# On some classes of weakly projective symmetric manifolds.

1. Introduction

As is well known, symmetric spaces play an important role in differential geometry. The study of Riemannian symmetric spaces was initiated in the late twenties by Cartan , who, in particular, obtained a classification of those spaces. Let ([M.sup.n], g), (n = dimM) be a Riemannian manifold, i.e., a manifold M with the Riemannian metric g, and let [nabla] be the Levi-Civita connection of ([M.sup.n], g). A Riemannian manifold is called locally symmetric  if [nabla]R = 0, where R is the Riemannian curvature tensor of ([M.sup.n], g). This condition of local symmetry is equivalent to the fact that at every point P [member of] M, the local geodesic symmetry F(P) is an isometry . The class of Riemannian symmetric manifolds is very natural generalization of the class of manifolds of constant curvature. During the last six decades the notion of locally symmetric manifolds have been weakened by many authors in several ways to different extent such as conformally symmetric manifolds by Chaki and Gupta , recurrent manifolds introduced by Walker , conformally recurrent manifolds by Adati and Miyazawa , pseudo symmetric manifolds by Chaki , weakly symmetric manifolds by Tamassy and Binh  etc.

A non-flat Riemannian manifold ([M.sup.n], g)(n > 2) is called weakly symmetric  if the curvature tensor R of type (0,4) satisfies the condition

([[nabla].sub.X]R)(Y, Z, U, V) = A(X)R(Y, Z, U, V) + B(Y)R(X, Z, U, V) + C(Z)R(Y, X, U, V) + D(U)R(Y, Z, X, V) + E(V)R(Y, Z, U, X), (1.1)

where R(Y, Z, U, V) = g(R(Y, Z)U, V), R is the curvature tensor of type (1,3) and A, B, C, D and E are 1-forms respectively which are non-zero simultaneously. Such a manifold is denoted by [(WS).sub.n]. It was proved in  that the 1-forms must be related as follows

B = C and D = E.

That is, the weakly symmetric manifold is characterized by the condition

([[nabla].sub.X]R)(Y, Z, U, V) = A(X)R(Y, Z, U, V) + B(Y)R(X, Z, U, V) + B(Z)R(Y, X, U, V) + D(U)R(Y, Z, X, V) + D(V)R(Y, Z, U, X). (1.2)

The 1-forms A,B and D are called the associated 1-forms. If in (1.2) the 1-form A is replaced by 2A; B and D are replaced by A, then the manifold ([M.sup.n], g) reduces to a pseudo symmetric manifold in the sense of Chaki .

Again if A = B = D = 0, the manifold defined by (1.2) reduces to a symmetric manifold in the sense of Cartan. The existence of a [(WS).sub.n] was proved by Prvanovic  and a concrete example is given by De and Bandyopadhyay (, ).

Weakly symmetric manifolds have been studied by several authors (, , , , , , , , , , , ) and many others.

Let [[rho].sub.1], [[rho].sub.2] and [[rho].sub.3] are the basic vectors corresponding to the 1-forms A, B and D respectively, that is

g(X, [[rho].sub.1]) = A(X), g(X, [[rho].sub.2]) = B(X) and g(X, [[rho].sub.3]) = D(X) and g(X, [[rho].sub.4]) = E(X). (1.3)

In 1993 Tamassy and Binh  introduced the notion of weakly Ricci symmetric manifolds. A non-flat Riemannian manifold ([M.sup.n], g)(n > 2) is called weakly Ricci symmetric if its Ricci tensor S of type (0,2) is not identically zero and satisfies the condition

([[nabla].sub.X]S)(Y, Z) = A(X)S (Y,Z) + B (Y)S (X,Z) + C (Z)(Y,X), (1.4)

where A, B, C are three non-zero 1-forms, and [nabla] denotes the operator of covariant differentiation with respect to the metric g. Such an n-dimensional manifold is denoted by [(WRS).sub.n].

