{f_{xx}}{{\left(f_x\right)}^2}{\partial }_x{f}_x={\partial }_x\bar{\gamma }\left(x\right)\leftrightarrow \frac{f_{xxx}}{f_x}-{\left(\frac{f_{xx}}{f_x}\right)}^2={\partial }_x\bar{\gamma }(\; x\; )$

Substracting the second from the first as a way to eliminate $\bar{\gamma }$, we obtain a linear relation involving only the components of the nonlinear Spencer operator in a coherent way with the theory of nonlinear systems, namely:

$\frac{1}{f_x}\left({\partial }_x{f}_{xx}-f_{xxx}\right)-\frac{f_{xx}}{{\left(f_x\right)}^2}\left({\partial }_x{f}_x-f_{xx}\right)=0$

At first sight it does not seem possible to know whether we have a linear combination of the components of ${\chi }_2$ or of the components of ${\sigma }_2$. However, if we come back to the original situation $f_q^{-1}\left(\omega \right)=\bar{\omega }$, we have eliminated $j_1\left(\bar{\gamma }\right)$ over the source and we are thus only left with $j_1\left(\gamma \right)$ over the target. Hence it can only depend on ${\sigma }_2$ and we find indeed the striking relation:

$-\frac{1}{f_x}\left[\frac{1}{f_x}\left({\partial }_x{f}_{xx}-f_{xxx}\right)-\frac{f_{xx}}{{\left(f_x\right)}^2}\left({\partial }_x{f}_x-f_{xx}\right)\right]\frac{1}{{\partial }_{x}f}={\sigma }_{yy,y}=0$

provided by the simple second order Medolaghi equation $\gamma =0\Rightarrow {\eta }_{yy}=0$ over the target. It is essential to notice that no classical technique can provide these results which are essentially depending on the nonlinear Spencer operator and are thus not known today.

EXAMPLE 4.16: The above methods can be applied to any explicit example. The reader may treat as an exercise the case of the pseudogroup of isometries of a non degenerate metric which can be found in any textbook of continuum mechanics or elasticity theory, though in a very different framework with methods only valid for tensors. With the previous notations, let $\omega \in S_2{T}^{\mathrm{<mo>\; *\; </mo>}}$ with $det\left(\omega \right)\ne 0$ and consider the following nonlinear system ${\omega }_{kl}\left(f\left(x\right)\right){\partial }_i{f}^k\left(x\right){\partial }_j{f}^l\left(x\right)={\bar{\omega }}_{ij}\left(x\right)$ with $1\le i\mathrm{,}j\mathrm{,}k\mathrm{,}l\le n$. One obtains therefore:

$\delta {\bar{\omega }}_{ij}={\bar{\omega }}_{rj}{\partial }_i{\xi }^r+{\bar{\omega }}_{ir}{\partial }_i{\xi }^u+{\xi }^r{\partial }_r{\bar{\omega }}_{ij}={\partial }_i{f}^k{\partial }_j{f}^l\left({\omega }_{ul}\frac{\partial {\eta }^u}{\partial y^k}+{\omega }_{ku}\frac{\partial {\eta }^u}{\partial y^l}+{\eta }^u\frac{\partial {\omega }_{kl}}{\partial y^u}\right)$

and thus the same recapitulating formulas linking the source and target variations:

$\delta \bar{\omega }=\mathcal{L}\left(\xi \right)\bar{\omega }\stackrel{\left(j_1\left(f\right)\right)}{\to }\delta \bar{\omega }={\partial }_i{f}^k{\partial }_j{f}^l{\left(\mathcal{L}\left(\eta \right)\omega \right)}_{kl}$

It is also difficult to compute or compare the variational formulas over the source and target in the nonlinear Spencer sequence, even when $m=n=1$ and $q=0,1$ ( [29] ).

EXAMPLE 4.17: Let us prove that the explicit computation of the gauge transformation is at the limit of what can be done with the hand, even when $m=n=1,q=1$. We have successively:

${\chi }_{,x}=\frac{{\partial }_{x}f}{f_x}-1,{\chi }_{x,x}=\frac{1}{f_x}\left({\partial }_x{f}_x-\frac{{\partial }_{x}f}{f_x}{f}_{xx}\right)$

$f^{\prime }\left(x\right)=g\left(f\left(x\right)\right)\Rightarrow {f^{\prime }}_x=g_y{f}_x,{f^{\prime }}_{xx}=g_{yy}{\left(f_x\right)}^2+g_y{f}_{xx}$

and thus:

