ON THE EQUIVALENCE OF

VOL. 76 1998 NO. 2

ON THE EQUIVALENCE OF

RICCI-SEMISYMMETRY AND SEMISYMMETRY

BY

KADRI A R S L A N (BURSA), YUNUS C . E L I K (K UTAHYA), ¨ RYSZARD D E S Z C Z (WROC lAW)

RIDVAN E Z E N T A S. (BURSA)

Dedicated to the memory of our friend Dr. S . ahnur Yaprak

Introduction. A semi-Riemannian manifold (M, g), n = dim M ≥ 3, is said to be semisymmetric [28] if

(1) R · R = 0

holds on M . It is well known that the class of semisymmetric manifolds in- cludes the set of locally symmetric manifolds (∇R = 0) as a proper subset.

Recently the theory of Riemannian semisymmetric manifolds has been pre- sented in the monograph [1]. It is clear that every semisymmetric manifold satisfies

(2) R · S = 0.

The semi-Riemannian manifold (M, g), n ≥ 3, satisfying (2) is called Ricci- semisymmetric. There exist non-semisymmetric Ricci-semisymmetric man- ifolds. However, under some additional assumptions, (1) and (2) are equiv- alent for certain manifolds. For instance, we have the following statement.

Remark 1.1. (1) and (2) are equivalent on every 3-dimensional semi- Riemannian manifold as well as at all points of any semi-Riemannian mani- fold (M, g), of dimension ≥ 4, at which the Weyl tensor C of (M, g) vanishes (see e.g. [15, Lemma 2]). In particular, (1) and (2) are equivalent for every conformally flat manifold.

It is a long standing question whether (1) and (2) are equivalent for hy- persurfaces of Euclidean spaces; cf. Problem P 808 of [27] by P. J. Ryan, and references therein. More generally, one can ask the same question for hypersurfaces of semi-Riemannian space forms. It was proved in [29] that

1991 Mathematics Subject Classification: 53B20, 53B30, 53B50, 53C25, 53C35, 53C80.

Key words and phrases: semisymmetric manifolds, pseudosymmetry type manifolds, hypersurfaces.

[279]

(1) and (2) are equivalent for hypersurfaces which have positive scalar cur- vature in Euclidean space E ⁿ⁺¹ , n ≥ 3. In [26] this result was generalized to hypersurfaces of E ⁿ⁺¹ , n ≥ 3, which have non-negative scalar curvature and also to hypersurfaces of constant scalar curvature. [26] also proves that (1) and (2) coincide for hypersurfaces of Riemannian space forms with non-zero constant sectional curvature.

Further, in [24] it was proved that (1) and (2) are equivalent for hypersur- faces of E ⁿ⁺¹ , n ≥ 3, under the additional global condition of completeness.

In [4] it was shown that (1) and (2) are equivalent for Lorentzian hyper- surfaces of a Minkowski space E ⁿ⁺¹ 1 , n ≥ 4. [4] also proves that (1) and (2) are equivalent for para-K¨ ahler hypersurfaces of a semi-Euclidean space E ^2m+1 s , m ≥ 2. The problem of equivalence of (1) and (2) was solved in the 4-dimensional case. More precisely, we have the following statement.

Theorem 1.1 ([3, Theorem 4.1]). (1) and (2) are equivalent for hyper- surfaces of semi-Riemannian spaces of constant curvature N ⁵ (c).

The problem of equivalence of (1) and (2) for hypersurfaces with pseu- dosymmetric Weyl tensor of semi-Euclidean spaces was considered in [5].

Theorem 1.2 ([5, Theorem 4.1]). Let M be a Ricci-semisymmetric hy- persurface of a semi-Euclidean space E ⁿ⁺¹ s of index s, n ≥ 4. If M has pseudosymmetric Weyl tensor then (1) holds on the set U S consisting of all points of M at which the Ricci tensor S of M is not proportional to the metric tensor of M .

Hypersurfaces with pseudosymmetric Weyl tensor were studied in [7], [20] and [21]. In particular, the following curvature property of semisym- metric hypersurfaces was found.

Theorem 1.3 ([20, Theorem 7.3(ii)]; [21, Theorem 4.1]). Every semisym- metric hypersurface M isometrically immersed in a semi-Euclidean space E ⁿ⁺¹ s , n ≥ 4, is a hypersurface with pseudosymmetric Weyl tensor.

