Minimum Resolution of the Minkowski, Schwarzschild and Kerr Differential Modules

J.-F. Pommaret

CERMICS, Ecole des Ponts ParisTech, France.

DOI: 10.4236/jmp.2022.134036                

Abstract

Our recent arXiv preprints and published papers on the solution of the Riemann-Lanczos and Weyl-Lanczos problems have brought our attention on the importance of revisiting the algebraic structure of the Bianchi identities in Riemannian geometry. We also discovered in the meantime that, in our first GB book of 1978, we had already used a new way for studying the compatibility conditions (CC) of an operator that may not be necessarily formally integrable (FI) in order to construct canonical formally exact differential sequences on the jet level. The purpose of this paper is to prove that the combination of these two facts clearly shows the specific importance of the Spencer operator and the Spencer δ-cohomology, totally absent from mathematical physics today. The results obtained are unavoidable because they only depend on elementary combinatorics and diagram chasing. They also provide for the first time the purely intrinsic interpretation of the respective numbers of successive first, second, third and higher order generating CC. However, if they of course agree with the linearized Killing operator over the Minkowski metric, they largely disagree with recent publications on the respective numbers of generating CC for the linearized Killing operator over the Schwarzschild and Kerr metrics. Many similar examples are illustrating these new techniques, providing in particular a few resolutions in which the orders of the successive operators may go “up and down” surprisingly, like in the conformal situation for various dimensions.

Keywords

Formal Integrability, Involutivity, Compatibility Conditions, Spencer Operator, Janet Sequence, Spencer Sequence, Differential Module, Homological Algebra, Extension Module

1. Introduction

The present study is mainly local and we only use standard notations of differential geometry. For simplicity, we shall also adopt the same notation for a vector bundle $\left(E\mathrm{,}F\mathrm{,}\cdots \right)$ and its set of sections $\left(\xi \mathrm{,}\eta \mathrm{,}\zeta \mathrm{,}\cdots \right)$. Now, if X is the ground manifold with dimension n and local coordinates $\left(x^1\mathrm{,}\cdots \mathrm{,}{x}^n\right)$ and E is a vector bundle over X with local coordinates $\left(x\mathrm{,}y\right)$, we shall denote by $J_q\left(E\right)$ the q-jet bundle of E with local coordinates $\left(x\mathrm{,}{y}_q\right)$ and sections ${\xi }_q$ transforming like the q-derivatives $j_q\left(\xi \right)$ of a section $\xi ={\xi }_0$ of E. If F with section $\eta$ is another vector bundle over X and $\Phi \mathrm{<mo>\; :\; </mo>}{J}_q\left(E\right)\to F$ is an epimorphism with kernel the linear system $R_q\subset J_q\left(E\right)$, we shall associate the differential operator $\mathcal{D}=\Phi \circ j_q\mathrm{<mo>\; :\; </mo>}E\to F\mathrm{<mo>\; :\; </mo>}\xi \to \eta$ and set $\Theta =ker\left(\mathcal{D}\right)$. All the operators considered will be locally defined over a differential field K with n derivations $\left({\partial }_1\mathrm{,}\cdots \mathrm{,}{\partial }_n\right)$ and we shall indicate the order of an operator under its arrow. It is well known and we shall provide many explicit examples, that, if we want to solve, at least locally the linear inhomogeneous system $\mathcal{D}\xi =\eta$, one usually needs compatibility conditions (CC) of the form ${\mathcal{D}}_1\eta =0$ defined by another differential operator ${\mathcal{D}}_1\mathrm{<mo>\; :\; </mo>}F=F_0\to F_1\mathrm{<mo>\; :\; </mo>}\eta \to \zeta$ that may be of high order in general but still locally defined over K. However, two types of “phenomena” can arise for exhibiting such CC but, though they can be quite critical in actual practice, we do not know any other reference on the possibility to solve them effectively because most people rely on the work of E. Cartan.

1) As shown in ( [1], Introduction) or ( [2] ) with the Janet system $\left({\xi }_{33}-x^2{\xi }_{11}=0,{\xi }_{22}=0\right)$ over the differential field $K=\mathbb{Q}\left(x\right)$ and in ( [3] ), it may be possible to find no CC of order one, no CC of order two, one CC of order three, then nothing new but one additional CC of order six and so on with no way to know when to stop. For the fun, when we started computer algebra around 1990, we had to ask a special permit to the head of our research department for running the computer a full night and were not even able after a day to go any further on. Hence, a first basic problem is to establish a preliminary list of generating CC and know their maximum order.

2) Once the previous problem is solved, we do know a generating ${\mathcal{D}}_1$ of order $q_1$ and may start anew with it in order to obtain a generating ${\mathcal{D}}_2$ of order $q_2$ and so on as a way to work out a differential sequence. Contrary to what can be found in the Poincaré sequence for the exterior derivative where all the successive operators are of order one, things may not be so simple in actual practice and “jumps” may appear, that is the orders may go up and down in a apparently surprising manner that only the use of “acyclicity” through the Spencer cohomology can explain. As we shall see with more details in the case of the conformal Killing operator of order 1, the successive orders are $\left(\mathrm{1,3,1}\right)$ when $n=3$, $\left(\mathrm{1,2,2,1}\right)$ when $n=4$, $\left(\mathrm{1,2,1,2,1}\right)$ when $n=5$ ( [4] ).

A we have shown in our seven books, the only possibility to escape from these two types of problems is to start with an involutive operator $\mathcal{D}$ and construct in an intrinsic way two canonical differential sequences, namely the linear Janet sequence ( [5], p. 185, 391 for a global definition):

$0\to \Theta \to E\underset{q}{\stackrel{\mathcal{D}}{\to }}{F}_0\underset{1}{\stackrel{{\mathcal{D}}_1}{\to }}{F}_1\underset{1}{\stackrel{{\mathcal{D}}_2}{\to }}\cdots \underset{1}{\stackrel{{\mathcal{D}}_{n-1}}{\to }}{F}_{n-1}\underset{1}{\stackrel{{\mathcal{D}}_n}{\to }}{F}_n\to 0$ (1)

and the linear Spencer sequence ( [5], p. 185 for a global definition):

$0\to \Theta \stackrel{j_q}{\to }{C}_0\underset{1}{\stackrel{D_1}{\to }}{C}_1\underset{1}{\stackrel{D_2}{\to }}{C}_2\underset{1}{\stackrel{D_3}{\to }}\cdots \underset{1}{\stackrel{D_{n-1}}{\to }}{C}_{n-1}\underset{1}{\stackrel{D_n}{\to }}{C}_n\to 0$ (2)

As in both cases, the central operator is the Spencer operator but not the exterior derivative, contrary to what is done in ( [6] [7] [8] ) and the corresponding references, in particular ( [6], Ref. [8] ), we do not agree on the effectivity of their definition of “involutivity” ( [6], pp. 1608-1609). In fact, the most important property of theses two sequences is that they are formally exact on the jet level as follows. Introducing the (composite) r- prolongation by means of the formal derivatives $d_i$:

${\rho }_r\left(\Phi \right)\mathrm{<mo>\; :\; </mo>}{J}_{q+r}\left(E\right)\to J_r\left(J_q\left(E\right)\right)\to J_r\left(F_0\right)\mathrm{<mo>\; :\; </mo>}\left(x\mathrm{,}{y}_{q+r}\right)\to \left(x\mathrm{,}{z}_{\nu }=d_{\nu }\Phi \mathrm{,0}\le \left|\nu \right|\le r\right)$

with kernel $R_{q+r}={\rho }_r\left(R_q\right)=J_r\left(R_q\right)\cap J_{q+r}\left(E\right)\subset J_r\left(J_q\left(E\right)\right)$, we have the long exact sequences:

$0\to R_{q+r}\to J_{q+r}\left(E\right)\to J_r\left(F_0\right)$ (3)

$0\to R_{q+q_1+r}\to J_{q+q_1+r}\left(E\right)\to J_{q_1+r}\left(F_0\right)\to J_r\left(F_1\right)$ (4)

$0\to R_{q+q_1+q_2+r}\to J_{q+q_1+q_2+r}\left(E\right)\to J_{q_1+q_2+r}\left(F_0\right)\to J_{q_2+r}\left(F_1\right)\to J_r\left(F_2\right)$ (5)

and so on till the similar ones stopping at $J_r\left(F_n\right)\mathrm{,}\forall r\ge 0$. As shown by the counterexample exhibited in ( [9], p. 119-126), all these sequences may be absolutely useful till the last one. We shall also define the symbol $g_q=R_q\cap S_q{T}^{*}\otimes E$ and its r-prolongations $g_{q+r}={\rho }_r\left(g_q\right)$ only depends on $g_q$ in a purely algebraic way, that is no differentiation is involved. On the contrary, we shall say that $R_q$ or $\mathcal{D}$ is formally integrable (FI) if $R_{q+r}$ is a vector bundle $\forall r\ge 0$ and all the epimorphisms ${\pi }_{q+r}^{q+r+1}\mathrm{<mo>\; :\; </mo>}{J}_{q+r+1}\left(E\right)\to J_{q+r}\left(E\right)\mathrm{<mo>\; :\; </mo>}\left(x\mathrm{,}{y}_{q+r+1}\right)\to \left(x\mathrm{,}{y}_{q+r}\right)$ are inducing epimorphisms $R_{q+r+1}\to R_{q+r}$ of constant rank $\forall r\ge 0$, which is a true purely differential property.

Of course, for people familiar with functional analysis, the definition of $\Theta$ could seem strange and uncomplete as it is not clear where to look for solutions. In our opinion (See [10] and review Zbl 1079.93001) it is mainly for this reason that differential modules or simply D-modules have been introduced but we shall explain why such a procedure leads in fact to a (rather) vicious circle as follows. Working locally for simplicity with $dim\left(E\right)=m$, $dim\left(F\right)=p$, we may turn the definition backwards by introducing the non-commutative ring $D=K\left[d_1\mathrm{,}\cdots \mathrm{,}{d}_n\right]=K\left[d\right]$ of differential polynomials $\left(P\mathrm{,}Q\mathrm{,}\cdots \right)$ with coefficients in K. Then, instead of acting on the “left” of column vectors of sections by differentiations as in the previous differential setting, we shall use the same operator matrix still denoted by $\mathcal{D}$ but now acting on the “right” of row vectors by composition. Introducing the canonical projection onto the residual module M, we obtain the exact sequence $D^p\underset{q}{\stackrel{\mathcal{D}}{\to }}{D}^m\to M\to 0$ of differential modules also called “free resolution” of M because $D^m$ and $D^p$ are clearly free differential modules. However, as D is filtred by the order of operators, then $I=im\left(\mathcal{D}\right)\subset D^m$ is filtred too and, as we shall clearly see on the motivating examples, the induced filtration of $M=D^m/I$ can only been obtained in any applications if and only if $R_q$ or $\mathcal{D}$ is FI. Accordingly, all the difficulty will be to use the following key theorem (For Spencer cohomology and acyclicity or involutivity, see [1] [2] [5] [9] - [14] ):

THEOREM 1.1: There is a finite Prolongation/Projection (PP) algorithm providing two integers $r\mathrm{,}s\ge 0$ by successive increase of each of them such that the new system $R_{q+r}^{\left(s\right)}={\pi }_{q+r}^{q+r+s}\left(R_{q+r+s}\right)$ has the same solutions as $R_q$ but is FI with a 2-acyclic or involutive symbol and first order CC. The order of a generating ${\mathcal{D}}_1$ is thus bounded by $r+s+1$ as we used $r+s$ prolongations.

EXAMPLE 1.2: In the Janet example we have $R_2\to R_3^{\left(1\right)}\to R_4^{\left(2\right)}\to R_5^{\left(2\right)}$ with $8<11<12=12$ and $di{m}_K\left(M\right)=12\Rightarrow r{k}_D\left(M\right)=0$. The final system is trivially involutive because it is FI with a zero symbol, a fact highly not evident a prori because it needs 5 prolongations and the maximum order of the CC is thus equal to $3+2+1=6$. We obtain therefore a minimum resolution of the form $0\to D\underset{4}{\stackrel{{\mathcal{D}}_2}{\to }}{D}^2\underset{6}{\stackrel{{\mathcal{D}}_1}{\to }}{D}^2\underset{2}{\stackrel{\mathcal{D}}{\to }}D\to M\to 0$ (See the introduction of [1] or [2] for details).

When a system is FI, we have a projective limit $R=R_{\infty }\to \cdots \to R_q\to \cdots \to R_1\to R_0$.

As we are dealing with a differential field K, there is a bijective correspondence:

$M_q=ho{m}_K\left(R_q\mathrm{,}K\right)\iff R_q=ho{m}_K\left(M_q\mathrm{,}K\right)$ (6)

and we obtain the injective limit $0\subseteq M_0\subseteq M_1\subseteq \dots \subseteq M_q\subseteq \cdots \subseteq M_{\infty }=M$ providing the filtration of M. We have in particular $d_i{M}_q\subseteq M_{q+1}$ and $M=D{M}_q$ for $q\gg 0$.

THEOREM 1.3: $R=ho{m}_K\left(M\mathrm{,}K\right)$ is a differential module for the Spencer operator.

Proof: As the ring D is generated by K and $T=\left\{a^i{d}_i|a^i\in K\right\}$, we just need to define:

$\begin{array}{l}\left(af\right)\left(m\right)=a\left(f\left(m\right)\right)=f\left(am\right),\\ \left(d_{i}f\right)\left(m\right)={\partial }_i\left(f\left(m\right)\right)-f\left(d_{i}m\right),\forall a\in K\mathrm{,}\forall m\in M\mathrm{,}\forall d_i\in T\mathrm{,}\forall f\in R\end{array}$

and obtain $d_{i}a=a{d}_i+{\partial }_{i}a$ in the operator sense. Choosing $m\in M$ to be the residue of $d_{\mu }{y}^k=y_{\mu }^k$ and setting $f\left(y_q\right)={\xi }_q:f\left(y_{\mu }^k\right)={\xi }_{\mu }^k\in K$, we obtain in actual practice exactly the Spencer operator: $d\mathrm{<mo>\; :\; </mo>}R\to T^{\mathrm{<mo>\; *\; </mo>}}\otimes R\mathrm{<mo>\; :\; </mo>}f\to d{x}^i\otimes d_{i}f$ with ${\left(d_{i}f\right)}_{\mu }^k={\partial }_i{\xi }_{\mu }^k-{\xi }_{\mu +1_i}^k$ or $d{\xi }_{q+1}=j_1\left({\xi }_q\right)-{\xi }_{q+1}$ or simply $d=\partial -\delta$ with a slight abuse of language. We notice that a “section” ${\xi }_q\in R_q$ has in general, particularly for the non-commutative case (See [4] for examples), nothing to do with a “solution”, a concept missing in ( [6] [7] [8] ).

As we shall see in the motivating examples, once a differential module M or the dual system $R=ho{m}_K\left(M\mathrm{,}K\right)$ is given, there may be quite different differential sequences or quite different resolutions and the problem will be to choose the one that could be the best in the application considered. During the last world war, many mathematicians discovered that a few concepts, called extension modules, were not depending on the sequence used in order to compute them but only on M. A (very) delicate theorem of (differential) homological algebra even proves that no others can exist ( [15] ). Let us explain in a way as simple as possible these new concepts.

As a preliminary crucial definition, if $P=a^{\mu }{d}_{\mu }\in D$, we shall define its (formal) adjoint by the formula $ad\left(P\right)={\left(-1\right)}^{\left|\mu \right|}{d}_{\mu }{a}^{\mu }$ where we have set $\left|\mu \right|={\mu }_1+\cdots +{\mu }_n$ whenever $\mu =\left({\mu }_1\mathrm{,}\cdots \mathrm{,}{\mu }_n\right)$ is a multi-index. Such a definition can be extended by linearity in order to define the formal adjoint $ad\left(\mathcal{D}\right)$ to be the transposed operator matrix obtained after taking the adjoint of each element. The main property is that $ad\left(PQ\right)=ad\left(Q\right)ad\left(P\right),\forall P,Q\in D\Rightarrow ad\left({\mathcal{D}}_1\circ \mathcal{D}\right)=ad\left(\mathcal{D}\right)\circ ad\left({\mathcal{D}}_1\right)$.

EXAMPLE 1.4: With ${\partial }_{22}\xi ={\eta }^2,{\partial }_{12}\xi ={\eta }^1$ for $\mathcal{D}$, we get ${\partial }_1{\eta }^2-{\partial }_2{\eta }^1=\zeta$ for ${\mathcal{D}}_1$. Then $ad\left({\mathcal{D}}_1\right)$ is defined by ${\mu }^2=-{\partial }_1\lambda ,{\mu }^1={\partial }_2\lambda$ while $ad\left(\mathcal{D}\right)$ is defined by $\nu ={\partial }_{12}{\mu }^1+{\partial }_{22}{\mu }^2$ but the CC of $ad\left({\mathcal{D}}_1\right)$ are generated by ${\nu }^{\prime }={\partial }_1{\mu }^1+{\partial }_2{\mu }^2$. In the operator framework, we have the differential sequences:

$\begin{array}{rrrrr}\hfill \xi & \hfill \stackrel{\mathcal{D}}{\to }& \hfill \eta & \hfill \stackrel{{\mathcal{D}}_1}{\to }& \hfill \zeta \\ \hfill \nu & \hfill \stackrel{ad\left(\mathcal{D}\right)}{\leftarrow }& \hfill \mu & \hfill \stackrel{ad\left({\mathcal{D}}_1\right)}{\leftarrow }& \hfill \lambda \\ \hfill & \hfill \swarrow & \hfill & \hfill & \hfill \\ \hfill {\nu }^{\prime }& \hfill & \hfill & \hfill & \hfill \end{array}$ (7)

where the upper sequence is formally exact at $\eta$ but the lower sequence is not formally exact at $\mu$.

Passing to the module framework, we obtain the sequences:

$\begin{array}{rrrrrr}\hfill D& \hfill \stackrel{{\mathcal{D}}_1}{\to }& \hfill D^2& \hfill \stackrel{\mathcal{D}}{\to }& \hfill D& \hfill \to M\to 0\\ \hfill D& \hfill \stackrel{ad\left({\mathcal{D}}_1\right)}{\leftarrow }& \hfill D^2& \hfill \stackrel{ad\left(\mathcal{D}\right)}{\leftarrow }& \hfill D& \hfill \end{array}$ (8)

where the lower sequence is not exact at $D^2$. The “extension modules” have been introduced in order to study this kind of “gaps”.

Therefore, it may be important or useful to prove that certain extension modules vanish, that is $ad\left(\mathcal{D}\right)$ generates the CC of $ad\left({\mathcal{D}}_1\right)$ whenever ${\mathcal{D}}_1$ generates the CC of $\mathcal{D}$. Such a problem is even essential for checking controllability in control theory ( [2] ) but we also remind the reader that it is not so easy to exhibit the CC of the Maxwell or Morera parametrizations when $n=3$ and that a direct checking for $n=4$ should be strictly impossible ( [16] ). It has been proved by L. P. Eisenhart in 1926 (Compare to [5] ) that the solution space $\Theta$ of the Killing system has $n\left(n+1\right)/2$ infinitesimal generators $\left\{{\theta }_{\tau }\right\}$ linearly independent over the constants if and only if $\omega$ had constant Riemannian curvature, namely zero in our case. As we have a Lie group of transformations preserving the metric, the three theorems of Sophus Lie assert than $\left[{\theta }_{\rho },{\theta }_{\sigma }\right]=c_{\rho \sigma }^{\tau }{\theta }_{\tau }$ where the structure constants c define a Lie algebra $\mathcal{G}$. We have therefore $\xi \in \Theta \iff \xi ={\lambda }^{\tau }{\theta }_{\tau }$ with ${\lambda }^{\tau }=cst$. Hence, we may replace the Killing system by the system ${\partial }_i{\lambda }^{\tau }=0$, getting therefore the differential sequence:

$0\to \Theta \to {\wedge }^0{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes \mathcal{G}\stackrel{d}{\to }{\wedge }^1{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes \mathcal{G}\stackrel{d}{\to }\cdots \stackrel{d}{\to }{\wedge }^n{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes \mathcal{G}\to 0$ (9)

which is the tensor product of the Poincaré sequence for the exterior derivative by the Lie algebra $\mathcal{G}$. Finally, as the extension modules do not depend on the resolution used and that most of them do vanish because the Poincaré sequence is self adjoint (up to sign), that is $ad\left(d\right)$ generates the CC of $ad\left(d\right)$ at any position, exactly like d generates the CC of d at any position. We invite the reader to compare with the situation of the Maxwell equations in electromagnetism ( [13] ). However, we have proved in ( [17] [18] [19] [20] ) why neither the Janet sequence nor the Poincaré (de Rham in USA!) sequence can be used in physics and must be replaced by another resolution of $\Theta$ called Spencer sequence (See [14] for details and compare to [21] ).

