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Journal of Geometry and Physics 62 (202) 657 674 Contents lists availale at SciVerse ScienceDirect Journal of Geometry and Physics journal homepage: www.elsevier.com/locate/jgp Projective versus metric structures Paweł Nurowski Instytut Fizyki Teoretycznej, Uniwersytet Warszawski, ul. Hoza 69, Warszawa, Poland a r t i c l e i n f o a s t r a c t Article history: Availale online 29 April 20 Keywords: Projective structures Metric structures Levi-Civita connection We present a numer of conditions which are necessary for an n-dimensional projective structure (M, [ ]) to include the Levi-Civita connection of some metric on M. We provide an algorithm, which effectively checks whether a Levi-Civita connection is in the projective class and, which finds this connection and the metric, when it is possile. The article also provides asic information on invariants of projective structures, including the treatment via the Cartan normal projective connection. In particular we show that there are a numer of Fefferman-like conformal structures, defined on a suundle of the Cartan undle of the projective structure, which encode the projectively invariant information aout (M, [ ]). 20 Elsevier B.V. All rights reserved.. Projective structures and their invariants.. Definition of a projective structure A projective structure on an n-dimensional manifold M is an equivalence class of torsionless connections [ ] with an equivalence relation identifying every two connections ˆ and for which ˆ X Y = X Y + A(X)Y + A(Y)X, X, Y TM, () with some -form A on M. Two connections from a projective class have the same unparameterised geodesics in M, and the converse is also true: two torsionless connections have the same unparameterised geodesics in M if they elong to the same projective class. The main purpose of this article is to answer the following question: When does a given projective class of connections [ ] on M include a Levi-Civita connection of some metric g on M? This prolem has a long history; see e.g. [ 3]. It was recently solved in dim M = 2 in a eautiful paper [4], which also, in its last section, indicates how to treat the prolem in dim M 3. In the present paper we follow [4] and treat the prolem in full generality 2 in dim M 3. For doing this we need the invariants of projective structures. The system of local invariants for projective structures was constructed y Cartan [6] (see also [7]). We riefly present it here for completeness (see e.g. [8 0] for more details). For our purposes it is convenient to descrie a connection in terms of the connection coefficients Γ a c associated with any frame (X a ) on M. This is possile via the formula: a X = Γ c a X c, a := Xa. E-mail address: nurowski@fuw.edu.pl. Research supported y Polish Ministry of Research and Higher Education, grants NN20 607540 and NN202 04838. 2 I have een recently informed y Dunajski that the prolem is also eing considered y him and Casey [5]. 0393-0440/$ see front matter 20 Elsevier B.V. All rights reserved. doi:0.06/j.geomphys.20.04.0

658 P. Nurowski / Journal of Geometry and Physics 62 (202) 657 674 Given a frame (X a ) these relations provide a one-to-one correspondence etween connections and the connection coefficients Γ a c. In particular, a connection is torsionless iff Γ c a Γ c a = θ c ([X a, X ]), where (θ a ) is a coframe dual to (X a ), θ (X a ) = δ a. Moreover, two connections ˆ and are in the same projective class iff there exists a coframe in which ˆΓ c a = Γ c a + δc a A + δ c A a, for some -form A = A a θ a. In the following, rather than using the connection coefficients, we will use a collective oject Γ a = Γ a c θ c, which we call connection -forms. In terms of them the projective equivalence reads: ˆΓ a = Γ a + δa A + A θ a. (2).2. Projective Weyl, Schouten and Cotton tensors Now, given a projective structure [ ] on M, we take connection -forms (Γ i j ) of a particular representative. Because of no torsion we have: dθ a + Γ a θ = 0. The curvature of this connection Ω a = dγ a + Γ a c Γ c, which defines the curvature tensor R a cd via: Ω a = 2 Ra cd θ c θ d, (3) (4) is now decomposed onto the irreducile components with respect to the action of GL(n, R) group: Ω a = W a + θ a ω + δ a θ c ω c. Here W a is endomorphism-valued 2-form: (5) W a = 2 W a cd θ c θ d, which is totally traceless: W a a = 0, W a ac = 0, and has all the symmetries of R a cd. Quantity ω a is a covector-valued -form. It defines a tensor P a via ω = θ a P a. The tensors W a cd and P a are called the (projective) Weyl tensor, and the (projective) Schouten tensor, respectively. They are related to the curvature tensor R a cd via: R a cd = W a cd + δa c P d δ a d P c 2δ a P [cd]. In particular, we have also the relation etween the Schouten tensor P a and the Ricci tensor R a = R c ac, which reads: P a = n R (a) n + R [a]. One also introduces the Cotton tensor Y ca, which is defined via the covector-valued 2-form (6) (7) Y a = 2 Y caθ θ c, (8) y Y a = dω a + ω Γ a. Note that Y ca is antisymmetric in {c}. (9)

P. Nurowski / Journal of Geometry and Physics 62 (202) 657 674 659 Now, comining Eqs. (3) (5), (8) and (9), we get the Cartan structure equations: dθ a + Γ a θ = 0 dγ a + Γ a c Γ c = W a + θ a ω + δ a θ c ω c dω a + ω Γ a = Y a. (0) It is convenient to introduce the covariant exterior differential D, which on tensor-valued k-forms acts as: DK a...a r... s = dk a...a r... s + i Γ a i a K a...a...a r... s i Γ i K a...a r... s. This, in particular satisfies the Ricci identity: D 2 K a...a r... s = i Ω a i a K a...a...a r... s i Ω i K a...a r... s. () This identity will e crucial in the rest of the paper. Using D we can write the first and the third Cartan structure equation in respective compact forms: Dθ a = 0, Dω a = Y a. (2) Noting that on tensor-valued 0-forms we have: DK a...a r... s = θ c c K a...a r... s, and comparing with definition (6) one sees that the second Eq. (2) is equivalent to: Y ca = 2 [ P c]a. (3).3. Bianchi identities We now apply D on oth sides of the Cartan structure Eqs. (0) and use the Ricci formula () to otain the Bianchi identities. Applying D on the first of (0) we get 0 = D 2 θ a = Ω a θ, i.e., tensorially: R a [cd] = 0. This, ecause the Weyl tensor has the same symmetries as R a cd, also means that W a [cd] = 0. (4) Next, applying D on the second of (0) we get: DW a = θ a Y + δ a θ c Y c. This, when written in terms of the tensors W a cd and Y ac, reads: a W d + ec cw d + ea W d = eca δd a Y ce + δ d c Y ae + δ d Y cae + δ d e (Y ac + Y ca + Y ca ). (5) This, when contracted in {ad}, and compared with (4), implies in particular that: d W d ac = (n 2)Y ca (6) and Y [ac] = 0. (7) Thus when n > 2 the Cotton tensor is determined y the divergence of the Weyl tensor. It is also worthwhile to note that ecause of (7) the identity (5) simplifies to: a W d ec + cw d ea + W d eca = δd a Y ce + δ d c Y ae + δ d Y cae. (8) Another immediate ut useful consequence of the identity (7) is [a P c] = 0. (9) This fact suggests an introduction of a 2-form β = 2 P [a]θ a θ.

