Lobachevskii Journal of Mathematics

http://ljm.ksu.ru
ISSN 1818-9962

Vol. 27, 2007, 3–13 \begin{matrix}c \\ ◯ \end{matrix} K. K. Baishya, S. Eyasmin, and A. A. Shaikh

K. K. Baishya, S. Eyasmin, and A. A. Shaikh

ON ϕ-RECURRENT GENERALIZED (k, μ)-CONTACT METRIC MANIFOLDS

(submitted by M. A. Malakhaltsev)

Abstract. The aim of the present paper is to introduce a type of contact metric manifolds called ϕ-recurrent generalized (k, μ)-contact metric manifolds and to study their geometric properties. The existence of such manifolds is ensured by a non-trivial example.

1. Introduction

In 1995 Blair, Koufogiorgos, and Papantoniou [5] introduced the notion of (k, μ)-contact metric manifolds, where k and μ are real constants, and a full classification of such manifolds was given by E. Boeckx [6]. Assuming k, μ be smooth functions, T. Koufogiorgos and C. Tsichlias introduced the notion of generalized (k, μ)-contact metric manifolds and gave several examples [8].

The notion of local symmetry of a Riemannian manifold has been weakened by many authors in several ways to a different extent. As a weaker version of local symmetry, T. Takahashi [9] introduced the notion of local ϕ-symmetry on a Sasakian manifold. Recently, Shaikh etl. [2] studied the locally ϕ-symmetric (k, μ)-contact metric manifolds and proved that such a manifold exists whereas a locally ϕ-symmetric generalized

contact metric manifold..

2000 Mathematical Subject Classification. 53C05, 53C15, 53C25.

Key words and phrases. (k, μ)-contact metric manifold, generalized (k, μ)- contact metric manifold, locally ϕ-symmetric and locally ϕ-recurrent (k, μ)-

(k, μ)-contact metric manifold does not exist. Extending the notion of local ϕ-symmetry, in the present paper we introduce the notion of locally ϕ-recurrent (k, μ)-contact metric manifolds and locally ϕ-recurrent generalized (k, μ)-contact metric manifolds. In [8], the authors proved that the generalized (k, μ)-contact metric manifolds exist only for dimension 3 and hence we confined ourselves to the study of 3-dimensional generalized (k, μ)-contact metric manifolds. The (k, μ)-contact metric manifold is of our special interest as it contains both the Sasakian and non-Sasakian cases. The paper is organized as follows. Section 2 is concerned with contact metric manifolds. Section 3 deals with (k, μ)-contact metric manifolds and section 4 is the discussion of generalized (k, μ)-contact metric manifolds. In section 5 we study locally ϕ-recurrent (k, μ)-contact metric manifolds. Section 6 is devoted to the study of locally ϕ-recurrent generalized (k, μ)-contact metric manifolds and proved that such a manifold is either flat or an η-Einstein manifold. Finally, we construct an example of a locally ϕ-recurrent generalized (k, μ)-contact metric manifold which is neither locally symmetric nor locally ϕ-symmetric.

2. Contact metric manifolds

A contact manifold is a C^{∞} manifold M^{2n+1} equipped with a global 1-form η such that η ∧ (dη)^{n} ≠ 0 everywhere on M^{2n+1}. Given a contact form η it is well known that there exists a unique vector field ξ, called the characteristic vector field of η, such that η(ξ)=1 and dη(X, ξ) = 0 for every vector field X on M^{2n+1}. A Riemannian metric is said to be associated metric if there exists a tensor field ϕ of type (1,1) such that

dη(X, Y) = g(X, ϕY), η(X) = g(X, ξ),  (2.1)
ϕξ = 0,  η(ϕX) = 0, ϕ^{2}X = −X + η(X)ξ,  (2.2)
g(ϕX, ϕY) = g(X, Y) − η(X)η(Y)  (2.3)

for all vector fields X,Y on M^{2n+1}. Then the structure (ϕ, ξ, η, g) on M^{2n+1} is called a contact metric structure and the manifold M^{2n+1} equipped with such structure is called a contact metric manifold [3].

