projective structure in 4-dimensional manifolds with positive definite metrics

Journal of Geometry and Physics 62 (2012) 449–463

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Projective structure in 4-dimensional manifolds with positive definitemetricsGraham Hall, Zhixiang Wang ∗

Institute of Mathematics, University of Aberdeen, Aberdeen AB24 3UE, Scotland, UK

a r t i c l e i n f o

Article history:Received 25 April 2011Accepted 19 October 2011Available online 12 November 2011

Keywords:ProjectiveHolonomyGeodesicsPositive definite metric

a b s t r a c t

This paper considers the situation on a 4-dimensional manifold admitting two metricconnections, one ofwhich is compatiblewith a positive definitemetric, andwhich have thesame unparametrised geodesics. It shows how, in many cases, the relationship betweenthese connections and metrics can be found. In many of these cases, the connectionsare found to be necessarily equal. The general technique used is that based on a certainclassification of the curvature tensor together with holonomy theory.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

There has beenmuch recent interest in the problem of projective relatedness between connections on amanifold, that is,the relationship between connections on a manifold which have the same family of unparametrised geodesics [1–12]. Thisstudy, like many others, covers the situation when the connections are metric connections. It will concentrate on the casewhen one of the metrics involved is positive definite. Studies similar to the present one, but which cover the case when oneof the metrics is of Lorentz signature and which are important in Einstein’s general theory of relativity (in particular, for theprinciple of equivalence in that theory) can be found in [1–3,5,8,9,12].

In Section 2, a brief review of the general theory of projective structure will be given. Section 3 will be given over toestablishing some algebraic preliminaries. This will include some results concerning the vector space of 2-forms relevant tothis paper together with a pointwise classification of the curvature tensor. Section 4 is devoted to introducing the holonomytheory required and this will involve listing the subalgebras of the relevant orthogonal algebra, o(4). In Section 5, someimportant operational techniques which will be required, are collected together. Section 6 contains the main results of thepaper. They will give a description of the solutions to the projective problem for each possible holonomy group and yield acomplete answer in all but the most general case. Some consequent remarks, including a result on projective symmetry, aregiven in Section 7.

Throughout the paper, unless stated otherwise, M will denote a 4-dimensional smooth, connected, Hausdorff manifoldand g a smooth, positive definite metric on M with Levi-Civita connection ∇ . This structure will be denoted by (M, g) or(M, g,∇). Interestwill then centre on anothermetric g ′ onM , of arbitrary signature, andwith Levi-Civita connection∇

′, thatis, on the structure (M, g ′) or (M, g ′,∇ ′). If ∇ and ∇

′ have the same family of unparametrised geodesics, they will be calledprojectively related. Some of the calculations are more conveniently done in (coordinate) component notation, especiallythose involving contractions. All coordinate neighbourhoods are assumed to be connected.

∗ Corresponding author. Tel.: +44 01224273753.E-mail addresses: [email protected] (G. Hall), [email protected] (Z. Wang).

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450 G. Hall, Z. Wang / Journal of Geometry and Physics 62 (2012) 449–463

2. Projective structure

Suppose that (M, g) is projectively related to (M, g ′). ThenM necessarily admits a smooth, global 1-formψ such that, inany coordinate domain, the Christoffel symbols Γ and Γ ′ of ∇ and ∇

′, respectively, satisfy [13–15].

Γ ′abc − Γ a

bc = δabψc + δacψb (1)

and, conversely, if (1) holds in any coordinate domain for some global 1-form,ψ , (M, g) and (M, g ′) are projectively related.It is clear that ∇ = ∇

′ if and only if ψ is identically zero on M . Since ∇ and ∇′ are metric connections, ψ can be shown to

be an exact 1-form onM (see, e.g. [15]) and so ψ = dχ for some smooth function χ onM . Eq. (1) can, by using the identity∇

′g ′= 0, be written in the equivalent form

g ′

ab;c = 2g ′

abψc + g ′

acψb + g ′

bcψa (2)

where a semi-colon denotes a covariant derivative with respect to ∇ .Thus the problem considered has been reduced to solving (2) for g ′ andψ . However, the problem can, to some extent, be

simplified by adopting the Sinjukov transformation [10], (see also [4,6–8,11]). This technique involves introducing anothernon-degenerate second order tensor a and another 1-form λ on M to replace g ′ and ψ , respectively, and which are definedin terms of them by

aab = e2χg ′cdgacgbd λa = −e2χψbg ′bcgac(⇒ λa = −aabψb) (3)

where an abuse of notation has been used in that g ′ab denotes the contravariant components of g ′ (and not the tensor g ′

abwith indices raised using g) so that g ′

acg′cb

= δba . Then (3) may be inverted to give

g ′ab= e−2χacdgacgbd ψa = −e−2χλbgbcg ′

ac . (4)

The condition (2) for projective relatedness can now be shown, from (3) and (4), to be equivalent to Sinjukov’s equation [10]

aab;c = gacλb + gbcλa (5)

fromwhich it easily follows that λ is an exact 1-form, being the global gradient of 12aabg

ab. The object now becomes to solve(5) for a and λ on the original geometry (M, g,∇). With a and λ thus found, one first defines a type (2, 0) tensor a−1 on Mwhose components are, at each m ∈ M , the inverse matrix of aab (aaca−1cb

= δab). Then one defines a related type (0, 2)

tensor onM by a−1ab = gacgbda−1cd. Finally, one defines a global function χ =

12 log |

det gdet a | and a global, exact 1-formψ ≡ dχ

on M . Then g ′

ab = e2χa−1ab , which is a global metric on M , and ψ together satisfy (4) and constitute the required solution of

(2) onM .Here, it is useful to note that, using (5) and applying the Ricci identity to a, one finds (see, e.g. [4])

(aab;cd − aab;dc =)aaeRebcd + abeRe

acd = gacλbd + gbcλad − gadλbc − gbdλac (6)

where Rabcd are the components of the curvature tensor, Riem, associated with ∇ and λab ≡ λa;b (=λba since λ is exact) are

the components of ∇λ.

3. Preliminary algebraic results

Form ∈ M , letΛm denote the 6-dimensional vector space of 2-forms atm. The liberty of identifying the tensor type (2, 0),(1, 1) and (0, 2) representations of members ofΛm will be taken because of the isomorphisms between them resulting fromthe metric g(m) (raising and lowering indices). Members of Λm are referred to as bivectors and will sometimes be writtenin component form; F ∈ Λm, F ↔ F ab(= −F ba). There is a natural metric P on Λm for which the inner product P(F ,G) ofF ,G ∈ Λm is F abGab = PabcdF abGcd, with Pabcd =

12 (gacgbd − gadgbc)(m). This metric has signature (+,+,+,+,+,+) and

P(F , F)(= FabF ab) is denoted by |F |. The symbol ∗ will denote the usual duality operator onΛm and since, for this signature,∗∗

F = F for each F ∈ Λm, themap F →∗

F is a linear, self adjoint isomorphismΛm → Λm with respect to P whose eigenvaluesare either ± 1.

Any F ∈ Λm has even (matrix) rank. Thus the rank of any non-zero member ofΛm is two or four. If rank F = 2, F is calledsimple and if four, it is called non-simple. If F is simple, one may write F ab

= paqb − qapb (or F = p ∧ q), where p and q aremembers of the tangent space TmM to M at m and which may be chosen to be orthogonal. In this case, the 2-dimensionalsubspace (2-space) of TmM spanned by p and q is independent of the choice of p and q and called the blade of F . Sometimesit is also denoted by p ∧ q. The following result is useful and believed to be known (see e.g. [16]).

Lemma 1. The condition that F is simple is equivalent to the statement that∗

F is simple (and then the blades of F and∗

F areorthogonal complements of each other) and also to any of the conditions

(i)∗

F abF ab= 0, (ii)

∗

F acF cb= 0 (iii) Fa[bFcd] = 0 (7)

where square brackets enclosing indices denote the usual skew-symmetrisation of those indices.

G. Hall, Z. Wang / Journal of Geometry and Physics 62 (2012) 449–463 451

If F ∈ Λm is non-simple, a standard result shows that there exists an orthonormal basis x, y, z, w ∈ TmM and α, β ∈ Rsuch that α = 0 = β and

F ab= α(xayb − yaxb)+ β(zawb

− wazb) (⇔ F = α(x ∧ y)+ β(z ∧ w)). (8)

In this case, the eigenvectors of F (with respect to g) are x ± iy with corresponding eigenvalues ±iα, and z ± iw withcorresponding eigenvalues ±iβ . A special subclass of non-simple bivectors in Λm will be important in what is to followand are those for which an eigenvalue degeneracy for F occurs, that is, for which α = ±β . If x, y, z, w are chosen so that

(x ∧ y)∗ = (z ∧ w) this subclass consists of precisely the ± 1 eigenspaces+

Sm and−

Sm of the duality map, F →∗

F , where

+

Sm = {F ∈ Λm :∗

F = F}−

Sm = {F ∈ Λm :∗

F = −F} (9)

where+

Sm (respectively,−

Sm) corresponds to α = β (respectively, α = −β) in (8). It is then convenient to define the subsetSm ≡+

Sm ∪−

Sm ofΛm and it is noted that F ∈ Λm \Sm if and only if F and∗

F are independent members of Λm. Each F ∈ Λm

may be written as F =12 (F +

∗

F)+ 12 (F −

∗

F) and then, since+

Sm ∩−

Sm is the trivial subspace ofΛm, it follows thatΛm is the

direct sum,Λm =+

Sm ⊕−

Sm. It is also easily checked that under matrix commutation, denoted by [ ],Λm is a Lie algebra, that+

Sm and−

Sm are 3-dimensional Lie subalgebras ofΛm isomorphic to the Lie algebra su(2) (that is, to o(3)) and that if F ∈+

Sm

and G ∈−

Sm their Lie product [F ,G] = 0 (and soΛm =+

Sm ×−

Sm).With the above choice of x, y, z, w, the bivectors (x ∧ y + z ∧ w), (x ∧ z + w ∧ y) and (x ∧ w + y ∧ z) give a basis for

+

Sm and (x ∧ y − z ∧ w), (x ∧ z − w ∧ y) and (x ∧ w − y ∧ z) give a basis for−

Sm. It is also true that if F ∈+

Sm and if G ∈−

Sm,P(F ,G) = 0.