${{\chi }^{\prime }}_{,x}=\frac{{\partial }_x{f}^{\prime }}{{f^{\prime }}_x}-1=\frac{{\partial }_{y}g{\partial }_{x}f}{g_y{f}_x}-1=\left({\chi }_{,y}+1\right)\frac{{\partial }_{x}f}{f_x}-1=\frac{{\partial }_{x}f}{f_x}{\chi }_{,y}+\left(\frac{{\partial }_{x}f}{f_x}-1\right)$

$\begin{array}{c}{{\chi }^{\prime }}_{x,x}=\frac{1}{{f^{\prime }}_x}\left({\partial }_x{f^{\prime }}_x-\frac{{\partial }_x{f}^{\prime }}{{f^{\prime }}_x}{f^{\prime }}_{xx}\right)\\ =\frac{1}{g_y{f}_x}\left({\partial }_y{g}_y\left({\partial }_{x}f\right)f_x+g_y{\partial }_x{f}_x\right)-\frac{{\partial }_{y}g{\partial }_{x}f}{g_y{f}_x}\left(g_{yy}{\left(f_x\right)}^2+g_y{f}_{xx}\right)\\ =\frac{1}{g_y}{\partial }_y{g}_y{\partial }_{x}f+\frac{{\partial }_x{f}_x}{f_x}-\frac{{\partial }_{y}g{\partial }_{x}f{g}_{yy}}{{\left(g_y\right)}^2}-\frac{{\partial }_{y}g{\partial }_{x}f{f}_{xx}}{g_y{\left(f_x\right)}^2}\\ =\left({\partial }_{x}f{\chi }_{y,y}-\frac{{\partial }_{x}f{f}_{xx}}{{\left(f_x\right)}^2}{\chi }_{,y}\right)+\frac{1}{f_x}\left({\partial }_x{f}_x-\frac{{\partial }_{x}f}{f_x}{f}_{xx}\right)\end{array}$

Setting $f_2=i{d}_2+t{\xi }_2+\cdots$ and passing to the limit when $t\to 0$, we finally obtain:

$\begin{array}{l}\delta {\chi }_{,x}=\left({\partial }_x\xi -{\xi }_x\right)+\left(\xi {\partial }_x{\chi }_{,x}+{\chi }_{,x}{\partial }_x\xi -{\chi }_{,x}{\xi }_x\right)\\ \delta {\chi }_{x,x}=\left({\partial }_x{\xi }_x-{\xi }_{xx}\right)+\left(\xi {\partial }_x{\chi }_{x,x}+{\chi }_{x,x}{\partial }_x\xi -{\chi }_{,x}{\xi }_{xx}\right)\end{array}$

If we use the standard euclidean metric $\omega =1\Rightarrow \gamma =0$, we may thus introduce the pure 1-form $\alpha ={\chi }_{x,x}+\gamma {\chi }_{,x}$. We should consider the defining formula ${{\chi }^{\prime }}_1=f_2^{-1}\circ {\chi }_1\circ j_1\left(f_1\right)+\bar{D}{f}_2$ and have to introduce the additional term $f_2^{-1}\left(\gamma \right){\chi }_{\mathrm{,}x}$ which is only leading to the additional infinitesimal term $\left(L\left({\xi }_2\right)\gamma \right){\chi }_{,x}={\xi }_{xx}{\chi }_{,x}$ because $\gamma =0$. We finally obtain:

$\delta \alpha =\delta {\chi }_{x,x}+{\xi }_{xx}{\chi }_{,x}+\gamma \delta {\chi }_{,x}=\left({\partial }_x{\xi }_x-{\xi }_{xx}\right)+\left(\xi {\partial }_x\alpha +\alpha {\partial }_x\xi \right)$

and this result can be easily extended to an arbitrary dimension with the formula:

${\alpha }_i={\chi }_{r,i}^r+{\gamma }_{sr}^s{\chi }_{,i}^r\Rightarrow {\left(\delta \alpha \right)}_i=\left({\partial }_i{\xi }_r^r-{\xi }_{ri}^r\right)+\left({\xi }^r{\partial }_r{\alpha }_i+{\alpha }_r{\partial }_i{\xi }^r\right)$

Comparing this procedure with the one we have adopted in the previous exampes, we have:

${\chi }_{,x}=\frac{{\partial }_{x}f}{f_x}-1=A-1\Rightarrow \delta {\chi }_{,x}=\frac{{\partial }_x\delta f}{f_x}-\frac{{\partial }_{x}f}{{\left(f_x\right)}^2}\delta f_x=\frac{1}{f_x}\left(\frac{\partial \eta }{\partial y}-{\eta }_y\right){\partial }_{x}f$

