PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 63 (77), 1998, 115–130

ON A CERTAIN EXTENSION OF THE CLASS OF SEMISYMMETRIC MANIFOLDS

Ryszard Deszcz and Marian Hotloś

Dedicated to Professor Witold Roter on his 65th birthday

Communicated by Mileva Prvanović

Abstract. We study curvature properties of semi-Riemannian manifolds satisfying a new condition of pseudosymmetry type. Basing on obtained results we construct non-trivial examples of such manifolds.

1. Introduction

Let (M, g) be a connected n-dimensional, n ≥ 3, semi-Riemannian manifold of class C^{∞}. We denote by ∇, R̃, R, C, S and κ the Levi-Civita connection, the curvature operator, the Riemann-Christoffel curvature tensor, the Weyl conformal curvature tensor, the Ricci tensor and the scalar curvature of (M, g), respectively.

A semi-Riemannian manifold (M, g) is said to be semisymmetric [18] if

R · R = 0

holds on M. As a proper generalization of locally symmetric spaces (∇R = 0) semisymmetric manifolds were studied by many authors. In the Riemannian case, Z. I. Szabó obtained in the early eighties a full intrinsic classification of semisymmetric Riemannian manifolds [18]. Very recently theory of Riemannian semisymmetric manifolds has been presented in the monograph [1]. The profound investigation of several properties of semisymmetric manifolds, gave rise to their next generalization: the pseudosymmetric manifolds.

A semi-Riemannian manifold (M, g) is said to be pseudosymmetric [10] if at every point of M the following condition is satisfied:

AMS Subject Classification (1991): Primary 53B20, 53B30; Secondary 53C25, 53C50.

Keywords: semisymmetric manifolds, pseudosymmetry type conditions, conformal deformations.

(∗)_{1} the tensors R · R and Q(g, R) are linearly dependent. This condition is equivalent to the relation

R · R = L_{R} Q(g, R)

on the set U_{R} = \{x ∈ M | R − \frac{κ}{n(n−1)} G ≠ 0 at x\}, where L_{R} is some function on U_{R}. The definitions of the tensors used will be given in Section 2. There exist various examples of pseudosymmetric manifolds which are non-semisymmetric and a review of results on pseudosymmetric manifolds is given in [9] (see also [V]).

It is easy to see that if (∗)_{1} holds on a semi-Riemannian manifold (M, g), n ≥ 4, then at every point of M the following condition is satisfied: (∗)_{2} the tensors R · C and Q(g, C) are linearly dependent. The converse statement is not true [8] (cf. Example 3.1).

A semi-Riemannian manifold (M, g), n ≥ 4, is called Weyl-pseudosymmetric if at every point of M the condition (∗)_{2} is fulfilled. If a manifold (M, g) is Weylpseudosymmetric then the relation

R · C = L_{C} Q(g, C)

holds on the set U_{C} = {x ∈ M | C ≠ 0 at x}, where L_{C} is some function on U_{C}.

It is easy to see that at every point of pseudosymmetric Einstein manifold the following condition is fulfilled: (∗)_{3} the tensors R · R − Q(S, R) and Q(g, C) are linearly dependent. It is known that every hypersurface M,dim M ≥ 4, immersed isometrically in a semi-Riemannian space of constant curvature realizes (∗)_{3} ([13]). More precisely, the following relation R · R − Q(S, R) = −\frac{(n−2)κ̃}{n(n+1)} Q(g, C) holds on M, where κ̃ is the scalar curvature of the ambient space. Recently, pseudosymmetric manifolds satisfying (∗)_{3} were investigated in [12]. Semi-Riemannian manifolds realizing (∗)_{1}– (∗)_{3} and other conditions of this kind, described in [9] or [V], are called manifolds of pseudosymmetry type.

The present paper concerns with semi-Riemannian manifolds satisfying the new condition of pseudosymmetry type: (∗) the tensors R · C and Q(S, C) are linearly dependent at every point of M. This condition is equivalent to the relation

(1)  R · C = L Q(S, C)

on the set U = {x ∈ M | Q(S, C) ≠ 0 at x}, for some function L on U, called the associated function of M. It is clear that every semisymmetric manifold satisfies (∗). The converse statement is not true (see Example 5.1).

In Section 2 of this paper we fix the notations and present auxiliary lemmas. In Section 3 we consider manifolds satisfying the equality Q(S, C) = 0 and we prove that such manifolds are pseudosymmetric. In Section 4 we investigate manifolds satisfying (1) and admitting a 1-form a such that the cyclic sum ∑_{X,Y,Z} a(X)C̃(Y, Z) = 0. We prove that the associated function of such manifold must be equal to 1/(n − 1) or 1/(n − 2). Applying this result, we find in Section 5 the necessary and sufficient condition for a metric ḡ with harmonic Weyl tensor C̄ conformal to an essentially conformally symmetric metric g to satisfy (1). As a consequence of these considerations, we give an example of a manifold realizing (1) with L = 1/(n − 2) which is not pseudosymmetric. Finally, Section 6 contains some results on concircular changes of metrics satisfying (1).

