Conformally symmetric semi-Riemannian manifolds

Conformally symmetric semi-Riemannian manifolds ^夽

Young Jin Suh

^a,∗

, Jung-Hwan Kwon

^b

, Hae Young Yang

^a

aDepartment of Mathematics, Kyungpook University, Taegu 702-701, Korea

bDepartment of Mathematics Education, Taegu University, Taegu 705-714, Korea

Received 14 March 2005; received in revised form 25 April 2005; accepted 11 May 2005 Available online 1 July 2005

Abstract

In this paper we introduce the concept of conformal curvature-like tensor on a semi-Riemannian manifold, which is weaker than the notion of conformal curvature tensor defined on a Riemannian manifold. By such kind of conformal curvature-like tensor we give a complete classification of con- formally symmetric semi-Riemannian manifolds with generalized non-null stress energy tensor.

© 2005 Elsevier B.V. All rights reserved.

MSC:Primary 53C40; Secondary 53C15

Keywords:Weyl curvature tensor; Conformally symmetric; Conformal-like curvature tensor; Semi-Riemannian manifold

1. Introduction

LetMbe ann(≥4)-dimensional semi-Riemannian manifold with a metric tensorgand a Riemannian connection∇and letR(resp.Sorr) be the Riemannian curvature tensor (resp.

the Ricci tensor or the scalar curvature) onM. Any two self-adjoint (1,1) tensor fieldsA,B

夽This work was supported by grant Proj. No. R14-2002-003-01001-0 from Korea Research Foundation, 2005.

∗Corresponding author. Tel.: +82 53 950 5315; fax: +82 53 950 6306.

E-mail addresses:yjsuh@mail.knu.ac.kr (Y.J. Suh), jhkwon@biho.daegu.ac.kr (J.-H, Kwon).

0393-0440/$ – see front matter © 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.geomphys.2005.05.005

on a semi-Riemannian manifold (M, g) we define Kulkrani–Nomizu tensor productA⊗B of EndΛ²TMin such a way that

(A⊗B)(X, Y)=¹₂(AX∧BY +BY∧AY).

Then the conformal curvature tensorCof a semi-Riemannian manifold (M, g) acting on two-forms is given by

C=R−2(n−1)⁻¹id⊗S+(n−1)⁻¹(n−2)⁻¹rid⊗id,

where id denotes the identity tensor of type (1,1) on (M, g). The conformal curvature tensor Cshould be the trace free part of the Riemannian curvature tensorRin above orthogonal decomposition, that is, Ric(C)=0, and isconformally invariant. Moreover, the conformal curvature tensor C, if n is at least 4, vanishes if and only if the metric is conformally flat.

Such aconformal flatnessis equivalent to the vanishing of the Weyl conformal curvature tensor in dimension not less than 4. This should be an interesting subject, because there are many other examples ofconformally flatmanifolds which are not spaces of constant curvature, and because of its important applications to physics (see[6,8,9]).

Now the componentsCijklof the conformal curvature tensorCcan be written by C_ijkl=R_ijkl− 1

n−2(S_ilg_jk−S_ikg_jl+S_jkg_il−S_jlg_ik)

+ r

(n−1)(n−2)(g_ilg_jk−g_ikg_jl), (1.1)

whereR_ijkl (resp.S_ij) denotes the components of the curvature tensorR(resp. the Ricci tensorS) onM.

We say thatMisconformally symmetricif the conformal curvature tensorCis parallel, that is∇C=0. Such kind of conformally flat or conformally symmetric semi-Riemannian manifolds have been studied by Besse [1], Bourguignon [2], Derdzi´nski and Shen [4], Ryan[10], Simon[15], Weyl[17,18], Yano[19], Yano and Bochner[20]. More generally, conformally symmetric semi-Riemannian manifolds with indices 0 and 1 are investigated by Derdzi´nski and Roter[3]. In particular, in semi-Riemannian manifolds with index 0, which are also said to be Riemannian manifolds, Derdzi´nski and Roter[3]and Miyazawa [7]proved the following

Theorem A. Ann(≥4)-dimensional conformally symmetric manifold is conformally flat or locally symmetric.

In particular, Derdzi´nski and Roter[3]investigated the structure of analytic conformally symmetric indefinite Riemannian manifold of index 1 which is neitherconformally flatnor locally symmetric.

The symmetric tensorKof type (0,2) with componentsK_ij is called theWeyl tensor, if it satisfies

Kijl−Kilj= 1

2(n−1)(klgij−kjgil), (1.2)

wherek=TrKandKijl (resp.kj) are components of the covariant derivative∇K(resp.

∇k).

