RIEMANN COMPATIBLE TENSORS. 1. Introduction

Transcription

1 RIEMANN COMPATIBLE TENSORS CARLO ALBERTO MANTICA AND LUCA GUIDO MOLINARI Abstract. Derdzinski and Shen s theore on the restrictions posed by a Codazzi tensor on the Rieann tensor holds ore generally when a Rieann-copatible tensor exists. Several properties are shown to reain valid in this broader setting. Rieann copatibility is equivalent to the Bianchi identity of the new Codazzi deviation tensor, with a geoetric significance. The general properties are studied, with their iplications on Pontryagin fors. Exaples are given of anifolds with Rieann-copatible tensors, in particular those generated by geodesic appings. Copatibility is extended to generalized curvature tensors, with an application to Weyl s tensor and general relativity. 1. Introduction The Rieann tensor R ijk and its contractions, R kl = R kl and R = g kl R kl, are the fundaental tensors to describe the local structure of a Rieannian anifold (M n, g) of diension n. In a rearkable theore [9, 3] Derdzinski and Shen showed that the existence of a nontrivial Codazzi tensor poses strong constraints on the structure of the Rieann tensor. Because of their geoetric relevance, Codazzi tensors have been studied by several authors, such as Berger and Ebin [1], Bourguignon [4], Derdzinski [7, 8], Derdzinski and Shen [9], Ferus [10], Sion [28]; a copendiu of results is found in Besse s book [3]. Recently, we showed [21] that the Codazzi differential condition (1.1) i b jk j b ik = 0, sufficient for the theore to hold, can be replaced by the ore general notion of Rieann-copatibility, which is instead algebraic: Definition 1.1. A syetric tensor b ij is Rieann copatible (R-copatible) if: (1.2) b i R jkl + b j R kil + b k R ijl = 0. With this requireent, we proved the following extension of Derdzinski and Shen s theore: Date: 10 sept revised Matheatics Subject Classification. Priary 53B20, Secondary 53B21. Key words and phrases. Codazzi tensor, Rieann tensor, Rieann copatibility, generalized curvature tensor, geodesic apping, Pontryagin fors. 1

2 2 C. A. MANTICA AND L. G. MOLINARI Theore 1.2. [21] Suppose that a syetric R-copatible tensor b ij exists. Then, if X, Y and Z are three eigenvectors of the atrix b r s at a point of the anifold, with eigenvalues λ, µ and ν, one has R ijkl X i Y j Z k = 0 provided that both λ and µ are different fro ν. The concept of copatibility allows for a further extension of the theore, where the Rieann tensor R is replaced by a generalized curvature tensor K, and b is required to be K-copatible [21]. This paper studies the properties of Rieann copatibility, and its iplications on the geoetry of the anifold. In section 2 R-copatibility is shown to be equivalent to the Bianchi identity of a new tensor, the Codazzi deviation. In section 3 the irreducible coponents of the covariant derivative of a syetric tensor are classified in a siple anner, based on the decoposition into traceless ters. This is helpful in the study of various structures suited for R-copatibility. The general properties of Rieann copatibility are presented in section 4. In section 5 several properties of anifolds in the presence of a Rieann copatible tensor that were obtained by Derdzinski and Shen and Bourguignon for anifolds with a Codazzi tensor, are recovered. In particular, it is shown that R-copatibility iplies pureness, a property of the Rieann tensor introduced by Maillot that iplies the vanishing of Pontryagin fors. Manifolds that carry R-copatible tensors are presented in section 6; interesting exaples are generated by geodesic appings, that induce etric tensors that are R- copatible. Finally, in section 7, K-tensors and K-copatibility are presented, with applications to the standard curvature tensors. In the end, an application to general relativity is entioned, that will be discussed fully elsewhere. 2. The Codazzi deviation tensor and R-copatibility Since Codazzi tensors are Rieann copatible, for a non-codazzi differentiable syetric tensor field b it is useful to define its deviation fro the Codazzi condition. This tensor satisfies an unexpected relation that generalizes Lovelock s identity for the Rieann tensor, and shows that Rieann copatibility is a condition for closedness of certain 2-fors. Definition 2.1. The Codazzi deviation of a syetric tensor b kl is (2.1) C jkl =: j b kl k b jl

