Projective relatedness and conformal flatness

Cent. Eur. J. Math. • 10(5) • 2012 • 1763-1770DOI: 10.2478/s11533-012-0087-6

Central European Journal of Mathematics

Projective relatedness and conformal flatness

Research Article

Graham Hall1∗

1 University of Aberdeen, Aberdeen AB24 3UE, Scotland, United Kingdom

Received 13 November 2011; accepted 24 May 2012

Abstract: This paper discusses the connection between projective relatedness and conformal flatness for 4-dimensionalmanifolds admitting a metric of signature (+,+,+,+) or (+,+,+,−). It is shown that if one of the manifolds isconformally flat and not of the most general holonomy type for that signature then, in general, the connectionsof the manifolds involved are the same and the second manifold is also conformally flat. Counterexamples areprovided which place limitations on the potential strengthening of the results.

MSC: 53A20, 53A30

Keywords: Projective Structure • Conformal flatness • Holonomy© Versita Sp. z o.o.

1. Introduction and notation

Let M be a smooth, connected, Hausdorff, 4-dimensional manifold admitting a smooth metric of signature either(+,+,+,+) (positive definite) or (+,+,+,−) (Lorentz). With the usual manifold topology on M, M is necessarilyparacompact. Let ∇ be the associated Levi-Civita connection on M and suppose that (M,g) is conformally flat (thatis, the Weyl tensor C associated with ∇ and g vanishes on M) and non-flat (that is, the curvature tensor, Riem, from∇ does not vanish over any non-empty open subset of M). Now let g′ be another metric on M with any allowablesignature and with Levi-Civita connection ∇′ and assume that ∇ and ∇′ (or g and g′) are projectively related on M,that is, the unparametrised geodesics on M arising from ∇ and ∇′ coincide. The object of this paper is to consider theholonomy group, Φ, of ∇ on M and to show (for either of the above signatures for g) that in all cases for Φ except themost general one, ∇ = ∇′ and that with one exception for each signature, (M,g′) is also conformally flat. It is alsoshown that the first of these results is, for each signature, the “best possible” in the sense that it fails if Φ is of themost general holonomy type. It is also pointed out that in some cases when ∇ = ∇′, the signature of g′ may not bethe same as that chosen for g and could be (+,+,−,−) in addition to the above two signatures (up to sign).∗ E-mail: [email protected]

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Since (M,g) is conformally flat one may relate the components (Ra)bcd of Riem, the components Rab ≡ (Rc)acb of theassociated Ricci tensor, Ricc, and the associated Ricci scalar R ≡ Rabgab by(0 =)Cabcd = Rabcd − Eabcd −

R12 (gacgbd − gadgbc), (1)where

Eabcd = 12 (R̃acgbd − R̃adgbc + R̃bdgac − R̃bcgad) (2)

and R̃ab ≡ Rab − Rgab/4 are the components of the tracefree Ricci tensor, R̃icc. The tensor E in (2) satisfies Eabcd =−Ebacd = −Eabdc = Ecdab and (Ec)acb = R̃ab at each m ∈ M and so the condition E(m) = 0 is equivalent to thecondition R̃icc(m) = 0 and hence to the Einstein space condition at m.Let TmM and ΛmM denote, respectively, the tangent space and the vector space of all 2-forms (bivectors) to M at m.The representations of ΛmM as type (0, 2), type (1, 1) or type (2, 0) tensors at m will be used interchangeably becauseof the obvious and natural isomorphisms which arise between them from g(m). If ∗ denotes the usual Hodge dualityoperator on ΛmM, the double dual satisfies ∗∗F = F in the positive definite case and ∗∗

F = −F in the Lorentz case.Then the Ruse–Lanczos identity, see, e.g. [2], shows that the tensor E with duals taken on both the right and left sidessatisfies ∗E∗ = −E in the positive definite case and ∗E∗ = E in the Lorentz case (and so in either case ∗E = −E∗).A bivector F ∈ ΛmM with components Fab = −Fba is called simple if the matrix Fab (necessarily of even matrix ranksince it is skew-symmetric) has rank 2. This is equivalent to the statement that there exist p, q ∈ TmM such thatFab = paqb − qapb. The 2-dimensional subspace (2-space) of TmM spanned by p and q is then uniquely determinedby F and called the blade of F . Then F is simple if and only if ∗F is simple and the blades of F and ∗

F are orthogonalto each other. If g is positive definite, the blades of F and ∗F are also complementary (that is, their union spans TmM)whereas for Lorentz signature such blades are either complementary or each is a null 2-space and they intersect in asingle null direction. In the above notation, if F is simple, F and its blade are sometimes denoted by p ∧ q. If F is notsimple it is called non-simple.