2. Preliminaries

Let (M, g) be an n-dimensional, n ≥ 3, semi-Riemannian manifold. A tensor B̃ of type (1, 3) on M is said to be a generalized curvature tensor [16], if

∑_{X_{1},X_{2},X_{3}} B̃(X_{1}, X_{2})X_{3} = 0,
B̃(X_{1}, X_{2}) + B̃(X_{2}, X_{1}) = 0,
B(X_{1}, X_{2}, X_{3}, X_{4}) = B(X_{3}, X_{4}, X_{1}, X_{2}),

where B(X_{1}, X_{2}, X_{3}, X_{4}) = g(B̃(X_{1}, X_{2})X_{3}, X_{4}). The Ricci tensor Ric(B̃) of B̃ is the trace of the linear mapping X_{1} → B̃(X_{1}, X_{2})X_{3}. For a generalized curvature tensor B̃ we define the scalar curvature κ(B̃) by

κ(B̃) = ∑_{i=1}^{n} ϵ_{i} Ric(B̃)(E_{i}, E_{i}), ϵ_{i} = g(E_{i}, E_{i}) ,

where E_{1},…, E_{n} is an orthonormal basis. Let the tensor G be defined by

G(X_{1}, X_{2}, X_{3}, X_{4}) = g((X_{1} ∧ X_{2})X_{3}, X_{4}) ,
(X_{1} ∧ X_{2})X_{3} = g(X_{2}, X_{3})X_{1} − g(X_{1}, X_{3})X_{2}.

Further, we define the Weyl curvature tensor C(B̃) associated with B̃ by

C(B̃)(X_{1}, X_{2}, X_{3}, X_{4}) = B(X_{1}, X_{2}, X_{3}, X_{4}) + \frac{κ(B̃)}{(n − 1)(n − 2)}G(X_{1}, X_{2}, X_{3}, X_{4})

− \frac{1}{n − 2}(g((R̃ic(B̃)X_{1} ∧ X_{2})X_{3}, X_{4}) − g((R̃ic(B̃)X_{1} ∧ X_{2})X_{4}, X_{3})) ,

where the tensor field R̃ic(B̃) is defined by Ric(B̃)(X, Y) = g(R̃ic(B̃)X, Y). For an (0, 2)-tensor field A on (M, g) we define the endomorphism X ∧_{A} Y of Ξ(M) by (X ∧_{A} Y)Z = A(Y, Z)X −A(X, Z)Y, where X, Y, Z ∈ Ξ(M). In particular we have X ∧_{g} Y = X ∧ Y. For an (0, k)-tensor field T, k ≥ 1, an (0, 2)-tensor field A and a generalized curvature tensor B̃ on (M, g) we define the tensors B · T and Q(A, T) by

(B · T)(X_{1},…, X_{k}; X, Y) = −T (B̃((X, Y)X_{1}, X_{2},…, X_{k}) − ⋯
− T (X_{1},…, X_{k−1}, B̃(X, Y)X_{k}) ,
Q(A, T)(X_{1},…, X_{k}; X, Y) = −T ((X ∧_{A} Y)X_{1}, X_{2},…, X_{k}) − ⋯
− T (X_{1},…, X_{k−1}, (X ∧_{A} Y)X_{k}) ,

where X, Y, Z, X_{1}, X_{2},… ∈ Ξ(M). Putting in the above formulas

B̃(X, Y)Z = R̃(X, Y)Z = ∇_{X}∇_{Y} Z − ∇_{Y} ∇_{X}Z − ∇_{[X,Y]}Z ,

T = R or T = C, A = g or A = S, we obtain the tensors R · R, Q(g, R), Q(S, R), R · C, Q(g, C) and Q(S, C), respectively.

Let (M, g) be a semi-Riemannian manifold covered by a system of charts {W; x^{k}}. We denote by g_{ij}, R_{hijk}, S_{ij}, S_{i}^{j} = g^{jk}S_{ik}, G_{hijk} = g_{hk}g_{ij} − g_{hj}g_{ik} and

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

+ \frac{κ}{(n − 1)(n − 2)}G_{hijk} (2)

the local components of the metric tensor g, the Riemann–Christoffel curvature tensor R, the Ricci tensor S, the Ricci operator S̃, the tensor G and the Weyl tensor C, respectively.

At the end of this section we present some results which will be used in the next sections. Let g be a metric on a manifold M and let ḡ be another metric on M conformally related to g, i.e., ḡ = exp(2p)g, where p is a nonconstant function on M. When Ω is a quantity formed with respect to g, we denote by \overline{Ω} the similar quantity formed with respect to ḡ. We shall use the following general formulas for conformally related metrics (cf. [20]):

ḡ_{ij} = exp(2p)g_{ij}, ḡ^{ij} = exp(−2p)g^{ij}, (3)

S̄_{ij} = S_{ij} − (n − 2)P_{ij} − (Δ_{2}p + (n − 2)Δ_{1}p))g_{ij}, (4) κ̄ = exp(−2p)(κ − (n − 1)(2Δ_{2}p + (n − 2)Δ_{1}p)), (5)

R̄_{hijk} = exp(2p)(R_{hijk} − U_{hijk}), (6)

C̄_{ijk}^{h} = C_{ijk}^{h}, C̄_{hijk} = exp(2p)C_{hijk}, (7)

