Bulletin T.CXXXI de l’Académie serbe des sciences et des arts − 2005 Classe des Sciences mathématiques et naturelles Sciences mathématiques, N\underline{o} 30

ON SOME PRODUCT CONFORMALLY FLAT LOCALLY DECOMPOSABLE RIEMANNIAN MANIFOLDS

MILEVA PRVANOVIĆ

(Presented at the 2nd Meeting, held on March 25, 2005)

A b s t r a c t. We investigate product conformally flat locally decomposable Riemannian manifolds whose curvature tensor can be expressed in the terms of Ricci tensors only.

AMS Mathematics Subject Classification (2000): 53C15

Key Words: locally decomposable Riemannian manifold, product conformally flat manifold

The object of the paper.

The Riemannian manifolds (M, g) whose Riemannian curvature tensor satisfies the condition

R(X, Y, Z, W) = L[ρ(X, W)ρ(Y, Z) − ρ(X, Z)ρ(Y, W)],  (∗)

where ρ(X, W) is the Ricci tensor and L is some scalar function, appears in the investigations of some authors ([1],[2],[3]). D. Kowalczyk proved in [1] that conformally flat manifold satisfies the condition (*) on

U_{R} = \{p ∈ M | R − \frac{κ}{n(n − 1)}γ ≠ 0 at p\}, if and only if

L = \frac{n − 1}{(n − 2)κ}, ρ = \frac{κ}{n − 1}g + βw ⊗ w, w ∈ T_{p}(U), β ∈ R.

Here n = dimM, κ is the scalar curvature and

γ(X, Y, Z, W) = g(X, W)g(Y, Z) − g(X, Z)g(Y, W).

The object of the present paper is to generalize this result to Riemannian manifolds endowed with product structure. More precisely, we investigate product conformally flat locally decomposable Riemannian manifolds whose curvature tensor can be expressed in the terms of the Ricci tensors only.

1. Preliminary results

Let (M, g, f) be a locally decomposable Riemannian manifold. This means that the Riemannian manifold (M, g) is endowed with product structure f, f^{2} = id., such that

F (X, Y) = g(fX, Y) = g(X, fY), ▽f = 0,

for all X, Y ∈ T_{p}(M), where T_{p}(M) is the tangent vector space of M at p ∈ M, and ▽ is the operator of the Levi-Civita connection. M being the locally product manifold M_{1} × M_{2}, we denote dim M_{1} = p, dim M_{2} = q. Then n = dim M = p + q, and tr f = r = p − q. We suppose p > 2, q > 2.

Let e_{i}, i = 1, 2, …, n be an orthonormal basis of T_{p}. We define the Ricci tensor ρ, ∼ Ricci tensor ρ̃ and the scalar curvatures κ and κ̃ by

ρ(X, Y) = ∑_{i=1}^{n} R(e_{i}, X, Y, e_{i}), ρ̃(X, Y) = ∑_{i=1}^{n} R(fe_{i}, X, Y, e_{i}),

κ = ∑_{i=1}^{n} ρ(e_{i}, e_{i}), κ̃ = ∑_{i=1}^{n} ρ̃(e_{i}, e_{i}).

Both ρ and ρ̃ are symmetric. Moreover, as a consequence of ▽f = 0, we have

ρ(fX, fY) = ρ(X, Y), ρ̃(fX, fY) = ρ̃(X, Y), ρ(fX, Y) = ρ̃(X, Y).

Now, let us suppose that

R(X, Y, Z, W) =

= L_{1}[ρ(X,W)ρ(Y,Z)−ρ(X,Z)ρ(Y,W)+ρ̃(X,W)ρ̃(Y,Z)−ρ̃(X,Z)ρ̃(Y,W)] (1.1)

+L_{2}[ρ(X,W)ρ̃(Y,Z)+ρ̃(X,W)ρ(Y,Z)−ρ(X,Z)ρ̃(Y,W)−ρ̃(X,Z)ρ(Y,W)], where L_{1} and L_{2} are some scalar functions such that L_{1}^{2} − L_{2}^{2} ≠ 0. With respect to the local coordinates, (1.1) can be rewritten as follows

\begin{matrix}R_{ijhk} = & L_{1}[ρ_{ik}ρ_{jh} − ρ_{ih}ρ_{jk} + ρ̃_{ik}ρ̃_{jh} − ρ̃_{ih}ρ̃_{jk}] \\ +L_{2}[ρ_{ik}ρ̃_{jh} + ρ̃_{ik}ρ_{jh} − ρ_{ih}ρ̃_{jk} − ρ̃_{ih}ρ_{jk}], \end{matrix}  (1.2)

from which we have

\begin{matrix}ρ_{ik} − L_{1}(κρ_{ik} + κ̃ρ̃_{ik}) − L_{2}(κ̃ρ_{ik} + κρ̃_{ik}) = −2L_{1}ρ_{ia}ρ_{k}^{a} − 2L_{2}ρ_{ia}ρ̃_{k}^{a}, \\ ρ̃_{ik} − L_{1}(κρ̃_{ik} + κ̃ρ_{ik}) − L_{2}(κρ_{ik} + κ̃ρ̃_{ik}) = −2L_{2}ρ_{ia}ρ_{k}^{a} − 2L_{1}ρ_{ia}ρ̃_{k}^{a}. \end{matrix} (1.3)