Our main result (see Theorem 5.2) is related Theorem 1.2. Namely, we prove that if (M, g), dim M ≥ 4, is a Riemannian Ricci-semisymmetric man- ifold with pseudosymmetric Weyl tensor, satisfying

(3) R · R = Q(S, R),

then (1) holds on U S . Theorem 1.2, in the case when the ambient space is a Euclidean space, is an immediate consequence of Theorem 5.2. We recall that every hypersurface M isometrically immersed in a semi-Euclidean space E ⁿ⁺¹ s , n ≥ 3, satisfies (3) ([18, Corollary 3.1]).

The paper is organized as follows. In Section 2 we fix the notations

and give precise definitions of the symbols used. Moreover, we give a short

presentation of classes of semi-Riemannian manifolds satisfying curvature

conditions of pseudosymmetry type. In Section 3 we give preliminary results.

Finally, in Sections 4 and 5 we present our main results.

2. Certain curvature conditions. Let (M, g) be a connected n-di- mensional, n ≥ 3, semi-Riemannian manifold of class C ^∞ . We define on M the endomorphisms e R(X, Y ) and X ∧ Y by

R(X, Y )Z = [∇ e _X , ∇ Y ]Z − ∇ _{[X,Y ]} Z, (X ∧ Y )Z = g(Y, Z)X − g(X, Z)Y,

where ∇ is the Levi-Civita connection of (M, g) and X, Y, Z ∈ Ξ(M ), Ξ(M ) being the Lie algebra of vector fields on M . Furthermore, we define the Riemann–Christoffel curvature tensor R and the (0, 4)-tensor G of (M, g) by

R(X 1 , X 2 , X 3 , X 4 ) = g( e R(X ₁ , X 2 )X 3 , X 4 ), G(X 1 , X 2 , X 3 , X 4 ) = g((X 1 ∧ X ₂ )X 3 , X 4 ).

We denote by S and κ the Ricci tensor and the scalar curvature of (M, g), respectively. For a (0, k)-tensor field T on M , k ≥ 1, we define the (0, k + 2)- tensors R · T and Q(g, T ) by

(R · T )(X 1 , . . . , X k ; X, Y ) = ( e R(X, Y ) · T )(X ₁ , . . . , X k )

= −T ( e R(X, Y )X 1 , X 2 , . . . , X k ) − . . . − T (X 1 , . . . , X k−1 , e R(X, Y )X k ), Q(g, T )(X 1 , . . . , X k ; X, Y ) = ((X ∧ Y ) · T )(X 1 , . . . , X k )

= −T ((X ∧ Y )X 1 , X 2 , . . . , X k ) − . . . − T (X 1 , . . . , X k−1 , (X ∧ Y )X k ).

A semi-Riemannian manifold (M, g) is said to be pseudosymmetric ([11], [30]) if

(∗) 1 the tensors R · R and Q(g, R) are linearly dependent at every point of M . This is equivalent to the equality

(4) R · R = L R Q(g, R)

holding on

U R =

 x ∈ M

R − κ

n(n − 1) G 6= 0 at x

 ,

for some function L R on U R . It is clear that every semisymmetric mani-

fold is pseudosymmetric. The condition (∗) 1 arose in the study of totally

umbilical submanifolds of semisymmetric manifolds as well as when consid-

ering geodesic mappings of semisymmetric manifolds ([11], [30]). There exist

pseudosymmetric manifolds which are non-semisymmetric. For instance, in

[12, Example 3.1 and Theorem 4.1] it was shown that the warped product

S ^p × _F S ^n−p , p ≥ 2, n − p ≥ 1, of the standard spheres S ^p and S ^n−p , with

a certain function F , is such a manifold.

A semi-Riemannian manifold (M, g) is said to be Ricci-pseudosymmetric ([8], [16]) if

(∗) 2 the tensors R · S and Q(g, S) are linearly dependent

at every point of M . Thus (M, g) is Ricci-pseudosymmetric if and only if

(5) R · S = L S Q(g, S)

on

U S =

 x ∈ M

S − κ

n g 6= 0 at x

 ,

for some function L S on U S . Note that U S ⊂ U _R . It is clear that if (∗) 1 holds at x then so does (∗) 2 . The converse is not true. E.g. every warped product M 1 × _F M 2 , dim M 1 = 1, dim M 2 = n − 1 ≥ 3, of a manifold (M 1 , g) and a non-pseudosymmetric Einstein manifold (M 2 , e g) is a non- pseudosymmetric, Ricci-pseudosymmetric manifold (cf. [16, Remark 3.4] and [13, Theorem 4.1]).