¯∇_{r}C̄_{ijk}^{r} = ∇_{r}C_{ijk}^{r} + (n − 3)p_{r}C_{ijk}^{r}, (8)

where

Δ_{1}p = g^{ij}p_{i}p_{j} = ⟨dp, dp⟩, Δ_{2}p = g^{ij}∇_{j}p_{i},
U_{hijk} = g_{hk}P_{ij} − g_{hj}P_{ik} + g_{ij}P_{hk} − g_{ik}P_{hj} + Δ_{1}p (g_{hk}g_{ij} − g_{hj}g_{ik}),

P_{ij} and p_{i} are local components of the tensors P = ∇dp − dp ⊗ dp and dp, respectively. Using (3), (6) and (7) we also have

exp(−2p)(R̄ · C̄)_{hijklm} = (R · C)_{hijklm} − Δ_{1}pQ(g, C)_{hijklm} − Q(P, C)_{hijklm}
− P_{m}^{r}(g_{hl}C_{rijk} + g_{il}C_{hrjk} + g_{jl}C_{hirk} + g_{kl}C_{hijr})

+ P_{l}^{r}(g_{hm}C_{rijk} + g_{im}C_{hrjk} + g_{jm}C_{hirk} + g_{km}C_{hijr}).

Lemma 2.1. [5, Lemma 1] Let a tensor A_{lmhs_{1}...s_{N}} of type (0, N + 3) be symmetric in (l, m) and skew-symmetric in (m, h). Then A_{lmhs_{1}...s_{N}} = 0.

Lemma 2.2. [17] We define the metric g in ℝ^{n} by the formula

(10)  ds^{2} = Q(dx^{1})^{2} + k_{αβ}dx^{α}dx^{β} + 2dx^{1}dx^{n},

where α, β = 2,…, n − 1, [k_{αβ}] is a symmetric and nonsingular matrix consisting of constants, and Q is independent of x^{n}. The only components of ∇ and C, not identically zero are those related to:

Γ_{11}^{α} = −\frac{1}{2}k^{αω}Q_{.ω}, Γ_{11}^{n} = \frac{1}{2}Q_{.1}, Γ_{1γ}^{n} = \frac{1}{2}Q_{.γ}, (11)

C_{1λμ1} = \frac{1}{2}Q_{.λμ} − \frac{1}{2(n − 2)}k_{λμ}(k^{βω}Q_{.βω}), (12)

where [k^{λμ}] = [k_{λμ}]^{−1} and the dot denotes partial differentiation with respect to coordinates.

Lemma 2.3. [11, Theorem 1] Let B̃ be a generalized curvature tensor at x ∈ M such that the condition ∑_{X,Y,Z} ω(X)B̃(Y, Z) = 0 is satisfied for B̃ and a covector ω at x, where X, Y, Z ∈ T_{x}(M), Σ denotes the cyclic sum. If ω ≠ 0 then B · B = Q(Ric(B̃), B) at x

Lemma 2.4. [2, Proposition 4.1] Let (M, g), dim M ≥ 3, be a semi-Riemannian manifold. Let A be a nonzero symmetric (0, 2)-tensor and B̃ a generalized curvature tensor at a point x of M satisfying the condition Q(A, B) = 0. Moreover, let V be a vector at x such that the scalar ρ = a(V) is nonzero, where a is a covector defined by a(X) = A(X, V), X ∈ T_{x}(M).

(i) If the tensor A−(1/ρ) a⊗a vanishes, then the relation ∑_{X,Y,Z}a(X)B̃(Y, Z) = 0 holds at x, where X, Y, Z ∈ T_{x}(M).

(ii) If the tensor A − (1/ρ) a ⊗ a is nonzero, then the relation

ρB(X, Y, Z, W) = λ(A(X, W)A(Y, Z) − A(X, Z)A(Y, W))

holds at x, where λ ∈ ℝ and X, Y, Z, W ∈ T_{x}(M).

Moreover, in both cases B · B = Q(Ric(B̃), B) at x.

Lemma 2.5. [14, Theorems 1 and 2] Let (M,g) be a Weyl-pseudosymmetric semi-Riemannian manifold satisfying the condition ∑_{X,Y,Z} a(X)C̃(Y, Z) = 0, where a is a 1-form on M. If a ≠ 0 and C ≠ 0 at a point x ∈ M, then the following relations are satisfied at x:

L_{C} = \frac{κ}{n(n − 1)} , S(W, C̃(X, Y)Z) = \frac{κ}{n} C(X, Y, Z, W),

Q(S − \frac{κ}{n}g, C) = 0, R · R = L_{C}Q(g, R) .

Lemma 2.6. [12, Theorem 4.2] Let (M, g) be a semi-Riemannian manifold with the curvature tensor of the form

R(X, Y, Z, W) = ϕ(S(X, W)S(Y, Z) − S(X, Z)S(Y, W)) + ηG(X, Y, Z, W)

+ μ(S(X, W)g(Y, Z) + S(Y, Z)g(X, W) − S(X, Z)g(Y, W) − S(Y, W)g(X, Z))

at x ∈ M, where X, Y, Z, W ∈ T_{x}(M) and ϕ, μ, η ∈ ℝ. If C ≠ 0 and S −(κ/n) g ≠ 0 at x, then the following equalities hold at x:

R · R = L_{R}Q(g, R), L_{R} = \frac{μ}{ϕ} ((n − 2)μ − 1) − η(n − 2),

R · R = Q(S, R) + (L_{R} + \frac{μ}{ϕ})Q(g, C).

