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Journal of Geometry and Physics 60 (2010) 1–7
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Journal of Geometry and Physics
journal homepage: www.elsevier.com/locate/jgp
Some remarks on the converse of Weyl’s conformal theoremGraham Hall ∗Department of Mathematics, University of Aberdeen, Aberdeen, AB24 3UE, Scotland, UK
a r t i c l e i n f o
Article history:Received 6 July 2009Accepted 4 August 2009Available online 15 August 2009
MSC:53C1553C2153C2553C50
Keywords:Weyl tensorConformal metrics
a b s t r a c t
One ofWeyl’s classical theorems states that a certain tensor, theWeyl tensor, is unchangedwhen the metric from which it is constructed is replaced by another metric conformallyrelated to it. This paper explores the converse of this theorem.
© 2009 Elsevier B.V. All rights reserved.
1. Introduction and notation
LetM be a smooth, n-dimensional, Hausdorff manifoldwith n ≥ 3 and smoothmetric g , of arbitrary signature, associatedLevi-Civita connection ∇ and corresponding type (1,3) curvature tensor Riem, with components denoted by Rabcd. Theresulting Ricci tensor, Ricc , has components Rab ≡ Rc acb and R ≡ Rabgab, is the Ricci scalar. The associated type (1,3)Weylconformal tensor C [1] has components denoted by Cabcd, where
Cabcd = Rabcd +1n− 2
(δadRbc − δacRbd + gbcRad − gbdRac)+R
(n− 1)(n− 2)(δacgbd − δadgbc). (1)
The Weyl tensor is not defined for the case n = 2 and for the case n = 3, C , although still defined by (1), is identically zero.[This latter case stimulated the introduction of the Schouten tensor [2] (see also, [3]) in place of Weyl’s tensor and will notbe discussed further here.] Thus, for the remainder of this paper, it is assumed that n ≥ 4. For m ∈ M , TmM denotes thetangent space to M at m and ΛmM the 12n(n − 1)-dimensional vector space of tensor type (2,0) 2-forms at m. If F ∈ ΛmMwith components F ab(= −F ba), its rank is the matrix rank of F ab and for F 6= 0 this rank is a positive even integer. If the rankof F is equal to 2, F is called simple and then F ab = paqb − qapb for p, q ∈ TmM . For such a simple F , although p and q are notuniquely determined by F , the 2-dimensional subspace (2-space) of TmM spanned by p and q is uniquely determined by Fand is called the blade of F . It will be denoted by p ∧ q.Weyl was able to show that, although the definition of C in (1) clearly reveals its dependence on the metric g , C only
actually depends on the conformal class of g in the sense that if g ′ is another smooth metric on M which is conformallyrelated to g , so that g ′ = φg for some smooth function φ : M → R, then the Weyl tensor C ′ constructed from g ′ as C isconstructed from g in (1) is equal to C , C ′ = C . This paper studies a possible converse to this theorem.
∗ Tel.: +44 1224 272748; fax: +44 1224 272607.E-mail addresses: [email protected], [email protected].
0393-0440/$ – see front matter© 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.geomphys.2009.08.002
2 G. Hall / Journal of Geometry and Physics 60 (2010) 1–7
2. A possible converse of Weyl’s theorem
Because of the possibility of making arbitrary coordinate transformations one can deduce little about the metric in anopen region at each point of which C vanishes. Thus let it be agreed that the manifold and metric pair (M, g) be called non-conformally flat if C does not vanish over any non-empty open subset ofM (in the natural manifold topology onM), that is,C is not zero at each point of some open dense subset of M . Then, perhaps, the most natural possible question regarding aconverse of Weyl’s theorem is; if g and g ′ are smooth metrics of arbitrary signature on a manifold M (as described above) ofdimension n ≥ 4 whose Weyl tensors are equal on M and non-zero over some open dense subset of M , are g and g ′ conformallyrelated?. It turns out that for all but one of the infinite number of possible choices of n and the signature of g , the answer isno. To see this, first consider the situation when the manifoldM is some appropriate connected open subset of R4 and g isa global Lorentz metric of signature (−,+,+,+) onM given in a global coordinate system u, v, x, y by
ds2 = 2dudv + H(u, x, y)du2 + dx2 + dy2 (2)