We are now in a position to tell about the unpleasant story that has motivated such a paper. In October 23-27, 2017, I was invited to lecture at the Albert Einstein Institute (AEI, Potsdam/Berlin). Though the series of lectures was already planned and written (arXiv: 1802.09610 published in [18] ), the day before the first lecture the group leader decided that I should lecture on compatibility conditions (CC). I suddenly understood that General Relativity (GR) at AEI was no longer a Science but became a Religion that does not admit any criticism by lecturing visitors, with a similar comment for Gauge Theory (GT) ( [1] [13] [14] ). The following elementary example will explain the title of this paper and the problems raised by its content in such a framework.

With $n=2$, $D=\mathbb{Q}\left(a\right)\left[d_1,d_2\right]$, y an indeterminate and a an arbitrary constant parameter, let us consider the second order system $R_2\subset J_2\left(E\right)$ defined by the PD equations $y_{22}=0$, $y_{12}+a{y}_1=0$. When $a=0$, we have the involutive system $y_{22}=0$, $y_{12}=0$ defined over $D=\mathbb{Q}\left[d_1,d_2\right]$ and the minimum resolution $0\to D\underset{1}{\to }{D}^2\underset{2}{\to }D\to M\to 0$ already considered with one first order CC.

However, if $a\ne 0$, after a few crossed derivatives, we may obtain the successive strict inclusions $R_2^{\left(2\right)}\subset R_2^{\left(1\right)}\subset R_2$ providing first the intermediate subsystem $y_{22}=0$, $y_{12}=0$, $y_1=0$ and then the final involutive subsystem $y_{22}=0$, $y_{12}=0$, $y_{11}=0$, $y_1=0$. Not only the final subsystem is surprisingly no longer depending on the parameter but the corresponding minimum resolution, namely $0\to D\underset{2}{\to }{D}^2\underset{2}{\to }D\to M\to 0$, is quite different as it now involves one second order CC because $\left(d_{12}+a{d}_1\right)d_{22}-d_{22}\left(d_{12}+a{d}_1\right)=0$. One can also consider the linear inhomogeneous second order system $y_{22}=u$, $y_{12}=v-aw$, $y_{11}=w_1$, $y_1=w$ with $a^{2}w=u_1-v_2+av$ and discover that the 4 canonical CC determined by the corresponding Janet tabular are in fact generated by a single second order CC (See [1] [3] and [5] for other more sophisticated explicit examples).

As a kind of training exercise, I solved the PP problem for the linearized Killing operator over the Minkowski and Schwarzschild (S) metrics with parameter (a) in ( [22] ) and over the Kerr (K) metric with parameters $\left(a\mathrm{,}m\right)$ in ( [23] ). As I hope to have convinced the reader that the previous example is exactly similar, my claim in this paper is that the search for CC has nothing to do with GR and is a purely mathematical problem of “Formal Integrability”.

After this long introduction, the content of the paper will become clear:

In Section 2 we provide the mathematical tools from homological algebra and differential geometry needed for finding the generating CC of various orders.

Then, Section 3 will provide motivating examples in order to illustrate these new concepts.

They are finally applied to the Killing systems for the S and K metrics in Section 4 in such a way that the results obtained, though surprising they are, cannot be avoided because they will only depend on diagram chasing and elementary combinatorics. They largely disagree with ( [6] [7] [8] ) because the techniques used in these papers are not intrinsic. As the final involutive systems do not depend any longer on the S or K parameters like in the above example, the worst conclusion concerns the physical usefulness of solving such a problem but… this is surely another story!

2. Mathematical Tools

A) HOMOLOGICAL ALGEBRA

We now need a few definitions and results from homological algebra ( [2] [10] [15] ). In all that follows, $A\mathrm{,}B\mathrm{,}C\mathrm{,}\cdots$ are modules over a ring or vector spaces over a field and the linear maps are making the diagrams commutative. We introduce the notations $rk=rank$, $nb=number$, $dim=dimension$, $ker=kernel$, $im=image$, $coker=cokernel$. When $\Phi \mathrm{<mo>\; :\; </mo>}A\to B$ is a linear map (homomorphism), we may consider the so-called ker/coker exact sequence where $coker\left(\Phi \right)=B/im\left(\Phi \right)$:

$0\to ker\left(\Phi \right)\to A\stackrel{\Phi }{\to }B\to coker\left(\Phi \right)\to 0$

In the case of vector spaces over a field k, we successively have:

$\begin{array}{l}rk\left(\Phi \right)=dim\left(im\left(\Phi \right)\right)\mathrm{,}dim\left(ker\left(\Phi \mathrm{<mo\; stretchy="false">\; )\; </mo>}\right)\right)=dim\left(A\right)-rk\left(\Phi \right)\mathrm{,}\\ dim\left(coker\left(\Phi \right)\right)=dim\left(B\right)-rk(\; \Phi \; )\end{array}$

with $dim\left(coker\left(\Phi \right)\right)=nb\left(compatibilityconditions\right)$. We obtain thus by substraction:

$dim\left(ker\left(\Phi \right)\right)-dim\left(A\right)+dim\left(B\right)-dim\left(coker\left(\Phi \right)\right)=0$

In the case of modules, using localization, we may replace the dimension by the rank and obtain the same relations because of the additive property of the rank. The following result is essential:

SNAKE THEOREM 2.A.1: When one has the following commutative diagram resulting from the two central vertical short exact sequences by exhibiting the three corresponding horizontal ker/coker exact sequences:

$\begin{array}{ccccccccccc}& & 0& & 0& & 0& & & & \\ & & \downarrow & & \downarrow & & \downarrow & & & & \\ 0& \to & K& \to & A& \to & A^{\prime }& \to & Q& \to & 0\\ & & \downarrow & & \downarrow \Phi & & \downarrow {\Phi }^{\prime }& & \downarrow & & \\ 0& \to & L& \to & B& \to & B^{\prime }& \to & R& \to & 0\\ & & \downarrow & & \downarrow \Psi & & \downarrow {\Psi }^{\prime }& & \downarrow & & \\ 0& \to & M& \to & C& \to & C^{\prime }& \to & S& \to & 0\\ & & & & \downarrow & & \downarrow & & \downarrow & & \\ & & & & 0& & 0& & 0& & \end{array}$ (10)

then there exists a connecting map $M\to Q$ both with a long exact sequence:

$0\to K\to L\to M\to Q\to R\to S\to 0.$

Proof: We start constructing the connecting map by using the following succession of elements:

$\begin{array}{ccccccc}& & a& \cdots & a^{\prime }& \to & q\\ & & ⋮& & \downarrow & & ⋮\\ & & b& \to & b^{\prime }& \cdots & 0\\ & & \downarrow & & ⋮& & \\ m& \to & c& \cdots & 0& & \end{array}$

Indeed, starting with $m\in M$, we may identify it with $c\in C$ in the kernel of the next horizontal map. As $\Psi$ is an epimorphism, we may find $b\in B$ such that $c=\Psi \left(b\right)$ and apply the next horizontal map to get $b^{\prime }\in B^{\prime }$ in the kernel of ${\Psi }^{\prime }$ by the commutativity of the lower square. Accordingly, there is a unique $a^{\prime }\in A^{\prime }$ such that $b^{\prime }={\Phi }^{\prime }\left(a^{\prime }\right)$ and we may finally project $a^{\prime }$ to $q\in Q$. The map is well defined because, if we take another lift for c in B, it will differ from b by the image under $\Phi$ of a certain $a\in A$ having zero image in Q by composition. The remaining of the proof is similar and left to the reader as an exercise. The above explicit procedure will not be repeated.

We may now introduce cohomology theory through the following definition:

DEFINITION 2.A.2: If one has any sequence $A\stackrel{\Phi }{\to }B\stackrel{\Psi }{\to }C$, then one may introduce $coboundary=im\left(\Phi \right)\subseteq ker\left(\Psi \right)=cocycle\subseteq B$ and the cohomology at B is the quotient cocycle/coboundary.

THEOREM 2.A.3: The following commutative diagram where the two central vertical sequences are long exact sequences and the horizontal lines are ker/coker exact sequences:

$\begin{array}{ccccccccccccc}& & 0& & 0& & 0& & & & & & \\ & & \downarrow & & \downarrow & & \downarrow & & & & & & \\ 0& \to & K& \to & A& \to & A^{\prime }& \to & Q& \to & 0& & \\ & & \downarrow & & \downarrow \Phi & & \downarrow {\Phi }^{\prime }& & \downarrow & & & & \\ 0& \to & L& \to & B& \to & B^{\prime }& \to & R& \to & 0& & \\ \cdots & \cdots & \downarrow & \cdots & \downarrow \Psi & \cdots & \downarrow {\Psi }^{\prime }& \cdots & \downarrow & \cdots & \cdots & \cdots & cut\\ 0& \to & M& \to & C& \to & C^{\prime }& \to & S& \to & 0& & \\ & & \downarrow & & \downarrow \Omega & & \downarrow {\Omega }^{\prime }& & \downarrow & & & & \\ 0& \to & N& \to & D& \to & D^{\prime }& \to & T& \to & 0& & \\ & & & & \downarrow & & \downarrow & & \downarrow & & & & \\ & & & & 0& & 0& & 0& & & & \end{array}$ (11)

induces an isomorphism between the cohomology at M in the left vertical column and the kernel of the morphism $Q\to R$ in the right vertical column.

Proof: Let us “cut” the preceding diagram into the following two commutative and exact diagrams by taking into account the relations $im\left(\Psi \right)=ker\left(\Omega \right)$, $im\left({\Psi }^{\prime }\right)=ker\left({\Omega }^{\prime }\right)$:

$\begin{array}{ccccccccccc}& & 0& & 0& & 0& & & & \\ & & \downarrow & & \downarrow & & \downarrow & & & & \\ 0& \to & K& \to & A& \to & A^{\prime }& \to & Q& \to & 0\\ & & \downarrow & & \downarrow \Phi & & \downarrow {\Phi }^{\prime }& & \downarrow & & \\ 0& \to & L& \to & B& \to & B^{\prime }& \to & R& \to & 0\\ & & \downarrow & & \downarrow \Psi & & \downarrow {\Psi }^{\prime }& & & & \\ 0& \to & cocycle& \to & im\Psi & \to & im{\Psi }^{\prime }& & & & \\ & & & & \downarrow & & \downarrow & & & & \\ & & & & 0& & 0& & & & \end{array}$

$\begin{array}{ccccccc}& & 0& & 0& & 0\\ & & \downarrow & & \downarrow & & \downarrow \\ 0& \to & cocycle& \to & ker\Omega & \to & ker{\Omega }^{\prime }\\ & & \downarrow & & \downarrow & & \downarrow \\ 0& \to & M& \to & C& \to & C^{\prime }\\ & & \downarrow & & \downarrow \Omega & & \downarrow {\Omega }^{\prime }\\ 0& \to & N& \to & D& \to & D^{\prime }\\ & & & & \downarrow & & \downarrow \\ & & & & 0& & 0\end{array}$

Using the snake theorem, we successively obtain:

$\begin{array}{cccc}\Rightarrow & \exists & 0\to K\to L\stackrel{\Psi }{\to }cocycle\to Q\to R& exact\\ \Rightarrow & \exists & 0\to coboundary\to cocycle\to ker\left(Q\to R\right)\to 0& exact\\ \Rightarrow & & cohomologyatM\simeq ker\left(Q\to R\right)& \end{array}$

B) DIFFERENTIAL GEOMETRY

Comparing the sequences obtained in the previous examples, we may state:

DEFINITION 2.B.1: A differential sequence is said to be formally exact if it is exact on the jet level composition of all the prolongations involved. A formally exact sequence is said to be strictly exact if all the operators/systems involved are FI (See [1] [5] [11] [14] [24] [25] for more details). A strictly exact sequence is called canonical if all the operators/systems are involutive. Forty years ago, we did provide the link existing between the only known canonical sequences, namely the Janet and Spencer sequences ( [5], See in particular the pages 185 and 391).

With canonical projection ${\Phi }_0=\Phi \mathrm{<mo>\; :\; </mo>}{J}_q\left(E\right)\Rightarrow J_q\left(E\right)/R_q=F_0$, the various prolongations are described by the following commutative and exact “introductory diagram” often used in the sequel:

$\boxed{\begin{array}{ccccccccc}& 0& & 0& & 0& & & \\ & \downarrow & & \downarrow & & \downarrow & & & \\ 0\to & g_{q+r+1}& \to & S_{q+r+1}{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes E& \stackrel{{\sigma }_{r+1}\left(\Phi \right)}{\to }& S_{r+1}{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes F_0& \to & h_{r+1}& \to 0\\ & \downarrow & & \downarrow & & \downarrow & & \downarrow & \\ 0\to & R_{q+r+1}& \to & J_{q+r+1}\left(E\right)& \stackrel{{\rho }_{r+1}\left(\Phi \right)}{\to }& J_{r+1}\left(F_0\right)& \to & Q_{r+1}& \to 0\\ & \downarrow & & \downarrow & & \downarrow & & \downarrow & \\ 0\to & R_{q+r}& \to & J_{q+r}\left(E\right)& \stackrel{{\rho }_r\left(\Phi \right)}{\to }& J_r\left(F_0\right)& \to & Q_r& \to 0\\ & & & \downarrow & & \downarrow & & \downarrow & \\ & & & 0& & 0& & 0& \end{array}}$ (12)

Chasing along the diagonal of this diagram while applying the standard “snake” lemma, we obtain the useful “long exact connecting sequence” also often used in the sequel:

$\boxed{0\to g_{q+r+1}\to R_{q+r+1}\to R_{q+r}\to h_{r+1}\to Q_{r+1}\to Q_r\to 0}$ (13)

which is thus connecting in a tricky way FI (lower left) with CC (upper right).

We finally recall the “fundamental diagram I” that we have presented in many books and papers, relating the (upper) canonical Spencer sequence to the (lower) canonical Janet sequence, that only depends on the left commutative square $\mathcal{D}=\Phi \circ j_q$ with $\Phi ={\Phi }_0$ when one has an involutive system $R_q\subseteq J_q\left(E\right)$ over E with $dim\left(X\right)=n$ and $j_q\mathrm{<mo>\; :\; </mo>}E\to J_q\left(E\right)$ is the derivative operator up to order q while the epimorphisms ${\Phi }_1\mathrm{,}\cdots \mathrm{,}{\Phi }_n$ are successively induced by $\Phi$:

$\boxed{\begin{array}{ccccccccccccc}& & & & & 0& & 0& & & & 0& \\ & & & & & \downarrow & & \downarrow & & & & \downarrow & \\ & 0& \to & \Theta & \stackrel{j_q}{\to }& C_0& \stackrel{D_1}{\to }& C_1& \stackrel{D_2}{\to }& \cdots & \stackrel{D_n}{\to }& C_n& \to 0\\ & & & & & \downarrow & & \downarrow & & & & \downarrow & \\ & 0& \to & E& \stackrel{j_q}{\to }& C_0\left(E\right)& \stackrel{D_1}{\to }& C_1\left(E\right)& \stackrel{D_2}{\to }& \cdots & \stackrel{D_n}{\to }& C_n\left(E\right)& \to 0\\ & & & \parallel & & \downarrow {\Phi }_0& & \downarrow {\Phi }_1& & & & \downarrow {\Phi }_n& \\ 0\to & \Theta & \to & E& \stackrel{\mathcal{D}}{\to }& F_0& \stackrel{{\mathcal{D}}_1}{\to }& F_1& \stackrel{{\mathcal{D}}_2}{\to }& \cdots & \stackrel{{\mathcal{D}}_n}{\to }& F_n& \to 0\\ & & & & & \downarrow & & \downarrow & & & & \downarrow & \\ & & & & & 0& & 0& & & & 0& \end{array}}$ (14)

This result will be used in order to compare the M, S and K metrics when $n=4$ but it is important to notice that this whole diagram does not depend any longer on the parameter (m) of S or on the parameters $\left(a\mathrm{,}m\right)$ of K ( [22] [23] ).

PROPOSITION 2.B.2: If $R_q\subset J_q\left(E\right)$ and $R_{q+1}\subset J_{q+1}\left(E\right)$ are two systems of respective orders $q$ and $q+1$, then $R_{q+1}\subset {\rho }_1\left(R_q\right)$ if and only if ${\pi }_q^{q+1}\left(R_{q+1}\right)\subset R_q$ and $d{R}_{q+1}\subset T^{\mathrm{<mo>\; *\; </mo>}}\otimes R_q$.

Proof: First we notice that necessarily we must have ${\pi }_q^{q+1}\left(R_{q+1}\right)\subset R_q$ because, as ${\rho }_1\left(R_q\right)$ may not project onto $R_q$, it is nevertheless defined by (maybe) more equations of strict order q than $R_q$. Now, if ${\xi }_{q+1}\in R_{q+1}\subset J_{q+1}\left(E\right)$ is such that $d{\xi }_{q+1}\in T^{\mathrm{<mo>\; *\; </mo>}}\otimes R_q$, then ${\xi }_q\in R_q\Rightarrow j_1\left({\xi }_q\right)\in J_1\left(R_q\right)$. As $J_1\left(R_q\right)$ is an affine bundle over $R_q$ modelled on $T^{\mathrm{<mo>\; *\; </mo>}}\otimes R_q$ (or simply $T^{\mathrm{<mo>\; *\; </mo>}}\otimes R_q\subset J_1\left(R_q\right)$) and $J_{q+1}\left(E\right)\subset J_1\left(J_q\left(E\right)\right)$, we have thus ${\xi }_{q+1}=j_1\left({\xi }_q\right)-d{\xi }_{q+1}\in J_1\left(R_q\right)\cap J_{q+1}\left(E\right)={\rho }_1\left(R_q\right)$.

The converse way is similar.

The next key idea has been discovered in ( [5] ) as a way to define the so-called Janet bundles and thus for a totally different reason.

DEFINITION 2.B.3: Let us “cut” the preceding introductory diagram by means of a central vertical line and define ${R^{\prime }}_r=im\left({\rho }_r\left(\Phi \right)\right)\subseteq J_r\left(F_0\right)$ with ${R^{\prime }}_0=F_0$. Chasing in this diagram, we notice that ${\pi }_r^{r+1}\mathrm{<mo>\; :\; </mo>}{J}_{r+1}\left(E\right)\to J_r\left(E\right)$ induces an epimorphism ${\pi }_r^{r+1}\mathrm{<mo>\; :\; </mo>}{R^{\prime }}_{r+1}\to {R^{\prime }}_r\mathrm{,}\forall r\ge 0$. However, a chase in this diagram proves that the kernel of this epimorphism is not $im\left({\sigma }_{r+1}\left(\Phi \right)\right)$ unless $R_q$ is FI (care). For this reason, we shall define it to be exactly ${g^{\prime }}_{r+1}$.

THEOREM 2.B.4: ${R^{\prime }}_{r+1}\subseteq {\rho }_1\left({R^{\prime }}_r\right)$ and $dim\left({\rho }_1\left({R^{\prime }}_r\right)\right)-dim\left({R^{\prime }}_{r+1}\right)$ is the number of new generating CC of order $r+1$.

Proof: First of all, we have the following commutative and exact diagram obtained by applying the Spencer operator to the top long exact sequence:

$\begin{array}{ccccccccc}0\to & R_{q+r+1}& \to & J_{q+r+1}\left(E\right)& \to & J_{r+1}\left(F_0\right)& \to & Q_{r+1}& \to 0\\ & \downarrow d& & \downarrow d& & \downarrow d& & \downarrow d& \\ 0\to & T^{\mathrm{<mo>\; *\; </mo>}}\otimes R_{q+r}& \to & T^{\mathrm{<mo>\; *\; </mo>}}\otimes J_{q+r}\left(E\right)& \to & T^{\mathrm{<mo>\; *\; </mo>}}\otimes J_r\left(F_0\right)& \to & T^{\mathrm{<mo>\; *\; </mo>}}\otimes Q_r& \to 0\end{array}$

“Cutting” the diagram in the middle as before while using the last definition, we obtain the induced map ${R^{\prime }}_{r+1}\stackrel{d}{\to }{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes {R^{\prime }}_r$ and the first inclusion follows from the last proposition. Such a procedure cannot be applied to the top row of the introductory diagram through the use of $\delta$ instead of d because of the comment done on the symbol in the last definition.