When g has Lorentz signature then for any F ∈ ΛmM, F and ∗F are independent members of ΛmM. When g ispositive definite the situation is a little bit different; if F is simple then again F and ∗

F are independent but this is notnecessarily the case if F is non-simple. For this latter signature the problem lies with the 3-dimensional subspaces±Sm ≡ {F ∈ ΛmM : ∗F = ±F} of ΛmM with respect to which ΛmM decomposes as the vector space sum ΛmM = +

Sm⊕−Sm.Any non-zero member of the set S̃m ≡ +

Sm∪−Sm is non-simple and any F ∈ ΛmM \ S̃m is independent of ∗F . In the Lorentzcase each of ±Sm is the trivial subspace. (A similar (non-trivial) complex decomposition of ΛmM exists in the Lorentz casebut will not be needed here.) More details on these matters can be found in [3, 10]. The vector space ΛmM admits ametric denoted by P and defined by P(F,G) = (gacgbd−gadgbc)FabGcd/2 (= FabGab). This metric has positive definitesignature (+,+,+,+,+,+) if g(m) is positive definite and signature (+,+,+,−,−,−) if g(m) is Lorentz.It turns out that the exact details of holonomy theory and of the theory of projective relatedness will not be required (onlysome general points, in fact, are needed). Thus for brevity it is remarked that further details on projective relatednesscan be found in [5–9, 11, 13–15, 17] (see the short summary in Section 3) and on holonomy theory in [3, 4, 12]. Anynecessary results will be indicated and referenced when they are required.

2. Preliminary results

The first result concerns the simple members of ΛmM and is independent of signature. The proof can be (mostly) foundin [3, 10].Lemma 2.1.If U is a subspace of ΛmM all of whose non-zero members are simple, dimU ≤ 3.

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It is noted that 3-dimensional subspaces of ΛmM all of whose non-zero members are non-simple, exist. In the positivedefinite case either of the subspaces ±Sm gives an example. In the Lorentz case one chooses an orthonormal basis

x, y, z, t ∈ TmM with −g(m)(t, t) = g(m)(x, x) = g(m)(y, y) = g(m)(z, z) = 1 and easily checks that the bivectorsz ∧ t + x ∧ y, y ∧ t + z ∧ x and x ∧ t + y ∧ z span such a subspace V of ΛmM. Since dim ΛmM = 6, any 4-dimensionalsubspace of ΛmM must intersect ±Sm or V non-trivially and hence the restriction dimU ≤ 3 (but with any of thesedimensions being possible for U).For the next result one introduces the linear maps f and f̃ : ΛmM → ΛmM given by f : Fab → (Rab)cdF cd (the curvaturemap) and f̃ : Fab → (Eab)cdF cd; which are often abbreviated by f : F → RF and f̃ : F → EF .Lemma 2.2.Let m ∈ M and suppose that E(m) 6= 0. Then if F ∈ ΛmM is an eigen(bi)vector of f̃ with non-zero eigenvalue α (so thatEF = αF ), F is necessarily simple.

Proof. For either signature suppose EF = αF , α 6= 0. Then ∗F abEabcdF cd = α

∗F abFab. But since ∗E = −E∗ thisequation becomes ∗EabcdFabF cd = −E∗abcdFabF cd = −EabcdFab

∗F cd = −α ∗F abFab (since Eabcd = Ecdab). It follows that

∗FabFab = 0 which is equivalent to F being simple, see e.g. [3, 10].Lemma 2.3.For either signature, if m ∈ M and E(m) 6= 0, rank f̃ is even at m.