3 RIEMANN COMPATIBLE TENSORS 3 Siple properties are: C jkl = C kjl and C jkl + C klj + C ljk = 0. The following identity holds in general, and relates the Bianchi differential cobination for C to the Rieann copatibility of b: Proposition 2.2. (2.2) i C jkl + j C kil + k C ijl = b i R jkl + b j R kil + b k R ijl Proof. i C jkl + j C kil + k C ijl = [ i, j ]b kl + [ k, i ]b jl + [ j, k ]b il = b l (R ijk + R kij + R jki ) + b i R jkl + b j R kil + b k R ijl and the first ter vanishes by the first Bianchi identity. Reark 2.3. The identity (2.2) holds true if b ij is replaced by b ij = b ij + χa ij, where a ij is a Codazzi tensor and χ a scalar field. Then: C jkl = C jkl (a kl j a jl k )χ. The deviation tensor is associated to the 2-for C l = 1 2 C jkldx j dx k. The closedness condition 0 = DC l = 1 2 ic jkl dx i dx j dx k (D is the exterior covariant derivative) is the second Bianchi identity for the Codazzi deviation: i C jkl + j C kil + k C ijl = 0. This gives a geoetric picture of Rieann copatibility: Theore 2.4. b ij is Rieann copatible if and only if DC l = 0. Reark 2.5. The Codazzi deviation of the Ricci tensor is, by the contracted second Bianchi identity: C jkl =: j R kl k R jl = R jkl. For the Ricci tensor the identity (2.2) identifies with Lovelock s identity [17] for the Rieann tensor: (2.3) i R jkl + j R kil + k R ijl = R i R jkl R j R kil R k R ijl. A Veblen-like identity holds, that corresponds to (4.2) (For b ij = R ij it specializes to Veblen s identity for the divergence of the Rieann tensor [19]): Proposition 2.6. (2.4) i C jlk + j C kil + k C lji + l C ikj = b i R jlk + b j R kil + b k R lji + b l R ikj

4 4 C. A. MANTICA AND L. G. MOLINARI Proof. Write four versions of equation (2.2) with cyclically peruted indices i, j, k, l and su up. Then siplify by eans of the first Bianchi identity for the Rieann tensor and the cyclic identity C jkl + C klj + C ljk = Irreducible coponents for j b kl and R-copatibility We begin with a siple procedure to classify the O(n) invariant coponents of the tensor j b kl. They will guide us in the study of R-copatibility. If b is the Ricci tensor, this siple construction reproduces the seven equations linear in i R jk, invariant for the O(n) group, that are enuerated and discussed in Besse s treatise Einstein Manifolds [3]. For a syetric tensor b kl with j b kl 0, the tensor j b kl can be decoposed into O(n) invariant ters, where B 0 jkl is traceless (B0 jk j = B 0 kj j = 0) [13, 16]: (3.1) j b kl = B 0 jkl + A jg kl + B k g jl + B l g jk (3.2) A j = (n + 1) jb 2 b j, B n 2 j = jb n b j + n 2 n 2 + n 2 The traceless tensor can then be written as a su of orthogonal coponents [17]: (3.3) B 0 jkl = 1 3 [ B 0 jkl + Bklj [ ljk] B0 + B 0 3 jkl Bkjl] 0 1 [ + B 0 3 jlk Bljk] 0 The orthogonal subspaces classify the O(n) invariant equations that are linear in j b kl. The trivial subspace: j b kl = 0. The subspace I (we follow Gray s notation, [12]) where B 0 jkl = 0: j b kl = A j g kl + B k g jl + B l g jk. The copleent I is characterized by A j, B j = 0 i.e. j b kl is traceless. This gives two invariant equations: j b j l = 0, and j b = 0. Since j b kl = B 0 jkl, the structure of B0 specifies two orthogonal subspaces, so that I = A B. In A: In B: j b kl + k b lj + l b jk = 0. j b kl k b jl = 0. The subspace I A contains tensors with traceless part j b kl A j g kl B k g jl B l g jk that satisfies the cyclic condition: [ j b kl 1 n + 2 ( jb + 2 b j)g kl ] + cyclic = 0.