Now, using only the definition of the prolongation for the system and its symbol, we have the following commutative and exact diagram:

$\begin{array}{ccccccccc}& 0& & 0& & 0& & 0& \\ & \downarrow & & \downarrow & & \downarrow & & \downarrow & \\ 0\to & {\rho }_1\left({g^{\prime }}_r\right)& \to & S_{r+1}{T}^{*}\otimes F_0& \stackrel{{\sigma }_1\left(\Psi \right)}{\to }& T^{*}\otimes Q_r& \to & {Q^{\prime }}_1& \to 0\\ & \downarrow & & \downarrow & & \downarrow & & \parallel & \\ 0\to & {\rho }_1\left({R^{\prime }}_r\right)& \to & J_{r+1}\left(F_0\right)& \stackrel{{\rho }_1\left(\Psi \right)}{\to }& J_1\left(Q_r\right)& \to & {Q^{\prime }}_1& \to 0\\ & \downarrow & & \downarrow & & \downarrow & & \downarrow & \\ 0\to & {R^{\prime }}_r& \to & J_r\left(F_0\right)& \stackrel{\Psi }{\to }& Q_r& \to & 0& \\ & \downarrow & & \downarrow & & \downarrow & & & \\ & 0& & 0& & 0& & & \end{array}$

and obtain the following commutative and exact diagram:

$\begin{array}{ccccccc}& 0& & 0& & 0& \\ & \downarrow & & \downarrow & & \downarrow & \\ 0\to & {g^{\prime }}_{r+1}& \to & {\rho }_1\left({g^{\prime }}_r\right)& \to & A& \to 0\\ & \downarrow & & \downarrow & & \parallel & \\ 0\to & {R^{\prime }}_{r+1}& \to & {\rho }_1\left({R^{\prime }}_r\right)& \to & A& \to 0\\ & \downarrow & & \downarrow & & \downarrow & \\ 0\to & {R^{\prime }}_r& =& {R^{\prime }}_r& \to & 0& \\ & \downarrow & & \downarrow & & & \\ & 0& & 0& & & \end{array}$

The computation of $y=dim\left(A\right)=dim\left({\rho }_1\left({R^{\prime }}_r\right)\right)-dim\left({R^{\prime }}_{r+1}\right)$ only depends on $x=dim\left({Q^{\prime }}_1\right)$ and is rather tricky as follows (See the motivating examples):

$dim\left(Q_r\right)=dim\left(R_{q+r}\right)+dim\left(J_r\left(F_0\right)\right)-dim\left(J_{q+r}(\; E\; )\right)$

$dim\left({\rho }_1\left({R^{\prime }}_r\right)\right)=dim\left(J_{r+1}\left(F_0\right)\right)+x-dim\left(J_1\left(Q_r\right)\right)$

$dim\left({R^{\prime }}_{r+1}\right)=dim\left(J_{q+r+1}\left(E\right)\right)-dim\left(R_{q+r+1}\right)$

As we shall see with the motivating examples and with the S or K metrics, the computation is easier when the system is FI but can be much more difficult when the system is not FI.

However, the number of linearly independent CC of order $r+1$ coming from the CC of order r is $dim\left(J_1\left(Q_r\right)\right)-x$ while the total number of CC of order $r+1$ is:

$\begin{array}{c}dim\left(Q_{r+1}\right)=dim\left(R_{q+r+1}\right)+dim\left(J_{r+1}\left(F_0\right)\right)-dim\left(J_{q+r+1}\left(E\right)\right)\\ =dim\left(J_{r+1}\left(F_0\right)\right)-dim\left({R^{\prime }}_{r+1}\right)\end{array}$

The number of new CC of strict order $r+1$ is equal to y because $dim\left(J_{r+1}\left(F_0\right)\right)$ disappears by difference. For a later use in GR, we point out the fact that, if the given system $R_q\subset J_q\left(E\right)$ depends on parameters that must be contained in the ground differential field K (only (m) for the S metric but $\left(a\mathrm{,}m\right)$ for the K metric), all the dimensions considered may highly depend on them even if the underlying procedure is of course the same.

As an alternative proof, we may say that the number of CC of strict order $r+1$ obtained from the CC of order r is equal to $dim\left(S_{r+1}{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes F_0\right)-dim\left({\rho }_1\left({g^{\prime }}_r\right)\right)$ while the total number of CC of order $r+1$ is equal to $dim\left(S_{r+1}{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes F_0\right)-dim\left({g^{\prime }}_{r+1}\right)$. The number of new CC of strict order $r+1$ is thus also equal to $y=dim\left({\rho }_1\left({g^{\prime }}_r\right)\right)-dim\left({g^{\prime }}_{r+1}\right)$ because $dim\left(S_{r+1}{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes F_0\right)$ also disappears by difference. However, unless $R_q$ is FI, we have in general ${g^{\prime }}_r\ne im\left({\sigma }_r\left(\Phi \right)\right)$ and it thus better to use the systems rather than their symbols.

COROLLARY 2.B.5: The system ${R^{\prime }}_r\subset J_r\left(F_0\right)$ becomes FI with a 2-acyclic or involutive symbol and ${R^{\prime }}_{r+1}={\rho }_1\left({R^{\prime }}_r\right)\subset J_{r+1}\left(F_0\right)$ when r is large enough.

Proof: According to the last diagram, we have ${g^{\prime }}_{r+1}\subseteq {\rho }_1\left({g^{\prime }}_r\right)$ and ${g^{\prime }}_{r+1}$ is thus defined by more linear equations than ${\rho }_1\left({g^{\prime }}_r\right)$. We are facing a purely algebraic problem over commutative polynomial rings and well known noetherian arguments are showing that ${g^{\prime }}_{r+1}={\rho }_1\left({g^{\prime }}_r\right)$ or, equivalently, $y=0$ when r is large enough. Chasing in the last diagram, we obtain therefore ${R^{\prime }}_{r+1}={\rho }_1\left({R^{\prime }}_r\right)$ for r large enough and ${R^{\prime }}_r$ is a vector bundle because because $R_{q+r}$ is a vector bundle. If we denote by $M^{\prime }$ the differential module obtained from the system ${R^{\prime }}_r\subset J_r\left(F_0\right)$ exactly like we have denoted by M the differential module obtained from the system $R_q\subset J_q\left(E\right)$, we have the short exact sequence $0\to M^{\prime }\to D^m\to M\to 0$. Accordingly, $M^{\prime }\simeq I\subset D^m$ is a torsion-free differential module and there cannot exist any specialization as an epimorphism $M^{\prime }\to M^{″}\to 0$ with $r{k}_D\left(M^{\prime }\right)=r{k}_D\left(M^{″}\right)$ because the kernel should be a torsion differential module and thus should vanish. This comment is strengthening the fact that the knowledge of M and thus of I can only be done through Theorem 1.1. Therefore, if $\left(r\mathrm{,}s\right)$ are the ones produced by this theorem, then the order of the CC system must be $r+s+1$. We obtain $3+2+1=6$ for the Janet system with systems ${R^{\prime }}_r$ of successive dimensions 2, 8, 20, 39, 66, 102, 147 and ask the reader to find $dim\left({R^{\prime }}_7\right)=202$ (Hint: [1] ).

We are now ready for working out the generating CC $D_1\mathrm{<mo>\; :\; </mo>}{F}_0\to F_1$ and start afresh in a simpler way because this new operator is FI (Compare to [5], Proposition 2.9, p 173). However, contrary to what the reader could imagine, it is precisely at this point that troubles may start and the best example is the conformal Killing operator. Indeed, it is known that the order of the generating CC for a system of order q which is FI is equal to $s+1$ if the symbol $g_{q+s}$ becomes 2-acyclic before becoming involutive. This fact will be illustrated in a forthcoming motivating example but we recall that the conformal Killing symbol ${\hat{g}}_1\subset T^{\mathrm{<mo>\; *\; </mo>}}\otimes T$ is such that ${\hat{g}}_2$ is 2-acyclic when $n\ge 4$ while ${\hat{g}}_3=0$, a fact explaining why the Weyl operator is of order 2 but the Bianchi-type operator is also of order 2, a result still neither known nor even acknowledged today ( [4] [9] ).

3. Motivating Examples

We now provide three motivating examples in order to illustrate both the usefulness and the limit of the previous procedure.

EXAMPLE 3.1: With $m=1,n=3,K=\mathbb{Q}$, we revisit the nice example of Macaulay ( [26] ) presented in ( [3] ), namely the homogeneous second order linear system $R_2\subset J_2\left(E\right)$ defined by ${\xi }_{33}=0$, ${\xi }_{13}-{\xi }_2=0$ which is far from being formally integrable. We let the reader prove the strict inclusions $R_2^{\left(2\right)}\subset R_2^{\left(1\right)}\subset R_2\subset J_2\left(E\right)$ with successive dimensions $6<7<8<10$. The respective symbols are involutive but only the final system $R_2^{\left(2\right)}$ is involutive. It follows that the generating CC of the operator defined by $R_2$ are at most of order 3 but there is indeed only one single generating second order CC ( [3] ). Elementary combinatorics allows to prove the formulas $dim\left(g_{r+2}\right)=r+4$, $dim\left(R_{r+2}\right)=4r+8$, $\forall r\ge 0$. We have the short exact sequences:

$0\to R_2\to J_2\left(E\right)\to F_0\to \mathrm{0,}0\to 8\to 10\to 2\to 0$

$0\to R_3\to J_3\left(E\right)\to J_1\left(F_0\right)\to \mathrm{0,}0\to 12\to 20\to 8\to 0$

and the following commutative diagram:

$\begin{array}{ccccccccc}& 0& & 0& & 0& & 0& \\ & \downarrow & & \downarrow & & \downarrow & & \downarrow & \\ 0\to & g_5& \to & S_5{T}^{*}\otimes E& \to & \boxed{S_3{T}^{*}\otimes F_0}& \to & T^{*}\otimes F_1& \to 0\\ & \downarrow & & \downarrow & & \downarrow & & \parallel & \\ 0\to & R_5& \to & J_5\left(E\right)& \to & J_3\left(F_0\right)& \to & J_1\left(F_1\right)& \to 0\\ & \downarrow & & \downarrow & & \downarrow & & \downarrow & \\ 0\to & R_4& \to & J_4\left(E\right)& \to & J_2\left(F_0\right)& \to & F_1& \to 0\\ & & & \downarrow & & \downarrow & & \downarrow & \\ & & & 0& & 0& & 0& \end{array}$

$\begin{array}{ccccccccc}& 0& & 0& & 0& & 0& \\ & \downarrow & & \downarrow & & \downarrow & & \downarrow & \\ 0\to & 7& \to & 21& \to & 20& \to & 3& \to 0\\ & \downarrow & & \downarrow & & \downarrow & & \downarrow & \\ 0\to & 20& \to & 56& \to & 40& \to & 4& \to 0\\ & \downarrow & & \downarrow & & \downarrow & & \downarrow & \\ 0\to & 16& \to & 35& \to & 20& \to & 1& \to 0\\ & & & \downarrow & & \downarrow & & \downarrow & \\ & & & 0& & 0& & 0& \end{array}$

First of all, we have ${R^{\prime }}_0=F_0$, ${R^{\prime }}_1=J_1\left(F_0\right)$, $dim\left({R^{\prime }}_2\right)=35-16=19$, $dim\left({R^{\prime }}_3\right)=56-20=36$.

It follows that we have successively:

$dim\left({\rho }_1\left({R^{\prime }}_0\right)\right)-dim\left({R^{\prime }}_1\right)=8-8=0\Rightarrow$ 0 CC of order 1.

$dim\left({\rho }_1\left({R^{\prime }}_1\right)\right)-dim\left({R^{\prime }}_2\right)=20-19=1\Rightarrow$ 1 new CC of order 2.

$dim\left({\rho }_1\left({R^{\prime }}_2\right)\right)-dim\left({R^{\prime }}_3\right)=36-36=0\Rightarrow$ 0 new CC of order 3 and so on with:

$dim\left({R^{\prime }}_{r+3}\right)=dim\left(J_{r+5}\left(E\right)\right)-dim\left(R_{r+5}\right)=\left(r+6\right)\left(r+7\right)\left(r+8\right)/6-\left(4r+20\right)$

$\begin{array}{c}dim\left({\rho }_1\left({R^{\prime }}_{r+2}\right)\right)=dim\left(J_{r+3}\left(F_0\right)\right)-dim\left(J_{r+1}\left(F_1\right)\right)\\ =2\left(r+4\right)\left(r+5\right)\left(r+6\right)/6-\left(r+2\right)\left(r+3\right)\left(r+4\right)/6\end{array}$

and check that $dim\left({R^{\prime }}_{r+3}\right)=dim\left({\rho }_1\left({R^{\prime }}_{r+2}\right)\right)=\left(r+4\right)\left(r^2+17r+54\right)/6$, $\forall r\ge 0$.

Then, counting the dimensions, it is easy to check that the two prolongation sequences are exact on the jet level but that the upper symbol sequence is not exact at $S_3{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes F_0$ with coboundary space of imension $21-7=14$, cocycle space of dimension $20-3=17$ and thus cohomology space of dimension $17-14=3$ that is $dim\left(R_4/R_4^{\left(1\right)}\right)$ as we check that $7-20+16-3=0$. The reader may use the snake theorem to find this result directly through a chase not evident at first sight.

We have then $dim\left(R_{r+2}^{\left(1\right)}\right)=dim\left(R_{r+3}\right)-dim\left(g_{r+3}\right)=3r+7$ and similarly $dim\left(g_{r+2}^{\left(1\right)}\right)=r+3$ leading to $dim\left(R_{r+2}^{\left(2\right)}\right)=dim\left(R_{r+3}^{\left(1\right)}\right)-dim\left(g_{r+3}^{\left(1\right)}\right)=2r+6$ with $dim\left(g_{r+2}^{\left(2\right)}\right)=2$, $\forall r\ge 0$. This result is of course coherent with the fact that the involutive system with the same solutions as $R_2$ is $R_2^{\left(2\right)}$ which is defined by ${\xi }_{33}=0$, ${\xi }_{23}=0$, ${\xi }_{22}=0$, ${\xi }_{13}-{\xi }_2=0$.

EXAMPLE 3.2: With $m=1,n=3,q=2,K=\mathbb{Q}$ and the commutative ring $D=K\left[d_1,d_2,d_3\right]$ of PD operators with coefficients in K, we revisit another example of Macaulay ( [26] ), namely the homogeneous second order formally integrable linear system $R_2\subset J_2\left(E\right)$ defined in operator form by $P\xi \equiv {\xi }_{33}=0$, $Q\xi \equiv {\xi }_{23}-{\xi }_{11}=0$, $R\xi \equiv {\xi }_{22}=0$ and an epimorphism $R_2\to J_1\left(E\right)\to 0$. As for the systems, we have $dim\left(R_2\right)=7$, $dim\left(R_{r+3}\right)=8$, $\forall r\ge 0$. As for the symbols, we have $dim\left(g_2\right)=3$, $dim\left(g_3\right)=1$, $g_{r+4}=0$, $\forall r\ge 0$. This finite type system has the very particular feature that $g_3$ is 2-acyclic but not 3-acyclic (thus involutive) with the short exact δ-sequence:

$3\times 1=3=1\times 3\Rightarrow 0\to {\wedge }^2{T}^{*}\otimes g_3\stackrel{\delta }{\to }{\wedge }^3{T}^{*}\otimes g_2\to 0$

and we have the three linearly independent equations:

$\{\begin{array}{l}{\xi }_{\mathrm{11,123}}={\xi }_{\mathrm{111,23}}+{\xi }_{\mathrm{112,31}}+{\xi }_{\mathrm{113,12}}={\xi }_{\mathrm{111,23}}\\ {\xi }_{\mathrm{12,123}}={\xi }_{\mathrm{112,23}}+{\xi }_{\mathrm{122,31}}+{\xi }_{\mathrm{123,12}}={\xi }_{\mathrm{111,12}}\\ {\xi }_{\mathrm{13,123}}={\xi }_{\mathrm{113,23}}+{\xi }_{\mathrm{123,31}}+{\xi }_{\mathrm{133,12}}={\xi }_{\mathrm{111,31}}\end{array}$

Collecting these results, we get the two following commutative and exact diagrams:

$\begin{array}{ccccccccc}& & & 0& & 0& & 0& \\ & & & \downarrow & & \downarrow & & \downarrow & \\ & 0& \to & S_4{T}^{*}\otimes E& \to & S_2{T}^{*}\otimes F_0& \to & F_1& \to 0\\ & \downarrow & & \downarrow & & \downarrow & & \parallel & \\ 0\to & R_4& \to & J_4\left(E\right)& \to & J_2\left(F_0\right)& \to & F_1& \to 0\\ & \downarrow & & \downarrow & & \downarrow & & \downarrow & \\ 0\to & R_3& \to & J_3\left(E\right)& \to & J_1\left(F_0\right)& \to & 0& \\ & \downarrow & & \downarrow & & \downarrow & & & \\ & 0& & 0& & 0& & & \end{array}$

$\begin{array}{ccccccccc}& & & 0& & 0& & 0& \\ & & & \downarrow & & \downarrow & & \downarrow & \\ & 0& \to & S_5{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes E& \to & S_3{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes F_0& \to & T^{\mathrm{<mo>\; *\; </mo>}}\otimes F_1& \to 0\\ & \downarrow & & \downarrow & & \downarrow & & \downarrow & \\ 0\to & R_5& \to & J_5\left(E\right)& \to & J_3\left(F_0\right)& \to & J_1\left(F_1\right)& \to 0\\ & \downarrow & & \downarrow & & \downarrow & & \downarrow & \\ 0\to & R_4& \to & J_4\left(E\right)& \to & J_2\left(F_0\right)& \to & F_1& \to 0\\ & \downarrow & & \downarrow & & \downarrow & & \downarrow & \\ & 0& & 0& & 0& & 0& \end{array}$

We obtain from these diagrams ${R^{\prime }}_1=J_1\left(F_0\right)\Rightarrow {g^{\prime }}_1=T^{*}\otimes F_0$, ${\rho }_1\left({R^{\prime }}_1\right)=J_2\left(F_0\right)\Rightarrow {R^{\prime }}_2\subset {\rho }_1\left({R^{\prime }}_1\right)$ with a strict inclusion because $27<30$ and we have at least $30-27=3$ generating second order CC. However, from the second diagram, we obtain $dim\left({\rho }_1\left({R^{\prime }}_2\right)\right)=60-12=48=56-8=dim\left({R^{\prime }}_3\right)$ and thus ${R^{\prime }}_3={\rho }_1\left({R^{\prime }}_2\right)$, a result showing that there are no new generating CC of order 3.

As $dim\left(E\right)=1$, we have $S_q{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes E\simeq S_q{T}^{\mathrm{<mo>\; *\; </mo>}}$ and the commutative diagram of δ-sequences:

$\begin{array}{ccccccccc}& 0& & 0& & 0& & & \\ & \downarrow & & \downarrow & & \downarrow & & & \\ 0\to & S_6{T}^{*}& \to & T^{*}\otimes S_5{T}^{*}& \to & {\wedge }^2{T}^{*}\otimes S_4{T}^{*}& \to & {\wedge }^3{T}^{*}\otimes S_3{T}^{*}& \to 0\\ & \parallel & & \parallel & & \parallel & & \downarrow & \\ 0\to & {g^{\prime }}_4& \to & T^{*}\otimes {g^{\prime }}_3& \to & {\wedge }^2{T}^{*}\otimes {g^{\prime }}_2& \to & {\wedge }^3{T}^{*}\otimes {g^{\prime }}_1& \to 0\\ & \downarrow & & \downarrow & & \downarrow & & \downarrow & \\ & 0& & 0& & 0& & 0& \end{array}$

Using the fact that the upper sequence is known to be exact and $dim\left({g^{\prime }}_1\right)=9<10=dim\left(S_3{T}^{\mathrm{<mo>\; *\; </mo>}}\right)$, an easy chase proves that the lower sequence cannot be exact and thus ${g^{\prime }}_2$ cannot be 2-acyclic.

The generating CC of ${\mathcal{D}}_1$ is thus a second order operator ${\mathcal{D}}_2\mathrm{<mo>\; :\; </mo>}{F}_1\to F_2$ where $F_2$ is defined by the long exact prolongation sequence:

$0\to R_6\to J_6\left(E\right)\to J_4\left(F_0\right)\to J_2\left(F_1\right)\to F_2\to 0$

or by the long exact symbol sequence (by chance if one refers to the previous example!):

$0\to S_6{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes E\to S_4{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes F_0\to S_2{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes F_1\to F_2\to 0$

showing that $dim\left(F_2\right)=dim\left(S_6{T}^{\mathrm{<mo>\; *\; </mo>}}\right)-3dim\left(S_4{T}^{\mathrm{<mo>\; *\; </mo>}}\right)+3dim\left(S_2{T}^{\mathrm{<mo>\; *\; </mo>}}\right)=28-45+18=1$ in a coherent way with ( [9] [14] ).