On the other hand, in Weyl[17,18]it can be easily seen that the Ricci tensor is equal to the Weyl tensor when we only consider ann(≥4)-dimensional conformally flat Riemannian manifold.

Now as a generalization of conformal curvature tensor we introduce a new notion of conformal curvature-like tensorB(T, U), which is defined in[12–14]due to the present authors. It was given as follows:

LetT be any curvature-like tensor (see its define in Section2, in detail) and letUbe any symmetric tensor of type (0,2) satisfying 2C₁₂(∇U)=C₂₃(∇U), whereC₁₂andC₂₃ denote the metric contraction defined by 2

i_i∇U(E_i, E_i, X)=

i_i∇U(X, E_i, E_i) for any vectorXatxand for any orthonormal basis{Ej}for the tangent spaceTxMtoMatx.

Then let us define the tensorB=B(T, U) with componentsBijklsuch that B_ijkl=T_ijkl− 1

n−2(U_ilg_jk−U_ikg_jl+U_jkg_il−U_jlg_ik)

+ u

(n−1)(n−2)(g_ilg_jk−g_ikg_jl) (1.3)

whereu=TrU. Then such a tensorB=B(T, U) is said to be theconformal curvature-like tensor, which is a general extension of the usual conformal curvature tensorCin(1.1). For this kind of conformal curvature-like tensorB(T, U) we also define the notion ofconformally flatorconformally symmetricaccording asB=0 or∇B=0 respectively.

The Ricci-like tensor Ric(B) ofBis defined by tr{Z→B(Z, X)Y}. Then the components B_ijof Ric(B) is defined byB_ij =

k_kB_kijk. Thus by(1.3), its components can be written asB_ij =T_ij−U_ij, whereT_ij =

k_kT_kijk.

Now the tensor Ric(B)−_n^bgdefined on a semi-Riemannian manifold is said to be the generalized stress energytensor for the conformal curvature-like tensorB=B(R, U), where b=Tr(Ric(B))=C12(Ric(B)). The physical meaning of such kind of stress energy tensor can be explained in more detail as follows:

In general relativity there can be no universal a priori geometry, since for any spacetime the Einstein equation already determines the stress energy tensorT, which is given by

T = 1 8π

Ric−1

2Sg

,

whereS=C₁₂(Ric) denotes the scalar curvature. This is an Einstein equation between the stress energy tensor in physics and the Ricci curvature in differential geometry of spacetime.

Thus a given spacetime can be used to model matter only in the unlikely case thatThappens to be a physically realistic stress energy tensor (See[6,8,9]).

When the curvature-like tensor T(resp. the symmetric tensorU) mentioned above is equal to the curvature tensorR(resp. the Ricci tensorS), thenB(R, S) can be identified with the conformal curvature tensorC. Moreover, a semi-Riemannian manifoldMis said to be locally symmetricif its derivative of the curvature tensorRvanishes, that is,∇R=0.

Now in this paper we want to make a generalization of Theorem A in the direction of semi-Riemannian manifolds with symmetric conformal curvature-like tensor. In order to do this we need a geometric physical condition, that is, thegeneralized non-null stress energy tensor which is weaker than the notion ofstress energy tensorgiven in B.O’Neill[8,9].

That is, we show the following

Theorem.Let M be ann(≥4)-dimensional semi-Riemannian manifold with generalized non-null stress energy tensor. Let U be the symmetric tensor inD²Mand its trace∇uis non-null and letB=B(R, U)be the conformal curvature-like tensor. If it is conformally symmetric for the conformal curvature-like tensorB(R, U),then it is locally symmetric, conformally flat or∇U,∇U =0.

In the proof of our Theorem, we have used a usefulCorollary 5.1given in Section5and Lemma 6.3and Theorem 6.1 given in Section6. Now we will give its brief outline of the proof as follows:

InCorollary 5.1, under the assumption that∇uis non-null we have proved the scalar- like curvatureuis constant. Moreover, inLemma 6.3, if the conformal curvature-like tensor B=B(R, U) is symmetric on a semi-Riemannian manifoldM, that is∇B=0, we have proved that the generalized non-null stress energy tensor vanishes whenMis not locally symmetric. By using such results, in Theorem 6.4 we are able to show thatMis conformally flat whenMis not locally symmetric.

IfMis a Riemannian manifold, the result∇U,∇U =0 in our Theorem implies that the symmetric tensorUis parallel onM. From this together with the assumption of conformal symmetry∇B=0 we can assert that∇R=0, that isMis locally symmetric.

2. Preliminaries

Let M be an n(≥2)-dimensional semi-Riemannian manifold of index s (0≤s≤n) equipped with semi-Riemannian metric tensor∇and letR(resp.Sorr) be the Riemannian curvature tensor (resp. the Ricci tensor or the scalar curvature) onM.

