Matter and Ricci collineations

The infinitesimal transformations that leave invariant a two-covariant symmetric tensor are studied. The interest of these symmetry transformations lays in the fact that this class of tensors includes the energy-momentum and Ricci tensors. We find that in most cases the class of infinitesimal generators of these transformations is a finite dimensional Lie algebra but also, in some cases exhibiting a higher degree of degeneracy, this class is infinite dimensional and may fail to be a Lie algebra.


Introduction
The interest in the study of symmetries in General Relativity is long-standing. Some of them, namely isometries and affine transformations and their infinitesimal counterparts, Killing vector fields and affine vector fields, are well understood since long ago [1], [2], [3], [4].
In the last twenty years, there has been an steady interest in curvature collineations, Ricci collineations and even matter (Einstein) collineations [5], [6], [7], [8]. Their infinitesimal counterparts, namely collineation fields, are characterized by the vanishing of the Lie derivative of the curvature tensor (resp., the Ricci or the energy-momentum tensor). Collineation fields are thus an extension of the aforementioned Killing fields and affine fields in that every Killing vector field is an affine vector filed which in turn is a curvature collineation field and also a Ricci and a matter collineation field.
However it is well known that collineation fields present new features. Indeed, contrarily to the case of Killing and affine fields, the class C of curvature (resp., Ricci and matter) collineation fields is a real vector space which may be infinite dimensional; this is due to the dependence on arbitrary functions, which also results in the fact that a collineation field needs not to be smooth and, as a consequence, C may not be a Lie algebra [7].
We shall here concentrate in infinitesimal Ricci and matter collineations. Our results are also useful in the study of curvature collineations because any of them is necessarily a Ricci collineation too.
In our view, although most recent work on the subject the spacetime metric (from which the Ricci tensor is derived) is given a significant presence in the approach to the problem, paying attention to the metric is rather hindering than helpful.
Given a 4-manifold M and a smooth field of symmetric 2-covariant tensors T , we shall concentrate on finding the class C T of vector fields X such that L X T = 0 and try to find out whether the number of dimensions of C T is finite, whether X is smooth and whether C T is a Lie algebra.
The answer to these questions depend, but not exclusively, on the rank of T . Particularly, if rank T = 4, T itself can be taken as a non-degenerate metric tensor and the collineation equation is actually a Killing equation and, as it is well known [1], C T is a Lie subalgebra of X (M), the class of smooth vector fields, and dim C T ≤ 10.
For rank T < 4, we come across an assorted casuistry which depends not only on the rank of T but also on the derivatives of T . We aim to set up a classification of the tensor fields T according to its class C T of collineation fields. The first variable to consider is the rank and different methods and techniques are suitable for different ranks, e. g. for rank three tensors the method is more similar to that used in studing the Killing fields whereas techniques imported from simplectic mechanics are best suited for rank one tensors. Whimsical as it could seem, the order in which the different ranks are presented here is dictated by their progressive degree of difficulty.
The classification we obtain is rather simple if only the generic, i. e. less degenerate, cases are considered. However as the degree of degeneracy increases, an intricated mess of cases and subcases arises. This is why we clos the paper with a Summary section.

Collineation fields
Let T be a 2-covariant symmetric smooth tensor field on a 4-manifold M. A T -collineation field (shortly, a collineation field) is a vector field X such that Notice that the definiton requires that X is of class C 1 at least but in general it does not guarantee that X is smooth.
Furthermore, as a consequence of the fact that [L X , L Y ] = L [X,Y] , provided that X and Y are of class C 2 at least, it is obvious that the class of smooth collineation fields is a Lie algebra.
The case rank T = 4 having been discussed, and finished off, in the Introduction, we shall assume Let us now expand the Lie derivatives of any φ ν as Including this and (2) it easily follows that equation (1) is equivalent to that is, the matrix M ν β is an so(r, s)-valued function on M.
The differential system associated to T is It is a differential system of constant rank 4 − m (see [9], sections V.3 and V.4) and the associated T is said to be holonomous if its associated Pfaff system is integrable and, in such a case, local charts (y 1 , . . . y 4 ) exist such that φ α = a α β (y b ) dy β -see ref. [9], Lemma V.4.10. In terms of these coordinates, where T αβ (y b ) = η µν a µ α (y b ) a ν β (y b ).