The relations (1.3) imply

\begin{matrix}ρ_{ia}ρ_{k}^{a} = −\frac{1}{2} (\frac{L_{1}}{L_{1}^{2} − L_{2}^{2}} − κ) ρ_{ik} + \frac{1}{2} (\frac{L_{2}}{L_{1}^{2} − L_{2}^{2}} + κ̃) ρ̃_{ik}, \\ ρ̃_{ia}ρ_{k}^{a} = −\frac{1}{2} (\frac{L_{1}}{L_{1}^{2} − L_{2}^{2}} − κ) ρ̃_{ik} + \frac{1}{2} (\frac{L_{2}}{L_{1}^{2} − L_{2}^{2}} + κ̃) ρ_{ik}. \end{matrix}  (1.4)

The product conformal curvature tensor is defined by Tachibana in [4]. If this tensor vanishes, then

\begin{matrix}R(X, Y, Z, W) = \\ \frac{1}{4} & \{g(X, W)[Aρ(Y, Z)−Bρ̃(Y, Z)]+g(Y, Z)[Aρ(X, W)−Bρ̃(X, W)] \\ −g(X, Z)[Aρ(Y, W)−Bρ̃(Y, W)]−g(Y, W)[Aρ(X, Z)−Bρ̃(X, Z)] \\ +F (X, W)[Aρ̃(Y, Z)−Bρ(Y, Z)]+F (Y, Z)[Aρ̃(X, W)−Bρ(X, W)] \\ −F (X, Z)[Aρ̃(Y, W)−Bρ(Y, W)]−F (Y, W)[Aρ̃(X, Z)−Bρ(X, Z)]\} \\ −\frac{1}{2}(Cκ−Dκ̃)G(X, Y, Z, W)+ \frac{1}{2}(Dκ−Cκ̃)G(fX, Y, Z, W), \end{matrix} (1.5)

where

\begin{matrix}A = \frac{4(n − 4)}{(n − 4)^{2} − r^{2}} , B = \frac{4r}{(n − 4)^{2} − r^{2}} , \\ C = \frac{2[(n − 2)(n − 4) + r^{2}]}{[(n−4)^{2} −r^{2}][(n−2)^{2} −r^{2}]}, & D = \frac{4r(n − 3)}{[(n−4)^{2} −r^{2}][(n−2)^{2} −r^{2}]}, \\ G(X, Y, Z, W) = g(X, W)g(Y, Z) − g(X, Z)g(Y, W) \\ +F (X, W)F (Y, Z) − F (X, Z)F (Y, W). \end{matrix} (1.6)

64 Mileva Prvanović

We note that (n − 4)^{2} − r^{2} ≠ 0, (n − 2)^{2} − r^{2} ≠ 0, because of p > 2, q > 2.

Thus, if (M, g, f) satisfies (1.1) and its product conformal curvature tensor vanishes, then

4L_{1}[ρ(X,W)ρ(Y,Z)−ρ(X,Z)ρ(Y,W)+ρ̃(X,W)ρ̃(Y,Z)−ρ̃(X,Z)ρ̃(Y,W)] +4L_{2}[ρ(X,W)ρ̃(Y,Z)+ρ̃(X,W)ρ(Y,Z)−ρ(X,Z)ρ̃(Y,W)−ρ̃(X,Z)ρ(Y,W)] = = A[g(X,W)ρ(Y,Z)+g(Y,Z)ρ(X,W)−g(X,Z)ρ(Y,W)−g(Y,W)ρ(X,Z) +F (X,W)ρ̃(Y,Z)+F (Y,Z)ρ̃(X,W)−F (X,Z)ρ̃(Y,W)−F (Y,W)ρ̃(X,Z)] (1.7) −B[g(X,W)ρ̃(Y,Z)+g(Y,Z)ρ̃(X,W)−g(X,Z)ρ̃(Y,W)−g(Y,W)ρ̃(X,Z) +F (X,W)ρ(Y,Z)+F (Y,Z)ρ(X,W)−F (X,Z)ρ(Y,W)−F (Y,W)ρ(X,Z)] −2(Cκ−Dκ̃)G(X,Y,Z,W)+2(Dκ−Cκ̃)G(fX,Y,Z,W).

