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Georgian Mathematical Journal Volume 16 (2009), Number 2, 295–304

THE STRUCTURE OF SOME CLASSES OF CONTACT METRIC MANIFOLDS ´ MOHIT KUMAR DWIVEDI, LUIS M. FERNANDEZ, AND MUKUT MANI TRIPATHI

Abstract. We study the conformal curvature tensor and the contact conformal curvature tensor in Sasakian and/or K-contact manifolds. We find a necessary and sufficient condition for a Sasakian manifold to be ϕ-conformally flat. We also find some necessary conditions for a K-contact manifold to be ϕ-contact conformally flat. Then we give a structure theorem for ϕ-contact conformally flat Sasakian manifolds. It is also proved that a Sasakian manifold cannot be ξ-contact conformally flat. 2000 Mathematics Subject Classification: 53C25, 53D10, 53D15. Key words and phrases: K-contact manifold, Sasakian manifold, conformal curvature tensor, contact conformal curvature tensor.

1. Introduction Weyl ([8], [9]) introduced a generalized curvature tensor on a Riemannian manifold, which vanishes whenever the metric is (locally) conformally equivalent to a flat metric; for this reason he called it the conformal curvature tensor of the metric. Schouten [7] showed that for the dimension > 3 the converse is true. Let M be an m-dimensional (m ≥ 3) Riemannian manifold equipped with a Riemannian metric g. The Weyl conformal curvature tensor is defined as a map C : T M × T M × T M → T M such that C(X, Y )Z = R(X, Y )Z 1 − {S(Y, Z)X − S(X, Z)Y + hY, ZiQX − hX, ZiQY } m−2 r + {hY, Zi X − hX, Zi Y } (m − 1) (m − 2) for all X, Y, Z ∈ T M , where h , i is the inner product of the Riemannian metric g, S is the Ricci tensor, Q is the Ricci operator, defined by S(X, Y ) = g(QX, Y ), and r = tr(S) is the scalar curvature. Let M be an almost contact metric manifold equipped with an almost contact metric structure (ϕ, η, ξ, g). Since at each point p ∈ M the tangent space Tp M can be decomposed into the direct sum Tp M = ϕ (Tp M ) ⊕ {ξp }, where {ξp } is the 1-dimensional linear subspace of Tp M generated by ξp , the conformal curvature tensor C is a map C : Tp M × Tp M × Tp M → ϕ (Tp M ) ⊕ {ξp }, c Heldermann Verlag www.heldermann.de ISSN 1072-947X / $8.00 / °

p ∈ M.

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It is natural to consider the following particular cases: (1) the projection of the image of C in ϕ(Tp M ) is zero; (2) the projection of the image of C in {ξp } is zero; and (3) the projection of the image of C|ϕ(Tp M )×ϕ(Tp M )×ϕ(Tp M ) in ϕ(Tp M ) is zero. An almost contact metric manifold satisfying the cases (1), (2) and (3) is said to be conformally symmetric [10], ξ-conformally flat [11] and ϕ-conformally flat [3], respectively. In [10], it is proved that a conformally symmetric K-contact manifold is locally isometric to the unit sphere. In [11], it is proved that a Kcontact manifold is ξ-conformally flat if and only if it is an η-Einstein Sasakian manifold. In [3], some necessary conditions for a K-contact manifold to be ϕ-conformally flat are proved. In [5], Kitahara, Matsuo and Pak defined a new tensor field on a Hermitian manifold which is conformally invariant and studied its some properties. By using the Boothby-Wang’s fibration [2], Jeong, Lee, Oh and Pak constructed the contact conformal curvature tensor [4] on a contact metric manifold from the new tensor field defined in [5]. Motivated by these studies, in this paper we study the conformal curvature tensor and the contact conformal curvature tensor in Sasakian and/or K-contact manifolds. The paper is organized as follows. Section 2 contains preliminaries. In section 3, we prove a necessary and sufficient condition for a Sasakian manifold to be ϕ-conformally flat (see Theorem 3.1). In section 4 we consider three cases of the contact conformal curvature tensor, analogous to those of the conformal curvature tensor, and give the definitions of quasi contact conformally flat, ξ-contact conformally flat and ϕ-contact conformally flat almost contact metric manifolds. We find some necessary conditions for a K-contact manifold to be ϕ-contact conformally flat. Then we give a structure theorem for ϕ-contact conformally flat Sasakian manifolds (see Theorem 4.2). Finally, we prove that a Sasakian manifold cannot be ξ-contact conformally flat (see Theorem 4.4). 2. Preliminaries Let M be an almost contact metric manifold of dimension (2n + 1) equipped with an almost contact metric structure (ϕ, ξ, η, g) consisting of a (1, 1) tensor field ϕ, a vector field ξ, a 1-form η and a Riemannian metric g. Then ϕ2 = − I + η ⊗ ξ,