Now we can choose a local field{E_j} = {E₁, . . ., E_n}of orthonormal frames on a neigh- borhood ofM. Here and in the sequel, the Latin small indicesi, j, k, . . ., run from 1 ton.

With respect to the semi-Riemannian metric we haveg(E_j, E_k)=_jδ_jk, where _j= −1 or1 according as 0≤j≤sors+1≤j≤n.

Let{θ_j},{θ_ij}and{Θ_ij}be the canonical form, the connection form and the curvature form onM, respectively, with respect to the field{E_j}of orthonormal frames. Then we have

the structure equations dθ_i+

j

_jθ_ij∧θ_j=0, θ_ij+θ_ji=0, dθ_ij+

k

_kθ_ik∧θ_kj=Θ_ij,

Θ_ij = −1 2

k,l

_klR_ijklθ_k∧θ_l,

where_ij...k =_ij. . ._k andR_ijkl denotes the components of the Riemannian curvature tensorRofM(See[9,11,12,14]).

Now, letCbe the conformal curvature tensor with componentsC_ijklonM, which is given by

C_ijkl=R_ijkl− 1

n−2(_jS_ilδ_jk−_jS_ikδ_jl+_iS_jkδ_il−_iS_jlδ_ik)

+ r

(n−1)(n−2)ij(δilδjk−δikδjl), (2.1) whereSij denotes the components of the Ricci tensorSwith respect to the orthonormal frame field{E_j}.

Remark 2.1. IfMis Einstein, the conformal curvature tensorCsatisfies Cijkl=Rijkl− r

n(n−1)ij(δilδjk−δikδjl).

This yields that the conformal curvature tensor of an Einstein Riemannian manifold is the concircular curvature one[20]. In particular, ifMis a space of constant curvature, the conformal curvature tensor vanishes identically.

Let D^rM be the vector bundle consisting of differentiable r-forms and DM = _n

r=0D^rM, whereD⁰Mis the algebra of differentiable functions onM. For any tensor fieldKinD^rMthe componentsK_ijklhof the covariant derivative∇KofKare defined by (for simplicity, we consider the caser=4)

h

_hK_ijklhθ_h=dK_ijkl−

h

_h(K_hjklθ_hi+K_ihklθ_hj+K_ijhlθ_hk+K_ijkhθ_hl).

Now we denote byTM the tangent bundle ofM. LetTbe a quadrilinear mapping of TM×TM×TM×TMintoRsatisfying the curvature-like properties:

(a) T(X, Y, Z, U)= −T(Y, X, Z, U)= −T(X, Y, U, Z), (b) T(X, Y, Z, U)=T(Z, U, X, Y),

(c) T(X, Y, Z, U)+T(Y, Z, X, U)+T(Z, X, Y, U)=0.

ThenTis called thecurvature-like tensoronM(see B.O. N’eill[8], for example). For an orthonormal frame{E_j}, letT_ijklbe the components ofTassociated with the orthonor-

mal frame. Accordingly, the componentsT_ijklare given byT_ijkl=T(E_i, E_j, E_k, E_l). The components ofTcorresponding to the conditions (a), (b) and (c) are given by respectively

Tijkl= −Tjikl= −Tijlk, (2.2)

T_ijkl=T_klij=T_lkji, (2.3)

T_ijkl+T_jkil+T_kijl=0. (2.4)

If the componentsT_ijklof a tensorTinD⁴M= ⊗⁴T^∗Msatisfy(2.2), (2.3) and (2.4), then it becomes a curvature-like tensor. LetFMbe the ring consisting of all smooth functions on Mand letT_r^sMbe the module overFMconsisting of all tensor fields of type (r, s) defined on M. LetH =H(M, g) be the vector subbundle inD³Mwhich, at any pointxinM, consists of all trilinear mappingξofTxMintoRsuch thatξ(X, Y, Z)=ξ(X, Z, Y) for any vectors atxand

2

r

_rξ(E_r, E_r, X)=

r

_rξ(X, E_r, E_r)

for any vectorXatxand for any orthonormal basis{Ej}for the tangent spaceTxMtoMat x.

For any integersaandbsuch that 1≤a < b≤sthe metric contraction reduced byaand bis denoted byCab:T_s^rM→T_s^r₋₂Mwith respect to the orthonormal frame{E_j}. In terms of the metric contraction, the sectionξinC^∞(H) satisfies thatξ(X, Y, Z) is symmetric with respect toYandZand 2C12(ξ)=C23(ξ).

