PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 67(81) (2000), 103–111

SEMISYMMETRY AND RICCI-SEMISYMMETRY FOR HYPERSURFACES OF SEMI-EUCLIDEAN SPACES

Marta Da̧browska, Filip Defever, Ryszard Deszcz, and Dorotea Kowalczyk

Communicated by Mileva Prvanović

Abstract. In the context of P.J. Ryan’s problem on the equivalence of the conditions R · R = 0 and R · S = 0 for hypersurfaces, we prove that there is indeed equivalence for hypersurfaces of semi-Euclidean spaces in any dimension, under an additional curvature condition of semisymmetric type.

1. Introduction

A semi-Riemannian manifold (M, g), dim M ≥ 3, is called semisymmetric [13] if

R · R = 0 , (1)

holds on M. It is well known that the class of semisymmetric manifolds includes the set of locally symmetric manifolds (∇R = 0) as a proper subset. A semi-Riemannian manifold (M, g), dim M ≥ 3, is said to be Ricci-semisymmetric, if the following condition is satisfied

R · S = 0 . (2)

Again, the class of Ricci-semisymmetric manifolds includes the set of Ricci-sym-metric manifolds (∇S = 0) as a proper subset. It is clear that every semisymmetric manifold is Ricci-semisymmetric. The converse statement is however not true, as can be seen for instance from the material in [6].

Although the conditions (1) and (2) do not coincide for manifolds in general, it is a long standing question whether the conditions R · R = 0 and R · S = 0 are equivalent for hypersurfaces of Euclidean spaces; cf. Problem P808 of Ryan [11] and references therein. Whereas for n = 3 this equivalence follows immediately, for n > 3 we have the following results. It had been proved in [12] that (1) and (2) are equivalent for hypersurfaces which have positive scalar curvature in a Euclidean space 𝔼^{n+1}, n > 3. In [10] this result was generalized to hypersurfaces of a Euclidean space 𝔼^{n+1}, n > 3, which have nonnegative scalar curvature and also to hypersurfaces of constant scalar curvature. [10] also proves that (1) and (2) coincide for hypersurfaces of Riemannian space forms with nonzero constant sectional curvature. Further, in [9] it was proved that (1) and (2) are equivalent for hypersurfaces of a Euclidean space 𝔼^{n+1}, n > 3, under the additional global condition of completeness. In [2], it has been shown that the conditions (1) and (2) are equivalent for hypersurfaces of the Euclidean space 𝔼^{5}. In [1] a negative answer to the above mentioned question was given for hypersurfaces of a Euclidean space 𝔼^{n+1}, n ≥ 5. Indeed, [1] gives an example of a hypersurface M^{5} of 𝔼^{6} which satisfies R · S = 0, but which is not semisymmetric; this proves that both concepts are not equivalent for hypersurfaces of Euclidean spaces in general.

1991 Mathematics Subject Classification. Primary 53B20, 53B30, 53B50.

The second author is a postdoctoral researcher F.W.O.-Vlaanderen, Belgium.

Although the fundamental question has now been solved, a number of new questions can be raised. Indeed, one may e.g. ask for a classification of the Riccisemisymmetric hypersurfaces of the Euclidean spaces which are not semisymmetric. One can also consider the more general problem, whether (1) and (2) are equivalent for hypersurfaces of a semi-Riemannian space form N^{n+1}(c). For example, [3] proves that there is indeed equivalence for all hypersurfaces of a 5-dimensional semi-Riemannian space form, thus generalizing the result of [2]; in [4] it was shown that (1) and (2) are equivalent for Lorentzian hypersurfaces of a Minkowski space 𝔼_{1}^{n+1}, n ≥ 4. [4] also proves that (1) and (2) are equivalent for para-Kähler hypersurfaces of a semi-Euclidean space 𝔼_{s}^{2m+1}, m ≥ 2.

In order to tackle such questions, it is necessary to pursue more insight into the differences and look for an improved description and characterisation of the similarities of such hypersurfaces; one possibility for doing so is searching for sufficient conditions on hypersurfaces for both concepts (1) and (2) to be equivalent; at the same time, this narrows down the set of hypersurfaces where differences can occur. In this respect, [5] proved that (1) and (2) are equivalent for hypersurfaces of a semi-Euclidean space 𝔼_{s}^{n+1} which satisfy the curvature condition of pseudosymmetric type C · C = LQ(g, C). In the present paper, we prove a similar result w.r.t. a supplementary condition of semisymmetric type; more precisely:

Theorem 1.1. For hypersurfaces of a semi-Euclidean space 𝔼_{s}^{n+1}, n ≥ 4, which satisfy the curvature condition C·R = 0, the conditions of semisymmetry and Riccisemisymmetry are equivalent.