Proof. If g(m) is positive definite f̃ is a non-trivial self adjoint map on the vector space ΛmM, the latter havingpositive definite metric P. Thus there exists F ∈ ΛmM satisfying EF = αF , α 6= 0, and which is simple by Lemma 2.2.Taking duals, using ∗E = −E∗, gives E ∗F = −α ∗F . Since F is simple, F and ∗F are independent and so rank f̃ ≥ 2. Then

P(F, ∗F ) = 0 and so with U = 〈F, ∗F〉 denoting the span of F and ∗F in ΛmM consider the restriction of f̃ to the orthogonalcomplement, U⊥, of U which is an invariant subspace of f̃ . If rank f̃ > 2 there exists G ∈ U⊥ such that EG = βG and

E∗G = −β ∗G, β 6= 0, and with G and ∗

G simple by Lemma 2.2. Clearly F , ∗F and G are independent and P(G, ∗G) = 0,and so F, ∗F,G and ∗G are independent. Hence rank f̃ ≥ 4. If rank f̃ > 4 a similar argument shows that rank f̃ = 6 andso f̃ has even rank at m.If g(m) is Lorentz the associated signature for the metric on Λm does not guarantee a non-zero eigenvalue for f̃ and adifferent approach is required. If rank f̃ = 6 at m the result is proved. Otherwise there exists F ∈ ΛmM with EF = 0and so E ∗F = 0 with F and ∗

F independent. Thus rank f̃ ≤ 4. If rank f̃ < 4 one chooses G independent of F and ∗F with

EG = 0 and E∗G = 0, and F,

∗F,G and ∗

G independent. The independence of these four bivectors follows by writingaF + b

∗F + cG + d

∗G = 0 for a, b, c, d ∈ R. Taking duals of this equation gives a ∗F − bF + c

∗G − dG = 0. Now c = 0implies a = b = d = 0 since F, ∗F and G are independent. If c 6= 0 elimination of ∗G between these two equations givesthe contradiction that c2 + d2 = 0. Thus rank f̃ ≤ 2. Another similar step shows that if rank f̃ < 2, f̃ is trivial, so acontradiction is obtained and the result follows. (It is remarked here that only the result that rank f̃ ≥ 2 at m is requiredfor what is to follow. This is instant in the positive definite case and (more easily) proved in the Lorentz case by notingthat from the condition E(m) 6= 0 it follows there exist non-trivial G,H ∈ Λm such that EG = H and so E ∗G = − ∗Hwith H and ∗

H, whether simple or not, independent. Thus rank f̃ ≥ 2 at m.)Now the conformally flat condition may be introduced.Lemma 2.4.Let (M,g) be conformally flat and with g of either of the assumed signatures. If m ∈ M and E(m) 6= 0, rank f ≥ 2 at m.

Proof. If the kernel of f , ker f , at m is trivial, rank f = 6 and the result follows. Otherwise any non-zero F ∈ ker fgives RF = 0 and hence, from (1) with C = 0, EF = −RF/6. Taking duals gives E∗F = R

∗F/6. If R(m) 6= 0,

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Lemma 2.2 shows that F is simple. Thus kerf is a subspace of ΛmM each member of which is simple. Then Lemma 2.1shows that dim ker f ≤ 3 and hence that rank f ≥ 3 at m. If R(m) = 0, Riem(m) = E(m) and Lemma 2.3 shows thatrank f = rank f̃ ≥ 2 at m.It is remarked that since the range space of the curvature map f at any m ∈ M is a subspace of the infinitesimalholonomy algebra, it is a subspace of the holonomy algebra of (M,g).3. Projective structure

Suppose that (M,g) is projectively related to (M,g′). Then M necessarily admits a smooth, global 1-form ψ such that,in any coordinate domain, the Christoffel symbols Γ and Γ′ of ∇ and ∇′, respectively, satisfy [1]Γ′abc − Γabc = δabψc + δac ψb, (3)

and, conversely, if (3) holds in any coordinate domain for some global 1-form, ψ, (M,g) and (M,g′) are projectivelyrelated. It is clear that ∇ = ∇′ if and only if ψ is identically zero on M. Since ∇ and ∇′ are metric connections, ψcan be shown to be an exact 1-form on M, see, e.g. [1], and so ψ = dχ for some smooth function χ on M. Equation (3)can, by using the identity ∇′g′ = 0, be written in the equivalent formg′ab;c = 2g′abψc + g′acψb + g′bcψa, (4)

where a semi-colon denotes a covariant derivative with respect to ∇.Thus the problem considered is reduced to that of solving (4) for g′ and ψ. In practice, one usually makes the Sinjukovtransformation, replacing the second order tensor g′ by another such tensor a (the Sinjukov tensor) and the 1-form ψby another exact 1-form, λ, and rewrites (4) in terms of a and λ to get the Sinjukov equation [17]. This latter equation(usually) turns out to be more manageable than (4) and when solved for a and λ, one can invert it to recover the originaltensors g′ and ψ. Such details, however, will not be needed here.4. The main results