Let us put

Γ(X,Y,Z,W) = = ρ(X,W)ρ(Y,Z)−ρ(X,Z)ρ(Y,W)+ρ̃(X,W)ρ̃(Y,Z)−ρ̃(X,Z)ρ̃(Y,W),

M(X,Y,Z,W) = = g(X,W)ρ(Y,Z)+g(Y,Z)ρ(X,W)−g(X,Z)ρ(Y,W)−g(Y,W)ρ(X,Z) +F (X,W)ρ̃(Y,Z)+F (Y,Z)ρ̃(X,W)−F (X,Z)ρ̃(Y,W)−F (Y,W)ρ̃(X,Z). Then (1.7) can be rewritten in the form

\begin{matrix}4L_{1}Γ(X,Y,Z,W)+4L_{2}Γ(fX,Y,Z,W) = A M(X,Y,Z,W) \\ −B M(fX,Y,Z,W)−2(Cκ−Dκ̃)G(X,Y,Z,W)+2(Dκ−Cκ̃)G(fX,Y,Z,W). \end{matrix}

(1.8)

Putting into (1.8) fX instead of X, we obtain

\begin{matrix}4L_{2}Γ(X, Y, Z, W)+4L_{1}Γ(fX, Y, Z, W) =−B M(X, Y, Z, W) \\ +A M(fX, Y, Z, W)+2(Dκ−Cκ̃)G(X, Y, Z, W)−2(Cκ−Dκ̃)G(fX, Y, Z, W). \end{matrix}

The last two equations imply

\begin{matrix}Γ(X, Y, Z, W)−\frac{α}{4}M(X, Y, Z, W)−\frac{β}{4} M(fX, Y, Z, W) = \\ =−\frac{L_{1}(Cκ−Dκ̃)+L_{2}(Dκ−Cκ̃)}{2(L_{1}^{2}−L_{2}^{2})} G(X, Y, Z, W) \\ +\frac{L_{1}(Dκ−Cκ̃)+L_{2}(Cκ−Dκ̃)}{2(L_{1}^{2}−L_{2}^{2})} G(fX, Y, Z, W), \end{matrix}  (1.9)

On some product conformally flat locally decomposable Riemannian manifolds 65

where

α = \frac{AL_{1}+BL_{2}}{L_{1}^{2}−L_{2}^{2}} , β =−\frac{AL_{2}+BL_{1}}{L_{1}^{2}−L_{2}^{2}} .  (1.10)

2. The operator Q(X, Y, Z, W, U, V)

We define the operator Q(X,Y,Z,W,U,V) by

Q(X,Y,Z,W,U,V) =

ρ(X,U)R(V,Y,Z,W)+ρ(Y,U)R(X,V,Z,W)+ρ(Z,U)R(X,Y,V,W)

+ρ(W,U)R(X,Y,Z,V)+ρ(X,fU)R(fV,Y,Z,W)+ρ(Y,fU)R(X,fV,Z,W)

+ρ(Z,fU)R(X,Y,fV,W)+ρ(W,fU)R(X,Y,Z,fV)−ρ(X,V)R(U,Y,Z,W)

−ρ(Y,V)R(X,U,Z,W)−ρ(Z,V)R(X,Y,U,W)−ρ(W,V)R(X,Y,Z,U) (2.1) −ρ(X,fV)R(fU,Y,Z,W)−ρ(Y,fV)R(X,fU,Z,W)−ρ(Z,fV)R(X,Y,fU,W)

−ρ(W,fV)R(X,Y,Z,fU).

Applying (2.1) to (1.7) and using (1.2) and (1.5), we get, after some simple but long calculation,

[P ρ(X,U)−Sρ̃(X,U)]G(V,Y,Z,W)+[P ρ(Y,U)−Sρ̃(Y,U)]G(X,V,Z,W) +[P ρ(Z,U)−Sρ̃(Z,U)]G(X,Y,V,W)+[P ρ(W,U)−Sρ̃(W,U)]G(X,Y,Z,V) −[Sρ(X,U)−P ρ̃(X,U)]G(fV,Y,Z,W)−[Sρ(Y,U)−P ρ̃(Y,U)]G(X,fV,Z,W) −[Sρ(Z,U)−P ρ̃(Z,U)]G(X,Y,fV,W)−[Sρ(W,U)−P ρ̃(W,U)]G(X,Y,Z,fV) −[P ρ(X,V)−Sρ̃(X,V)]G(U,Y,Z,W)−[P ρ(Y,V)−Sρ̃(Y,V)]G(X,U,Z,W) (2.2) −[P ρ(Z,V)−Sρ̃(Z,V)]G(X,Y,U,W)−[P ρ(W,V)−Sρ̃(W,V)]G(X,Y,Z,U) +[Sρ(X,V)−P ρ̃(X,V)]G(fU,Y,Z,W)+[Sρ(Y,V)−P ρ̃(Y,V)]G(X,fU,Z,W) +[Sρ(Z,V)−P ρ̃(Z,V)]G(X,Y,fU,W)+[Sρ(W,V)−P ρ̃(W,V)]G(X,Y,Z,fU) = 0,

where

P = αA − βB − 8(Cκ − Dκ̃), S = αB − βA − 8(Dκ − Cκ̃). (2.3)