660 P. Nurowski / Journal of Geometry and Physics 62 (202) 657 674 Since β is a scalar 2-form we have: dβ = Dβ = D 2 P [a]θ a θ = 2 (DP [a])θ a θ = 2 ( cp [a] )θ c θ a θ = 2 ( [cp a] )θ c θ a θ = 0. Thus, due to the Bianchi identity (9) and the first structure Eq. (2), the 2-form β is closed. Finally, applying D on the last Cartan equation (0) we get DY a + ω W a = 0. This relates the first derivatives of the Cotton tensor to a ilinear comination of the Weyl and the Schouten tensors: a Y cd + c Y ad + Y cad = P ae W e dc + P ew e dac + P cew e da. (20).4. Gauge transformations It is a matter of checking that if we take another connection ˆ from the projective class [ ], i.e., if we start with connection -forms ˆΓ i j related to Γ i j via ˆΓ a = Γ a + δa A + A θ a, then the asic ojects ω a, W a and Y a transform as: ˆω a = ω a DA a + AA a ˆβ = β da Ŵ a = W a Ŷ a = Y a + A W. a Equivalently: (2) ˆΓ a c = Γ a c + δa c A + δ a A c ˆP a = P a a A + A a A ˆP [a] = P [a] [a A ] (22) Ŵ a cd = W a cd Ŷ ac = Y ac + A d W d ca. This in particular means that the Weyl tensor is a projectively invariant oject. We also note that the 2-form β transforms modulo addition of a total differential. Corollary.. Locally in every projective class [ ] there exists a torsionless connection 0 for which the Schouten tensor is symmetric, P a = P (a). Proof. We know that due to the Bianchi identities (9) the 2-form β encoding the antisymmetric part of P a is closed, dβ = 0. Thus, using the Poincaré lemma, we know that there exists a -form Υ such that locally β = dυ. It is therefore sufficient to take A = Υ and ˆΓ a = Γ a + δa Υ + θ a Υ, to get ˆβ = 0, y the second relation in (2). This proves that in the connection ˆΓ a projectively equivalent to Γ a, we have ˆP[a] = 0. Remark.2. Note that if Γ a is a connection for which P a is symmetric then it is also symmetric in any projectively equivalent connection for which A = dφ, where φ is a function. Definition.3. A suclass of projectively equivalent connections for which the Schouten tensor is symmetric is called a special projective class. Mutatis mutandis we have: Corollary.4. Locally every projective class contains a special projective suclass. This suclass is given modulo transformations (2) with A eing a gradient, A = dφ.

P. Nurowski / Journal of Geometry and Physics 62 (202) 657 674 66 Corollary.5. The curvature Ω a of any connection from a special projective suclass of projective connections [ ] is traceless, Ω a = a 0. Proof. For the connections from a special projective suclass we have P a = P a. Hence θ a ω a = θ a P a θ = 0, and Ω a = W a + θ a ω. Thus Ω a a = W a a + θ a ω a = 0, ecause the Weyl form W a is traceless. Remark.6. We also remark that in dimension n = 2 the Weyl tensor of a projective structure is identically zero. In this dimension the Cotton tensor provides the lowest order projective invariant (see the last equation in (22)). In dimension n = 3 the Weyl tensor is generically nonzero, and may have as much as fifteen independent components. It is also generically nonzero in dimensions higher than three. Given an open set U with coordinates (x a ) surely the simplest projective structure [ ] is the one represented y the connection a = x a. This is called the flat projective structure on U. The following theorem is well known [6,7]: Theorem.7. In dimension n 3 a projective structure [ ] is locally projectively equivalent to the flat projective structure if and only if its projective Weyl tensor vanishes identically, W a cd 0. In dimension n = 2, a projective structure [ ] is locally projectively equivalent to the flat projective structure if and only if its projective Schouten tensor vanishes identically, Y ac 0..5. Cartan connection Ojects (θ a, Γ, ω c d) can e collected to the Cartan connection on an H principal fier undle H P M over (M, [ ]). Here H is a sugroup of the SL(n +, R) group defined y: A a 0 H = SL(n +, R) = a, A a GL(n, R), A a (R n ), a = det(a a ). Using (θ a, Γ, ω c d) we define an sl(n +, R)-valued -form Γ A = a n + Γ c c δa θ a ω + d. n + Γ c c This can e also written as A = ˆΓ a ˆΓ c c n + δa A ˆθ a ˆω n + ˆΓ c c, from which, knowing, one can deduce the transformation rules (θ a, Γ c, ω d) (ˆθ a, ˆΓ c, ˆω d); see e.g. [0]. Note that when the coframe θ a is fixed, i.e., when A a = δa, these transformations coincide with (2) and (2); the aove setup extends these transformations to the situation when we allow the frame to change under the action of the GL(n, R) group. The form A defines an sl(n +, R) Cartan connection on H P M. Its curvature R = da + A A, satisfies Ŵ R = W a a 0 0 =, Y 0 Ŷ 0 and consists of the 2-forms W a, Y as defined in (0). In particular we have Ŵ a = 2 Ŵ a cd ˆθ c ˆθ d, and Ŷ a = 2 Ŷac ˆθ ˆθ c, where Ŵ a cd and Ŷ ac are the transformed Weyl and Cotton tensors. Note that the (n + n 2 + n) -forms (ˆθ a, ˆΓ c, ˆω d) constitute a coframe on the (n 2 + 2n)-dimensional undle H P M; in particular these forms are linearly independent at each point of P. They satisfy the transformed Cartan structure equations dˆθ a + ˆΓ a ˆθ = 0 d ˆΓ a + ˆΓ a c ˆΓ c = Ŵ a + ˆθ a ˆω + δ a ˆθ c ˆω c d ˆω a + ˆω ˆΓ a = Ŷ a. (23)