Consider first the case when g is Lorentz so that (M,g) is a space-time. The potential holonomy algebras are thesubalgebras of the Lorentz algebra o(1, 3) and these are labelled according to a scheme in [16] (see also [3, 4]) asR1, . . . , R15. Of these, R5 cannot occur as a holonomy algebra for (M,g) (it is a 1-dimensional subalgebra of o(1, 1)×o(2)which is neither o(1, 1) nor o(2) and is generated by a non-simple bivector and thus fails to satisfy the algebraic Bianchiidentity on Riem [3]) but each of the others can. Here R1 is the case when (M,g) is flat and R15 = o(1, 3) is the mostgeneral situation. The exact details of these types are not required and the following brief summary will suffice. Since(M,g) is assumed non-flat it cannot be of type R1 and since it is taken, in addition, to be conformally flat it can beshown that its holonomy type must be one of the 2-dimensional types R7 (= o(1, 1)×o(2)) and R8 (an abelian subalgebraspanned by two null bivectors), or one of the 3-dimensional types R10 (= o(1, 2)) and R13 (= o(3)), or the well-known4-dimensional type R14, or the 6-dimensional most general type R15 [3, 4]. If g′ is another metric on M projectivelyrelated to g, it is known, quite generally, that if (M,g) has holonomy type R7 or R8 then ∇ =∇′ and that if (M,g) hasholonomy type R10 or R13 then either ∇ =∇′ or the curvature rank (that is, the rank of f) is ≤ 1 on M. Finally, if theholonomy type is R14 then either ∇ =∇′, or rank f = 1 on some non-empty open subset of M, or the Weyl tensor is ofPetrov type D (and hence nowhere zero) on some non-empty open subset of M [8, 9].In the case when g is positive definite a study of the (well-known) subalgebras of o(4) leads (after some deletions similarto the deletion of the R5 Lorentz case) to the following possibilities for the holonomy algebra each of which can occur.The notation is from [10] and the superscripts + and − arise from the decomposition of the vector space of 2-forms into

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the vector space sum of self dual and anti-self dual members. These types are labeled S0, S1 (= o(2)), S2 (= o(2)×o(2)),S3 (= o(3)), +

S3 (= su(2)), −S3 (= su(2)), +S4 (= u(2)), −S4 (= u(2)) and S6 (= o(4)), with S0 corresponding to the case when(M,g) is flat and S6 the 6-dimensional most general case. The algebra S1 is 1-dimensional, S2 is 2-dimensional, S3, +

S3and −S3 are 3-dimensional, and +S4 and −S4 are 4-dimensional. Since (M,g) is assumed non-flat, it cannot have holonomytype S0 and if the holonomy type is S1 a consideration of the infinitesimal holonomy algebra shows that rank f ≤ 1on M. If g′ is another metric on M projectively related to g and if (M,g) has holonomy type S2, +

S3 or −S3, it can beshown [10] that ∇ =∇′. However, if (M,g) has holonomy type S3, +S4 or −S4 then either ∇ =∇′ or rank f ≤ 1 on M.

Theorem 4.1.Let M be a 4-dimensional, connected, Hausdorff manifold admitting a metric g of signature (+,+,+,+) or (+,+,+,−).Suppose (M,g) is non-flat and conformally flat. Suppose also that g′ is another metric on M of arbitrary signature andwhich is projectively related to g. Then either(i) ∇ =∇′, or,(ii) (M,g) is of holonomy type R15 (Lorentz) or S6 (positive definite). This possibility can only occur if (M,g) admits

a region of non-zero constant curvature.