The projective curvature tensor is an important tensor from the differential geometric point of view. Let M be an n-dimensional Riemannian manifold. If there exists a one-to-one correspondence between each coordinate neighbourhood of M and a domain in Euclidean space, then M is said to be locally projectively flat. For n [greater than or equal to] 3, M is locally projectively flat if and only if the well known projective curvature tensor P vanishes. Here P is defined by 

P(Y, Z)U = R(Y, Z) U - 1/n - 1 [S(Z, U)Y - S(Y, U)Z], (1.5)

for all Y, Z, U [member of] T(M), where R is the curvature tensor and S is the Ricci tensor. In fact M is projectively flat if and only if it is a constant curvature . Thus the projective curvature tensor is the measure of the failure of a Riemannian manifold to be of constant curvature.

Now (1.5) can be expressed as

P(Y, Z, U, V) = R(Y, Z, U, V) - 1/n - 1 [S(Z, U)g(Y, V) - S(Y, U)g(Z, V)], (1.6)

where P(Y, Z, U, V) = g(P(Y, Z)U, V). Since the projective curvature tensor does not satisfy all the properties of Riemannian curvature tensor, therefore weakly projective symmetric manifold is characterized by the condition

([[nabla].sub.X]P)(Y, Z, U, V) = A(X)P(Y, Z, U, V) + B(Y)P(X, Z, U, V) + C(Z)P(Y, X, U, V) + D(U)P(Y, Z, X, V) + E(V)P(Y, Z, U, X), (1.7)

where the 1-forms A, B, C, D and E are not zero simultaneously. Such a manifold is denoted by [(WPS).sub.n]. In a recent paper Shaikh and Hui  proved that in a [(WPS).sub.n], B = C. Hence (1.7) can be expressed as

([[nabla].sub.X]P)(Y, Z, U, V) = A(X)P(Y, Z, U, V) + B(Y)P(X, Z, U, V) + B(Z)P(Y, X, U, V) + D(U)P(Y, Z, X, V) + E(V)P(Y, Z, U, X). (1.8)

where P(Y, Z, U, V) = g(P(Y, Z)U, V).

Recently, Mantica and Molinari  have studied weakly-Z-symmetric manifolds. On the otherhand, Mantica and Suh (, ) have studied pseudo-Z-symmetric Riemannian manifolds with harmonic curvature tensors, pseudo-Q-symmetric Riemannian manifolds. Moreover Mantica and Suh investigated deeply pseudo-Z-symmetric spacetimes . Motivated by the above studies in the present paper we have studied a type of non-flat Riemannian manifold defined by (1.7) and (1.8).

The paper is organized as follows:

After preliminaries, in Section 3, some properties of the 1-forms of a [(WPS).sub.n] have been studied. In Section 4, we study conformally flat [(WPS).sub.n]. Section 5 deals with the property of a [(WPS).sub.n], D = E and D([[rho].sub.3]) [not equal to] 0, with [[rho].sub.3] as a unit torse-forming vector field. Section 6 is devoted to the study of a [(WPS).sub.4] perfect fluid spacetime. Finally, we give an example of a [(WPS).sub.n].

2. Preliminaries

Let S and r denote the Ricci tensor of type (0,2) and the scalar curvature respectively and L denotes the symmetric endomorphism of the tangent space at each point corresponding to the Ricci tensor S, that is,

g(LX,Y) = 5 (X,Y). (2.1)

In this section, some formulas are derived, which will be useful to the study of [(WPS).sub.n]. Let {[e.sub.i]} be an orthonormal basis of the tangent space at each point of the manifold where 1 [less than or equal to] i [less than or equal to] n.

From (1.5) we can easily verify that the tensor P satisfies the following properties:

i) P (Y,Z)U = -P (Z,Y)U,

ii) P(Y, Z)U + P(Z, U)Y + P(U, Y)Z = 0. (2.2)

Also from (1.6) we have

[[summation].sup.n.sub.i=1] P(Y, Z, [e.sub.i], [e.sub.i]) = 0 = [[summation].sup.n.sub.i=1] P([e.sub.i], [e.sub.i], U, V) = [[summation].sup.n.sub.i=1] P([e.sub.i], Z, U, [e.sub.i]) (2.3)

and

[[summation].sup.n.sub.i=1] P (Y, [e.sub.i], [e.sub.i], V) = n/n - 1 [S(Y, V) - r/n g(Y,V)], (2.4)

where r = [[summation].sup.n.sub.i=1] S([e.sub.i], [e.sub.i]) is the scalar curvature.