Given a contact metric manifold M^{2n+1}(ϕ, ξ, η, g) we define a (1, 1) tensor field h by h = \frac{1}{2}£_{ξ}ϕ, where £ denotes the Lie differentiation. Then h is symmetric and satisfies hϕ = −ϕh. Thus, if λ is an eigenvalue of h with eigenvector X, −λ is also an eigenvalue with eigenvector ϕX. Also we have T r.h = T r.ϕh = 0 and hξ = 0. Moreover, if ∇ denotes the Riemannian connection of g, then the following relation holds:

∇_{X}ξ = −ϕX − ϕhX.  (2.4)

The vector field ξ is a Killing vector with respect to g if and only if h = 0. A contact metric manifold M^{2n+1}(ϕ, ξ, η, g) for which ξ is a Killing vector is said to be a K-contact manifold. A contact structure on M^{2n+1} gives rise to an almost complex structure on the product M^{2n+1} × R. If this almost complex structure is integrable, the contact metric manifold is said to be Sasakian. Equivalently, a contact metric manifold is Sasakian if and only if the relation

R(X, Y)ξ = η(Y)X − η(X)Y

holds for all X, Y, where R denotes the curvature tensor of the manifold. We shall now state a result which will be used later on.

Lemma 2.1. ([4]) Let M^{2n+1}(ϕ, ξ, η, g) be a contact metric manifold with R(X, Y)ξ=0 for all vector fields X,Y tangent to M. Then M is locally isometric to the Riemannian product E^{n+1}(0) × S^{n}(4).

3. (k, μ)-Contact metric manifolds

For a contact metric manifold M^{2n+1}(ϕ, ξ, η, g), the (k, μ)-nullity distribution is

p → N_{p}(k, μ) = [Z ∈ T_{p}M : R(X, Y)Z = k{g(Y, Z)X − g(X, Z)Y }
+ μ{g(Y, Z)hX − g(X, Z)hY }]

for any X, Y ∈ T_{p}M, k, μ are real numbers. Hence, if the characteristic vector field ξ belongs to the (k, μ)-nullity distribution, then we have

R(X, Y)ξ = k[η(Y)X − η(X)Y] + μ[η(Y)hX − η(X)hY].  (3.1)

Thus a contact metric manifold satisfying relation (3.1) is called a (k, μ)- contact metric manifold [5]. In particular, if μ = 0, then the notion of (k, μ)-nullity distribution reduces to the notion of k-nullity distribution, introduced by S. Tanno [7]. A (k, μ)-contact metric manifold is Sasakian if and only if k = 1. In a (k, μ)-contact metric manifold the following relations hold ([1], [5]):

h^{2} = (k − 1)ϕ^{2},  k ≤ 1,  (3.2)
(∇_{X}ϕ)(Y) = g(X + hX, Y)ξ − η(Y)(X + hX),  (3.3)
(∇_{X}h)(Y) = {(1 − k)g(X, ϕY) + g(X, hϕY)}ξ  (3.4)
+η(Y)[h(ϕX + ϕhX)] − μη(X)ϕhY ,
R(ξ, X)Y = k[g(X, Y)ξ − η(Y)X] + μ[g(hX, Y)ξ − η(Y)hX], (3.5)

η(R(X, Y)Z) = k[g(Y, Z)η(X) − g(X, Z)η(Y)]  (3.6)
+μ[g(hY, Z)η(X) − g(hX, Z)η(Y)],
S(X, ξ) = 2nkη(X),  (3.7)
Qϕ − ϕQ = 2[2(n − 1) + μ]hϕ,  (3.8)
S(X, Y) = [2(n − 1) − nμ]g(X, Y) + [2(n − 1) + μ]g(hX, Y)(3.9)
+[2(1 − n) + n(2k + μ)]η(X)η(Y), n ≥ 1,
r = 2n(2n − 2 + k − nμ),  (3.10)
S(ϕX, ϕY) = S(X, Y) − 2nkη(X)η(Y) − 2(2n − 2 + μ)g(hX, Y), (3.11)

where S is the Ricci tensor of type (0, 2), Q is the Ricci-operator, i.e., g(QX, Y) = S(X, Y) and r is the scalar curvature of the manifold. From (2.4), it follows that

(∇_{X}η)(Y) = g(X + hX, ϕY).  (3.12)

4. Generalized (k, μ)-contact metric manifolds

A generalized (k, μ)-contact metric manifold M^{3}(ϕ, ξ, η, g) is a (k, μ)- contact metric manifold in which k and μ are smooth functions on M. In [8] the authors proved that a generalized (k, μ)-contact metric manifold does not exist for dimension greater than three. Hence the generalized (k, μ)-contact metric manifold exists for dimension three and several examples are given in [8]. In a generalized (k, μ)-contact metric manifold M^{3}(ϕ, ξ, η, g), the relations (3.2), (3.5)-(3.11) hold ([2], [8]) and also the following relations hold :

ξk = 0,  (4.1)
ξr = 0,  (4.2)
h grad μ = gradk.  (4.3)

Definition 4.1. A generalized (k, μ)-contact metric manifold is said to be η-Einstein if its Ricci tensor S is of the form

S = αg + βη ⊗ η  (4.4)

where α and β are smooth functions on the manifold.