662 P. Nurowski / Journal of Geometry and Physics 62 (202) 657 674.6. Fefferman metrics In Ref. [], with any point equivalence class of second order ODEs y = Q (x, y, y ), we associated a certain 4-dimensional manifold P/ equipped with a conformal class of metrics of split signature [g F ], whose conformal invariants encoded all the point invariants of the ODEs from the point equivalent class. By analogy with the theory of 3-dimensional CR structures we called the class [g F ] the Fefferman class. The manifold P from P/ was a principal fier undle H P N over a 3-dimensional manifold N, which was identified with the first jet space J of an ODE from the equivalence class. The undle P was eight dimensional, and H was a 5-dimensional paraolic sugroup of SL(3, R). For each point equivalence class of ODEs y = Q (x, y, y ), the Cartan normal conformal connection of the corresponding Fefferman metrics [g F ], was reduced to a certain sl(3, R) Cartan connection A on P. The two main components of the curvature of this connection were the two classical asic point invariants of the class y = Q (x, y, y ), namely: and w = D 2 Q y y 4DQ yy DQ y y Q y + 4Q y Q yy 3Q y y Q y + 6Q yy, w 2 = Q y y y y. If oth of these invariants were nonvanishing the Cartan undle that encoded the structure of a point equivalence class of ODEs y = Q (x, y, y ) was just H P N with the Cartan connection A. The nonvanishing of w w 2, was reflected in the fact that the corresponding Fefferman metrics were always of the Petrov type N N, and never selfdual nor antiselfdual. In the case of w w 2 0, the situation was more special [0]: the Cartan undle H P N was also defining a Cartan undle H P M, over a 2-dimensional manifold M, with the 6-dimensional paraolic sugroup H of SL(3, R) as the structure group. The manifold M was identified with the solution space of an ODE representing the point equivalent class. Furthermore the space M was naturally equipped with a projective structure [ ], whose invariants were in one-to-one correspondence with the point invariants of the ODE. This one-to-one correspondence was realised in terms of the sl(3, R) connection A. This, although initially defined as a canonical sl(3, R) connection on H P N, in the special case of w w 2 0 ecame the sl(3, R)-valued Cartan normal projective connection of the structure (M, [ ]) on the Cartan undle H P M. In such a case the corresponding Fefferman class [g F ] on P/ ecame selfdual or antiselfdual depending on which of the invariants w or w 2 vanished. What we have overlooked in the discussions in [0,] was that in the case of w 2 0, w 0 we could have defined two, a priori different Fefferman classes [g F ] and [g F ]. As we see elow the construction of these classes totally relies on the fact that we had a canonical projective structure [ ] on M. Actually we have the following theorem. Theorem.8. Every n-dimensional manifold M with a projective structure [ ] uniquely defines a numer n of conformal metrics [g a ], each of split signature (n, n), and each defined on its own natural 2n-dimensional suundle P a = P/( a ) of the Cartan projective undle H P M. Proof. Given (M, [ ]) we will construct the pair (P a, [g a ]) for each a =,..., n. We use the notation of Section.5. Let (X a, X c, X d ) e a frame of vector fields on P dual to the coframe (ˆθ a, ˆΓ c, ˆω d). This means that X a ˆθ = δ a, X a ˆΓ c d = δa d δc, X a ˆω = δ a, (24) at each point, with all other contractions eing zero. We now define a numer of n ilinear forms ĝ a on P defined y ĝ a = ˆΓ a 2 ˆΓ c c n + δa ˆθ + ˆθ ˆΓ a 2 ˆΓ c c n + δa, or ĝ a = 2 ˆΓ a 2 ˆΓ c c n + δa ˆθ, for short. In this second formula we have used the classical notation, such as for example in g = g a θ a θ, which areviates the symmetrised tensor product of two -forms λ and µ on P to λ µ + µ λ = 2λµ. We note that the formula for ĝ a, when written in terms of the Cartan connection A, reads 3 : ĝ a = 2A a µ Aµ n+, where the index µ is summed over µ =,..., n, n +. Indeed: 2A a µ Aµ = n+ 2 ˆΓ a ˆΓ c c n + δa ˆθ + 2ˆθ a ˆΓ c c n + = 2 ˆΓ a 2 ˆΓ c c n + δa ˆθ = ĝ a. 3 Compare with the defining formula for G in [].

P. Nurowski / Journal of Geometry and Physics 62 (202) 657 674 663 The ilinear forms ĝ a are degenerate on P. For each fixed value of the index a, a =,..., n, they have n 2 degenerate directions spanned y (X, Z c D ), where, c =,..., n and D =,..., n without D = a. The n(n ) vector fields Z c D are defined to e Z c D = X c D 2 n X d d δc D. Oviously (X, Z c D ) annihilate all θ s. Also oviously all X s annihilate all ˆΓ a all ˆΓ a ˆΓ c n+ c δa s we extend the definition of Z c D s to Z c f = X c f 2 n X d d δc f, where now f =,..., n. For these we get: Z c d ˆΓ a 2 ˆΓ h h n + δa = δ c δa. d ˆΓ c n+ c δa s. To see that all Z c D s annihilate Thus, if d a we see that each Z c d annihilates ˆΓ a 2 ˆΓ h n+ h δa. Hence n(n ) directions Z a D are degenerate directions for ĝ a. Another oservation is that the n 2 degenerate directions (X, Z c D ) form an integrale distriution. This is simplest to see y considering their annihilator. At each point this is spanned y the 2n -forms ˆθ, ˆΓ (a) 2 ˆΓ h n+ h δ(a), where the index (a) in rackets says that it is the fixed a which is not present in the range of indices D. Now using (23) it is straightforward to see that the forms (ˆθ, ˆτ (a) ) = ˆθ, ˆΓ (a) 2 ˆΓ h n+ h δ(a) dˆθ a ˆθ ˆθ n = 0, dˆτ (a) ˆτ (a) ˆτ (a) n ˆθ ˆθ n = 0. satisfy the Froenius condition Thus the n 2 -dimensional distriution spanned y (X, Z c D ) is integrale. Now, using (23) we calculate the Lie derivatives of ĝ a with respect to the directions (X, Z c D ). It is easy to see that: and L X ĝ a = 0 L Z c dĝ a = δ a dĝ c + 2 n δc dĝ a. The last equation means also that L Z c Dĝ a = 2 n δc Dĝ a. Thus, the ilinear form ĝ a transforms conformally when Lie transported along the integrale distriution spanned y (X, Z c D ). Now, for each fixed a =,..., n, we introduce an equivalence relation a on P, which identifies points on the same integral leaf of Span(X, Z c D ). On the 2n-dimensional leaf space P a = P/( a ) the n 2 degenerate directions for ĝ a are squeezed to points. Since the remainder of ĝ a is given up to a conformal rescalling on each leaf, the ilinear form ĝ a descends to a unique conformal class [g a ] of metrics, which on P a have split signature (n, n). Thus, for each a =,..., n we have constructed the 2n-dimensional split signature conformal structure (P a, [g a ]). It follows from the construction that P a may e identified with any 2n-dimensional sumanifold Pa of P, which is transversal to the leaves of Span(X, Z c D ). The conformal class [g a ] is represented on each P a y the restriction g a = ĝ a Pa. This completes the proof of the theorem. One can calculate the Cartan normal conformal connection for the conformal structures (P a, g a ). This is a lengthy, ut straightforward calculation. The result is given in the following theorem. Theorem.9. In the null frame (ˆτ (a), ˆθ c ) the Cartan normal conformal connection for the metric ĝ a is given y: ˆΓ d d 0 ˆω c 0 G = n + ˆθ d ˆR(a) dc ˆτ (a) ˆΓ e + ˆΓ d d n + δe ˆθ f 0 ˆΓ f c 0 ˆθ e ˆτ (a) c ˆω ˆΓ d d n + δf c 0 n + ˆΓ d d.