We put W = V = e_{i} into (2.2). Summing up, we get

\begin{matrix}[P ρ(X,U) − Sρ̃(X,U)][(n − 2)g(Y,Z) + rF (Y,Z)] \\ −[P ρ(Y,U) − Sρ̃(Y,U)][(n − 2)g(X,Z) + rF (X,Z)] \\ +[P ρ̃(X,U) − Sρ(X,U)][(n − 2)F (Y,Z) + rg(Y,Z)] \\ −[P ρ̃(Y,U) − Sρ(Y,U)][(n − 2)F (X,Z) + rg(X,Z)] \\ −2[P ρ(X,Z) − Sρ̃(X,Z)]g(Y,U) + 2[P ρ(Y,Z) − Sρ̃(Y,Z)]g(X,U) \\ −2[P ρ̃(X,Z) − Sρ(X,Z)]F (Y,U) + 2[P ρ̃(Y,Z) − Sρ(Y,Z)]F (X,U) \\ −(P κ − Sκ̃)G(X,Y,Z,U) − (P κ̃ − Sκ)G(fX,Y,Z,U) = 0. \end{matrix} (2.4)

Now, we put Y = Z = e_{i} into (2.4). Summing up, we obtain

\begin{matrix}[n(n − 4) + r^{2}][P ρ(X, U) − Sρ̃(X, U)] \\ +2r(n − 2)[P ρ̃(X, U) − Sρ(X, U)] \\ −[(n − 4)(P κ − Sκ̃) + r(P κ̃ − Sκ)]g(X, U) \\ −[(n − 4)(P κ̃ − Sκ) + r(P κ − Sκ̃)]F (X, U) = 0. \end{matrix}  (2.5)

Putting into (2.5) fX instead of X, we find

\begin{matrix}[n(n − 4) + r^{2}][P ρ̃(X, U) − Sρ(X, U)] \\ +2r(n − 2)[P ρ(X, U) − Sρ̃(X, U)] \\ −[(n − 4)(P κ − Sκ̃) + r(P κ̃ − Sκ)]F (X, U) \\ −[(n − 4)(P κ̃ − Sκ) + r(P κ − Sκ̃)]g(X, U) = 0. \end{matrix}  (2.6)

According to our supposition, (n − 4)^{2} − r^{2} ≠ 0. Thus, from (2.5) and (2.6), we get

(P^{2}−S^{2}) \{ρ(X,U)−\frac{1}{n^{2}−r^{2}} [(nκ−rκ̃)g(X,U)−(rκ−nκ̃)F (X,U)]\} = 0. (2.7)

The relation (2.7) shows that

ρ(X, U) = \frac{1}{n^{2} − r^{2}} [(nκ − rκ̃)g(X, U) − (rκ − nκ̃)F (X, U)],  (2.8)

or

P^{2} − S^{2} = 0.  (2.9)

The condition (2.8) can be rewritten in the form ρ(X, U) = λg(X, U) + μF (X, U).

On some product conformally flat locally decomposable Riemannian manifolds 67

Substituting this into (1.1), we find

\begin{aligned}R(X, Y, Z, W) & = [(λ^{2} + μ^{2})L_{1} + 2λμL_{2}]G(X, Y, Z, W) \\ & +[2λμL_{1} + (λ^{2} + μ^{2})L_{2}]G(fX, Y, Z, W), \end{aligned}

that is, (M, g, f) is manifold of \underline{almost constant curvature}.

The condition (2.9) will be examined in the sections 3, 4 and 5.

3. The case P = S ≠ 0

According to (2.3), this condition implies

(α + β)(A − B) = 8(C − D)(κ + κ̃).

But, in view of (1.6), we have

A − B = \frac{4}{(n − 4) + r} , C − D = \frac{2}{[(n − 4) + r][(n − 2) + r]}.

Thus

α + β = \frac{4(κ + κ̃)}{(n − 2) + r} .  (3.1)

On the other hand, in view of (1.10), we have

α + β = \frac{AL_{1} + BL_{2} − AL_{2} − BL_{1}}{L_{1}^{2} − L_{2}^{2}} = \frac{4}{[(n − 4) + r](L_{1} + L_{2})}.

This and (3.1) give

L_{1} + L_{2} = \frac{1}{(κ + κ̃)} \frac{[(n − 2) + r]}{[(n − 4) + r]}.  (3.2)

In the case P = S, (2.5) reduces to

ρ(X, U) − ρ̃(X, U) = \frac{κ − κ̃}{n − r}[g(X, U) − F (X, U)]  (3.3)

or, in the local coordinates

ρ_{ij} − ρ̃_{ij} = \frac{κ − κ̃}{n − r}(g_{ij} − F_{ij}).  (3.4)

From (3.4), we find

(ρ_{ia} − ρ̃_{ia})ρ_{j}^{a} = (\frac{κ − κ̃}{n − r})^{2} (g_{ij} − F_{ij}).  (3.5)

On the other hand, from (1.4) we get

(ρ_{ia} − ρ̃_{ia})ρ_{j}^{a} = −\frac{1}{2} [\frac{1}{L_{1} − L_{2}} − (κ − κ̃)] (\frac{κ − κ̃}{n − r}) (g_{ij} − F_{ij}).