If F is a non-simple bivector inΛm\Sm, it follows from the distinctness of the eigenvalues in (8) that F uniquely determinesthe orthogonal pair of 2-spaces x ∧ y and z ∧ w and these will be called the canonical blades of F . If, however, F ∈ Sm theeigenvalue degeneracy α = ±β in (8) means that there is, in this sense, no uniquely determined blade pair. In fact, for

F ∈+

Sm one may write F in the form (8), with α = β , in the orthonormal tetrads x, y, z, w and x′, y′, z ′, w′ only under thefollowing conditions

F = α[(x′∧ y′)+ (z ′

∧ w′)] = α[(x ∧ y)+ (z ∧ w)] (10)x′

= K(x + az − bw), y′= K(y + bz + aw),

z ′= K(z − ax − by), w′

= K(w − ay + bx)

K = (1 + a2 + b2)−12 (a, b ∈ R)

(apart from the trivial exchanges (x∧ y) ↔ (z ∧w) [17], but note some typographical errors in this reference one of whichis corrected by (10)). Any such pair x′

∧ y′ and z ′∧w′ will be referred to as a canonical pair of blades (blade pair) for F . Similar

remarks apply to−

Sm. One thus achieves a ‘‘2-parameter’’ collection of canonical blade pairs for each member ofSm [It is also

remarked that, as a consequence of this, if G ∈+

Sm with G = A+ B for simple bivectors A and B it does not follow that∗

A = B.

To see this, let A and B be any simple bivectors such that∗

A−A and B−∗

B are the samemember of−

Sm. Then A+ B ∈+

Sm. (and

similarly for G ∈−

Sm).]. It is also clear from (10) that if two distinct 2-spaces atm are such that each appears as a member of

a canonical blade pair of F atm, then their span equals TmM . Further, if F1, F2 ∈+

Sm and F1 and F2 are each non-zero and havea common canonical blade then they have a common pair of canonical blades and each can be written as in (8) with α = βand hence they are proportional (and so they have the same family of canonical blades). Again, similar comments apply to−

Sm. This can be clarified, geometrically, by the following argument. Let F ∈+

Sm and G ∈−

Sm with |F | = α > 0, |G| = β > 0

and put H ≡ F + κG (κ ∈ R). Since∗

F = F and∗

G = −G and P(F ,G) = 0 one finds from Lemma 1 that H is simple if and only

if∗

HabHab= 0 which, in turn, is equivalent to κ = ±( α

β)12 . Thus H ≡ F + ( α

β)12 G and

∗

H ≡ F − ( αβ)12 G are simple and duals

of each other. Choosing an orthonormal tetrad x, y, z, w at m such that, say, H =√α(x ∧ y) and

∗

H =√α(z ∧ w) one can

then write

F =

√α

2(x ∧ y + z ∧ w), G =

√β

2(x ∧ y − z ∧ w) (11)

and this gives the following lemma.

Lemma 2. If F ∈+

Sm and G ∈−

Sm with |F | = α > 0 and |G| = β > 0, then F ± ( αβ)12 G are simple and duals of each other. Thus

F and G may be written as in (11) and each has the blades of H and∗

H as a pair of canonical blades. The bivectors H and∗

H and


their non-zero multiples are the only simple members in the span (in Λm) of F and G and if F ′∈

+

Sm and G′∈

−

Sm lead, by the

above construction, to the same bivectors H and∗

H (up to H → γH and∗

H → γ∗

H for γ ∈ R) then the spans of F and G, F ′ and

G′ and H and∗

H are equal and so F ′ and G′ are non-zero multiples of F and G, respectively.

If one defines an equivalence relation ∼ on+

Sm \ {0} by F1 ∼ F2 ⇔ F2 = σ F1 (0 = σ ∈ R), and similarly for−

Sm \ {0}, there is

a one-to-one correspondence between ((+

Sm \ {0})/ ∼)× ((−

Sm \ {0})/ ∼) and the set of all orthogonal pairs of 2-spaces ofTmM (that is, the quotient space of the Grassmann manifold G(2,R4) of all 2-spaces of TmM and which consists of all such

orthogonal pairs of 2-spaces). Then for a fixedmember of (+

Sm \ {0})/ ∼ containing F , the collection of canonical blade pairs

for F exhibited in (10) includes exactly one canonical blade pair for each member of−

Sm \ {0}. One simply chooses G ∈−

Sm

and applies Lemma 2 to F and G to get the blade pairs associated with H and∗

H .

Lemma 3. Suppose V is a subspace of Λm, all of whose non-zero members are simple. Then either (i) V is 1-dimensional andspanned by a simple bivector, (ii) V is 2-dimensional and spanned by two (simple) bivectors F and G such that the blades of

F and G (equivalently, the blades of∗

F and∗

G) intersect in a 1-dimensional subspace of TmM or (iii) V is 3-dimensional and

spanned by three bivectors F , G and H such that either, (a), the blades of F , G and H, or, (b), the blades of∗

F ,∗

G and∗

H, intersectin a 1-dimensional subspace of TmM. It is clear that, in each of the cases (i)–(iii), should a subspace V of Λm be spanned by thebivectors described above, each of its (non-zero) members is simple. Any subspace of Λm of dimension ≥ 4 contains both simpleand non-simple members.

Proof. Part (i) is clear. For part (ii), let F and G be simple bivectors spanning V . The conditions that F , G and F + λG(0 = λ ∈ R) are simple, give from the second condition of Lemma 1,

Gab

∗

F bc + Fab

∗

Gbc = 0. (12)

Since F is simple, there exists 0 = k ∈ TmM such that Fabkb = 0, (that is, k is in the blade of∗

F ). Then a contraction of (12)

with ka leads to (Gabka)∗

F bc = 0. If Gabka = 0, then k is in the intersection of the blades of

∗

F and∗

G and so the blades of∗

Fand

∗

G (and hence of F and G) have a common non-zero member. If Gabkb ≡ k′a

= 0, then∗

F abk′b= 0 and k′ is in the blades

of F and G. So the blades of F and G and hence the blades of∗

F and∗

G have a common non-zero member. For (iii), let F , G andH be simple bivectors spanning V . Then the pairs (F ,G), (F ,H) and (G,H) span subspaces V1, V2 and V3, respectively, of Vsatisfying the conditions of part (ii) above. So there exists non-zero members k1, k2 and k3 of TmM which lie, respectively,

in the blades of∗

F and∗

G,∗

F and∗

H , and∗

G and∗

H . If the span of (k1, k2, k3) is 1-dimensional, then clearly a common non-zero

member of the blades of∗

F∗

G and∗

H exists, and conversely. If the span of (k1, k2, k3) is 2-dimensional, say with k1 and k2independent, then

∗

F is a multiple of k1 ∧ k2 and so k3 is a member of the blade of∗

F . Hence k3 spans the intersection of the

blades of∗

F ,∗

G and∗

H and one is led to a contradiction to the dimension of the span of (k1, k2, k3). If the span of (k1, k2, k3)

is 3-dimensional,∗

F ,∗

G and∗

H are proportional to k1 ∧ k2, k1 ∧ k3 and k2 ∧ k3, respectively, and the unique 1-dimensionalsubspace of TmM orthogonal to the span of (k1, k2, k3) is the intersection of the blades of F , G andH . The lemma follows aftera proof that if V has dimension ≥ 4, it must contain both simple and non-simple members. This follows by first noting that

Λm admits a 3-dimensional subspace+

Sm which contains only non-simple members. The intersection of this subspace with

V gives a non-trivial subspace of V and+

Sm and hence V must contain non-simple members. Similarly, one sees that V mustcontain simple members by intersecting it with one of the 3-dimensional subspaces ofΛm containing only simple membersconstructed above. �

The final two results required are closely related and, since they have been given before [16,17], they will be stated onlybriefly. Let F ∈ Λm so that the components of F satisfy gacF c

b + gbcF ca = 0 atm. One now asks which symmetric tensors at

m other than (multiples of) g(m) satisfy this equation.

Lemma 4. Let h be a symmetric tensor type (0, 2) tensor at m, let F ∈ Λm and consider the following equation at m.

hacF cb + hbcF c

a = 0. (13)

(i) If F is simple, its blade is an eigenspace of h with respect to g(m) (that is, there exists µ ∈ R such that for each non-zerok ∈ TmM with k in the blade of F , habkb = µgabkb = µka).

(ii) If F is non-simple and F ∈ Sm, each of the (uniquely determined) canonical pair of blades of F is an eigenspace of h (butwith possibly different eigenvalues).

(iii) If F ∈Sm, there exists a canonical blade pair of F , each ofwhich is an eigenspace of h (butwith possibly different eigenvalues).