However, taking into account the formulas $\eta =\xi {\partial }_{x}f$ and ${\eta }_y={\xi }_x+\frac{{\partial }_x{f}_x}{f_x}\xi$, we also get:

$\begin{array}{c}\delta {\chi }_{,x}=\frac{1}{f_x}\left({\partial }_x\xi {\partial }_{x}f+\xi {\partial }_{xx}f\right)-\frac{{\partial }_{x}f}{{\left(f_x\right)}^2}\left({\xi }_x{f}_x+\xi {\partial }_x{f}_x\right)\\ =A\left({\partial }_x\xi -{\xi }_x\right)+\xi {\partial }_x{\chi }_{,x}\\ =\left({\partial }_x\xi -{\xi }_x\right)+\left(\xi {\partial }_x{\chi }_{,x}+{\chi }_{,x}{\partial }_x\xi -{\chi }_{,x}{\xi }_x\right)\end{array}$

Working over the target is more difficult. Indeed, we have successively ( care to the first step):

${\sigma }_{,y}=\frac{f_x}{{\partial }_{x}f}-1\Rightarrow \delta {\sigma }_{,y}+\eta \frac{\partial {\sigma }_{,y}}{\partial y}=\frac{\delta f_x}{{\partial }_{x}f}-\frac{f_x}{{\left({\partial }_{x}f\right)}^2}{\partial }_x\delta f=-\frac{f_x}{{\left({\partial }_{x}f\right)}^2}\left(\frac{\partial \eta }{\partial y}-{\eta }_y\right)$

$\begin{array}{c}\delta {\sigma }_{\mathrm{,}y}=-\left[\frac{f_x}{{\partial }_{x}f}\left(\frac{\partial \eta }{\partial y}-{\eta }_y\right)+\eta \frac{\partial {\sigma }_{\mathrm{,}y}}{\partial y}\right]\\ =-\left[\left(\frac{\partial \eta }{\partial y}-{\eta }_y\right)+\left(\eta \frac{\partial {\sigma }_{\mathrm{,}y}}{\partial y}+{\sigma }_{\mathrm{,}y}\frac{\partial \eta }{\partial y}-{\sigma }_{\mathrm{,}y}{\eta }_y\right)\right]\end{array}$

More generaly, we let the reader prove that the variation of ${\sigma }_q$ over the target (respectively the source) is described by “minus” the same formula as the variation of ${\chi }_q$ over the source (respectively the target). In any case, the reader must not forget that the word “variation” just means that the section $f_{q+1}$ is changed, not that the source is moved. Accordingly, getting in mind this example and for simplicity, we shall always prefer to work with vertical bundles over the source, closely following the purely mathematical definitions, contrary to Weyl ( [3], §28, formulas (17) to (27), p 233-236). The reader must be now ready for comparing the variations of ${\chi }_{x\mathrm{,}x}$ and ${\sigma }_{y\mathrm{,}y}$.

In order to conclude this section, we provide without any proof two results and refer the reader to ( [7] ) for details.

PROPOSITION 4.18: Changing slightly the notation while setting ${\sigma }_{q-1}={\bar{D}}^{\prime }{\chi }_q$, we have:

${{\chi }^{\prime }}_q=f_{q+1}^{-1}\circ {\chi }_q\circ j_1\left(f\right)+\bar{D}{f}_{q+1}\Rightarrow {{\sigma }^{\prime }}_{q-1}=f_q^{-1}\circ {\sigma }_{q-1}\circ j_1(\; f\; )$

where $f_q^{-1}$ acts on $J_{q-1}\left(T\right)$ and $j_1\left(f\right)$ acts on ${\wedge }^2{T}^{\mathrm{<mo>\; *\; </mo>}}$. It follows that gauge transformations exchange the solutions of ${\bar{D}}^{\prime }$ among themselves.

COROLLARY 4.19: Denoting by $\mathcal{C}\left(\right)$ the cyclic sum, we have the so-called Bianchi identity:

$D{\sigma }_{q-1}\left(\xi \mathrm{,}\eta \mathrm{,}\zeta \right)+\mathcal{C}\left(\xi \mathrm{,}\eta \mathrm{,}\zeta \right)\left\{{\sigma }_{q-1}\left(\xi \mathrm{,}\eta \right)\mathrm{,}{\chi }_{q-1}\left(\zeta \right)\right\}=0$

5. Applications

Before studying in a specific way electromagnetism and gravitation, we shall come back to Example 4.10 and provide a technical result which, though looking like evident at first sight, is at the origin of a deep misunderstanding done by the brothers Cosserat and Weyl on the variational procedure used in the study of physical problems (Compare to [14] ).