Remark 2.1. In [10, Theorem 4] it was shown that (∗) 1 and (∗) 2 are equivalent on the subset U S of a 4-dimensional warped product M 1 × _F M 2 . In particular, (1) and (2) are equivalent on the subset U S of a 4- dimensional warped product M 1 × _F M 2 . We also note that there exist non- semisymmetric Einsteinian 4-dimensional warped products M 1 × _F M 2 , e.g.

the Schwarzschild spacetimes as well as the Kerr spacetimes. Moreover, the Schwarzschild spacetimes are pseudosymmetric manifolds.

For any X, Y ∈ Ξ(M ) we define the endomorphism e C(X, Y ) by C(X, Y ) = e e R(X, Y ) − 1

n − 2



X ∧ e SY + e SX ∧ Y − κ

n − 1 X ∧ Y

 . The Ricci operator e S and the Weyl conformal curvature tensor C of (M, g) are defined by

g( e SX, Y ) = S(X, Y ), C(X 1 , X 2 , X 3 , X 4 ) = g( e C(X ₁ , X 2 )X 3 , X 4 ).

Now we define the (0, 6)-tensor C · C by

(C · C)(X 1 , X 2 , X 3 , X 4 ; X, Y ) = ( e C(X, Y ) · C)(X ₁ , X 2 , X 3 , X 4 )

= −C( e C(X, Y )X ₁ , X 2 , X 3 , X 4 ) − . . . − C(X 1 , X 2 , X 3 , e C(X, Y )X ₄ ).

A semi-Riemannian manifold (M, g), n ≥ 4, is said to be a manifold with pseudosymmetric Weyl tensor ([11], [23], [30]) if

(∗) 3 the tensors C · C and Q(g, C) are linearly dependent

at every point of M . The manifold (M, g) has pseudosymmetric Weyl tensor

if and only if

(6) C · C = L C Q(g, C) on

U C = {x ∈ M | C 6= 0 at x},

for some function L C on U C . It is known that every warped product M 1 × _F M 2 , dim M 1 = dim M 2 = 2, satisfies (∗) 3 ([10, Theorem 2]). An example of a 4-dimensional Riemannian manifold satisfying (∗) 3 , which is not a warped product, was found in [23]. Manifolds satisfying (∗) 1 and (∗) 3 were investi- gated in [23].

For a symmetric (0, 2)-tensor A we define the endomorphism X ∧ A Y of Ξ(M ) by (X ∧ A Y )Z = A(Y, Z)X − A(X, Z)Y . Furthermore, for a (0, k)- tensor field T , k ≥ 1, and the tensor field A we define the tensor Q(A, T ) by

Q(A, T )(X 1 , . . . , X k ; X, Y ) = ((X ∧ A Y ) · T )(X 1 , . . . , X k )

= −T ((X ∧ A Y )X 1 , X 2 , . . . , X k ) − . . . − T (X 1 , . . . , X k−1 , (X ∧ A Y )X k ).

In particular, in this way we obtain the (0, 6)-tensor field Q(S, R).

Semi-Riemannian manifolds satisfying (∗), (∗) 1 , (∗) 2 or (∗) 3 are called manifolds of pseudosymmetry type ([11], [30]). We finish this section with some examples of Ricci-pseudosymmetric manifolds.

Example 2.1. It is known that the Cartan hypersurfaces in the sphere S ⁿ⁺¹ (c), n = 3, 6, 12 or 24, are compact, minimal hypersurfaces with con- stant principal curvatures −(3c) ^1/2 , 0, (3c) ^1/2 having the same multiplicity.

More precisely, the Cartan hypersurfaces are the tubes of constant radius over the standard Veronese embeddings i : FP ² → S ^3d+1 (c) → E ^3d+2 , d = 1, 2, 4, 8, of the projective plane FP ² in the sphere S ^3d+1 (c) in E ^3d+2 , where F = R (real numbers), C (complex numbers), Q (quaternions) or O (Cayley numbers), respectively. The Cartan hypersurfaces satisfy certain curvature condition of pseudosymmetry type. In [22, Theorem 1] it was shown that ev- ery Cartan hypersurface in S ⁿ⁺¹ (c), n = 6, 12, 24, is a non-pseudosymmetric, Ricci-pseudosymmetric manifold with non-pseudosymmetric Weyl tensor satisfying the relations

R · S = e κ

n(n + 1) Q(g, S), R · R − Q(S, R) = − (n − 2) e κ

n(n + 1) Q(g, C)

on M , where e κ is the scalar curvature of S ⁿ⁺¹ (c). The Cartan hypersurface in S ⁴ (c) is a pseudosymmetric manifold satisfying

R · R = e κ

12 Q(g, R).

3. Preliminary results. Let (M, g), n ≥ 3, be a semi-Riemannian man- ifold covered by a system of coordinate neighbourhoods {U ; x ^h }. We denote by g ij , R hijk , S ij and C hijk the local components of the tensors g, R, S and C, respectively. Further, we denote by S _ij ² = S ip S _j ^p and S _j ^k = g ^ks S js

the local components of the tensor S ² defined by S ² (X, Y ) = S( e SX, Y ), X, Y ∈ Ξ(M ), and of the Ricci operator e S, respectively.