From (1.6) it follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.5)

3. Some properties of the 1-forms of a [(WPS).sub.n] (n > 2)

Let ([M.sup.n], g) be a [(WPS).sub.n]. Using (1.6) in (1.8) we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Contracting (3.1) over Y and V we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

or,

B(R(X, Z)U) - 1/n - 1 {B(X)S(Z, U) - B(Z)S(X, U)} -E(R(U, X)Z) - 1/n - 1 {E(X)S(Z, U) - E(LU)g(X, Z)} = 0. (3.2)

Contracting (3.2) over X and U we get

B (LZ) = r/n B(Z). (3.3)

Replacing Z by X in (3.3) we get

B(LX) = r/n B(X), (3.4)

which can be written as

S (X, [[rho].sub.2]) = r/n g(X, [[rho].sub.2]). (3.5)

Hence we have the following theorem:

Theorem 3.1. In a [(WPS).sub.n], r/n is an eigen value of the Ricci tensor S corresponding to the eigenvector [[rho].sub.2] by (1.3).

In a recent paper Shaikh and Hui  proved that

(B + E)(LX) = r/n (B + E)(X). (3.6)

From (3.4) and (3.6) we can conclude that

E (LX) = r/n E(X), (3.7)

which can be written as

(X, [[rho].sub.4]) = r/n g(X, [[rho].sub.4]). (3.8)

Hence we have the following theorem:

Theorem 3.2. In a [(WPS).sub.n], r/n is an eigenvalue of the Ricci tensor S corresponding to the eigenvector [[rho].sub.4] defined by (1.3).

Now contracting (3.1) over U and V we get

D(R(Y, Z)X) - 1/n - 1 {D(y)S(X, Z) - D(Z)S(X,Y)} - E(R(Y, Z)X) - 1/n - 1 {E(LZ)g(X, Y) - E(LY)g(X, Z)} = 0. (3.9)

Again contracting (3.9) over X and Z we get

D(LY) = r/n D(Y). (3.10)

Replacing Y by X from (3.10) we get

D(LX) = r/n D(X). (3.11)

or,

S (X, [[rho].sub.3]) = r/n g(X, [[rho].sub.3]). (3.12)

Thus we have the following theorem:

Theorem 3.3. In a [(WPS).sub.n], r/n is an eigenvalue of the Ricci tensor S corresponding to the eigenvector [[rho].sub.3] defined by (1.3).

If possible we assume that in a [(WPS).sub.n] D = E. Then from (3.9) we get

-D(Y)S(X, Z) + D(Z)S(X, Y) -D(LZ)g(X, Y) + D(LY)g(X, Z) = 0. (3.13)

Now using (3.11) in (3.13) we get

D(Y)S(X, Z) - D(Z)S(X, Y) + r/n D(Z)g(X, Y) - r/n D(Y)g(X, Z) = 0.

or,

D(Y)[S(X, Z) - r/n g(X,Z)] - D(Z)[S(X,Y) - r/n g(X,Y)] = 0. (3.14)

Putting Y = [[rho].sub.3] in (3.14) we get

D([[rho].sub.3])[S(X, Z) - r/n g(X, Z)] - D(Z)D(LX) + r/n D(X)D(Z) = 0. (3.15)

Again using (3.11) in (3.15) we get

D([[rho].sub.3])[S(X, Z) - r/n g(X, Z)] = 0. (3.16)

Now let D([[rho].sub.3]) [not equal to] 0 then from (3.16) we get

S(X, Z) = r/n g(X,Z). (3.17)

Hence in this case the [(WPS).sub.n] is an Einstein manifold. Thus we can state the following theorem:

Theorem 3.4. If in a [(WPS).sub.n], D = E and D([[rho].sub.3]) = 0, then the manifold reduces to an Einstein manifold.

Contracting (3.1) over Z and U we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

or,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3.18)

Let in this [(WPS).sub.n] r is a non-zero constant and the manifold is (WRS)n with the same 1-forms A, B and E then from (3.18) we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3.19)

Again contracting (3.19) over Y and V we get

-rA(X) - r/n B(X) + 1/n B(LX) + 1/n D(LX) - r/n E(X) = 0. (3.20)

Using (3.4) and (3.11) in (3.20) we get

[n.sup.2] A(X) + (n - 1)B(X) - D(X) + nE(X) = 0. (3.21)

Thus we have the following theorem:

Theorem 3.5. If a [(WPS).sub.n] with non-zero constant scalar curvature is also [(WRS).sub.n] with the same 1-forms A, B and E then [n.sup.2] A(X) + (n - 1)B(X) - D(X) + nE(X) = 0 holds for all X.