Let U be a symmetric tensor of type (0,2) in D²M with components U_ij(=U_ji)= U(E_i, E_j). The symmetric tensorUinD²Mis called theWeyl tensorif its components of the covariant derivative∇UofUsatisfy

U_ijk− 1

2(n−1)u_kiδ_ij =U_ikj− 1

2(n−1)u_jiδ_ik (2.5)

whereu=C12U. In particular, ifuis constant, thenUis called theCodazzi tensor. Now we define the covariant derivative ∇U of the symmetric tensor U in such a way that

∇U(X, Y, Z)= ∇XU(Y, Z). SinceUis symmetric, so is∇Uwith respect toYandZ. More- over, we know that

∇E_kU(Ei, Ej)= ∇U(Ek, Ei, Ej)=Uijk(=Ujik).

Then by(2.5)and the expression of∇Uit can be easily seen that

k

_kU_kjk= 1 2u_j, whereu_j=

l_lU_llj =

l_l∇U(E_j, E_l, E_l). This meansC₁₂(∇U)=C₂₃(∇U)/2. Ac- cordingly, we know that∇Uis the section of the bundleH. For such a pair (T, U), we define the tensorB=B(T, U) with componentsB_ijklby

B_ijkl=T_ijkl− 1

n−2(_iU_jkδ_il−_iU_jlδ_ik+_jU_ilδ_jk−_jU_ikδ_jl)

+ u

(n−1)(n−2)ij(δilδjk−δikδjl), (2.6) which is said to be theconformal curvature-like tensorforTandU. The Ricci-like tensor Ric(B) ofBis defined by tr{Z→B(Z, X)Y}. Then the componentsB_ij of Ric(B) is given byB_ij=

k_kB_kijk. By(2.6), we get

B_ij=T_ij−U_ij. (2.7)

Remark 2.2. For a given semi-Riemannian manifoldMwith Riemannian connection∇, there exist so many kind of pairs (T, U) for the curvature-like tensorTand the symmetric tensor ofD²Msuch that∇Uis contained inC^∞(H). Among them the most popular pair is (R, S). In particular, letUbe the Weyl tensor and letKbe the parallel symmetric tensor inD²M. ThenU+Kis also the Weyl tensor.

For an orthonormal frame{Ej}, letθ0= {θj}, θ= {θij}andΘ= {Θij}be the canonical form, the connection form and the curvature form onM. In the same way as we associate a curvature formΘto the Riemannian curvature tensorR, we associate a two-formΦT = {Φij} to the curvature-like tensorTin the following

Φ_ij =

k,l

_klT_ijklθ_k∧θ_l, (2.8)

which is the analogue to the curvature formΘexcept for the coefficient. So it is called the curvature-like formfor the curvature-like tensorT. The canonical formθ₀= {θ_j}can be regarded as a vector inRⁿand the connection formθ= {θ_ij}and the curvature-like form Φ_T = {Φ_ij}can be regarded asn×nskew-symmetric matrices. We call the equation

dΦ_T =Φ_T∧θ−θ∧Φ_T (2.9)

thesecond Bianchi equationfor the curvature-like formΦ_T. Then it is not difficult to assert the following proposition:

Proposition 2.1. On the semi-Riemannian manifold M,let T be the curvature-like tensor inD⁴Mon M. Then the following assertions are equivalent:

(a) The curvature-like formΦ_T satisfies the second BianchiEq.(2.9).

(b) Its components satisfy

Tijklh+Tijlhk+Tijhkl=0. (2.10)

Now letTbe the curvature-like tensor inD⁴Mand letΨ_T be the associated curvature- like form withT. The mappingδ:∧⁴T_∗M→∧³T_∗Mdefined by the divergence:δ(Ψ_T)=

−(C15∇T), whereCab is the metric contraction defined byC_ab:T_s^rM→T_s^r₋₂M. This is a

generalization of the well-known differential operators onR³. For the orthonormal frame {Ej}, in terms of coordinates, the components ofδ(ΨT) is given by

δ(Ψ_T)_ijk= −

l

_lT_lijkl.

Ifδ(Ψ_T)=0, then the formΨ_T is said to becoclosed.

Remark 2.3. On the Riemannian manifold (M, g) with Riemannian connection∇, it has a formal adjoint∇^∗:T^∗M×T_s^rM→T_s^rMdefined as follows: for any vector fieldsX₁, . . . , X_r and anyαinT^∗M×T_s^rM,∇^∗αis given by

∇^∗α(X1, . . ., Xr)= −

k

k∇E_kα(Ek, X1, . . ., Xr),

where{E_j}is the orthonormal frame. Namely,∇^∗α(X₁, . . ., X_r) is the opposite of the trace with respect togof theD^sMvalued two-form

(X, Y)−→∇Xα(Y, X1, . . ., Xr).