where H : M → R is a smooth function of u, v and x satisfying the condition that at least one of the functions∂2H/∂x2 − ∂2H/∂y2 and ∂2H/∂x∂y does not vanish over any non-empty open subset of M [4,5]. The covector field du isglobal, nowhere zero and null onM and is, in fact, covariantly constant with respect to the Levi-Civita connection associatedwith g . The choice ofH in (2) ensures that theWeyl tensor is nowhere zero over some open dense subsetU ofM . Now chooseany global smooth function ψ : M → R which depends only on u, ψ(u), and which is not identically zero onM and hencenot identically zero on U . Then define the smooth metric g ′ onM in the above coordinates by g ′ = g +ψdu⊗ du. It can bechecked that g ′ also has Lorentz signature onM and that g and g ′ have the sameWeyl tensors onM . Hence theWeyl tensorsof g and g ′ are equal, but g and g ′ are not conformally related. This takes care of the n = 4 and Lorentz signature case. Nowtake the metric product of the above pair (M, g)with (M ′, h)whereM ′ is a manifold of dimension n ≥ 1 and h is a smoothflatmetric of arbitrary signature onM ′. The resultingmetric product spaces given by (M×M ′, g⊗h) and (M×M ′, g ′⊗h) areeasily checked to have the sameWeyl tensor (and which is nowhere zero over the open dense subset U×M ′ ofM×M ′) butthe metrics g ⊗ h and g ′⊗ h are not conformally related onM ×M ′. This counter-example to the above suggested converseto Weyl’s theorem applies for all dimensions ≥4 and all indefinite (that is, not positive or negative definite) signatures upto sign except the single case when n = 4 and the signature is (+,+,−,−). The result in the 4-dimensional Lorentz caseabove was given in [6].Next, let M be a 4-dimensional manifold with positive definite metric g such that g has vanishing Ricci tensor (the Ricci
flat condition) but such that its curvature tensor is nowhere zero over an open dense subset V ofM . Then, for n ≥ 1, letM ′ bean n-dimensional manifold with a smooth flat metric h of arbitrary signature. Consider the metric product (M ×M ′, g ⊗ h).Because of the vanishing of the Ricci tensor of each metric of the above product, it is easily checked that the Ricci tensorof the product metric vanishes and hence that the Weyl tensor and curvature tensor of the product are equal. This showsthat the Weyl tensor is nowhere zero over the open dense subset V ×M ′ ofM ×M ′. Now replace the metric h by any othersmooth flat metric h′ on M ′ which is not conformally related to h and consider the metric product (M × M ′, g ⊗ h′). Thislatter product is easily checked to have the same curvature tensor as the former product and hence it also has zero Riccitensor. Thus these two metric products have the same Weyl tensors (non-vanishing over the open dense subset V × M ′ ofM × M ′) but their metrics are not conformally related. [Attention is drawn to the difference between the technique usedfor constructing counter-examples in this paragraph and in the previous one. This is necessitated by the lack of a pair ofnon-conformally related metrics in the 4-dimensional, positive definite case with the same Weyl tensor (see Section 3 andthe Appendix)]. In particular, if h is chosen to be positive definite, it follows that the suggested converse to Weyl’s theoremis false in the cases when n ≥ 5 and g is of positive definite signature.The case when dimM = 4 and g has signature (−−++) is easily resolved. Consider themetric g in (2) with the positive
sign before the dy2 term changed to a negative sign. Then the resulting metric is of signature (−−++). Suppose a metricg ′ is then constructed from g exactly as in the Lorentz signature example and that H is chosen so that one of the functions∂2H/∂x2+∂2H/∂y2 and ∂2H/∂x∂y does not vanish over any non-empty open subset ofM . Then g and g ′ are not conformallyrelated and have the same Weyl tensor which does not vanish over any non-empty open subset ofM .The previous three paragraphs show that the suggested converse to Weyl’s theorem fails, for n ≥ 5, for any signature
and also for n = 4 if the signature is either (−+++) or (−−++). The only unresolved case is thus when n = 4 and thesignature is (+,+,+,+). This case will be dealt with in Section 3. Even though this suggested converse fails, one can getsome idea of the ‘‘power’’ of the Weyl tensor to determine a conformal class of metrics in certain situations. To see this letM be a manifold of dimension n ≥ 4 with metric g of arbitrary signature and, for m ∈ M , consider the linear map f fromΛmM to the vector space Tm(1, 1) of type (1,1) tensors atm determined by the Weyl tensor C and given, in components, by(cf [6,7])