5 RIEMANN COMPATIBLE TENSORS 5 The subspace I B contains tensors with traceless part that satisfies the Codazzi condition: [ j b kl 1 n 1 ( jb b j)g kl ] = [ k b jl 1 n 1 ( kb b k)g jl ] Accordingly, the Codazzi deviation tensor has the (unique) decoposition in irreducible coponents (3.4) C jkl = C 0 jkl + λ j g kl λ k g jl, where C 0 is traceless. Eq.(2.2) becoes (3.5) λ j = A j B j = jb b j n 1 b i R jkl + b j R kil + b k R ijl = i C 0 jkl + jc 0 kil + kc 0 ijl +g il ( j λ k k λ j ) + g jl ( k λ i i λ k ) + g kl ( i λ j j λ i ) There are only two orthogonal invariant cases: - C 0 jkl = 0, then b is R copatible if and only if λ is closed. If b is the Ricci tensor, this requireent gives nearly conforally syetric (NCS) n anifolds, that were introduced by Roter [27]. - j b b j = 0 then b is R copatible if and only if C = C 0 satisfies the second Bianchi identity. If b is the Ricci tensor, this corresponds to j R = 0. Reark 3.1. The decoposition (3.4) for the deviation of the Ricci tensor turns out to be (3.6) C jkl = n 2 n 3 C 1 jkl + 2(n 1) [g kl j R g jl k R] where C jkl is the conforal curvature tensor, or Weyl s tensor. In this case the λ covector is closed. 4. Rieann copatibility: general properties The existence of a Rieann copatible tensor has various iplications. A first one is the existence of a new generalized curvature tensor. This leads to the generalization of the Derdzinski-Shen theore and other relations that were obtained for Codazzi tensors. We need the definition, fro Kobayashi and Noizu s book [15]: Definition 4.1. A tensor K ijl is a generalized curvature tensor (or, briefly, a K-tensor) if it has the syetries of the Rieann curvature tensor: a) K ijkl = K jikl = K ijlk, b) K ijkl = K klij, c) K ijkl + K jkil + K kijl = 0 (first Bianchi identity).

6 6 C. A. MANTICA AND L. G. MOLINARI It follows that the tensor K jk =: K jk is syetric. In ref. [21], Lea 2.2, we proved this interesting fact: Theore 4.2. If b is R-copatible then K ijkl =: R ijpq b p kb q l is a K-tensor. The next result rearks the relevance of the local basis of eigenvectors of the Ricci tensor. Another syetric contraction of the Rieann tensor was introduced by Bourguignon [4]: (4.1) Rij =: b pq R pijq. Theore 4.3. If b is R-copatible then: 1) b i R j b j R i = 0, 2) b i Rj b j Ri = 0 Proof. The first identity is proven by transvecting (1.2) with g kl. The second one is a restateent of the syetry of the tensor K ij. Reark 4.4. A) Identities 1 and 2 are here obtained directly fro R- copatibility. Bourguignon [4] obtained the fro Weitzenböck s forula for Codazzi tensors, and Derdzinski and Shen [9] fro their theore. B) As the syetric atrices b ij, R ij, R ij coute, they share at each point of the anifold an orthonoral set of n eigenvectors. C) If b is a syetric tensor that coutes with a Rieann copatible b, then it can be shown that R ij =: b pq R pijq coutes with b. Finally, this Veblen-type identity holds: Proposition 4.5. If b is R-copatible, then: (4.2) b i R jlk + b j R kil + b k R lji + b l R ikj = 0 Proof. Write four versions of equation (1.2) with cyclically peruted indices i, j, k, l and su up, and use the first Bianchi identity. 5. Pure Rieann tensors and Pontryagin fors Rieann copatibility and nondegeneracy of the eigenvalues of b iply directly that the Rieann tensor is pure and Pontryagin fors vanish. We quote two results fro Maillot s paper [18]: Definition 5.1. In a Rieann anifold M n, the Rieann curvature tensor is pure if at each point of the anifold there is an orthonoral basis of n tangent vectors X(1),...,X(n), X(a) i X(b) i = δ ab, such that the tensors X(a) i X(b) j =: X(a) i X(b) j X(a) j X(b) i, a < b, diagonalize it: (5.1) R ij l X(a) i X(b) j = λ ab X(a) l X(b)