2. Preliminaries

Let (M, g), n = dim M ≥ 3, be a connected semi-Riemannian manifold of class C^{∞} and let ∇ be its Levi-Civita connection. We define on M the endomorphisms X ∧_{A} Y, R(X, Y) and C(X, Y) by

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

C(X, Y) = R(X, Y) − \frac{1}{n − 2}(X ∧_{g} SY + SX ∧_{g} Y − \frac{κ}{n − 1}X ∧_{g} Y ) ,

where the Ricci operator S is defined by S(X, Y) = g(X, SY), S is the Ricci tensor, κ the scalar curvature, A a symmetric (0, 2)-tensor and X, Y, Z ∈ Ξ(M), Ξ(M) being the Lie algebra of vector fields of M. Next, we define the tensor G, the Riemann-Christoffel curvature tensor R and the Weyl conformal curvature tensor C of (M, g) by

G(X_{1}, X_{2}, X_{3}, X_{4}) = g((X_{1} ∧_{g} X_{2})X_{3}, X_{4}) ,
R(X_{1}, X_{2}, X_{3}, X_{4}) = g(R(X_{1}, X_{2})X_{3}, X_{4}) ,
C(X_{1}, X_{2}, X_{3}, X_{4}) = g(C(X_{1}, X_{2})X_{3}, X_{4}) .

For a (0, k)-tensor T, k ≥ 1, and a symmetric (0, 2)-tensor A, we define the (0, k+2)- tensors R · T and Q(A, T) by

(R · T)(X_{1}, …, X_{k}; X, Y) = (R(X, Y) · T)(X_{1}, …, X_{k})

= −T (R(X, Y)X_{1}, X_{2}, …, X_{k}) − ⋯ − T (X_{1}, …, X_{k−1}, R(X, Y)X_{k}) ,

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}) .

Putting in the above formulas T = R, T = S, T = C or T = G and A = g or A = S, we obtain the tensors R · R, R · S, R · C, Q(g, R), Q(g, C), Q(S, R), and Q(S, C) respectively. The tensors C · R and C · C we define in the same way as the tensor R · R; the tensor C · S is defined in the same way as the tensor R · S. The (0, 2)-tensor S^{2} is defined by S^{2}(X, Y) = S(SX, Y), X, Y ∈ Ξ(M).

A semi-Riemannian manifold (M, g), n ≥ 3, is said to be semisymmetric [13] if R · R = 0 holds on M. Curvature conditions involving tensors of the form R · T only are called curvature conditions of semisymmetric type; other examples are e.g. the Ricci-semisymmetric space (R · S = 0).

Manifolds satisfying curvature conditions involving tensors of both the form R · T and Q(A, T) are called manifolds of pseudosymmetric type.

For example, we have semi-Riemannian manifolds (M, g), n ≥ 4, satisfying at every point the following condition

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

the condition (∗) is satisfied on a manifold (M, g) if and only if

C · R = L Q(g, C) (3)

holds on U_{C} = {x ∈ M | C ≠ 0 at x}, where L is a function on U_{C}. Other examples are the manifolds with pseudosymmetric Weyl tensor (C ·C = LQ(g, C)), and the Ricci-generalized pseudosymmetric manifolds (R · R = Q(S, R)). For more information on the geometric motivation for the introduction of the concept of pseudosymmetry and a survey of various properties, including also applications to the general theory of relativity, we refer to the papers [6] and [14].

3. Proof of the results

The proof of Theorem 3.1 follows from results established in Proposition 3.1 and Proposition 3.2 which we prove first. Whereas Theorem 3.1 applies to hypersurfaces of a semi-Euclidean space 𝔼_{s}^{n+1}, Proposition 3.1 and Proposition 3.2 are more generally valid for semi-Riemannian manifolds subjected to suitable additional conditions.