f : F ab → CabcdF cd (F ∈ ΛmM). (3)
Denoting the range space of f by Bm, it is clear from the algebraic symmetries of C that, if g ′ is another smooth metric onMwhose Weyl tensor is the same as that of g , then for each G ∈ Bm,
gacGc b + gbcGc a = 0 g ′acGcb + g ′bcG
ca = 0. (4)
G. Hall / Journal of Geometry and Physics 60 (2010) 1–7 3
Thus G is skew-self-adjoint with respect to g and g ′. Regarding g as the original fixed metric it follows from (4) that if themember Gab ≡ Gacgcb ofΛmM is simple then its blade is an eigenspace of g ′ with respect to g , that is, for each k ∈ TmM in thisblade, g ′abk
b= λgabkb with the eigenvalue λ ∈ R independent of k [6–8]. To see this let Gab = paqb − qapb for p, q ∈ TmM
and use the original metric g to raise and lower indices. Then the second equation in (4) gives
p′aqb − q′
apb + p′
bqa − q′
bpa = 0 (5)
where p′a = g ′abpb and similarly for q. Now, independently of the signature of g , one can choose r ∈ TmM such that
raqa 6= 0 = rapa. A contraction of (5) with rb then shows that p′a is a linear combination of pa and qa and similarly so isq′a. Substituting these linear combinations back into (5) gives the desired result. Thus significant information regarding thealgebraic relationship between g and g ′ is available at each point ofM from the map f provided by C(m). If one can achievethe situation where TmM becomes an eigenspace of g ′ with respect to g , then g ′ and g are conformally related at m. [It isrecalled here that, since g ′ is symmetric, eigenvectors of g ′ corresponding to distinct eigenvalues are (g-) orthogonal.]In the case n = 4 and with the original metric g of Lorentz signature (the case of General Relativity) one can say more
[6–8]. If, as above, a member ofΛmM is not simple then it can be written uniquely as a sum of two simple members ofΛmMwhose blades are g-orthogonal complements of each other (see, e.g. [9]) and may be called the canonical blade pair of this2-form. It can then be shown, in a similar way to that given above, that each blade is an eigenspace of g ′ with respect to g [6].(This result depends on the Lorentz signature of g as will be seen in the next section.) The consequences for the possibleconverse ofWeyl’s theorem can then be resolved by an algebraic study of C(m) [6]. To see this, it is first noted that the Hodgedual operator, denoted by the symbol ∗, satisfies, in this case, the following equations for F ∈ ΛmM and the Weyl tensor C(see for example [10]). The symbol ε denotes the usual alternating (pseudo-)tensor.
∗
∗
F = −F ∗C∗ = −C (⇒∗ C = C∗) (6)∗
F ab = gacgbd(12εcdef F ef
)∗Cabcd =
12εabef C ef cd C∗abcd =
12εcdef Cabef .
Now one may, using g , uniquely identify the map f in (3) with the linear map∼
fg : ΛmM → ΛmM given by F ab → Gab =
gebCaecdF cd = CabcdF cd. From (6) it follows that if G is in the range of∼
fg then so also is∗
G (because if G =∼
fg(F),∗
G =∼
fg(∗
F)).
Since, for this dimension and signature and for F ∈ ΛmM , F and∗
F are independent members of ΛmM , it is easily checked
that, for C(m) 6= 0, the dimension of the range of∼
fg , equal in dimension to that of f , is a positive even integer (see, forexample, [6]). An application of the italicised algebraic result following (4) then reveals that if this range dimension at m(≤6) is 4 or 6, TmM is an eigenspace of g ′ with respect to g and so g ′ and g are conformally related atm, but that this resultmay fail if this range dimension is 2. These cases based on range dimension may be conveniently rephrased in terms of thewell-known Petrov classification of the Weyl tensor [11]. Then, the range dimension at m equals 4 if and only if the Petrovtype atm is either I, II, III or Dwhilst it is equal to 2 atm if and only if the Petrov type atm is N. Thus, for C(m) 6= 0, g ′ andg are conformally related at m if the Petrov type at m is not N. In fact, if the Petrov type is N the result above can fail; theexample in (2) is of this Petrov type at each point of the open dense subset ofM on which C does not vanish. The followingtheorem can now be given.