The final result requires a little preparation. The curvature tensor, Riem, arising from ∇ onM and with components Rabcd

gives rise at each m ∈ M to the curvature map f : Λm → Λm given in components by f : F ab→ Rab

cdF cd. The rank of thismap (the dimension of the range space, rg(f ), of f ) is called the curvature rank (of (M, g)) at m. This leads to a (pointwise)classification of Riem of (M, g) at each m ∈ M into five mutually exclusive and exhaustive curvature classes and whichdepends on the curvature rank and the nature of rg(f ) at m [16].Class A

This occurs when the curvature is none of the other classes given below. For this class, the curvature rank at m is 2, 3, 4,5 or 6.Class B

This occurs when dim rg(f ) = 2 and when rg(f ) is spanned by a dual pair of simple bivectors F and∗

F . In this case, Riemmay be written atm as (using the algebraic identity Ra[bcd] = 0 to remove cross terms)

Rabcd = αFabFcd + β∗

F ab∗

F cd (14)

for α, β ∈ R, α = 0 = β .Class C

In this case, dim rg(f ) = 2 or 3 and rg(f ) is spanned by independent simple bivectors F and G (or F , G and H) with the

property that there exists 0 = r ∈ TmM such that r lies in the blades of∗

F and∗

G (or∗

F ,∗

G and∗

H). Thus Fabrb = Gabrb(=Habrb) = 0 and r is then unique up to a multiplicative non-zero real number.Class D

In this case, dim rg(f ) = 1. If rg(f ) is spanned by the bivector F then, atm,

Rabcd = αFabFcd (15)

for 0 = α ∈ R and Ra[bcd] = 0, then implies that Fa[bFcd] = 0 and hence, from Lemma 1, that F is necessarily simple.Class O

In this case Riem vanishes atm.In general, the curvature class will vary from point to point onM , subject to certain topological restrictions. The following

lemma, whose proof is similar to that in the Lorentz case (see [16] p. 393), summarises the situation.

Lemma 5. Let A, . . . ,O, denote the subsets of M at each point of which the curvature class is, respectively, A, . . . ,O. Then

(i) If rg(f ) is of dimension at least 4 , it must contain a simple and a non-simple member (Lemma 3) and the curvature class atm is A.

(ii) The subsets A, A ∪ B, A ∪ B ∪ C and A ∪ B ∪ C ∪ D are open subsets of M in the usual manifold topology on M.(iii) One may disjointly decompose M as

= A ∪ int B ∪ int C ∪ intD ∪ intO ∪ Z (16)

where int denotes the interior operator in the manifold topology on M and Z is defined by the disjointness of thedecomposition and is closed and, from part (ii), can be checked to have empty interior, int Z = ∅.

(iv) When (M, g) has curvature class C or D at m, there exists non-trivial solutions k ∈ TmM of the equation

Rabcdkd = 0 (17)

these solutions being, in the above notation, just the non-zero multiples of r in the class C case and the non-zero members of

the blade of∗

F in the class D case. However, in the class A and B cases, no such solutions exist.

From this classification scheme, one can state the following theorem (whose proof is also known in the Lorentz case [16]and again the proof here is essentially the same).

Theorem 1. For the above geometry (M, g), let m ∈ M and let h be a non-zero second order, symmetric, type (0, 2) tensor at msatisfying haeRe

bcd + hbeReacd = 0.

(i) If the curvature class of (M, g) at m is D and u, v ∈ TmM span the 2-space at m orthogonal to F in (15) (that is, u∧ v is the

blade of∗

F), there exists ρ,µ, ν, λ ∈ R such that, at m,

hab = ρgab + µuaub + νvavb + λ(uavb + vaub). (18)

(ii) If the curvature class of (M, g) at m is C, there exists r ∈ TmM (the vector appearing in the above definition of class C) andρ, λ ∈ R such that, at m,

hab = ρgab + λrarb. (19)


(iii) If the curvature class of (M, g) at m is B, there exists an orthonormal tetrad x, y, z, w such that, in the notation of (14),

F = x ∧ y and∗

F = z ∧ w and ρ, λ ∈ R such that, at m (and making use of the associated completeness relation for thistetrad),

hab = ρgab + λ(xaxb + yayb) = (ρ + λ)gab − λ(zazb + wawb). (20)

(iv) If the curvature class of (M, g) at m is A, there exists ρ ∈ R such that, at m,

hab = ρgab. (21)

It is also noted, from part (iv) above, that if (M, g) is of class A everywhere and if h is a non-trivial symmetric tensorsatisfying the conditions of Theorem 1 then h is a (conformal) multiple of g and use of the Bianchi identity reveals that theconformal factor is a non-zero constant. (This was given in the Lorentz case in [16]. The proof in the present case is similarand, in fact, easier.) In this sense, if (M, g) is of class A everywhere, Riem uniquely determines its metric up to units anduniquely determines its (Levi-Civita) connection.

4. Holonomy theory

Let Φ denote the holonomy group of (M, g) (more precisely of ∇). Then Φ is a Lie group with Lie (holonomy) algebradenoted by φ. If Φ0 denotes the associated restricted holonomy group, then Φ0 is the identity component of Φ . In fact,Φ and Φ0 are Lie subgroups of O(4), and φ a Lie subalgebra of o(4). Thus the possible holonomy algebras of (M, g) arejust the subalgebras of o(4). It is useful to note that, by a consideration of the infinitesimal holonomy algebra [18], rg(f ) is

isomorphic to a subspace of φ. Now the Lie algebraΛm and its subalgebras+

Sm,−

Sm and subset Sm will for simplicity, and up to

isomorphism, sometimes be denoted byΛ,+

S ,−

S and S. So φ is a subalgebra of the algebra product+

S ×−

S which is o(3)×o(3).The possibilities for φ are known but will be briefly described here for ease of notation later on. First, it is noted that the

natural projections p : Λ →+

S and q : Λ →−

S are Lie algebra homomorphisms with kernels−

S and+

S , respectively. Now

let A be a Lie subalgebra of+

S ×−

S = o(3)× o(3). Then p(A) and q(A) are subalgebras of+

S and−

S , respectively, of dimension0, 1 or 3. If dim A = 0, A will be said to be of type S0. If dim A = 1, A will be said to be of type S1 (if A is spanned by a

simple member of Λ) and SNS1 ,+

S1 and−

S1 (if A is spanned by a non-simple member of Λ \ S, of+

S and of−

S , respectively). If

dim A = 2, p(A) and q(A) are necessarily 1-dimensional and spanned, respectively, by F ∈+

S and G ∈−

S . Using ⟨⟩ to denote‘‘spanned by’’ one has p(A) = ⟨F⟩ and q(A) = ⟨G⟩ and A is the product ⟨F⟩ × ⟨G⟩, that is, A = ⟨F ,G⟩. Thus, from Lemma 2,one may choose an orthonormal basis x, y, z, w such that A = ⟨x ∧ y, z ∧w⟩. This type will be denoted by S2. If dim A = 3,

A =+

S and A =−

S are possibilities and will be denoted, respectively, by+

S3 and−

S3. Otherwise, p(A) and q(A) are non-trivial

subalgebras of+

S and−

S , respectively, and so, since dim A = 3, either p(A) =+

S or q(A) =−

S . Suppose p(A) =+

S . Then p is

an isomorphism A →+

S . It follows that dim q(A) = 1, otherwise the kernel of q would be a 2-dimensional subalgebra of A

and hence of+

S . So q(A) =−

S . It follows that p and q are isomorphisms and that p ◦ q−1 is an isomorphism−

S →+

S . This map

preserves the inner product | | on Λ and so if a ∈ A, p(a) = F , q(a) = G, p ◦ q−1(G) = F , (F ∈+

S , G ∈−

S) then |F | = |G|

and a = F + G is simple by Lemma 2. Thus all members of A are simple and since dim A = 3, Lemma 3 (part (iii)(b)) maybe used to show that, since A is a Lie algebra (rather than just a subspace ofΛm), there exists an orthonormal basis x, y, z, w

such that A = ⟨x ∧ y, x ∧ z, y ∧ z⟩. This type is denoted by S3. Now suppose that dim A = 4. Certainly A = ⟨+

S, G⟩, G ∈−

S

and A = ⟨F ,−

S⟩, F ∈+

S are possibilities. Otherwise, one must have p(A) =+

S and q(A) =−

S . In this case, since dim A = 4,(the restriction to A of) p and q have non-trivial kernels and so A must contain a non-trivial subalgebra isomorphic to a

subalgebra of+

S and also one isomorphic to a subalgebra of−

S . It follows that+

S ⊕{0} ⊂ A and {0} ⊕−

S ⊂ A and hence one

has the contradiction A = Λ. So the only possibilities in the case dim A = 4 are A = ⟨+

S, G⟩ with G ∈−

S and A = ⟨F ,−

S⟩ with

F ∈+

S , as above, and are denoted by+

S4 and−

S4, respectively. That dim A = 5 is impossible is well-known and follows easily

from the above. If dim A = 6, A =+

S ⊕−

S = Λm which is o(4)(= o(3)× o(3)) and this case is denoted by the label S6. These

are the only possibilities for the holonomy algebra φ. However, the cases SNS1 ,+

S1 and−

S1 cannot occur as a holonomy algebra.This is because if any of them could occur (M, g) is not flat and so there exists m ∈ M such that Riem does not vanish at m.Then a consideration of the infinitesimal holonomy algebra atm shows that Riem is of class D atm so that (15) holds with Fsimple. But F must generate the holonomy algebra and the result follows.

If φ is of type S1 (isomorphic to o(2)), holonomy theory shows [19] (see also [16]) that each m ∈ M admits an openneighbourhood U and two independent vector fields, u and v on U such that u and v are covariantly constant (parallel),

∇u = ∇v = 0. Thus if∗

F = u ∧ v, F and∗

F are smooth and ∇F = ∇∗

F = 0 on U . A consideration of the infinitesimal


holonomy algebra shows that Riem is, at each m ∈ M , of curvature class O or D and so Riem satisfies (15), with F as above,over U for smooth α : U → R.