η(ξ) = 1,

ϕξ = 0,

η ◦ ϕ = 0,

(2.1)

hX, Y i = hϕX, ϕY i + η(X)η(Y ), (2.2) for all X, Y ∈ T M , where h , i is the inner product of the Riemannian metric g. From (2.1) and (2.2) we easily get hX, ϕY i = − hϕX, Y i , hX, ξi = η(X), ® ϕ2 X, ϕ2 Y = hϕX, ϕY i = hX, Y i − η (X) η (Y ) , hϕ2 X, ϕY i = hϕX, Y i for all X, Y ∈ T M . ­

An almost contact metric manifold is

(2.3) (2.4) (2.5)

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(1) a contact metric manifold if hX, ϕY i = dη (X, Y ) for all X, Y ∈ T M ; (2) a K-contact manifold if ∇ξ = −ϕ, where ∇ is Levi-Civita connection; and (3) a Sasakian manifold if (∇X ϕ)Y = hX, Y i ξ − η(Y )X for all X, Y ∈ T M . In a Sasakian manifold M equipped with the structure (ϕ, ξ, η, g), the following relations are well known: R (ξ, X) Y = hX, Y i ξ − η (Y ) X,

(2.6)

R (X, Y ) ξ = η (Y ) X − η (X) Y,

(2.7)

S (X, ξ) = 2nη (X) ,

(2.8)

S (ϕX, ϕY ) = S (X, Y ) − 2nη (X) η (Y ) ,

(2.9)

S (ϕX, Y ) = − S (X, ϕY )

(2.10)

for all X, Y ∈ T M . A Sasakian manifold M is a Sasakian space form M (c) with constant ϕsectional curvature c if its curvature tensor is given by R (X, Y ) Z =

c+3 {hY, Zi X − hX, Zi Y } 4 c−1 + {hϕY, Zi ϕX − hϕX, Zi ϕY − 2 hϕX, Y i ϕZ} 4 − hY, Zi η (X) ξ + hX, Zi η (Y ) ξ + η (X) η (Z) Y − η (Y ) η (Z) X} (2.11)

for all X, Y, Z ∈ T M . For more details we refer to [1]. 3. ϕ-Conformally Flat Sasakian Manifolds Let M be an almost contact metric manifold of dimension > 3. Then M is ϕ-conformally flat [3] if and only if hC (ϕX, ϕY ) ϕZ, ϕW i = 0,

X, Y, Z, W ∈ T M.

(3.1)

In [3], some necessary conditions for a K-contact manifold to be ϕ-conformally flat are proved. In the following theorem we find a necessary and sufficient condition for a Sasakian manifold to be ϕ-conformally flat. Theorem 3.1. Let M be a Sasakian manifold of dimension (2n + 1) > 3. Then M is ϕ-conformally flat if and only if it is locally isometric to the unit sphere S 2n+1 (1). Proof. If a K-contact manifold is ϕ-conformally flat then it is known that [3] hR (ϕX, ϕY ) ϕZ, ϕW i =

r − 4n {hϕY, ϕZi hϕX, ϕW i 2n (2n − 1) − hϕX, ϕZi hϕY, ϕW i} . (3.2)

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In a Sasakian manifold, in view of (2.6) and (2.7) we can verify ­ ¡ 2 ¢ ® R ϕ X, ϕ2 Y ϕ2 Z, ϕ2 W = hR (X, Y ) Z, W i − hY, Zi η (X) η (W ) + hX, Zi η (Y ) η (W ) + hY, W i η (X) η (Z) − hX, W i η (Y ) η (Z)

(3.3)

for all X, Y, Z, W ∈ T M . Replacing X, Y, Z, W by ϕX, ϕY, ϕZ, ϕW respectively in (3.2) and using (3.3) and (2.4) we get r − 4n R (X, Y ) Z = {hY, Zi X − hX, Zi Y } 2n (2n − 1) r − 2n (2n + 1) {hY, Zi η (X) ξ − hX, Zi η (Y ) ξ − 2n (2n − 1) + η (Y ) η (Z) X − η (X) η (Z) Y } (3.4) for all X, Y, Z ∈ T M . For a Sasakian manifold, we have the following general formula (see Lemma 7.1 in [1]): hR (X, Y ) Z, ϕW i + hR (X, Y ) ϕZ, W i = − dη (X, Z) hY, W i + dη (X, W ) hY, Zi +dη (Y, Z) hX, W i − dη (Y, W ) hX, Zi .