Theorem 1. (i) Let M be a 4-dimensional, smooth, connected, Hausdorff manifold with smooth Lorentz metric g. If the subset ofpoints where the associated Weyl tensor is either zero or of Petrov type N has empty interior, then any other smooth metric g ′ onM with the same Weyl tensor as g is conformally related to g (and hence also has Lorentz signature, up to sign, on M).(ii) Let M be an n-dimensional connected Hausdorff manifold with a smooth metric g of arbitrary signature. Suppose there
exists an open dense subset U of M at each point m of which there exists a basis e1, . . . , en of TmM with the property that the(n− 1) simple members ek ∧ ek+1 for k = 1, . . . , (n− 1) are in Bm. Then any other smooth metric g ′ on M which has the sameWeyl tensor as g on M is conformally related to g on M (and so has the same signature as g, up to sign, on M).
Proof. Part (i) was discussed earlier and a proof can be found in [6]. The proof of part (ii) follows from the work above since,now, e1 ∧ e2, e2 ∧ e3, . . . , en−1 ∧ en are eigenspaces of g ′ with respect to g at m and so TmM is also such an eigenspace, foreachm ∈ M . Thus g ′ and g are conformally related at each point of U . Since U is open and dense inM , it follows that g ′ andg are conformally related onM . �
It is clear that the power of the Weyl tensor to determine its conformal class depends on the spaces BmM . But it alsodepends on the signature of the original metric g . This latter dependence is less obvious and will be clarified in the nextsection.
3. The 4-dimensional positive definite case
Now let M be 4-dimensional and let g be a positive definite metric on M . The Weyl tensor C and any F ∈ ΛmM satisfy(using a similar convention to that employed earlier)
4 G. Hall / Journal of Geometry and Physics 60 (2010) 1–7
∗
∗
F = F ∗C∗ = C (⇒∗ C = C∗). (7)
If F is simple then so also is∗
F and the blades of F and∗
F are g-orthogonal complements of each other. In this case F and∗
F
are independent members ofΛmM . Now define two subspaces+
Sm and−
Sm ofΛmM by
+
Sm = {F ∈ ΛmM :∗
F = F}−
Sm = {F ∈ ΛmM :∗
F = −F} (8)
and define, also, the subset∼
Sm ofΛmM by∼
Sm =+
Sm ∪−
Sm. Clearly,+
Sm ∩−
Sm = {0} where 0 is the zero member ofΛmM . Now
F ∈ ΛmM ⇒ F = 12 (F +
∗
F)+ 12 (F −
∗
F) and so each F ∈ ΛmM may be written, uniquely, as the sum of a member of+
Sm and
a member of−
Sm. From this it is easily checked thatΛmM =+
Sm⊕−
Sm and that+
Sm and−
Sm are 3-dimensional and isomorphic.
If F ∈ ΛmM(F 6= 0) and∗
F = λF (λ ∈ R) then the first equation in (7) shows that λ = ±1 and so, for any F ∈ ΛmM , F
and∗
F are independent if and only if F ∈ ΛmM \∼
Sm. If F ∈ ΛmM \∼
Sm and F is not simple, the usual eigenproblem for thelinear map TmM → TmM associated with F in the usual way reveals four distinct (necessarily imaginary) eigenvalues ±iaand±ib (a, b ∈ R, 0 6= a 6= ±b 6= 0) and corresponding eigenvectors x± iy (for±ia) and z ± iw (for±ib). where x, y, z, wmay be chosen as an orthonormal basis of TmM . The 2-spaces x∧ y and z ∧w are then orthogonal invariant 2-spaces of theabove linear map derived from F . These are the only two such invariant 2-spaces and will also be referred to as the canonicalpair of blades of F . Then F may be written as
F = a(x ∧ y)+ b(z ∧ w) (a, b ∈ R, 0 6= a 6= ±b 6= 0) (9)
with the basis x, y, z, w, chosen such that∗
(x ∧ y) = (z ∧w),∗
(y ∧ z) = (x ∧w) and∗
(x ∧ z) = (w ∧ y). If, however, F ∈∼
Sm(and is thus necessarily non-simple) then a = ±b in the above and a single conjugate pair ±ia of eigenvalues results forF . The consequent multiplicity of (complex) eigenvectors means that no unique canonical pair of blades exists but rather
infinitely many such pairs. In this case F may be written in the form (9) with a = b (if F ∈+