Proposition 3.1. Let (M, g), n ≥ 4, is a semi-Riemannian manifold satisfying C · R = L Q(g, C) on U_{C}, then the following relation is satisfied on U_{C}:

C · C = L Q(g, C) . (4)

Moreover, on the set U_{C}, we also have that

(5)  C · S = 0 ,
(6)  R · S = \frac{1}{n − 2} Q(g, D) ,

where

D = S^{2} − \frac{κ}{n − 1} S . (7)

Proof. The local components of the (0, 6)-tensor C · R are given by

(C · R)_{hijklm} = g^{pq}(R_{pijk}C_{qhlm} + R_{hpjk}C_{qilm} + R_{hipk}C_{qjlm} + R_{hijp}C_{qklm}) . (8)

Contracting (8) with g^{ij} we get

g^{ij}(C · R)_{hijklm} = (C · S)_{hklm} . (9)

Recall now that the local components of the (0, 6)-tensor Q(g, C) are given by

Q(g, C)_{hijklm} = g_{hl}C_{mijk} + g_{il}C_{hmjk} + g_{jl}C_{himk} + g_{kl}C_{hijm}
(10)  −g_{hm}C_{lijk} − g_{im}C_{hljk} − g_{jm}C_{hilk} − g_{km}C_{hijl} .

Next, contracting the relation

(C · R)_{hijklm} = L Q(g, C)_{hijklm} (11)

with g^{ij} and using (9) and the identity g^{ij}Q(g, C)_{hijklm} = 0 we get (5). Substituting the expression for the components of the Weyl conformal curvature tensor

C_{hijk} = R_{hijk} − \frac{1}{n − 2} (g_{hk}S_{ij} + g_{ij}S_{hk} − g_{hj}S_{ik} − g_{ik}S_{hj})

− \frac{κ}{(n − 1)(n − 2)} (g_{hk}g_{ij} − g_{hj}g_{ik}) (12)

into (5) gives (6). Further, we note that the following identity holds on M

(C · C)_{hijklm} = (C · R)_{hkijlm}

− \frac{1}{n − 2} (g_{ij}(C · S)_{hklm} − g_{ik}(C · S)_{hjlm} + g_{hk}(C · S)_{ijlm} − g_{hj}(C · S)_{iklm}) ,

where

(13) (C · R)_{hijklm} =

(R · R)_{hijklm} − \frac{1}{n − 2} Q(S, R)_{hijklm} + \frac{κ}{(n − 1)(n − 2)} Q(g, R)_{hijklm}

− \frac{1}{n − 2} (g_{hl}A_{mijk} − g_{hm}A_{lijk} − g_{il}A_{mhjk} + g_{im}A_{lhjk}

+ g_{jl}A_{mkhi} − g_{jm}A_{lkhi} − g_{kl}A_{mjhi} + g_{km}A_{ljhi}) ,

A_{mijk} = S_{m}^{p}R_{pijk} . (14)

and S_{m}^{p} = g^{rp}S_{mr}. Applying in this (3) and (5), we obtain (4). This finishes the proof of Proposition 3.1. □

Before proceeding, we derive a number of useful formulas which will find application in the next propositions; we organize them into the following lemma.

Lemma 3.1. For a semi-Riemannian manifold (M, g), n ≥ 4, satisfying C ·R = L Q(g, C) on U_{C}, the following relations hold on U_{C}:

A_{hijk} + A_{ihjk} = \frac{1}{n − 2} (g_{hj}D_{ik} + g_{ij}D_{hk} − g_{hk}D_{ij} − g_{ik}D_{hj}) , (15)

g^{hm}Q(S, R)_{hijklm} = −A_{iljk} − κ R_{lijk} + S_{kl}S_{ij} − S_{jl}S_{ik} , (16)

g^{hm}Q(g, C)_{hijklm} = −(n − 1) C_{lijk} , (17)

B_{ij} = S^{rs}R_{rijs} = −\frac{1}{n − 2} (S_{ij}^{2} − κ S_{ij}) . (18)

where D_{ij} are the local components of the (0, 2)-tensor D, defined by (7).