7 RIEMANN COMPATIBLE TENSORS 7 Theore 5.2. If a Rieannian anifold has pure Rieann curvature tensor, then all Pontryagin fors vanish. Consider the aps on tangent vectors, built with the Rieann tensor, ω 4 (X 1...X 4 ) = R ija b R klb a (X i 1 Xj 2)(X k 3 Xl 4 ), ω 8 (X 1...X 8 ) = R ija b R klb c R nc d R pqd a (X i 1 Xj 2 ) (Xp 7 Xq 8 ), They are antisyetric under exchange of vectors in the single pairs, and for cyclic perutation of pairs. The Pontryagin fors [25] Ω 4k result fro total antisyetrization of ω 4k : Ω 4k (X 1...X 4k ) = P ( 1)P ω 4k (X i1...x i4k ) where P is the perutation taking (1...4k) to (i 1...i 4k ). Ω 4k = 0 if two vectors repeat, interediate fors Ω 4k 2 vanish identically. Pontryagin fors on generic tangent vectors are linear cobinations of fors evaluated on basis vectors. If the Rieann tensor is pure, all Pontryagin fors on the basis of eigenvectors of the Rieann tensor vanish. For exaple, if X, Y, Z, W are orthogonal: ω 4 (XY ZW) = λ XY λ ZW (X a Y b )(Z b W a ) = 0 and Ω 4 (XY ZU) = 0. A consequence of the extended Derdzinski-Shen theore 1.2 is the following: Theore 5.3. If a syetric tensor field b ij exists that is R-copatible and has distinct eigenvalues at each point of the anifold, then the Rieann tensor is pure and all Pontryagin fors vanish. Proof. At each point of the anifold the syetric atrix b ij (x) is diagonalized by n tangent orthonoral vectors X(a), with distinct eigenvalues. Since b is R-copatible, theore 1.2 holds and, because of antisyetry of R in first two indices: 0 = R ij kl X(a) i X(b) j X(c) k, a b c. This eans that all colun vectors of the atrix V (a, b) kl = R kl ij X(a) i X(b) j are orthogonal to vectors X(c) i.e. they belong to the subspace spanned by X(a) and X(b). Because of antisyetry in indices k, l, it is necessarily V (a, b) = λ ab X(a) X(b), i.e. the Rieann tensor is pure. This property has been checked by Petersen [26] in various exaples with rotationally invariant etrics, by giving explicit orthonoral fraes such that R(e i, e j )e k = 0.