If φ is of type S2 (isomorphic to o(2)× o(2)), eachm ∈ M admits an open neighbourhood U and smooth bivector fields F

and∗

F on U such that ∇F = ∇∗

F = 0 on U . (In fact, two independent complex recurrent 1-forms exist on U whose real and

imaginary parts span F and∗

F .) Riem is, at each m ∈ M , of curvature class O, D or B and satisfies (14) over U with F and∗

F asabove for functions α, β : U → R. [There is a technical problem here in that if Riem is of class B at each point of U it seemsonly possible to guarantee the smoothness of α and β over the open subset of U consisting of those points where α = βand also over the interior of those points where α = β (the union of these two subsets is then open dense in U). See, forexample, [16,8]. However, this is not relevant here.]

If φ is of type S3 (isomorphic to o(3)), each m ∈ M admits an open neighbourhood U and a non-trivial, covariantlyconstant vector fieldw so that ∇w = 0 on U . Riem is, at eachm ∈ U , of curvature class O, D or C and (17) is satisfied at eachm ∈ U by k = w(m).

If φ is of type+

S3 (isomorphic to su(2)), each m ∈ M admits an open neighbourhood U such that (U, g) is a Kaehler

manifold and U admits a covariantly constant bivector field F , ∇F = 0, such that F(m) ∈−

Sm at eachm ∈ U , that is, F = −∗

Fon U [18]. (In fact, it can be checked that U may be chosen so that three such bivector fields exist, each satisfying these

conditions and which are pointwise independent on U .) Similar remarks apply when the type of φ is−

S3. In each case, aconsideration of the infinitesimal holonomy algebra shows that Riem is of curvature class O or A (dim rg (f ) ≥ 2) at eachm ∈ U .

If φ is of type+

S4 (isomorphic to u(2)), each m ∈ M admits an open neighbourhood U such that (U, g) is a Kaehler

manifold and again U admits a covariantly constant bivector field F(= −∗

F) on U such that ∇F = 0 [18]. In this case, Riem

is of curvature class O, D, B or A and φ may be spanned by {F1, F2, F3,G} with {F1, F2, F3} a spanning set for+

S3 and G ∈−

S3.

Then, using Lemma 2, this may be replaced by the equivalent spanning set {F1, F2,H,∗

H}whereH and∗

H are simple bivectorsarising, as in Lemma 2, from F3 and G. The impossibility of curvature class C at anym ∈ M follows from the discussion (and

notation) around (10) since, because no members of+

S3 are simple, the only simple members of rg (f ) at m ∈ M are of the

form F ′+ Gwith F ′

∈+

S . Hence their blades are members of some canonical blade pair of G, from Lemma 2. Thus their spanequals TmM and this contradicts the curvature class C condition atm.

In each of the above cases, one may choosem and U such that Riem vanishes nowhere on U .It is perhaps worth remarking at this point that the local nature of the holonomy algebra can be characterised in terms

of local covariantly constant vector and bivector fields. This can be pieced together from the above discussion together withthe following brief remarks. Let U be a connected open neighbourhood in M which is not flat, let B be the collection of allsmooth, covariantly constant bivectors on U and form ∈ M , let Bm = {F(m) : F ∈ B}. Clearly B is a Lie algebra isomorphic

to the Lie subalgebra Bm ofΛm. Also, if F ∈ B its type (that is, whether it is simple, in+

S3, in−

S3, or non-simple and not in S3)is the same at each point of U . It is also useful to notice that for 0 = F ∈ Λm its centraliser, C(F) = {J ∈ Λm : [F , J] = 0},is a subalgebra of Λm which can easily be checked to be 2-dimensional unless F is a member of Sm in which case it is 4-dimensional. It then follows that if dim C(F) = 2, C(F) is spanned by a simple bivector and its dual and that for any bivector

F ∈ Λm, C(F) = C(∗

F) and [F ,∗

F ] = 0 (cf, Lemma1). Now, supposing throughout that dimB = 0, if F ∈ B so also is∗

F and so if

B is 1-dimensional, F ∈+

S3 or F ∈−

S3. Otherwise dimB ≥ 2 and eitherB is isomorphic to+

S3 or−

S3 and hence 3-dimensional

or B contains independent members F and∗

F (satisfying [F ,∗

F ] = 0 on U). Also, if F ∈ B, the Ricci identity on F implies thatF(m) commutes with each member of rg (f ) at each m ∈ U and so Bm ⊂ C(G) for each G ∈ rg (f ) and any G ∈ rg (f ) atm commutes with each member of Bm. So either there exists G ∈ rg (f ) with G ∈ S(⇒ dim C(G) = 2 ⇒ dimB = 2) or

rg (f ) is a subspace of+

S or of−

S . With the proviso that dimB ≥ 2, it follows that in this case, dimB = 3 [It is remarked that

if F ∈ B with F ∈ S, one may choose a smooth tetrad of orthonormal vector fields x, y, z, w on U so that F = γH + δ∗

Hwith H ≡ x ∧ y,

∗

H ≡ z ∧ w and γ and δ nowhere-zero functions on U satisfying γ 2= δ2. The facts that F ,

∗

F ∈ B (and so

|F |, P(F ,∗

F), |H|(= 2), |∗

H |(= 2) and P(H,∗

H)(= 0) are constant) then show that γ and δ are constant. It then follows that

γ F − δ∗

F = (γ 2− δ2)H and so H,

∗

H ∈ B and hence B is 2-dimensional, being spanned by H and∗

H .]. Finally, let P denotethe vector space of covariantly constant vector fields on U . Since U is not flat, the Ricci identity shows that dimP ≤ 2and if it is 2 and spanned by X and Y , X ∧ Y and its dual are simple members of B and so dimB = 2. If dimP = 1, Bcannot contain any non-simple (and hence any non-trivial) members since if X ∈ P and F ∈ B is non-simple, the vectorfield with components F a

bXb is a (non-zero) member of P independent of X . These remarks give a complete local holonomybreakdown in the sense that the pairs (dimP, dimB) for U not flat can only be (2, 2), (1.0), (0, 1), (0, 2), (0, 3) and (0, 0)

and which give, respectively, the holonomy types, S1, S3,±

S4, S2,±

S3 and S6.


In summary, the only possible holonomy types may be grouped and labelled as S0, S1, S2, S3,+

S3,−

S3,+

S4,−

S4 and S6. Their2-form representation (in terms of an orthonormal basis x, y, z, w) and possible curvature classes (after a consideration ofthe infinitesimal holonomy algebra) are listed below.S0. This is the trivial case when (M, g) is flat. The curvature class is O everywhere.S1. This subalgebra is 1-dimensional and spanned by a simple 2-form x ∧ y. At eachm ∈ M , the curvature class is O or D.S2. This subalgebra is 2-dimensional and spanned by simple 2-forms x ∧ y and z ∧w. At eachm ∈ M , the curvature class isO, D or B.S3. This subalgebra is 3-dimensional and spanned by simple 2-forms x ∧ y, x ∧ z and y ∧ z. At each m ∈ M , the curvatureclass is O, D or C.+

S3. This subalgebra is of the form+

S . At eachm ∈ M , the curvature class is O or A.−

S3. This subalgebra is of the form−

S . At eachm ∈ M , the curvature class is O or A.+

S4. This subalgebra is spanned by+

S and G for some Gwith 0 = G ∈−

S . At eachm ∈ M , the curvature class is O, D, B or A.−

S4. This subalgebra is spanned by−

S and F for some F with 0 = F ∈+

S . At eachm ∈ M , the curvature class is O, D, B or A.S6. This subalgebra is just so(4). The curvature class is unrestricted (except for reasons of continuity).

It is remarked that if (M, g) admits a covariantly constant bivector field F , the Ricci identity gives

FebReacd + FaeRe

bcd = 0 (⇒ −FebRed + F aeRebad = 0) (22)

where Rab ≡ Rcacb are the components of the (symmetric) Ricci tensor, Ricc , and the implication follows after a contraction

of the first equation with gac . The second equation in (22) shows, after some elementary index juggling, that RcaF cb is skew

symmetric in the indices a and b. Thus (13) holdswith h = Ricc. It follows fromLemma4 that, should a non-trivial covariantlyconstant bivector field exist over some open regionU ofM , there will be an eigenvalue degeneracy in Ricc overU and shouldany covariantly constant non-simple bivector field exist on U the Segre type of Ricc is {(11)(11)} on U . In particular, if the

holonomy type of (M, g) is+

S3 or−

S3 the comments above show that Ricc has Segre type {(1111)} on M and hence (M, g) isan Einstein space.

Regarding the actual existence of metrics of each of the possible holonomy types, any flat metric on M yields holonomytype S0 and it is clear that, by taking appropriate metric products, metrics of holonomy types S1, S2 and S3 can easily be

constructed. The existence of metrics of holonomy type+

S3 and−

S3 can be seen from the K3 surfaces which admit Ricci-flat

Kaehler metrics [19]. Metrics of holonomy type+

S4 and−

S4 will be studied later and these are Kaehler metrics [19].

5. Mathematical techniques

A fewmoremathematical lemmas are required at this point in order to handle the imposition of the projective conditionon an alternative (metric) connection projectively related to ∇ onM .