This, together with (3.5) gives

L_{1} − L_{2} = \frac{n − r}{[(n − 2) − r]} · \frac{1}{(κ − κ̃)}.  (3.6)

Finally, (3.2) and (3.6) imply

\begin{matrix}L_{1} = \frac{1}{2} \{\frac{1}{(κ + κ̃)} \frac{[(n − 2) + r]}{[(n − 4) + r]} + \frac{1}{(κ − κ̃)} \frac{(n − r)}{[(n − 2) − r]}\}, \\ L_{2} = \frac{1}{2} \{\frac{1}{(κ + κ̃)} \frac{[(n − 2) + r]}{[(n − 4) + r]} − \frac{1}{(κ − κ̃)} \frac{(n − r)}{[(n − 2) − r]}\} \end{matrix}  (3.7)

Now, let us consider the left hand side of (1.9). Explicitly, it is ρ(X,W)ρ(Y,Z)−ρ(X,Z)ρ(Y,W)+ρ̃(X,W)ρ̃(Y,Z)−ρ̃(X,Z)ρ̃(Y,W)

−\frac{α}{2}[g(X,W)ρ(Y,Z)+g(Y,Z)ρ(X,W)−g(X,Z)ρ(Y,W)−g(Y,W)ρ(X,Z)

+F (X,W)ρ̃(Y,Z)+F (Y,Z)ρ̃(X,W)−F (X,Z)ρ̃(Y,W)−F (Y,W)ρ̃(X,Z)]

−\frac{β}{2} [F (X,W)ρ(Y,Z)+g(Y,Z)ρ̃(X,W)−F (X,Z)ρ(Y,W)−g(Y,W)ρ̃(X,Z)

+g(X,W)ρ̃(Y,Z)+F (Y,Z)ρ(X,W)−g(X,Z)ρ̃(Y,W)−F (Y,W)ρ(X,Z)].

(3.8)

But, in view of (3.3), we have

ρ̃(X, Y) = ρ(X, Y) − \frac{κ − κ̃}{n − r}[g(X, Y) − F (X, Y)].

Substituting this into (3.8), we obtain, after some calculation, the following expression for (1.9):

2 \{ρ(X,W)−\frac{κ−κ̃}{2(n−r)}[g(X,W)−F (X,W)]−\frac{α+β}{8} [g(X,W)+F (X,W)]\} ×

× \{ρ(Y,Z)−\frac{κ−κ̃}{2(n−r)}[g(Y,Z)−F (Y,Z)]−\frac{α+β}{8} [g(Y,Z)+F (Y,Z)]\}

−2 \{ρ(X,Z)−\frac{κ−κ̃}{2(n−r)}[g(X,Z)−F (X,Z)]−\frac{α+β}{8} [g(X,Z)+F (X,Z)]\} ×

× \{ρ(Y,W)−\frac{κ−κ̃}{2(n−r)}[g(Y,W)−F (Y,W)]−\frac{α+β}{8} [g(Y,W)+F (Y,W)]\} =

= \{−\frac{L_{1}(Cκ−Dκ̃)+L_{2}(Dκ−Cκ̃)}{2(L_{1}^{2}−L_{2}^{2})}−\frac{1}{2} (\frac{κ−κ̃}{n−r})^{2}+\frac{(α+β)^{2}}{32}

+\frac{α−β}{4} (\frac{κ−κ̃}{n−r})\} G(X,Y,Z,W) (3.9)

+\{\frac{L_{1}(Dκ−Cκ̃)+L_{2}(Cκ−Dκ̃)}{2(L_{1}^{2}−L_{2}^{2})}+\frac{1}{2} (\frac{κ−κ̃}{n−r})^{2}+\frac{(α+β)^{2}}{32} −

−\frac{α−β}{4} (\frac{κ−κ̃}{n−r})\} G(fX,Y,Z,W).

In view of (1.6), (1.10) and (3.7), we can see that

\frac{L_{1}(Cκ − Dκ̃) + L_{2}(Dκ − Cκ̃)}{2(L_{1}^{2} − L_{2}^{2})} = \frac{(κ − κ̃)^{2}}{2(n − r)[(n − 4) − r]} + \frac{(κ + κ̃)^{2}}{2[(n − 2) + r]^{2}} ,

\frac{L_{1}(Dκ − Cκ̃) + L_{2}(Cκ − Dκ̃)}{2(L_{1}^{2} − L_{2}^{2})} = \frac{(κ − κ̃)^{2}}{2(n − r)[(n − 4) − r]} − \frac{(κ + κ̃)^{2}}{2[(n − 2) + r]^{2}} ,

\frac{1}{4} (\frac{κ − κ̃}{n − r}) (α − β) = (\frac{κ − κ̃}{n − r})^{2} \frac{[(n − 2) − r]}{[(n − 4) − r]}.