Collineations of a rank tensor
If rankT = 3, it is obvious that T is holonomous and local charts exist such that the expressions (5) hold. We write N := ∂ 4 and T = T αβ dx α ⊗ dx β , with det T αβ = 0 and then decompose the collineation field as It is obvious that T (N, _) = 0, which implies that L f N T = f L N T and therefore, equation (1) amounts to The projections of this over N and over the submanifolds y 4 = constant respectively yield where K := 1 2 ∂ 4 T αβ dy α ⊗ dy β or, in components, where ∇ is the Levi-Civita connection for the non-degenerate metric T αβ on the hypersurfaces for some appropriate f ?
If K αβ = 0, i. e. L N T = 0, the answer is obvious because it reduces to a Killing equation in 3 dimensions. The collineation field is then X = Z + f N, where f is arbitrary and Z is a Killing vector for the non-degenerate metric T in each submanifold y 4 constant.
If K αβ = 0, things are not so simple. Let us start from equations (8) -that are equivalent to (7)-, the second of them implies that The integrability conditions imply new equations on Ω αβ and f . These can be derived by means of the Lie derivative of a connection -see ref. [2], section I. 4 . We have that Then, as for the Levi-Civita connection ∇ µ T κλ = 0, from (10) it easily follows that also where T αλ T λµ = δ α µ .
What has been done so far amounts to analysing the integrability conditions derived from the commutation of ∇ µ and ∇ ν . Let us now turn to the commutation of ∇ ν and N = ∂ 4 .
The first of equations (7) implies that [L Z , L N ] = 0 which, applied to the second equation (7) yields Similarly, by applying the commutation relation [L Z , L N ] = 0 to equations (12), (14) and to the whole above hierarchy, we should obtain further algebraic relations connecting Z λ , Ω µν , f ,ḟ , f |µ , f |µ , f µν , etc, which we shall not write explicitely.
Also, applying ∂ 4 to (9) and including (8) we have that and, using (15) it follows that Turning now back to equation (15), unlessK µ β ∝ K µ β , it permits to derive f as a linear function of Z λ and Ω µν . Indeed, ifK µ β is not proportional to K µ β , it exists M α µ such that M α µ K µ α = 0 and M α µK µ α = 1; therefore (If there are more than one independent matrix M α β fulfilling the above trace equalities, it will result in constraints connecting Z α and Ω µβ .) Substituting then this f in equations (8), (13) and (16) we obtain a closed partial differential system on Z α and Ω µν (with no extra functions). If it is integrable, each solution is parametrized by six real numbers, namely Z α (0) and Ω µν (0), i. e. the values of the unknowns at one point. The above mentioned hierarchy of integrability conditions then act as constraints on these parameters and the number of dimensions of the collineation algebra C T is at most six.
If, on the contrary,K µ β = bK µ β , then equation (15) implies that which allows to deriveḟ as a linear function of Z α , Ω µν and f . Indeed, as K µ Now, applying ∂ 4 to both sides of equation (13), we obtain that (see the Appendix) where W λ µκ is a linear function of Z α and Ω µν . In some cases this permits to derive f α as a linear function of Z α , Ω µβ and f . Indeed, K λα K µκ − K µβ K β κ T αλ can be seen as a linear map from the 4-dimensional space f α into the 4× 10 space W λ µκ and it can be inverted whenever (a) it is injective, which only fails to happen if K α µ K µ β = 0 or K α µ ∝ δ α µ , and (b) the right hand side W λ µκ fulfills some conditions, i. e. some linear constraints on Z α , Ω µβ and f . This f α , written as a linear function of Z α , Ω µβ and f , together with (8), (9), (13) and (17), yields a partial differential system on the variables Z α , Ω µν and f . If it is integrable, each solution is parametrized by seven real numbers, namely Z α (0), Ω µν (0) and f (0), the values of the variables at a point. The above mentioned hierarchy of integrability conditions are to be taken as constraints on these parameters and the number of dimensions of the collineation algebra C T is at most seven.

Collineations of a rank 1 tensor
If rank T = 1, it can be written locally as T = ± φ ⊗ φ , φ ∈ Λ 1 M , and the collineation condition L X T = 0 is equivalent to L X φ = 0, which means that, locally, a function f exists such that which is a linear system on X whose compatibility depends on f and on the ranks of the differential forms φ and dφ. The general solution I f can be written as is the general solution of the homogeneous system.