Sm) or a = −b (if F ∈−
Sm) ininfinitely many ways (that is with infinitely many distinct choices of the blades (x ∧ y) and (z ∧ w)). For example, witha = b = 1,
(x ∧ y)+ (z ∧ w) = (p ∧ q)+ (r ∧ s) (10)
with p = 1√2(x + z), q = 1
√2(y + w)r = 1
√2(x − z) and s = 1
√2(y − w) an orthonormal basis for TmM . But the blades
(x∧ y) and (z ∧w) are each different from the blades (p∧ q) and (r ∧ s). Recalling the map∼
f g , it is then clear from (7) that∼
f (+
Sm) ⊂+
Sm and that∼
f (−
Sm) ⊂−
SmNow let g ′ be any other metric onM whose (type (1,3)) Weyl tensor coincides with the (type (1,3)) Weyl tensor C of the
positive definite metric g . Then (4) holds for each G in the range of f . Let∼
G ∈ ΛmM be the associatedmember in the range of∼
f g , so that∼
G =∼
f g(H) for someH ∈ ΛmM . If∼
G is simple, its blade is an eigenspace of g ′with respect to g . Now suppose that Gis non-simple. Since g is positive definite and g ′ is symmetric, g ′ is diagonalisable over R (with respect to g). So let r ∈ TmMbe a (g-)unit eigenvector of g ′ with eigenvalue λ ∈ R with λ 6= 0 since g ′ is non-degenerate. Then g ′abr
b= λgabrb = λra
and a contraction of the second equation in (4) with rarb and use of the first equation in (4) then shows that the non-zeromember r∗ ∈ TmM , where r∗a = Gabrb, is (g-orthogonal to r and) an independent eigenvector of g ′ also with eigenvalue λ.Now choose, as one can, another unit eigenvector s of g ′ atm orthogonal to r and r∗ and with eigenvalueµwithµ 6= 0. Onethen obtains, as before, the associated eigenvector s∗, where s∗a = Gabsb, also with eigenvalue µ and such that s and s∗ areorthogonal. It then follows, after a contraction of the second equation in (4) with rasb and use of the definitions of r∗ and s∗,that s∗ and r are orthogonal, since s and r∗ are, and then that r, r∗, s and s∗ are independent. To see this latter result writeαr + βr∗ + γ s+ δs∗ = 0 for α, β, γ , δ ∈ R and take respective inner products (with respect to g) on each side with r ands to get α = γ = 0. There remains Gab(βrb + δsb) = 0 from which β = δ = 0 follows from the non-degeneracy of G. Soeither g ′ is a multiple of g or λ 6= µ and a unique (g-orthogonal) pair of 2-dimensional eigenspaces of g ′ arise. In the lattercase, after normalisation of the tetrad r, r ′, s, s′ to give an orthonormal basis x, y, z andw one finds the expression
g ′ab = λ(xaxb + yayb)+ µ(zazb + wawb). (11)
for g ′. On substituting (11) and the general expression for G in terms of the above orthonormal tetrad (assumed ordered
such that∗
(x ∧ y) = (z ∧ w)) into (4) one finds
G = A(x ∧ y)+ B(z ∧ w) (A, B ∈ R). (12)
Now, if A 6= B, the 2-spaces x∧ y and z∧w are uniquely determined by G if one requires an expression like (12). If however,A = B, then (12) can be rewritten in infinitely many ways in the form
G. Hall / Journal of Geometry and Physics 60 (2010) 1–7 5
G = A(x ∧ y)+ B(z ∧ w) (A, B ∈ R) (13)
provided only that x, y, z and w constitute an orthonormal tetrad given in terms of the tetrad x, y, z andw by
x = ν(x+ az + bw), y = ν(y− bz + aw) (14)z = ν(z − ax+ by), w = ν(w + bx+ ay)
where ν = (1 + a2 + b2)−12 . In any case, the above shows that an orthonormal tetrad x, y, z and w exists in which g ′ and
Gmay be simultaneously written as in (11) and (12) and which may be called the eigenspace representation of G and g ′. Thepair of 2-spaces so determined is unique.
Lemma 1. Let V ⊂ ΛmM denote the range space of∼
fg at m ∈ M (so that if V ⊂∼
Sm then V ⊂+
Sm or V ⊂−
Sm). Suppose V 6= {0}
(that is, C(m) 6= 0). Then dim V ≥ 2. If, in addition, V 6⊂∼
Sm (and this is necessarily the case if dim V ≥ 4) there exists a basis for
V , no member of which is in∼
Sm and which contains a simple member together with its dual.