Setting $dx=d{x}^1\wedge \cdots \wedge d{x}^n$ for simplicity while using Lemma 4.11 and the fact that the standard Lie derivative is commuting with any diffeomorphism, we obtain at once:

$y=f\left(x\right)\Rightarrow dy=det\left({\partial }_i{f}^k\left(x\right)\right)dx=\Delta \left(x\right)dx$

$\begin{array}{l}\eta =T\left(f\right)\xi \Rightarrow \mathcal{L}\left(\eta \right)dy=\mathcal{L}\left(\xi \right)\left(\Delta \left(x\right)dx\right)\\ \Rightarrow \delta \Delta =\Delta di{v}_y\left(\eta \right)=\Delta di{v}_x\left(\xi \right)+{\xi }^r{\partial }_r\Delta \end{array}$

The interest of such a presentation is to provide the right correspondence between the source/target and the Euler/Lagrange choices. Indeed, if we use the way followed by most authors up to now in continuum mechanics, we should have source = Lagrange, target = Euler, a result leading to the conservation of mass $dm=\rho dy={\rho }_{0}dx=dx$ when ${\rho }_0$ is the original initial mass per unit volume. We may set ${\rho }_0=1$ and obtain therefore $\rho \left(f\left(x\right)\right)=1/\Delta \left(x\right)$, a choice leading to:

$\delta \rho +{\eta }^k\frac{\partial \rho }{\partial y^k}=-\frac{1}{{\Delta }^2}\delta \Delta \Rightarrow \delta \rho =-\rho \frac{\partial {\eta }^k}{\partial y^k}-{\eta }^k\frac{\partial \rho }{\partial y^k}=-\rho \frac{\partial {\xi }^r}{\partial x^r}\Rightarrow \delta \rho =-\frac{\partial \left(\rho {\eta }^k\right)}{\partial y^k}$

but the concept of “variation” is not mathematically well defined, though this result is coherent with the classical formulas that can be found for example in ( [4] [9] ) or ( [3], (17) and (18) p 233, (20) to (21) p 234, (76) p 289, (78) p 290) where “points are moved”.

On the contrary, if we adopt the unusual choice source = Euler, target = Lagrange, we should get $\rho \left(x\right)=\Delta \left(x\right)$, a choice leading to $\delta \rho =\delta \Delta$ and thus:

$\delta \rho =\rho \frac{\partial {\eta }^k}{\partial y^k}=\rho \frac{\partial {\xi }^r}{\partial x^r}+{\xi }^r{\partial }_r\rho ={\partial }_r\left(\rho {\xi }^r\right)$

which is the right choice agreeing, up to the sign, with classical formulas but with the important improvement that this result becomes a purely mathematical one, obtained from a well defined variational procedure involving only the so-called “vertical” machinery. This result fully explains why we had doubts about the sign involved in the variational formulas of ( [4], p. 383) but without being able to correct them at that time. We may finally revisit Lemma 4.11 in order to obtain the fundamental identity over the source:

$\frac{\partial }{\partial x^i}\left(\Delta \left(x\right)\frac{\partial g^i}{\partial y^k}\left(f\left(x\right)\right)\right)\equiv \mathrm{0,}\forall x\in X$

which becomes the conservation of mass when $n=4$ and $k=4$.

In addition, as many chases will be used through many diagrams in the sequel, we invite the reader not familiar with these technical tools to consult the books ( [30] [31] ) that we consider as the best references for learning about homological algebra. A more elementary approach can be found in ( [32] ) that has been used during many intensive courses on the applications of homological algebra to control theory. As for differential homological algebra, one of the most difficult tools existing in mathematics today, and its link with applications, we refer the reader to the various references provided in ( [33] ).

Finally, for the reader interested by a survey on more explicit applications, we particularly refer to ( [2] [34] [35] [36] ) for analytical mechanics and hydrodynamics, ( [5] [37] [38] ) for coupling phenomenas, ( [36] [39] [40] ) for the foundations of Gauge Theory, ( [36] [41] ) for the foundations of General Relativity.