To study the compatibility of (19), we invoke the following corollary of Darboux theorem -see [9], Theorem VI.4.1-Theorem 1 Given φ ∈ Λ 1 (M ), they exist a canonical coordinate system p 1 , p 2 , q 1 , q 2 and a function ψ, such that φ = dψ + e 1 p 1 dq 1 + e 2 p 2 dq 2 , with e 1 ≥ e 2 e 1 , e 2 = 0, 1 (20) A remark on notation is appropriate: hereon a stroke means partial derivative, so v |a := ∂ a v := ∂v/∂x a , a = 1, . . . 4; particularly in canonical coordinates (q i , p j ), we shall write Writing now X and df in canonical coordinates, we have Then, substituting this and (20) into (19.a), we obtain that the latter amounts to According to the values of e 1 and e 2 , different cases are possible, which we shall analyse separately: Then Ω := dφ is a symplectic form and e 1 = e 2 = 1.
In this case the class of the differential form φ is 4 -see [9], Section VI.1.3-and Darboux theorem states more precisely that canonical local charts exist such that ψ = 0, that is which, by Euler theorem, means that f (q i , p j ) is an homogeneous function of the first degree in the variables p j . The general collineation field is thus with dψ ∧ dp 1 ∧ dq 1 = 0. In this case, local charts of canonical coordinates exist such that dφ = dq 2 + dp 1 ∧ dq 1 , i. e. e 2 = 0 and e 1 = 1. Combining then equations (22) and (21), we obtain that The component X 2 is not determined and the general collineation field is where f (p 1 , q 1 ) and X 2 (p i , q j ) are arbitrary functions of their respective variables.
In this case φ is integrable and a local chart exists such that φ = p 1 dq 1 combining then equations (22) and (24), we obtain that There is no constraint on the components X 2 and X 2 and the general collineation field, which contains three arbitrary functions, namely F (q 1 ), X 2 (p i , q j ) and X 2 (p i , q j ). x 1 = ψ, the general collineation field is with X ν (x a ) arbitrary.

Collineations of a rank 2 tensor
In what follows it will be helpful to consider the 2-forms dφ α and the exterior products Under a T -rotation (4) we have that and, as det(R α β ) = ±1, it follows thatΥ Now, let Ω ∈ Λ 4 M be a volume tensor (Ω = 0) and let us define l α by Υ α = l α Ω. The relation (29) implies thatl α := ±R α β l β and, as R α β is a T -rotation, we have that As a consequence, unless η αβ = diag(1, −1) and Υ 1 = Υ 2 , we can allways perform a T -rotation such that one of the exterior products Υ α vanishes. (We can label the 1-forms φ β so that this is Υ 1 .) Therefore, T can be classified in one of the following types:
Proof: The first expression in (30) follows immediately from Υ 1 = 0 -see ref. [9], Chapter V, Proposition 4.12 . Then the fact that The 1-forms ψ α are determined apart from the gauge freedom: We now write dφ 2 = P α β ψ α ∧φ β +a φ 1 ∧φ 2 +m ψ 1 ∧ψ 2 and, including that Σ 11 = −2φ 1 ∧φ 2 ∧ψ 1 ∧ψ 2 , we have from (31) that Under the gauge transformations (32) the components of dφ 2 change according to: whereas m = −2l and P α α = 2r are gauge invariant. We can therefore choose the gauge matrix B βα so that the traceless part P ′α β vanishes. That is, the base 1-forms ψ α can be chosen so that P α β = r δ α β and the second and third expressions in (30) follow immediately. Notice also that the above choices exhaust the gauge freedom. ✷ As rank T = 2, equation (3) reads ǫ νβ = −ǫ βν , ǫ 12 = 1 and b is a function. Therefore it follows that Then for Υ α we have that L X Υ 1 = b Υ 2 and, as Υ 1 = 0 and Υ 2 = 0, it follows that b = 0 which, substituted in (33) yields On their turn, these equations imply that L X dφ α = 0 which lead to L X Σ αβ = L X Υ α = 0 and Including this and equation (30), L X dφ α = 0 implies that whence it easily follows that If this partial differential system is integrable, each solution is parametrized by the values X b 0 at one point. Therefore the dimension of the collineation algebra for type 2.I.a tensors is at most 4.