664 P. Nurowski / Journal of Geometry and Physics 62 (202) 657 674 Its curvature R = dg + G G is given y: 0 0 Ŷ c 0 R = 0 Ŵ e Ŝ c Ŷ 0 0 Ŵ f c 0, 0 0 0 0 where Ŝ c = ˆθ d ( ˆDˆR(a) dc = ˆθ d ( ˆDŴ (a) d c (a) ˆτ s Ŵ s ) dc ˆτ (a) s Ŵ s dc ) + δ(a) Ŷc δ (a) c Ŷ. 2. When a projective class includes a Levi-Civita connection? 2.. Projective structures of the Levi-Civita connection Let us now assume that an n-dimensional manifold M is equipped with a (pseudo-)riemannian metric ĝ. We denote its Levi-Civita connection y ˆ. The Levi-Civita connection ˆ defines its projective class [ ] with connections such that () holds. Now, with the Levi-Civita representative ˆ of [ ] we can define its curvature ˆΩ a, as in (4), and decompose it into the projective Weyl and Schouten tensors Ŵ a, cd ˆPa, as in (5): ˆΩ a = Ŵ a + θ a ˆω + δ a θ c ˆω c. However, since now M has an additional metric structure ĝ = ĝ a θ a θ, with the inverse ĝ a such that ĝ a ĝ c = δ c a, another decomposition of the curvature is possile. This is the decomposition onto the metric Weyl and Schouten tensors W a cd, P a, given y: ˆΩ a = W a + ĝ ac ĝ d ωc θ d + θ a ω. The tensor counterparts of formulae (25) (26) are respectively: ˆR a cd = Ŵ a cd + δa c ˆP d δ a d ˆP c 2δ a ˆP [cd] ˆR a cd = W a cd + δa c P d δ a d P c + ĝ d ĝ ae P ec ĝ c ĝ ae P ed. To find relations etween the projective and the metric Weyl and Schouten tensors one compares the right-hand sides of (27). For example, ecause of the equality on the left-hand sides of (27), the projective and the Levi-Civita Ricci tensors are equal: (25) (26) (27) ˆR d = ˆRa ad = R d. Thus, via (7), we get ˆP a = R a. n Further relations etween the projective and Levi-Civita ojects can e otained y recalling that: where R a = (n 2) P a + ĝ a P, P= ĝ a P a, (28) and that the Levi-Civita Ricci scalar is given y: R= ĝ a R a. After some algera we get the following proposition. Proposition 2.. The projective Schouten tensor ˆPa for the Levi-Civita connection ˆ is relatedto the metric Schouten tensor P a via: ˆP a = P a (n )(n 2) G a, where G a is the Einstein tensor for the Levi-Civita connection: G a = R a 2 ĝa R.

P. Nurowski / Journal of Geometry and Physics 62 (202) 657 674 665 The projective Weyl tensor Ŵ a cd for the Levi-Civita connection ˆ is related to the metric Weyl tensor W a cd via: Ŵ a = cd W a + cd n 2 (ĝ dĝ ae R ec ĝ c ĝ ae R ed) + (n )(n 2) (δa c R d δ a R d R c) + (n )(n 2) (δa dĝc δ a cĝd). (29) In particular we have the following corollary: Corollary 2.2. The projective Schouten tensor ˆPa of the Levi-Civita connection ˆ is symmetric ˆP a = ˆPa. Moreover, the projective Weyl tensor W a cd of any connection from the projective class [ ] of a Levi-Civita connection satisfies ĝ ae ĝ c W e cd = ĝ deĝ c W e ca. (30) Proof. The first part of the corollary is an immediate consequence of the fact that the metric Schouten tensor of the Levi- Civita connection as well as the Einstein tensor are symmetric. The second part follows from the relation (29), which yields: (n )ĝ ae ĝ c Ŵ e cd = n R ad+ R ĝ ad. Since R a is symmetric we get ĝ ae ĝ c Ŵ e = cd ĝ deĝ c Ŵ e ca. But according to the fourth transformation law in (22) the Weyl tensor is invariant under the projective transformations, Ŵ a = cd W a cd. Thus (30) holds, for all connections from the projective class of ˆ. This ends the proof. The aove corollary is oviously related to the question in the title of this Section. It gives the first, very simple, ostruction for a projective structure [ ] to include a Levi-Civita connection of some metric. We reformulate it to the following theorem. Theorem 2.3. A necessary condition for a projective structure (M, [ ]) to include a connection ˆ, which is the Levi-Civita connection of some metric ĝ a, is an existence of a symmetric nondegenerate ilinear form g a on M, such that the Weyl tensor W a cd of the projective structure satisfies g ae g c W e cd = g deg c W e ca, (3) with g a eing the inverse of g a, g ac g c = δ a. If the Levi-Civita connection ˆ from the projective class [ ] exists, then its corresponding metric ĝ a must e conformal to the inverse g a of some solution g a of Eq. (3), i.e., ĝ a = e 2φ g a, for a solution g a of (3) and some function φ on M. As an example we consider a projective structure [ ] on a 3-dimensional manifold M parameterised y three real coordinates (x, y, z). We choose a holonomic coframe (θ, θ 2, θ 3 ) = (dx, dy, dz), and generate a projective structure from the connection -forms 0 adz ady Γ a = dz 0 dx, with a = a(z), = (z), c = c(z), (32) cdy cdx 0 via (2). It is easy to calculate the projective Weyl forms W a, and the projective Schouten forms ω, for this connection. They read: 2 c dx dy 0 a dy dz W a = 0 2 c dx dy dx dz, 2 c dy dz 2 c dx dz 0 and ω a = cdx + 2 c dy, acdy + 2 c dx, adz.

