On conformal curvature tensor of (∈)-para Sasakian manifolds

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On conformal curvature tensor of (²)-para Sasakian

manifolds

S. S. Shukla and D. D. Singh

(Received November 24, 2010; Revised November 13, 2011)

Abstract. In this article, we study (²)-para Sasakian manifolds with con-formal curvature tensor. In this context, we consider concon-formally flat, quasi-conformally flat and quasi-quasi-conformally semi-symmetric (²)-para Sasakian man-ifolds. It is proved that (²)-para Sasakian manifold Mn(n > 3) is conformally flat if and only if it is locally isometric to a pseudo hyperbolic space Hνn(1) or to a pseudo sphere Sn

ν(1).

AMS 2010 Mathematics Subject Classiﬁcation. 53C15,53C25,53C50.

Key words and phrases. Almost paracontact manifold, (²)-para Sasakian

man-ifold, η-Einstein manman-ifold, conformally ﬂat, conformally ﬂat, quasi-conformally semi-symmetric.

§1. Introduction

In Riemannian geometry, one of the basic interest is curvature properties and to what extent these determine the manifold itself. One of the important curvature properties is conformal ﬂatness.

The Weyl conformal curvature tensor C of type (1,3) of an n-dimensional (n > 3) Riemannian manifold is given by

(1.1) C(X, Y )Z = R(X, Y )Z− 1

n− 2[S(Y, Z)X− S(X, Z)Y

+ g(Y, Z)QX− g(X, Z)QY ]

+ r

(n− 1)(n − 2)[g(Y, Z)X− g(X, Z)Y ], where R, S, Q and r are the Riemannian curvature tensor of type (1,3), the Ricci tensor of type (0,2), the Ricci operator and the scalar curvature of the

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manifold respectively. If C vanishes identically, then the manifold is called con-formally flat. As a generalization of the notion of conformal flatness, K.Yano and S.Sawaki [11] introduced the notion of the quasi-conformal curvature ten-sor eC, which is defined as

(1.2) C(X, Y )Z = aR(X, Y )Z + b[S(Y, Z)Xe − S(X, Z)Y

+ g(Y, Z)QX− g(X, Z)QY ] −r n · a n− 1+ 2b ¸ [g(Y, Z)X−g(X, Z)Y ], where a and b are constants. If a = 1 and b =−_n₋₂1 , then eC = C. Thus the

conformal curvature tensor C is a particular case of the tensor eC.

An n-dimensional (n > 3) manifold is called quasi-conformally ﬂat if eC = 0

identically. Such a quasi-conformally ﬂat manifold is either conformally ﬂat if

a6= 0 or Einstein if a = 0 and b 6= 0 [1]. Several geometers ([1],[2],[3],[5],[9])

studied contact metric manifolds with quasi-conformal curvature tensor. On the other hand, in [10], authors introduced the notion of an (²)-para Sasakian manifold by associating a semi-Riemannian metric to an almost para-contact structure and gave several examples to ensure the existence of such manifolds.

In present article, we study (²)-para Sasakian manifolds with conformal and quasi-conformal curvature tensor. In section 2, we review some basic for-mulae and definitions for (²)-para Sasakian manifolds. Section 3 is devoted to conformally flat (²)-para Sasakian manifolds. In section 4, we study quasi-conformally flat and quasi-quasi-conformally semi-symmetric (²)-para Sasakian man-ifolds. It is proved that such manifolds are η-Einstein.

§2. (²)-para Sasakian manifolds

An n-dimensional differentiable manifold M is called an (²)-para Sasakian manifold [10], if it admits a (1,1) tensor field φ, a vector field ξ, a 1-form η and an indefinite metric g on M satisfying

(2.1) φ2= I − η ⊗ ξ,

(2.2) η(ξ) = 1, φξ = 0, η◦φ = 0,

(2.3) g(φX, φY ) = g(X, Y )−²η(X)η(Y ),

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for all X, Y, Z ∈ T (M), where T (M) denotes the Lie algebra of vector ﬁelds on M and ² is 1 or -1 according as ξ is spacelike or timelike.

In particular, if the metric g is positive deﬁnite, i.e. ² = 1, then M is usual para Sasakian manifold ([7], [8]). If g is Lorentzian, i.e. ² = −1 and ξ is replaced by−ξ, then M is a Lorentzian para Sasakian manifold [4].