8 8 C. A. MANTICA AND L. G. MOLINARI 6. Structures for Rieann copatibility Soe differential structures are presented that yield Rieann copatibility. Of particular interest are geodesic appings, which leave the condition for R-copatibility for-invariant, and generate R-copatible tensors. Other exaples where b is the Ricci tensor are discussed in [20, 22] 6.1. Pseudo-K-syetric anifolds. They are characterized by a generalized curvature tensor K such that ([5, 23]) i K jkl = 2A i K jkl + A j K ikl + A k K jil + A l K jki + A K jkli, The tensor b jk =: K jk is syetric. It is R copatible if its Codazzi deviation C ikl = A i b kl A k b il +3A K ikl fulfills the second Bianchi identity. This is ensured by A being concircular, i.e. i A = A i A + γ g i Generalized Weyl tensors. A Rieannian anifold is a (NCS) n [27] if the Ricci tensor satisfies j R kl k R jl = 1 2(n 1) [g kl j R g jl k R]. As such, the Ricci tensor is the Weyl tensor, and the left-hand side is its Codazzi deviation. This condition, by (3.6), is equivalent to C jkl = 0. This suggests a class of deviations of a syetric tensor b with C 0 jkl = 0 in (3.4): (6.1) C jkl = λ j g kl λ k g jl Proposition 6.1. b is R-copatible if and only if λ i is closed. Proof. Transvect (3.5) with g kl and obtain: b i R j + b j R i = (n 2)( i λ j j λ i ). Then b coutes with the Ricci tensor if and only if λ is closed and, by the previous equation, b is R-copatible. An exaple is provided by spaces with (6.2) j b kl = A j g kl + B k g jl + B l g jk, where C jkl = λ j g kl λ k g jl with λ j = A j B j. Sinyukov anifolds [29] are of this sort, with b ij being the Ricci tensor itself Geodesic appings. Rieann copatible tensors arise naturally in the study of geodesic appings, i.e. appings that preserve geodesic lines [24, 11]. Their iportance arise fro the fact that Sinyukov anifolds are (NCS) n anifolds and they always adit a nontrivial geodesic apping. Geodesic appings preserve Weyl s projective curvature tensor [29]. We show that they also preserve the for of the copatibility relation.

9 RIEMANN COMPATIBLE TENSORS 9 A ap f : (M n, g) (M n, g) is geodesic if and only if Christoffel sybols are related by Γ k ij = Γ k ij + δi k X j + δj k X i where, on a Rieannian anifold, X is closed ( i X j = j X i ). The condition is equivalent to: (6.3) k g jl = 2X k g jl + X j g kl + X l g kj which has the for (6.2). The corresponding relation between Rieann tensors is (6.4) R jkl = R jkl + δ j P kl δ k P jl where P kl = k X l X k X l is the deforation tensor. The syetry P kl = P lk is ensured by closedness of X. Proposition 6.2. Geodesic appings preserve R-copatibility: (6.5) b i R jkl + b j R kil + b k R ijl = b i R jkl + b j R kil + b k R ijl where b is a syetric tensor. Proof. Let s show that the difference of the two sides is zero. Eq.(6.4) gives: b i (δ j P kl δ k P jl) + b j (δ k P il δ i P kl) + b k (δ i P jl δ j P il) = b ij P kl b ik P jl + b jk P il b ji P kl + b ki P jl b kj P il = 0 Since g is trivially R-copatible (first Bianchi identity), for invariance iplies: Corollary 6.3. g is R-copatible. 7. Generalized curvature tensors. Several results that are valid for the Rieann tensor with a Rieann copatible tensor, extend to generalized curvature tensors K ijkl (hereafter referred to as K-tensors) with a K-copatible syetric tensor b jk. The classical curvature tensors are K-tensors. The copatibility with the Ricci tensor is then exained. Definition 7.1. A syetric tensor b ij is K-copatible if (7.1) b i K jkl + b j K kil + b k K ijl = 0. The etric tensor is always K-copatible, as (7.1) then coincides with the first Bianchi identity for K. Proposition 7.2. If K ijl is a K-tensor and b kl is K-copatible, then ˆK ijkl =: K ijrs b k r b l s is a K-tensor. We quote without proof the extension of Derdzinski and Shen theore for generalized curvature tensors [21]:

10 10 C. A. MANTICA AND L. G. MOLINARI Theore 7.3. Suppose that K ijkl is a K-tensor, and a syetric K- copatible tensor b ij exists. Then, if X, Y and Z are three eigenvectors of the atrix b r s at a point x of the anifold, with eigenvalues λ, µ and ν, it is X i Y j Z k K ijkl = 0 provided that both λ and µ are different fro ν. Proposition 7.4. If b is K copatible, and b coutes with a tensor h, then the syetric tensor K kl =: K jkl h j coutes with b. Proof. Multiply relation of K copatibility for b by h kl. The last ter vanishes for syetry. The reaining ters give the null coutation relation. In ref.[19] (prop.2.4) we proved that a generalization of Lovelock s identity (2.3) holds for certain K-tensors, that include all classical curvature tensors: Proposition 7.5. Let K jkl be a K-tensor with the property (7.2) K jkl = α R jkl + β (a kl j a jl k )ϕ, where α, β are non zero constants, ϕ is a real scalar function and a kl is a Codazzi tensor. Then: (7.3) i K jkl + j K kil + k K ijl = α(r i R jkl + R j R kil + R k R ijl ) ABC curvature tensors. A class of curvature tensors with the structure (7.2) are the ABC curvature tensors. They are cobinations of the Rieann tensor and its contractions (A, B, C are constants unless otherwise stated): (7.4) K jkl = R jkl + A(δ j R kl δ k R jl ) + B(R j g kl R k g jl ) The canonical curvature tensors are of this sort: +C(Rδ j g kl Rδ k g jl ) Conforal tensor C ijkl : A = B = 1 n 2, C = 1 (n 1)(n 2) ; Conharonic tensor N ijkl : A = B = 1 n 2, C = 0; Projective tensor: P ijkl : A = 1 n 1, B = C = 0; Concircular tensor: Cijkl : A B = 0, C = 1 n(n 1). ABC tensors are generalized curvature tensors (in the sense of Kobayashi and Noizu, Def. 4.1) only for A = B. If A B the (0,4) tensor is not antisyetric in the last two indices. Proposition 7.6. Let K jkl be an ABC tensor (A, B, C ay be scalar functions) and b ij a syetric tensor;

11 RIEMANN COMPATIBLE TENSORS 11 1) if b is R-copatible then b is K-copatible. 2) if b is K-copatible and B 1 then b is R-copatible. n 2 Proof. The following identity holds for ABC tensors and a syetric tensor b: (7.5) b i K jkl + b j K kil + b k K ijl = b i R jkl + b j R kil + b k R ijl +B [g kl (b i R j b j R i ) + g il (b j R k b k R j ) + g jl (b k R i b i R k )]. 1) by theore 4.3, if b is R-copatible then it coutes with the Ricci tensor, and K-copatibility follows. 2) if b is K-copatible it coutes with K ij. Contraction with g kl gives: b i K j b j K i = (b i R j b j R i )[1 B(n 2)], then, if B 1, b coutes with the Ricci tensor and by (7.5) it is n 2 R-copatible. The first stateent of the proposition was proven for A = B in [21], Prop Proposition 7.7. Let K be an ABC tensor with constant A 1 and B. If (7.6) i K jkl + j K kil + k K ijl = 0 then the Ricci tensor is K-copatible. Proof. If A and B are constants, one evaluates (7.7) K jkl = (1 A) R jkl (B + 2C) (g kl j R g jl k R), Lovelock s identity (2.3) for the Rieann tensor iplies (7.8) i K jkl + j K kil + k K ijl = (1 A)(R i R jkl + R j R kil + R k R ijl ). In the right-hand side the Rieann tensor can be replaced by the tensor K by (7.5) written for the Ricci tensor. Sufficient conditions are: K is haronic, K is recurrent (with closed recurrency 1-for, see eq.(3.13) in [19]). Note that prop. 7.7 reains valid for the Weyl conforal tensor, which is traceless.