Proof. From (6), by (14), we get (15). Summing (15) cyclically in h, j, k we obtain

A_{hijk} + A_{jikh} + A_{kijh} = 0 . (19)

Contracting now Q(S, R)_{hijklm} and Q(g, C)_{hijklm} with g^{hm} and applying (19) we obtain (16) and (17), respectively. Furthermore, contracting (15) with g^{hk} and using (14) and S^{rs} = g^{rp}S_{p}^{s}, we get (18). This finishes the proof of Lemma 3.1. □

Proposition 3.2. Let (M, g), n ≥ 4, be a semi-Riemannian manifold satisfying on the set U_{C} the conditions C · R = L Q(g, C) and

R · R − Q(S, R) = L_{2} Q(g, C) , (20)

where L and L_{2} are functions on U_{C}, then the following relation holds on U_{C}:

R · S = 0 . (21)

Moreover, on the set U_{C}, we also have that

(a) S^{2} = \frac{κ}{n − 1} S , (b) tr (S^{2}) = \frac{κ^{2}}{n − 1} . (22)

Proof. Applying in (13) the relations (3) and (20) we obtain

−\frac{n − 3}{n − 2} Q(S, R)_{hijklm} = \frac{κ}{(n − 1)(n − 2)} Q(g, R)_{hijklm} + (L_{2} − L) Q(g, C)_{hijklm}

− \frac{1}{n − 2}(g_{hl}A_{mijk} − g_{hm}A_{lijk} − g_{il}A_{mhjk} + g_{im}A_{lhjk}

+ g_{jl}A_{mkhi} − g_{jm}A_{lkhi} − g_{kl}A_{mjhi} + g_{km}A_{ljhi}) . (23)

Contracting (23) with g^{hm} and using (16) and (17) we find

(n − 3) A_{iljk} − (n − 1) A_{lijk} = −(n − 2) κ R_{lijk} − (n − 1)(n − 2) (L_{2} − L) C_{lijk}
(24)  + (n − 3) (S_{lk}S_{ij} − S_{jl}S_{ik}) + \frac{κ}{n − 1}(g_{lk}S_{ij} − g_{jl}S_{ik}) + g_{jl}B_{ik} − g_{kl}B_{ij} ,

whence

(n − 3) (A_{iljk} + A_{lijk}) − 2(n − 2) A_{lijk} = −(n − 2)κ R_{lijk}
− (n − 2)(n − 1)(L_{2} − L) C_{lijk} + (n − 3) (S_{lk}S_{ij} − S_{jl}S_{ik})

+ \frac{κ}{n − 1}(g_{lk}S_{ij} − g_{jl}S_{ik}) + g_{jl}B_{ik} − g_{hl}B_{ij} . (25)

This, in view of (15), turns into

−2(n − 2) A_{lijk} = −\frac{n − 3}{n − 2} Q(g, D)_{lijk} − (n − 2)κ R_{lijk}
− (n − 2)(n − 1)(L_{2} − L) C_{lijk} + (n − 3) (S_{lk}S_{ij} − S_{jl}S_{ik})

+ \frac{κ}{n − 1} (g_{lk}S_{ij} − g_{jl}S_{ik}) + g_{jl}B_{ik} − g_{hl}B_{ij} . (26)

From this, after symmetrization in l, i, it follows that

−\frac{2}{n − 2} Q(g, D) = −\frac{κ}{n − 1} Q(g, S) + Q(g, B) ,

which, by making use of (7) and (18) reduces to Q(g, D) = 0. Now (15) reduces to (21). Further, from (21) we get S^{rs}R_{rijs} = S_{ij}^{2}. Comparing this with (18) we get (22)(a), and consequently also (22)(b). This finishes the proof of Proposition 3.2. □

Theorem 3.1. Let M be a Ricci-semisymmetric hypersurface of a semi-Euclid-ean space 𝔼_{s}^{n+1}, n ≥ 4. If C · R = LQ(g, C) is satisfied on M then M is a semisymmetric manifold.

Proof. It is well known that every hypersurface M of a semi-Riemannian space of constant curvature N^{n+1}(c), n ≥ 4, fulfils a particular curvature condition of pseudosymmetric type [8]. More precisely,

R · R − Q(S, R) = −\frac{n − 2}{n(n + 1)} κ̃ Q(g, C) (27)

holds on M, where κ̃ is the scalar of the ambient space. When the ambient space is a semi-Euclidean space 𝔼_{s}^{n+1}, n ≥ 4, then the scalar curvature κ̃ = 0 and the hypersurface M fulfils

R · R = Q(S, R) . (28)

First, we deal now with the question on the subset U_{C} where C ≠ 0. Applying Proposition 3.1 learns that the condition C · C = LQ(g, C) holds on U_{C}. In view of (28), the assumptions of Proposition 3.2 are also satisfied; hence U_{C} is a Ricci-semisymmetric manifold. Following Theorem 4.1 of [5] a Ricci-semisymmetric hypersurface of a semi-Euclidean space which satisfies C · C = LQ(g, C) is in fact semisymmetric; this establishes the result on the set U_{C}.