666 P. Nurowski / Journal of Geometry and Physics 62 (202) 657 674 With this information in mind it is easy to check that g a = fa g 2 0 g 2 f 0, 0 0 g 33 (33) with some undetermined functions f = f (x, y, z), g 2 = g 2 (x, y, z), g 33 = g 33 (x, y, z), satisfies (3). Thus the connection Γ a may, in principle, e the Levi-Civita connection of some metric ĝ a. According to Theorem 2.3 we may expect that the inverse of this g a is proportional to ĝ a. 2.2. Comparing natural projective and (pseudo-)riemannian tensors Proposition 2. in an ovious way implies the following corollary: Corollary 2.4. The Levi-Civita connection ˆ of a metric ĝ a has its projective Schouten tensor equal to the Levi-Civita one, ˆP a = P a, if and only if its Einstein (hence the Ricci) tensor vanishes. If this happens ˆPa 0, and oth the projective and the Levi-Civita Weyl tensors are equal, Ŵ a cd = W a cd. Now we answer the question whether there are Ricci non-flat metrics having equal projective and Levi-Civita Weyl tensors. We use (29). The requirement that Ŵ a = cd W a cd yields the following proposition. Proposition 2.5. The Levi-Civita connection ˆ of a metric ĝ a has its projective Weyl tensor equal to the Levi-Civita one, Ŵ a = cd W a cd, if and only if its Levi-Civita Ricci tensor satisfies where M acd ef M acd ef R ef = 0, = ĝ ac δ e d δf ĝ adδ e c δf + ĝ adĝ c ĝ ef ĝ ac ĝ d ĝ ef + (n )(ĝ d δ e a δf c ĝ cδ e a δf d ). (34) One easily checks that the Einstein metrics, i.e., the metrics for which R a = Λĝ a, satisfy (34). Therefore we have the following corollary: Corollary 2.6. The projective and the Levi-Civita Weyl tensors of Einstein metrics are equal. In particular, all conformally flat Einstein metrics (metrics of constant curvature) are projectively equivalent. It is interesting to know if there are non-einstein metrics satisfying condition (34). 2.3. Formulation a la Roger Liouville In this susection we shall link our work with the approach of []. If is in the projective class of the Levi-Civita connection ˆ of a metric ĝ we have: 0 = ˆDĝa = Dĝ a 2Aĝ a A a θ c ĝ c A θ c ĝ ac, for some -form A = A a θ a. Thus the condition that a torsionless connection is projectively equivalent to the Levi-Civita connection of some metric, is equivalent to the existence of a pair (ĝ a, A a ) such that Dĝ a = 2Aĝ a + θ c (A a ĝ c + A ĝ ac ), with an invertile symmetric tensor ĝ a. Dually this last means that a torsionless connection is projectively equivalent to a Levi-Civita connection of some metric, iff there exists a pair (ĝ a, A a ) such that Dĝ a = 2Aĝ a A c (θ ĝ ca + θ a ĝ c ), with an invertile ĝ a. The unknown A can e easily eliminated from these equations y contracting with the inverse ĝ a : a ĝadĝ A = 2(n + ), (35)

P. Nurowski / Journal of Geometry and Physics 62 (202) 657 674 667 so that the if an only if condition for to e in a projective class of a Levi-Civita connection ˆ is the existence of ĝ a such that 2(n + )Dĝ a = 2(ĝ cd Dĝ cd )ĝ a + (ĝ ef c ĝ ef )(θ ĝ ca + θ a ĝ c ), ĝ ac ĝ c = δ c. This is an unpleasant-to-analyse, nonlinear system of PDEs, for the unknown ĝ a. It follows that it is more convenient to discuss the equivalent system (35) for the unknowns (ĝ a, A a ), which we will do in the following. The aim of this susection is to prove the following theorem: Theorem 2.7. A torsionless connection on an n-dimensional manifold M is projectively equivalent to a Levi-Civita connection ˆ of a metric ĝ a if and only if its projective class [ ] contains a special projective suclass [ ] whose connections satisfy the following: for every [ ] there exists a nondegenerate symmetric tensor g a and a vector field µ a on M such that c g a = µ a δ c + µ δ a c, or, which is the same, there exists a nondegenerate g a and µ a such that: Dg a = µ a θ + µ θ a. (36) Proof. If ˆ is the Levi-Civita connection of a metric ĝ = ĝ a θ a θ, we consider connections associated with ˆ via (), in which A = dφ, with aritrary functions (potentials) on M. This is a special class of connections, since the projective Schouten tensor ˆPa for ˆ is symmetric (see Corollary 2.2), and the transformation (22) with gradient As, preserves the symmetry of the projective Schouten tensor (see Remark.2). Any connection from this special class satisfies (35) with A = dφ, and therefore is characterised y the potential φ, = (φ). We now take the inverse ĝ a of the metric ĝ a, ĝ ac ĝ c = δ a, and rescale it to g a = e 2f ĝ a, where f is a function on M. Using (35) with A = dφ, after a short algera, we get: Thus taking Dg a = 2(dφ df )g a ( c φ)(θ g ca + θ a g c ). f = φ + const, for each = (φ) from the special class [ ], we associate g a = e 2f ĝ a satisfying Dg a = ( c φ)(θ g ca + θ a g c ). Defining µ a = A c g ca = e 2f ( c φ)ĝ ca we get (36). Oviously g a is symmetric and nondegenerate since ĝ a was. The proof in the opposite direction is as follows: We start with (, g a, µ a ) satisfying (36). In particular, connection is special, i.e., it has symmetric projective Schouten tensor and, y Corollary.5, its curvature satisfies Ω a a = 0. Since g a is invertile, we have a symmetric g a such that g ac g c = δ a. We define A = g a µ θ a. Contracting with (36) we get: (37) g a Dg a = 2A, or A = 2 g adg a. Now this last equation implies that: da = 2 Dg a Dg a 2 g ad 2 g a. This compared with the Ricci identity D 2 g a = Ω a c g c + Ω c g ac, the defining Eq. (35), and its dual yields Dg a = g ac g d (µ c θ d + µ d θ c ), da = Ω a a = 0. Thus the -form A defined y (37) is locally a gradient of a function φ 0 on M, A = dφ 0. The potential φ 0 is defined y (, g a, µ a ) up to φ 0 φ = φ 0 + const, A = dφ.

668 P. Nurowski / Journal of Geometry and Physics 62 (202) 657 674 We use it to rescale the inverse g a of g a. We define ĝ a = e 2φ g a. This is a nondegenerate symmetric tensor on M. Using our definitions we finally get Dĝ a = 2dφĝ a e 2φ g ac g d (µ c θ d + µ d θ c ) = 2Aĝ a + A a ĝ c θ c + A ĝ ac θ c. This means that the new torsionless connection ˆ defined y (), with A as aove, satisfies ˆDĝ a = Dĝ a 2Aĝ a A a ĝ c θ c A ĝ ac θ c = 0, and thus is the Levi-Civita connection for a metric ĝ = ĝ a θ a θ. Since A = dφ this shows that in the special projective class defined y there is a Levi-Civita connection ˆ. This completes the proof. We also have the following corollary, which can e traced ack to Roger Liouville [], (see also [4,2,2,3]): Corollary 2.8. A projective structure [ ˆ ] on an n-dimensional manifold M contains a Levi-Civita connection of some metric if and only if at least one special connection in [ ˆ ] admits a solution to the equation c g a n + δa c dg d n + δ c dg ad = 0. (38) with a symmetric and nondegenerate tensor g a. Proof. We use Theorem 2.7. If (, g a, µ a ) satisfies (36) it is a simple calculation to show that (38) holds. The other way around: if (38) holds for a special connection and an invertile g a, then defining µ a y µ a = n+ dg ad we get c g a = µ a δ + c µ δ a c, i.e., Eq. (36), after contracting with θ c. Now, if we take any other special connection ˆ, then it is related to via ˆ X (Y) = X (Y) + X(φ)Y + Y(φ)X. Rescalling g a to ĝ a = e 2φ g a one checks that ˆ c ĝ a n+ δa ˆ c d ĝ d n+ δ ˆ c d g ad = 0. Thus in any special connection ˆ we find an invertile ĝ a = e 2φ g a with ˆµ a = ˆ n+ d ĝ ad satisfying ˆ c ĝ a = ˆµ a δ + c ˆµ δ a c. Remark 2.9. It is worthwhile to note that ˆµ a and µ as in the aove proof are related y ˆµ a = e 2φ (µ a + g da d φ). 2.4. Prolongation and ostructions In this section, given a projective structure [ ], we restrict it to a corresponding special projective suclass. All the calculations elow are performed assuming that a is in this special projective suclass. We will find consequences of the necessary and sufficient conditions (36) for this special class to include a Levi-Civita connection. Applying D on oth sides of (36), and using the Ricci identity () we get as a consequence: Ω a g ac + Ω c a g a = Dµ c θ + Dµ θ c. (39) This expands to the following tensorial equation: δ d aµ c δ a dµ c + δ c d aµ δ c a dµ = R ead g ec + R c ead g e. (40) Now contracting this equation in {ac} we get: a µ = δ a ρ P acg c n W cda g cd (4) with some function ρ on M. This is the prolonged equation (36). It can e also written as: Dµ = ρθ ω c g c n W cda g cd θ a. (42) Applying D on oth sides of this equation, after some manipulations, one gets the equation for the function ρ: a ρ = 2P a µ + 2 n Y acg c. (43) This is the last prolonged equation implied y (36). It can e also written as: Dρ = 2ω µ + 2 n Y acg c θ a. (44)