5. Locally ϕ-recurrent (k, μ)-contact metric manifolds

Definition 5.1. ([9]) A (k, μ)-contact metric manifold is said to be locally ϕ-symmetric in the sense of Takahashi if the relation

ϕ^{2}((∇_{W} R)(X, Y)Z) = 0  (5.1)

holds for all vector fields X, Y, Z, W orthogonal to ξ.

Definition 5.2. A (k, μ)-contact metric manifold M^{2n+1}(ϕ, ξ, η, g) is said to be locally ϕ-recurrent if and only if there exists a non-zero 1-form A such that

ϕ^{2}((∇_{W} R)(X, Y)Z) = A(W)R(X, Y)Z  (5.2)

for all vector fields X, Y, Z, W tangent to M, where A(X) = g(X, ρ).

If the 1-form A vanishes identically and the vector fields X, Y, Z, W are orthogonal to ξ, then the manifold reduces to a locally ϕ-symmetric manifold in the sense of Takahashi.

Theorem 5.1. Let M^{2n+1}(ϕ, ξ, η, g) be a locally ϕ-recurrent (k, μ)- contact metric manifold. Then any one of the following holds:

(i) The manifold is locally isometric to the Riemannian product E^{n+1}(0)× S^{n}(4), including a flat contact metric structure for n = 1. (ii) The manifold is locally ϕ-symmetric in the sense of Takahashi. (iii) The characteristic vector field ξ and the associated vector field ρ of the 1-form of recurrence are codirectional.

Proof. In a locally ϕ-recurrent (k, μ)-contact metric manifold the relation (5.2) holds. Then, by virtue of (2.2), we obtain

−(∇_{W} R)(X, Y)Z + η((∇_{W} R)(X, Y)Z)ξ = A(W)R(X, Y)Z. (5.3)

Taking the inner product on both sides of (5.3) by any vector field U, we get

\begin{matrix}−g((∇_{W} R)(X, Y)Z, U) + η((∇_{W} R)(X, Y)Z)η(U) \\ = A(W)g(R(X, Y)Z, U). \end{matrix}  (5.4)

Let {e_{i} : i = 1, 2, …, 2n + 1} be an orthonormal basis of the tangent space at any point of the manifold. Then setting X = U = e_{i} in (5.4) and taking summation over i, 1 ≤ i ≤ 2n + 1, we get

−(∇_{W} S)(Y, Z) + η((∇_{W} R)(ξ, Y)Z) = A(W)S(Y, Z).  (5.5)

Plugging Z by ξ in (5.5) we obtain, by virtue of the skew-symmetry property of the curvature tensor, that

−(∇_{W} S)(Y, ξ) = A(W)S(Y, ξ).  (5.6)

In view of (3.7) and (3.12), (5.6) reduces to

2nkg(Y, ϕW + ϕhW) − S(Y, ϕW + ϕhW) = 2nkA(W)η(Y). (5.7)

Setting Y = ξ in (5.7) we get by virtue of (2.2) and (3.7) that

kA(W) = 0,

which yields either k = 0, or A(W) = 0 for all vector fields W tangent to M.