(3.5)

Now applying (3.4) in (3.5) we obtain r − 4n {hY, Zi hX, ϕW i − hX, Zi hY, ϕW i 2n (2n − 1) + hY, ϕZi hX, W i − hX, ϕZi hY, W i} 2 r − 4n − 2n + {η (X) η (Z) hY, ϕW i − η (Y ) η (Z) hX, ϕW i 2n (2n − 1) + − η (Y ) η (W ) hX, ϕZi − η (X) η (W ) hY, ϕZi} = − hX, ϕZi hY, W i + hX, ϕW i hY, Zi + hY, ϕZi hX, W i − hY, ϕW i hX, Zi (3.6) for all X, Y, Z ∈ T M . Contracting the above equation with respect to X and W , we get (r − 2n (2n + 1)) (n − 1) hY, ϕZi = 0. (3.7) Since n > 1, from (3.7) we get r = 2n (2n + 1) , which makes (3.4) R (X, Y ) Z = hY, Zi X − hX, Zi Y, so that the manifold is of constant curvature 1. The converse is straightforward.

¤

An almost contact metric manifold is said to be η-Einstein [1] if its Ricci tensor S is of the form S(Y, Z) = a hY, Zi + bη(Y )η(Z),

Y, Z ∈ T M.

(3.8)

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A (2n + 1)-dimensional Sasakian manifold is η-Einstein if and only if ³ r ´ ³ r ´ S(Y, Z) = − 1 hY, Zi + 2n + 1 − η(Y )η(Z), Y, Z ∈ T M. (3.9) 2n 2n From (3.4) it is easy to obtain (3.9). Therefore we have the following Corollary 3.2. A ϕ-conformally flat Sasakian manifold is always η-Einstein. Since a K-contact manifold is ξ-conformally flat if and only if it is an ηEinstein Sasakian manifold [11], therefore in view of Corollary 3.2 we can state Corollary 3.3. A ϕ-conformally flat Sasakian manifold is always ξ-conformally flat. 4. ϕ-contact conformally flatness In a (2n + 1)-dimensional Sasakian manifold (M, ϕ, ξ, η, g) the contact conformal curvature tensor C0 is defined by [4] C0 (X, Y ) Z = R (X, Y ) Z 1 © + −S (Y, Z) ϕ2 X + S (X, Z) ϕ2 Y 2n + hϕY, ϕZi QX − hϕX, ϕZi QY + S (ϕX, Z) ϕY − S (ϕY, Z) ϕX + 2S (ϕX, Y ) ϕZ + hϕX, Zi QY − hϕY, Zi QX + 2 hϕX, Y i QZ} µ ¶ 1 (n + 2) r 2 + 2n − n − 2 + 2n (n + 1) 2n × {hϕY, Zi ϕX − hϕX, Zi ϕY − 2 hϕX, Y i ϕZ} µ ¶ 1 (3n + 2) r + n+2− (hY, Zi X − hX, Zi Y ) 2n (n + 1) 2n µ ¶ (3n + 2) r 1 2 − 4n + 5n + 2 − 2n (n + 1) 2n × {η (Y ) η (Z) X − η (X) η (Z) Y + η (X) hY, Zi ξ − η (Y ) hX, Zi ξ} (4.1) for all X, Y, Z ∈ T M , where R is the curvature tensor, S is the Ricci tensor, Q is the Ricci operator and r is the scalar curvature. Analogously to the consideration of conformal curvature tensor, for the map C0 : Tp M × Tp M × Tp M → ϕ (Tp M ) ⊕ {ξp },

p ∈ M.

we have the following three natural cases: (1) the projection of the image of C0 in ϕ(Tp M ) is zero, that is, h C0 (X, Y )Z, ϕW i = 0,

X, Y, Z, W ∈ T M,

(4.2)

(2) the projection of the image of C0 in {ξp } is zero, that is, C0 (X, Y )ξ = 0,

X, Y ∈ T M,

(4.3)

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(3) the projection of the image of C0 |ϕ(Tp M )×ϕ(Tp M )×ϕ(Tp M ) in ϕ(Tp M ) is zero, that is, h C0 (ϕX, ϕY )ϕZ, ϕW i = 0,

X, Y, Z, W ∈ T M.