P. Nurowski / Journal of Geometry and Physics 62 (202) 657 674 669 Thus we have the following theorem [2]: Theorem 2.0. Eq. (38) admits a solution for g a if and only if the following system Dg c = µ c θ + µ θ c Dµ = ρθ ω c g c n W cda g cd θ a (45) Dρ = 2ω µ + 2 n Y acg c θ a, has a solution for (g a, µ c, ρ). Simple ostructions for having solutions to (45) are otained y inserting Dµ from (42) into the integraility conditions (39), or what is the same, into (40). This insertion, after some algera, yields the following proposition. Proposition 2.. Eq. (42) is compatile with the integraility conditions (39) (40) only if g a satisfies the following algeraic equation: where T [ed] c af g af = 0, (46) T [ed] c af = 2 δc (a W f )ed + 2 δ (a W c f )ed + n W c (af )[e δ d] + n W (af )[e δc d]. (47) Remark 2.2. Note that although the integraility condition (46) was derived in the special gauge when the connection was special, it is gauge independent. This is ecause the condition involves the projectively invariant Weyl tensor, and ecause it is homogeneous in g a. For each pair of distinct indices [ed] the tensor T [ed] c af provides a map S 2 M κ a T [ed] κ a = T [ed] a cd κ cd S 2 M, (48) which is an endomorphism T [ed] of the space S 2 M of symmetric 2-tensors on M. It is therefore clear that Eq. (46) has a nonzero solution for g a only if each of these endomorphisms is singular. Therefore we have the following theorem (see also the last section in [4]): Theorem 2.3. A necessary condition for a projective structure [ ] to include a Levi-Civita connection of some metric g is that all the endomorphisms T [ed] : S 2 M S 2 M, uilt from its Weyl tensor, as in (47), have nonvanishing determinants. In dimension n 3 this gives in general n(n ) ostructions to metrisaility. 2 Remark 2.4 (Puzzle). Note that here we have I = n(n ) ostructions, whereas the naive count, as adapted from [4], yields 2 I = 4 (n4 7n 2 6n + 4). For n = 3, we see that we constructed I = 3 invariants, wheres I says that there is only one. Why? Remark 2.5. Note that Remark 2.2 enaled us to use any connection from the projective class, not only the special ones, in this theorem. Further integraility conditions for (36) may e otained y applying D on oth sides of (42) and (44). Applying it on (42), using again the Ricci identity (), after some algera, we get the following proposition. Proposition 2.6. The integraility condition D 2 µ = Ω a µa, for (g a, µ c, ρ) satisfying (45), is equivalent to: S [ae] cd g cd n + 4 = 2 W + cae W [ae]c µ c, (49) where the tensor S [ae] cd is given y: S [ae] cd = n 2 2 Y ea(cδ d) + (cw d)ea + W (cd)[e;a]. Here, in the last term, for simplicity of the notation, we have used the semicolon to denote the covariant derivative, e f = f ;e. Remark 2.7. Note that in dimension n = 2, where W a cd 0, the integraility conditions (46) and (49) are automatically satisfied.

670 P. Nurowski / Journal of Geometry and Physics 62 (202) 657 674 The last integraility condition D 2 ρ = 0 yields: Proposition 2.8. The integraility condition D 2 ρ = 0, for (g a, µ c, ρ), satisfying (45) is equivalent to: U [a]cd g cd = n + 3 2 Y acµ c, where the tensor U [a](cd) reads: U [a]cd = [a Y ](cd) + W e (cd)[a P ]e. (50) Remark 2.9. For the sufficiency of conditions (46), (49) and (50) see Remark 4.. 3. Metrisaility of a projective structure check list Here, ased on Theorems 2.3, 2.7, 2.0 and 2.3 and Propositions 2., 2.6 and 2.8, we outline a procedure how to check if a given projective structure contains a Levi-Civita connection of some metric. The procedure is valid for the dimension n 3. Given a projective structure (M, [ ]) on an n-dimensional manifold M: () calculate its Weyl tensor W a cd and the corresponding operators T [ed] as in (48). If at least one of the determinants τ ed = det(t [ed] ), e < d =, 2,..., n, is not zero the projective structure (M, [ ]) does not include any Levi-Civita connection. (2) If all the determinants τ ed vanish, find a special connection 0 in [ ], and restrict to a special projective suclass [ 0 ] [ ]. (3) Now taking any connection from [ 0 ] calculate the Weyl, (symmetric) Schouten, and Cotton tensors, and the tensors T [ed] c af, S [ae] cd, U [a]cd of Propositions 2., 2.6 and 2.8. (4) Solve the linear algeraic equations (46), (49) and (50) for the unknown symmetric tensor g a and vector field µ a. (5) If these equations have no solutions, or the n n symmetric matrix g a has vanishing determinant, then (M, [ ]) does not include any Levi-Civita connection. (6) If Eqs. (46), (49) and (50) admit solutions with nondegenerate g a, find the inverse g a of the general solution for g a, and check whether Eq. (30) is satisfied. If this equation cannot e satisfied y restricting the free functions in the general solution g a of Eqs. (46), (49) and (50), then (M, [ ]) does not include any Levi-Civita connection. (7) If (30) may e satisfied, restrict the general solution g a of (46), (49) and (50) to only g a s satisfying (30), and insert (g a, µ a ), with such g a s and the most general µ a solving (46), (49) and (50), in Eqs. (45). (8) Find the general solution to Eqs. (45) for (g a, µ a, ρ), with (g a, µ a ) from the ansatz descried in point (7). (9) If the solution for such (g a, µ a, ρ) does not exist, or the symmetric tensor g a is degenerate, then (M, [ ]) does not include any Levi-Civita connection. (0) Otherwise find the inverse g a of g a from the solution (g a, µ a, ρ), and solve for a function φ on M such that dφ = g a µ a θ. () The metric ĝ = e 2φ g a θ a θ has the Levi-Civita connection ˆ which is in the special projective class [ 0 ] [ ]. 4. 3-dimensional examples Example. Here, as the first example, we consider a 3-dimensional projective structure (M, [ ]) with the projective class represented y the connection -forms: 2 adx 4 dy 4 dx 0 Γ a = 4 ady 4 adx + 2 dy 0. (5) cdy 4 adz cdx 4 dz 4 adx 4 dy The 3-manifold M is parameterised y (x, y, z), and a = a(z), = (z), c = c(z) are sufficiently smooth real functions of z. In addition we assume that a 0, 0, c const.