Now, from (2.3), we have

(2.5) g(X, φY ) = g(φX, Y ), g(X, ξ) = ²η(X),

for all X, Y ∈ T (M). From (2.5), we obtain (2.6) g(ξ, ξ) = ²,

which shows that the structure vector ﬁeld ξ is never lightlike. The funda-mental (0,2) symmetric tensor of an (²)-para Sasakian structure is deﬁned by Φ(X, Y ) = g(X, φY ), for all X, Y ∈ T (M). For an (²)-para Sasakian manifold [6], we have

(2.7) ∇ξ = ²φ,

(2.8) Φ(X, Y ) = g(φX, Y ) = ²g(∇Xξ, Y ) = (∇Xη)Y.

An (²)- para Sasakian manifold is called η-Einstein if its Ricci tensor S satisﬁes the condition

(2.9) S(X, Y ) = Ag(X, Y )+Bη(X)η(Y ),

where A and B are smooth functions and X, Y are any vector ﬁelds on M . If

B = 0, then it becomes Einstein manifold.

For an n-dimensional (²)-para Sasakian manifold, the Riemannian curva-ture tensor R and Ricci tensor S satisfy the following relations [6]:

(2.10) R(X, Y )ξ = η(X)Y−η(Y )X,

(2.11) R(X, ξ)ξ = η(X)ξ−X,

(2.12) R(ξ, X)Y =−²g(X, Y )ξ+η(Y )X,

(2.13) R(X, Y, U, ξ) =−η(X)g(Y, U)+η(Y )g(X, U),

(2.14) η(R(X, Y )U ) =−²η(X)g(Y, U)+²η(Y )g(X, U),

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§3. Conformally ﬂat (²)-para Sasakian manifolds

Lemma 1. Let R be the Riemannian curvature tensor of an (²)-para Sasakian manifold M , then (3.1) R(X, Y )φZ−φR(X, Y )Z = ²{g(X, Z)φY −g(Y, Z)φX + g(X, φZ)Y − g(Y, φZ)X} + 2²{g(Z, φX)η(Y )ξ +g(Y, φX)η(Z)ξ − g(Z, φY )η(X)ξ − g(X, φY )η(Z)ξ} + 2{η(Y )η(Z)φX − η(X)η(Z)φY }.

Proof. By using (2.4), we have

(3.2) ∇XφY = φ∇XY−g(X, Y )ξ−²η(Y )X+2²η(X)η(Y )ξ.

In view of (3.2), we obtain

(3.3) R(X, Y )φZ =∇X{φ∇YZ− g(Y, Z)ξ − ²η(Z)Y + 2²η(Y )η(Z)ξ} − ∇Y {φ∇XZ− g(X, Z)ξ − ²η(Z)X + 2²η(X)η(Z)ξ}

−©φ∇_{[X,Y ]}Z− g([X, Y ], Z)ξ − ²η(Z)[X, Y ] + 2²η([X, Y ])η(Z)ξª.

On simpliﬁcation, the equation (3.3) becomes

R(X, Y )φZ = φR(X, Y )Z + ²{g(X, Z)φY − g(Y, Z)φX +g(Z, φY )X− g(Z, φX)Y } + 2²{g(Z, φX)η(Y )ξ + g(Y, φX)η(Z)ξ − g(Z, φY )η(X)ξ − g(X, φY )η(Z)ξ} + 2{η(Y )η(Z)φX − η(X)η(Z)φY }, which implies (3.1).

Proposition 2. Let M be an n-dimensional (n > 3) (²)-para Sasakian man-ifold. If M is conformally ﬂat, then M is a space of constant curvature −².

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Proof. Let us suppose that M is conformally ﬂat, then the Riemannian

cur-vature tensor is given by

(3.4) R(X, Y )Z = 1

n− 2{S(Y, Z)X − S(X, Z)Y + g(Y, Z)QX − g(X, Z)QY }

+ r

(n− 1)(n − 2){g(X, Z)Y − g(Y, Z)X} .