We have thus obtained the following formally exact differential sequence which is nevertheless not a Janet sequence because $R_2$ is FI but not involutive as $g_2$ is finite type with $g_4=0$:

$0\to \Theta \to E\underset{2}{\stackrel{\mathcal{D}}{\to }}{F}_0\underset{2}{\stackrel{{\mathcal{D}}_1}{\to }}{F}_1\underset{2}{\stackrel{{\mathcal{D}}_2}{\to }}{F}_2\to 0$

$0\to \Theta \to 1\to 3\underset{2}{\to }3\to 1\to 0$

$0\to D\to D^3\to D^3\to D\stackrel{p}{\to }M\to 0$

Surprisingly, the situation is even quite worst if we start with $R_3\subset J_3\left(E\right)$ which has nevertheless a 2-acyclic symbol $g_3$ which is not 3-acyclic (thus involutive because $n=3$). Indeed, we know from the second section or by repeating the previous procedure for this new third order operator $\mathcal{D}$ that the generating CC are described by a first order operator ${\mathcal{D}}_1$. However, the symbol of this operator is only 1-acyclic but not 2-acyclic (exercise). Hence, one can prove that the corresponding CC are described by a new second order operator ${\mathcal{D}}_2$ which is involutive… by chance, giving rise to a Janet sequence with first order operators as follows ${\mathcal{D}}_3\mathrm{,}{\mathcal{D}}_4\mathrm{,}{\mathcal{D}}_5$ ( [9], p 119-125):

$0\to \Theta \to 1\underset{3}{\stackrel{\mathcal{D}}{\to }}12\underset{1}{\stackrel{{\mathcal{D}}_1}{\to }}21\underset{2}{\stackrel{{\mathcal{D}}_2}{\to }}46\underset{1}{\stackrel{{\mathcal{D}}_3}{\to }}72\underset{1}{\stackrel{{\mathcal{D}}_4}{\to }}48\underset{1}{\stackrel{{\mathcal{D}}_5}{\to }}12\to 0$

One could also finally use the involutive system $R_4\subset J_4\left(E\right)$ in order to construct the canonical Janet sequence and consider the first order involutive system $R_5\subset J_1\left(R_4\right)$ in order to obtain the canonical Spencer sequence with $C_r={\wedge }^r{T}^{*}\otimes R_4$ and dimensions $\left(\mathrm{8,24,24,8}\right)$:

$0\to \Theta \stackrel{j_4}{\to }{C}_0\underset{1}{\stackrel{D_1}{\to }}{C}_1\underset{1}{\stackrel{D_2}{\to }}{C}_2\underset{1}{\stackrel{D_3}{\to }}{C}_3\to 0$

To recapitulate, this example clearly proves that the differential sequences obtained largely depend on whether we use $R_2\mathrm{,}{R}_3$ or $R_4$ but also whether we look for a sequence of Janet or Spencer type.

We invite the reader to treat similarly the example ${\xi }_{33}-{\xi }_{11}=0$, ${\xi }_{23}=0$, ${\xi }_{22}-{\xi }_{11}=0$.

EXAMPLE 3.3: In our opinion, the best striking use of acyclicity is the construction of differential sequences for the Killing and conformal Killing operators which are both defined over the ground differential field $K=\mathbb{Q}$ for the Minkowski metric in dimension 4 or the Euclidean metric in dimension 5. We have indeed ( [9] [20] ):

$0\to \Theta \to 4\underset{1}{\stackrel{\mathcal{D}}{\to }}10\underset{2}{\stackrel{{\mathcal{D}}_1}{\to }}20\underset{1}{\stackrel{{\mathcal{D}}_2}{\to }}20\underset{1}{\stackrel{{\mathcal{D}}_3}{\to }}6\to 0$

with $E=T,F_0=S_2{T}^{*}$ and, successively, the Killing, Riemann and Bianchi operators acting on the left of column vectors. The differential module counterpart over $D=K\left[d\right]$ is the resolution of the differential Killing module M:

$0\to D^6\underset{1}{\stackrel{{\mathcal{D}}_3}{\to }}{D}^{20}\underset{1}{\stackrel{{\mathcal{D}}_2}{\to }}{D}^{20}\underset{2}{\stackrel{{\mathcal{D}}_1}{\to }}{D}^{10}\underset{1}{\stackrel{\mathcal{D}}{\to }}{D}^4\stackrel{p}{\to }M\to 0$

with the same operators as before but acting now on the right of row vectors by composition.

The conformal situation for $n=4$ is quite unexpected with a second order Bianchi-type operator:

$0\to \Theta \to 4\underset{1}{\stackrel{\mathcal{D}}{\to }}9\underset{2}{\stackrel{{\mathcal{D}}_1}{\to }}10\underset{2}{\stackrel{{\mathcal{D}}_2}{\to }}9\underset{1}{\stackrel{{\mathcal{D}}_3}{\to }}4\to 0$

$0\to D^4\underset{1}{\stackrel{{\mathcal{D}}_3}{\to }}{D}^9\underset{2}{\stackrel{{\mathcal{D}}_2}{\to }}{D}^{10}\underset{2}{\stackrel{{\mathcal{D}}_1}{\to }}{D}^9\underset{1}{\stackrel{\mathcal{D}}{\to }}{D}^4\stackrel{p}{\to }M\to 0$

The conformal situation for $n=5$ is even quite different with the conformal differential sequence:

$0\to \Theta \to 5\underset{1}{\stackrel{\mathcal{D}}{\to }}14\underset{2}{\stackrel{{\mathcal{D}}_1}{\to }}35\underset{1}{\stackrel{{\mathcal{D}}_2}{\to }}35\underset{2}{\stackrel{{\mathcal{D}}_3}{\to }}14\underset{1}{\stackrel{{\mathcal{D}}_4}{\to }}5\to 0$

Though these results and “jumps” highly depend on acyclicity, in particular the fact that the conformal symbol ${\hat{g}}_2$ is 2-acyclic for $n=4$ but 3-acyclic for $n\ge 5$, and have been confirmed by computer algebra, they are still neither known nor acknowledged ( [4] [9] ).

4. Applications

Considering the classical Killing operator $\mathcal{D}:\xi \to \mathcal{L}\left(\xi \right)\omega =\Omega \in S_2{T}^{*}=F_0$ where $\mathcal{L}\left(\xi \right)$ is the Lie derivative with respect to $\xi$ and $\omega \in S_2{T}^{\mathrm{<mo>\; *\; </mo>}}$ is a nondegenerate metric with $det\left(\omega \right)\ne 0$. Accordingly, it is a lie operator with $\mathcal{D}\xi =0,\mathcal{D}\eta =0\Rightarrow \mathcal{D}\left[\xi ,\eta \right]=0$ and we denote simply by $\Theta \subset T$ the set of solutions with $\left[\Theta \mathrm{,}\Theta \right]\subset \Theta$. Now, as we have explained many times, the main problem is to describe the CC of $\mathcal{D}\xi =\Omega \in F_0$ in the form ${\mathcal{D}}_1\Omega =0$ by introducing the so-called Riemann operator ${\mathcal{D}}_1\mathrm{<mo>\; :\; </mo>}{F}_0\to F_1$. We advise the reader to follow closely the next lines and to imagine why it will not be possible to repeat them for studying the conformal Killing operator. Introducing the well known Levi-Civita isomorphism $j_1\left(\omega \right)=\left(\omega \mathrm{,}{\partial }_x\omega \right)\simeq \left(\omega \mathrm{,}\gamma \right)$ by defining the Christoffel symbols ${\gamma }_{ij}^k=\frac{1}{2}{\omega }^{kr}\left({\partial }_i{\omega }_{rj}+{\partial }_j{\omega }_{ir}-{\partial }_r{\omega }_{ij}\right)$ where $\left({\omega }^{rs}\right)$ is the inverse matrix of $\left({\omega }_{ij}\right)$ and the formal Lie derivative, we get the second order system $R_2\subset J_2\left(T\right)$:

$\{\begin{array}{l}{\Omega }_{ij}\equiv {\left(L\left({\xi }_1\right)\omega \right)}_{ij}={\omega }_{rj}\left(x\right){\xi }_i^r+{\omega }_{ir}\left(x\right){\xi }_j^r+{\xi }^r{\partial }_r{\omega }_{ij}\left(x\right)=0\\ {\Gamma }_{ij}^k\equiv {\left(L\left({\xi }_2\right)\gamma \right)}_{ij}^k={\xi }_{ij}^k+{\gamma }_{rj}^k\left(x\right){\xi }_i^r+{\gamma }_{ir}^k\left(x\right){\xi }_j^r+{\gamma }_{ir}^k\left(x\right){\xi }_j^r-{\gamma }_{ij}^r\left(x\right){\xi }_r^k+{\xi }^r{\partial }_r{\gamma }_{ij}^k\left(x\right)=0\end{array}$

with sections ${\xi }_2\mathrm{<mo>\; :\; </mo>}x\to \left({\xi }^k\left(x\right)\mathrm{,}{\xi }_i^k\left(x\right)\mathrm{,}{\xi }_{ij}^k\left(x\right)\right)$ transforming like $j_2\left(\xi \right)\mathrm{<mo>\; :\; </mo>}x\to \left({\xi }^k\left(x\right)\mathrm{,}{\partial }_i{\xi }^k\left(x\right)\mathrm{,}{\partial }_{ij}{\xi }^k\left(x\right)\right)$. The system $R_1\subset J_1\left(T\right)$ has a symbol $g_1\simeq {\wedge }^2{T}^{\mathrm{<mo>\; *\; </mo>}}\subset T^{\mathrm{<mo>\; *\; </mo>}}\otimes T$ depending only on $\omega$ with $dim\left(g_1\right)=n\left(n-1\right)/2$ and is finite type because its first prolongation is $g_2=0$. It cannot be thus involutive and we need to use one additional prolongation. Indeed, using one of the main results to be found in ( [1] [5] [9] [10] [14] ), we know that, when $R_1$ is FI, then the CC of $\mathcal{D}$ are of order $s+1$ where s is the number of prolongations needed in order to get a 2-acyclic symbol, that is $s=1$ in the present situation, a result that should lead to CC of order 2 if $R_1$ were FI. However, it is known that $R_2$ is FI, thus involutive, if and only if $\omega$ has constant Riemannian curvature, a result first found by L.P. Eisenhart in 1926 which is only a particular example of the Vessiot structure equations discovered b E. Vessiot in 1903 ( [27] ), though in a quite different setting (See [1] [5] [9] [14] for an explicit modern proof and compare to the references ( [22] [23] ) of ( [6] ).

We may introduce the (formal) linearization $\Gamma \in S_2{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes T$ of the Christoffel symbols by linearizing the relations ${\omega }_{kr}{\gamma }_{ij}^k=\frac{1}{2}\left({\partial }_i{\omega }_{rj}+{\partial }_j{\omega }_{ir}-{\partial }_r{\omega }_{ij}\right)$ in such a way that $\Omega =0\Rightarrow \Gamma =0$ with:

${\omega }_{kr}{\Gamma }_{ij}^k=\frac{1}{2}\left(d_i{\Omega }_{rj}+d_j{\Omega }_{ir}-d_r{\Omega }_{ij}\right)-{\gamma }_{ij}^k{\Omega }_{kr}$

We may also introduce the Riemann tensor ${\rho }_{l\mathrm{,}ij}^k$ and its (formal) linearization:

$R_{l\mathrm{,}ij}^k\equiv {\left(L\left({\xi }_1\right)\rho \right)}_{l\mathrm{,}ij}^k=-{\rho }_{l\mathrm{,}ij}^s{\xi }_s^k+{\rho }_{s\mathrm{,}ij}^k{\xi }_l^s+{\rho }_{l\mathrm{,}sj}^k{\xi }_i^s+{\rho }_{l\mathrm{,}is}^k{\xi }_j^s+{\xi }^r{\partial }_r{\rho }_{l\mathrm{,}ij}^k$

in order to obtain the Ricci tensor ${\rho }_{ij}={\rho }_{i,rj}^r={\rho }_{ji}$ and its linearization:

$R_{ij}=R_{i,rj}^r={\rho }_{rj}{\xi }_i^r+{\rho }_{ir}{\xi }_j^r+{\xi }^r{\partial }_r{\rho }_{ij}=R_{ji}$

allowing to introduce the Einstein tensor ${ϵ}_{ij}={\rho }_{ij}-\frac{1}{2}{\omega }_{ij}{\omega }^{rs}{\rho }_{rs}={\rho }_{ij}-\frac{1}{2}{\omega }_{ij}\rho$ with linearization:

$E_{ij}\equiv {\left(L\left({\xi }_1\right)ϵ\right)}_{ij}=R_{ij}-\frac{1}{2}{\omega }_{ij}{\omega }^{rs}{R}_{rs}-\frac{1}{2}\rho {\Omega }_{ij}+\frac{1}{2}{\omega }_{ij}{\omega }^{ru}{\omega }^{sv}{\rho }_{rs}{\Omega }_{uv}$

and we must notice (care) that the linearization of $\rho ={\omega }^{rs}{\rho }_{rs}$ is $R={\omega }^{rs}{R}_{rs}-{\rho }_{rs}{\omega }^{ru}{\omega }^{sv}{\Omega }_{uv}$.

These formulas become particularly simple when $\omega$ is a solution of Einstein equations in vacuum, that is when ${\epsilon }_{ij}=0\iff {\rho }_{ij}=0\Rightarrow \rho =0$.

LEMMA 4.1: When $n=4$ and the fixed euclidean metric for simplicity, we have the useful formula:

$E_{00}=-\left(R_{\mathrm{12,12}}+R_{\mathrm{13,13}}+R_{\mathrm{23,23}}\right)$

Proof: We have $2E_{00}=2R_{00}-\left(R_{00}+R_{11}+R_{22}+R_{33}\right)=R_{00}-\left(R_{11}+R_{22}+R_{33}\right)$

and $R_{00}=\left(R_{\mathrm{10,10}}+R_{\mathrm{20,20}}+R_{\mathrm{30,30}}\right)$. However, we have also:

$\begin{array}{l}{R}_{11}=R_{\mathrm{01,01}}+R_{\mathrm{21,21}}+R_{\mathrm{31,31}}\\ R_{22}=R_{\mathrm{02,02}}+R_{\mathrm{12,12}}+R_{\mathrm{32,32}}\\ R_{33}=R_{\mathrm{03,03}}+R_{\mathrm{13,13}}+R_{\mathrm{23,23}}\end{array}$

Summing, we obtain $\left(R_{11}+R_{22}+R_{33}\right)=\left(R_{\mathrm{01,01}}+R_{\mathrm{02,02}}+R_{\mathrm{03,03}}\right)+2\left(R_{\mathrm{12,12}}+R_{\mathrm{13,13}}+R_{\mathrm{23,23}}\right)$. It follows that $E_{00}=-\left(R_{\mathrm{12,12}}+R_{\mathrm{13,13}}+R_{\mathrm{23,23}}\right)$ and the three other $E_{ii}$ are obtained by circular permutations of $\left(\mathrm{0,1,2,3}\right)$. We let the reader treat the general situation as an exercise.

A) MINKOWSKI METRIC:

We have considered this situation in many books or papers and refer the reader to our arXiv page or to the recent references ( [22] [28] ). All the operators are first order between the vector bundles $E=T$, $F_0=T^{\mathrm{<mo>\; *\; </mo>}}\otimes T/g_1\simeq S_2{T}^{\mathrm{<mo>\; *\; </mo>}}$, $F_1=H^2\left(g_1\right)$, $F_2=H^3\left(g_1\right)$, $F_3=H^4\left(g_1\right)$ that are only depending on $g_1$ with dimensions 4, 10, 20, 20, 6 when $n=4$ and Euler-Poincaré characteristic $r{k}_D\left(M\right)=4-10+20-20+6=0$. The case of an arbitrary n, provided in ( [20] ), depends on various chases in commutative diagrams that will be exhibited later on for comparing the respective dimensions. This is not a Janet sequence because $R_1$ is FI but $g_1$ is not involutive.

B) SCHWARZSCHILD METRIC:

With the standard Boyer-Lindquist local coordinates $\left(x^0=t,x^1=r,x^2=\theta ,x^3=\varphi \right)$ and a constant parameter m, we may introduce the field of constants $k=\mathbb{Q}\left(m\right)$ and all the systems or differential modules considered in the sequel will be defined over the ground differential field $K=k\left(t,r,\sin \left(\theta \right),\cos \left(\theta \right)\right)$ with differential structure obtained by setting ${\partial }_2\sin \left(\theta \right)=\cos \left(\theta \right)$, ${\partial }_2\cos \left(\theta \right)=-\sin \left(\theta \right)$ together with ${\sin }^2\left(\theta \right)+{\cos }^2\left(\theta \right)=1$ instead of using the so-called “rational coordinates” ( [23] ). With speed of light $c=1$ and $A=1-\frac{m}{r}$, we shall introduce the diagonal Schwarzschild metric $\omega =\left(A\left(r\right),-1/A\left(r\right),-r^2,-r^2{\sin }^2\left(\theta \right)\right)$ with $det\left(\omega \right)=-r^4{\sin }^2\left(\theta \right)$. Following closely the motivating examples already presented, our challenge is to prove that the purely mathematical formal study of the corresponding Killing system $R_1\subset J_1\left(T\right)$ can be achieved as a simple exercise of formal integrability, with no extra physical technical tool, contrary to ( [6] [7] [8] ). As the computations will be explicitly done, the numbers of CC obtained will bring serious doubts about the validity of the results obtained in the above references, later confirmed with the K metric. First of all we obtain easily the following 10 first order Killing equations $mod\left(\Omega \right)$:

$R_1\subset J_1\left(T\right)\{\begin{array}{lll}{\Omega }_{33}\hfill & \to \hfill & \boxed{{\xi }_3^3}+\frac{1}{r}{\xi }^1+\cot \left(\theta \right){\xi }^2=0\hfill \\ {\Omega }_{23}\hfill & \to \hfill & \boxed{{\xi }_3^2}+{\sin }^2\left(\theta \right){\xi }_2^3=0\hfill \\ {\Omega }_{13}\hfill & \to \hfill & \boxed{{\xi }_3^1}+A{r}^2{\sin }^2\left(\theta \right){\xi }_1^3=0\hfill \\ {\Omega }_{03}\hfill & \to \hfill & \boxed{{\xi }_3^0}-\frac{r^2}{A}{\sin }^2\left(\theta \right){\xi }_0^3=0\hfill \\ {\Omega }_{22}\hfill & \to \hfill & \boxed{{\xi }_2^2}+\frac{1}{r}{\xi }^1=0\hfill \\ {\Omega }_{12}\hfill & \to \hfill & \boxed{{\xi }_2^1}+A{r}^2{\xi }_1^2=0\hfill \\ {\Omega }_{02}\hfill & \to \hfill & \boxed{{\xi }_2^0}-\frac{r^2}{A}{\xi }_0^2=0\hfill \\ {\Omega }_{11}\hfill & \to \hfill & \boxed{{\xi }_1^1}-\frac{m}{2A{r}^2}{\xi }^1=0\hfill \\ {\Omega }_{01}\hfill & \to \hfill & \boxed{{\xi }_0^1}-A^2{\xi }_1^0=0\hfill \\ {\Omega }_{00}\hfill & \to \hfill & \boxed{{\xi }_0^0}+\frac{m}{2A{r}^2}{\xi }^1=0\hfill \end{array}$

where we have framed the leading jets.

This is a finite type system because we get ${\Gamma }_{ij}^k\equiv {\xi }_{ij}^k+\cdots =0$ with only one prolongation!

The only 9 non-zero Christoffel symbols on 40 are:

$\boxed{\begin{array}{lll}{\gamma }_{00}^1=\frac{mA}{2r^2},\hfill & {\gamma }_{01}^0=\frac{m}{2A{r}^2},\hfill & {\gamma }_{11}^1=-\frac{m}{2A{r}^2}\hfill \\ {\gamma }_{12}^2=\frac{1}{r},\hfill & {\gamma }_{13}^3=\frac{1}{r},\hfill & {\gamma }_{22}^1=-Ar,\hfill \\ {\gamma }_{23}^3=\cot \left(\theta \right),\hfill & {\gamma }_{33}^1=-Ar{\sin }^2\left(\theta \right),\hfill & {\gamma }_{33}^2=-\sin \left(\theta \right)\cos (\; \theta \; )\hfill \end{array}}$

We obtain for example:

$\begin{array}{l}{\Gamma }_{22}^1\equiv {\xi }_{22}^1+\left(1-\frac{3m}{2r}\right){\xi }^1=0,\\ {\Gamma }_{33}^1\equiv {\xi }_{33}^1+\sin \left(\theta \right)\cos \left(\theta \right){\xi }_2^1+\left(1-\frac{3m}{2r}\right){\sin }^2\left(\theta \right){\xi }^1=0\end{array}$

$\Rightarrow {\Gamma }_{33}^1-{\sin }^2\left(\theta \right){\Gamma }_{22}^1\equiv {\xi }_{33}^1+\sin \left(\theta \right)\cos \left(\theta \right){\xi }_2^1-{\sin }^2\left(\theta \right){\xi }_{22}^1=0$

after only one prolongation (care).