4. Conformally flat [(WPS).sub.n](n > 3)

In this section we assume that the manifold [(WPS).sub.n] is conformally flat. Then divC = 0 where C denotes the Weyl's conformal curvature tensor and 'div' denotes divergence. Hence we have 

([[nabla].sub.X]S)(Y,Z) - ([[nabla].sub.Z]S)(X, Y) = 1/2(n - 1) [g(Y,Z)dr(X) - g(X, Y)dr(Z)]. (4.1)

Now replacing V by Z in (3.18) we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4.2)

Contracting (4.2) over X and Z we get

(n - 2/2n) dr(Y) = A(LY)/ - r/n A(Y) + E(LY) - r/n E(Y). (4.3)

Replacing Y by X in (4.3) we get

(A + E)(LX) - r/n (A + E)(X) = (n - 2)/2n dr(X). (4.4)

Again using (4.2) in (4.1) we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4.5)

Contracting (4.5) over Y and Z we get

(A - D - E)(LX) - r/n (A - D - E)(X) = n - 2/2n dr(X). (4.6)

From (4.4) and (4.6) we get

(D + 2E)(LX) = r/n (D + 2E)(X). (4.7)

Now (4.7) can be rewritten as

S(X, [[rho].sub.3] + 2[[rho].sub.4]) = r/n g(X, [[rho].sub.3] + 2[[rho].sub.4]), (4.8)

where [[rho].sub.3] and [[rho].sub.4] defined by (1.3). Also (4.7) can be rewritten as

S (X, [[rho].sub.5]) = r/n g(X, [[rho].sub.5]), (4.9)

where we assume that

[[rho].sub.5] = [[rho].sub.3] + [[rho].sub.4]. (4.10)

Thus we have the following theorem:

Theorem 4.1. In a conformally flat [(WPS).sub.n](n > 3), r/n is an eigen value of the Ricci tensor S corresponding to the eigen vector [[rho].sub.5] defined by (4-10).

Let ([M.sup.n], g) be a conformally flat [(WPS).sub.n](n > 3). Now using (3.7) and (3.11) in (4.6) we get

A(LX) - r/n A(X) = n - 2/2n dr(X). (4.11)

Again if in the [(WPS).sub.n] the scalar curvature r is constant then from (4.11) we get

A(LX) = r/n A(X), (4.12)

which can be written as

S (X, [[rho].sub.1]) = r/n g(X, [[rho].sub.1]), (4.13)

where [[rho].sub.1] is defined by (1.3). Thus we have the following theorem:

Theorem 4.2. In a conformally flat [(WPS).sub.n](n > 3), if the scalar curvature r is constant then r/n is an eigenvalue of the Ricci tensor S corresponding to the eigen vector [[rho].sub.1] defined by (1.3).

Also for a conformally flat [(WPS).sub.n](n > 3) if (4.12) holds then from (4.11) we can conclude that r = constant. Hence we have the following theorem:

Corollary 4.1. In a conformally flat [(WPS).sub.n](n > 3), the scalar curvature r is constant if and only if (4-12) holds.

5. The vector field [[rho].sub.3] as a torse-forming vector field

In this section we suppose that [[rho].sub.3] is a unit torse-forming vector field  defined by (1.3) and given by

[[nabla].sub.X] [[rho].sub.3] = [lambda]X + [omega](X)[[rho].sub.3], (5.1)

where [lambda] is a non-zero scalar and [omega] is a non-zero 1-form, called respectively the scalar and 1-form of the vector field [[rho].sub.3].

Now if in a [(WPS).sub.n] D = E and D([[rho].sub.3]) [not equal to] 0 then from Theorem 3.4 we can conclude that the manifold is Einstein and so we can use (3.17). Hence from (3.17) we get

([[nabla].sub.X]S)(Y, [[rho].sub.3]) = 0. (5.2)

But

([[nabla].sub.X]S)(Y, [[rho].sub.3]) = [[nabla].sub.X] S(Y, [[rho].sub.3]) - S([[nabla].sub.X]Y, [[rho].sub.3]) - S(Y, [[nabla].sub.X][[rho].sub.3]).