12 12 C. A. MANTICA AND L. G. MOLINARI 8. Weyl-copatibility and General Relativity In general relativity, the Ricci tensor is related to the energy-oentu tensor by the Einstein equation: R jl = 1 2 Rg jl + kt jl with scalar curvature R = 2kT/(n 2) (T = T k k). The contracted second Bianchi identity gives R jkl = k ( k T jl j T kl ) (g jl k R g kl j R). Let K be an ABC tensor, with constant A, B, C. Its divergence (7.7) can be expressed in ters of the gradient of the trace of the energy oentu tensor T ij. In the sae way Einstein s equations and (7.8) give an equation which is local in the energy oentu tensor: (8.1) i K jkl + j K kil + k K ijl = (1 A)k (T i K jkl + T j K kil + T k K ijl ). The Weyl tensor C jkl is the traceless part of the Rieann tensor, and it is an ABC tensor. There are advantages in discussing General Relativity by taking the Weyl tensor as the fundaental geoetrical quantity [2, 14, 6]. The first equation (7.7) C jkl = k n 3 n 2 [ k T jl j T kl + 1 ] n 1 (g jl k T g kl j T) is reported in textbooks, such as De Felice [6], Hawking Ellis [14], Stephani [30], and in the paper [2]. Instead, a further derivation yields a Bianchi-like equation for the divergence, eq.(8.1), which contains no derivatives of the sources (8.2) i C jkl + j C kil + k C ijl = k n 3 n 2 (T ic jkl + T j C kil + T k C ijl ). It can be viewed as a condition for Weyl-copatibility for the energy oentu tensor. In view of prop.7.4 and the previous equation, the following holds: Proposition 8.1. If T ij is Weyl-copatible, the syetric tensor C kl =: T j C jkl coutes with T ij. In 4 diensions, given a tie-like velocity field u i, Weyl s tensor is projected in longitudinal (electric) and transverse (agnetic) tensorial coponents [2] E kl = u j u C jkl, H kl = 1 4 uj u (ǫ pqjk C pq l + ǫ pqjl C pq k)

13 RIEMANN COMPATIBLE TENSORS 13 that solve equations that reseble Maxwell s equations with source. Therefore, the tensor E kl = C kl can be viewed as a generalized electric field. It coincides with the standard definition if T ij = (p + ρ)u i u j + pg ij (perfect fluid). The generalized agnetic field is H kl = 1 4 T j (ǫ pqjk C pq l+ǫ pqjl C pq k). Proposition 8.2. If T kl is Weyl copatible then H kl = 0. Proof. Fro the condition for Weyl copatibility we obtain ǫ ijkp [T i C jk l+ T j C ki l + T k C ij l] = 0. The first and the second ter are odified as follows: ǫ ijkp T i C jk l = ǫ kijp T k C ij l = ǫ ijkp T k C ij l ǫ ijkp T j C ki l = ǫ jkip T k C ij l = ǫ ijkp T k C ij l. Then, since the su becoes ǫ ijkp T k C ij l = 0, the agnetic part of Weyl s tensor is zero. References [1] M. Berger and D. Ebin, Soe characterizations of the space of syetric tensors on a Rieannian anifold, J. Differential Geoetry 3, (1969) [2] E. Bertschinger and A. J. S. Hailton, Lagrangian evolution of the Weyl tensor, Astroph. J (1994) [3] Besse A. L., Einstein Manifolds, Springer (1987) [4] J. P. Bourguignon, Les variétés de diension 4 à signature non nulle dont la courbure est haronique sont d Einstein, Invent. Math. 63 n.2, (1981) [5] U. C. De, On weakly syetric structures on Rieannian anifolds, Facta Universitatis, Series; Mechanics, Autoatic Control and Robotics vol. 3 n.14, (2003) [6] F. De Felice and C. J. S. Clarke, Relativity on curved anifolds, Cabridge (2001) [7] A. Derdzinski, Soe rearks on the local structure of Codazzi tensors, in Global differential geoetry and global analysis, Lecture Notes Math. n.838, , Springer-Verlag (1981) [8] A. Derdzinski, On copact Rieannian anifolds with haronic curvature, Math. Ann. 259, (1982) [9] A. Derdzinski and C. L. Shen, Codazzi tensor fields, curvature and Pontryagin fors, Proc. London Math. Soc. 47 n , (1983) [10] D. Ferus, A reark on Codazzi tensors in constant curvature spaces, in Global differential geoetry and global analysis, Lecture Notes Math. n.838, 257, Springer- Verlag (1981) [11] S. Forella, On soe class of nearly conforally syetric anifolds, Colloq. Math. 68 n.1, (1995) [12] A. Gray, Einstein-like anifolds which are not Einstein, Geo. Ded. 7 n.3, (1978) [13] M. Haeresh, Group theory and its application to physical probles, Dover reprint (1989). [14] S. W. Hawking and G. F. R. Ellis, The large scale structure of space tie, Cabridge (1973) [15] S. Kobayashi and K. Noizu, Foundations of differential geoetry, Vol. 1, Interscience, New York (1963) [16] D. Krupka, The trace decoposition proble, Beitrage Algebra Geo 36 n.2, (1995)