Again, changing W, X, Y cyclically in (5.3) and then adding the results, we obtain

\begin{matrix}−[(∇_{W} R)(X, Y)Z + (∇_{X}R)(Y, W)Z + (∇_{Y} R)(W, X)Z] \\ +[η((∇_{W} R)(X, Y)Z) + η((∇_{X}R)(Y, W)Z) + η((∇_{Y} R)(W, X)Z)]ξ \\ = A(W)R(X, Y)Z + A(X)R(Y, W)Z + A(Y)R(W, X)Z, \end{matrix} ; (5.8)

which, by virtue of Bianchi identity, yields

A(W)η(R(X, Y)Z) + A(X)η(R(Y, W)Z) + A(Y)η(R(W, X)Z) = 0. (5.9)

In view of (3.6), (5.9) reduces to

A(W) [k{g(Y, Z)η(X) − g(X, Z)η(Y)} + μ {g(hY, Z)η(X)(5.10) −g(hX, Z)η(Y)}] + A(X) [k{g(W, Z)η(Y) − g(Y, Z)η(W)} +μ{g(hW, Z)η(Y) − g(hY, Z)η(W)}] + A(Y) [k {g(X, Z)η(W)

nonumber −g(W, Z)η(X)} + μ{g(hX, Z)η(W) − g(hW, Z)η(X)}] = 0. (5.11)

Setting Y = Z = e_{i} in (5.10) and taking summation over i, 1 ≤ i ≤ 2n+1, we get

(2n − 1)k[A(W)η(X) − A(X)η(W)] + μ[A(hX)η(W) − A(hW)η(X)] = 0. (5.12)

Substituting X by ξ in (5.12), we have

(2n − 1)k[A(W) − A(ξ)η(W)] − μA(hW) = 0.  (5.13)

If k = 0, then (5.13) yields either μ = 0, or A(hW) = 0. Thus for k = 0 = μ, (3.1) reduces to R(X, Y)ξ = 0 for all X, Y and hence, by virtue of Lemma 2.1, the manifold under consideration is locally isometric to the Riemannian product E^{n+1}(0)×S^{n}(4) and, for n = 1, the manifold is a flat contact metric manifold.

Again, for k = 0 and A(hW) = 0, we have A(W) = A(ξ)η(W), which can be written as A(W)η(ξ) = A(ξ)η(W). This implies that the vector field ξ and ρ associated to the 1-form A are codirectional. Finally, if A(W) = 0, (for k ≠ 0) for all W, then (5.2) reduces to (5.1) and hence the manifold under consideration is locally ϕ-symmetric in the sense of Takahashi. This proves the theorem.

6. Locally ϕ-recurrent generalized (k, μ)-contact metric manifolds

Definition 6.1. A generalized (k, μ)-contact metric manifold is said to be locally ϕ-recurrent if and only if relation (5.2) holds.

In particular, if A vanishes, then a generalized (k, μ)-contact metric manifold is said to be a locally ϕ-symmetric manifold.

Theorem 6.1. A locally ϕ-recurrent generalized (k, μ)-contact metric manifold M^{3}(ϕ, ξ, η, g) is either a flat contact metric manifold or an η-Einstein manifold.

Proof. Proceeding similarly to the proof of Theorem 5.1, we can easily show that in a generalized (k, μ)-contact metric manifold the relation (5.6) holds, and hence in view of (3.7) and (3.12), we obtain

(∇_{W} S)(Y, ξ) = 2(W k)η(Y) − 2kg(Y, ϕW + ϕhW) + S(Y, ϕW + ϕhW). (6.1)

Using (6.1) in (5.6) we get

− 2(W k)η(Y) + 2kg(Y, ϕW + ϕhW) − S(Y, ϕW + ϕhW) = 2kA(W)η(Y). (6.2)

Replacing Y by ϕY in (6.2) we have

2kg(ϕY, ϕW + ϕhW) − S(ϕY, ϕW + ϕhW) = 0.  (6.3)

By virtue of (2.2) and (3.11), it follows from (6.3) that

S(W, Y) + S(Y, hW) = −2[k − μ]g(Y, W)  (6.4)
−2[k − μ]g(Y, hW)
+[4k + 2(k − 1)μ]η(W)η(Y).

Again, Replacing W by hW in (6.4) and then using (2.2) and (3.2), we obtain

S(W, Y) + S(Y, hW) − kS(Y, W) = −2[k − μ]g(Y, hW)  (6.5)
+2(k − 1)[k − μ]g(Y, W)
−2(k − 1)[2k − μ]η(W)η(Y).

Subtracting (6.5) from (6.4), it follows that either k = 0, or

S(W, Y) = −2kg(W, Y) + 4kη(W)η(Y).  (6.6)

The relation (6.6) implies that the manifold under consideration is an η-Einstein manifold.