These relations, together with (3.1), imply

\begin{matrix}\frac{L_{1}(Cκ−Dκ̃)+L_{2}(Dκ−Cκ̃)}{2(L_{1}^{2}−L_{2}^{2})}+\frac{1}{2} (\frac{κ−κ̃}{n−r})^{2}−\frac{(α+β)^{2}}{32} −\frac{α−β}{4} (\frac{κ−κ̃}{n−r}) = 0, \\ \frac{L_{1}(Dκ−Cκ̃)+L_{2}(Cκ−Dκ̃)}{2(L_{1}^{2}−L_{2}^{2})}+\frac{1}{2} (\frac{κ−κ̃}{n−r})^{2}+\frac{(α+β)^{2}}{32} −\frac{α−β}{4} (\frac{κ−κ̃}{n−r}) = 0, \end{matrix}

70 Mileva Prvanović

because of which, (3.9) reduces to

\{ρ(X,W)−\frac{κ−κ̃}{2(n−r)}[g(X,W)−F (X,W)]−\frac{α+β}{8} [g(X,W)+F (X,W)]\} ×

× \{ρ(Y,Z)−\frac{κ−κ̃}{2(n−r)}[g(Y,Z)−F (Y,Z)]−\frac{α+β}{8} [g(Y,Z)+F (Y,Z)]\}

−\{ρ(X,Z)−\frac{κ−κ̃}{2(n−r)}[g(X,Z)−F (X,Z)]−\frac{α+β}{8} [g(X,Z)+F (X,Z)]\} ×

× \{ρ(Y,W)−\frac{κ−κ̃}{2(n−r)}[g(Y,W)−F (Y,W)]−\frac{α+β}{8} [g(Y,W)+F (Y,W)]\} = 0.

This yields

ρ(X,Y)−\frac{κ−κ̃}{2(n−r)}[g(X,Y)−F (X,Y)]−\frac{α+β}{8} [g(X,Y)+F (X,Y)] = θv(X)v(Y),

where v(X) is l-form such that v(fX)=v(X) and θ is a scalar function. Without loss of the generality, we can suppose that ∑_{i=1}^{n} v(e_{i})v(e_{i}) = 1. Then

θ = −\frac{κ + κ̃}{(n − 2) + r} ,

and therefore

\begin{aligned}ρ(X,Y) = & \frac{1}{2} [\frac{κ+κ̃}{(n−2)+r} +\frac{κ−κ̃}{n−r}] g(X,Y) \\ & +\frac{1}{2} [\frac{κ+κ̃}{(n−2)+r} −\frac{κ−κ̃}{n−r}] F (X,Y)−\frac{κ+κ̃}{(n−2)+r} v(X)v(Y). \end{aligned}  (3.10)
4. The case P = −S ≠ 0

This condition, in view of (2.3) and (1.6), gives

α − β = \frac{4(κ − κ̃)}{(n − 2) − r} .  (4.1)

On the other hand, in view of (1.10),

α − β = \frac{4}{[(n − 4) − r]} · \frac{1}{(L_{1} − L_{2})}.

On some product conformally flat locally decomposable Riemannian manifolds 71

Thus

L_{1} − L_{2} = \frac{(n − 2) − r}{(n − 4) − r} · \frac{1}{(κ − κ̃)}.  (4.2)

If P = −S, (2.5) reduces to

ρ(X, U) + ρ̃(X, U) = \frac{κ + κ̃}{n + r}[g(X, U) + F (X, U)],  (4.3)

or, in the local coordinates,

ρ_{ij} + ρ̃_{ij} = \frac{κ + κ̃}{n + r}(g_{ij} + F_{ij}).

Therefore,

(ρ_{ia} + ρ̃_{ia})ρ_{j}^{a} = (\frac{κ + κ̃}{n + r})^{2} (g_{ij} + F_{ij}).  (4.4)

On the other hand, using (1.4), we find

(ρ_{ia} + ρ̃_{ia})ρ_{j}^{a} = \frac{1}{2} [−\frac{1}{L_{1} + L_{2}} + (κ + κ̃)] (\frac{κ + κ̃}{n + r}) (g_{ij} + F_{ij}).  (4.5)

The equations (4.4) and (4.5) show that

(\frac{κ + κ̃}{n + r}) (g_{ij} + F_{ij}) = \frac{1}{2} [−\frac{1}{L_{1} + L_{2}} + (κ + κ̃)] (g_{ij} + F_{ij}),

from which we find

L_{1} + L_{2} = \frac{n + r}{[(n − 2) + r]} \frac{1}{(κ + κ̃)}.  (4.6)

From (4.2) and (4.6), we get

\begin{matrix}L_{1} = \frac{1}{2} [\frac{1}{(κ − κ̃)} \frac{(n − 2) − r}{[(n − 4) − r]} + \frac{1}{(κ + κ̃)} \frac{n + r}{[(n − 2) + r]}] , \\ L_{2} = \frac{1}{2} [−\frac{1}{(κ − κ̃)} \frac{(n − 2) − r}{[(n − 4) − r]} + \frac{1}{(κ + κ̃)} \frac{n + r}{[(n − 2) + r]}] . \end{matrix} (4.7)

According (4.3),

ρ̃(X, U) = −ρ(X, U) + \frac{κ + κ̃}{n + r}[g(X, U) + F (X, U)].