The integrability conditions of (37) put some further constraints on the parameters X b 0 . These integrability conditions are obtained by taking the exterior derivative and read L X dφ a = 0 or, in terms of the coefficients C a bc , XC a bc = 0 Locally this amounts to [XC a bc ] 0 = 0, which is an algebraic constraint on X b 0 , plus d (XC a bc ) = 0. Using the fact that d and L X commute, the latter is equivalent to: Iterating this procedure, we obtain that (38) implies that which is an infinite homogeneous linear system on the parameters X b 0 . Provided that its rank is not greater than 4, the codimension of the collineation algebra for type 2.I.a tensors is precisely this rank, otherwise T admits no collineation fields.
which is equivalent to: dX a = X e C a ec + f δ a 4 δ 3 c φ c , where as before X = X a Y a and C a bc are the commutation coefficients in this base. This is a first order partial differnetial system on the unknowns X a but, due to the occurrence of the unknown function f , it is not in closed form. However, in some cases the integrability conditions could help to determine f . The integrability condition for the equation (43), a = 4, yields where the fact that dφ a = − 1 2 C a bc dφ b ∧ dφ c has been included. Now, if φ 3 ∧ dφ 3 = 0, we can obtain f = f (X c ), which closes the differential system (43). If it is integrable, then the solution depends on the four real parameters X a 0 , which are submitted to the hyerarchy of constraints that follow from the full integrability conditions of the system (43), and dim C ≤ 4.
If, on the contrary, φ 3 ∧ dφ 3 = 0, then it exists ψ such that dφ 3 = φ 3 ∧ ψ and equation (44) implies that M = φ 3 ∧ µ, for some µ. Moreover, the integrability condition for equation (44) leads to or, separating f in all terms, which, provided that the right hand side does not vanish, permits to derive f = f (X a ), which closes the partial differential system (44); therefore dim C ≤ 4.
We do not analise the highly non-generic case that neither equation (44) nor equation (45) can be solved for f , which would require furhter study.
The exterior derivative of the latter yields dr i. e. dφ 1 = 0 and locally a function y exists such that φ 1 = dy. The condition (41) then implies that Xy = C, constant, and two cases must be considered depending on whether s does vanish or not: 2.I.b.0.nd If s = 0, then dφ 2 is simplectic and we can apply the results in section 4, case 1.nd.
Using canonical coordinates, φ 2 = p i dq i , i = 1, 2, and X = −{f, _} , where p i f |i = f . As a consequence, f is a solution of the partial differential system: In order to study its integrability, consider the minimal integrable submodule H ⊂ X (M) containing P = p j ∂ j and Y = {y, _}. It is obvious that 2 ≤ dim H ≤ 4 and that df ∈ H ⊥ . Therefore, 2.I.b.0.d If s = 0, then dφ 2 ∧ dφ 2 = 0 but, as Υ 2 = 0, we also have that dφ 2 ∧ φ 2 = 0, the results in section 4, case 1.d apply and canonical coordinates can be chosen such that φ 2 = dq 2 +p 1 dq 1 , φ 1 = dp 2 and with where the condition i X φ 1 = C has been included.
(b) In this case det(P αβ ) = 0 and therefore P αβ = v α w β . Then the SL(2) gauge can be chosen so The value of det(P αβ ) is T -frame dependent. Indeed, by a T -rotation we have that where R α ν is a O(1, 1) matrix valued function. Using that we obtain whence it easily follows that dR α µ ∧ dR β ν = 0.
Particularly we have that: and, asΥ 1 = e ζ Υ 1 , this amounts to e 3ζ (t +s − 2r) which finally leads to where dζ = ζ a φ a .

The integrability conditions
must also be considered.
If u = 0, the first of these equations means that L X u φ 3 = −2bu φ 3 + A α φ α . Besides, the integrability conditions (54) can be further exploited to obtain that Then equations (56) to (58) follow immediately.
If on the contrary u = 0, the first of equations (60) is identically satisfied and the other implies equation (55). ✷ Subtype 2.N.1: This corresponds to u = 0 and equations (56) to (58) hold. We take the basê , and equations (53) and (56) to (58) can be written as where a, c = 1 . . . 4, the only nonvanishing U a c are U 1 2 = U 2 1 = 1, U 3 3 = U 4 2 = −2 and U 4 4 = 3, and a cbφ c ∧φ b , and X =X a Y a . The first of equations (64) then reads dX a = X eĈ a ec + bU a c + f δ a 4 δ 3 c φ c . Therefore, equation (64) is a partial differential system on the unknownsX a and b.