3. Manifolds with vanishing tensor field Q(S,C)

Theotem 3.1. Let (M,g), dim M ≥ 4, be a semi-Riemannian manifold satisfying at a point x of M the equality Q(S, C) = 0. If S ≠ 0 and C ≠ 0 at x, then the relation

(13)  R · R = \frac{κ}{n − 1}Q(g, R)

holds at x.

Proof. It is easy to verify that the following identity is satisfied on M

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

− \frac{1}{n − 2}(g_{hl}S_{mr}C_{ijk}^{r} − g_{hm}S_{lr}C_{ijk}^{r} − g_{il}S_{mr}C_{hjk}^{r} + g_{im}S_{lr}C_{hjk}^{r}

+ g_{jl}S_{mr}C_{khi}^{r} − g_{jm}S_{lr}C_{khi}^{r} − g_{kl}S_{mr}C_{jhi}^{r} + g_{km}S_{lr}C_{jhi}^{r}).

According to Lemma 2.4, we may consider two cases (we will use notations of the mentioned lemma):

(i) S = (1/ρ) a ⊗ a. In this case we have a_{l}C_{hijk} + a_{h}C_{iljk} + a_{i}C_{lhjk} = 0, which implies a_{r}C_{ijk}^{r} = 0 and consequently S_{ir}C_{hjk}^{r} = 0. Thus the equation C · C = 0, which follows from Lemma 2.3, and our assumption turns (14) into

R · C = −\frac{κ}{(n − 1)(n − 2)}Q(g, C).

Applying now Lemma 2.5 we obtain κ = 0 and next R · R = 0.

(ii) S − (1/ρ) a ⊗ a ≠ 0. In this case we have ρC_{hijk} = λ(S_{hk}S_{ij} − S_{hj}S_{ik}). This equation, in virtue of (2) leads to

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

− \frac{κ}{(n − 1)(n − 2)}G_{hijk}.

Applying now Lemma 2.6 we obtain (13), which completes the proof.

From the above theorem it follows

Corollary 3.1. Let (M, g), dim M ≥ 4, be an analytic semi-Riemannian manifold with nonzero tensors S and C. If the equality Q(S, C) = 0 is fulfilled on M, then (M,g) is pseudosymmetric manifold satisfying (13).

On the other hand, manifolds realizing (∗) for which Q(S, C) ≠ 0, i.e., manifolds fulfilling (1), may be pseudosymmetric or not. This fact illustrates the following

Example 3.1. Let (M, g) be the 4-dimensional manifold defined in [4, Lemme 1.1] As it was shown in [4] (see Lemme 1.1 and Remarqué 1.5), (M, g) is a nonconformally flat and non-semisymmetric, Weyl-semisymmetric manifold, i.e., the tensors C and R · R are nonzero and the condition R · C = 0 holds on M. From these facts it follows that (M, g) is a non-pseudosymmetric manifold.

(i) Let V be a connected subset of the set W = {x ∈ M | u(x) ≠ 0}, where u is the function defined in [4, Lemme 1.1]. By formula (10) of [4] we have W = U_{C}. The scalar curvature κ of (M, g) satisfies the equality ([4, Lemme 1.1(iv)] κ = u, which implies that the Ricci tensor S of (M, g) is nonzero at every point of V. Using now Theorem 3.1 and the fact that the tensors S and C and the scalar curvature κ are nonzero at every point of V we can easily conclude that the tensor Q(S, C) is nonzero at every point of V. Thus we have on V the following equality:

R · C = L Q(S, C) with L = 0 .

(ii) We consider now on V the conformal deformation g → ḡ = (1/u^{2}) g of the metric g, where u > 0 or u < 0 on V. It is known that the manifold (V, ḡ) is an Einstein manifold [4, Lemme 1.1(viii)], i.e., S̄ = (κ̄/4) ḡ holds on V. Moreover, as it was shown in [8] (see Example 3) the relation

(15)  R̄ · R̄ = −\frac{1}{12}(u^{3} − pq)Q(ḡ, R̄)

holds on V, where R̄ is the Riemann-Christoffel curvature tensor of the metric ḡ and p, q are some constants. Evidently, if the Ricci tensor S̄ vanishes at a point x ∈ V, then Q(S̄, C̄) = 0 holds at x and, of course, the condition (∗) is fulfilled at x. If at a point x ∈ M we have S̄ ≠ 0, then (15) turns into

R̄ · C̄ = −\frac{u^{3} − pq}{3κ̄} Q(S̄, C̄) .

Thus the manifold (V, ḡ) realizes (∗).

Since the equality Q(S, C) = 0 at x leads to the condition (∗)_{1} at x, we restrict our considerations in the remaining sections to the set U.

4. Manifolds satisfying some curvature conditions

Theorem 4.1. Let (M, g), dim M ≥ 4, be a semi-Riemannian manifold satisfying (∗) and the following condition

(16)  ∑_{X,Y,Z} a(X)C̃(Y, Z) = 0

for a 1-form a. If a ≠ 0 and Q(S, C) ≠ 0 at a point x ∈ M, then L = 1/(n − 2) or L = 1/(n − 1).