Proof. Let U = {m ∈ M : E(m) 6= 0} and V = {m ∈ M : E(m) = 0}, where E is the tensor (2) for (M,g). DecomposeM disjointly as M = U ∪ intV ∪ Z , where int denotes the interior operator in the manifold topology on M, U and intVare open and Z is a closed subset of M defined by the disjointness of the decomposition. It is easily checked thatintZ = ∅. If intV 6= ∅ the tensor E vanishes (and hence the Einstein space condition holds) on intV and since (M,g)is conformally flat each (necessarily open) component of intV is, with the metric induced from g, of constant curvature.Since (M,g) is non-flat any such curvature constant on these components is non-zero and so the range space of f hasdimension 6 on intV . Since this range space is a subspace of the holonomy algebra, this holonomy algebra is R15 or S6.The possibility of a signature change in the metric here is noted.Now suppose intV = ∅ so that U is an open dense subset of M. Then Lemma 2.4 shows that rank f ≥ 2 at eachpoint of U . Since (M,g) is conformally flat, it then follows from the discussion immediately preceding the theorem that∇ =∇′ on each component of U , hence on the open dense subset U of M, and hence on M.It is remarked that this result is in a sense “best possible”. This can be seen in the Lorentz case from the (conformally flat)generic Friedmann–Robertson–Walker–Lemaitre (FRWL) cosmological models which are of holonomy type R15. Such aspace-time may be non-trivially (that is, ∇ 6= ∇′) projectively related to another space-time (but which is necessarilystill an FRWL model [5]). A similar (analogous) positive definite example of holonomy type S6 with the same propertiescan also be constructed [10].One might now ask whether the condition that (M,g) is conformally flat/non-flat and projectively related to (M,g′)forces (M,g′) also to be conformally flat/non-flat. The final theorem reveals a more or less clean answer to this question.[It is remarked first that if (M,g) is flat (and hence conformally flat) then (M,g′) is of constant curvature (and henceconformally flat).]Theorem 4.2.Let M be a 4-dimensional, connected, Hausdorff manifold admitting a metric g of signature (+,+,+,+) or (+,+,+,−).Suppose (M,g) is non-flat and conformally flat. Suppose also that g′ is another metric on M of arbitrary signature andwhich is projectively related to g. If (M,g) is not of holonomy type R15 or S6 then (M,g′) is also non-flat and it is alsoconformally flat unless the holonomy type of (M,g) is R7 or S2 in which case (M,g′) is conformally flat if and only ifg′ = cg (0 6= c ∈ R).Proof. The restriction that (M,g) is not of holonomy type R15 or S6 together with Theorem 4.1 shows that ∇ = ∇′and that the tensor E is non-vanishing over an open dense subset U of M (to avoid non-empty open regions of constantcurvature which, in turn, would contradict the non-flat condition or the holonomy restriction on (M,g)). Since (M,g) isnon-flat and ∇ =∇′, (M,g′) also is non-flat.

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If (M,g′) admits a non-empty open (connected) subset V on which Riem′ vanishes then (V , g) and (V , g′) are projectivelyrelated and since the latter is of (zero) constant curvature it follows that (V , g) is also of constant curvature [1] andhence satisfies the Einstein space condition on V (that is, E vanishes on V ). But U ∩ V is non-empty and open (sinceU is open and dense in M) and a contradiction is achieved. It follows that (M,g′) is non-flat.If the holonomy type of (M,g) is either R14, ±S3 or ±S4, it can be shown from Lemma 2.4 and the discussion immediatelypreceding Theorem 4.1 that the equality ∇ = ∇′ implies that g′ = cg (0 6= c ∈ R) [3, 8–10] and so g′ is a conformallyflat metric on M. So suppose the holonomy type is either S3, R10 or R13 (it being convenient to deal with thesecases together). Then, from holonomy theory, each point m of the open dense subset U defined above admits an open,connected coordinate neighbourhood V ⊂ U on which a non-zero ∇-covariantly constant vector field k exists, ∇k = 0,and which is spacelike for type R10 and timelike for type R13. In addition, V can be chosen to be a metric product I×Nfor an open interval I of R and a 3-dimensional connected submanifold N orthogonal to k . Also, since ∇ = ∇′, theprojectively related metric g′ is of the same holonomy type as g and admits a similar local decomposition with k beingalso ∇′-covariantly constant [6]. Choose coordinates x1 = x, x2 = y, x3 = z and x0 = w so that the metric g takes theform (with Greek indices running 1, 2, 3)