By (3.4) and by putting X = Z = ξ and W = X in g(R(X, Y )Z, W ), we obtain

(3.5) (n−1)S(X, Y ) = {(n − 1)² + r} g(X, Y )−² {r + ²n(n − 1)} η(X)η(Y ). Moreover, using (3.4) and (3.5), we have

(3.6) R(X, Y )φZ−φR(X, Y )Z = 1

(n− 1)(n − 2){2(n − 1)² + r} {g(Y, φZ)X

− g(X, φZ)Y − g(Y, Z)φX + g(X, Z)φY }

+{r+²n(n−1)}{g(X, φZ)η(Y )ξ−g(Y, φZ)η(X)ξ + ²η(X)η(Z)φY − ²η(Y )η(Z)φX}.

Now, using (3.1) in (3.6), we obtain

(3.7) {r + ²n(n − 1)} {g(X, φZ)Y − g(Y, φZ)X}

+{²(n − 1)(n − 4) − r} {g(X, Z)φY − g(Y, Z)φX}

−{²(n−1)(3n−4)+r}{g(X, φZ)η(Y )ξ−g(Y, φZ)η(X)ξ}

+{(n − 1)(3n − 4) + r²}{η(Y )η(Z)φX − η(X)η(Z)φY } +2{²(n−1)(n−2)} {g(Y, φX)η(Z)ξ − g(X, φY )η(Z)ξ} = 0. Putting X = e1, Y = e2 and Z = φe2 in (3.7), where {ξ, e1, φe1, ..., en, φen} is

an orthonormal basis of TpM at p∈ M, we get

(3.8) r =−²n(n−1).

So, using (3.8) in (3.5), we have

(3.9) S(X, Y ) =−²(n−1)g(X, Y ).

Thus, M is an Einstein manifold. Now, from (3.9), we have

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By substituting (3.8), (3.9) and (3.10) in (3.4), we get

(3.11) R(X, Y )Z =−² {g(Y, Z)X − g(X, Z)Y } ,

which shows that M is a space of constant curvature −².

Thus, in view of corollary (5.13) [10], we can state the following:

Corollary 1. Let M be an n-dimensional (n > 3) (²)-para Sasakian manifold. Then following statements are equivalent:

(I). M is conformally ﬂat. (II). M is symmetric. (III). M is semi-symmetric. (IV). M satisﬁes R(ξ, X).R = 0.

An n-dimensional (n > 3) conformally flat (²)-para Sasakian manifold is of constant curvature −². Also a manifold of constant curvature is conformally flat. Thus an (²)-para Sasakian manifold is conformally flat if and only if it is a space of constant curvature −². Consequently in this case M is locally isometric to a pseudo hyperbolic space H_νn(1) or pseudo sphere S_νn(1) according as ξ is spacelike or timelike [10].

Thus we have the following:

Proposition 3. Let M be an n-dimensional (n > 3) (²)-para Sasakian mani-fold. Then it is conformally ﬂat if and only if it is locally isometric to a pseudo hyperbolic space H_νn(1) or to a pseudo sphere S_νn(1).

§4. (²)-para Sasakian manifolds with quasi-conformal curvature

tensor

Let M (φ, ξ, η, g, ²) be an n-dimensional quasi-conformally semi-symmetric (²)-para Sasakian manifold, i.e.

(4.1) R(X, Y ). eC = 0,

which implies that

(4.2) R(X, Y ) eC(U, V )W− eC(R(X, Y )U, V )W − eC(U, R(X, Y )V )W − eC(U, V )R(X, Y )W = 0.

Putting X = ξ in (4.2), we obtain

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− eC(U, R(ξ, Y )V )W − eC(U, V )R(ξ, Y )W = 0.

In view of (2.12), the above expression gives

(4.4) −C(U, V, W, Y )+g(Y, U )η( e´e C(ξ, V )W )

+ g(Y, V )η( eC(U, ξ)W ) + g(Y, W )η( eC(U, V )ξ)

+ ²η(Y )η( eC(U, V )W )− ²η(U)η( eC(Y, V )W ) − ²η(V )η( eC(U, Y )W )− ²η(W )η( eC(U, V )Y ) = 0,

whereC(U, V, W, Y ) = g( e´e C(U, V )W, Y ).

Now, from equation (1.2), we have

(4.5) η( eC(X, Y )Z) = aη(R(X, Y )Z) + b[S(Y, Z)η(X)− S(X, Z)η(Y ) + g(Y, Z)η(QX)− g(X, Z)η(QY )] −r n · a n− 1+ 2b ¸ [g(Y, Z)η(X)−g(X, Z)η(Y )]. Taking X = Y in the above equation, we get

(4.6) η( eC(X, X)Z) = 0.