Lemma 6. Suppose that ∇ and ∇′ are projectively related metric connections on M with associated metrics g and g ′ (and with

g positive definite) so that the material of Section 2 holds. If F is a simple bivector in the kernel of the curvature map, ker f , from∇ at m ∈ M, the blade of F is a 2-dimensional eigenspace of the symmetric type (0, 2) tensor ∇λ at m whilst if F ∈ Λm \Smis a non-simple bivector in ker f at m the canonical blade pair of F give two g-orthogonal 2-dimensional eigenspaces of ∇λ. IfF ∈Sm is in ker f , then there exists a canonical blade pair of F whose members are g-orthogonal eigenspaces of ∇λ at m. If ker fis such that TmM is an eigenspace of ∇λ at each point of some non-empty open subset U of M then, on U,

(a) λa;b = cgab (b) λdRdabc = 0 (c) aaeRe

bcd + abeReacd = 0 (23)

where c is constant in (a). Then λ is a homothetic (co)vector field on U and so if λ vanishes over some non-empty open subsetof U it vanishes on U. Further, if λ is proper homothetic on U and if it vanishes at some m ∈ U, Riem vanishes on some open

neighbourhood of m. For holonomy types S1, S2, S3,+

S3,−

S3,+

S4 and−

S4, ker f is such that TmM is an eigenspace of ∇λ at each pointof M and so the lemma applies to all holonomy types except the most general one, S6. Also, if the curvature class at each point ofsome non-empty open subset U of M is B, C, or D, the lemma and hence (23) holds on U.

Proof. Most of the proof can be found in [8] except for the part regarding the zeros of λ which can be found in [18,16].The final two sentences are proved by noting that rg (f ) is a subspace of the infinitesimal holonomy algebra and henceisomorphic to a subspace of φ and this facilitates the computation of ker f . �

The equations in parts (b) and (c) of (23) may be solved algebraically for λ and a at any m ∈ M if the curvature class isknown atm [16]. Part (c) is also usefully related to the curvature class at the appropriate point (see Theorem 1).

The next lemma seems to have appeared independently in several places [6,7,9] and states how the equations controllingthe Sinjukov tensor a and the associated 1-form λmay be reduced to a first order differential system.


Lemma 7. The tensor a, 1-form λ and function Ψ ≡ λa;a satisfy a first order system of differential equations on M. Thus a globalsolution of Sinjukov’s equation (5) is uniquely determined if a, λ and Ψ are given at some m ∈ M. In particular, if (a, λ) and(a′, λ′) are global solutions of (5) on M and if a = a′ on some non-empty open subset of M, then a = a′ and λ = λ′ on M.

Proof. Although the proof has been given elsewhere, the equations needed for it are required later in the paper and thuswill be briefly mentioned here and are taken from [9]. A contraction of (6) with gac leads to

4λa;b = Ψ gab + aecRebcd − abeRed (24)

where Ψ = λa;a. Then a covariant derivative of (24) and use of (5) gives

4λb;df = gbdΨ,f + aecRebcd;f + λeRebfd + λcRfbcd − abeRed;f − λbRdf − gbf λeRe

d (25)

where a comma denotes, in component form, a partial derivative. Finally, one achieves

3Ψ,d = aec(Rec;d − 2Red;c)− 10λeRed. (26)

The system of differential equations alluded to is now clear and the lemma follows, since, if a is given on some non-emptyopen subset U ofM , (5) uniquely determines λ on U . �

6. Projective structure and holonomy types

In this section, (M, g,∇) is as before and ∇′ is a metric connection on M projectively related to ∇ and with compatible

metric g ′ of arbitrary signature.

Theorem 2. Let (M, g) be a 4-dimensional, smooth, connected, Hausdorff manifold with smooth, positive definite metric g andassociated Levi-Civita connection ∇ . Suppose g ′ is any other smooth metric on M of arbitrary signature and with Levi-Civitaconnection ∇

′ and that ∇ and ∇′ are projectively related. If (M, g) has holonomy type S0, it is flat and (M, g ′) is of constant

curvature (and of holonomy type S0 or S6 if g ′ is positive definite but g ′ could also be of signature (+,+,+,−) or (+,+,−,−)).

If (M, g) has holonomy type S1, S2,+

S3 or−

S3, then ∇ = ∇′. The relationship between g and g ′ can be found.

Proof. For the first statement, one notes that if ∇ and ∇′ are projectively related and one is of constant curvature, then

so is the other [15]. The result now follows, since, if (M, g ′) has constant non-zero curvature and g ′ is positive definite, itscurvature rank is 6 at each point of M and hence, by appealing to infinitesimal holonomy theory, the holonomy type is S6.The possibility of alternative signatures is clear.

Next, and asmentioned in Lemma 6, (23) applies in each of the remaining cases and so λ is a global homothetic (co)vectorfield onM . Also there existsm ∈ M , and hence a coordinate neighbourhoodU ofm, onwhich Riem never vanishes. Eq. (23)(c)holds at eachm ∈ M in all cases.

For holonomy type S1, m and U may be chosen so that there are smooth, independent, orthogonal, unit, covariantly

constant (parallel) vector fields u and v on U and then∗

F = u ∧ v and F are smooth bivector fields on U satisfying

∇F = 0, ∇∗

F = 0. The Ricci identity gives Rabcdud

= Rabcdv

d. On U , Riem is of curvature class D and satisfies (15) withF as above and α : U → R which is easily checked to be smooth and nowhere zero on U . Then (23)(c) and Theorem 1(i)show that, on U ,

aab = µgab + νuaub + ρvavb + σ(uavb + vaub) (27)

for real-valued functionsµ, ν,ρ andσ onU which, following contractionswith gab, uaub, vavb and uavb are seen to be smoothonU . At anym ∈ U , onemay augment u(m) and v(m)with p, q ∈ TmM to an orthonormal tetrad atm. Finally, on substituting(27) into (5) and contracting successively with paqb, uapb and vapb, one finds that λaua

= λava

= λapa = λaqa = 0 at m.It follows that λ vanishes on U and hence on M . Thus, from (4), ψ ≡ 0 on M and so, from (1), ∇ ′

= ∇ . The relationshipbetween g ′ and g can be found by first using Theorem 1(i) to see that, on U , the components of g ′

ab equal an expression likethe right hand side of (27). The condition ∇

′g ′= ∇g ′

= 0 then shows that µ, ν, ρ and σ are constants (chosen consistentwith the non-degeneracy of g ′) but which can be regulated to make the signature of g ′ any of the three possibilities for thisdimension. A global relation between g ′ and g may be similarly achieved if the holonomy group is connected (for example,ifM is simply connected).

For holonomy type S2, the curvature class is O, D or B at eachm ∈ M and a consideration of rank shows that the subsetsB and B ∪ D are open in M with the latter non-empty. For any m ∈ M , one may choose an open neighbourhood U of m and

smooth, simple bivector fields F and∗

F on U satisfying ∇F = ∇∗

F = 0 and (14). On U , since F is smooth, one may choosesmooth, orthogonal vector fields u and v on U such that F = u ∧ v and with u a unit vector field. The condition ∇F = 0then shows that v has constant size with respect to g and hence one may re-choose v and F so that u and v are orthogonalunit vector fields on U , ∇F = 0 and F = u ∧ v. Now the condition ∇F = 0 and the orthogonality of u and v imply that


∇u = v⊗ s and ∇v = −u⊗ s for some smooth 1-form s on U . Hence, ∇(u⊗ u+ v⊗ v) = 0 on U . If B = ∅, one may choosem ∈ U ⊂ B. Then (23)(c), and Theorem 1(iii) show that one may take, on U ,

aab = µgab + ν(uaub + vavb) (28)

for smooth functionsµ and ν on U . So on substituting (28) into (5) and performing the same contractions as on the previouscase, one again achieves the result that λ = 0 on U and hence on M and so ∇ = ∇

′. If B = ∅, then D is non-empty andopen and one may choose m ∈ U ⊂ D. Then (23)(c) and Theorem 1(i) show that an expression like (27) holds for a on Uand a substitution into (5) followed by contractions, as before, show that λ = 0 on U , hence on M and thus ∇ = ∇

′. Therelationship between g ′ and g on U can again be found, this time from Theorem 1(iii), as in the previous case with g ′

ab beingas in the right hand side of (28) with µ and ν constants. The signature of g ′ is either (+,+,+,+) or (+,+,−,−).

If the holonomy type is+

S3, there exists m ∈ M and a coordinate neighbourhood U of m such that Riem is everywhereof curvature class A on U . Then (23)(c) and Theorem 1(iv) show that a = µg on U for a smooth function µ : U → R.A substitution into (5) and contractions first with gab (to get λa = 2µa on U) and then with tatb with t chosen so that0 = t ∈ TmM and taλa = 0 (to get λ = 0 at m) again show that λ vanishes on U and hence on M and so ∇ = ∇

′.

Similar comments apply to the holonomy type−

S3. Alternatively, since the curvature class is A on U , (23)(b) admits onlythe trivial solution for λ (Section 3) and again the result follows. There is yet another proof in this case. As remarked atthe end of Section 4, (M, g) is necessarily an Einstein space and the result in this case is known by alternative methods [3,4]. Theorem 1(iv) and the remarks following it show that g ′ and g are conformally related on U with constant conformalfactor. �


′ and that ∇ and ∇′ are projectively related. If (M, g) has holonomy type S3 and if there exists m ∈ M at which the

curvature rank is ≥ 2 (equivalently, the curvature class is C), then ∇ = ∇′. This result fails if the curvature rank is ≤ 1 at each

m ∈ M.