Proof. Suppose that dim V = 1 with V spanned by F ∈ ΛmM . Then from the definitions of f and∼
fg and the algebraicsymmetries of C , one has, in components, Cabcd = λFabFcd at m where Cabcd ≡ gaeC ebcd, Fab = gacgbdF cd and 0 6= λ ∈ R.But then, using square brackets to denote the usual skew-symmetrisation of indices, Ca[bcd] = 0 and so Fa[bFcd] = 0 and thislatter equation is equivalent to F being simple. [Just choose r, s ∈ TmM with Fabrasb 6= 0 as one always can, and contract it
with rasb.] Now F ∈ V ⇒∗
F ∈ V and so, since F is simple, F and∗
F are independent members of V . This gives dimV ≥ 2 and a
contradiction. It follows that dim V ≥ 2. If, in addition, V is not contained in∼
Sm (which is clearly true if dimV ≥ 4) suppose
0 6= F ∈ V and F is not a member of∼
Sm. If F is simple then F and∗
F are simple, independent members of V . If F is not simple,
say F = cP + dQ for simple P,Q ∈ ΛmM with orthogonal blades and with∗
P = Q and∗
Q = P and c, d ∈ Rwith c2 6= d2 (c.f.
(9)), then cF − d∗
F(= (c2 − d2)P) and its dual are simple, independent members of V . So whether F ∈ V is simple or notone can find two independent, simple members of V . On extending these to a basis for V , one can always replace any basis
member which happens to be in∼
Sm by that which is obtained by adding to it one of the simple members already found in
this basis, and which is then not in∼
Sm. �
Theorem 2. Let M be a 4-dimensional, smooth, Hausdorff manifold admitting a smooth positive definite metric g. SupposeC(m) 6= 0 for each point in some open dense subset U of M. If g ′ is any other smooth metric on M whose Weyl tensor coincideswith that of g, then g ′ and g are conformally related on M.
Proof. Let m ∈ U and let V be the range space of∼
f at m. Since C(m) 6= 0, the lemma applies and only two cases emerge
for consideration. The first is when dim V ≥ 2 and V 6⊂∼
Sm and the second is when dim V ≥ 2 and V ⊂∼
Sm. In the first
case the lemma reveals an orthonormal basis x, y, z, w for TmM such that F = x ∧ y and∗
F = z ∧ w are members of a basis
for V which contains no members of∼
Sm. If dim V = 2 then F and∗
F span V and, lowering indices with g , one has from thealgebraic symmetries of C atm
Cabcd = αFabFcd + β∗
Fab∗
Fcd+γ (Fab∗
Fcd+∗
Fab Fcd) (15)
where α, β, γ ∈ R. The condition Cabad = 0 shows that α = β = 0 and the condition Ca[bcd] = 0, on contraction with xayb,
reveals that γ = 0. Thus, this case cannot occur. If dim V ≥ 3 (and V 6⊂∼
Sm) the lemma shows that one may choose a basis
for V containing F and∗
F , as above, and also G ∈ V \∼
Sm. Earlier results then show that the blades of F and∗
F are eigenspaces
of g ′ as is the blade (or canonical pair of blades) of G. Since F ,∗
F and G are independent it is clear that the eigenvalues of g ′
associated with the (orthogonal) blades of F and∗
F must be equal and hence that g ′ and g are conformally related atm.
In the second case, one has V ⊂∼
Sm and dim V = 2 or 3. Suppose that V ⊂+
Sm (the case V ⊂−
Sm is similar). Thus one
has the existence of (at least) two independent members F1 and F2 of+
Sm in V . If g ′ is not a multiple of g , g ′ determines itstwo eigenspaces as explained earlier and so g ′, F1 and F2 can then be written simultaneously as in (11) and (12) with A = Bin (12). It follows that F1 and F2 are multiples of each other and a contradiction follows. The conclusion is that g ′ and g areconformally related on U and hence onM since U is open and dense inM . �
The final conclusion of this analysis is that if M is a smooth, connected, Hausdorff manifold of dimension n ≥ 4 and g and g ′are smoothmetrics of arbitrary signature onM whose type (1,3)Weyl tensors are equal onM and non-zero over some non-emptyopen subset of M , then g and g ′ are only necessarily conformally related if n = 4 and g (and hence g ′) is, up to sign, of positivedefinite signature.