(4.4)

We say that an almost contact metric manifold is quasi contact conformally flat, ξ-contact conformally flat or ϕ-contact conformally flat when it satisfies (4.2), (4.3) or (4.4), respectively. In a (2n+1)-dimensional almost contact metric manifold M, if {e1 , . . . , e2n , ξ} is a local orthonormal basis of vector fields in M , then {ϕe1 , . . . , ϕe2n , ξ} is also a local orthonormal basis. It is easy to verify that 2n X

hei , ei i =

2n X

i=1

hϕei , ϕei i = 2n,

i=1

2n 2n X X hei , ZiS(Y, ei ) = hϕei , ZiS(Y, ϕei ) = S(Y, Z) − S(Y, ξ)η(Z) i=1

(4.5)

(4.6)

i=1

for all Y, Z ∈ T M . In particular, in view of η ◦ ϕ = 0 we get 2n X

2n X hei , ϕZiS(Y, ei ) = hϕei , ϕZiS(Y, ϕei ) = S(Y, ϕZ)

i=1

(4.7)

i=1

for all Y, Z ∈ T M . Moreover, if M is a K-contact manifold then it is known that R (X, ξ) ξ = X − η (X) ξ, X ∈ T M. (4.8) and S (ξ, ξ) = 2n.

(4.9)

From (4.9) we get 2n X

S(ei , ei ) =

i=1

2n X

S(ϕei , ϕei ) = r − 2n.

(4.10)

i=1

In a K-contact manifold we also get hR(ξ, Y )Z, ξi = hϕY, ϕZi ,

Y, Z ∈ T M.

(4.11)

Consequently, 2n 2n X X hR(ei , Y )Z, ei i = hR(ϕei , Y )Z, ϕei i = S(Y, Z) − hϕY, ϕZi, i=1

(4.12)

i=1 2n X

hR(ei , ϕY )ϕZ, ei i =

i=1

2n X

hR(ϕei , ϕY )ϕZ, ϕei i

i=1

= S (ϕY, ϕZ) − hϕY, ϕZi . First we prove the following:

(4.13)

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Theorem 4.1. Let M be a K-contact manifold of dimension 2n + 1. If M is ϕ-contact conformally flat, then r = 2n,

(4.14)

S (Y, Z) = 2nη(Y )η(Z)

(4.15)

for all Y, Z ∈ T M ; and hR(ϕX, ϕY )ϕZ, ϕW i =

n {hϕX, ZihϕY, W i − hϕY, ZihϕX, W i n+1 + 2hϕX, Y ihϕZ, W i} 1 + {hϕY, ϕZihϕX, ϕW i n+1 −hϕX, ϕZihϕY, ϕW i} (4.16)

for all X, Y, Z, W ∈ T M . Proof. Let M be a (2n + 1)-dimensional K-contact manifold. From (4.1) we have h C0 (ϕX, ϕY ) ϕZ, ϕW i = hR (ϕX, ϕY ) ϕZ, ϕW i 1 + {S (ϕY, ϕZ) hϕX, ϕW i − S (ϕX, ϕZ) hϕY, ϕW i 2n + hϕY, ϕZi S (ϕX, ϕW ) − hϕX, ϕZi S (ϕY, ϕW ) ¡ ¢ ¡ ¢ + S ϕ2 X, ϕZ hϕY, W i − S ϕ2 Y, ϕZ hϕX, W i ¡ ¢ + 2S ϕ2 X, ϕY hϕZ, W i + hϕX, Zi S (ϕY, ϕW ) − hϕY, Zi S (ϕX, ϕW ) + 2 hϕX, Y i S (ϕZ, ϕW )} µ ¶ 1 (n + 2) r 2 + 2n − n − 2 + × 2n (n + 1) 2n × {hϕY, Zi hϕX, W i − hϕX, Zi hϕY, W i − 2 hϕX, Y i hϕZ, W i} µ ¶ (3n + 2) r 1 n+2− × + 2n (n + 1) 2n × (hϕY, ϕZi hϕX, ϕW i − hϕX, ϕZi hϕY, ϕW i) (4.17) for all X, Y, Z, W ∈ T M . If {e1 , . . . , e2n , ξ} is a local orthonormal basis of vector fields in M , then from (4.17) and using (4.12), (4.5), (4.7), (4.10) and (4.6) gives 2n X