Let U and S be the (0, 4)-tensor fields on (M, g) defined by

U (X 1 , X 2 , X 3 , X 4 ) = g(X 1 , X 4 )S(X 2 , X 3 ) − g(X 1 , X 3 )S(X 2 , X 4 ) (7)

+ g(X 2 , X 3 )S(X 1 , X 4 ) − g(X 2 , X 4 )S(X 1 , X 3 ), S(X 1 , X 2 , X 3 , X 4 ) = S(X 1 , X 4 )S(X 2 , X 3 ) − S(X 1 , X 3 )S(X 2 , X 4 ).

(8)

Lemma 3.1. The following identities hold on any semi-Riemannian man- ifold (M, g), n ≥ 3:

Q(g, U ) = −Q(S, G), (9)

Q(S, U ) = −Q(g, S), (10)

Q(g, C) = Q(g, R) + 1

n − 2 Q(S, G), (11)

Q(S, C) = Q(S, R) − 1

n − 2 Q(S, U ) + κ

(n − 1)(n − 2) Q(S, G), (12)

Q(S, C) = Q(S, R) + 1

n − 2 Q(g, S) + κ

(n − 1)(n − 2) Q(S, G).

(13)

P r o o f. The identities (9) and (10) are immediate consequences of the definitions of the tensors G, U and S. (11) was shown in [2, Remark 2.1].

Using the definition of the Weyl tensor C we easily get (12). Finally, putting (10) in (12) we obtain (13).

Lemma 3.2. Let (M, g), n ≥ 4, be a semi-Riemannian manifold satisfying (14) R( e S(X), Y, Z, W ) = τ R(X, Y, W, Z),

where τ is a function on M and X, Y, Z, W ∈ Ξ(M ). Then (15) C · C = R · R − 1

n − 2 Q(S, R) + 1 n − 2

 κ n − 1 − τ



Q(g, C).

P r o o f. First of all, it is easy to verify that (14) implies R · S = 0,

(16)

S ² = τ S.

(17)

Now, using (16), we get

(18) R · C = R · R.

Further, we can check that the following identities hold on any semi-Rieman- nian manifold:

(C · C) hijklm = (R · C) hijklm + 1 n − 2 Q

 κ

n − 1 g − S, C



hijklm

(19)

− 1

n − 2 (g hl S _m ^p C pijk − g hm S _l ^p C pijk

− g _il S _m ^p C phjk + g im S _l ^p C phjk + g jl S _m ^p C pkhi

− g _jm S _l ^p C pkhi − g _kl S _m ^p C pjhi + g km S _l ^p C pjhi ), S _m ^p C pijk = S _m ^p R pijk − 1

n − 2 (S mk S ij − S _mj S ik ) (20)

− 1

n − 2 (g ij S _mk ² − g _ik S _mj ² )

+ κ

(n − 1)(n − 2) (g ij S mk − g _ik S mj ).

The relation (20), by (14) and (17), turns into S _m ^p C pijk = τ R mijk − 1

n − 2 (S mk S ij − S _mj S ik ) (21)

− τ

n − 2 (g ij S mk − g _ik S mj )

+ κ

(n − 1)(n − 2) (g ij S mk − g ik S mj ).

Applying (18) and (21) in (19) we find C · C = R · R + κ

(n − 1)(n − 2) Q(g, C) − 1

n − 2 Q(S, C) (22)

− τ

n − 2 Q(g, R) + 1

(n − 2) ² Q(g, S)

+ 1

(n − 2) ²

 κ

n − 1 − τ



Q(S, G).

This, by making use of (13), gives C · C = R · R − 1

n − 2 Q(S, R) + κ

(n − 1)(n − 2) Q(g, C) (23)

− τ

n − 2 Q(g, R) − τ

(n − 2) ² Q(S, G), which, by (11), yields (15). Our lemma is thus proved.