Next, we can remove the restriction C ≠ 0. Indeed, it is well known that on every semi-Riemannian manifold (M, g), n ≥ 4, the conditions: R · R = 0 and R · S = 0 are equivalent on the set where the Weyl conformal curvature vanishes. This finishes the proof of Theorem 3.1. □

We now strengthen Theorem 3.1 by proving that the function L necessarilly has to vanish in the given circumstances. For the technicalities of the next proposition, we work on the set U = {x ∈ M | C ≠ 0 and S ≠ 0}; it will work out that this will not cause obstructions for the conclusion.

Proposition 3.3. Let (M, g), n ≥ 4, be a hypersurface of a semi-Euclidean space 𝔼_{s}^{n+1} satisfying C · R = L Q(g, C) on U_{C}, then the function L vanishes on U.

Proof. Consider a point x ∈ U where the function L is nonzero. We note that (3) and (5) can be presented in the following form

(C · R)_{hijklm} = L Q(g, C)_{hijklm} , (29)

S_{h}^{p}C_{pijk} + S_{i}^{p}C_{phjk} = 0 , (30)

respectively. From (29) we get

(C · R)_{hijklp}S_{m}^{p} + (C · R)_{hijkmp}S_{l}^{p} = L (Q(g, C)_{hijklp}S_{m}^{p} + Q(g, C)_{hijkmp}S_{l}^{p}) ,

which by making use of (8) and (30), reduces to

Q(g, C)_{hijklp}S_{m}^{p} + Q(g, C)_{hijkmp}S_{l}^{p} = 0 .

From this, by a application of (5) and (10), we get

S_{hl}C_{mijk} + S_{il}C_{hmjk} + S_{jl}C_{himk} + S_{kl}C_{hijm}
+ S_{hm}C_{lijk} + S_{im}C_{hljk} + S_{jm}C_{hilk} + S_{km}C_{hijl}
− g_{hm}S_{l}^{p}C_{pijk} + g_{im}S_{l}^{p}C_{phjk} − g_{jm}S_{l}^{p}C_{pkhi} + g_{km}S_{l}^{p}C_{pjhi}

− g_{hl}S_{m}^{p}C_{pijk} + g_{il}S_{m}^{p}C_{phjk} − g_{jl}S_{m}^{p}C_{pkhi} + g_{kl}S_{m}^{p}C_{pjhi} = 0 . (31)

Further, from (30) it follows that

S^{pq}C_{hpqk} = 0 and S_{h}^{p}C_{pijk} + S_{j}^{p}C_{pikh} + S_{k}^{p}C_{pihj} = 0 . (32)

Furthermore, using (12) and (22), we get

S_{h}^{p}C_{pijk} = S_{h}^{p}R_{pijk} − \frac{1}{n − 2} (S_{hk}S_{ij} − S_{hj}S_{ik}) . (33)

Contracting now (31) with g^{lh} and applying (30) and (32) we find

S_{m}^{p}C_{pijk} = \frac{κ}{n} C_{mijk} , (34)

which, by transvection with S_{h}^{m} and making use of (22), yields

κ S_{h}^{p}C_{pijk} = 0 . (35)

From (34) and (35) it follows that

κ = 0 , (36)

holds at x. Now (33) reduces to

S_{h}^{p}R_{pijk} = \frac{1}{n − 2} (S_{hk}S_{ij} − S_{hj}S_{ik}) . (37)

Using (18) and (22), and in view of (36), we get

−B_{ij} = \frac{1}{n − 2}S_{ij}^{2} = \frac{tr(S^{2})}{n(n − 2)}g_{ij} = 0 .

Since L_{2} = 0, and in view of (36), (26) therefore reduces to

A_{lijk} = −\frac{n − 1}{2}LC_{lijk} − \frac{n − 3}{2(n − 2)}(S_{ij}S_{lk} − S_{lj}S_{ik}) . (38)

Next, comparing the right sides of (37) and (38) we obtain

S_{hk}S_{ij} − S_{hj}S_{ik} = −(n − 2) L C_{hijk} . (39)

From this we obtain

S_{ij}(R · S)_{hklm} + S_{hk}(R · S)_{ijlm} − S_{ik}(R · S)_{hjlm} − S_{hj}(R · S)_{iklm}

= −(n − 2) L (R · C)_{hijklm} ,

which, by making use of (4) and (21), implies L^{2}Q(g, C) = 0. Since Q(g, C) is nonzero at x, the last equality implies that L = 0. This finishes the proof of Proposition 3.3. □

Since at points where C = 0, R · S = 0 is always equivalent to R · R = 0, and since at points where S = 0, C · R = 0 implies R · R = 0, Theorem 3.1 together with Proposition 3.3 give

Theorem 3.2. Let M be a Ricci-semisymmetric hypersurface of a semi-Euclid-ean space 𝔼_{s}^{n+1}, n ≥ 4. If C · R = 0 is satisfied on M then M is a semisymmetric manifold.