A) POINCARE, WEYL AND CONFORMAL GROUPS

When constructing inductively the Janet and Spencer sequences for an involutive system $R_q\subset J_q\left(E\right)$, we have to use the following commutative and exact diagrams where we have set $F_0=J_q\left(E\right)/R_q$ and used a diagonal chase:

$\boxed{\begin{array}{ccccccc}& 0& & 0& & 0& \\ & \downarrow & & \downarrow & & \downarrow & \\ 0\to & \delta \left({\wedge }^{r-1}{T}^{*}\otimes g_{q+1}\right)& \to & {\wedge }^r{T}^{*}\otimes R_q& \to & C_r& \to 0\\ & \downarrow & & \downarrow & & \downarrow & \\ 0\to & \delta \left({\wedge }^{r-1}{T}^{*}\otimes S_{q+1}{T}^{*}\otimes T\right)& \to & {\wedge }^r{T}^{*}\otimes J_q\left(E\right)& \to & C_r\left(E\right)& \to 0\\ & \downarrow & & \downarrow & \searrow & \downarrow & \\ 0\to & {\wedge }^r{T}^{*}\otimes R_q+\delta \left({\wedge }^{r-1}{T}^{*}\otimes S_{q+1}{T}^{*}\otimes E\right)& \to & {\wedge }^r{T}^{*}\otimes F_0& \to & F_r& \to 0\\ & \downarrow & & \downarrow & & \downarrow & \\ & 0& & 0& & 0& \end{array}}$

It follows that the short exact sequences $0\to C_r\to C_r\left(E\right)\stackrel{{\Phi }_r}{\to }{F}_r\to 0$ are allowing to define the Janet and Spencer bundles inductively. If we consider two involutive systems $0\subset R_q\subset {\hat{R}}_q\subset J_q\left(E\right)$, it follows that the kernels of the induced canonical epimorphisms $F_r\to {\hat{F}}_r\to 0$ are isomorphic to the cokernels of the canonical monomorphisms $0\to C_r\to {\hat{C}}_r\subset C_r\left(E\right)$ and we may say that Janet and Spencer play at see-saw because we have the formula $dim\left(C_r\right)+dim\left(F_r\right)=dim\left(C_r\left(E\right)\right)$.

When dealing with applications, we have set $E=T$ and considered systems of finite type Lie equations determined by Lie groups of transformations. Accordingly, we have obtained in particular $C_r={\wedge }^r{T}^{*}\otimes R_2\subset {\wedge }^r{T}^{*}\otimes {\hat{R}}_2={\hat{C}}_r\subset C_r\left(T\right)$ when comparing the classical and conformal Killing systems, but these bundles have never been used in physics. However, instead of the classical Killing system $R_1\subset J_1\left(T\right)$ defined by the infinitesimal first order PD Lie equations $\Omega \equiv \mathcal{L}\left(\xi \right)\omega =0$ and its first prolongations $R_2\subset J_2\left(T\right)$ defined by the infinitesimal additional second order PD Lie equations $\Gamma \equiv \mathcal{L}\left(\xi \right)\gamma =0$ or the conformal Killing system ${\hat{R}}_2\subset J_2\left(T\right)$ defined by $\Omega \equiv \mathcal{L}\left(\xi \right)\omega =2A\left(x\right)\omega$ and $\Gamma \equiv \mathcal{L}\left(\xi \right)\gamma =\left({\delta }_i^k{A}_j\left(x\right)+{\delta }_j^k{A}_i\left(x\right)-{\omega }_{ij}{\omega }^{ks}{A}_s\left(x\right)\right)\in S_2{T}^{*}\otimes T$ but we may also consider the formal Lie derivatives for geometric objects:

${\Omega }_{ij}\equiv {\left(L\left({\xi }_1\right)\omega \right)}_{ij}\equiv {\omega }_{rj}{\xi }_i^r+{\omega }_{ir}{\xi }_j^r+{\xi }^r{\partial }_r{\omega }_{ij}=0$

${\Gamma }_{ij}^k\equiv {\left(L\left({\xi }_2\right)\gamma \right)}_{ij}^k\equiv {\xi }_{ij}^k+{\gamma }_{rj}^k{\xi }_j^r+{\gamma }_{ir}^k{\xi }_j^r-{\gamma }_{ij}^r{\xi }_k^r+{\xi }^r{\partial }_r{\gamma }_{ij}^k=0$