Then, using r as a summation index, we shall see that we have in general for the linearization of the Riemann and Ricci tensors:

$R_{kl\mathrm{,}ij}\equiv {\rho }_{rl\mathrm{,}ij}{\xi }_k^r+{\rho }_{kr\mathrm{,}ij}{\xi }_l^r+{\rho }_{kl\mathrm{,}rj}{\xi }_i^r+{\rho }_{kl\mathrm{,}ir}{\xi }_j^r+{\xi }^r{\partial }_r{\rho }_{kl\mathrm{,}ij}\ne 0$

$R_{ij}\equiv {\rho }_{rj}{\xi }_i^r+{\rho }_{ir}{\xi }_j^r+{\xi }^r{\partial }_r{\rho }_{ij}\ne 0$

The only 6 non-zero components of the Riemann tensor are:

$\boxed{\begin{array}{lll}{\rho }_{01,01}=+\frac{m}{r^3},\hfill & {\rho }_{02,02}=-\frac{mA}{2r},\hfill & {\rho }_{03,03}=-\frac{mA{\sin }^2\left(\theta \right)}{2r}\hfill \\ {\rho }_{12,12}=+\frac{m}{2Ar},\hfill & {\rho }_{13,13}=+\frac{m{\sin }^2\left(\theta \right)}{2Ar},\hfill & {\rho }_{23,23}=-mr{\sin }^2(\; \theta \; )\hfill \end{array}}$

but we must not forget hat we have indeed ${\rho }_{ij}=0$ for the 10 components of the Ricci tensor, in particular ${\rho }_{ii}=0$ for the diagonal components with $i=0,1,2,3$. We have in particular:

$\begin{array}{c}{\rho }_{22}={\rho }_{2,02}^0+{\rho }_{2,12}^1+{\rho }_{2,32}^3\\ =\frac{1}{A}{\rho }_{02,02}-A{\rho }_{12,12}-\frac{1}{r^2{\sin }^2\left(\theta \right)}{\rho }_{23,23}\\ =-\frac{m}{2r}-\frac{m}{2r}+\frac{m}{r}=0\end{array}$

We also obtain $mod\left(\Omega \right)$:

$\begin{array}{l}{R}_{01,01}\equiv 2{\rho }_{01,01}\left({\xi }_0^0+{\xi }_1^1\right)+{\xi }^r{\partial }_r\left({\rho }_{01,01}\right)={\xi }^1{\partial }_1{\rho }_{01,01}\\ \Rightarrow \boxed{R_{01,01}\equiv -\frac{3m}{r^4}{\xi }^1=0\Rightarrow {\xi }^1=0}\end{array}$

and similarly:

$R_{01,02}\equiv {\rho }_{01,01}{\xi }_2^1+{\rho }_{02,02}{\xi }_1^2+{\xi }^r{\partial }_r{\rho }_{01,02}=\frac{3m}{2r^3}{\xi }_2^1=0\Rightarrow {\xi }_2^1=0$

$R_{02,12}\equiv {\rho }_{12,12}{\xi }_0^1+{\rho }_{02,02}{\xi }_1^0+{\xi }^r{\partial }_r{\rho }_{02,12}=\frac{m}{2rA}\left({\xi }_0^1-A^2{\xi }_1^0\right)=0$

$\boxed{\begin{array}{c}{R}_{23,12}\equiv {\rho }_{21,12}{\xi }_3^1+{\rho }_{23,32}{\xi }_1^3+{\xi }^r{\partial }_r{\rho }_{23,12}\\ =-\frac{m}{2rA}{\xi }_3^1+mr{\sin }^2\left(\theta \right){\xi }_1^3\\ =-\frac{3m}{2rA}{\xi }_3^1=0\end{array}}$

$\Rightarrow R_{01,03}=\frac{3m}{2r^3}{\xi }_3^1=-\frac{A}{r^2}{R}_{23,12}$

${\rho }_{01,23}=0\Rightarrow R_{01,23}={\rho }_{01,23}\left({\xi }_0^0+{\xi }_1^1+{\xi }_2^2+{\xi }_3^3\right)+{\xi }^r{\partial }_r{\rho }_{01,23}=0$

and so on, in order to avoid using computer algebra. However, the main consequence of this remark is to explain the existence of the 15 second order CC. Indeed, denoting by “~” a linear proportional dependence $mod\left(\Omega \right)$, we have the successive three cases:

$\boxed{\begin{array}{cc}\left(R_{00}\mathrm{,}{R}_{11}\mathrm{,}{R}_{22}\mathrm{,}{R}_{33}\right)& R_{\mathrm{01,01}}\sim R_{\mathrm{02,02}}\sim R_{\mathrm{03,03}}\sim R_{\mathrm{12,12}}\sim R_{\mathrm{13,13}}\sim R_{\mathrm{23,23}}\to {\xi }^1=0\\ & \\ \left(R_{12}\right)& R_{\mathrm{01,02}}\sim R_{\mathrm{13,23}}\to {\xi }_2^1=0\\ \left(R_{13}\right)& R_{\mathrm{01,03}}\sim R_{\mathrm{12,23}}\to {\xi }_3^1=0\\ \left(R_{02}\right)& R_{\mathrm{01,12}}\sim R_{\mathrm{03,23}}\to {\xi }_2^0=0\\ \left(R_{03}\right)& R_{\mathrm{01,13}}\sim R_{\mathrm{02,23}}\to {\xi }_3^0=0\\ & \\ \left(R_{01}\mathrm{,}{R}_{23}\right)& R_{\mathrm{01,23}}\to \mathrm{0,}{R}_{\mathrm{02,03}}\to \mathrm{0,}{R}_{\mathrm{02,12}}\to \mathrm{0,}{R}_{\mathrm{02,13}}\to \mathrm{0,}{R}_{\mathrm{03,13}}\to \mathrm{0,}{R}_{\mathrm{12,13}}\to 0\end{array}}$

as a way to obtain the 5 equalities to zero on the right and thus a total of $20-5=15$ second order CC obtained by elimination. However, the present partition $15=5+4+6$ is quite different from the partition $15=10+5$ used by the authors quoted in the Introduction which is obtained by taking into account the vanishing assumption of the 10 components of the Ricci tensor. As such a result questions once more the mathematical foundations of general relativity, in particular the existence of gravitational waves, we provide a few additional technical comments.

The main point is a tricky formula which is not evident at all. Indeed, using the well known properties of the Lie derivative, we have the following geometric objects (not necessarily tensors) and their linearizations (generally tensors):

${\omega }_{ij}\to {\Omega }_{ij}\in S_2{T}^{\mathrm{<mo>\; *\; </mo>}}\mathrm{,}{\gamma }_{ij}^k\to {\Gamma }_{ij}^k\in S_2{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes T\mathrm{,}$

${\rho }_{kl,ij}\to R_{kl,ij}\in {\wedge }^2{T}^{*}\otimes T^{*}\otimes T,{\beta }_{kl,ijr}\to B_{kl,ijr}\in {\wedge }^3{T}^{*}\otimes T^{*}\otimes T$

Then, using r as a summation index, we shall see that we have in general:

$R_{kl\mathrm{,}ij}\equiv {\rho }_{rl\mathrm{,}ij}{\xi }_k^r+{\rho }_{kr\mathrm{,}ij}{\xi }_l^r+{\rho }_{kl\mathrm{,}rj}{\xi }_i^r+{\rho }_{kl\mathrm{,}ir}{\xi }_j^r+{\xi }^r{\partial }_r{\rho }_{kl\mathrm{,}ij}\ne 0$

${\rho }_{kl,ij}={\omega }_{kr}{\rho }_{l,ij}^r\Rightarrow R_{kl,ij}={\omega }_{kr}{R}_{l,ij}^r+{\rho }_{l,ij}^r{\Omega }_{kr}\Rightarrow {\omega }^{rs}{R}_{ri,sj}=R_{ij}+{\omega }^{rs}{\rho }_{i,rj}^t{\Omega }_{st}$

${\rho }_{ij}={\rho }_{i,rj}^r\Rightarrow R_{ij}=R_{i,rj}^r\ne {\omega }^{rs}{R}_{ri,sj}$

We prove these results using local coordinates and the formal Lie derivative obtained while replacing $j_1\left(\xi \right)$ by ${\xi }_1$ (See [1] [5] [9] [14] for details). First of all, from the tensorial property of the Riemann tensor and the Killing equations ${\Omega }_{us}={\omega }_{ku}{\xi }_s^k+{\omega }_{ks}{\xi }_u^k+{\xi }^r{\partial }_r{\omega }_{us}$, we have:

$R_{l,ij}^k\equiv {\left(L\left({\xi }_1\right)\rho \right)}_{l,ij}^k=-{\rho }_{l,ij}^s{\xi }_s^k+{\rho }_{s,ij}^k{\xi }_l^s+{\rho }_{l,sj}^k{\xi }_i^s+{\rho }_{l,is}^k{\xi }_j^s+{\xi }^r{\partial }_r{\rho }_{l,ij}^k$

$\begin{array}{c}{\omega }_{ku}\left(-{\rho }_{v\mathrm{,}ij}^s{\xi }_s^k+{\xi }^r{\partial }_r{\rho }_{v\mathrm{,}ij}^s\right)={\rho }_{v\mathrm{,}ij}^s{\omega }_{ks}{\xi }_u^k+\left({\xi }^r{\partial }_r{\omega }_{us}\right){\rho }_{v\mathrm{,}ij}^s+{\omega }_{ku}{\xi }^r{\partial }_r{\rho }_{v\mathrm{,}ij}^k-{\rho }_{v\mathrm{,}ij}^s{\Omega }_{su}\\ ={\rho }_{sv\mathrm{,}ij}{\xi }_u^s+{\xi }^r{\partial }_r{\rho }_{uv\mathrm{,}ij}-{\rho }_{v\mathrm{,}ij}^s{\Omega }_{su}\end{array}$

and thus ${\omega }_{ku}{R}_{v,ij}^k=R_{uv,ij}-{\rho }_{v,ij}^s{\Omega }_{su}$.

${\rho }_{1,11}^1=0\Rightarrow {\rho }_{11}={\rho }_{1,01}^0+{\rho }_{1,21}^2+{\rho }_{1,31}^3=\frac{1}{A}{\rho }_{01,01}-\frac{1}{r^2}{\rho }_{12,12}-\frac{1}{r^2{\sin }^2\left(\theta \right)}{\rho }_{13,13}=0$

We have for example, in this particular case:

$\begin{array}{l}{\rho }_{1,01}^0=\frac{1}{A}{\rho }_{01,01}=\frac{m}{A{r}^3}\\ \Rightarrow R_{1,01}^0={\left(\mathcal{L}\left(\xi \right)\rho \right)}_{1,01}^0=2{\rho }_{1,01}^0{\xi }_1^1+{\xi }^1{\partial }_1{\rho }_{1,01}^0=\frac{2m}{A{r}^3}{\xi }_1^1+{\xi }^1{\partial }_1\left(\frac{m}{A{r}^3}\right)\end{array}$

The only use of $R_{\mathrm{01,01}}$ is allowing to get ${\xi }_1=0$ in the previous list, but we have also exactly:

${\omega }^{rs}{R}_{r1,s2}=\frac{1}{A}{R}_{01,02}-\frac{1}{r^2{\sin }^2\left(\theta \right)}{R}_{31,32}=-\frac{m}{2r^3}{\Omega }_{12}=R_{12}+{\omega }^{11}{\rho }_{1,12}^2{\Omega }_{12}\Rightarrow R_{12}=0$

The use of $R_{\mathrm{01,02}}$ or $R_{\mathrm{13,23}}$ is allowing to get ${\xi }_2^1=0$ in the previous list with:

$R_{1,02}^0=-\frac{3m}{2r^3}{\xi }_{1,2}+\frac{m}{2r^3}{\Omega }_{12},R_{1,32}^3=+\frac{3m}{2r^3}{\xi }_{1,2}-\frac{m}{r^3}{\Omega }_{12}$

and thus also exactly:

${\omega }^{rs}{\rho }_{1,r2}^t{\Omega }_{st}=-{\omega }^{11}{\omega }^{22}{\rho }_{12,12}{\Omega }_{12}=-\frac{m}{2r^3}{\Omega }_{12}\Rightarrow R_{12}=R_{1,02}^0+R_{1,32}^3+\frac{m}{2r^3}{\Omega }_{12}=0$

It follows that the 4 central second order CC of the list successively amounts to $R_{12}=0,R_{13}=0,R_{02}=0,R_{03}=0$, a result breaking the intrinsic/coordinate-free interpretation of the 10 Einstein equations and the situation is even worst for the other components of the Ricci tensor. Indeed, $R_{01}$ and $R_{23}$ only depend on the vanishing of $R_{\mathrm{02,12}}\mathrm{,}{R}_{\mathrm{03,13}}$ and $R_{\mathrm{02,03}}\mathrm{,}{R}_{\mathrm{12,13}}$ among the bottom CC of the list, while the diagonal terms $R_{00}\mathrm{,}{R}_{11}\mathrm{,}{R}_{22}\mathrm{,}{R}_{33}$ only depend, as we just saw, on the 6 non zero components of the Riemann tensor. We have thus obtained the totally unusual partition $10=4+4+2$ along the successive blocks of the former list with:

$\left\{R_{ij}\right\}=\left\{R_{00},R_{11},R_{22},R_{33}\right\}+\left\{R_{12},R_{13},R_{02},R_{03}\right\}+\left\{R_{01},R_{23}\right\}$

Finally, we notice that $R_{01,23}=0,R_{02,31}=0\Rightarrow R_{03,12}=0$ from the identity in ${\wedge }^3{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes T^{\mathrm{<mo>\; *\; </mo>}}$:

$R_{01,23}+R_{02,31}+R_{03,12}=0$

and there is no way to have two identical indices in the first jets appearing through the (formal) Lie derivative just described. As for the third order CC, setting ${\xi }_1^1=\frac{A^{\prime }}{2A}{\xi }^1\in j_2\left(\Omega \right)$, we have at least the first prolongations of the previous second order CC to which we have to add the three new generating ones:

$\boxed{d_1{\xi }^1-{\xi }_1^1=0,d_2{\xi }^1-{\xi }_2^1=0,d_3{\xi }^1-{\xi }_3^1=0}$

provided by the Spencer operator, leading to the crossed terms $d_i{\xi }_j^1-d_j{\xi }_i^1=0$ for $i,j=1,2,3$ because the Spencer operator is not FI.

Setting now ${\xi }^1=U,{\xi }_2^1=V_2,{\xi }_3^1=V_3,{\xi }_2^0=W_2,{\xi }_3^0=W_3$ with $\left(U\mathrm{,}V\mathrm{,}W\right)\in j_2\left(\Omega \right)$, we have to look for the CC of the system $R_1^{\left(1\right)}=R_1$ already presented, then the system $R_1^{\left(2\right)}$ with $dim\left(R_1^{\left(2\right)}\right)=5$ and finally $R_1^{\left(3\right)}$ with $dim\left(R_1^{\left(3\right)}\right)=4$ which is formally integrable but not involutive because it is of finite type. Beside the only zero order equation ${\xi }^1=U$, we have the following 15 first order ones:

$\boxed{\begin{array}{l}{\xi }_0^0=-\frac{A^{\prime }}{2A}U,{\xi }_1^0=\frac{1}{A^2}{d}_{0}U,{\xi }_2^0=W_2,{\xi }_3^0=W_3,\\ {\xi }_0^1=d_{0}U,{\xi }_1^1=\frac{A^{\prime }}{2A}U,{\xi }_2^1=V_2,{\xi }_3^1=V_3,\\ {\xi }_0^2=\frac{A}{r^2}{W}_2,{\xi }_1^2=-\frac{1}{A{r}^2}{V}_2,{\xi }_0^3=\frac{A}{r^2{\sin }^2\left(\theta \right)}{W}_3,{\xi }_1^3=-\frac{1}{A{r}^2{\sin }^2\left(\theta \right)}{V}_3,\\ {\xi }_2^2=-\frac{1}{r}U,{\xi }_3^2+{\sin }^2\left(\theta \right){\xi }_2^3=0,{\xi }_3^3+\cot \left(\theta \right){\xi }^2=-\frac{1}{r}U\end{array}}$

Among the CC we must have $d_2{V}_3-d_3{V}_2=0$ which is among the differential consequences of the Spencer operator as we saw but we must also have $d_2{W}_3-d_3{W}_2=0$ and both seem to be new third order CC, together with the CC obtained by eliminating ${\xi }^2$ and ${\xi }^3$ from the three last equations after two prolongations as in ( [23] ):

$d_3{V}_3+\sin \left(\theta \right)\cos \left(\theta \right)V_2+{\sin }^2\left(\theta \right)d_2{V}_2+2{\sin }^2\left(\theta \right)U=0$

However, things are not so simple, even if we have in mind that $\left(V\mathrm{,}W\right)\in j_2\left(\Omega \right)$, because the central sign in the previous formula is opposite to the sign found after one prolongation in the formula:

${\xi }_{33}^1+\sin \left(\theta \right)\cos \left(\theta \right){\xi }_2^1-{\sin }^2\left(\theta \right){\xi }_{22}^1=0$

and it is at this moment that we need introduce new differential geometric methods!

First of all, we have:

${\rho }_{l,ij}^k={\partial }_i{\gamma }_{lj}^k-{\partial }_j{\gamma }_{li}^k+{\gamma }_{lj}^r{\gamma }_{ri}^k-{\gamma }_{li}^r{\gamma }_{rj}^k$

and thus, because $\Gamma \in S_2{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes T$ is a tensor::

$\begin{array}{c}{R}_{l\mathrm{,}ij}^k=d_i{\Gamma }_{lj}^k-d_j{\Gamma }_{li}^k+{\gamma }_{lj}^r{\Gamma }_{ri}^k-{\gamma }_{li}^r{\Gamma }_{rj}^k+{\gamma }_{ri}^k{\Gamma }_{lj}^r-{\gamma }_{rj}^k{\Gamma }_{li}^r\\ =\left(d_i{\Gamma }_{lj}^k-{\gamma }_{li}^r{\Gamma }_{rj}^k+{\gamma }_{ri}^k{\Gamma }_{lj}^r\right)-\left(d_j{\Gamma }_{li}^k-{\gamma }_{lj}^r{\Gamma }_{ri}^k+{\gamma }_{rj}^k{\Gamma }_{li}^r\right)\\ =\left(d_i{\Gamma }_{lj}^k-{\gamma }_{li}^r{\Gamma }_{rj}^k-{\gamma }_{ji}^r{\Gamma }_{lr}^k+{\gamma }_{ri}^k{\Gamma }_{lj}^r\right)-\left(d_j{\Gamma }_{li}^k-{\gamma }_{lj}^r{\Gamma }_{ri}^k-{\gamma }_{ij}^r{\Gamma }_{lr}^k+{\gamma }_{rj}^k{\Gamma }_{li}^r\right)\\ ={\nabla }_i{\Gamma }_{lj}^k-{\nabla }_j{\Gamma }_{li}^k\end{array}$

by introducing the covariant derivative $\nabla$. We recall that ${\nabla }_r{\omega }_{ij}=0,\forall r,i,j$ or, equivalently, that $\left(id,-\gamma \right):\xi \in T\to \left({\xi }^k,{\xi }_i^k=-{\gamma }_{ir}^k{\xi }^r\right)\in R_1$ is a $R_1$ -connection with ${\omega }_{sj}{\gamma }_{ir}^s+{\omega }_{is}{\gamma }_{jr}^s={\partial }_r{\omega }_{ij}$, a result allowing to move down the index k in the previous formulas (See [9] for more details).

We may thus take into account the Bianchi identities implied by the cyclic sums on $\left(ijr\right)$

${\beta }_{kl,ijr}\equiv {\nabla }_r{\rho }_{kl,ij}+{\nabla }_i{\rho }_{kl,jr}+{\nabla }_j{\rho }_{kl,ri}=0\iff \beta \equiv \underset{cycl}{\Sigma }\left(\partial \rho -\gamma \rho \right)=0$

and their respective linearizations $B_{kl,ijr}=0$ as described below. We shall see later on that $\beta$ and B are sections of the vector bundle $F_2$ defined by the short exact sequence:

$0\to F_2\to {\wedge }^3{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes g_1\stackrel{\delta }{\to }{\wedge }^4{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes T\to 0$

with $\begin{array}{c}dim\left(F_2\right)=\left(n\left(n-1\right)\left(n-2\right)/6\right)\left(n\left(n-1\right)/2\right)-\left(n\left(n-1\right)\left(n-2\right)\left(n-3\right)/24\right)n\\ =n^2\left(n^2-1\right)\left(n-2\right)/24\end{array}$ because $dim\left(g_1\right)=n\left(n-1\right)/2$ for any nondegenerate metric, that is $24-4=20$ when $n=4$.

Such results cannot be even imagined by somebody not aware of the δ-acyclicity ( [1] [9] [13] ).

We have the linearized cyclic sums of covariant derivatives both with their respective symbolic descriptions, not to be confused with the non-linear corresponding ones:

$\begin{array}{l}{B}_{kl\mathrm{,}rij}\equiv {\nabla }_r{R}_{kl\mathrm{,}ij}+{\nabla }_i{R}_{kl\mathrm{,}jr}+{\nabla }_j{R}_{kl\mathrm{,}ri}=0mod\left(\Gamma \right)\\ \iff \underset{cycl}{\Sigma }\left(dR-\gamma R-\rho \Gamma \right)=0\\ \iff B\equiv \underset{cycl}{\Sigma }\left(\nabla R\right)=\underset{cycl}{\Sigma }\left(\rho \Gamma \right)\end{array}$

In order to recapitulate these new concepts obtained after one, two or three prolongations, we have successively $\omega \to \gamma \to \rho \to \beta$ and the respective linearizations $\Omega \to \Gamma \to R\to B$.

The 24 Bianchi identities are related by the 4 linear relations like $B_{01,023}-B_{02,013}+B_{03,012}=0$ when $n=4$ because $B_{00,123}=0$. These relations are existinging between the 24 components of the Lanczos tensor because $B\in F_2\subset ker\left(\delta \right)$ in the previous short exact sequence ( [20] ).