14 14 C. A. MANTICA AND L. G. MOLINARI [17] D. Lovelock and H. Rund, Tensors, differential fors and variational principles, reprint Dover Ed. (1988) [18] H. Maillot, Sur les variétés Rieanniennes à operateur de courbure pur, C. R. Acad. Sci. Paris Ser. A 278, (1974) [19] C. A. Mantica and L. G. Molinari, A second order identity for the Rieann tensor and applications, Colloq. Math. 122 n.1, (2011); arxiv: [ath.dg]. [20] C. A. Mantica and L. G. Molinari, Weakly Z-syetric anifolds, Acta Math. Hung. 135 n.1-2, (2012); arxiv: [ath.dg]. [21] C. A. Mantica and L. G. Molinari, Extended Derdzinski-Shen theore for curvature tensors, subitted to Colloq. Math. (2012), arxiv: [ath.dg]. [22] C. A. Mantica and J. Y. Suh, Pseudo Z syetric Rieannian anifolds with haronic curvature tensors, Int. J. Geo. Meth. Mod. Phys. 9 n.1, (21 pages) (2012) [23] C. A. Mantica and J. Y. Suh, The closedness of soe generalized 2-fors on a Rieannian anifold I, subitted to Publ. Math. Debrecen (2012) [24] J. Mikes, Geodesic Mappings of affine-connected and Rieannian spaces, Journal of Matheatical Science 78 (3) (1996) [25] M. Nakahara, Geoetry, Topology and Physics, Ada Hilger, Bristol (1990) [26] Petersen P., Rieannian Geoetry, Springer (2006) [27] W. Roter, On a generalization of conforally syetric etrics, Tensor (N.S.) (1987) [28] U. Sion, Codazzi tensors, in Global differential geoetry and global analysis, Lecture Notes Math. n.838, , Springer-Verlag (1981) [29] N. S. Sinyukov, Geodesic apping of Rieannian spaces, Nauka, Moscow (1979) [30] H. Stephani, D. Kraer, M. MacCallu, C. Hoenselaers and E. Hertl, Exact Solutions of Einstein s Field Equations, Cabridge University Press, 2nd ed. (2003) I.I.S. Lagrange, Via L. Modignani 65, 20161, Milano, Italy E-ail address: carloalberto.antica@libero.it Physics Departent, Universitá degli Studi di Milano and I.N.F.N. sezione di Milano, Via Celoria 16, Milano, Italy. E-ail address: luca.olinari@unii.it