We may now introduce the intermediate differential system ${\tilde{R}}_2\subset J_2\left(T\right)$ defined by $\mathcal{L}\left(\xi \right)\omega =2A\left(x\right)\omega$ and $\Gamma \equiv \mathcal{L}\left(\xi \right)\gamma =0$, for the Weyl group obtained by adding the only dilatation with infinitesimal generator $x^i{\partial }_i$ to the Poincaré group. We have the relations $R_1\subset {\tilde{R}}_1={\hat{R}}_1$ and the strict inclusions $R_2\subset {\tilde{R}}_2\subset {\hat{R}}_2$ when $R_2={\rho }_1\left(R_1\right)$, ${\tilde{R}}_2={\rho }_1\left({\tilde{R}}_1\right)$, ${\hat{R}}_2={\rho }_1\left({\hat{R}}_1\right)$ but we have to notice that we must have ${\partial }_{i}A-A_i=0$ for the conformal system and thus $A_i=0\Rightarrow A=cst$ if we do want to deal again with an involutive second order system ${\tilde{R}}_2\subset J_2\left(T\right)$. However, we must not forget that the comparison between the Spencer and the Janet sequences can only be done for involutive operators, that is we can indeed use the involutive systems $R_2\subset {\tilde{R}}_2$ but we have to use ${\hat{R}}_3$ even if it is isomorphic to ${\hat{R}}_2$. Finally, as ${\hat{g}}_2\simeq T^{\mathrm{<mo>\; *\; </mo>}}$ and ${\hat{g}}_3=0,\forall n\ge 3$, the first Spencer operator ${\hat{R}}_2\stackrel{D_1}{\to }{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes {\hat{R}}_2$ is induced by the usual Spencer operator ${\hat{R}}_3\stackrel{D}{\to }{T}^{*}\otimes {\hat{R}}_2:\left(0,0,{\xi }_{rj}^r,{\xi }_{rij}^r=0\right)\to \left(0,{\partial }_{i}0-{\xi }_{ri}^r,{\partial }_i{\xi }_{rj}^r-0\right)$ and thus projects by cokernel onto the induced operator $T^{\mathrm{<mo>\; *\; </mo>}}\to T^{\mathrm{<mo>\; *\; </mo>}}\otimes T^{\mathrm{<mo>\; *\; </mo>}}$. Composing with $\delta$, it projects therefore onto $T^{*}\stackrel{d}{\to }{\wedge }^2{T}^{*}:A\to dA=F$ as in EM and so on by using the fact that $D_1$ and d are both involutive or the composite epimorphisms ${\hat{C}}_r\to {\hat{C}}_r/{\tilde{C}}_r\simeq {\wedge }^r{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes \left({\hat{R}}_2/{\tilde{R}}_2\right)\simeq {\wedge }^r{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes {\hat{g}}_2\simeq {\wedge }^r{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes T^{\mathrm{<mo>\; *\; </mo>}}\stackrel{\delta }{\to }{\wedge }^{r+1}{T}^{\mathrm{<mo>\; *\; </mo>}}$. The main result we have obtained is thus to be able to increase the order and dimension of the underlying jet bundles and groups as we have ( [29] ):

$POINCAREGROUP\subset WEYLGROUP\subset CONFORMALGROUP$

that is $10<11<15$ when $n=4$ and our aim is now to prove that the mathematical structures of electromagnetism and gravitation only depend on the second order jets.

With more details, the Killing system $R_2\subset J_2\left(T\right)$ is defined by the infinitesimal Lie equations in Medolaghi form with the well known Levi-Civita isomorphism $\left(\omega \mathrm{,}\gamma \right)\simeq j_1\left(\omega \right)$ for geometric objects:

$\{\begin{array}{l}{\Omega }_{ij}\equiv {\omega }_{rj}{\xi }_i^r+{\omega }_{ir}{\xi }_j^r+{\xi }^r{\partial }_r{\omega }_{ij}=0\\ {\Gamma }_{ij}^k\equiv {\gamma }_{rj}^k{\xi }_i^r+{\gamma }_{ir}^k{\xi }_j^r-{\gamma }_{ij}^r{\xi }_r^k+{\xi }^r{\partial }_r{\gamma }_{ij}^k=0\end{array}$

We notice that $R_2\left(\bar{\omega }\right)=R_2\left(\omega \right)\iff \bar{\omega }=a\omega ,a=cst,\bar{\gamma }=\gamma$ and refer the reader to ( [27] ) for more details about the link between this result and the deformation theory of algebraic structures. We also notice that $R_1$ is formally integrable and thus $R_2$ is involutive if and only if $\omega$ has constant Riemannian curvature along the results of L. P. Eisenhart ( [26] ). The only structure constant c appearing in the corresponding Vessiot structure equations is such that $\bar{c}=c/a$ and the normalizer of $R_1$ is $R_1$ if and only if $c\ne 0$. Otherwise $R_1$ is of codimension 1 in its normalizer ${\tilde{R}}_1$ as we shall see below by adding the only dilatation. In all what follows, $\omega$ is assumed to be flat with $c=0$ and vanishing Weyl tensor.