Proof. For the first part, there existsm ∈ M and a coordinate neighbourhood, U , ofm on which Riem is everywhere of classC and which admits a smooth unit vector fieldwwhich is covariantly constant,∇w = 0. The conditions of Lemma 6 leadingto (23) hold at each point of M and so (23)(c) and Theorem 1(ii) give, on U ,

aab = µgab + νwawb (29)

for functions µ and ν, U → R, which are easily checked to be smooth. At each m ∈ U , one may augment w(m) withx, y, z ∈ TmM to an orthonormal tetrad at m. Then a substitution of (29) into (5) and contractions with xayb and zawb at mshow, as before, that λ vanishes on U and hence onM . Thus ∇ = ∇

′.For the second part, examples (M, g) can be constructed which are of holonomy type S3 and of curvature class D (that

is, of curvature rank equal to 1) everywhere and on which an alternative connection ∇′ with compatible metric g ′ exists

with ∇ and ∇′ projectively related but ∇ = ∇

′. In general, ∇ ′ can be shown to have holonomy type S6 but special caseswhere it has holonomy type S3 exist (and these are the only possible holonomy types for ∇

′ — see Theorem 4). A similarconstruction to this has been carried out in the Lorentz case [8] but which goes through essentially without change in thiscase also. Thus no further details need be given here. Suffice it to say that, in the local context in which the solution willnow be described and given that ∇

′ has holonomy type S3, the examples below are essentially the only such examples andare given in a notation similar to that in [8] (in which one needs only to make appropriate choices of ϵ1 and ϵ2 in Section 7of that paper). More precisely, if (M, g) is of holonomy type S3 and an alternative connection ∇

′ with compatible metric g ′

exists with ∇ and ∇′ projectively related and different but each of holonomy type S3, then eachm ∈ M is of (∇−) curvature

class D or O and D is open and non-empty and any m ∈ D admits a coordinate neighbourhood U ⊂ D with coordinatesz, w, x1, x2 such that g takes the form

ds2 = dw2+ dz2 + z2hαβdxαdxβ (α, β = 1, 2) (30)

where the functions hαβ are independent of z andw and constitute a positive definite matrix at each point of U . The metricg ′ then takes the form (up to a constant multiplicative factor and coordinate restrictions to avoid degeneracies)

ds2 = k(w)[k(w)(1 + cz2)dw2+ dz2 − 2cz(cw + d)−1dwdz + z2hαβdxαdxβ ] (31)

where k(w) = c(cw + d)−2 and c and d are constants. It can be checked that g ′ is of (∇ ′−) curvature class D on an open

connected subset ofU where it is defined and has holonomy type S3 on this open submanifold. These last two results perhapsfollows most easily by a coordinate transformation of (31) into a form identical to that of g in (30) (for details, see [12]). Thegeneral solution (31) is controlled by the single function k and two arbitrary constants (up to the limitations imposed by achoice of signature for, and the non-degeneracy of, g ′). �

It is remarked that in the general case where (M, g ′) has holonomy type S6, it necessarily admits a (local or global) Killingvector field. This follows from the fact that ∂/∂w is a Killing vector field for g on U in (30) and from the result that any


(local or global) Killing vector field in (M, g) automatically gives rise to a similar one in (M, g ′) if ∇ and ∇′ are projectively

related [5]. [It should be noted that in Theorem 1(ii) of this reference, Y is in fact, a global g ′-Killing vector field on M andnot a local one, as stated, since the functions φ and ρ used there are global (smooth) functions onM .]


′ and that ∇ and ∇′ are projectively related. Then

(a) If (M, g) has holonomy type+

S4 or−

S4, then either ∇ = ∇′ and g ′

= κg for κ constant or (M, g ′) has holonomy type S6.(b) If (M, g) has holonomy type S3 with curvature rank ≤ 1 at each m ∈ M, then either (M, g ′) has holonomy type S3 or

holonomy type S6.

Proof. (a) From previous remarks (Section 4) together with Lemma 5, if (M, g) has holonomy type+

S4 or−

S4, thenM may bedisjointly decomposed first as M = A ∪ B ∪ D ∪ O with A, A ∪ B and A ∪ B ∪ D open in M and second (since A is open) asM = A ∪ int B ∪ intD ∪ intO ∪ Z with Z closed and int Z = ∅. Also ker f at any m ∈ M contains (at least) two independent

members of−

Sm (or+

Sm) and hence from Lemmas 2 and 4, ker f is such that TmM is an eigenspace of ∇λ. Then Lemma 6and (23) hold and λ is a homothetic (co)vector field on M . If A is non-empty, then since A is open and (23)(b) has no non-trivial solutions at any point of A, λ = 0 on A and hence on M . [Alternatively, the vanishing of λ at any point of A gives acontradiction from Lemma 6 since, then Riem would be forced to vanish at that point (and in some neighbourhood of it).]Similar remarks apply to A ∪ B. Thus either ∇ = ∇

′ or A = B = ∅. Thus in order to get a solution for which ∇ = ∇′, one

must haveM = D∪ Owith D open inM and non-empty. Now (M, g ′) cannot have holonomy type S1, S2,+

S3 or−

S3 or type S3with curvature rank 2 or 3 at some m ∈ M . This is immediate from Theorems 2 and 3, since then ∇ = ∇

′ and the assumed

holonomy type of (M, g) is contradicted. Thus (M, g ′)must be of holonomy type+

S4 or−

S4 or S6 or of holonomy type S3 withcurvature rank ≤ 1 at each m ∈ M . Suppose that (M, g ′) is not of holonomy type S6 and that ∇

′= ∇ . Then M decomposes

not only asM = D ∪ O, as above, but also asM = D′∪ O′, where D′ and O′ are the subsets ofM which are of curvature class

D and O, respectively, with respect to ∇′ and where D′ is open in M . This follows from the second part of Theorem 3. Now

the connections ∇ and ∇′ have coefficients which, in any coordinate system, satisfy (1) and their corresponding curvature

tensor components then satisfy in an obvious notation (see, e.g. [15])

R′abcd = Ra

bcd + δadψbc − δacψbd (32)

whereψab ≡ ψa;b−ψaψb are the components of the tensor∇ψ−ψ⊗ψ and, as before, a semi-colon denotes, in componentform, a ∇-covariant derivative. Now the curvature class conditions on ∇ and ∇

′ reveal that, at anym ∈ M , there exists non-zero members p, q, r, s ∈ TmM with p and q independent and r and s independent such that

Rabcdpd = Ra

bcdqd = R′abcdrd = R′a

bcdsd = 0. (33)

Now (32) contracted with pbrc and use of (33) together with a simple rank argument in the resulting equation gives, first,ψabparb = 0 and, finally, ψabpb = 0. Then (32) contracted first with pb gives R′a

bcdpb = 0 and then with pc gives ψab = 0. Itfollows that ∇ψ = ψ ⊗ψ onM . Nowψ is exact and is thus the global gradient of a smooth function χ : M → R,ψ = dχ .It then follows that ∇(e−χψ) = 0 and thus (e−χψ) is a global (∇−) covariantly constant 1-form on M . But then (1) showsthat ∇

′(eχψ) = 0 on M and so eχψ is a global (∇ ′−) covariantly constant 1-form on M . It follows that either ∇ = ∇

′ or, ifψ is not the zero 1-form on M , each of ∇ and ∇

′ admits a global nowhere-zero 1-form on M and one has the contradiction

that ∇ is not of holonomy type+

S4. Thus if the holonomy type of (M, g) is+

S4 (respectively,−

S4), and ∇′ is not of holonomy

type S6,∇ = ∇′ and the holonomy type of (M, g ′) is

+

S4 (respectively,−

S4). Further, given that∇ = ∇′, (13) is satisfied for any

F ∈ φ and with h = g and h = g ′. It now follows from Lemma 4 that, at eachm ∈ M , g ′ and g are conformally related. Thusg ′ and g are conformally related on M , g ′

= κg , for some smooth function κ on M . But then the conditions ∇g = ∇g ′= 0

show that, sinceM is connected, κ is constant onM . The result follows.For part (b), if the holonomy type of (M, g) is S3 with curvature rank≤ 1 at eachm ∈ M , thenM decomposes asM = D∪O

as in the previous case and, from the previous two theorems and the first part of this one, the holonomy type of (M, g ′) isS3 or S6. Supposing (M, g ′) is not of holonomy type S6. Then it is of type S3 with curvature rank ≤ 1 at each m ∈ M . Thus, italso decomposes into curvature classes in the same way as does (M, g) and the argument is essentially the same as that inpart (a) of this theorem. However, one can say a little more in this case because if ∇ = ∇

′, the argument in part (a) revealsthat M admits a global, nowhere zero, covariantly constant 1-form with respect to each of ∇ and ∇

′. (This holonomy type,in general, only guarantees a local covariantly constant vector field unless Φ is connected as would follow, for example, ifM were simply connected.) �


The following consequence of these results can now be stated.

Corollary 1. Let (M, g) be a 4-dimensional, smooth, connected, Hausdorff manifold with smooth, positive definite metric g andassociated Levi-Civita connection ∇ . Suppose g ′ is any other smooth metric on M of arbitrary signature and with Levi-Civitaconnection ∇

′ and that ∇ and ∇′ are projectively related. Then either ∇ and ∇

′ have the same holonomy type or one of these

holonomy types is S6 (and the other is either+

S4, or−

S4, or S3 and, in each case, with curvature rank ≤1 at each point of M, or S0).In the event that each is of holonomy type S3 and ∇ = ∇

′, then each admits a global, nowhere zero, covariantly constant 1-form.

Now consider the case when (M, g) has holonomy type+

S4 (the case−

S4 is similar) and the situation when (M, g) isprojectively related to (M, g ′) with ∇ = ∇

′. Then, from Theorem 4(a), (M, g ′) has holonomy type S6 and M may bedecomposed, as above, in the formM = D∪Owith respect to∇ . LetU be a coordinate neighbourhood ofm inD,m ∈ U ⊂ D.