For the exterior differentiald:D^rM→D^r+1Mlet us denote byδ:D^rM→D^r−1Mits formal adjoint. For the orthonormal base{Ej}ofTxMat any pointx, the components ofδ(Ψ_T) are given by

δ(ΨT(T))(X, Y, Z)= −

k

k∇E_kΨT(T)(Ek, X, Y, Z).

Accordingly, the above operatorδon semi-Riemannian manifolds is a formal analogue of the adjoint operator to the exterior differentialdon a Riemannian manifold. See Besse[1], for example.

The semi-Riemannian manifold (M, g) is said to haveharmonic-like curvatureforTif δ(ΨT)=0. In particular, ifT =R, then we say that (M, g) hasharmonic curvature( See Besse[1]).

Now, the concept of Ricci-like tensors for the curvature-like tensor on the semi- Riemannian manifold is introduced. LetM be ann-dimensional semi-Riemannian man- ifold with semi-Riemannian metric gand curvature like tensor Twith componentsT_ijkl. The tensor Ric(T) associated with the curvature-like tensorTis defined by

Ric(T)(X, Y)=trace{Z→T(Z, X)Y},

whereT(Z, X)Yis a vector field defined byT(X, Y, Z, W)=g(T(X, Y)Z, W) for any vector fieldsX, Y, ZandW. Then Ric(T) is called theRicci-like tensorforT. By the definition, the Ricci-like tensor Ric(T) ofTis a symmetric tensor of type (0,2) and its componentsT_ij are given byT_ij=

k_kT_kijk. Moreover, by(2.3)we know thatT_ij =T_ji. The scalar-like

curvaturetassociated withTis defined byt=C12(Ric(T))=

j,kjkTkjjk. If we follow new extended formulas in a semi-Riemannian manifold, we are able to show the following theorem in[12], which gives an extension of the paper[14]discussed in its Riemannian version,

Theorem 2.1. Let M be ann(≥4)-dimensional semi-Riemannian manifold and let T be the curvature-like tensor and let U be the symmetric tensor inD²M.If any two assertions in the following are satisfied, then another one holds true:

(a) T satisfies the second Bianchi identity,

(b) The conformal curvature-like tensorB=B(T, U)satisfies the second Bianchi identity, (c) U is the Weyl tensor.

3. Weyl tensors

LetMbe ann(≥2)-dimensional semi-Riemannian manifold of index 2s, 0≤s≤n, with Riemannian connection∇and letR(resp.Sorr) be the Riemannian curvature tensor (resp.

the Ricci tensor or the scalar curvature) onM.

Now letTbe a curvature-like tensor and letUbe a symmetric tensor of type (0,2) such that 2C12(∇U)=C23(∇U). In this section we are going to prove some lemmas concerned with such a symmetric tensorU. From the conformal curvature-like tensor B=B(T, U) given in(2.6), we know thatB=B(T, U) is the curvature-like tensor for the pair (T, U).

Moreover, we have

Ric(B)=Ric(T)−U. (3.1)

In fact, putting i=l, multiplying i and summing up with respect to the index l in (2.6)we getB_jk =T_jk−U_jk, whereT_jkandB_jkare components of the Ricci-like tensors Ric(T)=C14(T) and Ric(B)=C14(B).

Now let us putt=C12(Ric(T)), b=C12(Ric(B)) andu=C12(U). Then by(3.1)we have

b=t−u. (3.2)

Remark 3.1. For the conformal curvature tensorC, the Ricci-like tensor Ric(C) satisfies Ric(C)=0.

When the symmetric tensorUinD²Mis the Weyl tensor, we are going to prove some lemmas, which will be used in Section5, as follows:

Lemma 3.1. Let M be ann(≥2)-dimensional semi-Riemannian manifold. If U is the Weyl tensor,we obtain

r

_r(R_riklU_rj+R_riljU_rk+R_rijkU_rl)=0. (3.3)

Proof. By thedefinition (2.5)we have U_ijk−U_ikj= 1

2(n−1)(u_kiδ_ij−u_jiδ_ik).