Lemma 3.3. If (M, g), n ≥ 4, is a semi-Riemannian Ricci-pseudo- symmetric manifold then at any point x ∈ U S ⊂ M ,

(R · S) hijk = L S (g hj S ik + g ij S hk − g _hk S ij − g _ik S hj ),

(24)

S _h ^p R pijk + S _j ^p R pikh + S _k ^p R pihj = 0, (25)

(R · S ² ) hijk = L S (g hj S _ik ² + g ij S _hk ² − g hk S _ij ² − g ik S _hj ² ), (26)

S _ph ² R ^p _ijk + S _pj ² R ^p _ikh + S _pk ² R ^p _ihj = 0, (27)

A ij = S ^pq R pijq = S _ij ² − nL _S S ij + κL S g ij . (28)

Moreover , if (3) is satisfied at x then

(29) S _ij ² = λS ij + ((n − 1)L S − κ)L S g ij , λ ∈ R, holds at x.

P r o o f. (24) is an immediate consequence of (5). Summing cyclically (24) in h, j, k and using the identity

(30) (R · S) hijk = S _h ^p R pijk + S _i ^p R phjk , we obtain (25). We note that

(R · S ² ) hijk = S _h ^p (S _p ^a R aijk + S _i ^a R apjk ) + S _i ^p (S _p ^a R ahjk + S _h ^a R apjk ), whence

(R · S ² ) hijk = S _h ^p (R · S) pijk + S _i ^p (R · S) phjk .

Substituting here (24) we easily get (26). (27) follows from (26). Contracting (24) with g ^hk and using (30) we obtain (28). From (28) we get

(R · S) pqhk R ^p _ij ^q + S ^pq (R · R) pijqhk = (R · S ² ) ijhk − nL _S (R · S) ijhk . Substituting here (3), (5), (24), (26) and

Q(S, R) pijqhk = S ph R kijq + S ih R pkjq + S jh R pikq + S qh R pijk

(31)

− S _pk R hijq − S _ik R phjq − S _jk R pihq − S _qk R pijh , we obtain

L S (S kp R ^p _jih + S kp R ^p _ijh − S hp R ^p _jik − S hp R ^p _ijk )

− S _kp ² R ^p _jih − S _kp ² R ^p _ijh + S _hp ² R ^p _jik + S _hp ² R ^p _ijk + S ih A jk + S jh A ik − S _ik A jh − S _jk A ih

= L S (g ih S _jk ² + g jh S _ik ² − g _ik S _jh ² − g _jk S _ih ² )

− nL ² _S (g ih S jk + g jh S ik − g _ik S jh − g _jk S ih ).

Applying (25) and (27) we find

−L _S (R · S) ijhk + (R · S ² ) ijhk + S ih A jk + S jh A ik − S _ik A jh − S _jk A ih

= L S (g ih S _jk ² + g jh S _ik ² − g _ik S _jh ² − g _jk S _ih ² )

− nL ² _S (g ih S jk + g jh S ik − g ik S jh − g jk S ih ),

which, by making use of (24), (26) and (28), turns into

L S ((n − 1)L S − κ)(g _ih S jk + g jh S ik − g _ik S jh − g _jk S ih )

= S ik S _jh ² + S jk S _ih ² − S ih S _jk ² − S jh S _ik ² . The last equality can be rewritten as

S jh (S _ik ² − ((n − 1)L _S − κ)L _S g ik ) + S ih (S _jk ² − ((n − 1)L _S − κ)L _S g jk )

− S _jk (S _ih ² − ((n − 1)L _S − κ)L _S g ih )

− S ik (S _jh ² − ((n − 1)L S − κ)L S g jh ) = 0, or briefly

Q(S, S ² − ((n − 1)L _S − κ)L _S g) = 0,

from which, in view of Lemma 2.4(i) of [18], we get (29), completing the proof.

4. Ricci-semisymmetric manifolds with κ = 0

Proposition 4.1. If (M, g), n ≥ 4, is a semi-Riemannian Ricci-pseudo- symmetric manifold satisfying (3) then, at any point x ∈ U S ⊂ M ,

(nL S −κ)S _m ^p R pijk = κL S (g mk S ij −g _jm S ik ) + nL ² _S (g ij S km −g _ki S jm ) (32)

+ (κL S − tr(S ² ))R mijk − κL ² _S G mijk

+ nL S (S km S ij − S _jm S ik ).

P r o o f. From (3) we have (R · R) hijklq = Q(S, R) hijklq , i.e.