The Weyl system ${\tilde{R}}_2\subset J_2\left(T\right)$ is defined by the infinitesimal Lie equations:

$\{\begin{array}{l}{\omega }_{rj}{\xi }_i^r+{\omega }_{ir}{\xi }_j^r+{\xi }^r{\partial }_r{\omega }_{ij}=2A\left(x\right){\omega }_{ij}\\ {\xi }_{ij}^k+{\gamma }_{rj}^k{\xi }_i^r+{\gamma }_{ri}^k{\xi }_j^r-{\gamma }_{ij}^r{\xi }_r^k+{\xi }^r{\partial }_r{\gamma }_{ij}^k=0\end{array}$

and is involutive if and only if ${\partial }_{i}A=0\Rightarrow A=cst$. Introducing for convenience the metric density ${\hat{\omega }}_{ij}={\omega }_{ij}/{\left(\left|det\left(\omega \right)\right|\right)}^{\frac{1}{n}}$, we obtain the Medolaghi form for $\left(\hat{\omega }\mathrm{,}\gamma \right)$ with $\left|det\left(\hat{\omega }\right)\right|=1$:

$\{\begin{array}{l}{\hat{\Omega }}_{ij}\equiv {\hat{\omega }}_{rj}{\xi }_i^r+{\hat{\omega }}_{ir}{\xi }_j^r-\frac{2}{n}{\hat{\omega }}_{ij}{\xi }_r^r+{\xi }^r{\partial }_r{\hat{\omega }}_{ij}=0\\ {\Gamma }_{ij}^k\equiv {\xi }_{ij}^k+{\gamma }_{rj}^k{\xi }_i^r+{\gamma }_{ri}^k{\xi }_j^r-{\gamma }_{ij}^r{\xi }_r^k+{\xi }^r{\partial }_r{\gamma }_{ij}^k=0\end{array}$

Finally, the conformal system ${\hat{R}}_2\subset J_2\left(T\right)$ is defined by the following infinitesimal Lie equations:

$\{\begin{array}{l}{\omega }_{rj}{\xi }_i^r+{\omega }_{ir}{\xi }_j^r+{\xi }^r{\partial }_r{\omega }_{ij}=2A\left(x\right){\omega }_{ij}\\ {\xi }_{ij}^k+{\gamma }_{rj}^k{\xi }_i^r+{\gamma }_{ri}^k{\xi }_j^r-{\gamma }_{ij}^r{\xi }_r^k+{\xi }^r{\partial }_r{\gamma }_{ij}^k={\delta }_i^k{A}_j\left(x\right)+{\delta }_j^k{A}_i\left(x\right)-{\omega }_{ij}{\omega }^{kr}{A}_r(\; x\; )\end{array}$

and is involutive if and only if ${\partial }_{i}A-A_i=0$ or, equivalently, if $\omega$ has vanishing Weyl tensor.

However, introducing again the metric density $\hat{\omega }$ while substituting, we obtain after prolongation and division by ${\left(\left|det\left(\omega \right)\right|\right)}^{\frac{1}{n}}$ the second order system ${\hat{R}}_2\subset J_2\left(T\right)$ in Medolaghi form and the Levi-Civita isomorphim $\left(\omega \mathrm{,}\gamma \right)\simeq j_1\left(\omega \right)$ restricts to an isomorphism $\left(\hat{\omega }\mathrm{,}\hat{\gamma }\right)\simeq j_1\left(\hat{\omega }\right)$ if we set:

${\hat{\gamma }}_{ij}^k={\gamma }_{ij}^k-\frac{1}{n}\left({\delta }_i^k{\gamma }_{rj}^r+{\delta }_j^k{\gamma }_{ri}^r-{\omega }_{ij}{\omega }^{ks}{\gamma }_{rs}^r\right)\Rightarrow {\hat{\gamma }}_{ri}^r=0\left(tr\left(\hat{\gamma }\right)=0\right)$

$\{\begin{array}{l}{\hat{\Omega }}_{ij}\equiv {\hat{\omega }}_{rj}{\xi }_i^r+{\hat{\omega }}_{ir}{\xi }_j^r-\frac{2}{n}{\hat{\omega }}_{ij}{\xi }_r^r+{\xi }^r{\partial }_r{\hat{\omega }}_{ij}=0\Rightarrow {\omega }^{ij}{\bar{\Omega }}^{ij}=0\\ {\hat{\Gamma }}_{ij}^k\equiv {\xi }_{ij}^k-\frac{1}{n}\left({\delta }_i^k{\xi }_{rj}^r+{\delta }_j^k{\xi }_{ri}^r-{\hat{\omega }}_{ij}{\hat{\omega }}^{kr}{\xi }_{rs}^s\right)+{\hat{\gamma }}_{rj}^k{\xi }_i^r+{\hat{\gamma }}_{ri}^k{\xi }_j^r-{\hat{\gamma }}_{ij}^r{\xi }_r^k+{\xi }^r{\partial }_r{\hat{\gamma }}_{ij}^k=0\Rightarrow {\hat{\Gamma }}_{ri}^r=0\end{array}$

Contracting the first equations by ${\hat{\omega }}^{ij}$ we notice that ${\xi }_r^r$ is no longer vanishing while, contracting in k and j the second equations, we now notice that ${\xi }_{ri}^r$ is no longer vanishing. It is also essential to notice that the symbols ${\hat{g}}_1$ and ${\hat{g}}_2$ only depend on $\omega$ and not on any conformal factor.