Proof. First of all we note that (16), which in local coordinates takes the form

(17)  a_{l}C_{hijk} + a_{j}C_{hikl} + a_{k}C_{hilj} = 0,

leads to

(18)  a_{r}a^{r} = 0 , a_{r}C_{ijk}^{r} = 0

and

(19)  C · C = 0

(cf. Lemma 2.3). In local coordinates the equation R · C = L Q(S, C) takes the form

R_{hlm}^{r}C_{rijk} + R_{ilm}^{r}C_{hrjk} + R_{jlm}^{r}C_{hirk} + R_{klm}^{r}C_{hijr}
= L(S_{hl}C_{mijk} − S_{hm}C_{lijk} + S_{il}C_{hmjk} − S_{im}C_{hljk} + S_{jl}C_{himk}

− S_{jm}C_{hilk} + S_{kl}C_{hijm} − S_{km}C_{hijl}). (20)

Transvecting (20) with a^{h}, in view of (18), we obtain

(21)  C_{rijk}R_{slm}^{r}a^{s} = L(d_{l}C_{mijk} − d_{m}C_{lijk}),

where d_{i} = a^{r}S_{ri}. Substituting (2) into (18) we have

R_{srlm}a^{s} = \frac{1}{n − 2}(d_{m}g_{rl} − d_{l}g_{rm} + a_{m}S_{rl} − a_{l}S_{rm}) − \frac{κ}{(n − 1)(n − 2)}(a_{m}g_{rl} − a_{l}g_{rm}).

The substitution of the above equality into (21) and making use of a_{m}C_{lijk} − a_{l}C_{mijk} = a_{i}C_{lmjk} , which follows from (17), yields

(22) ((n−2)L−1)(d_{m}C_{lijk} −d_{l}C_{mijk})=a_{m}S_{lr}C_{ijk}^{r} −a_{l}S_{mr}C_{ijk}^{r} + \frac{κ}{n − 1}a_{i}C_{mljk} .

Transvection of (22) with a^{m}, in virtue of (18), gives

((n − 2)L − 1)a^{r}d_{r}C_{lijk} = −a_{l}d_{r}C_{ijk}^{r} ,

which immediately implies d_{r}C_{ijk}^{r} = 0.

Contracting now (22) with g^{km} and using the above equality we have

(23)  S^{rs}C_{rijs} = 0.

Transvecting (17) with S_{p}^{l} we get d_{p}C_{hijk} = a_{k}C_{hijr}S_{p}^{r} −a_{j}C_{hikr}S_{p}^{r}. Substituting twice the above equality into (22) (taking suitable indices), we obtain

(24) (n − 2)L(a_{l}S_{mr}C_{ijk}^{r} − a_{m}S_{lr}C_{ijk}^{r})

= a_{i} (\frac{κ}{n − 1}C_{mljk} + ((n − 2)L − 1)) (S_{mr}C_{ljk}^{r} − S_{lr}C_{mjk}^{r}).

Hence, by cyclic permutation in m, j, k, we get

(25)  (n − 2)La_{l}T_{mijk} = ((n − 2)L − 1)a_{i}T_{mljk},

where T_{mijk} = S_{mr}C_{ijk}^{r} + S_{jr}C_{ikm}^{r} + S_{kr}C_{imj}^{r}. We assert that T_{mijk} = 0, i.e.,

(26)  S_{mr}C_{ijk}^{r} + S_{jr}C_{ikm}^{r} + S_{kr}C_{imj}^{r} = 0.

In fact, if L = 0 then we immediatey have T_{mijk} = 0. Assume now that L ≠ 0 at x. Using (25) we get

a_{l}T_{mijk} = αa_{i}T_{mljk} = α^{2}a_{l}T_{mijk},

where α = \frac{(n−2)L−1}{(n−2)L}. If α^{2} ≠ 1 at x, then we get (26). On the other hand the equality α^{2} = 1 is equivalent to (n − 2)L = 1/2. In this case (25) takes the form a_{l}T_{mijk} + a_{i}T_{mljk} = 0, which immediately leads to (26). The equalities (1), (14) and (19) imply

(L(n − 2) − 1)Q(S, C)_{hijklm} + \frac{κ}{n − 1}Q(g, C)_{hijklm}
= g_{hl}S_{mr}C_{ijk}^{r} − g_{hm}S_{lr}C_{ijk}^{r} − g_{il}S_{mr}C_{hjk}^{r} + g_{im}S_{lr}C_{hjk}^{r}

+ g_{jl}S_{mr}C_{khi}^{r} − g_{jm}S_{lr}C_{khi}^{r} − g_{kl}S_{mr}C_{jhi}^{r} + g_{km}S_{lr}C_{jhi}^{r}. (27)

Contracting (27) with g^{hl}, in virtue of (26) and (22), we obtain

(28)  L(n − 2)κC_{mijk} + (L(n − 2) − 1)S_{ir}C_{mjk}^{r} = (n − 1)S_{mr}C_{ijk}^{r}.

Symmetrizing this in m, i, we find (L(n − 2) − n)(S_{ir}C_{mjk}^{r} + S_{mr}C_{ijk}^{r}) = 0. If L(n − 2) ≠ n, then we have

(29)  S_{ir}C_{mjk}^{r} = −S_{mr}C_{ijk}^{r}.

