On a class of Ricci-recurrent manifolds

Stanisław Ewert-Krzemieniewski (Szczecin)

1 Introduction

Following Prvanivić ([P]), a semi-Riemannian manifold (M, g) will be called conformally quasi-recurrent if its Weyl conformal curvature tensor C satisfies

\begin{matrix}∇_{Z}C(X, Y, V, W) = w(Z)C(X, Y, V, W)+ \\ p(X)C(Z, Y, V, W) + p(Y)C(X, Z, V, W)+ \\ p(V)C(X, Y, Z, W) + p(W)C(X, Y, V, Z) \end{matrix}  (1)

for some 1-forms w, p. In condition considered originally by Prvanović w = 2p. However, the last relation together with (1) implies that for the tensor C ( and in fact for any generalized curvature tensor satisfying (1)) the second Bianchi identity must hold. The aim of this note is to give a classification of conformally quasi-recurrent manifolds in the sense of (1) which are simultaneously Ricci-recurrent, i. e. those the Ricci tensor S satisfies

∇S = b ⊗ S

for some 1-form b.

For a generalized curvature tensor B define B̃ by

g (B̃ (X, Y) V, W ) = B (X, Y, V, W) .

Then for a (0, k) tensor field T, k ≥ 1, and (0, 2) tensor field S we define the tensor fields B · T and Q(S, T) by the formulas

\begin{matrix}(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}) , \end{matrix}

\begin{matrix}Q (S, T) (X_{1}, ...X_{k}; X, Y) = \\ T ((X ∧_{S} Y) X_{1}, X_{2}, ..., X_{k}) + ... + T (X_{1}, ..., X_{k−1}, (X ∧_{S} Y) X_{k}) , \end{matrix}

where

(X ∧_{S} Y) Z = S(Y, Z)X − S(X, Z)Y.

If the tensors R·R and Q(S, R) are linearly dependent then the manifold is said to be Ricci-generalized pseudosymmetric one ([D-D]). It is obvious that any semisymmetric as well as any Ricci flat manifold is Ricci generalized pseudosymmetric. The manifold (M, g) is Ricci-generalized pseudosymmetric iff the relation

R · R = LQ(S, R)  (2)

holds on the set {x ∈ M, Q(S, R) (x) ≠ 0} , L being a function on M. Remark that the relation (2) with L = 1 is of particular importance.

All manifolds under consideration are assumed to be smooth Hausdorff connected and their metrics are not assumed to be definite.

2 Results

The first Lemma shows the difference between the 1-forms w and p :

Lemma 1 . Suppose that at a point of the manifold M relation (1) holds. Then

p_{r}C_{ijk}^{r} = 0,
w_{r}C_{ijk}^{r} = C_{ijk,r}^{r},

\begin{matrix}C_{hijk,[lm]} = Δw_{lm}C_{hijk} + p_{hm}C_{lijk} + p_{im}C_{hljk} + p_{jm}C_{hilk} + p_{km}C_{hijl}− \\ p_{hl}C_{mijk} − p_{il}C_{hmjk} − p_{jl}C_{himk} − p_{kl}C_{hijm}, \end{matrix}

where Δw_{lm} = w_{l,m} − w_{m,l}, p_{hm} = p_{h,m} − p_{h}p_{m} and comma denotes covariant differentiation.

From the so called Patterson identity we have

Proposition 2 . Let M be a 4-dimensional manifold with nowhere vanishing Weyl conformal curvature tensor C. If C satisfies (1), then M is conformally recurrent manifold, precisely

∇C = (w + 2p) ⊗ C.

In the sequel we shall assume the following hypothesis:

(H) M is a Ricci − recurrent manifold with nowhere vanishing W eyl conformal curvature tensor and Ricci tensor. Moreover, the W eyl conformal curvature tensor satisfies (1), p does not vanishes on a dense subset and the Ricci tensor is not parallel.

By hypothesis, M admits a covector field b satisfying

S_{ij,k} = b_{k}S_{ij}, S_{ij,kl} = (b_{k,l} + b_{k}b_{l})S_{ij}, S_{ij,[kl]} = Δb_{kl}S_{ij},

where Δb_{kl} = b_{k,l} − b_{l,k}.

Proposition 3 . Let M (dimM > 3) be a Ricci-recurrent manifold with non-parallel Ricci tensor and suppose that M admits a covector field p with properties:

i) p does not vanish on a dense subset of M;

ii) p_{r}C_{ijk}^{r} = 0 on M.

Then the scalar curvature of M vanishes.

Lemma 4 . Under hypothesis (H) relations

p_{r}S_{h}^{r} = 0,
db = 0
S_{mr}C_{ijk}^{r} = 0

hold on M.