6 G. Hall / Journal of Geometry and Physics 60 (2010) 1–7
Two immediate corollaries of this result can be given. First, letM and g satisfy the conditions of Theorem 2 and useL todenote a Lie derivative. Let X be aWeyl conformal vector field on M , that is, a global vector field on M satisfying LXC = 0.Next let ϕt : U → V be any local flow diffeomorphism of X with domain U and range V open connected submanifolds ofM .Then the pullback, ϕ∗t , of ϕt leads to a metric ϕ
∗t g on U with Weyl conformal tensor ϕ
∗t C(= C since X is a Weyl conformal
vector field). It follows from Theorem 2 that ϕ∗t g = κg (κ : U → R) for each ϕt . Thus X is a conformal vector field onM . Sincethe Lie algebra, C(M), of conformal vector fields on M is always finite-dimensional (for dim M ≥ 3) it follows that for dimM = 4 and g positive definite, the Lie algebra,WC(M) of Weyl conformal vector fields is also finite-dimensional (≤ 15). (Infact, since C(M) ⊂ WC(M), it follows in this case that C(M) = WC(M)). Second, suppose the above pair (M, g) is an Einsteinspace and letW be the type (1,3)Weyl projective tensor onM (see e.g.[3,12]). Then if X is aWeyl projective vector field onM(that is, a global vector field on M satisfying LXW = 0), X is a conformal vector field. This follows immediately from thefirst result above together with the fact that, quite generally for any dimension n (≥3) and metric signature,W = C onM ifand only if (M, g) is an Einstein space. To see this latter result note that the usual expression forW can be rewritten as
W abcd = Cabcd −1
(n− 1)(n− 2)[gadRbc − gac Rbd + (n− 1)(gbc Rad − gbdRac)] (16)
where R is the tracefree Ricci tensor with components Rab = Rab − Rngab. That R = 0 implies W = C is now clear and
the converse follows easily after an obvious contraction of (16). Thus, if dim M = 4 with g positive definite and (M, g) anEinstein space, satisfying the condition of Theorem 2 the Lie algebra, W (M), of Weyl projective vector fields equals C(M)(andWC(M)) and is also finite-dimensional (≤15).
4. Remarks and summary
It is, perhaps, instructive in comparing Theorem 1(i) (together with the preceding counter-examples) and Theorem 2to mention some differences in the vector space ΛmM for the positive definite, Lorentz and (− − ++) signatures in four
dimensions. First, if F ∈ ΛmM is simple then, in all cases, the blades of F and∗
F are orthogonal complements of each other and
for the Lorentz and positive definite cases, F and∗
F are independent members ofΛmM . However, for signature (−−++), asimple member F ofΛmM may be totally null, that is, its blade may be spanned by two independent orthogonal null vectors.
In this case F and∗
F are not independent, but equal, up to a sign. For non-simple F , F and∗
F are always independent for Lorentz
signature butmay not be in the other two cases (cf the set∼
Sm described earlier, and note that the relation∗
∗
F = F holds also for
signature (−−++)) and∗
∗
F = −F holds in the Lorentz case). Also, in the positive definite case and for any simple F , the union
of the blades of F and∗
F span TmM . This fails in the Lorentz case if and only if F is null (and the said span is 3-dimensional)and in the case of signature (− − ++) if and only if F is null or totally null (and the said spans are, respectively, 3- and2-dimensional). For Lorentz signature, any non-simple F ∈ ΛmM always uniquely determines its canonical blade pair and,if (4) is imposed, each becomes an eigenspace of g ′ [6,8]. This may fail in the other two cases (see Section 3 and note thatthe situation is similar in the case of signature (− − ++)). It is also recalled that, for the signature (− − ++), ∗C∗ = C .Finally, regarding the results in the final paragraph of Section 3, it is remarked that, in the 4-dimensional Lorentz case, theLie algebrasWC(M) andW (M) are not necessarily finite-dimensional (the latter even in the case when (M, g) is an Einsteinspace; see e.g. [6,13]). Similar comments can be shown to apply in the case of signature (−−++).
Acknowledgements
The author wishes to thank David Lonie and Matthias Lampe for several valuable discussions and suggestions.