P. Nurowski / Journal of Geometry and Physics 62 (202) 657 674 67 It can e checked that this connection is special. More specifically we have: 2 c dxy 3 8 a dxz + 3 4 dyz 8 dxz 8 dxy W a = 3 8 a dyz 2 c dxy + 4 a dxz 3 8 dyz 8 a dxy, acdxy 2 c dyz cdxy 2 c dxz 8 a dxz + 8 dyz where (dxy, dxz, dyz) is an areviation for (dx dy, dx dz, dy dz), and ω a = 3 6 a2 dx + 6 (8c + a)dy 8 a dz, 6 (8c + a)dx 3 6 2 dy 8 dz, 8 a dx 8 dy. Having these relations we easily calculate the ostructions τ [ed]. These are: and τ 3 = 9 892 (a ) 6, τ 23 = 9 892 ( ) 6, τ 2 = 3 28 c2 (c ) 2 (a a ) 2. This shows that (M, [ ]) may e metrisale only if a = const, = const. For such a and all the ostructions τ [ed] vanish. Assuming this we pass to point (4) of our procedure from Section 3. It follows that with our assumptions, the general solution of Eq. (46) is: g = g 22 = 0, g 3 = c c g 2, g 23 = ac c g 2. (52) Inserting this in (49) shows that its general solution is given y the aove relations for g a and µ = 4cc g 2, µ 2 = 4cc 2 (c ) 2 2 (c ag 2. (53) ) 2 The general solution (52) and (53) of (46) and (49) is compatile with the last integraility condition (50) if and only if the function c = c(z) defining our projective structure (M, [ ]) satisfies a third order ODE: c (3) c c + (c ) 2 2cc c = 0. (54) If this condition for c = c(z) is satisfied then (52) and (53) is the general solution of (46), (49) and (50). Moreover, it follows that the solution (52) and (53) also satisfies (30), and the tensor g a is nondegenerate for this solution provided that g 2 0. This means that (i) the projective structure (M, [ ]) with a 0, 0, c const may include a Levi-Civita connection only if (54) holds, and (ii) if it holds, the integraility conditions (46), (49) and (50) are all satisfied with the general solution (52) and (53), with g 2 0. We now pass to point (8) of the procedure from Section 3: assuming that (54) holds, we want to solve (45) for (g a, µ a ) satisfying (52) and (53). It follows that the {} component of the first of Eq. (45) gives a further restriction on the function c. Namely, if (g a, µ a ) are as in (52) and (53), then Dg = 2µ θ iff c c (c ) 2 = 0, i.e., iff c = c e c 2z, where c, c 2 are constants s.t. c c 2 0. Luckily this c satisfies (54). Looking at the next component, {2}, of the first Eq. (45), we additionally get dg 2 = 2 (adx + dy)g 2. And now, this is compatile with the {3} component of the first Eq. (45), if and only if = 0 or g 2 = 0. We have to exclude g 2 = 0, since in such a case g a is degenerate. On the other hand = 0 contradicts our assumptions aout the function. Thus, according to the procedure from Section 3, we conclude that (M, [ ]) with the connection represented y (5) with a 0 and c const never includes a Levi-Civita connection. Remark 4.. Note that this example shows that even if all the integraility conditions (30), (46), (49) and (50) are satisfied Eqs. (45) may have no solutions with nondegenerate g a. Thus conditions (46), (49) and (50) and (30) are not sufficient for the existence of a Levi-Civita connection in the projective class.

672 P. Nurowski / Journal of Geometry and Physics 62 (202) 657 674 Example 2. As a next example we consider the same 3-dimensional manifold M as aove, and equip it with a projective structure [ ] corresponding to Γ a as in (5), ut now assuming that the functions a = a(z) and = (z) satisfy a 0 and 0. For further convenience we change the variale c = c(z) to the new function h = h(z) 0 such that c(z) = h (z). When running through the procedure of Section 3, which enales us to say if such a structure includes a Levi-Civita connection, everything goes in the same way as in the previous example, up to Eqs. (53). Thus applying our procedure of Section 3 we get that the general solution to (46) and (49) is given y g = g 22 = g 3 = g 23 = µ = µ 2 = 0. It follows that this general solution to (46) and (49), automatically satisfies (50) and (30). Now, with g = g 22 = g 3 = g 23 = µ = µ 2 = 0, the first of Eq. (45) gives: g 2 = const, dg 33 = 2h g 2 dz, µ 3 = h g 2, and the second, in addition, gives: ρ = 2 3 h g 2. This makes the last of Eqs. (45) automatically satisfied. The only differential equation to e solved is dg 33 = 2h g 2 dz, which after a simple integration yields: g 33 = 2g 2 h. Thus we have 0 0 g a = g 2 0 0, 0 0 2h with the inverse 0 0 g a = 0 0 g 2, g 2 = const 0, h = h(z) 0. 0 0 2h Now, realising point (0) of the procedure of Section 3, we define A = g a µ a θ = h 2h dz = d log(h). (55) 2 This means that the potential φ = log(h), and that the metric ĝ 2 a whose Levi-Civita connection is in the projective class of 0 0 0 Γ a = 0 0 0, (56) h dy h dx 0 is given y 0 0 ĝ a = h g 2 0 0 h, g 2 = const 0, h = h(z) 0, 0 0 2h 2 or what is the same y: ĝ = 2hdxdy + dz 2 ), g 2 = const 0, h = h(z) 0. g 2 h 2 It is easy to check that in the coframe (θ, θ 2, θ 3 ) = (dx, dy, dz), the Levi-Civita connection -forms for the metric ĝ as aove is given y h 2h dz 0 h 2h dx ˆΓ a = 0 h 2h dz h 2h dy, h dy h dx h 2h dz which satisfies (2) with Γ a given y (56) and A given y (55).