From the above Lemma it follows Corollary 5 . Under hypothesis (H)

d(w + 2p) = 0

holds on M.

Making use of the Corollary 5 one can deduce

Proposition 6 . Assume that on a manifold (M, g) hypothesis (H) is satisfied.

If t = w − 2p = 0 on (M, g), then the manifold is conformally related to a nonconformally flat conformal symmetric one (M, Exp (2f) g). Conversely, if (M, g) is conformally related to a non-conformally flat conformally symmetric one, then w−2p = 0 .

Proposition 7 . Suppose that on a manifold M, (dimM > 4), hypothesis (H) is satisfied. If b_{l} ≠ 0 at a point x ∈ M, then on some neighbourhood of x there exists null (i.e. isotropic) parallel vector field

v_{i} = Exp[\frac{−1}{2}b] k_{i}, ∂_{i}b = b_{i},

related to the Ricci tensor by

S_{ij} = ϵ k_{i}k_{j}, |ϵ| = 1.

Define the tensor

\begin{matrix}Q(S, C)_{hiqtpj} = S_{hp}C_{jiqt} − S_{hj}C_{piqt} + S_{ip}C_{hjqt} − S_{ij}C_{hpqt}+ \\ S_{qp}C_{hijt} − S_{qj}C_{hipt} + S_{tp}C_{hiqj} − S_{tj}C_{hiqp}. \end{matrix}

It is well known, that if the scalar curvature vanishes and the rank of the Ricci tensor is one, then

Q(S, C)_{hiqtpj} = Q(S, R)_{hiqtpj}

Lemma 8 . Suppose that on a manifold M, (dimM > 4), hypothesis (H) is satisfied. Then

t_{r}b^{r} · Q(S, C)_{hiqtpj} = 0.

Lemma 9 . Suppose that on a manifold M, (dimM > 4), hypothesis (H) is satisfied and

t_{r}b^{r} = 0.

Then

(n − 4) b_{r}b^{r} = 0,
t_{l}C_{hijk} + t_{j}C_{hikl} + t_{k}C_{hilj} = 0

and

t_{l} (C_{tqlr}C_{jih}^{r} − C_{tqjr}C_{lih}^{r}) = 0

Proposition 10 . Suppose that on a manifold M, (dimM ≥ 4), hypothesis (H) is satisfied. If at x ∈ M, t_{r}b^{r} (x) ≠ 0 (hence b_{l}(x) ≠ 0), then on some neighbourhood of x the Riemann-Christoffel curvature tensor has the form

R_{hijk} = k_{i}k_{j}S_{hk} − k_{i}k_{k}S_{hj} + k_{h}k_{k}S_{ij} − k_{h}k_{j}S_{ik},

where S_{ij} = z^{r}z^{s}R_{rijs}, z^{r}k_{r} = 1.

As a consequence of Propositions 3, 7 and 10 we obtain

Corollary 11 .Suppose that on a manifold M, (dimM > 4), hypothesis (H) and and Q (S, C)_{hiqtpj} = 0 hold. If b_{l} ≠ 0 at a point x ∈ M, then on some neighbourhood of x the metric g is of the Walker type. Moreover, M is semi-symmetric, i.e. R · R = 0.

On the other hand, if Q (S, C)_{hiqtpj} ≠ 0, then Lemma 9 results in

Corollary 12 .Let on a manifold M, (dimM > 4), hypothesis (H) be satisfied. Suppose moreover that Q (S, C)_{hiqtpj} ≠ 0 and w − 2p ≠ 0. Then

(R · R)_{hiqtpj} = Q(S, R)_{hiqtpj}.

Thus we conclude with the following

Theorem 13 .A conformally quasi-recurrent and Ricci-recurrent manifold of dimension n > 4 with nowhere vanishing Weyl conformal curvature tensor and Ricci tensor non-conformally related to a conformally symmetric one must be necessary Riccigeneralized pseudosymmetric manifold.

References

[D-D] F. Defever, R. Deszcz, On semi-Riemannian manifolds satisfying the condition R · R = Q(S, R), Geometry and Topology of Submanifolds, 3(1991), 108-130, World Sci., Singapore.

[P] M. Prvanović, Conformally quasi-recurrent manifolds, Finsler and Lagrange spaces, Proc. 5th Natl. Semin., Hon. 60th Birthday R. Miron, Braşov/Rom. 1988, (1988), 321-328.

Institute of Mathematics

Technical University of Szczecin

al. Piastów 17, 70-310 Szczecin

POLAND

e-mail: ewert@arcadia.tuniv.szczecin.pl