72 Mileva Prvanović

Substituting this into (3.8), we find that (1.9), in the case P = −S ≠ 0, can be expressed in the form

\{ρ(X,W)−\frac{κ+κ̃}{2(n+r)}[g(X,W)+F (X,W)]−\frac{α−β}{8} [g(X,W)−F (X,W)]\} ×

× \{ρ(Y,Z)−\frac{κ+κ̃}{2(n+r)}[g(Y,Z)+F (Y,Z)]−\frac{α−β}{8} [g(Y,Z)−F (Y,Z)]\}

−\{ρ(X,Z)−\frac{κ+κ̃}{2(n+r)}[g(X,Z)+F (X,Z)]−\frac{α−β}{8} [g(X,Z)−F (X,Z)]\} ×

× \{ρ(Y,W)−\frac{κ+κ̃}{2(n+r)}[g(Y,W)+F (Y,W)]−\frac{α−β}{8} [g(Y,W)−F (Y,W)]\} = 0.

This equation implies

ρ(X,Y)−\frac{κ+κ̃}{2(n+r)}[g(X,Y)+F (X,Y)]−\frac{α−β}{8} [g(X,Y)−F (X,Y)] = φw(X)w(Y),

where w(X) is a l-form and φ is a scalar function. Without loss of the generality, we can suppose that ∑_{i+1}^{n} w(e_{i})w(e_{i}) = 1. Then

φ = −\frac{κ − κ̃}{(n − 2) − r}

and therefore

\begin{aligned}ρ(X, Y) = & \frac{1}{2} [\frac{κ+κ̃}{n+r}+\frac{κ−κ̃}{(n−2)−r} ] g(X, Y) \\ & +\frac{1}{2} [\frac{κ+κ̃}{n+r}−\frac{κ−κ̃}{(n−2)−r} ] F (X, Y)−\frac{κ−κ̃}{(n−2)−r} w(X)w(Y). \end{aligned}  (4.8)
5. The case P = S = 0

In this case we obtain from (2.3)

α = \frac{4[(n − 2)κ − rκ̃]}{(n − 2)^{2} − r^{2}}, β = \frac{4[(n − 2)κ̃ − rκ]}{(n − 2)^{2} − r^{2}}.  (5.1)

Thus

α + β = \frac{4}{(n − 2) + r} (κ + κ̃), α − β = \frac{4}{(n − 2) − r} (κ − κ̃).  (5.2)

On some product conformally flat locally decomposable Riemannian manifolds 73

On the other hand, in view of (1.10), we have

L_{1} + L_{2} = \frac{A − B}{α + β} , L_{1} − L_{2} = \frac{A + B}{α − β} .

Therefore

L_{1} = \frac{1}{2} [\frac{A − B}{α + β} + \frac{A + B}{α − β} ] , L_{2} = \frac{1}{2} [\frac{A − B}{α + β} − \frac{A + B}{α − β} ] .

Finally, taking into account (1.10) and (5.2), we obtain

\begin{matrix}L_{1} = \frac{1}{2} [\frac{1}{(κ + κ̃)} \frac{(n − 2) + r}{[(n − 4) + r]} + \frac{1}{(κ − κ̃)} \frac{(n − 2) − r}{[(n − 4) − r]}] \\ L_{2} = \frac{1}{2} [\frac{1}{(κ + κ̃)} \frac{(n − 2) + r}{[(n − 4) + r]} − \frac{1}{(κ − κ̃)} \frac{(n − 2) − r}{[(n − 4) − r]}] . \end{matrix}  (5.3)

Now, we note that (1.9) can be rewritten in the form

[ρ(X,W)−\frac{α}{4}g(X,W)−\frac{β}{4} F (X,W)][ρ(Y,Z)−\frac{α}{4}g(Y,Z)−\frac{β}{4} F (Y,Z)]

−[ρ(X,Z)−\frac{α}{4}g(X,Z)−\frac{β}{4} F (X,Z)][ρ(Y,W)−\frac{α}{4}g(Y,W)−\frac{β}{4} F (Y,W)]

+[ρ̃(X,W)−\frac{α}{4}F (X,W)−\frac{β}{4} g(X,W)][ρ̃(Y,Z)−\frac{α}{4}F (Y,Z)−\frac{β}{4} g(Y,Z)] (5.4)

−[ρ̃(X,Z)−\frac{α}{4}F (X,Z)−\frac{β}{4} g(X,Z)][ρ̃(Y,W)−\frac{α}{4}F (Y,W)−\frac{β}{4} g(Y,W)] =

= [\frac{α^{2}+β^{2}}{16} −\frac{L_{1}(Cκ−Dκ̃)+L_{2}(Dκ−Cκ̃)}{2(L_{1}^{2}−L_{2}^{2})}] G(X,Y,Z,W)

+[\frac{αβ}{8} +\frac{L_{1}(Dκ−Cκ̃)+L_{2}(Cκ−Dκ̃)}{2(L_{1}^{2}−L_{2}^{2})}] G(fX,Y,Z,W).