Differentiating covariantly, we get U_ijkl−U_ikjl= 1

2(n−1)(u_kliδ_ij−u_jliδ_ik). (3.4) Interchanging the indiceskandland substituting the resulting equation from the above one, we obtain

U_ijkl−U_ijlk+U_iljk−U_ikjl= 1

2(n−1)(u_jkiδ_il−u_jliδ_ik),

where we have used the propertyu_ijis symmetric with respect toiandj, becauseuis the function. So we have

The left hand side

(U_ijkl−U_ijlk)+(U_iljk−U_ilkj)+(U_ilkj−U_ikjl)

= (U_ijkl−U_ijlk)+(U_iljk−U_ilkj) +

U_iklj+ 1

2(n−1)(u_kjiδ_il−u_ljiδ_ik)

−U_ikjl

= −

r

_r(R_lkirU_rj+R_lkjrU_ir)−

r

_r(R_kjirU_rl+R_kjlrU_ir)

−

r

_r(R_jlirU_rk+R_jlkrU_ir)+ 1

2(n−1)(u_kjiδ_il−u_ljiδ_ik)

= −

r

(RlkirUrj+RjlirUrk+RkjirUrl)+ 1

2(n−1)(ukjiδil−uljiδik), where the second equality follows from(3.4)and the third equality can be derived from the Ricci identity for the tensorU_ij. Accordingly, we have(3.3), which completes the proof.

Lemma 3.2. Let M be an n(≥2)-dimensional semi-Riemannian manifold and let B= B(R, U)be the conformal curvature-like tensor for the Riemannian curvature tensor R and any symmetric tensor U inD²M.If U is the Weyl tensor,then we obtain

r

_r(B_riklU_rj+B_riljU_rk+B_rijkU_rl)=0, (3.5)

r

_rB_krU_rl=

r

_rB_lrU_rk. (3.6)

Proof. By the assumption of this Lemma, the components ofBare given by B_ijkl=R_ijkl− 1

n−2(_iU_jkδ_il−_iU_jlδ_ik+_jU_ilδ_jk−_jU_ikδ_jl)

+ u

(n−1)(n−2)_ij(δ_ilδ_jk−δ_ikδ_jl). (3.7) SubstitutingB_ijklinto the left hand side of(3.5)and using(3.3), we get the equation (3.5). Since the tensorU_ijis symmetric and the tensorB_ijklis skew-symmetric indicesiand j, we have

r,s_rsB_rsklU_rs=0. Puttingi=j, multiplying_iand summing up with respect toiin(3.5), from the above property together withB_ij =

r_rB_rijr, we get (3.6). This completes the proof.

4. Conformally symmetric

LetMbe ann(≥2)-dimensional semi-Riemannian manifold of index 2s, 0≤s≤n, with Riemannian connection∇and letR(resp.Sorr) be the Riemannian curvature tensor (resp.

the Ricci tensor or the scalar curvature) onM.

Now, letUbe a symmetric tensor of type (0,2) such that 2C12(∇U)=C23(∇U). We put u=C12U. For such a pair (R, U) we define a tensorB=B(R, U) with componentsB_ijkl with respect to the field{E_j}of orthonormal frames by

B_ijkl=R_ijkl− 1

n−2(_iU_jkδ_il−_iU_jlδ_ik+_jU_ilδ_jk−_jU_ikδ_jl)

+ u

(n−1)(n−2)ij(δilδjk−δikδjl). (4.1) We have then

Ric(B)=S−U. (4.2)

In fact, puttingi=l, multiplying_iand summing up with respect to the indexiin(4.1)we getB_jk=S_jk−U_jk, whereB_jkare components of the Ricci-like tensor Ric(B)=C14(B).

Now let us putb=C12(Ric(B))=Tr(Ric(B)) andu=C12(U)=TrU. Then by(4.2)we have

b=r−u. (4.3)

ByTheorem 2.1if the curvature-like tensorB=B(R, U) for the pair (R, U) satisfies the second Bianchi identity, thenUis the Weyl tensor. Of course the Ricci tensorSis the Weyl tensor. We remark that the restrictionn≥4 of the dimension is here necessary.

The semi-Riemannian manifoldM is said to beconformally symmetric, if the confor- mal curvature-like tensorBis parallel, i.e., if∇B=0. Concerned with such a conformal symmetry of the conformal curvature-like tensorBwe prove the following:

Lemma 4.1. Let M be ann(≥4)-dimensional semi-Riemannian manifold. If M is confor- mally symmetric for B,then

r

r(BriklUrjh+BriljUrkh+BrijkUrlh)=0. (4.4) Proof. The Riemannian curvature tensorRsatisfies the second Bianchi identity. On the other hand, since the conformal curvature-like tensorB(R, U) is parallel, it also satisfies the second Bianchi identity byProposition 2.1. Accordingly, the symmetric tensorUis the Weyl tensor byTheorem 2.1, because ofn≥4. ByLemma 3.2we have

r

r(BriklUrj+BriljUrk+BrijkUrl)=0,

from which together with the assumption thatBis parallel we have the equation(4.4). This completes the proof.