ds2 = εdw2 + gαβdxαdxβ,where the components gαβ of g are independent of w and are, in fact, the components of the metric h induced on N from gin the coordinate system x, y, z. Also, ε = ±1 (and is chosen together with the signature of h to suit the overall signatureconsidered) and k = ∂/∂w. The condition ∇k = 0 and the Ricci identity reveal that the curvature components R0αβγand the Ricci components R0α are zero. Writing down (1) in this coordinate system for the index choice a = 0, b = β,c = 0, d = δ and using (2) (or using (5) below) one can show that Rαβ = Rgαβ/3, where R is simultaneously theRicci scalar for g and h, and Rαβ the Ricci tensor components for h. It follows that (N,h) is an Einstein space andthen, since dimN = 3, (N,h) is of constant curvature. Thus, after reducing V and N, if necessary, (N,h) admits a6-dimensional Lie algebra of Killing vector fields, K (N). The members of K (N) give rise, in a natural way, to vectorfields on V and which constitute a 6-dimensional Lie algebra of Killing vector fields for g on V . But since g and g′ areprojectively related, these latter vector fields give rise to 6 independent Killing vector fields on V with respect to g′,see e.g. [5], and which, in turn, give rise to 6 independent Killing vector fields on N with respect to the metric h′ on Ninduced from g′. It follows that (N,h′) is of constant curvature and hence an Einstein space. It then follows that theWeyl tensor C ′ on (V , g′) vanishes and so (M,g′) is conformally flat. To see this note that, in the above system ofcoordinates, the curvature components, R ′αβγδ , of g′ equal the corresponding curvature components of the metric h′. Theconstant curvature condition on h′ then reveals that R ′αβγδ = R ′(h′αγh′βδ − h′αδh′βγ)/6 = R ′(g′αγg′βδ − g′αδg′βγ)/6. It thenfollows from (1) that the components C ′αβγδ , C ′0βγδ and C ′0β0δ of the Weyl tensor C ′ for g′ on V are all zero and then itfollows that C ′ = 0. [It is remarked here that the condition ∇ = ∇′ together with Lemma 2.4 show that rank f ≥ 2on the open dense subset U of M. It follows [3] that each m ∈ U admits a connected coordinate neighbourhood onwhich g′ab = agab + bkakb for a, b ∈ R with a 6= 0 6= a + bg(k, k). Of course k (more relevantly here, kakb) may notbe globally defined; this would require extra assumptions such as the holonomy group Φ being connected or M beingsimply connected, but it shows that (locally or globally) g and g′ may not be of the same signature.]If the holonomy type of (M,g) is R8, each m ∈ U admits a connected open coordinate neighbourhood V ⊂ U on whicha ∇-covariantly constant non-zero null vector field l is defined, ∇l = 0. Since ∇ = ∇′ it follows [3, 6] that, on V ,∇′l = 0 and g′ab = agab + blalb, a, b ∈ R, a 6= 0, and so g and g′ have the same signature on V . The null vectorfield may, after reducing V , if necessary, be extended to a g-null tetrad of smooth vector fields l, n, x, y on V satisfyingg(l, l) = g(n, n) = 0, g(x, x) = g(y, y) = g(l, n) = 1 on V and with all other inner products between tetrad memberszero. The curvature tensor for (M,g) on V may be written as

Rabcd = αFabFcd + β∗Fab

∗Fcd + γ (Fab ∗Fcd + ∗

FabFcd),where F = l ∧ x and ∗

F = l ∧ y, and where α, β and γ are smooth functions on V . Thus Rab = (α + β)lalb and R = 0on V . But ∇ =∇′ implies that, in an obvious notation, Riem′ = Riem and Ricc′ = Ricc. Since g′ab = a−1gab−blalb/a2one finds that R ′ = 0. The Ricci identity for l gives (R ′a)bcd ld = 0 and so R ′abcd = g′ae(R ′e)bcd = aRabcd and R ′ab =Rab = (α + β)lalb. The expression for the Weyl tensor for (M,g′) is