The following Proposition does not seem to be known:

PROPOSITION 5.A.1: $\left(id\mathrm{,}-\hat{\gamma }\right)$ is the only symmetric ${\hat{R}}_1$ -connection with vanishing trace.

Proof: Using a direct substitution, we have to study:

$-{\hat{\omega }}_{ir}{\hat{\gamma }}_{jt}^r-{\hat{\omega }}_{rj}{\hat{\gamma }}_{it}^r+\frac{2}{n}{\hat{\omega }}_{ij}{\hat{\gamma }}_{rt}^r+{\partial }_t{\hat{\omega }}_{ij}$

Multiplying by ${\left(\left|det\left(\omega \right)\right|\right)}^{\frac{1}{n}}$, we have to study:

$-{\omega }_{ir}{\hat{\gamma }}_{jt}^r-{\omega }_{rj}{\hat{\gamma }}_{it}^r+\frac{2}{n}{\omega }_{ij}{\hat{\gamma }}_{rt}^r+{\left(\left|det\left(\omega \right)\right|\right)}^{\frac{1}{n}}{\partial }_t{\hat{\omega }}_{ij}$

or equivalently:

$-{\omega }_{ir}{\hat{\gamma }}_{jt}^r-{\omega }_{rj}{\hat{\gamma }}_{it}^r+\frac{2}{n}{\omega }_{ij}{\hat{\gamma }}_{rt}^r+{\partial }_t{\omega }_{ij}-\frac{1}{n}{\omega }_{ij}{\left(\left|det\left(\omega \right)\right|\right)}^{-1}{\partial }_t\left(\left|det\left(\omega \right)\right|\right)$

that is to say:

$-{\omega }_{ir}{\hat{\gamma }}_{jt}^r-{\omega }_{rj}{\hat{\gamma }}_{it}^r+{\partial }_t{\omega }_{ij}-\frac{2}{n}{\omega }_{ij}{\gamma }_{st}^s$

Now, we have:

$-{\omega }_{ir}\left({\gamma }_{jt}^r-\frac{1}{n}\left({\delta }_j^r{\gamma }_{st}^s+{\delta }_t^r{\gamma }_{sj}^s-{\omega }_{jt}{\omega }^{ru}{\gamma }_{su}^s\right)\right)=-{\omega }_{ir}{\gamma }_{jt}^r+\frac{1}{n}{\omega }_{ij}{\gamma }_{st}^s+\frac{1}{n}{\omega }_{it}{\gamma }_{sj}^s-\frac{1}{n}{\omega }_{jt}{\gamma }_{si}^s$

Finally, taking into account that $\left(id\mathrm{,}-\gamma \right)$ is a $R_1$ -connection, we have:

$-{\omega }_{ir}{\gamma }_{jt}^r-{\omega }_{rj}{\gamma }_{it}^r+{\partial }_t{\omega }_{ij}=0$

Hence, collecting all the remaining terms, we are left with $\frac{2}{n}{\omega }_{ij}{\gamma }_{st}^s-\frac{2}{n}{\omega }_{ij}{\gamma }_{st}^s=0$.

As for the unicity, it comes from a chase in the commutative and exact diagram:

$\begin{array}{ccccccc}& 0& & 0& & 0& \\ & \downarrow & & \downarrow & & \downarrow & \\ 0\to & {\hat{g}}_2& \stackrel{\delta }{\to }& T^{*}\otimes {\hat{g}}_1& \stackrel{\delta }{\to }& {\wedge }^2{T}^{*}\otimes T& \to 0\\ & \downarrow & & \downarrow & & \parallel & \\ 0\to & S_2{T}^{*}\otimes T& \stackrel{\delta }{\to }& T^{*}\otimes T^{*}\otimes T& \stackrel{\delta }{\to }& {\wedge }^2{T}^{*}\otimes T& \to 0\\ & & & & & \downarrow & \\ & & & & & 0& \end{array}$

obtained by counting the respective dimensions with $dim\left({\hat{g}}_1\right)=\left(n\left(n-1\right)/2\right)+1=\left(n^2-n+2\right)/2$ and $dim\left({\hat{g}}_2\right)=n$ while checking that $-n+n$

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

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