Now we have

r,s_rsB_rsklU_rsh=0, becauseB_ijklis skew-symmetric with respect toi andjandU_ijhis symmetric with respect toiandj. Puttingi=j, multiplying_i, summing up with respect toiin(4.4)and applying the above property to the obtained equation, we have

r

rBkrUrlh=

r

rBlrUrkh. (4.5)

SinceUis the Weyl tensor, it satisfies

rrUkrr=uk/2. Puttingl=h, multiplyingl, summing up with respect tolin(4.4)and applying the property

r_rU_krr =u_k/2, we have

r,s

_rs(B_riksU_jrs−B_rijsU_krs)= −1 2

r

_rB_rijku_r. (4.6)

Puttingi=j, multiplying_iand summing up with respect toiin(4.6), we have

2

r,s

rsBrsUkrs=

r

rBkrur, (4.7)

because we see

rst_rstB_rtksU_trs=0.

Summing up the above formulas, we prove the following which will be useful in Section 5.

Lemma 4.2. Let M be ann(≥4)-dimensional semi-Riemannian manifold. If M is confor- mally symmetric for the conformal curvature-like tensorB(R, U),then we obtain

2(n−1)

r

r(BirUjkr−BkrUjir)=

r

rj(Birurδjk−Bkrurδij). (4.8) Proof. SinceMis conformally symmetric for the conformal curvature-like tensorB(R, U), we haveBijklh=0 andBijklhp=0. Accordingly, we have by(4.1)

Rijklh= 1

n−2(jUilhδjk−jUikhδjl+iUjkhδil−iUjlhδik)

− 1

(n−1)(n−2)uhij(δilδjk−δikδjl). (4.9)

On the other hand, by the Ricci identity, we get

r

_r(R_phirB_rjkl+R_phjrB_irkl+R_phkrB_ijrl+R_phlrB_ijkr)=0.

Differentiating covariantly and taking account ofB_ijklh=0, we have

r

_r(R_phirqB_rjkl+R_phjrqB_irkl+R_phkrqB_ijrl+R_phlrqB_ijkr)=0.

By(4.9)and the above equation we have

r

_r[{(_hU_prqδ_hi−_hU_piqδ_hr+_pU_hiqδ_pr−_pU_hrqδ_pi) +u_qhp(δ_prδ_hi−δ_piδ_hr)/(n−1)}B_rjkl+ {(_hU_prqδ_hj

−_hU_pjqδ_hr+_pU_hjqδ_pr−_pU_hrqδ_pj)+u_qhp(δ_prδ_hj−δ_pjδ_hr)/(n−1)}B_irkl + {(_hU_prqδ_hk−_hU_pkqδ_hr+_pU_hkqδ_pr−_pU_hrqδ_pk)

+u_qhp(δ_prδ_hk−δ_pkδ_hr)/(n−1)}B_ijrl+ {(_hU_prqδ_hl−_hU_plqδ_hr

+_pU_hlqδ_pr−_pU_hrqδ_pl)+u_qhp(δ_prδ_hl−δ_plδ_hr)/(n−1)}B_ijkr]=0. (4.10) Puttingh=i, multiplying_iand summing up with respect toiand taking account of the first Bianchi identity, we have

r

_r (n−2)B_rjklU_prq−

s

_sp(B_rjslδ_pk+B_sjkrδ_pl)U_rsq

+(B_rpklU_rjq+B_rjplU_rkq+B_rjkpU_rlq)

+(B_jlU_pkq−B_jkU_plq)

− 1

n−1u_qp(B_jlδ_pk−B_jkδ_pl)=0.

Taking account of(4.4), the above equation is deformed as

r

_r (n−1)B_rjklU_prq−

s

_sp(B_rjslδ_pk+B_sjkrδ_pl)U_rsq+B_rpklU_rjq

+(B_jlU_pkq−B_jkU_plq)− 1

n−1u_qp(B_jlδ_pk−B_jkδ_pl)=0. (4.11) Puttingl=qin the above equation, multiplyingl and summing up with respect tol, we get

r,s

rs (n−2)BrjksUprs−

t

tUrstp(Brjstδpk+Bsjkrδpt)

+(BrpksUrjs+BrjpsUrks+BrjkpUrss)

+

r

r{(BjrUpkr−BjkUprr)

− 1

n−1urp(Bjrδpk−Bjkδpr)} =0. (4.12) Taking account of the fact thatBijklis skew symmetric with respect tokandland using (2.5), we get the following relation:

r,s,t

_rstB_jrstU_rst = 1 2(n−1)

r

_rB_jru_r. (4.13)

In fact, we get

The left hand side= 1 2

r,s,t

rstBjrst(Urst−Urts)