On the other hand contracting (1) with g^{hk} we get the equality

L(S_{lr}C_{jim}^{r} + S_{mr}C_{jli}^{r} + S_{lr}C_{ijm}^{r} + S_{mr}C_{ilj}^{r}) = 0,

which, in virtue of (26), takes the form L(S_{ir}C_{jlm}^{r} + S_{jr}C_{ilm}^{r}) = 0. Thus in the case L(n − 2) = n we also have (29). Substituting (29) into (28) we obtain

(30)  LκC_{mijk} = (L + 1)S_{mr}C_{ijk}^{r}.

We shall show that L ≠ −1. Suppose that L = −1. Thus from (30) it follows that κ = 0 and (27) and (22) take the forms

(1 − n)Q(S, C)_{hijklm} = g_{hl}S_{mr}C_{ijk}^{r} − g_{hm}S_{lr}C_{ijk}^{r} − g_{il}S_{mr}C_{hjk}^{r} + g_{im}S_{lr}C_{hjk}^{r}

+ g_{jl}S_{mr}C_{khi}^{r} − g_{jm}S_{lr}C_{khi}^{r} − g_{kl}S_{mr}C_{jhi}^{r} + g_{km}S_{lr}C_{jhi}^{r} (31)

and

(32)  (1 − n)(d_{m}C_{lijk} − d_{l}C_{mijk}) = a_{m}S_{lr}C_{ijk}^{r} − a_{l}S_{mr}C_{ijk}^{r},

respectively. But using (29) we can rewrite the right hand side of the last equation as

−(a_{m}S_{ir}C_{ljk}^{r} − a_{l}S_{ir}C_{mjk}^{r}) = −S_{i}^{r}(a_{m}C_{rljk} − a_{l}C_{rmjk})
= −S_{i}^{r}a_{r}C_{mljk} = −d_{i}C_{mljk}.

Thus (32) takes the form

(n − 1)(d_{m}C_{lijk} − d_{l}C_{mijk}) = d_{i}C_{mljk} .

Hence, by standard calculation, we can obtain d_{i} = 0. Applying this to (32) we have a_{m}S_{l}^{r}C_{rijk} = a_{l}S_{m}^{r}C_{rijk} and, in virtue of (29),

a_{m}S_{i}^{r}C_{rljk} = −a_{m}S_{l}^{r}C_{rijk} .

We put A_{mlijk} = a_{m}S_{l}^{r}C_{rijk}. We see that the tensor A is symmetric with respect to m, l and antisymmetric with respect to i, l, which, in view of Lemma 2.1, implies A = 0. Hence S_{l}^{r}C_{rijk} = 0 and (31) implies now Q(S, C) = 0, a contradiction. Thus we have L ≠ −1 and we can rewrite (30) in the form

(33)  S_{mr}C_{ijk}^{r} = ϕC_{mijk}, where ϕ = \frac{Lκ}{L + 1}.

Substituting (33) into (24) and using (17) we find

(n − 2)Lϕa_{i}C_{mljk} = (2ϕ((n − 2)L − 1) + \frac{κ}{n − 1}) a_{i}C_{mljk},

which implies

(n − 2)Lϕ = (2ϕ((n − 2)L − 1) + \frac{κ}{n − 1})

and next

κ (\frac{L}{L + 1}((n − 2)L − 2) + \frac{1}{n − 1}) = 0.

We consider two cases:

(i) κ = 0. In this case from (33) we have S_{mr}C_{ijk}^{r} = 0 and taking into account (27), we obtain L = 1/(n − 2).

(ii) κ ≠ 0. In this case we get the following equation

L(n − 1)((n − 2)L − 2) + L + 1 = 0,

which has two solutions: L = 1/(n − 2) or L = 1/(n − 1). This completes the proof.

Corollary 4.1. Suppose that (M,g) satisfies the assumptions of the last theorem. If L = 1/(n − 1), then (M,g) is pseudosymmetric.

Proof. For L = 1/(n − 1) (33) takes the form S_{mr}C_{ijk}^{r} = (κ/n) C_{mijk}. Substituting this into (27) we find

Q(S, C) = \frac{κ}{n}Q(g, C).

Now (1) implies

R · C = \frac{κ}{n(n − 1)}Q(g, C),

which denotes that (M, g) is Weyl-pseudosymmetric at x. From Lemma 2.5 we conclude our assertion.

Remark 4.1. It will be shown in the next section that a manifold (M, g) with the associated fundamental function L = 1/(n − 2) need not be pseudosymmetric.

5. Conformal deformations of e.c.s. manifolds

A semi-Riemannian manifold (M, g) is said to be conformally symmetric if its Weyl conformal curvature tensor C satisfies the condition ∇C = 0. Conformally symmetric manifolds which are neither conformally flat nor locally symmetric are called essentially conformally symmetric (e.c.s. in short). It is known that every e.c.s. manifold is semisymmetric [6, Theorem 9].

Theorem 5.1. Let (M,g) be an e.c.s. manifold. Assume that M admits a function p such that ḡ = exp(2p)g is a metric with harmonic Weyl conformal curvature tensor C̄. Then:

(i) If (M, ḡ) satisfies the relation (1) and is not pseudosymmetric, then Δ_{2}p = 0. (ii) If Δ_{2}p = 0, then R̄ · C̄ = (1/(n − 2)) Q(S̄, C̄).