But, in the case P = S = 0, (5.1) and (5.3) hold good, because of which we have

\begin{matrix}\frac{α^{2} + β^{2}}{16} − \frac{L_{1}(Cκ − Dκ̃) + L_{2}(Dκ − Cκ̃)}{2(L_{1}^{2} − L_{2}^{2})} = 0 \\ \frac{αβ}{8} + \frac{L_{1}(Dκ − Cκ̃) + L_{2}(Cκ − Dκ̃)}{2(L_{1}^{2} − L_{2}^{2})} = 0 \end{matrix}

In other words, the equation (5.4), in the case P = S = 0, reduces to

\begin{matrix}[ρ(X,W)−\frac{α}{4}g(X,W)−\frac{β}{4} F (X,W)][ρ(Y,Z)−\frac{α}{4}g(Y,Z)−\frac{β}{4} F (Y,Z)] \\ −[ρ(X,Z)−\frac{α}{4}g(X,Z)−\frac{β}{4} F (X,Z)][ρ(Y,W)−\frac{α}{4}g(Y,W)−\frac{β}{4} F (Y,W)] \\ +[ρ̃(X,W)−\frac{α}{4}F (X,W)−\frac{β}{4} g(X,W)][ρ̃(Y,Z)−\frac{α}{4}F (Y,Z)−\frac{β}{4} g(Y,Z)] \\ −[ρ̃(X,Z)−\frac{α}{4}F (X,Z)−\frac{β}{4} g(X,Z)][ρ̃(Y,W)−\frac{α}{4}F (Y,W)−\frac{β}{4} g(Y,W)] = 0. \end{matrix} (5.5)

In view of (5.1), the equation (5.5) yields

\begin{matrix}ρ(X, Y) = \frac{(n − 2)κ − rκ̃}{(n − 2)^{2} − r^{2}}g(X, Y) − \frac{rκ − (n − 2)κ̃}{(n − 2)^{2} − r^{2}}F (X, Y) \\ +η[u(X)u(Y) + ũ(X)ũ(Y)] + ψ[ũ(X)u(Y) + u(X)ũ(Y)], \end{matrix} (5.6)

where η and ψ are some scalar functions, while u(X) is a l-form and ũ(X) =

u(fX). Without loss of the generality, we can suppose that ∑_{i=1}^{n} u(e_{i})u(e_{i}) = 2.

Then, putting into (5.6) X = Y = e_{i} and summing up, we find

η = −\frac{(n − 2)κ − rκ̃}{2[(n − 2)^{2} − r^{2}]}.

In a similar way, using ρ̃(X, Y), we get

ψ = \frac{rκ − (n − 2)κ̃}{2[(n − 2)^{2} − r^{2}]}.

Thus

\begin{aligned}ρ(X,Y) = & \frac{(n−2)κ−rκ̃}{(n−2)^{2}−r^{2}} \{g(X,Y)−\frac{1}{2} [u(X)u(Y)+ũ(X)ũ(Y)]\} \\ & −\frac{rκ−(n−2)κ̃}{(n−2)^{2}−r^{2}} \{F (X,Y)−\frac{1}{2} [ũ(X)u(Y)+u(X)ũ(Y)]\} . \end{aligned}  (5.7)

6. Conclusion

Summing up the results obtained in the sections 2, 3, 4 and 5, we can state the following

Theorem. Let (M, g, f) be a locally decomposable Riemannian manifold, dim M = n = p + q, tr f = r = p − q, p > 2, q > 2. If (M, g, f) is product conformally flat and the Riemannian curvature tensor satisfies the condition (1.1) where L_{1} and L_{2} are some scalar functions such that L_{1}^{2} −L_{2}^{2} ≠ 0, then there occurs one of the following cases:

(i) (M, g, f) is a manifold of almost constant curvature at the points where κ^{2} − κ̃^{2} ≠ 0,

(ii) the relations (3.7) and (3.10) are fulfilled, where v(X) is 1-form such

that ∑_{i=1}^{n} v(e_{i})v(e_{i}) = 1;

(iii) the relations (4.7) and (4.8) are fulfilled, where w(X) is 1-form such

that ∑_{i=1}^{n} w(e_{i})w(e_{i}) = 1;

(iv) the relations (5.3) and (5.7) are fulfilled, where u(X) is 1-form such

that ∑_{i=1}^{n} u(e_{i})u(e_{i}) = 2 and ũ(X) = u(fX).

REFERENCES

[1] D. K o w a l c z y k, On some subclass of semisymmetric manifolds, Soochow Journal of Mathematics, vol. 27 (2001), 445–461.

[2] C. M u r a t h a n, K. A r s l a n, R. D e s z c z, R. E z e n t o ş and C. Ö z g ü r, On some class of hypersurfaces of semi-Euclidean spaces, Publ. Math. Debrecen, 58 (2001), 587–604.

[3] W. R o t e r, A note on second order recurrent spaces, Bull. Acad. Polon. Sci., Ser. sci., math., astr. et phys. 12 (1964), 621–626.

[4] S. T a c h i b a n a, Some theorems on locally product Riemannian spaces, Tôhoku Math. J. 12 (1960), 281–292.

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