Although this PDS is not in closed form, due to the presence of an arbitrary function, the integrability conditions may help to determine f . Indeed, after a little algebra, the integrability condition for equation (64) with a = 4 becomes where Now, ifφ 3 ∧dφ 3 = 0, we have that fφ 3 ∧dφ 3 =φ 3 ∧M and we can obtain f = f (X c , b), which closes the differential system (64). If it is integrable, then the solution depends on the five real parameterŝ X a 0 and b 0 . Similarly as in previous cases, these parameters are submitted to the hierarchy of constraints that follow from the full integrability conditions, and dim C ≤ 5.
If, on the contrary,φ 3 ∧ dφ 3 = 0, then it exists ψ such that dφ 3 =φ 3 ∧ ψ and equation (65) implies that M ∧φ 3 = 0, that is a µ exists such that M =φ 3 ∧ µ. Moreover, the integrability condition for equation (65) leads to or, separating f in all terms, which, provided that the right hand side does not vanish, permits to derive f = f (X c , b) and then Similarly as in case 2.I.b.1 above, we do not consider the residual nongeneric subcase that neither equation (65) nor equation (66) can be solved for f .
Therefore φ 2 − φ 1 is integrable and, as Σ 11 = 0, dφ 1 is symplectic, canonical charts (p i , q j ), with i, j = 1, 2, and two functions u and y exist such that which, written in terms of u and y, implies that a one variable function B(u) exists such that where B ′ means the derivative.
From equation (53) we also have that L X φ 1 = bφ 2 , which in terms of canonical coordinates reads where we have written X = X j ∂ j + X i ∂ i (as in section 4). Their components are where u |i := ∂ i u and u |i := ∂ i u.
If we now write the components X j as X j = 1 z 2 ξp j + η r j , with p j := p j , r j := r j = (p 2 , −p 1 ) and z 2 : Case 2.N.01: If 1 + v l y |l ≡ y(1 + y |l u |l ) + p l y |l = 0, we can derive: Now equations (72) together with the first of equations (69) are to be taken as a partial differential system on the two unknowns X j . Using equation (75) and after a little algebra, this PDS can be written as Including now the decomposition (73), η can be derived from one of these equations whenever the remaining two equations then yielding a PDS to be fulfilled by the unknown ξ.
On the contrary, if none of the above inequalities hold, η is arbitrary, does not occur in the PDS and we are left with three equations on the unknown ξ.
In any case, the PDS looks like: with α running either from 1 to 2 (resp., 1 to 3). Using the commutation relations we then find the minimal integrable modulus H containing the fields H α .
The solution ξ then depends on an arbitrary function of 4 − dim H variables. The component η is either determined or arbitrary, depending on whether the inequalities (78) do hold or do not, and the components X j can be derived from (75).
Case 2.N.00: In case that y(1 + y |l u |l ) + p l y |l = 0, equation (74) implies the constraint and its general solution is the component ζ being arbitrary. Including these, equations (72) and (69) become Now, if u |i = 0, the first of these equations permits to obtain which substituted in equations (79), (81) and (82) yields a PDS to be fulfilled by ξ. The discussion about its solution is then similar to that in case 2.N.01 above.
If, on the contrary u |i = 0, after a little algebra equations (79), (81) and (82) yield: η = r l ∂ l ξ and the PDS: The discussion about the existence of a solution is then similar to that in case 2.N.01 above.

Type 2.H
In this case, Υ α = 0 and T is holonomous. Therefore coordinates x a , a = 1 . . . 4, exist such that Type 2.H tensors will be dealt in much the same way as rank 3 tensors. We first write the collineation field as It is obvious that T (N A , _) = 0, which implies that L f A N A T = f A L N A T and therefore, equation where K A := 1 2 ∂ A T αβ dx α ⊗ dx β or, in components, where ∇ is the Levi-Civita connection for the non-degenerate metric T αβ on the hypersurfaces And the successive integrability conditions that follow from the commutation relations for ∇ α and ∇ µ imply a hierarchy of new equations on Ω αβ and f , namely and so on, where we have included that, as the dimension is two, As for the commutation of the derivatives ∂ A and ∇ α applied to Z α , we readily obtain that: In the case 2.H.2 we have that Then equation (91) yields with L A and L B A are linear functions of Z α and Ω αβ , a "stroke" means "partial derivative" and A ′ = A. Equations (88) and (90) give all derivatives of Ω αβ in terms of Z α , Ω µν , f B and ∂ ν f B ,and equations (92) can be taken as a linear system of six equations for the six unknowns f B and ∂ A f B . If the matrix of the system has rank six, this is a Cramer's system and we can derive Substituting the above relations in equations (86), (88) and (90), we obtain a closed partial differential system on Z ν and Ω αβ whose solutions are parametrized by three real parameters, namely the values of Z ν and the skewsymmetric 2 × 2 matrix Ω αβ at one point.