4n−5 (2n−r) (n−2) S (ϕY, ϕZ) + hϕY, ϕZi 2 2n n i=1 ¢ ¡ ¢ ª 1 © ¡ − S ϕY, ϕ2 Z + 2S ϕ2 Y, ϕZ − (2n − r) hϕY, Zi (4.18) 2n for all Y, Z ∈ T M . If M is ϕ-contact conformally flat, then from (4.18) we get hC0 (ϕei , ϕY ) ϕZ, ϕei i =

0 =

(2n − r) (n − 2) 4n − 5 S (ϕY, ϕZ) + hϕY, ϕZi 2n n2 ¢ ¡ ¢ ª 1 © ¡ − S ϕY, ϕ2 Z + 2S ϕ2 Y, ϕZ − (2n − r) hϕY, Zi (4.19) 2n

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for all Y, Z ∈ T M . Interchanging Y and Z in (4.19) and subtracting the resulting equation from (4.19) we get ¡ ¢ ¡ ¢ S ϕY, ϕ2 Z − S ϕ2 Y, ϕZ + 2 (2n − r) hϕY, Zi = 0, Y, Z ∈ T M, (4.20) from which we get ¡ ¢ S ϕ2 Y, ϕ2 Z + S (ϕY, ϕZ) − 2 (2n − r) hϕY, ϕZi = 0,

Y, Z ∈ T M.

The above equation gives 2n 2n 2n X X ¡ 2 ¢ X 2 0= S ϕ ei , ϕ ei + S (ϕei , ϕei ) − 2 (2n − r) hϕei , ϕei i i=1

i=1

i=1

or 0 = (r − 2n) + (r − 2n) − 4n (2n − r) = 2 (2n + 1) (r − 2n) , which gives (4.14). Using (4.14) in (4.20) and (4.19) we get ¡ ¢ (4n − 5) S (ϕY, ϕZ) = 3S ϕY, ϕ2 Z

(4.21)

for all Y, Z ∈ T M . Putting ϕZ instead of Z in (4.21) and using ϕ3 = − ϕ we obtain ¡ ¢ (4n − 5) S ϕY, ϕ2 Z = − 3S (ϕY, ϕZ) . (4.22) From (4.21) and (4.22) we get ¡ 2 ¢ 8n − 20n + 17 S (ϕY, ϕZ) = 0. Since n > 1, the above equation gives S(ϕY, ϕZ) = 0. So, S(ϕY, ϕ2 Z) = 0 and hence S(ϕY, Z) = η(Z)S(ϕY, ξ). But, as it is well-known, the Ricci curvature tensor of a K-contact manifolds satisfies the condition S(X, ξ) = 2nη(X), for any X ∈ T M (see Proposition 7.2 in [1]). Therefore, S(ϕY, ξ) = 0 and, consequently, S(ϕY, Z) = 0. Going further, from (2.1) we deduce (4.15). Using (4.4), (4.14) and (4.15) in (4.17) we have (4.16). ¤ Theorem 4.2. Let M be a Sasakian manifold of dimension (2n + 1) > 3. Then the following statements are equivalent: (a) M is quasi contact conformally flat. (b) M is ϕ-contact conformally flat. (c) M is a Sasakian space form with ϕ-sectional curvature (1−3n) / (n+1). Proof. From (4.2) and (4.4) it is obvious that (a) implies (b). Now we assume that (b) is true and wish to prove (c). Replacing X, Y, Z, W by ϕX, ϕY , ϕZ, ϕW , respectively, in (4.16) and using (2.5), (2.4) and (3.3) we get R (X, Y ) Z =

1 {hY, Zi X − hX, Zi Y } n+1 n − {hϕY, Zi ϕX − hϕX, Zi ϕY − 2 hϕX, Y i ϕZ} n+1 − hY, Zi η (X) ξ + hX, Zi η (Y ) ξ + η (X) η (Z) Y − η (Y ) η (Z) X} (4.23)

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for all X, Y, Z ∈ T M , which in view of (2.11) shows that M is a Sasakian space form with ϕ-sectional curvature equal to (1 − 3n) / (n + 1). Thus the statement (c) follows from the statement (b). Finally, it is easy to check that (4.23) implies (4.2); thus the statement (c) implies the statement (a). This completes the proof. ¤ Proposition 4.3. In a K-contact manifold C0 (X, ξ) ξ = 0,

X ∈ T M.