If k = 0, then (6.4) reduces to

S(W, Y) + S(Y, hW) = 2μ[g(Y, W) + g(Y, hW) − η(W)η(Y)]. (6.7)

Again, if k = 0, then for n = 1, (3.9) takes the form

S(X, Y) = −μg(X, Y) + μg(hX, Y) + μη(X)η(Y),  (6.8)

which yields by setting Y = hY that

S(X, hY) = −μg(X, hY) + μg(X, Y) − μη(X)η(Y).  (6.9)

By virtue of (6.8) and (6.9), we obtain from (6.7) that either μ = 0, or hY = −Y + η(Y)ξ. If k = 0 = μ, then the manifold is a flat contact metric structure. Again, if k = 0 and hY = −Y + η(Y)ξ, then (6.8) yields

S(W, Y) = −2μg(Y, W) + 2μη(W)η(Y),

which implies that the manifold is an η-Einstein manifold. This proves the theorem.

Theorem 6.2. In a locally ϕ-recurrent generalized (k, μ)-contact metric manifold M^{3}(ϕ, ξ, η, g), the vector field associated to the 1-form of recurrence is given by

ρ = −\frac{1}{k} grad k, provided that k ≠ 0.

Proof. In a locally ϕ-recurrent generalized (k, μ)-contact metric manifold relation (6.2) holds. Setting Y = ξ in (6.2), we obtain A(W) = −\frac{1}{k}(W k) for k ≠ 0, which implies that ρ = −\frac{1}{k} grad k. This proves the theorem.

Theorem 6.3. In a locally ϕ-recurrent generalized (k, μ)-contact metric manifold M^{3}(ϕ, ξ, η, g), the characteristic vector field ξ and the vector field ρ associated to the 1-form A are orthogonal to each other provided that k ≠ 0.

Proof. In a locally ϕ-recurrent generalized (k, μ)-contact metric manifold relation (6.2) holds. Setting W = ξ in (6.2) and then using (4.1), we get A(ξ) = 0, i.e., g(ξ, ρ) = 0, provided that k ≠ 0. This proves the theorem.

We now give an example of a locally ϕ-recurrent generalized (k, μ)- contact metric manifold. Example. We consider the 3-dimensional manifold M = {(x, y, z) ∈ R^{3} : x ≠ 0}, where (x, y, z) are the standard coordinates in R^{3}. Let {E_{1}, E_{2}, E_{3}} be linearly independent global frame on M given by

E_{1} = \frac{2}{x} \frac{∂}{∂y} , E_{2} = 2\frac{∂}{∂x} − \frac{4z}{x} \frac{∂}{∂y} + xy \frac{∂}{∂z} , E_{3} = \frac{∂}{∂z} .

Let g be the Riemannian metric defined by

g(E_{1}, E_{3}) = g(E_{2}, E_{3}) = g(E_{1}, E_{2}) = 0,
g(E_{1}, E_{1}) = g(E_{2}, E_{2}) = g(E_{3}, E_{3}) = 1.

Let η be the 1-form defined by η(U) = g(U, E_{3}) for any U ∈ χ(M). Let ϕ be the (1, 1)-tensor field defined by ϕE_{1} = E_{2}, ϕE_{2} = −E_{1}, ϕE_{3} = 0. Then using the linearity of ϕ and g, we have η(E_{3}) = 1, ϕ^{2}U = −U + η(U)E_{3} and g(ϕU, ϕW) = g(U, W) − η(U)η(W) for any U, W ∈ χ(M). Moreover hE_{1} = −E_{1}, hE_{2} = E_{2} and hE_{3} = 0. Thus for E_{3} = ξ, (ϕ, ξ, η, g) defines a contact metric structure on M.

Let ∇ be the Levi-Civita connection with respect to the Riemannian metric g and R be the curvature tensor of g. Then we have

[E_{1}, E_{2}] = 2E_{3} + \frac{2}{x}E_{1}, [E_{1}, E_{3}] = 0, [E_{2}, E_{3}] = 2E_{1}.

Taking E_{3} = ξ and using Koszul formula for the Riemannian metric g, we can easily calculate

∇_{E_{1}}E_{3} = 0,  ∇_{E_{2}}E_{3} = 2E_{1},  ∇_{E_{3}}E_{3} = 0,
∇_{E_{3}}E_{1} = 0,  ∇_{E_{1}}E_{2} = \frac{2}{x}E_{1},  ∇_{E_{2}}E_{1} = −2E_{3},
∇_{E_{2}}E_{2} = 0,  ∇_{E_{3}}E_{2} = 0,  ∇_{E_{1}}E_{1} = −\frac{2}{x}E_{2}.