R pijk R ^p _hlq − R phjk R ^p _ilq + R pkhi R ^p _jlq − R pjhi R ^p _klq

= S hl R qijk + S il R hqjk + S jl R hiqk + S kl R hijq

− S _hq R lijk − S _iq R hljk − S _jq R hilk − S _kq R hijl . Transvecting this with S _m ^q we obtain

R pijk R ^p _hlq S _m ^q − R _phjk R ^p _ilq S _m ^q + R pkhi R ^p _jlq S _m ^q − R _pjhi R ^p _klq S _m ^q

= S hl S _m ^q R qijk + S il S _m ^q R hqjk + S jl S _m ^q R hiqk + S kl S _m ^q R hijq

− S _hm ² R lijk − S _im ² R hljk − S _jm ² R hilk − S _km ² R hijl . Symmetrizing in l, m and using (24) we get

L S (g hl S _m ^p R pijk + g hm S _l ^p R pijk − g il S _m ^p R phjk − g im S _l ^p R phjk

+ g jl S _m ^p R pkhi + g jm S _l ^p R pkhi − g _kl S _m ^p R pjhi − g _km S _l ^p R pjhi

− S hm R lijk − S hl R mijk + S im R lhjk + S il R mhjk

− S _jm R lkhi − S _jl R mkhi + S km R ljhi + S kl R mjhi )

= S jl S _m ^p R pkhi + S jm S _l ^p R pkhi − S _kl S _m ^p R pjhi − S _km S _l ^p R pjhi

− S _hm ² R lijk − S _hl ² R mijk − S _im ² R hljk − S _il ² R hmjk

− S _jm ² R hilk − S _jl ² R himk − S _km ² R hijl − S _kl ² R hijm .

Further, contracting with g ^lh and applying (25), (24) and (27), we find (33) L S (nS _m ^p R pijk − g _mj A ik + g mk A ij − κR _mijk

+ S jm S ik − S _km S ij − g _jm S ik − g _ij S mk + g mk S ij + g ik S mj )

= κS _m ^p R pijk + L S (S jm S ik + g ij S _mk ² − S _mk S ij − g _ik S _mj ² )

− S _jm A ik + S km A ij + S ik S _jm ² − S _ij S _km ²

− tr(S ² )R mijk − S _ip ² R ^p _mjk − S _mp ² R ^p _ijk . We note that (28), by (29), turns into

(34) A ij = (λ − nL S )S ij + (n − 1)L ² _S g ij .

Substituting (29) and (34) into (33) we get (30), completing the proof.

Proposition 4.2. Let (M, g), n ≥ 4, be a semi-Riemannian Ricci- pseudosymmetric manifold. If

(35) βR = nL ² _S U − κL ² _S G + nL S S, β ∈ R, at a point x ∈ U S ⊂ M , then, at x,

(36) β(R · R − L S Q(g, R)) = 0.

P r o o f. First of all we note that (35) implies (37) βR · R = nL ² _S R · U + nL S R · S.

We now prove that, at x,

R · U = L S Q(g, U ), (38)

R · S = L S Q(g, S).

(39) We have

(R · U ) hijklm = g ij (R · S) hklm − g _hj (R · S) iklm

+ g hk (R · S) ijlm − g ik (R · S) jhlm

= L S (g ij (g hl S km + g kl S hm − g _hm S kl − g _km S hl )

− g _hj (g il S km + g kl S im − g _im S kl − g _km S il ) + g hk (g il S jm + g jl S im − g _im S jl − g _jm S il )

− g _ki (g hl S jm + g jl S hm − g _hm S jl − g _jm S hl ))

= − L S Q(S, G) hijklm , or, briefly,

(40) R · U = −L S Q(S, G).

Applying (9) in (40) we get (38). Furthermore, we have (R · S) hijklm = S ij (R · S) hklm − S hj (R · S) iklm

+ S hk (R · S) ijlm − S _ik (R · S) jhlm

= L S (S ij (g hl S km + g kl S hm − g _hm S kl − g _km S hl )

− S _hj (g il S km + g kl S im − g _im S kl − g _km S il ) + S hk (g il S jm + g jl S im − g _im S jl − g _jm S il )

− S _ki (g hl S jm + g jl S hm − g _hm S jl − g _jm S hl ))

= L S Q(g, S) hijklm ,

i.e. (39) holds at x. Applying now (38) and (39) in (37) we get βR · R = nL ² _S (L S Q(g, U ) + Q(g, S)).

But on the other hand, from (35) we also obtain

βQ(g, R) = nL ² _S Q(g, U ) + nL S Q(g, S).

The last two relations complete the proof of our proposition.

Theorem 4.1. Let (M, g), n ≥ 4, be a semi-Riemannian manifold and let x ∈ U S ⊂ M .

(i) If (M, g) is a Ricci-pseudosymmetric manifold satisfying (3) then (32) is satisfied on U S .