Since semisymmetry always implies Ricci-semisymmetry, this leads to Theorem 1.1 as formulated in the Introduction.

Finally, we present an example of a semisymmetric hypersurface satisfying C · R = 0.

Example 3.1. Let (M, g) be a semi-Riemannian manifold defined in Example 4.1 of [7]. This manifold satisfies the following conditions: rank S = 1, κ = 0, S^{2} = 0, R · R = 0 and C(SX_{1}, X_{2}, X_{3}, X_{4}) = 0, for any vector fields X_{1}, …, X_{4} on M. From these relations it follows immediately R(SX_{1}, X_{2}, X_{3}, X_{4}) = 0, i.e. the tensor A with the local components A_{hijk}, defined by (14), is a zero tensor. Further, the manifold (M, g) can be realized as a hypersurface in a semi-Euclidean space ([7], Example 5.1). Thus we have on M: R · R = Q(S, R) = 0. Now we see that (13) reduces on M to C · R = 0.

References

[1] F. Defever, Solution to a problem of P.J. Ryan, preprint.

[2] F. Defever, R. Deszcz, Z. S.entürk, L. Verstraelen, and S.. Yaprak, On a problem of P.J. Ryan, Kyungpook Math. J. 37 (1997), 371–376.

[3] F. Defever, R. Deszcz, Z. S.entürk, L. Verstraelen, and S.. Yaprak, P.J. Ryan’s problem in semi-Riemannian space forms, Glasgow Math. J. 41 (1999), 271–281.

[4] F. Defever, R. Deszcz, and L. Verstraelen, On semisymmetric para-Kähler manifolds, Acta Math. Hung. 74 (1997), 7–17.

[5] F. Defever, R. Deszcz, L. Verstraelen, and S.. Yaprak, On the equivalence of semisymmetry and Ricci-semisymmetry for hypersurfaces, Indiana Math. J., in print.

[6] R. Deszcz, On pseudosymmetric spaces, Bull. Soc. Belg. Math. A44 (1992), 1–34.

[7] R. Deszcz, M. Głogowska, M. Hotloś, and Z. S.entürk, On certain quasi-Einstein semisymmetric hypersurfaces, Annales Univ. Sci. Budapest. 41 (1998), 153–166.

[8] R. Deszcz and L. Verstraelen, Hypersurfaces of semi-Riemannian conformally flat manifolds, Geometry and Topology of Submanifolds 3 (1991), 131–147.

[9] Y. Matsuyama, Complete hypersurfaces with R · S = 0 in 𝔼^{n+1}, Proc. Amer. Math. Soc. 88 (1983), 119–123.

[10] P.J. Ryan, Hypersurfaces with parallel Ricci tensor, Osaka J. Math. 8 (1971), 251–259.

[11] P.J. Ryan, A class of complex hypersurfaces, Colloquium Math. 26 (1972), 175–182.

[12] S. Tanno, Hypersurfaces satisfying a certain condition on the Ricci tensor, Tôhoku Math. J. 21 (1969), 297–303.

[13] Z.I. Szabó, Structure theorems on Riemannian spaces satisfying R(X, Y) · R = 0. I. The local version, J. Diff. Geom. 17 (1982), 531–582.

[14] L. Verstraelen, Comments on pseudo-symmetry in the sense of Ryszard Deszcz, Geometry and Topology of Submanifolds 6 (1994), 199–209.

Marta Da̧browska, Ryszard Deszcz,

and Dorota Kowalczyk  Filip Defever
Department of Mathematics  Departement Wiskunde
Agricultural University of Wrocław  Katholieke Universiteit Leuven
ul. Grunwaldzka 53  Celestijnenlaan 200 B
PL - 50-357 Wrocław  3001 Heverlee
Poland  Belgium

rysz@ozi.ar.wroc.pl               Filip.Defever@wis.kuleuven.ac.be

(Received 23 02 1999)