= 1 4(n−1)

s,t

_stB_jrst(u_Tδ_rs−u_sδ_rt)

= 1 4(n−1)

t

_t(B_jtu_t+B_jtu_t)

Making use of(4.13)and calculating straightforwardly, we can obtain

r

_rs

(n−2)B_rjksU_prs−B_rjksU_rsp+B_rpksU_jrs+B_rjpsU_krs +

r

_r(1

2B_rjkpu_r+B_jrU_pkr)

− 1

2(n−1) (n−3)B_jku_p+

r

_rpB_jru_rδ_pk

=0. (4.14)

Interchanging indicesjandkin(4.14)and subtracting the resulting equation from the original one, using(2.5) and (4.6), we have

(n−1)

r

_r(B_pjkr+B_jkpr−B_pkjr)u_r+2(n−1)

r

_r(B_jrU_pkr

−B_krU_pjr)−_p

r

_ru_r(B_jrδ_pk−B_krδ_pj)=0.

Making use of the first Bianchi identity for B, we have (4.8). This completes the

proof.

5. Scalar-like curvatures

Let M be ann(≥4)-diemnsional semi-Riemannian manifold of index s (0≤s≤n) equipped with semi-Riemannian metricgand Riemannian connection∇and letR(resp.Sor r) be the Riemannian curvature tensor (resp. the Ricci tensor or the scalar curvature) onM.

LetUbe the symmetric tensor inD²Mand letB=B(R, U) be the conformal curvature-like tensor. Thenu=TrUis called thescalar-like curvatureforB.

Now letCbe the conformal curvature tensor defined onM. Glodeck[5]and Tanno[16]

proved that any non-conformally flat conformal symmetric Riemannian manifold has the constant scalar curvature. We putb=Tr(Ric(B)). If∇B=0, then the functionbis constant onM. In this section we prove the following

Proposition 5.1. Let M be ann(≥4)-dimensional semi-Riemannian manifold and let U be the symmetric tensor inD²Mand letB=B(R, U)be the conformal curvature-like tensor.

If∇B=0,thenb∇u,∇u =0.

In order to prove this proposition we verify some lemmas step by step as follows:

Lemma 5.1. Under the situation ofProposition 5.1,we have

r

_rB_irU_rj=

r

_rB_jrU_ri, (5.1)

r

_rB_irU_rjk =

r

_rB_jrU_rik. (5.2)

Proof. SinceBis parallel, it satisfies the second Bianchi identity. Accordingly, byTheorem 2.1,Uis the Weyl tensor and it satisfies(3.5). Puttingi=j in(3.5), multiplying_i, and summing up with respect to the indexi, we get

r,s

_rsB_rsklU_rs−

r

_rB_lrU_rk+

r

_rB_krU_rl=0,

the first term of which vanishes identically onM. Thus we get

r

_rB_irU_rj =

r

_rB_jrU_ri.

Accordingly,(5.1)is satisfied. Because of∇B=0, we have∇Ric(B)=0, which means thatB_ijk=0. SoLemma 5.1is proved.

Lemma 5.2. Under the situation ofProposition 5.1,we have

r

_rB_iru_r=bu_i, (5.3)

∇u,∇uB_ij=bu_iu_j. (5.4)

Proof. By(2.5) and (4.8), we have

r

r(BirUjkr−BkrUijr)− 1 2(n−1)

r

rBirurjδjk−

r

rBkruriδij

=

r

_rB_ir

U_jrk+ 1

2(n−1)(u_rjδ_jk−u_kjδ_jr)

−

r

_rB_kr

U_jri+ 1

2(n−1)(u_rjδ_ji−u_ijδ_jr)

− 1 2(n−1)

r

rBirurjδjk−

r

rBkruriδij

= 1

2(n−1)(−B_iju_k+B_kju_i)=0,

where the second equality follows from(5.2). Accordingly, we can obtain

B_iju_k=B_kju_i, (5.5)

from which we have(5.3)and

r

_ru_ru_rB_ij=

r

_rB_jru_ru_i

This implies(5.4), which proves our assertion.

Lemma 5.3. Under the situation ofProposition 5.1,we have 2b∇u,∇u

r

_rU_riju_r =b∇u,∇uu_iu_j. (5.6)

Proof. By(4.7), (5.3) and (5.4), we have

2b

r,s

rsUirsurus=b

r

rururui.

On the other hand, the left side of the above equation can be reformed as

2b

r,s

rs

Ursi+ 1

2(n−1)(usrδri−uirδrs)

urus=2b

r,s

rsUrsiurus

by(2.5). From this together with the above equations, we get

2b

r,s

rsUrsiurus=b

r

rururui. (5.7)