(ii) If the conditions: (32) and nL S − κ 6= 0 are satisfied at x then R · S = L S Q(g, S) at x.

(iii) If the conditions: (32) and nL S − κ = 0 are satisfied at x then, at x,



tr(S ² ) − κ ² n



(R · R − L S Q(g, R)) = 0.

P r o o f. The proof of (i) is presented in Lemma 3.2. Next, (32), by sym- metrization in m, i, implies (5). Finally, (iii) is a consequence of Proposi- tion 4.2.

As an immediate consequence of Theorem 4.1(iii) we have the following.

Theorem 4.2. Let (M, g), n ≥ 4, be a Riemannian Ricci-semisymmetric manifold satisfying (3). If the scalar curvature κ of the manifold (M, g) vanishes on U S ⊂ M then R · R = 0 on U _S .

5. Ricci-semisymmetric manifolds with κ 6= 0. Let (M, g),

dim M ≥ 3, be a semi-Riemannian manifold. We denote by U κ the set of

all points of M at which the scalar curvature κ of (M, g) is non-zero. We

note that if (M, g), dim M ≥ 4, is a Ricci-semisymmetric manifold satisfying

(3) then, in view of Lemma 3.3 and Proposition 4.1, (29) and (32) take on

U S ∩ U _κ the forms

S ² = τ S, (41)

R( e S(X), Y, Z, W ) = τ R(X, Y, Z, W ), (42)

respectively, where X, Y, Z, W ∈ Ξ(M ) and

(43) τ = tr(S ² )

κ . Now Lemma 3.2 yields immediately the following.

Proposition 5.1. If (M, g), n ≥ 4, is a semi-Riemannian Ricci-semi- symmetric manifold satisfying (3) then

(44) C · C = n − 3

n − 2 R · R + 1 n − 2

 κ

n − 1 − τ



Q(g, C) on U S ∩ U _κ .

Theorem 5.1. Let (M, g), n ≥ 4, be a semi-Riemannian Ricci-semi- symmetric manifold satisfying (3). If (M, g) has pseudosymmetric Weyl tensor then R · R = 0 on U S ∩ U _κ .

P r o o f. Let x ∈ M . If x ∈ M − U C then our assertion follows from Remark 1.1. Let x ∈ U S ∩ U κ ∩ U C . Thus (44), by (6), turns into

(45) R · R = µQ(g, C),

which, by (2), gives

(46) R · C = µQ(g, C),

where

µ = L C − 1 n − 2

 κ n − 1 − τ

 . We consider two cases.

I. First we assume that dim M ≥ 5. Now (46), in view of [14, Theorem 1], implies R · R = µQ(g, R), whence

(47) R · S = µQ(g, S).

But this, by (2), yields µQ(g, S) = 0. Since x ∈ U S , the last relation gives µ = 0, which reduces (47) to (1).

II. We now assume that dim M = 4. It is well known that the follow- ing identity is satisfied for every 4-dimensional semi-Riemannian manifold (M, g) ([25]):

0 = g hm C lijk + g lm C ihjk + g im C hljk + g hj C likm + g lj C ihkm

(48)

+ g ij C hlkm + g hk C limj + g lk C ihmj + g ik C hlmj .

From this we get immediately

0 = g hm (R · C) lijkab + g lm (R · C) ihjkab + g im (R · C) hljkab

(49)

+ g hj (R · C) likmab + g lj (R · C) ihkmab + g ij (R · C) hlkmab

+ g hk (R · C) limjab + g lk (R · C) ihmjab + g ik (R · C) hlmjab , which, in virtue of (46), turns into

0 = µ(g hm Q(g, C) lijkab + g lm Q(g, C) ihjkab + g im Q(g, C) hljkab

(50)

+ g hj Q(g, C) likmab + g lj Q(g, C) ihkmab + g ij Q(g, C) hlkmab

+ g hk Q(g, C) limjab + g lk Q(g, C) ihmjab + g ik Q(g, C) hlmjab ).

But on the other hand, (45), by (3), gives

(51) Q(S, R) = µQ(g, C).

Applying (51) in (50) we obtain

0 = g hm Q(S, R) lijkab + g lm Q(S, R) ihjkab + g im Q(S, R) hljkab

(52)

+ g hj Q(S, R) likmab + g lj Q(S, R) ihkmab + g ij Q(S, R) hlkmab

+ g hk Q(S, R) limjab + g lk Q(S, R) ihmjab + g ik Q(S, R) hlmjab . Further, using the definition of the tensor Q(S, R) and (41) and (42), we can easily check the following identities on U S ∩ U _κ :

S _c ^p Q(S, R) pijkab = τ Q(S, R) cijkab , (53)

S _c ^p Q(S, R) hijkap = τ Q(S, R) hijkac . (54)

Transvecting now (52) with S _p ^h and using (53) and (54) we get 0 = S pm Q(S, R) lijkab + S pj Q(S, R) likmab + S pk Q(S, R) limjab

(55)

+ τ (g lm Q(S, R) ihjkab + g im Q(S, R) hljkab + g lj Q(S, R) ihkmab

+ g ij Q(S, R) hlkmab + g lk Q(S, R) ihmjab + g ik Q(S, R) hlmjab ).