C ′abcd = R ′abcd + 12 (g′adR ′bc − g′acR ′bd + g′bcR ′ad − g′bdR ′ac)+ R ′6 (g′acg′bd − g′adg′bc) (5)

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and so it follows that C ′ = aC = 0 on V . Thus C ′ vanishes on U and hence on M and (M,g′) is conformally flat.Finally suppose that (M,g) is conformally flat, non-flat and of holonomy type R7 or S2. The analysis is essentially thesame in each of these cases and so only the R7 case will be given. If (M,g) has holonomy type R7 and m ∈ M one maychoose a connected coordinate neighbourhood V ⊂ U which is a metric product of two 2-dimensional manifolds andon which a g-null tetrad of smooth vector fields l, n, x, y is defined with l and n recurrent. The vector field pairs (l, n)and (x, y) span smooth orthogonal distributions D1 and D2 (spanning the two 2-dimensional submanifolds) on V whosemetric product (with their metrics h1 and h2, respectively, induced from g) give (V , g). The metrics h1 and h2 have Riccitensors Ricc1 and Ricc2 satisfying Ricc1 = R1h1/2 and Ricc2 = R2h2/2, where R1 and R2 are the Ricci scalars of h1 andh2, respectively (it is noted that the Ricci scalar, R , of (V , g) satisfies R = R1 + R2). The conformally flat conditionon (M,g) leads, through an equation like (5), to the condition R = R1 + R2 = 0. The curvature tensor for V is [3]

Rabcd = −R12 FabFcd + R22 ∗Fab

∗Fcd, (6)

where F = l ∧ n and ∗Fab = x ∧ y. Since ∇ = ∇′, Riem′ = Riem, Ricc′ = Ricc and [3] g′ab = agab + b(lanb + nalb)with a, b ∈ R and a 6= 0 6= a+ b (note that in the analogous situation for holonomy type S2 there is a possibility of asignature change here). Then g′ab = a−1gab − b(lanb + nalb)/(a(a+ b)), Rab = R1(lanb + nalb)/2 + R2(xaxb + yayb)/2,

R ′ = g′abR ′ab = g′abRab = a−1R − bR1/(a(a + b)) = −bR1/(a(a + b)). Since ∇ = ∇′ a local decomposition similarto this one also applies to (M,g′) and so a necessary condition for conformal flatness of (M,g′) is that its Ricci scalarR ′ = 0. But if R1 vanishes over any non-empty open subset of V , Riem will vanish there also and this is in contradictionto the non-flatness assumption on (M,g). Thus if (M,g′) is conformally flat on V , b = 0 on V , hence on U and so on M.Conversely, if g′ = cg on M and (M,g) is conformally flat so also is (M,g′).It is remarked that the preservation of the conformally flat condition also holds for the (projectively related) FRWLmodels (and their positive definite counterparts) as described before Theorem 4.2.Acknowledgements

The author wishes to thank David Lonie and Zhixiang Wang for many helpful discussions. He also thanks the refereefor many instructive comments.

References

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[9] Hall G.S., Lonie D.P., Projective structure and holonomy in general relativity, Classical Quantum Gravity, 2011,28(8), # 083101[10] Hall G., Wang Z., Projective structure in 4-dimensional manifolds with positive definite metrics, J. Geom. Phys.,2012, 62(2), 449–463[11] Kiosak V., Matveev V.S., Complete Einstein metrics are geodesically rigid, Comm. Math. Phys., 2009, 289(1), 383–400[12] Kobayashi S., Nomizu K., Foundations of Differential Geometry, I, Interscience, New York–London, 1963[13] Mikeš J., Kiosak V., Vanžurová A., Geodesic Mappings of Manifolds with Affine Connection, Palacký UniversityOlomouc, Olomouc, 2008[14] Mikeš J., Vanžurová A., Hinterleitner I., Geodesic Mappings and Some Generalizations, Palacký University Olomouc,Olomouc, 2009[15] Petrov A.Z., Einstein Spaces, Pergamon, Oxford–Edinburgh–New York, 1969[16] Schell J.F., Classification of four-dimensional Riemannian spaces, J. Math. Phys., 1961, 2, 202–206[17] Sinyukov N.S., Geodesic Mappings of Riemannian Spaces, Nauka, Moscow, 1979 (in Russian)

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Projective relatedness and conformal flatness