Again, putting Z = ξ in equation (4.5) and using (2.10), (2.15), we obtain

(4.7) η( eC(X, Y )ξ) = 0.

By substituting Y = U in equation (4.4) and using (4.6), (4.7), we get

(4.8) −C(U, V, W, U )+g(U, U )η( e´e C(ξ, V )W )

+ g(U, V )η( eC(U, ξ)W )− ²η(W )η( eC(U, V )U ) = 0.

Now, putting U = ei in (4.8) and taking summation over i, 1 ≤ i ≤ n, where {e1, e2, ..., en−1, en = ξ} is an orthonormal basis of tangent space at each

point of the manifold, we obtain

(4.9) − n X i=1 ´_e C(ei, V, W, ei)+(n−1+²)η( eC(ξ, V )W ) + n X i=1 g(ei, V )η( eC(ei, ξ)W )− ²η(W ) n X i=1 η( eC(ei, V )ei) = 0.

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Moreover, from (2.14), we get (4.10) n X i=1 η(R(ei, V )ei) = ²(n−1)η(V ), (4.11) η(R(ξ, V )W ) =−²g(V, W )+η(V )η(W ) and (4.12) n X i=1 g(ei, V )η(R(ei, ξ)W ) =−η(V )η(W )+²g(V, W ).

Now, from (1.2), we have

(4.13) C(U, V, W, Y ) = a ´´e R(U, V, W, Y ) + b[S(V, W )g(U, Y )− S(U, W )g(V, Y ) + g(V, W )g(QU, Y )− g(U, W )g(QV, Y )] −r n · a n− 1+ 2b ¸ [g(V, W )g(U, Y )− g(U, W )g(V, Y )]. Putting U = Y = ei in (4.13) and taking summation over i, 1 ≤ i ≤ n, we

obtain (4.14) n X i=1 ´_e C(ei, V, W, ei) ={a + b(n − 3 + ²)} S(V, W ) + ½ br− r n µ a n− 1+ 2b ¶ (n− 2 + ²) ¾ g(V, W ).

Again, by using (2.5), (2.15) and (4.10), the equation (4.5) implies

(4.15) n X i=1 η( eC(ei, V )ei) ={a²(n−1)+b²(n−1)2−b²(n−1) − br + r n µ a n− 1+ 2b ¶ (n− 1)}η(V ). In view of (2.5), (2.15), (4.12) and (4.5), we get

(4.16) n X i=1 g(ei, V )η( eC(ei, ξ)W ) =−bS(V, W ) + ½ b²(n− 1) + a² + r n µ a n− 1+ 2b ¶¾ g(V, W )

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− ½ 2b(n− 1) + a + r² n µ a n− 1+ 2b ¶¾ η(V )η(W ) and (4.17) n X i=1 g(ei, V )η( eC(ei, ξ)W ) =−η( eC(ξ, V )W ).

Therefore, by using (4.14), (4.15), (4.16), (4.17) in equation (4.9), we obtain

(4.18) S(V, W ) = 1

(a− b)[− {b(r + (n − 1)) + ²(n − 2) (a + b(n − 1)) + a} g(V, W ) +{b(n − 1)(n − 2) + b²(r + (n − 1)) + ² (a + b(n − 1)) − a} η(V )η(W )], which can be written as

(4.19) S(V, W ) = Ag(V, W )+Bη(V )η(W ), where (4.20) A = 1 (a− b)[− {b(r + (n − 1)) + ²(n − 2) (a + b(n − 1)) + a}] and (4.21) B = 1 (a− b)[{b(n − 1)(n − 2) + b²(r + (n − 1)) + ² (a + b(n − 1)) − a}]. Hence, we can state the following:

Proposition 4. A quasi-conformally semi symmetric (²)-para Sasakian man-ifold is an η-Einstein manman-ifold.

As a quasi-conformally ﬂat (²)-para Sasakian manifold is quasi-conformally semi-symmetric, so we can state the following corollary:

Corollary 2. A quasi-conformally ﬂat (²)-para Sasakian manifold is an η-Einstein manifold.

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S. S. Shukla

Department of Mathematics, University of Allahabad, Allahabad-211002, India

E-mail : ssshukla−[email protected] D. D. Singh

Department of Mathematics, University of Allahabad, Allahabad-211002, India