Of course, some constraints will follow from the fact that F B A = ∂ A F B . These, together with the hierarchy of integrability conditions, will result in a homogeneous linear system of conditions on the parameters Z ν (0) and Ω αβ (0). Therefore C T is a Lie algebra and dim C T ≤ 3. (92) is not Cramer's, we can at least derive ∂ A f B =L B A (Z, Ω, f C ). Now, (88) and (90) give all derivatives of Ω αβ and therefore some integrability conditions will follow, namely

If the linear system
where W Bµ is a linear function of Z ν , Ω αβ and f B . This is to be seen as a linear system of four equations on the four unknowns ∂ ν f B and, in case that the rank is four, we can derive and equations (86), (88) and (90) yield a closed partial differential system on the unknowns Z ν , Ω αβ and f B . The general solution is a vector space whose dimension is at most five. Therefore C T is a Lie algebra and dim C T ≤ 5. Equation (91) also impies that ∂ 4 f 3 = 0 and that: If K α 3| β and ∂ 3 K α 3| β are independent, then we can derive and close the partial differential system (86), (88) and (90). Its solutions depending on the three real parameters Z ν (0) and Ω αβ (0), which are further constrained by the hierarchy of integrability conditions, the space T is a Lie algebra whose dimension is at most three. If, on the conbtrary, K α 3| β and ∂ 3 K α 3| β are not independent, then as K α 3| β = 0, we can at least derive ∂ 3 f 3 = F 3 (Z, Ω, f 3 ). Now the integrability conditions that follow from equations (88) and (90) yield a linear system of two equations on ∂ ν f 3 . Generically this is a Cramer's system and can be solved for to derive ∂ ν f 3 = F ν (Z, Ω, f ) and, together with equations (86), (88) and (90), finally close a partial differential system on Z ν , Ω αβ and f 3 . The general solution depends on four real parameters, namely Z ν (0), Ω αβ (0) and f 3 (0), which are further constrained by the hierarchy of integrability conditions. Therfore the space C (0) T is a Lie algebra whose dimension is at most four.

Summary
We finally present an outline of the classification of covariant second order tensors, according to their respective classes of collineation fields, C T , and summarize what has been proved along previous sections. As a rule, it seems that for generic cases C T is a finite dimensional Lie algebra, whereas in nongeneric cases, i. e. some equalities do hold, C T is not a Lie algebra and has an infinite number of dimensions.
Rank 4 tensors: T can be viewed as a non-degenerate metric on M, C T is the corresponding Killing algebra and dim C T ≤ 10.
Write then X = Z + f N, where T (N, ) = 0, and consider K αβ := L N T αβ .
C T is not a lie algebra but the subclass C 0 T = {X ∈ C T | Xy 4 = 0} is a Lie algebra that has at most six dimensions.
• If ∂ 4 K αβ is not proportional to K αβ , then C T is a Lie algebra and dimC T ≤ 6.
• If ∂ 4 K αβ ∝ K αβ , but K αβ K βµ = 0 and K αβ is not proportional to T αβ , then C T is a Lie algebra and dimC T ≤ 7.
Our analysis of two residual, degenerate cases, has been left incomplete and they probably involve arbitrary functions, i.e. C T is infinite dimensional. These cases correspond to ∂ 4 K αβ ∝ K αβ and, either K αβ K βµ = 0 or K αβ ∝ T αβ .