P. Nurowski / Journal of Geometry and Physics 62 (202) 657 674 673 Remark 4.2. Thus we have shown that the projective structure [ ] generated y the connection -forms (56) is metrisale, and that modulo rescalling, ĝ const ĝ, there is a unique metric, whose Levi-Civita connection is in the projective structure [ ]. Note that the metric ĝ has Lorentzian signature. Example 3. Now we continue with the example of a projective structure defined in Section 2. y formula (32). Calculating the projective Cotton tensor for this structure we find that it is projectively flat if and only if c = 0 and 2c + 3c = 0 and 2ca + 3ac = 0. This happens when a = = c = 0, ut also e.g. when c = z, = s z 3 2 and a = s2 z 3 2, with s, s 2 eing constants. If the structure is not projectively flat the most general nondegenerate solution to Eq. (46) is g 33 g a c a g 2 0 = g 2 g 33 c 0. (57) 0 0 g 33 It follows that if c = 0, projectively non-flat structures which are metrisale do not exist. In formula (57) we recognise (33) with f = g33 c. Looking for projectively non-flat structures, we now pass to Eq. (49). With g a as in (57) this, in particular, yields µ = µ 2 = 0 and a a = 0. Thus only the structures satisfying this last equation can e metrisale. In the following we assume that oth a and are not constant. Then = s a, with s a a constant. This solution satisfies all the other Eqs. (49) if and only if µ 3 = 2g 2 (2cc a + ac 2 ) + g 33 (a c c a ) 6a c. Now, with all these choices Eqs. (50) are also satisfied. Thus we may pass to the differential equation (36) for the remaining undetermined g a. It follows that these equations can e satisfied if and only if c = s 2 a with s 2 = const. Now, the remaining Eqs. (36) are satisfied provided that the unknown functions g 2 and g 33 satisfy: g 2 z = 2 s s 2 a g 33 and g 33 z = 2s 2 a g 2 (58) and are independent of the variales x and y. If g 2 and g 33 solve (58) then all the other Eqs. (45) are satisfied if and only if ρ = s a 2 g 33 + 2 3 s 2 a g 2. System (58) can e solved explicitly (the solution is not particularly interesting), showing that in this case also our procedure defined in Section 3 leads effectively to the solution of metrisaility prolem. Example 4. Our last example goes eyond three dimensions. It deals with the so-called (anti-)de Sitter spaces. Let X a e a constant vector, and η a e a nondegenerate symmetric n n constant matrix. We focus on an example when η a = diag(,...,,,..., ), with p + s, and q s. In U = {(x a ) R n η cd X c x d 0} we consider metrics ĝ of the form ĝ = η adx a dx (η cd X c x d ) 2. We analyse these metrics in an orthonormal coframe θ a = dxa η c X x c, (59) (60)

674 P. Nurowski / Journal of Geometry and Physics 62 (202) 657 674 in which ĝ = η a θ a θ. In the following we will use a convenient notation such that: η fg X f X g = η(x, X). We call the vector X timelike iff η(x, X) > 0, spacelike iff η(x, X) < 0, and null iff η(x, X) = 0. It is an easy exercise to find that in the coframe (60) the Levi-Civita connection -forms ˆΓ a associated with metrics (59) are: ˆΓ a = η d (X a θ d X d θ a ). Thus the Levi-Civita connection curvature, ˆΩ a = d ˆΓ a + ˆΓ a c ˆΓ c, is given y ˆΩ a = η(x, X)θ a θ d η d. This, in particular, means that the Levi-Civita curvature tensor, ˆRa R a, look, respectively, as: and ˆR a cd = η(x, X) (η cδ a d η dδ a c ), W a cd = 0, R d = ( n)η(x, X)η d. This proves the following proposition: Proposition 4.3. The metrics ĝ = η adx a dx (η cd X c x d ) 2 cd, the Levi-Civita Weyl tensor, W a cd, and the Ricci tensor are the metrics of constant curvature. Their curvature is totally determined y their constant Ricci scalar R= n( n)η(x, X). It is positive, vanishing or negative depending on the causal properties of the vector X. Hence if X is spacelike (U, ĝ) is locally the de Sitter space, if X is timelike (U, ĝ) is locally the anti-de Sitter space, and if X is null (U, ĝ) is flat. Using this proposition and Corollary 2.6 we see that metrics (59) are all projectively equivalent. This fact may have some relevance in cosmology, as discussed e.g. in [3 6]. We discuss this point in more detail in a separate paper [7]. References [] R. Liouville, Sur une classe d equations differentiells, parmi lequelles, in particulier, toutes celles des lignes geodesiques se trouvent comprises, C. R. Hed. Seances Acad. Sci. 05 (887) 062 064. [2] J. Mikes, Geodesic mappings of affine-connected and Riemannian spaces, J. Math. Sci. 78 (996) 3 333. [3] N.S. Sinjukov, Geodesic Mappings of Riemannian Spaces, Nauka, Moscow, 979, (in Russian). [4] R.L. Bryant, M. Dunajski, M. Eastwood, Metrisaility of two-dimensional projective structures, 2008. Availale from: arxiv:080.0300. [5] S. Casey, M. Dunajski, Metrisaility of path geometries, 200 (in preparation). [6] E. Cartan, Sur les varietes a connection projective, Bull. Soc. Math. France 52 (924) 205 24; E. Cartan, Oeuvres III (955) 825 862. [7] T.Y. Thomas, Announcement of a projective theory of affinely connected manifolds, Proc. Natl. Acad. Sci. (925) 588 589. [8] M.G. Eastwood, Notes on projective differential geometry, in: Symmetries and Overdetermined Systems of Partial Differential Equations, in: IMA Volumes in Mathematics and its Applications, vol. 44, Springer Verlag, 2007, pp. 4 60. [9] S. Koayashi, Transformation Groups in Differential Geometry, Springer, Berlin, 970. [0] E.T. Newman, P. Nurowski, Projective connections associated with second-order ODEs, Class. Quantum Gravity 20 (2003) 2325 2335. [] P. Nurowski, G.A.J. Sparling, Three-dimensional Cauchy Riemann structures and second-order ordinary differential equations, Class. Quantum Gravity 20 (2003) 4995 506. [2] M.G. Eastwood, V. Matveev, Metric connections in projective differential geometry, in: Symmetries and Overdetermined Systems of Partial Differential Equations, in: IMA Volumes in Mathematics and its Applications, vol. 44, Springer Verlag, 2007, pp. 339 350. [3] G.S. Hall, D.P. Lonie, The principle of equivalence and projective structure in spacetimes, Class. Quantum Gravity 24 (2007) 367 3636. [4] G.S. Hall, D.P. Lonie, The principle of equivalence and cosmological metrics, J. Math. Phys. 49 (2008) 022502. [5] G.S. Hall, D.P. Lonie, Projective equivalence of Einstein spaces in general relativity, Class. Quantum Gravity 26 (2009) 25009. 0pp. [6] G.S. Hall, D.P. Lonie, Holonomy and projective equivalence in 4-dimensional Lorentz manifolds, SIGMA Symmetry Integraility Geom. Methods Appl. 5 (2009) 066. 23 pages. [7] P. Nurowski, Is dark energy meaningless? Rend. del Semin. Mat. Univ. a Politech. di Torino 68 (200) 36 367.