With more details, we number the 20 linearly independent Bianchi identities as follows:

$\boxed{\begin{array}{c}\boxed{1}{B}_{01,012},\boxed{2}{B}_{01,013},\\ \boxed{3}{B}_{02,123},\boxed{4}{B}_{02,012},\boxed{5}{B}_{02,013},\boxed{6}{B}_{02,023},\\ \boxed{7}{B}_{03,123},\boxed{8}{B}_{03,012},\boxed{9}{B}_{03,013},\boxed{10}{B}_{03,023},\\ \boxed{11}{B}_{12,123},\boxed{12}{B}_{12,012},\boxed{13}{B}_{12,013},\boxed{14}{B}_{12,023},\\ \boxed{15}{B}_{13,123},\boxed{16}{B}_{13,012},\boxed{14}{B}_{13,013},\boxed{18}{B}_{13,023},\\ \boxed{19}{B}_{23,123},\boxed{20}{B}_{23,023}\end{array}}$

to which we add the 4 linearly dependent:

$\boxed{\boxed{21}{B}_{\mathrm{01,123}}\mathrm{,}\boxed{22}{B}_{\mathrm{01,023}}\mathrm{,}\boxed{23}{B}_{\mathrm{23,012}}\mathrm{,}\boxed{24}{B}_{\mathrm{23,013}}}$

We successively study a few situations without any, with one or with two vanishing linearized Riemann components, taking into account that the four Einstein equations are described by:

$\boxed{12}$, $\boxed{17}$, $\boxed{20}$ for the index 0, $\boxed{4}$, $\boxed{9}$, $\boxed{19}$ for the index 1, $\boxed{1}$, $\boxed{10}$, $\boxed{15}$ for the index 2, $\boxed{2}$, $\boxed{6}$, $\boxed{11}$ for the index 3.

BIANCHI $\boxed{1}$: $\boxed{B_{\mathrm{01,012}}\equiv {\nabla }_0{R}_{\mathrm{01,12}}+{\nabla }_1{R}_{\mathrm{01,20}}+{\nabla }_2{R}_{\mathrm{01,01}}=-\frac{3m}{2r^3}\left({\Gamma }_{02}^0+{\Gamma }_{12}^1\right)}$.

First of all, we have $R_{01,12}=-\frac{3m}{2r^3}{\xi }_2^0$, $R_{01,02}=\frac{3m}{2r^3}{\xi }_2^1$, $R_{01,01}=-\frac{3m}{r^4}{\xi }^1$ and obtain:

$\begin{array}{c}{\nabla }_0{R}_{01,12}=-\frac{3m}{2r^3}{\xi }_{02}^0-{\gamma }_{01}^0{R}_{01,02}\\ =-\frac{3m}{2r^3}\left({\xi }_{02}^0+\frac{m}{2A{r}^2}{\xi }_2^1\right)\\ =-\frac{3m}{2r^3}{\Gamma }_{02}^0\end{array}$

$\begin{array}{c}{\nabla }_1{R}_{01,20}=\left(-\frac{3m}{2r^3}{\xi }_{12}^1+\frac{9m}{2r^4}{\xi }_2^1\right)-\left(2{\gamma }_{01}^0+{\gamma }_{11}^1+{\gamma }_{12}^2\right)R_{01,20}\\ =-\frac{3m}{2r^3}\left({\xi }_{12}^1-\frac{m}{2A{r}^2}{\xi }_2^1\right)+\frac{6m}{r^4}{\xi }_2^1\\ =-\frac{3m}{2r^3}{\Gamma }_{12}^1+\frac{6m}{r^4}{\xi }_2^1\end{array}$

$\begin{array}{c}{\nabla }_2{R}_{01,01}=-\frac{3m}{r^4}{\xi }_2^1-2{\gamma }_{12}^2{R}_{02,01}\\ =-\frac{3m}{r^4}{\xi }_2^1-\frac{2}{r}\frac{3m}{2r^3}{\xi }_2^1\\ =-\frac{6m}{r^4}{\xi }_2^1\end{array}$

and we notice that ${\Gamma }_{02}^0+{\Gamma }_{12}^1={\xi }_{02}^0+{\xi }_{12}^1=d_2\left({\xi }_0^0+{\xi }_1^1\right)=0\Rightarrow B_{01,012}=0$. As a byproduct, we have $d_0{W}_2+\frac{m}{2A{r}^2}{V}_2=0$, $d_1{V}_2-\frac{m}{2A{r}^2}{V}_2=0\Rightarrow d_1{V}_2+d_0{W}_2=0$.

BIANCHI $\boxed{3}$: $\boxed{B_{02,123}\equiv {\nabla }_1{R}_{02,23}+{\nabla }_2{R}_{02,31}+{\nabla }_3{R}_{02,12}=\frac{3mA}{2r}{\Gamma }_{13}^0}$.

First of all, we have $R_{02,23}=\frac{3mA}{2r}{\xi }_3^0$, $R_{02,31}=0$, $R_{02,12}=0$, $R_{01,31}=\frac{3m}{2r^3}{\xi }_3^0$ and obtain:

$\begin{array}{c}{\nabla }_1{R}_{02,23}=d_1{R}_{02,23}-{\gamma }_{01}^r{R}_{r2,23}-{\gamma }_{12}^r{R}_{0r,23}-{\gamma }_{12}^r{R}_{02,r3}-{\gamma }_{13}^r{R}_{02,2r}\\ =\frac{3mA}{2r}{\xi }_{13}^0+\left(\frac{3m^2}{2r^3}-\frac{3mA}{2r^2}\right){\xi }_3^0-\left({\gamma }_{01}^0-\frac{3}{r}\right)R_{02,23}\\ =\frac{3mA}{2r}{\xi }_{13}^0-\left(\frac{12m}{2r^2}-\frac{27m^2}{4r^3}\right){\xi }_3^0\\ =\frac{3mA}{2r}{\Gamma }_{13}^0-\frac{9mA}{2r^2}{\xi }_3^0\end{array}$

$\begin{array}{c}{\nabla }_2{R}_{02,31}=0-{\gamma }_{02}^r{R}_{r2,31}-{\gamma }_{22}^r{R}_{0r,31}-{\gamma }_{23}^r{R}_{02,r1}-{\gamma }_{12}^r{R}_{02,3r}\\ =Ar{R}_{01,31}-\cot \left(\theta \right)R_{02,31}-\frac{1}{r}{R}_{02,32}\\ =\frac{3mA}{2r^2}{\xi }_3^0+\frac{3mA}{2r^2}{\xi }_3^0\\ =\frac{3mA}{r^2}{\xi }_3^0\end{array}$

$\begin{array}{c}{\nabla }_3{R}_{02,12}=0-{\gamma }_{03}^r{R}_{r2,12}-{\gamma }_{23}^r{R}_{0r,12}-{\gamma }_{13}^r{R}_{02,r2}-{\gamma }_{23}^r{R}_{02,1r}\\ =-{\gamma }_{23}^3{R}_{03,12}-{\gamma }_{13}^3{R}_{02,32}-{\gamma }_{23}^3{R}_{02,13}\\ =\frac{3mA}{2r^2}{\xi }_3^0\end{array}$

and we may use the fact that ${\Gamma }_{13}^0\equiv {\xi }_{13}^0-\frac{1}{Ar}\left(1-\frac{3m}{2r}\right){\xi }_3^0=0\Rightarrow d_1{W}_3-\frac{1}{Ar}\left(1-\frac{3m}{2r}\right)W_3=0$.

BIANCHI $\boxed{4}$: $\boxed{B_{02,012}\equiv {\nabla }_0{R}_{02,12}+{\nabla }_1{R}_{02,20}+{\nabla }_2{R}_{02,01}=\frac{3m}{2r^3}{\Gamma }_{22}^1}$.

First of all we have $R_{02,12}=0$, $R_{02,02}=\frac{3mA}{2r^2}{\xi }^1$, $R_{02,01}=\frac{3m}{2r^3}{\xi }_2^1$, $R_{12,12}=-\frac{3m}{2A{r}^2}{\xi }^1$ and obtain:

$\begin{array}{c}{\nabla }_0{R}_{02,12}=d_0{R}_{02,12}-{\gamma }_{00}^r{R}_{r2,12}-{\gamma }_{02}^r{R}_{0r,12}-{\gamma }_{01}^r{R}_{02,r2}-{\gamma }_{02}^r{R}_{02,1r}\\ =0-{\gamma }_{00}^1{R}_{12,12}-{\gamma }_{01}^0{R}_{02,02}\\ =\left(-\frac{mA}{2r^2}\right)\left(-\frac{3m}{2A{r}^2}\right){\xi }^1-\left(\frac{m}{2A{r}^2}\right)\left(\frac{3mA}{2r^2}\right){\xi }^1\\ =0\end{array}$

$\begin{array}{c}{\nabla }_1{R}_{02,20}=d_1\left(-\frac{3mA}{2r^2}{\xi }^1\right)-2{\gamma }_{01}^r{R}_{r2,20}-2{\gamma }_{12}^r{R}_{0r,20}\\ =-\frac{3mA}{2r^2}{\xi }_1^1-{\partial }_1\left(\frac{3mA}{2r^2}\right){\xi }^1+2{\gamma }_{01}^0{R}_{02,02}+2{\gamma }_{12}^2{R}_{02,02}\\ =\left(-\frac{3mA}{2r^2}\right)\left(\frac{m}{2A{r}^2}\right){\xi }^1-{\partial }_1\left(\frac{3mA}{2r^2}\right){\xi }^1+\frac{3m^2}{2r^4}{\xi }^1+\frac{3mA}{r^3}{\xi }^1\\ =-\frac{3m^2}{4r^4}{\xi }^1+\left(\frac{3m}{r^3}-\frac{3m^2}{r^4}+\frac{3m}{r^3}-\frac{3m^2}{r^4}\right){\xi }^1\\ =\frac{6m}{r^3}{\xi }^1-\frac{27m^2}{4r^4}{\xi }^1\end{array}$

$\begin{array}{c}{\nabla }_2{R}_{02,01}=\frac{3m}{2r^3}{\xi }_{22}^1-{\gamma }_{02}^r{R}_{r2,01}-{\gamma }_{22}^r{R}_{0r,01}-{\gamma }_{02}^r{R}_{02,r1}-{\gamma }_{12}^r{R}_{02,0r}\\ =\frac{3m}{2r^3}{\xi }_{22}^1-{\gamma }_{22}^1{R}_{01,01}-{\gamma }_{12}^2{R}_{02,02}\\ =\frac{3m}{2r^3}{\xi }_{22}^1-\frac{9mA}{2r^3}{\xi }^1\\ =\frac{3m}{2r^3}{\Gamma }_{22}^1-\frac{6m}{r^3}{\xi }^1+\frac{27m^2}{4r^4}{\xi }^1\end{array}$

This delicate checking proves that $B_{02,012}\equiv \frac{3m}{2r^3}{\Gamma }_{22}^1=0\Rightarrow d_2{V}_2+\left(1-\frac{3m}{2r}\right)U=0$ is a differential consequence of $R_{02,12}=0$. We let the reader prove as an exercise that $B_{03,013}\equiv \frac{3m}{2r^3}{\Gamma }_{33}^1=0\Rightarrow d_3{V}_3+\sin \left(\theta \right)\cos \left(\theta \right)V_2+\left(1-\frac{3m}{2r}\right){\sin }^2\left(\theta \right)U=0$ in order to recover $\boxed{19}$ by eliminating U.

BIANCHI $\boxed{6}$: $\boxed{B_{02,023}\equiv {\nabla }_0{R}_{02,23}+{\nabla }_2{R}_{02,30}+{\nabla }_3{R}_{02,02}=\frac{3mA}{2r}{\Gamma }_{03}^0}$.

First of all, we have $R_{02,23}=\frac{3mA}{2r}{\xi }_3^0$, $R_{02,30}=0$, $R_{02,02}=\frac{3mA}{2r^2}{\xi }^1$ and obtain:

$\begin{array}{c}{\nabla }_0{R}_{02,23}=d_0{R}_{02,23}-{\gamma }_{00}^r{R}_{r2,23}-{\gamma }_{02}^r{R}_{0r,23}-{\gamma }_{02}^r{R}_{02,r3}-{\gamma }_{03}^r{R}_{02,2r}\\ =d_0{R}_{02,23}-{\gamma }_{00}^1{R}_{12,23}\\ =\frac{3mA}{2r}{\xi }_{03}^0+\frac{3m^2}{4r^3}{\xi }_3^1\\ =\frac{3mA}{2r}{\Gamma }_{03}^0\end{array}$

$\begin{array}{c}{\nabla }_2{R}_{02,30}=0-{\gamma }_{22}^1{R}_{01,30}\\ =Ar{R}_{01,30}\\ =-\frac{3mA}{2r^2}{\xi }_3^1\end{array}$

${\nabla }_3{R}_{02,02}=\frac{3mA}{2r^2}{\xi }_3^1$

where ${\Gamma }_{03}^0\equiv {\xi }_{03}^0+\frac{m}{2A{r}^2}{\xi }_3^1=d_3\left({\xi }_0^0+\frac{m}{2A{r}^2}{\xi }^1\right)=0$ and $R_{12,23}=-\frac{3m}{2Ar}{\xi }_3^1$, $R_{01,03}=\frac{3m}{2r^3}{\xi }_3^1$.

Again, only this final result proves that $B_{02,023}\equiv \frac{3mA}{2r}{\Gamma }_{03}^0\Rightarrow d_0{W}_3+\frac{m}{2A{r}^2}{V}_3=0$ is a differential consequence of $R_{02,03}=0$.

BIANCHI $\boxed{12}$: $\boxed{B_{12,012}\equiv {\nabla }_0{R}_{12,12}+{\nabla }_1{R}_{12,20}+{\nabla }_2{R}_{12,01}=-\frac{3m}{2r^3}{\Gamma }_{22}^0}$.

First of all, ${\Gamma }_{22}^0\equiv {\xi }_{22}^0+\frac{r}{A}{\xi }_0^1$, $R_{12,12}=-\frac{3m}{2A{r}^2}{\xi }^1$, $R_{12,20}=0$, $R_{12,01}=-\frac{3m}{2r^3}{\xi }_2^0$ and we obtain:

$\begin{array}{c}{\nabla }_0{R}_{12,12}=d_0{R}_{12,12}-2{\gamma }_{01}^r{R}_{r2,12}-2{\gamma }_{02}^r{R}_{1r,12}\\ =d_0{R}_{12,12}-2{\gamma }_{01}^0{R}_{02,12}\\ =-\frac{3m}{2A{r}^2}{\xi }_0^1\end{array}$

$\begin{array}{c}{\nabla }_1{R}_{12,20}=0-\left({\gamma }_{11}^1+2{\gamma }_{12}^2+{\gamma }_{01}^0\right)R_{12,20}\\ =-2{\gamma }_{12}^2{R}_{12,20}\\ =0\end{array}$

$\begin{array}{c}{\nabla }_2{R}_{12,01}=-\frac{3m}{2r^3}{\xi }_{22}^0-{\gamma }_{12}^2{R}_{12,02}\\ =-\frac{3m}{2r^3}{\xi }_{22}^0\\ =-\frac{3m}{2r^3}{\Gamma }_{22}^0+\frac{3m}{2A{r}^2}{\xi }_0^1\end{array}$

a result leading to $d_2{W}_2+\frac{r}{A}{d}_{0}U=0$. Again, only the final sum has an intrinsic mathematical meaning with $B_{12,012}=-\frac{3m}{2r^3}{\Gamma }_{22}^0$.

BIANCHI $\boxed{22}$: $B_{01,023}\equiv {\nabla }_0{R}_{01,23}+{\nabla }_2{R}_{01,30}+{\nabla }_3{R}_{01,02}=0$

First of all, we have $R_{01,23}=0$, $R_{01,03}=\frac{3m}{2r^3}{\xi }_3^1$, $R_{01,02}=\frac{3m}{2r^3}{\xi }_2^1$, $R_{02,03}=0$ and, as ${\Omega }_{23}=0$:

$\begin{array}{c}{\Gamma }_{23}^1\equiv {\xi }_{23}^1+{\gamma }_{33}^1{\xi }_2^3+{\gamma }_{22}^1{\xi }_3^2-{\gamma }_{23}^3{\xi }_3^1\\ ={\xi }_{23}^1-Ar\left({\xi }_3^2+{\sin }^2\left(\theta \right){\xi }_2^3\right)-\cot \left(\theta \right){\xi }_3^1\\ ={\xi }_{23}^1-\cot \left(\theta \right){\xi }_3^1\end{array}$

$\begin{array}{c}{\Gamma }_{13}^2\equiv {\xi }_{13}^2+{\gamma }_{r3}^2{\xi }_1^r+{\gamma }_{1r}^2{\xi }_3^r-{\gamma }_{13}^r\\ ={\xi }_{13}^2+{\gamma }_{33}^2{\xi }_1^3+\left({\gamma }_{12}^2-{\gamma }_{13}^3\right){\xi }_3^2\\ ={\xi }_{13}^2-\sin \left(\theta \right)\cos \left(\theta \right){\xi }_1^3\end{array}$

$\begin{array}{c}{\Gamma }_{12}^3\equiv {\xi }_{12}^3+{\gamma }_{r2}^3{\xi }_1^r+{\gamma }_{1r}^3{\xi }_2^r-{\gamma }_{12}^r{\xi }_r^3\\ ={\xi }_{12}^3+{\gamma }_{23}^3{\xi }_1^3+\left({\gamma }_{13}^3-{\gamma }_{12}^2\right){\xi }_2^3\\ ={\xi }_{12}^3+\cot \left(\theta \right){\xi }_1^3\end{array}$

$\begin{array}{c}{\nabla }_0{R}_{01,23}=d_0{R}_{01,23}-{\gamma }_{00}^r{R}_{r1,23}-{\gamma }_{01}^r{R}_{0r,23}-{\gamma }_{02}^r{R}_{01,r3}-{\gamma }_{03}^r{R}_{01,2r}\\ =d_0{R}_{01,23}-{\gamma }_{00}^1{R}_{01,23}-{\gamma }_{01}^0{R}_{00,13}\\ =0\end{array}$

$\begin{array}{c}{\nabla }_2{R}_{01,30}=d_2{R}_{01,30}-{\gamma }_{12}^2{R}_{02,30}-{\gamma }_{23}^3{R}_{01,30}\\ =-\frac{3m}{2r^3}{\xi }_{23}^1+\frac{3m}{2r^3}\cot \left(\theta \right){\xi }_3^1\\ =-\frac{3m}{2r^3}\left({\xi }_{23}^1-\cot \left(\theta \right){\xi }_3^1\right)\\ =-\frac{3m}{2r^3}{\Gamma }_{23}^1\end{array}$

$\begin{array}{c}{\nabla }_3{R}_{01,02}=d_3{R}_{01,02}-{\gamma }_{13}^3{R}_{03,02}-{\gamma }_{23}^3{R}_{01,03}\\ =\frac{3m}{2r^3}{\xi }_{23}^1-\frac{3m}{2r^3}\cot \left(\theta \right){\xi }_3^1\\ =\frac{3m}{2r^3}\left({\xi }_{23}^1-\cot \left(\theta \right){\xi }_3^1\right)\\ =\frac{3m}{2r^3}{\Gamma }_{23}^1\end{array}$

We could also say that $d_2{R}_{01,03}=\frac{3m}{2r^3}\left(d_2{V}_3-{\xi }_{23}^1\right)+\frac{3m}{2r^3}{\xi }_{23}^1$ and obtain therefore finally the formula $B_{01,023}=0\Rightarrow d_2{V}_3-d_3{V}_2=0$ is a differential consequence of $R_{01,23}=0$.