Appendix
The construction of the second counter-example in Section 2 can be seen as a special case of the following result whichis, perhaps, of some interest in itself. Let M1 and M2 be smooth manifolds of dimensions n1 and n2, respectively admittingrespective smooth metrics g1 and g2 of arbitrary signature. LetM = M1 ×M2 and g = g1 ⊗ g2 and denote by p1 and p2 theusual smooth projections p1 : M → M1 and p2 : M → M2. If n1 ≥ 3 and n2 ≥ 3 and if C , C1 and C2 are the Weyl tensors of(M, g), (M1, g1) and (M2, g2), respectively, (M, g)will be said to satisfy theWeyl tensor projection property if the type (0, 4)tensors C , C1 and C2 corresponding to C , C1 and C2 (so that C has components Cabcd ≡ gaeCabcd, etc) satisfy p∗1C1 + p
∗
2C2 = C .
Lemma 2. With the above notation (and retaining the restriction n1 ≥ 3 and n2 ≥ 3) the Weyl projection property holdsif and only if each of (M1, g1) and (M2, g2) is an Einstein space with their associated Ricci scalars R1 and R2 satisfyingR1
n1(n1−1)+
R2n2(n2−1)
= 0.
Proof. No doubt there are many ways to prove such a result. A particularly easy way is a direct coordinate approach. Firstnote that Riem, Ricc (and, by definition, g) always satisfy such a projection property. So suppose theWeyl projection property
G. Hall / Journal of Geometry and Physics 60 (2010) 1–7 7
holds and choosem = (m1,m2) ∈ M withm1 ∈ M1 andm2 ∈ M2 and coordinate neighbourhoods A and B ofm1 inM1 andm2 in M2 and with coordinates x1, . . . , xn1 and xn1+1, . . . , xn1+n2 , respectively. Then in the associated product coordinatesystem x1, . . . , xn1+n2 in the coordinate neighbourhood A × B of m in M , the components Cabcd (equivalently Cabcd sinceg = g1 ⊗ g2) have the property that they can only be non-zero if the indices a,b,c and d all take values in either the setS1 ≡ {1, . . . , n1} or the set S2 ≡ {n1+1, . . . , n1+n2}. This imposes the restriction that CαBγD = 0 if α, γ ∈ S1 and B,D ∈ S2(essentially the only restriction, up to symmetry, as can be shown from (1)). But this restriction, when written out using (1),gives
(n− 1)[gαγ RBD + gBDRαγ ] − Rgαγ gBD = 0 (17)
and which must hold for all choices of α, B, γ and D. One may make a particular choice of α and γ such that gαγ 6= 0 andthen (17) shows that (M2, g2) is an Einstein space (and similarly so is (M1, g1)). Hence, Rαγ =
R1n1gαγ and RBD =
R2n2gBD where
R1 and R2 are the Ricci scalars of (M1, g1) and (M2, g2), respectively, (and R = R1 + R2). A back substitution into (17) thengives
R1n1(n1 − 1)
+R2
n2(n2 − 1)= 0. (18)
Conversely, suppose that (18) holds. To establish the Weyl projection property it is sufficient to show that, in the aboveproduct coordinates, the value of any Weyl tensor component in (1) with indices restricted to the set S1 equals thecorresponding component in (M1, g1), and similarly for (M2, g2). Since Riem, Ricc and g satisfy the projection property,these two tensor components differ only by the replacement of n by n1 and R by R1 (and similarly for (M2, g2)with the useof (18) and the fact that R = R1 + R2). This is easily, if tediously, checked to be the case. �
In the case when n1 = 2 and n2 ≥ 3, so that theWeyl tensor of (M1, g1) is not defined, it follows by a similar argument tothat given above, using (17), that p∗2C2 = C if and only if (M1, g1) and (M2, g2) are Einstein spaces (and the former necessarilyis) satisfying (18). In the casewhen n1 = 1 and n2 ≥ 3 it is similarly shown that p∗2C2 = C if and only if (M2, g2) is an Einsteinspace. An example in the case when n1 = 1, n2 = 3, occurs in the Einstein static (cosmological) model in general relativitywhere C2 = C = 0 and (M2, g2) is a proper Einstein space.Assuming the Weyl projection property, Eq. (18) shows that if n1 6= 1 6= n2 then R1 = 0⇔ R2 = 0 and that if one (and
hence both) of these hold then R = 0. Hence, if either of R1 and R2 is non-zero they both are and their ratio is, somewhatcuriously, a negative rational number. If, in addition, n1 6= n2, the conditions R1 = 0, R2 = 0 and R = 0 are equivalent.The relation between this lemma and the technique used to construct the second type of counter-example (the insistence
of the Ricci flat (Einstein space) condition) is now clear.
References
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