Then the range of the curvaturemap f at eachm′∈ U is a 1-dimensional subspace of ⟨

+

Sm′ ,G⟩ (G ∈−

Sm′ ) spanned by a smooth

simplemember H of it. Thus since each member of+

Sm′ and of−

Sm′ is non-simple, H(m′) is the sum of two non-zero membersone from each of these sets. Now one chooses U such that it admits a smooth non-simple 1-form F defined on U satisfying

F(m′) ∈−

Sm′ for each m′∈ U and ∇F = 0. Then Lemma 6 and (23) hold on U . If the constant c in (23)(a) is zero but λ is

not identically zero on U , U admits the independent, covariantly constant vector fields λa and F abλb. This means that theholonomy type ofU has reduced to type S1 and hence, from an earlier result, λ = 0 onU and hence, from Lemma 6, λ = 0 onM . Thus ∇ = ∇

′. On the other hand, if c is not zero, the vanishing of λ at any point of U and Lemma 6 reveal a contradictionto the class D assumption on U .

So suppose c = 0 and λ is nowhere zero on U . The bivector field H is smooth and nowhere zero on U and U may be

chosen so that it admits a smooth orthogonal tetrad of vector fields x, y, z, w with H = x ∧ y and∗

H = z ∧ w. Use of theRicci identity on F shows that (22) holds on U and hence that F commutes, in the Lie algebra product, with every member

of the range of the curvature map f , that is, with H . Thus (see Section 4) F is a linear combination of H and∗

H and hence a

multiple of H −∗

H and so F = ϵ(H −∗

H)with ϵ smooth on U . Since∇F = 0 and U is connected, |F |, and hence ϵ are constant

on U with ϵ = 0. So choose F = x ∧ y − z ∧w = H −∗

H on U . Then, on U , Riem takes the form (15) with bivector H and soRabcdzd = Rabcdw

d= 0.

Then remarks following (17) together with (23)(b) show that on U

λa = ρza + σwa (34)

for (clearly smooth) functions ρ and σ and then ((23)(c)) and Theorem 1(i) give

aab = µgab + βzazb + γwawb + δ(zawb + wazb) (35)

where, as before,µ, β , γ and δ are smooth functions on U . The calculation then proceeds by substituting (35) into (5) to get

µ,cgab + β,czazb + β(zazb;c + za;czb)+ γ,cwawb + γ (wawb;c + wa;cwb)

+ δ,c(zawb + wazb)+ δ(za;cwb + zawb;c + wa;czb + wazb;c) = gacλb + gbcλa. (36)

It is first noted that a contraction of (36) with xaxb gives µ,c = 0 and so, since U is connected, µ is constant on U .Then, since the tetrad members are everywhere of constant size and mutually orthogonal with respect to g , xaxa;b = 0,zawa;c +waza;c = 0 (and similarly for y, z andw), and successive contractions of (36) with za andwa give the following twoequations

β,czb + βzb;c + γ (zawa;c)wb + δ,cwb + δwb;c + δ(zawa;c)zb = zcλb + ρgbc (37)

β(waza;c)zb + γ,cwb + γwb;c + δ,czb + δ(waza;c)wb + δzb;c = wcλb + σgbc . (38)

Then contractions of (37) with zb, (38) withwb and (37) withwb give, using (34),

β,c + 2δ(zbwb;c) = 2ρzc (39)

γ,c + 2δ(wbzb;c) = 2σwc (40)

δ,c + wbzb;c(β − γ ) = σ zc + ρwc . (41)

Finally, one substitutes for the gradients of β , γ and δ from (39)–(41) into (37) and (38) and uses (34) to get the two relations

ρ(gbc − zbzc − wbwc) = β(zb;c − waza;cwb)+ δ(wb;c − zawa;czb) (42)

σ(gbc − zbzc − wbwc) = δ(zb;c − waza;cwb)+ γ (wb;c − zawa;czb). (43)


On U , Riem takes the form (15) with α smooth and nowhere zero and the bivector denoted by F in (15) replaced byH = x ∧ y. Now Riem satisfies the differential Bianchi identity which, in components, is Rab[cd;e] = 0. Thus on substituting(15), one finds

αHabH[cd;e] + αHab[;eHcd] + α[,eHcd]Hab = 0. (44)

Now HabHab = 2 on U and so HabHab;c = 0. Thus a contraction of (44) with Hab gives

2αH[cd;e] + 2α[,eHcd] = 0 (45)

and then, (44) and (45) give

Hab[;eHcd] = 0. (46)

Now the set U may be decomposed, disjointly, both as U = U1 ∪ U2, where U1 = {m′∈ U : α,a = 0} and U2 =

{m′∈ U : α,a = 0} and then as U = U1 ∪ intU2 ∪ W , where int denotes the interior operator in the induced topology on U

(andwhich is the same as that onM since U is open) andW is defined by the disjointness of the decomposition and is closed(inU) with empty interior. Henceforth, the argument is restricted to the open dense subsetU1∪ intU2 ofU and this (possiblyreduced) new U will be supposed to be a connected open subset of this open dense subset. This means that the (smooth)tetrad x, y, z, w on U may be chosen so that z(α)(= α,aza) = 0 on U . Then (46) contracted with zd gives Hab;czc = 0 andthen a contraction of (44) with zc gives

Hcb;azc + Hac;bzc = 0. (47)

Since ∇F = 0 on U , ∇(H −∗

H) = 0 and hence ∇H = ∇∗

H on U . It then follows that (47) holds for∗

H also, that is,

∗

Hcb;azc +∗

Hac;bzc = 0. (48)

Since∗

H = z ∧ w, (48) gives after a short calculation

wd;e − (zcwc;e)zd = we;d − (zcwc;d)ze (49)

and so the tensor S on U with components Sab = wa;b − (wc;bzc)za is symmetric on U . Now return to (42), multiply by Fab(F = x ∧ y − z ∧ w) and contract over the repeated index b. This gives

ρ(xayb − yaxb) = β(wa;b − wd;bzdza)− δ(za;b − zd;bwdwa) (50)

whilst a similar calculation on (43) gives

σ(xayb − yaxb) = δ(wa;b − wd;bzdza)− γ (za;b − zd;bwdwa). (51)

Noting how the tensor S appears with coefficients δ and γ , respectively, in (42) and (43), one sees from these equations thatif at any m ∈ U and hence on some open neighbourhood V of m either β or δ is not zero, the tensor T with componentsza;b − zc;bwcwa is also symmetric on V . But then, from (50) and (51), ρ = σ = 0 and hence from (34), λ = 0 on V . It thenfollows from Lemma 6 that λ = 0 on M and hence that ∇ = ∇

′. Thus, in order to achieve ∇ and ∇′ being different, one

must take β = δ = 0 and hence from (50), ρ = 0, on U . So λa = σwa on U and since λ cannot vanish on U , σ is nowherezero on U . It follows from (43) that γ is nowhere zero on U . Thus on U one has, using (35)

λ = σw a = µg + νλ⊗ λν =

γ

σ 2

(52)

with c = 0 andµ a non-zero constant. Then a substitution of the second equation of (52) into (5), a contraction with λb anda simple rank argument shows that ν(= 1

c ) is a non-zero constant. Thus the solution so far is (52) with µ and ν non-zeroconstants (the former to preserve the non-degeneracy of a).

With the above conditions on U , (41) reduces to γwbzb;a = −σ zc . Also, the conditions that β = 0 and that σ and γ arenowhere zero on U mean, from (51), that the tensor T is skew-symmetric on U . So

za;b − zd;bwdwa = zd;awdwb − zb;a ⇒ za;b + zb;a = wasb + sawb (53)

where sa = zc;awc= −

σγza. Now, (40) shows that γ,a = 2σwa and so with τ ≡

√|γ | one can check that Xa

≡ τ za is anowhere-zero Killing vector field onU , Xa;b+Xb;a = 0. Thus, in addition to the global homothetic vector field arising from λ, a(local) Killing vector field arises as a necessary consequence of the assumptionsmade so far onU . Further, the Killing bivectorK defined by K = ∇X with components K a

b = Xa;b satisfies the condition ∇K = 0 on U by virtue of the Killing condition

Xa;bc(= K a

b;c) = RabcdXd

= 0, the latter equality following from the condition Rabcdzd = 0. Thus the Killing bivector, which

cannot be zero (to avoid the existence of non-trivial covariantly constant vector fields on U , as discussed before) must bea non-zero constant multiple of the covariantly constant bivector F . This can be directly checked by writing out F in tetradform, as given earlier, and applying appropriate contractions to the equation ∇F = 0.


The above arguments reveal how to construct a coordinate expression for such a metric g , should one exist. That suchmetrics do exist can be seen by recourse to the following example. It is, in fact, a special case of a more general class ofmetrics with the appropriate properties constructed by David Lonie, to whom the authors wish to record their thanks. Eachmember of this general class admits a Killing vector field as the above theory demands. The special case alluded to above iswritten in coordinates x, y, z, w withw > 0 and is given by

ds2 = w2[dx2 + dz2 + (4x2 + 1)dy2 + 4xdydz] + dw2. (54)

This metric is of positive definite signature and the only non-vanishing component of Riem is R1212 = −4w2 (up tocomponent symmetry). Thus it is of curvature classD on its domain of definition. Thismetric admits the covariantly constantbivector F =

1w∂/∂w∧ ∂/∂z +

1w∂/∂x∧ ( 1

w∂/∂y−

2xw∂/∂z) and, in fact, two independent (commuting) Killing vector fields

arising from the ignorable coordinates y and z. The general expression for the metrics projectively related to g in (54) is, upto a constant conformal factor, given by

ds2 = (1 + bw2)−1{w2

[dx2 + dz2 + 4xdydz + (4x2 + 1)dy2] + (1 + bw2)−1dw2} (55)

where b is a constant chosen so that g ′ is well defined and non-degenerate. Again, g ′ admits two Killing vector fields, asexpected, and must be of holonomy type S6 from the above corollary. It is of positive definite signature.