Subtracting (52) we find

(S pm − τ g _pm )Q(S, R) lijkab + (S pj − τ g _pj )Q(S, R) likmab

+ (S pk − τ g _pk )Q(S, R) limjab = 0.

From this, by transvection with S _c ^m and making use of (41) and (53), we obtain

τ ((S pj − τ g _pj )Q(S, R) likcab + (S pk − τ g _pk )Q(S, R) licjab ) = 0.

Further, transvecting this with S _d ^j and using again (41) and (53), we find τ ² (S pk − τ g _pk )Q(S, R) licdab = 0,

or, briefly,

τ ² (S − τ g)Q(S, R) = 0.

Since x ∈ U S , this reduces to

(56) τ Q(S, R) = 0.

If τ is non-zero at a point x ∈ U S ∩ U _κ then (56) implies Q(S, R) = 0 and by (3) we get (1). If τ = 0 at x then (41), (42), (53) and (54) reduce to

S _ij ² = 0, (57)

S _m ^p R pijk = 0, (58)

S _a ^p Q(S, R) pijklm = 0, (59)

S _a ^p Q(S, R) hijkpm = 0, (60)

respectively. Further, from (51) we have

(61) Q(S, R) hijklm = µQ(g, C) hijklm . Transvecting this with S _c ^l and using (60) we get (62) µS _a ^p Q(g, C) hijkpm = 0.

If µ vanishes at x then (61) reduces to Q(S, R) = 0. Thus (3) implies again (1). If µ is non-zero at x then (62) yields

S _a ^p Q(g, C) hijkpm = 0, whence we obtain

0 = S ah C mijk − S ai C mhjk + S aj C mkhi − S ak C mjhi

− g mh S _a ^l C lijk + g mi S _a ^l C lhjk − g mj S _a ^l C lkhi + g mk S _a ^l C ljhi . Contracting this with g ^ah we find

0 = κC mijk + S _i ^p C pmjk + S _j ^p C pimk − S _k ^p C pimj

(63)

− S _m ^l C lijk + g mj S ^pq C pkiq − g mk S ^pq C pjiq . Since τ = 0, (21) reduces to

T mijk = S _m ^p C pijk = − 1

n − 2 (S mk S ij − S mj S ik ) (64)

+ κ

(n − 1)(n − 2) (g ij S mk − g _ik S mj ).

Contracting this with g ^mk and using (57) we obtain (65) P ij = S ^pq C pijq = − nκ

(n − 1)(n − 2) S ij + κ ²

(n − 1)(n − 2) g ij .

Let T and P be the tensors with local components T mijk and P ij defined

by (64) and (65), respectively. From (64) and (65), in virtue of (2), we obtain

R · T = 0 and R · P = 0, respectively. Applying the last two relations in (63)

we find κR · C = 0, and by the assumption that κ 6= 0, we have R · C = 0,

which reduces (46) to µQ(g, C) = 0. Evidently, if µ 6= 0 then Q(g, C) = 0,

which, in view of Lemma 1.1 of [6], implies C = 0, a contradiction. Thus we have µ = 0. Now (45) completes the proof.

Finally, using Theorems 4.2 and 5.1 we get our main result.

Theorem 5.2. Let (M, g), n ≥ 4, be a Riemannian Ricci-semisymmetric manifold satisfying (3). If (M, g) is a manifold with pseudosymmetric Weyl tensor then R · R = 0 on U S ⊂ M .

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Kadri Arslan and Ridvan Ezentas. Ryszard Deszcz

Department of Mathematics Department of Mathematics

Uludaˇ g University Agricultural University of Wroc law

Gor¨ ukle Kamp¨ us¨ u, Bursa, Turkey Grunwaldzka 53

E-mail: arslan@uu20.bim.uludag.edu.tr 50-357 Wroc law, Poland E-mail: rysz@ozi.ar.wroc.pl Yunus C . elik

Department of Mathematics Dumlupinar University K¨ utahya, Turkey

Received 21 July 1997;

revised 24 October 1997