We also check in particular:

$\begin{array}{c}{\nabla }_0{R}_{01,23}\Rightarrow {\rho }_{r1,23}{\Gamma }_{00}^r+{\rho }_{0r,23}{\Gamma }_{01}^r+{\rho }_{01,r3}{\Gamma }_{02}^r+{\rho }_{01,2r}{\Gamma }_{03}^r\\ \Rightarrow 0\end{array}$

$\begin{array}{c}{\nabla }_2{R}_{01,30}\Rightarrow {\rho }_{r1,30}{\Gamma }_{02}^r+{\rho }_{0r,30}{\Gamma }_{12}^r+{\rho }_{01,r0}{\Gamma }_{23}^r+{\rho }_{01,3r}{\Gamma }_{02}^r\\ \Rightarrow {\rho }_{03,30}{\Gamma }_{12}^3+{\rho }_{01,10}{\Gamma }_{23}^1\\ \Rightarrow \frac{mA}{2r}{\sin }^2\left(\theta \right){\Gamma }_{12}^3-\frac{m}{r^3}{\Gamma }_{23}^1\\ \Rightarrow \frac{mA}{2r}{\sin }^2\left(\theta \right){\xi }_{12}^3-\frac{m}{2r^3}\cot \left(\theta \right){\xi }_3^1-\frac{m}{r^3}{\Gamma }_{23}^1\end{array}$

$\begin{array}{c}\Rightarrow -d_2\left(\frac{m}{2r^3}{\xi }_3^1\right)-\frac{mA}{r}\sin \left(\theta \right)\cos \left(\theta \right){\xi }_1^3+\frac{m}{r^3}\cot \left(\theta \right){\xi }_3^1\\ -\frac{m}{2r^3}\cot \left(\theta \right){\xi }_3^1-\frac{m}{r^3}{\Gamma }_{23}^1\\ \Rightarrow -\frac{m}{2r^3}\left({\xi }_{23}^1-\cot \left(\theta \right){\xi }_3^1\right)-\frac{m}{2r^3}{\Gamma }_{23}^1\\ \Rightarrow -\frac{3m}{2r^3}{\Gamma }_{23}^1\end{array}$

$\begin{array}{c}{\nabla }_3{R}_{01,02}\Rightarrow {\rho }_{r1,02}{\Gamma }_{03}^r+{\rho }_{0r,02}{\Gamma }_{13}^r+{\rho }_{01,r2}{\Gamma }_{03}^r+{\rho }_{01,0r}{\Gamma }_{23}^r\\ \Rightarrow {\rho }_{02,02}{\Gamma }_{13}^2+{\rho }_{01,01}{\Gamma }_{23}^1\\ \Rightarrow -\frac{mA}{2r}{\Gamma }_{13}^2+\frac{m}{r^3}{\Gamma }_{23}^1\end{array}$

$\begin{array}{c}\Rightarrow -\frac{mA}{2r}\left({\xi }_{13}^2-{\sin }^2\left(\theta \right)\cot \left(\theta \right){\xi }_1^3\right)+\frac{m}{r^3}{\Gamma }_{23}^1\\ \Rightarrow \frac{m}{2r^3}\left({\xi }_{23}^1-\cot \left(\theta \right){\xi }_3^1\right)+\frac{m}{r^3}{\Gamma }_{23}^1\\ \Rightarrow \frac{3m}{2r^3}{\Gamma }_{23}^1\end{array}$

As a tricky exercise too, we advise the reader to treat similarly the case of $\boxed{13}+\boxed{16}$ or $\boxed{21}$ in order to obtain $B_{01,123}=0\Rightarrow d_2{W}_3-d_3{W}_2=0$ because $R_{01,23}=0$ and $R_{03,12}=0$ (care).

REMARK 4.B.1: Though a few conditions like $\boxed{d_2{V}_3-d_3{V}_2=0}$ $\boxed{22}$ or $\boxed{d_2{W}_3-d_3{W}_2=0}$ $\boxed{21}$ look like to be third order CC for $\Omega$, we have thus proved that they come indeed from the first prolongations of the second order CC. The same comment is also valid for a few other striking CC. Using previous results, we have successively $6$ other relations:

${\xi }_{12}^1+{\xi }_{02}^0=d_2\left({\xi }_1^1+{\xi }_0^0\right)=0\Rightarrow \boxed{d_1{V}_2+d_0{W}_2=0}\boxed{1}$

${\xi }_{13}^1+{\xi }_{03}^0=d_3\left({\xi }_1^1+{\xi }_0^0\right)=0\Rightarrow \boxed{d_1{V}_3+d_0{W}_3=0}\boxed{2}$

$\begin{array}{l}{\xi }_{33}^1+\sin \left(\theta \right)\cos \left(\theta \right){\xi }_2^1-{\sin }^2\left(\theta \right){\xi }_{22}^1=0\\ \Rightarrow \boxed{d_3{V}_3+\sin \left(\theta \right)\cos \left(\theta \right)V_2-{\sin }^2\left(\theta \right)d_2{V}_2=0}\boxed{19}\end{array}$

$\begin{array}{l}{\xi }_{33}^0+\sin \left(\theta \right)\cos \left(\theta \right){\xi }_2^0-{\sin }^2\left(\theta \right){\xi }_{22}^0=0\\ \Rightarrow \boxed{d_3{W}_3+\sin \left(\theta \right)\cos \left(\theta \right)W_2-{\sin }^2\left(\theta \right)d_2{W}_2=0}\boxed{20}\end{array}$

because $\boxed{d_2{W}_2+\frac{r}{A}{d}_{0}U=0}$ $\boxed{12}$ and

$\boxed{d_3{W}_3+\sin \left(\theta \right)\cos \left(\theta \right)W_2+\frac{r}{A}{\sin }^2\left(\theta \right)d_{0}U=0}$ $\boxed{17}$

${\xi }_{02}^1-A^2{\xi }_{12}^0=d_2\left({\xi }_0^1-A^2{\xi }_1^0\right)=0\Rightarrow \boxed{d_0{V}_2-A^2{d}_1{W}_2=0}\boxed{24}$

${\xi }_{03}^1-A^2{\xi }_{13}^0=d_3\left({\xi }_0^1-A^2{\xi }_1^0\right)=0\Rightarrow \boxed{d_0{V}_3-A^2{d}_1{W}_3=0}\boxed{23}$

From the 24 B, we have thus used 8 of them and are left with $24-8=16$ expressions involving the $4\times 4=16$ different first derivatives of the 4 functions $\left(V\mathrm{,}W\right)$, namely B $\boxed{3}$ to B $\boxed{18}$. Now, we notice that, among these 24 B, only 4 of them do contain three components $R_{kl\mathrm{,}ij}$ that are not vanishing for the S-metric, namely $\boxed{1}$, $\boxed{2}$, $\boxed{19}$ and $\boxed{20}$. They are providing the terms $d_r{E}_{rr}$ for $r=0,1,2,3$ in the divergence type condition for the linearized Einstein equations implied by the linearized Bianchi identities over the Schwarzschild metric. Accordingly, it does not seem possible to obtain any other third order CC apart from these 4 divergence conditions.

It remains to apply these results to the successive prolongations of the Killing equations, as we know from the intrinsic study achieved in ( [22] [23] ) that we have the successive Lie algebroids:

$R_1^{\left(4\right)}=R_1^{\left(3\right)}\subset R_1^{\left(2\right)}\subset R_1^{\left(1\right)}=R_1\subset J_1(\; T\; )$

with respective dimensions $4=4<5<10=10<20$ and $R_1^{\left(3\right)}$ does not depend any longer on the S-parameter m.

The challenge will be to prove that… the only knowledge of these numbers is sufficient!

In an equivalent way as $g_2=0\Rightarrow g_{r+2}=0,\forall r\ge 0$, we obtain successively:

$dim\left(R_1\right)=dim\left(J_1\left(T\right)\right)-dim\left(S_2{T}^{\mathrm{<mo>\; *\; </mo>}}\right)=20-10=10,$

$dim\left(R_2\right)=dim\left(J_2\left(T\right)\right)-dim\left(J_1\left(S_2{T}^{\mathrm{<mo>\; *\; </mo>}}\right)\right)=60-50=\mathrm{10,}$

$dim\left(R_3\right)=dim\left(R_2\right)-5=5\Rightarrow dim\left(R_{r+4}\right)=dim\left(R_4\right)=dim\left(R_3\right)-1=4$

and shall use these results from now on.

First of all, using the introductory diagram when $q=1,r=1$, we may apply the Spencer δ-map to the symbol top row in order to obtain the diagram:

$\begin{array}{ccccccccc}& & & 0& & 0& & & \\ & & & \downarrow & & \downarrow & & & \\ & 0& \to & S_3{T}^{*}\otimes T& \to & S_2{T}^{*}\otimes F_0& \to & \boxed{h_2}& \to 0\\ & & & \downarrow & & \downarrow & & & \\ & 0& \to & T^{*}\otimes S_2{T}^{*}\otimes T& \to & T^{*}\otimes T^{*}\otimes F_0& \to & 0& \\ & & & \downarrow & & \downarrow & & & \\ 0\to & \boxed{{\wedge }^2{T}^{*}\otimes g_1}& \to & {\wedge }^2{T}^{*}\otimes T^{*}\otimes T& \to & {\wedge }^2{T}^{*}\otimes F_0& \to & 0& \\ & \downarrow & & \downarrow & & \downarrow & & & \\ 0\to & {\wedge }^3{T}^{*}\otimes T& =& {\wedge }^3{T}^{*}\otimes T& \to & 0& & & \\ & \downarrow & & \downarrow & & & & & \\ & 0& & 0& & & & & \end{array}$

Using the Spencer δ-cohomology $H^r\left(g_1\right)=Z^r\left(g_1\right)/B^r\left(g_1\right)$ at $\cdots \to \boxed{{\wedge }^r{T}^{*}\otimes g_1}\to \cdots$, we obtain:

PROPOSITION 4.B.2: $h_2\simeq H^2\left(g_1\right)\Rightarrow n\left(n+1\right)/2\le dim\mathrm{<mo\; stretchy="false">\; (\; </mo>}{Q}_2\mathrm{<mo\; stretchy="false">\; )\; </mo>}\le n^2\left(n^2-1\right)/12$ whenever $n\ge 3$.

Proof: As there cannot be any CC of order one and thus $Q_1=0$, we have the long exact connecting sequence $0\to R_3\to R_2\to h_2\to Q_2\to 0$ and counting the dimensions with $F_0=S_2{T}^{*}$, we have:

$dim\left(Q_2\right)\le dim\left(h_2\right)=dim\left(S_2{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes S_2{T}^{\mathrm{<mo>\; *\; </mo>}}\right)-dim\left(S_3{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes T\right)=n^2\left(n^2-1\right)/12$

This result is confirmed by a circular chase proving that the left bottom δ-map is an epimorphism and a snake chase in the last diagram providing the short exact sequence:

$0\to h_2\to {\wedge }^2{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes g_1\stackrel{\delta }{\to }{\wedge }^3{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes T\to 0,0\to 20\to 36\to 16\to 0$

Indeed, as $det\left(\omega \right)\ne 0$ we may use the metric for providing an isomorphism $T\simeq T^{\mathrm{<mo>\; *\; </mo>}}\mathrm{<mo>\; :\; </mo>}\left({\xi }^r\right)\to \left({\xi }_i={\omega }_{ri}{\xi }^r\right)$ in such a way that $g_1\simeq {\wedge }^2{T}^{\mathrm{<mo>\; *\; </mo>}}$ is defined by ${\xi }_{i,j}+{\xi }_{j,i}=0$ for both the M, S and K metrics.

However, introducing the conformal Killing system of infinitesimal Lie equations with symbol ${\hat{g}}_1$ defined by the $\left(n\left(n+1\right)/2\right)-1$ linear equations ${\omega }_{rj}{\xi }_i^r+{\omega }_{ir}{\xi }_j^r-\frac{2}{n}{\omega }_{ij}{\xi }_r^r=0$ that do not depend on any conformal factor, we have the fundamental diagram II ( [1] [12] ):

$\boxed{\begin{array}{ccccccccc}& & & & & & & 0& \\ & & & & & & & \downarrow & \\ & & & & & 0& & S_2{T}^{\mathrm{<mo>\; *\; </mo>}}& \\ & & & & & \downarrow & & \downarrow & \\ & & & 0& \to & Z^2\left(g_1\right)& \to & H^2\left(g_1\right)& \to 0\\ & & & \downarrow & & \downarrow & & \downarrow & \\ & 0& \to & T^{\mathrm{<mo>\; *\; </mo>}}\otimes {\hat{g}}_2& \stackrel{\delta }{\to }& Z^2\left({\hat{g}}_1\right)& \to & H^2\left({\hat{g}}_1\right)& \to 0\\ & & & \downarrow & & \downarrow & & \downarrow & \\ 0\to & S_2{T}^{\mathrm{<mo>\; *\; </mo>}}& \stackrel{\delta }{\to }& T^{\mathrm{<mo>\; *\; </mo>}}\otimes T^{\mathrm{<mo>\; *\; </mo>}}& \stackrel{\delta }{\to }& {\wedge }^2{T}^{\mathrm{<mo>\; *\; </mo>}}& \to & 0& \\ & & & \downarrow & & \downarrow & & & \\ & & & 0& & 0& & & \end{array}}$

showing that we have the splitting sequence $0\to S_2{T}^{\mathrm{<mo>\; *\; </mo>}}\to H^2\left(g_1\right)\to H^2\left({\hat{g}}_1\right)\to 0$ providing a totally unusual interpretation of the successive Ricci, Riemann and Weyl tensors and the corresponding splitting. However, it must be noticed that the Weyl-type operator is of order 3 when $n=3$ because $n^2\left(n^2-1\right)/12-n\left(n+1\right)/2=n\left(n+1\right)\left(n+2\right)\left(n-3\right)/12$ but of order 2 for $n\ge 4$ ( [4] [9] ). Similar results could be obtained for the Bianchi-type operator as we shall see.

Using now the same procedure for the introductory diagram with $r=2$, we get the diagram:

$\begin{array}{ccccccccc}& & & 0& & 0& & 0& \\ & & & \downarrow & & \downarrow & & \downarrow & \\ & 0& \to & S_4{T}^{*}\otimes T& \to & S_3{T}^{*}\otimes F_0& \to & h_3& \to 0\\ & & & \downarrow & & \downarrow & \searrow & \downarrow & \\ & 0& \to & T^{*}\otimes S_3{T}^{*}\otimes T& \to & T^{*}\otimes S_2{T}^{*}\otimes F_0& \to & \boxed{T^{*}\otimes h_2}& \to 0\\ & & & \downarrow & & \downarrow & & & \\ & 0& \to & {\wedge }^2{T}^{*}\otimes S_2{T}^{*}\otimes T& \to & {\wedge }^2{T}^{*}\otimes T^{*}\otimes F_0& \to & 0& \\ & & & \downarrow & & \downarrow & & & \\ 0\to & \boxed{{\wedge }^3{T}^{*}\otimes g_1}& \to & {\wedge }^3{T}^{*}\otimes T^{*}\otimes T& \to & {\wedge }^3{T}^{*}\otimes F_0& \to & 0& \\ & \downarrow & & \downarrow & & \downarrow & & & \\ 0\to & {\wedge }^4{T}^{*}\otimes T& =& {\wedge }^4{T}^{*}\otimes T& \to & 0& & & \\ & \downarrow & & \downarrow & & & & & \\ & 0& & 0& & & & & \end{array}$

Using a snake chase and Theorem 3.2.3, we obtain the short exact sequence:

$0\to h_3\to T^{\mathrm{<mo>\; *\; </mo>}}\otimes h_2\to H^3\left(g_1\right)\to \mathrm{0,}0\to 60\to 80\to 20\to 0$

A chase around the upper south-east arrow on the right is leading to the following corollary where ${g^{\prime }}_2\subset S_2{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes F_0$ is the symbol of the system ${R^{\prime }}_2\subset J_2\left(F_0\right)$ which is the image of $J_3\left(T\right)$ and ${Q^{\prime }}_1$ is the cokernel of the central bottom map:

COROLLARY 4.B.3: There is a long exact connecting diagram:

$\begin{array}{ccccccccccc}& & 0& & 0& & & & & & \\ & & \downarrow & & \downarrow & & & & & & \\ 0& \to & S_4{T}^{*}\otimes T& \to & S_3{T}^{*}\otimes F_0& \to & T^{*}\otimes h_2& \to & H^3\left(g_1\right)& \to & 0\\ & & \downarrow & & \parallel & & \downarrow & & \downarrow & & \\ 0& \to & {\rho }_1\left({g^{\prime }}_2\right)& \to & S_3{T}^{*}\otimes F_0& \to & T^{*}\otimes Q_2& \to & {Q^{\prime }}_1& \to & 0\\ & & & & \downarrow & & \downarrow & & \downarrow & & \\ & & & & 0& & 0& & 0& & \end{array}$

allowing to use the Bianchi identities as $B\in F_2\simeq H^3\left(g_1\right)$ and we have $dim\left({Q^{\prime }}_1\right)\le dim\left(H^3\left(g_1\right)\right)$.

Proof: Using the notations of the introductory diagram and the fact that $Q_1=0$, we have the following two commutative and exact diagrams obtained by choosing $F_1=Q_2$ for the first, then $F_1=Q_3$ for the second and so on, in a systematic manner as in the motivating examples:

$\begin{array}{ccccccccc}& 0& & 0& & 0& & 0& \\ & \downarrow & & \downarrow & & \downarrow & & \downarrow & \\ 0\to & {\rho }_1\left({g^{\prime }}_2\right)& \to & S_3{T}^{*}\otimes F_0& \to & T^{*}\otimes Q_2& \to & {Q^{\prime }}_1& \to 0\\ & \downarrow & & \downarrow & & \downarrow & \searrow & \parallel & \\ 0\to & {\rho }_1\left({R^{\prime }}_2\right)& \to & J_3\left(F_0\right)& \to & J_1\left(Q_2\right)& \to & {Q^{\prime }}_1& \to 0\\ & \downarrow & & \downarrow & & \downarrow & & \downarrow & \\ 0\to & {R^{\prime }}_2& \to & J_2\left(F_0\right)& \to & Q_2& \to & 0& \\ & \downarrow & & \downarrow & & \downarrow & & & \\ & 0& & 0& & 0& & & \end{array}$

First, we have the short exact sequence $0\to R_2\to J_2\left(T\right)\to J_1\left(F_0\right)\to 0$ with $10-60+50=0$ and get ${R^{\prime }}_0=F_0$, ${R^{\prime }}_1=J_1\left(F_0\right)$ and $dim\left({\rho }_1\left({R^{\prime }}_0\right)\right)-dim\left({R^{\prime }}_1\right)=0$, that is no CC of order 1.

Now, using the long exact sequence:

$0\to R_3\to J_3\left(T\right)\to J_2\left(F_0\right)\to Q_2\to 0$

$\begin{array}{l}{R^{\prime }}_2\subset {\rho }_1\left({R^{\prime }}_1\right)={\rho }_1\left(J_1\left(F_0\right)\right)=J_2\left(F_0\right)\\ \Rightarrow dim\left({\rho }_1\left({R^{\prime }}_1\right)\right)-dim\left({R^{\prime }}_2\right)=150-135=dim\left(Q_2\right)=15\end{array}$

because $dim\left({R^{\prime }}_2\right)=dim\left(J_3\left(T\right)\right)-dim\left(R_3\right)=140-5=135$ and there are 15 second order CC.

Then, with $dim\left({Q^{\prime }}_1\right)=x$, we obtain by counting the dimensions:

$dim\left({\rho }_1\left({R^{\prime }}_2\right)\right)=dim\left(J_3\left(F_0\right)\right)+x-dim\left(J_1\left(Q_2\right)\right)=350+x-75=x+\mathrm{275,}$

$dim\left({R^{\prime }}_3\right)=dim\left(J_4\left(T\right)\right)-dim\left(R_4\right)=dim\left(J_3\left(F_0\right)\right)-dim\left(Q_3\right)=276$

$\Rightarrow y=dim\left({\rho }_1\left({R^{\prime }}_2\right)\right)-dim\left({R^{\prime }}_3\right)=x+275-276=x-1$

that is $y\ge 3$ because $x\ge 4$ and thus $x=4\Rightarrow y=3$ if we only take into account the 4 divergence condition of the Einstein equations. The situation will be worst for the Kerr metric with $y=6$.

After one prolongation, we get:

$\begin{array}{ccccccccc}& 0& & 0& & 0& & 0& \\ & \downarrow & & \downarrow & & \downarrow & & \downarrow & \\ 0\to & {\rho }_1\left({g^{\prime }}_3\right)& \to & S_4{T}^{*}\otimes F_0& \to & T^{*}\otimes Q_3& \to & {Q^{\prime }}_1& \to 0\\ & \downarrow & & \downarrow & & \downarrow & & \parallel & \\ 0\to & {\rho }_1\left({R^{\prime }}_3\right)& \to & J_4\left(F_0\right)& \to & J_1\left(Q_3\right)& \to & {Q^{\prime }}_1& \to 0\\ & \downarrow & & \downarrow & & \downarrow & & \downarrow & \\ 0\to & {R^{\prime }}_3& \to & J_3\left(F_0\right)& \to & Q_3& \to & 0& \\ & \downarrow & & \downarrow & & \downarrow & & & \\ & 0& & 0& & 0& & & \end{array}$

From this second diagram we obtain the commutative and exact diagram:

$\begin{array}{ccccccccc}& 0& & 0& & & & & \\ & \downarrow & & \downarrow & & & & & \\ 0\to & {R^{\prime }}_4& \to & J_4\left(F_0\right)& \to & Q_4& \to & 0& \\ & \downarrow & & \parallel & & \downarrow & & & \\ 0\to & {\rho }_1\left({R^{\prime }}_3\right)& \to & J_4\left(F_0\right)& \to & J_1\left(Q_3\right)& \to & {Q^{\prime }}_1& \to 0\\ & & & \downarrow & & & & & \\ & & & 0& & & & & \end{array}$

Indeed, setting again $dim\left({Q^{\prime }}_1\right)=x$, we obtain now similarly:

$dim\left({\rho }_1\left({R^{\prime }}_3\right)\right)=dim\left(J_4\left(F_0\right)\right)+x-dim\left(J_1\left(Q_3\right)\right)=700+x-370=x+\mathrm{330,}$

$dim\left({R^{\prime }}_4\right)=dim\left(J_5\left(T\right)\right)-dim\left(R_5\right)=dim\left(J_4\left(F_0\right)\right)-dim\left(Q_4\right)=500$

$\Rightarrow y=dim\left({\rho }_1\left({R^{\prime }}_3\right)\right)-dim\left({R^{\prime }}_4\right)=x+330-500=x-170$

that is $y=0\iff x=170$. We find exactly $dim\left(F_2\right)=170$ like in ( [22], p. 1996) and the condition $y=0$ just means that the CC of order 4 are generated by the CC of order 3, but we have only ${R^{\prime }}_4\subseteq {\rho }_1\left({R^{\prime }}_3\right)$ in general.