Proof. We assert that all e.c.s. manifolds satisfy the condition (16). Every e.c.s. manifold satisfies the condition ∑_{X,Y,Z} S(W, X)C̃(Y, Z) = 0 [7, Theorem 7]. This implies (16) with a ≠ 0 at any point at which S ≠ 0 and, in virtue of parallelity of C, everywhere on M. Since C is parallel and C̄ is harmonic ( ¯∇_{r}C̄_{ijk}^{r} = 0), the equality (8) leads to p_{r}C_{ijk}^{r} = 0, whence

(34)  P_{lr}C_{ijk}^{r} = 0.

Now (9) takes the form

(35)  exp(−2p)(R̄ · C̄) = −Δ_{1}p: Q(g, C) − Q(P, C).

Assume now that (M, ḡ) satisfies (1). Since (M, ḡ) also satisfies (16), so using Theorem 4.1 and Corollary 4.1 we can rewrite (35) in the form

Q(\frac{1}{n − 2}S̄, C) = −Δ_{1}p: Q(g, C) − Q(P, C).

Hence, in virtue of (4) and Q(S, C) = 0 [6, Lemma 7], we get Δ_{2}p: Q(g, C) = 0, which implies Δ_{2}p = 0 and ends the proof of (i).

Assume now that Δ_{2}p = 0. Substituting the equality

P = \frac{1}{n − 2}S − \frac{1}{n − 2}S̄ − Δ_{1}pg

into (35) and using Q(S, C) = 0, we easily obtain R̄ · C̄ = \frac{1}{n − 2}Q(S̄, C̄). This completes the proof.

Example 5.1. Let M = {x ∈ ℝ^{5} | x^{2} + x^{3} > 0} be endowed with the metric given by (10), where Q = (A: k_{λμ} + a_{λμ})x^{λ}x^{μ}. A is nonconstant function of x^{1} only and

[a_{λμ}] = [ \begin{matrix}1 & −1 & 0 \\ −1 & 1 & 0 \\ 0 & 0 & 2 \end{matrix} ] , [k_{λμ}] = [ \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & −1 \end{matrix} ] .

It is known that (M, g) is essentially conformally symmetric and Ricci-recurrent manifold [17]. Further, it is easy to see, in view of (11) and (12), that the function p(x) = x^{2} + x^{3} satisfies equations: p^{r}C _{rijk} = 0, Δ_{2}p = 0 and Δ_{1}p = 2. Thus, according to Theorem 5.1, the metric ḡ = exp(2p)g satisfies the condition (1). We assert that this metric cannot be pseudosymmetric. Conversely, suppose that ḡ is pseudosymmetric. Hence ḡ is Weyl-pseudosymmetric. Applying now Theorem 3.1 of [15], we get Q(P − (1/n) tr(P)g, C) = 0. But the only nonzero components of the tensor P are P_{11} and P_{22} = P_{23} = P_{33} = −1. This, in virtue of (11) and (12), leads to Q(P − (1/n) tr(P)g, C)_{221441} ≠ 0, a contradiction. Thus the metric ḡ is not pseudosymmetric and, consequently, it cannot be semisymmetric.

Remark 5.1. The 5-dimensional metric g, defined in the above example, can be easily extended on any dimension n > 5. Namely, we can enlarge matrices [k_{λμ}] and [a_{λμ}] such that the equality a_{λμ}k^{λμ} = 0 is still satisfied (this equality guaranties that the metric g is conformally symmetric).

6. Concircular changes of metrics satisfying (1)

Let g be a metric on a manifold M and let ḡ be another metric conformally related to g, i.e., ḡ = exp(2p)g, where p is a non-constant function on M. If the tensor P of conformal change of the metric, given by P = ∇(dp) − dp ⊗ dp, is proportional to g at every point of M, then this conformal change is called concircular.

Lemma 6.1. Let (M,g) be a semi–Riemannian manifold and let on M be given a concircular change of metric g ⟶ ḡ = exp(2p)g. Assume that the condition (1) is satisfied at a point x of M. Then:

(i) If L = 1/(n − 1), then R̄ · C̄ = (1/(n − 1)) Q(S̄, C̄). (ii) If κ̄ = exp(−2p)κ, then R̄ · C̄ = LQ(S̄, C̄) at x.

Proof. For concircular change of metric we have P = \frac{1}{n} tr(P)g, where tr(P) = Δ_{2}p − Δ_{1}p. Hence, in virtue of (9), we get

exp(−2p)R̄ · C̄ = R · C − Δ_{1}p Q(g, C) − 2\frac{tr(P)}{n}Q(g, C) = R · C − \frac{α}{n}Q(g, C),

where α = (n − 2)Δ_{1}p + 2Δ_{2}p = (exp(2p)κ̄ − κ)/(n − 1) (cf. (5)). Using now our assumption we obtain

(36)  exp(−2p)R̄ · C̄ = Q(LS − \frac{α}{n}g, C).

But, in virtue of (4), we have S̄ = S − \frac{(n − 1)α}{n}g and we can rewrite (36) in the form

R̄ · C̄ = LQ(S̄, C̄) + \frac{α}{n}(L(n − 1) − 1)Q(g, C̄).