From the above it can be easily seen that (ϕ, ξ, η, g) is a generalized (k, μ)-contact metric structure on M. Consequently M^{3}(ϕ, ξ, η, g) is a generalized (k, μ)-contact metric manifold with k = −\frac{2}{x} and μ = −\frac{2}{x}.

Using the above relations, we can easily calculate the non-vanishing components of the curvature tensor as follows :

R(E_{2}, E_{3})E_{2} = −\frac{4}{x}E_{1},  R(E_{2}, E_{3})E_{1} = \frac{4}{x}E_{2},  (6.10)

and the components which can be obtained from these by the symmetry properties. We shall now show that such a generalized (k, μ)-contact metric manifold is locally ϕ-recurrent. Since {E_{1}, E_{2}, E_{3}} from a basis of M^{3}, any vector field X ∈ χ(M) can be written as

X = a_{1}E_{1} + a_{2}E_{2} + a_{3}E_{3}

where a_{i} ∈ R^{+} (the set of all positive real nonumbers), i = 1, 2, 3. Thus the covariant derivatives of the curvature tensor are given by

(∇_{X}R)(E_{2}, E_{3})E_{1} = −\frac{8a_{2}}{x^{2}} E_{2}, (∇_{X}R)(E_{2}, E_{3})E_{2} = \frac{8a_{2}}{x^{2}} E_{1}.

This implies that

ϕ^{2}((∇_{X}R)(E_{2}, E_{3})E_{1}) = \frac{8a_{2}}{x^{2}} E_{2},  (6.11)
ϕ^{2}((∇_{X}R)(E_{2}, E_{3})E_{2}) = − \frac{8a_{2}}{x^{2}} E_{1}.

Let us now consider the non-vanishing 1-form

A(X) = \frac{2a_{2}}{a_{3}x},  (6.12)

at any point p ∈ M. From (6.10)–(6.12), it follows that

ϕ^{2}((∇_{X}R)(E_{2}, E_{3})E_{1}) = A(X)R(E_{2}, E_{3})E_{1}
and ϕ^{2}((∇_{X}R)(E_{2}, E_{3})E_{2}) = A(X)R(E_{2}, E_{3})E_{2}

This implies that the manifold under consideration is a locallyϕ-recurrent generalized (k, μ)-contact metric manifold, which is neither locally symmetric nor locally ϕ-symmetric. This leads to the following:

Theorem 6.4. There exists a 3-dimensional locally ϕ-recurrent generalized (k, μ)-contact metric manifold which is neither locally symmetric nor locally ϕ-symmetric in the sense of Takahashi.

References

[1] A. A. Shaikh and K. K. Baishya, On a contact metric manifold, Diff. Geom. Dynamical System, 8(2006), 253-261.

[2] A. A. Shaikh, K. Arslan, C. Murathan, and K. K. Baishya, On 3-dimensional generalized (k, μ)-contact metric manifold, Balkan J. of Geom. and its Applications, 11(2007).

[3] D. E. Blair, Contact manifolds in Riemannian geometry, Lecture Notes in Math. 509, Springer-Verlag, 1976.

[4] D. E. Blair, Two remarks on contact metric structures, Tohoku Math. J., 29(1977), 319 - 324.

[5] D. E. Blair, T. Koufogiorgos and B. J.Papantoniou, Contact metric manifolds satisfying a nullity condition, Israel J. of Math., 19(1995), 189-214.

[6] E. Boeckx, A full Classification of contact metric (k, μ)-spaces, Illinois J. Math., 44(2000), 212-219.

CONTACT METRIC MANIFOLDS 13

[7] S. Tanno, Ricci curvatures of contact Riemannian manifolds, Tohoku Math. J., 40(1988), 441-448.

[8] T. Koufogiorgos and C. Tsichlias, On the existence of new class of contact metric manifolds, Canad. Math. Bull., XX(Y)(2000), 1-8.

[9] T. Takahashi, Sasakian ϕ-symmetric spaces, Tohoku Math. J., 29(1977), 91-113.

A. A. Shaikh, K. K. Baishya and S. Eyasmin, Department of Mathematics, University of Burdwan, Golapbag, Burdwan-713104, West Bengal, India E-mail address: aask2003@yahoo.co.in, aask@epatra.com

Received June 9, 2007