With one more prolongation, applying again the δ-map to the top symbol sequence, we get the following commutative diagram:

$\begin{array}{ccccccc}& 0& & 0& & 0& \\ & \downarrow & & \downarrow & & \downarrow & \\ 0\to & S_6{T}^{*}\otimes T& \to & S_5{T}^{*}\otimes F_0& \to & h_5& \to 0\\ & \downarrow \delta & & \downarrow \delta & & \downarrow & \\ 0\to & T^{*}\otimes S_5{T}^{*}\otimes T& \to & T^{*}\otimes S_4{T}^{*}\otimes F_0& \to & T^{*}\otimes h_4& \to 0\\ & \downarrow \delta & & \downarrow \delta & & \downarrow & \\ 0\to & {\wedge }^2{T}^{*}\otimes S_4{T}^{*}\otimes T& \to & {\wedge }^2{T}^{*}\otimes S_3{T}^{*}\otimes F_0& \to & {\wedge }^2{T}^{*}\otimes h_3& \to 0\\ & \downarrow \delta & & \downarrow \delta & & \downarrow & \\ 0\to & {\wedge }^3{T}^{*}\otimes S_3{T}^{*}\otimes T& \to & {\wedge }^3{T}^{*}\otimes S_2{T}^{*}\otimes F_0& \to & {\wedge }^3{T}^{*}\otimes h_2& \to 0\\ & \downarrow \delta & & \downarrow \delta & & \downarrow & \\ 0\to & {\wedge }^4{T}^{*}\otimes S_2{T}^{*}\otimes T& \to & {\wedge }^4{T}^{*}\otimes T^{*}\otimes F_0& \to & 0& \\ & \downarrow & & \downarrow & & & \\ & 0& & 0& & & \end{array}$

where the right exact vertical column is $0\to 224\to 504\to 360\to 80\to 0$. It just remains to replace in the two upper right epimorphisms $h_5$ by $T^{\mathrm{<mo>\; *\; </mo>}}\otimes Q_4$ and $h_4$ by $Q_4$ along with the following commutative diagram where we have chosen $F_1=Q_4$:

$\begin{array}{rrrrr}\hfill 0& \hfill & \hfill 0& \hfill & \hfill \\ \hfill & \hfill \searrow & \hfill \downarrow & \hfill & \hfill \\ \hfill & \hfill & \hfill h_5& \hfill & \hfill \\ \hfill & \hfill & \hfill \downarrow & \hfill \searrow & \hfill \\ \hfill 0& \hfill \to & \hfill T^{*}\otimes h_4& \hfill \to & \hfill T^{*}\otimes Q_4\end{array}$

in order to obtain the long exact sequence $0\to S_6{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes T\to S_5{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes F_0\to T^{\mathrm{<mo>\; *\; </mo>}}\otimes Q_4$.

Finally, chasing in the following commutative and exact introductory diagram:

$\begin{array}{ccccccccccc}& & & 0& & 0& & 0& & 0& \\ & & & \downarrow & & \downarrow & & \downarrow & & \downarrow & \\ & 0& \to & S_6{T}^{*}\otimes T& \stackrel{{\sigma }_5\left(\Phi \right)}{\to }& S_5{T}^{*}\otimes F_0& \to & T^{*}\otimes Q_4& \to & {Q^{\prime }}_1& \to 0\\ & \downarrow & & \downarrow & & \downarrow & & \downarrow & & \parallel & \\ 0\to & R_6& \to & J_6\left(T\right)& \stackrel{{\rho }_5\left(\Phi \right)}{\to }& J_5\left(F_0\right)& \to & J_1\left(Q_4\right)& \to & {Q^{\prime }}_1& \to 0\\ & \downarrow & & \downarrow & & \downarrow & & \downarrow & & \downarrow & \\ 0\to & R_5& \to & J_5\left(T\right)& \stackrel{{\rho }_4\left(\Phi \right)}{\to }& J_4\left(F_0\right)& \to & Q_4& \to & 0& \\ & \downarrow & & \downarrow & & \downarrow & & \downarrow & & & \\ & 0& & 0& & 0& & 0& & & \end{array}$

we deduce that ${R^{\prime }}_5={\rho }_1\left({R^{\prime }}_4\right)$ is involutive with $dim\left({R^{\prime }}_5\right)=840-4=836$ and symbol ${g^{\prime }}_5\simeq S_6{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes T$.

Unhappily, the reader will check at once that a similar procedure cannot be applied in order to prove that ${R^{\prime }}_4={\rho }_1\left({R^{\prime }}_3\right)$. Indeed, if we still have a monomorphism $0\to h_4\to Q_4$ we do not have a monomorphism $h_3\to Q_3$ because now this map has a kernel of dimension equal to $dim\left(R_3/R_3^{\left(1\right)}\right)=5-4=1$ according to the corresponding long exact connecting sequence.

IT IS THUS NOT POSSIBLE TO PROVE THAT THERE ARE ONLY SECOND AND THIRD ORDER GENERATING CC IN A SIMPLE INTRINSIC WAY.

However, like in the first motivating example in which we should be waiting for third order CC but a direct computation was proving that only second order ones could be used, we have:

THEOREM 4.B.4: The CC of the first order operator $D\mathrm{<mo>\; :\; </mo>}T\to F_0$ are generated by a third order operator $D_1:F_0\to F_1=Q_3$ and we have thus ${R^{\prime }}_4={\rho }_1\left({R^{\prime }}_3\right)$.

Proof: With $F_0=S_2{T}^{*}$ and $F_1=Q_3$ while applying the Spencer operator, we obtain the following commutative diagram in which the two central vertical columns are locally exact ( [1] [5] ):

$\begin{array}{ccccccccc}& 0& & 0& & 0& & & \\ & \downarrow & & \downarrow ,& & \downarrow & & & \\ 0\to & \Theta & \to & T& \stackrel{\mathcal{D}}{\to }& \boxed{F_0}& \stackrel{{\mathcal{D}}_1}{\to }& F_1& \\ & \downarrow j_4& & \downarrow j_4& & \downarrow j_3& \searrow & \parallel & \\ 0\to & R_4& \to & J_4\left(T\right)& \to & J_3\left(F_0\right)& \to & F_1& \to 0\\ & \downarrow d& & \downarrow d& & \downarrow d& & & \\ 0\to & \boxed{T^{*}\otimes R_3}& \to & T^{*}\otimes J_3\left(T\right)& \to & T^{*}\otimes J_2\left(F_0\right)& & & \\ & \downarrow d& & \downarrow d& & & & & \\ 0\to & {\wedge }^2{T}^{*}\otimes R_2& \to & {\wedge }^2{T}^{*}\otimes J_2(\; T\; )\end{array}$

Chasing in this diagram by using the Snake lemma of the second section, we discover that the local exactness at $F_0$ of the top row is equivalent to the local exactness at $T^{\mathrm{<mo>\; *\; </mo>}}\otimes R_3$ of the left column.

Now, we have the commutative diagram:

$\begin{array}{cccccccc}& & & 0& & 0& & 0\\ & & & \downarrow & & \downarrow & & \downarrow \\ 0\to & \Theta & \stackrel{j_5}{\to }& R_5& \stackrel{d}{\to }& T^{*}\otimes R_4& \stackrel{d}{\to }& {\wedge }^2{T}^{*}\otimes R_3\\ & \parallel & & \downarrow & & \downarrow & & \downarrow \\ 0\to & \Theta & \stackrel{j_4}{\to }& R_4& \stackrel{d}{\to }& \boxed{T^{*}\otimes R_3}& \stackrel{d}{\to }& {\wedge }^2{T}^{*}\otimes R_2\\ & & & \downarrow & & \downarrow & & \downarrow \\ & & & 0& \to & T^{*}\otimes \left(R_3/R_3^{\left(1\right)}\right)& \stackrel{d}{\to }& {\wedge }^2{T}^{*}\otimes \left(R_2/R_2^{\left(1\right)}\right)\\ & & & & & \downarrow & & \downarrow \\ & & & & & 0& & 0\end{array}$

The top row is known to be locally exact as it is isomorphic to a part of the Poincaré sequence according to the commutative diagram with $R_4\simeq R_5\simeq R_6$:

$\begin{array}{cccccccc}& & & 0& & 0& & 0\\ & & & \downarrow & & \downarrow & & \downarrow \\ 0\to & \Theta & \stackrel{j_6}{\to }& R_6& \stackrel{d}{\to }& T^{*}\otimes R_5& \stackrel{d}{\to }& {\wedge }^2{T}^{*}\otimes R_4\\ & \parallel & & \downarrow & & \downarrow & & \downarrow \\ 0\to & \Theta & \stackrel{j_5}{\to }& R_5& \stackrel{d}{\to }& T^{*}\otimes R_4& \stackrel{d}{\to }& {\wedge }^2{T}^{*}\otimes R_3\\ & & & \downarrow & & \downarrow & & \\ & & & 0& & 0& & \end{array}$

The bottom row is purely algebraic as it is induced by the exact sequence obtained by applying the Spencer operator to the long exact connecting sequence and chasing along the south west diagonal:

$0\to h_4\stackrel{\delta }{\to }{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes h_3\stackrel{\delta }{\to }{\wedge }^2{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes h_2\to {\wedge }^4{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes g_1\to 0$

$0\to 126\to 240\to 120\to 6\to 0$

Changing the confusing notations used in ( [24] ), we prove that the bottom Spencer operator is injective. Indeed, we have the following representative parametric jets for the various Lie equations:

$dim\left(R_2\right)=10\Rightarrow \left\{{\xi }^0\mathrm{,}{\xi }^1\mathrm{,}{\xi }^2\mathrm{,}{\xi }^3\mathrm{,}{\xi }_0^1\mathrm{,}{\xi }_2^0\mathrm{,}{\xi }_3^0\mathrm{,}{\xi }_2^1\mathrm{,}{\xi }_3^1\mathrm{,}{\xi }_3^2\right\}$

$dim\left(R_3\right)=5\Rightarrow \left\{{\xi }^0\mathrm{,}{\xi }^2\mathrm{,}{\xi }^3\mathrm{,}{\xi }_0^1\mathrm{,}{\xi }_3^2\right\}\mathrm{,}dim\left(R_4\right)=4\Rightarrow \left\{{\xi }^0\mathrm{,}{\xi }^2\mathrm{,}{\xi }^3\mathrm{,}{\xi }_3^2\right\}$

$dim\left(R_3/R_3^{\left(1\right)}\right)=1\Rightarrow \left\{{\xi }_0^1\right\}\mathrm{,}dim\left(R_2/R_2^{\left(1\right)}\right)=5\Rightarrow \left\{{\xi }^1\mathrm{,}{\xi }_2^1\mathrm{,}{\xi }_3^1\mathrm{,}{\xi }_2^0\mathrm{,}{\xi }_3^0\right\}$

We also recall the definition of the Spencer operator $d\mathrm{<mo>\; :\; </mo>}{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes J_{q+1}\left(T\right)\to {\wedge }^2{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes J_q\left(T\right)$:

$\left({\xi }_{\nu ,i}^k\right)\to {\xi }_{\mu ,ij}^k=\left({\partial }_i{\xi }_{\mu ,j}^k-{\partial }_j{\xi }_{\mu ,i}^k+{\xi }_{\mu +1_j,i}^k-{\xi }_{\mu +1_i,j}^k\right)$

Accordingly, we may choose local coordinates $\left({\xi }_{\mathrm{0,}i}^1\right)$ for a representative and a representative of the image by d is for example $\left({\xi }_{\mathrm{,0}i}^1={\xi }_{\mathrm{0,}i}^1-{\xi }_{i\mathrm{,0}}^1\right)$. Now, as $dim\left(R_3/R_3^{\left(1\right)}\right)=4-3=1$, we may introduce the four local coordinates ${\xi }_{\mathrm{0,}i}^1$ and ${\xi }_{\mathrm{1,}i}^0$ such that ${\xi }_{0,i}^1-A^2{\xi }_{1,i}^0=0$, $\forall i=0,1,2,3$. We may also use the $6\times 5=30$ local coordinates $\left({\xi }_{\mathrm{,}ij}^1\mathrm{,}{\xi }_{\mathrm{2,}ij}^1\mathrm{,}{\xi }_{\mathrm{3,}ij}^1\mathrm{,}{\xi }_{\mathrm{2,}ij}^0\mathrm{,}{\xi }_{\mathrm{3,}ij}^0\right)$ in order to describe ${\wedge }^2{T}^{\mathrm{<mo>\; *\; </mo>}}\otimes \left(R_2/R_2^{\left(1\right)}\right)$. In the kernel of d, we have in particular

${\xi }_{,0i}^1={\xi }_{0,i}^1-{\xi }_{i,0}^1=0\Rightarrow {\xi }_{0,i}^1={\xi }_{i,0}^1=0,\forall i=1,2,3$ because ${\xi }_1^1=\frac{m}{2A{r}^2}{\xi }^1$ in $R_1$ but also ${\xi }_{,01}^0={\xi }_{0,1}^0-{\xi }_{1,0}^0=0\Rightarrow {\xi }_{1,0}^0=0\Rightarrow {\xi }_{0,0}^1=0$ because $\left\{{\xi }^0\right\}$ is among the parametric jets of $R_3$ and thus ${\xi }_{0,i}^1=0,\forall i=0,1,2,3$. The bottom Spencer operator is thus injective and the bottom sequence is thus exact. A circular chase ends the proof: If $b\in T^{\mathrm{<mo>\; *\; </mo>}}\otimes R_3$ is killed by d, then its projection $c\in T^{\mathrm{<mo>\; *\; </mo>}}\otimes \left(R_3/R_3^{\left(1\right)}\right)$ is also killed by d and is such that $c=0$. Accordingly, $\exists a\in T^{\mathrm{<mo>\; *\; </mo>}}\otimes R_4$ with image b under the monomorphism $T^{\mathrm{<mo>\; *\; </mo>}}\otimes R_4\to T^{\mathrm{<mo>\; *\; </mo>}}\otimes R_3$ and such that $da=0$. We may thus find $e\in R_5$ and $f\in R_4$ because $R_5\simeq R_4$ with $a=de\Rightarrow$

Conflicts of Interest

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

References

[1]	Pommaret, J.-F. (1994) Partial Differential Equations and Group Theory. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-2539-2
[2]	Pommaret, J.-F. (2005) 5 Algebraic Analysis of Control Systems Defined by Partial Differential Equations. In: Lamnabhi-Lagarrigue, F., Loría, A. and Panteley, E., Eds., Advanced Topics in Control Systems Theory, Lecture Notes in Control and Information Sciences Vol. 311, Springer, London, 155-223. https://doi.org/10.1007/11334774_5
[3]	Pommaret, J.-F. (2019) Generating Compatibility Conditions and General Relativity. Journal of Modern Physics, 10, 371-401. (arXiv: 1811.12186). https://doi.org/10.4236/jmp.2019.103025
[4]	Pommaret, J.-F. (2021) The Conformal Group Revisited. Journal of Modern Physics, 12, 1822-1842. (arXiv:2006.03449). https://doi.org/10.4236/jmp.2021.1213106
[5]	Pommaret, J.-F. (1978/1983) Systems of Partial Differential Equations and Lie Pseudogroups. Gordon and Breach, New York; Russian translation, MIR, Moscow.
[6]	Aksteiner, S., Andersson L., Backdahl, T., Khavkine, I. and Whiting, B. (2021) Compatibility Complex for Black Hole Spacetimes. Communications in Mathematical Physics, 384, 1585-1614. (arXiv: 1910.08756). https://doi.org/10.1007/s00220-021-04078-y
[7]	Aksteiner, S. and Backdahl, T. (2019) New Identities for Linearized Gravity on the Kerr Spacetime. Physical Review D, 99, Article ID: 044043. (arXiv: 1601.06084). https://doi.org/10.1103/PhysRevD.99.044043
[8]	Aksteiner, S. and Backdahl, T. (2018) All Local Gauge Invariants for Perturbations of the Kerr Spacetime. Physical Review Letters, 121, Article ID: 051104. (arXiv: 1803.05341). https://doi.org/10.1103/PhysRevLett.121.051104
[9]	Pommaret, J.-F. (2016) Deformation Theory of Algebraic and Geometric Structures. Lambert Academic Publisher (LAP), Saarbrucken, Germany. In: Dubreil, S.P. and Malliavin, M.-P. (Eds), Topics in Invariant Theory, Springer Lecture Notes in Mathematics, 244-254. https://arxiv.org/abs/1207.1964
[10]	Pommaret, J.-F. (2001) Partial Differential Control Theory. Kluwer, Dordrecht. (Zbl 1079.93001).
[11]	Goldschmidt, H. (1968) Prolongations of Linear Partial Differential Equations: I Inhomogeneous equations. Annales scientifiques de l’école Normale Supérieure, 1, 617-625. https://doi.org/10.24033/asens.1173
[12]	Pommaret, J.-F. (1983) Differential Galois Theory. Gordon and Breach, New York.
[13]	Pommaret, J.-F. (1988) Lie Pseudogroups and Mechanics. Gordon and Breach, New York.
[14]	Pommaret, J.-F. (2018) New Mathematical Methods for Physics. Mathematical Physics Books, Nova Science Publishers, New York, 150 p.
[15]	Rotman, J.J. (1979) An Introduction to Homological Algebra, (Pure and Applied Mathematics). Academic Press, Cambridge.
[16]	Pommaret, J.-F. (2016) Airy, Beltrami, Maxwell, Einstein and Lanczos Potentials revisited. Journal of Modern Physics, 7, 699-728. (arXiv: 1512.05982). https://doi.org/10.4236/jmp.2016.77068
[17]	Pommaret, J.-F. (2010) Parametrization of Cosserat Equations. Acta Mechanica, 215, 43-55. https://doi.org/10.1007/s00707-010-0292-y
[18]	Pommaret, J.-F. (2019) The Mathematical Foundations of Elasticity and Electromagnetism Revisited. Journal of Modern Physics, 10, 1566-1595. (arXiv: 1802.02430). https://doi.org/10.4236/jmp.2019.1013104
[19]	Pommaret, J.-F. (2021) Minimum Parametrization of the Cauchy Stress Operator. Journal of Modern Physics, 12, 453-482. (arXiv: 2101.03959). https://doi.org/10.4236/jmp.2021.124032
[20]	Pommaret, J.-F. (2021) Homological Solution of the Lanczos Problems in Arbitrary Dimensions. Journal of Modern Physics, 12, 829-858. (arXiv: 1807.09122). https://doi.org/10.4236/jmp.2021.126053
[21]	Khavkine, I. (2017) The Calabi Complex and Killing Sheaf Cohomology. Journal of Geometry and Physics, 113, 131-169. https://doi.org/10.1016/j.geomphys.2016.06.009
[22]	Pommaret, J.-F. (2018) Minkowski, Schwarzschild and Kerr Metrics Revisited. Journal of Modern Physics, 9, 1970-2007. (arXiv: 1805.11958v2). https://doi.org/10.4236/jmp.2018.910125
[23]	Pommaret, J.-F. (2020) A Mathematical Comparison of the Schwarzschild and Kerr Metrics, Journal of Modern Physics. Journal of Modern Physics, 11, 1672-1710. (arXiv: 2010.07001). https://doi.org/10.4236/jmp.2020.1110104
[24]	Pommaret, J.-F. (2021) Differential Correspondences and Control Theory. Advances in Pure Mathematics, 11, 835-882. (arXiv: 2107.08797). https://doi.org/10.4236/apm.2021.1111056
[25]	Pommaret, J.-F. (2015) Relative Parametrization of Linear Multidimensional Systems. Multidimensional Systems and Signal Processing, 26, 405-437. https://doi.org/10.1007/s11045-013-0265-0
[26]	Macaulay, F.S. (1916) The Algebraic Theory of Modular Systems, Cambridge Tract. Cambridge University Press, Cambridge, 19. https://doi.org/10.3792/chmm/1263317740
[27]	Vessiot, E. (1903) Théorie des Groupes Continus. Annales scientifiques de l’École Normale Supérieure, 20, 411-451 (Can Be Found in Numdam). https://doi.org/10.24033/asens.529
[28]	Pommaret, J.-F. (2013) The Mathematical Foundations of General Relativity Revisited. Journal of Modern Physics, 4, 223-239. (arXiv: 1306.2818). https://doi.org/10.4236/jmp.2013.48A022