Hence we easily get our assertions, which completes the proof.

Proposition 6.1. Let (M,g) be a semi-Riemannian manifold satisfying the condition (1) and let on M be given a concircular change of metric g ⟶ ḡ = exp(2p)g. Assume that ḡ also satisfies (1), i.e.,

(37)  R̄ · C̄ = L̄Q(S̄, C̄).

If L = L̄ at x, then L = 1/(n − 1) or κ̄ = exp(−2p)κ at x.

Proof. Using (1), (9) and (37) we have

Q(L̄S̄ − LS + \frac{α}{n}g, C) = 0,

where α = (n − 2)Δ_{1}p + 2Δ_{2}p = (exp(2p)κ̄ − κ)/(n − 1). Hence, in virtue of the relation

(38)  S̄ = S − \frac{(n − 1)α}{n}g,

which follows from (4), we get

(39)  Q(A, C) = 0, where A = S(L̄ − L) − \frac{α}{n}(L̄(n − 1) − 1)g.

Because L̄ = L, the above equality implies L̄ = 1/(n − 1) or α = 0 and we have the situation described in the previous lemma. This completes the proof.

Theorem 6.1. Let (M,g) be a semi-Riemannian manifold satisfying the condition (1) and let on M be given a concircular change of metric g ⟶ ḡ = exp(2p)g. Assume that ḡ also satisfies (1) with the associated function L̄. If L ≠ L̄ at x, then the following equation

(40)  κ(L̄ + 1)(L(n − 1) − 1) = exp(2p) κ̄ (L + 1)(L̄(n − 1) − 1)

holds at x. Moreover, metrics g and ḡ are pseudosymmetric at x.

Proof. In the same manner as in the proof of the previous proposition we get the equality (39). We shall consider two cases:

(I) A = 0. In this case we have

S = \frac{α(L̄(n − 1) − 1)}{n(L̄ − L)}g, R · C = L\frac{κ}{n}Q(g, C).

So the metric g is Einsteinian and Weyl-pseudosymmetric and consequently, pseudosymmetric. In virtue of (38) ḡ is also Einsteinian. Pseudosymmetry of ḡ follows immediately from Theorem 5.1 of [3].

(II) A ≠ 0. According to Lemma 2.4 we have two possibilities:

(i) A = (1/ρ) a ⊗ a. Since the covector a satisfies the relation (17) we can apply Theorem 4.1. Thus we have L = 1/(n−1) or L = 1/(n−2). If L = 1/(n−1), then, in virtue of Lemma 6.1, we have L̄ = L, a contradiction. If L = 1/(n − 2), then also L̄ = 1/(n − 2) (because L̄ = 1/(n − 1) implies L = 1/(n − 1)), a contradiction.

(ii) A − (1/ρ) a ⊗ a ≠ 0. In this case we have

(41)  ρC_{hijk} = λ(A_{hk}A_{ij} − A_{hj}A_{ik}).

Contracting (41) with g^{hk} we get A_{ir}A_{j}^{r} = tr(A)A_{ij}, where tr(A) = κ(L̄ − L) − α(L̄(n − 1) − 1). Substituting (39) into the above equality we get

S_{ir}A_{j}^{r} = ϕA_{ij}, where ϕ = κ − \frac{α(n − 1)}{n(L̄ − L)}(L̄(n − 1) − 1).

Transvecting (41) with S_{l}^{r} we obtain S_{l}^{r}C_{rijk} = ϕC_{lijk}. Substitution of this equality into (14), in virtue of (19) and (1), leads to

(L(n − 2) − 1)Q(S, C) = (ϕ − \frac{κ}{n − 1})Q(g, C) = (\frac{(n − 2)κ}{n − 1} − \frac{(n − 1)β}{n(L̄ − L)}) Q(g, C),

where β = α(L̄(n − 1) − 1).

On the other hand (39) implies Q(S, C) = \frac{β}{n(L̄ − L)} Q(g, C). Substituting this relation into the previous one we get

(42)  β(L + 1) = \frac{nκ}{n − 1}(L̄ − L),

which can be rewritten in the form (40).

In the same manner as in the proof of Theorem 3.1 we get that the metric g is pseudosymmetric. Moreover, L_{R} = κ/(n − 1) − β/n(L̄ − L) = βL/n(L̄ − L) (in view of (42)). Pseudosymmetry of ḡ we obtain as in the case (I). This completes the proof.

References

[1] E. Boeckx, O. Kowalski, L. Vanhecke, Riemannian Manifolds of Conullity Two, World Scientific, River Edge, New Jersey, 1996.

[2] F. Defever, R. Deszcz, On semi-Riemannian manifolds satisfying the condition R·R = Q(S, R), in: Geometry and Topology of Submanifolds, III, World Scientific, River Edge, New Jersey, 1991, pp. 108–130.

[3] J. Deprez, R. Deszcz, L. Verstraelen, Examples of pseudosymmetric conformally flat warped products, Chinese J. Math., 17 (1989), 51–65.

[4] A. Derdziński, Examples de métriques de Kaehler et d’Einstein autoduales sur le plan complexe, in: Géométrie riemannianne en dimension 4 (Séminaire Arthur Besse 1978/79), Cedic/Fernand Nathan, Paris 1981, 334–346.