Rank 1 tensors:
We write T = φ ⊗ φ and distinguish several cases: Type 1.nd dφ is simplectic and, in canonical coordinates (q i , p j ), the collineation fields are where f (q i , p j ) is homogeneous and of first degree on the "momenta" p j . Type 1.d Characterised by dφ ∧ dφ = 0 and dφ ∧ φ = 0. Then coordinates (q i , p j ) exist such that where f (p 1 , q 1 ) and X 2 (p i , q j ) are arbitrary.
where F (q 1 ), X 2 (p i , q j ) and X 2 (p i , q j ) are arbitrary functions. Type 1.d0 In this case dφ = 0 and coordinates x a , a = 1 . . . 4 exist such that where C is a constant and the three functions X i (q j , p l ) are arbitrary.

Type 2.I.b
There is a T -frame in which Σ 11 = Υ 1 = 0 and Υ 2 = 0. Then, by Proposition 2, it exists a base, φ α , α = 1 . . . 4, in which the differential forms dφ α have the canonical expression (40). Two cases arise according to the values of v α : then it results that C T is a Lie algebra and dim C T ≤ 4. Otherwise X might contain arbitrary functions and therefore C T is not a Lie algebra has an infinite number of dimensions.   Subtype 2.N.1 If u = 0, X is the solution of the partial differential system (64), which involves an arbitrary f . Thus the system is not closed and, provided that eitherφ 3 ∧dφ 3 = 0 or that equation (66) can be solved for f , the class C T is a Lie algebra and dim C T ≤ 5.
In the residual nongeneric case thatφ 3 ∧ dφ 3 = 0 and that equation (66) cannot be solved for f , the partial differential system might not close and X might contain arbitrary functions. Therefore C T might not a Lie algebra and have an infinite number of dimensions.
An arbitrary one variable function B(u) appears.
Case 2.N.01 Characterized by y(1 + u |l y |l ) + p l y |l = 0. The components X j are determined by equation (75). As for the components X j , if one of the inequalities (78) holds, then η := p 1 X 2 −p 2 X 1 is determined in terms of ξ := p j X j , which is a solution of a 2-equations linear partial differential system and, provided that it is integrable, ξ is determined up to the addition of an arbitrary function of two variables at most.
If no inequality (78) holds, then η is arbitrary and ξ is a solution of a 3-equations linear partial differential system and, provided that it is integrable, ξ is determined up to the addition of an arbitrary function of one variable at most.
Case 2.N.00 This is characterized by y(1 + u |l y |l ) + p l y |l = 0 and if u |l = 0, the components X j are given by (80) and (83). The component η is determined by (89) and the component ξ is a solution of equations (79) and (82), a linear partial differential system and, provided that it is integrable, ξ is determined up to the addition of an arbitrary function of two variables at most.
If on the contrary u |l = 0, then X j are given by (80) and include an arbitrary function, besides η = p 1 ∂ 2 ξ −p 2 ∂ 1 ξ and ξ is a solution of the linear partial differential system (84) and, provided that it is integrable, it is determined up to the addition of an arbitrary function of one variable at most.
Therefore the class of collineation fields is infinite dimensional and is not a Lie algebra.
Writing then X = Z + f A ∂ A , with Z = Z ν ∂ ν , and K A|αβ := 1 2 ∂ A T αβ , A = 3, 4, T is a Lie algebra whose dimension is at most 3 and f B are arbitrary functions.
Case 2.H.2 If K A|αβ = 0, A = 3, 4, then • either the system (92) is Cramer's and C T is a Lie algebra whose dimension is at most three • or else, if the system (93) is Cramer's, C T is a Lie algebra whose dimension is at most five.
We have left unsolved the case when neither (92) nor (93) are Cramer's systems. This is a highly nongeneric instance that would require further study. T is a Lie algebra whose dimension is at most three or else, the analogous of equation (93) is a Cramer's system and C (0) T is a Lie algebra whose dimension is at most four. In both instances f 4 is an arbitrary function. We have neither considered the nongeneric case in which neither (92)  Appendix: The derivation of equation (18) To derive equation (18) we apply ∇ µ and ∂ 4 respectively to equations (16) and (13) and then substract. Other facts that must be taken into account are that and that: In case thatK α µ = bK α µ , it follows that ∂ 4 f K α µ = (ḟ + bf )K α µ and ∂ 4 (f K µκ ) = (ḟ + bf )K µκ + 2f K ακ K α µ Using all that, after a little algebra we finally obtain which, asḟ + bf is a function of Z, is a linear function of Z α , Ω µν and f .