Therefore, using (3.17) we obtain

r/n ([[nabla].sub.X] D)(Y) + S (Y, [[nabal].sub.X][[rho].sub.3]) = 0. (5.3)

By virtue of (5.1) we get from (5.3)

r/n ([[nabla].sub.X] D)(Y) + S(Y, [lambda]X + [omega](X)[[rho].sub.3]) = 0.

or,

r/n ([[nabla].sub.X] D)(Y) + [lambda]S(Y, X) + u(X)S (Y,pa) = 0. (5.4)

Using (3.16) in (5.4) we get

r/n ([[nabla].sub.X] D)(Y) + [lambda]S (Y, X) + r/n [omega](X)D(Y) = 0. (5.5)

Putting Y = [[rho].sub.3] in (5.5) we get

([[nabla].sub.X]D)([[rho].sub.3]) + [lambda]D(X) + [omega](X) = 0, (5.6)

since [[rho].sub.3] is a unit vector.

But

([[nabla].sub.X]D)([[rho].sub.3]) = D([[nabla].sub.X] [[rho].sub.3]), (5.7)

since [[rho].sub.3] is a unit vector.

Hence using (5.1) in (5.7) we get

([[nabla].sub.X]D)([[rho].sub.3]) = [lambda]D(X) + [omega](X). (5.8)

From (5.8) and (5.6) we get

[omega](X) = -[lambda]D(X). (5.9)

or

[lambda] = -[omega]([[rho].sub.3]). (5.10)

Hence (5.1) can be rewritten using (5.10) as

[[nabla].sub.X][[rho].sub.3] = -[omega]([[rho].sub.3])X + [omega](X)[[rho].sub.3].

Therefore [[nabla].sub.[rho]3] [[tau].sub.3] = 0. Thus we have the following:

Theorem 5.1. If in a [(WPS).sub.n], D = E and D([[rho].sub.3]) [not equal to] 0, the vector field [[rho].sub.3] is a unit torse-forming vector field, then the integral curves of the vector [[rho].sub.3] are

geodesics.

6. Application of [(WPS).sub.4]) perfect fluid spacetime with D = E and D([[rho].sub.3]) [not equal to] 0

A semi-Riemannian four-dimensional manifold ([M.sup.4], g) with Lorentzian metric g with signature (-, +, +, +) is called weakly projective symmetric spacetime if its projective curvature tensor satisfies (1.7) and (1.8), where the vector field [[rho].sub.3] is related by g(X, [[rho].sub.3]) = D(X) and also D = E and D([[rho].sub.3]) [not equal to] 0. In this section we consider [(WPS).sub.4] relativistic spacetime that is, a 4-dimensional [(WPS).sub.4] Lorentzian manifold as a perfect fluid spacetime with cosmological constant A in which the associated vector field [[rho].sub.3] is the velocity vector field of the fluid.

For a perfect fluid spacetime, we have the Einstein's equation with cosmological constant  as

S(X, Y) - r/2 g(X, Y) + [lambda]g(X,Y) = kT(X, Y), (6.1)

where k is the Einstein's gravitational constant, T is the energy momentum tensor of type (0,2) given by

T(X, Y) = ([sigma] + p)D(X)D(Y) + pg(X,Y), (6.2)

where [sigma] and p as the energy density and isotropic pressure of the fluid respectively and D being given by g(X, [[rho].sub.3]) = D(X) for all X, [[rho].sub.3] is the flow vector field of the fluid such that g([[rho].sub.3], [[rho].sub.3]) = -1. Using (6.2) in (6.1) we get

S(X, Y) - r/2 g(X, Y) + [lambda]g(X, Y) = k[([sigma] + p)D(X)D(Y) + pg(X, Y)]. (6.3)

Putting Y = [[rho].sub.3] in (6.3) and using (3.17) we have

[sigma] = r - 4[lambda]/4k. (6.4)

Taking a frame field and contracting (6.3) over X and Y we get

-r + 4[lambda] = k(3p - [alpha]). (6.5)

Again if we use (6.4) in (6.5) for [(WPS).sub.4] perfect fluid spacetime, we get

p = -r + 4[lambda]/4k. (6.6)