In conclusion, an example is given of a geometry (M, g)which is of holonomy type S6 and a geometry (M, g ′) projectivelyrelated to it with ∇

′ different from ∇ and also of holonomy type S6. This example is just the positive definite version of anearlier example (originally with Lorentz signature—the Friedmann–Robertson–Walker–Lemaitre metric from cosmology)given in [5,6]. The calculations involved are rather similar to those in [5] and are omitted. [The Lorentz argument drewon some results regarding orbits of Killing symmetries but which are easily shown to be true in the positive definitecase also.] The manifold is taken as a product M = I × H , where I is an open interval of R coordinatised by t and H a3-dimensional connected manifold admitting a global coordinate system r, θ, φ. The metric g on M is then given in theglobal coordinate system t, r, θ, φ by

ds2 = dt2 + R(t)2[dr2 + f 2(r)(dθ2 + sin2 θdφ2)] (56)

where the function f depends on the sign of the curvature of the (constant curvature) hypersurfaces of constant t and thefunctionR is chosen to avoid (M, g)being locally of constant curvature or admitting anynon-trivial local covariantly constantvector fields. It is then of holonomy type S6. The general solution to the projectively related condition (2) is then, up to aconstant conformal factor,

ds2 = q2(R)dt2 + q(R)R(t)2[dr2 + f 2(r)(dθ2 + sin2 θdφ2)] (57)

where q(R) = (1 + eR2)−1 and e is a constant. The Levi-Civita connections ∇ and ∇′ of g and g ′, respectively, are not, in

general, equal. The choice of g ′,given g , is controlled by the single function q and a single arbitrary constant e. The holonomytype of g ′ is necessarily S6. To see this, one notes that, from Corollary 1, it is only necessary to show that Riem′ cannot beof rank 1. Now, g ′ admits a Killing isotropy (at least) o(3) at each point of its domain and so, if the curvature rank is 1 at mand G ∈ Λm is the (necessarily simple—see (15)) 2-form spanning the range of the curvature map at m, at least one otherindependent member of G ∈ Λm (generated by the isotropy action on G) must lie in this curvature range. This contradictionshows that the curvature rank is at least 2 everywhere and the result is proved.

7. Further remarks

One remark which may be made concerns the differentiability of solutions to the projective related problem (and isindependent of the dimension and signature considered). The general approach in Section 2 was to assume that M wassmooth throughout and to take g (and hence ∇) smooth. Then, in addition, the projectively related metric g ′ (and hence∇

′) and ψ were taken as smooth also and one proceeded with the solution. But now the question is, with g smooth andg ′ projectively related to g , what level of differentiability is forced upon g ′ and ψ? Suppose that the components g ′

ab andψa (and hence g ′ab) and the function χ are C2. Then (3), (5), (6) and (24)–(26) make sense and (3) shows that aab and λaare C2. Then (5) shows that the aab are C3 and (26) reveals that Ψ is C3 and then (24) shows that the λa are C3. Then (5)shows that the aab are C4. On repeating this process, one sees that a and λ are smooth on M . Using the relations (4) to findg ′ and ψ one notes that the function χ , which is unique up to an additive constant, can be taken as (see, for example, [8,9])χ =

12 log |

det gdet a | which is smooth and so g ′ and ψ are smooth. Thus to obtain smoothness of the solution g ′, ψ and χ , it is

sufficient to assume that these quantities are C2.A second remark concerns projective symmetry on (M, g). With (M, g) as before, let X be a global, smooth vector field on

M . Then X is called projective if each local flow diffeomorphism φt associated with X and with domain U and range φt(U)maps (unparametrised) geodesics of U (with respect to the restriction of ∇ to U) to (unparametrised) geodesics of φt(U)(with respect to the restriction of ∇ to φt(U)). Further, X is called affine if it is projective and if, in addition, each map φtpreserves affine parameters. If X is projective but not affine, it is called proper projective. If X is proper projective, there existsa non-empty open (connected) subset U ⊂ M on which a local flow φt of X is defined and which is projective but does not


preserve affine parameters. Then the metrics g and φ∗t g , as metrics on U , can be seen to be projectively related but their

respective Levi-Civita connections on U are not equal. This shows that if a proper projective vector field exists on M , someconnected open subset U ofM with its induced metric and connection structure fails to satisfy the holonomy requirementsof Theorem 2, (the first part of) Theorem 3 and the appropriate parts of Theorem 4 and Corollary 1. This gives the followingtheorem.

Theorem 5. Let M be a 4-dimensional, smooth, connected, Hausdorff manifold with smooth positive definite metric g andassociated Levi-Civita connection∇ and suppose that M is non-flat, that is, the associated curvature tensor Riem does not vanish

over any non-empty open subset of M. Let X be a projective vector field on M. If (M, g) either has (i) holonomy type S1, S2,+

S3

or−

S3, or, (ii) holonomy type S3,+

S4 or−

S4 and, in addition, there exists no m ∈ M at which the curvature class is D (equivalently,there exists no m ∈ M at which the curvature rank is 1), then X is affine.

Proof. It suffices to show that each open subset U of M , with its restricted metric from g , has one of the holonomy typeslisted in this theorem. The result then follows from Theorems 2–4 and Corollary 1. Recalling the non-flat condition on (M, g)(so that U is not flat), it is easily checked that the holonomy type of (U, g)must necessarily be contained in that list and theresult follows. �

As a final remark, consider the equivalence relation on the collection of all smooth symmetric (but not necessarilymetric)connections on a certain manifoldM of arbitrary dimension given by projective relatedness. The following question is theninteresting. When does a particular equivalence class contain a Levi-Civita connection of some metric on M? Some progresson this problem (and further references regarding its history) can be found in [20]. The problem studied in the presentpaper can be reformulated in terms of uniqueness theorems regarding the existence of such Levi-Civita connections andthe uniqueness of the associated positive definite metrics in the case when dimM = 4. From the results presented in thispaper it is clear that, within a (projective) equivalence class, the same Levi-Civita connection may arise frommore than onemetric (other than constant multiples of it) and may arise from metrics of different signature. The former remark is clearfrom the above results and the latter one is easily seen, for example, from the holonomy types S1, S2 and S3 by utilising aproduct space for the example and changing certain signs in themetric. Similar remarks apply to the case of Lorentz signature[9,12].

Acknowledgements

The authorswish to record their gratitude to Dr. David Lonie for his help in constructing themetrics discussed in Section 6and for many valuable conversations and useful remarks on the topic of this paper.

References

[1] A.Z. Petrov, Einstein Spaces, Pergamon, 1969.[2] G.S. Hall, D.P. Lonie, The principle of equivalence and projective structure in spacetimes, Classical Quantum Gravity 24 (2007) 3617.[3] G.S. Hall, D.P. Lonie, Projective equivalence of Einstein spaces in general relativity, Classical Quantum Gravity 26 (2009) 125009.[4] V. Kiosak, V. Matveev, Complete Einstein metrics are geodesically rigid, Commun. Math. Phys. 289 (2009) 383.[5] G.S. Hall, D.P. Lonie, The principle of equivalence and cosmological metrics, J. Math. Phys. 49 (2008) 022502.[6] J. Mikes, I. Hinterleitner, V.A. Kiosak, in: J.-M. Alini, A. Fuzfa (Eds.), On the Theory of Geodesic Mappings of Einstein Spaces and their Generalizations,

in the Albert Einstein Centenary International Conference, American Institute of Physics, 2006.[7] J. Mikes, V. Kiosak, A. Vanzurova, Geodesic mappings of manifolds with affine connection, Olomouc, 2008.[8] G.S. Hall, D.P. Lonie, Holonomy and projective equivalence in 4-dimensional Lorentz manifolds, Sigma 5 (2009) 066.[9] G.S. Hall, D.P. Lonie, Projective structure and holonomy in 4-dimensional Lorentz manifolds, J. Geom. Phys. (2010) doi:10.1016/j.geomphys.

2010.10.07.[10] N.S. Sinyukov, Geodesic Mappings of Riemannian Spaces, Nauka, Moscow, 1979, MR0552022, Zbl 0637.53020 (in Russian).[11] J. Mikes, A. Vanzurova, I. Hinterleitner, Geodesic mappings and some generalisations, Olomouc, 2009.[12] G.S. Hall, D.P. Lonie, Projective structure and holonomy in general relativity, Classical Quantum Gravity 28 (2011) 083101.[13] T. Levi-Civita, Sulle trasformazioni delle equazioni dinamiche, Ann. Mat. Milano. 24 (1886) 255.[14] T.Y. Thomas, Differential Invariants of Generalised Spaces, Cambridge, 1934.[15] L.P. Eisenhart, Riemannian Geometry, Princeton, 1966.[16] G.S. Hall, Symmetries and Curvature Structure in General Relativity, World Scientific, 2004.[17] G.S. Hall, Some remarks on the converse of Weyl’s conformal theorem, J. Geom. Phys. (2009) doi:10.1016/j.geomphys.2009.08.002.[18] S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, vol. 1, Interscience, New York, 1963.[19] A. Besse, Einstein Manifolds, Springer, 1980.[20] P. Nurowski, University of Warsaw, 2010 (Preprint).

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projective structure in 4-dimensional manifolds with positive definite metrics

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