Proof. Let M be a (2n + 1)-dimensional K-contact manifold. Using (2.3) in (4.1) we get C0 (X, Y ) ξ = R (X, Y ) ξ 1 © + −S (Y, ξ) ϕ2 X + S (X, ξ) ϕ2 Y +2 hϕX, Y i Qξ} 2n µ ¶ (3n + 2) r 1 + n+2− {η (Y ) X − η (X) Y } 2n (n + 1) 2n µ ¶ 1 (3n + 2) r 2 − 4n + 5n + 2 − {η (Y ) X − η (X) Y } 2n (n + 1) 2n for all X, Y ∈ T M . Putting Y = ξ in the above equation and using (4.9), (2.1) and (4.8) we get C0 (X, ξ) ξ = (X − η (X) ξ) + (X − η (X) ξ) µ ¶ 1 (3n + 2) r + n+2− (X − η (X) ξ) 2n (n + 1) 2n µ ¶ 1 (3n + 2) r 2 − 4n + 5n + 2 − (X − η (X) ξ) 2n (n + 1) 2n = 0. This completes the proof.

¤

It is known that every contact metric manifold with vanishing contact conformal curvature tensor is a Sasakian space form [6]. But we prove the following: Theorem 4.4. A Sasakian manifold M cannot be ξ-contact conformally flat. Proof. From (4.1), we get C0 (X, Y ) ξ = 2 hϕX, Y i ξ,

X, Y ∈ T M,

(4.24)

where (2.7), (2.8), (2.10), (2.1) and (2.3) have been used. From (4.24) we see that M cannot be ξ-contact conformally flat. ¤ Acknowledgements The authors are thankful to the referee for the critical remarks concerning the improvement of this paper. The first author is also thankful to University Grants Commission (UGC), New Delhi, for financial support in the form of Junior Research Fellowship. The second author is partially supported by the

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PAI project (Junta de Andaluc´ıa, Spain, 2008) and by the MEC-FEDER grant MTM 2007-61284 (MEC, Spain, 2007). References 1. D. E. Blair, Riemannian geometry of contact and symplectic manifolds. Progress in Mathematics, 203. Birkh¨ auser Boston, Inc., Boston, MA, 2002 2. W. M. Boothby and H. C. Wang, On contact manifolds. Ann. of Math. (2) 68(1958), 721–734. ´ ndez, M. Ferna ´ ndez, and G. Zhen, The structure 3. J. L. Cabrerizo, L. M. Ferna of a class of K-contact manifolds. Acta Math. Hungar. 82(1999), No. 4, 331–340. 4. J. C. Jeong, J. D. Lee, G. H. Oh, and J. S. Pak, On the contact conformal curvature tensor. Bull. Korean Math. Soc. 27(1990), No. 2, 133–142. 5. H. Kitahara, K. Matsuo, and J. S. Pak, A conformal curvature tensor field on Hermitian manifolds. J. Korean Math. Soc. 27(1990), No. 1, 7–17. 6. J. S. Pak and Y. J. Shin, A note on contact conformal curvature tensor. Commun. Korean Math. Soc. 13(1998), No. 2, 337–343. ¨ 7. J. A. Schouten, Uber die konforme Abbildung n-dimensionaler Mannigfaltigkeiten mit quadratischer Maßbestimmung auf eine Mannigfaltigkeit mit euklidischer Maßbestimmung. Math. Z. 11(1921), No. 1-2, 58–88. 8. H. Weyl, Reine Infinitesimalgeometrie. Math. Z. 2(1918), No. 3-4, 384–411. 9. H. Weyl, Zur Infinitesimalgeometrie: Einordnung der projektiven und der konformen Auffassung. G¨ ott. Nachr. 1921, 99–112. 10. G. Zhen, On conformal symmetric K-contact manifolds. Chinese Quart. J. Math. 7(1992), 5–10. ´ ndez, and M. Ferna ´ ndez, On ξ-conformally 11. G. Zhen, J. L. Cabrerizo, L. M. Ferna flat contact metric manifolds. Indian J. Pure Appl. Math. 28(1997), No. 6, 725–734.

(Received 9.03.2007; revised 1.07.2008) Authors’ addresses: M. K. Dwivedi Department of Mathematics and Astronomy, Lucknow University Lucknow 226 007, India E-mail: [email protected] L. M. Fern´andez Department of Geometry and Topology, Faculty of Mathematics University of Sevilla, Apartado de Correos 1160, 41080 - Sevilla, Spain E-mail: [email protected] M. M. Tripathi Department of Mathematics and Astronomy, Lucknow University Lucknow 226 007, India Present Address: Department of Mathematics, Banaras Hindu University Varanasi 221 005, India E-mail: [email protected]