Since a [(WPS).sub.n] with D = E and D([[rho].sub.3]) [not equal to] 0 is an Einstein manifold so for n > 2 the scalar curvature r of the manifold will be constant. Then from (6.4) and (6.6) we can conclude that p and [sigma] are constants. Also from (6.4) and (6.6) we get [sigma] + [rho] = 0, which means the fluid behaves as a cosmological constant . This is also termed as phantom barrier . Now in a cosmology we know such a choice [sigma] = -p lead to rapid expansion of the spacetime which is now termed as inflation. Thus we have the following:

Theorem 6.1. In a [(WPS).sub.n] spacetime with the condition D = E and D([[rho].sub.3]) [not equal to] 0, the matter distribution is perfect fluid whose velocity vector field is [[rho].sub.3] de fined by (1.3), then the spacetime represents inflation. In this case the isotropic pressure p and the energy density [sigma] are constant. Also the fluid behaves as a cosmological constant. This is also termed as a phantom barrier.

7. Example of a [(WPS).sub.4]

In this section we give an example of [(WPS).sub.n], with the non-zero scalar curvature.

Example 7.1. Let ([R.sup.4], g) be a 4-dimensional Riemannian manifold endowed with the Riemannian metric g given by

[ds.sup.2] = [g.sub.ij] [dx.sup.i][dx.sup.j] = [x.sup.2][[([dx.sup.1]).sup.2] + [([dx.sup.2]).sup.2]] + [([dx.sup.3]).sup.2] + [x.sup.4][([dx.sup.4]).sup.2], (7.1)

where (i,j = 1, 2, 3, 4), [x.sup.1] and [x.sup.2] are non-zero. Here the only non-vanishing components of the Christoffel symbols and the curvature tensors are respectively

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the components obtained by the symmetry properties. The non-vanishing components of the Ricci tensors are:

[R.sub.11] = 1/2[([x.sup.2]).sup.2] - 1/4[([x.sup.1]).sup.2], [R.sub.22] = -1/2[([x.sup.2]).sup.2], [R.sub.44] = [R.sub.12] = -1/4[x.sup.1][x.sup.2],

It can be easily shown that the scalar curvature r of this ([R.sup.4], g) is - 1/2[([x.sup.1]).sup.2][x.sup.2] [[not equal to] 0, which is non-vanishing and non-constant. Therefore the non-vanishing components of the projective curvature tensor and their covariant derivatives are respectively:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let us choose the associated 1-forms as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7.2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7.3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7.4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7.5)

at any point x [member of] [R.sup.4]. Now the equation (1.8) reduces to the equations

[P.sub.1221,2] = [A.sub.2][P.sub.1221] + [B.sub.1][P.sub.2221] + [B.sub.2][P.sub.1221] + [D.sub.2][P.sub.1221] + [E.sub.1][P.sub.1222], (7.6)

[P.sub.1441,1] = [A.sub.1][P.sub.1441] + [B.sub.1][P.sub.1441] + [B.sub.4][P.sub.1141] + [D.sub.4][P.sub.1411] + [E.sub.1][P.sub.1441], (7.7)

[P.sub.11442,2] = [A.sub.2][P.sub.1442] + [B.sub.1][P.sub.2442] + [B.sub.4][P.sub.1242] + [D.sub.4][P.sub.1422] + [E.sub.2][P.sub.1442], (7.8)

since, for the other cases (1.8) holds trivially.

By (7.2), (7.3), (7.4) and (7.5) we get the following relation for the right hand side (R.H.S.) and the left hand side (L.H.S.) of (7.6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By similar argument it can be shown that (7.7) and (7.8) are true. So, [R.sup.4] is a [(WPS).sub.n] whose scalar curvature is non-zero and non-constant and the manifold ([R.sup.4], g) is neither protectively flat nor protectively symmetric.

Acknowledgement. The author is thankful to the referees for their valuable suggestions towards the improvement of this paper.

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Prajjwal Pa ([dagger])

Chakdaha Co-operative Colony Vidyayatan(H.S), P.O.-Chakdaha, Dist-Nadia, West Bengal, India

Received February 4, 2014, Accepted May 13, 2014.

* 2000 Mathematics Subject Classification. Primary 53C25.

([dagger]) E-mail: prajjwalpal@yahoo.in