Some Curvature Conditions on a Para-Sasakian Manifold with Canonical Paracontact Connection

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Volume 2012, Article ID 395462,24pages doi:10.1155/2012/395462

Research Article

Some Curvature Conditions on a Para-Sasakian Manifold with Canonical Paracontact Connection

Bilal Eftal Acet,

¹

Erol Kılıc¸,

²

and Selcen Y ¨uksel Perktas¸

¹

1Department of Mathematics, Arts and Science Faculty, Adıyaman University, 02040 Adıyaman, Turkey

2Department of Mathematics, Arts and Science Faculty, ˙In¨on ¨u University, 44280 Malatya, Turkey

Correspondence should be addressed to Selcen Y ¨uksel Perktas¸,sperktas@adiyaman.edu.tr Received 2 October 2012; Accepted 2 December 2012

Academic Editor: Jerzy Dydak

Copyrightq2012 Bilal Eftal Acet et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study canonical paracontact connection on a para-Sasakian manifold. We prove that a Ricci-flat para-Sasakian manifold with respect to canonical paracontact connection is anη-Einstein manifold. We also investigate some properties of curvature tensor, conformal curvature tensor,W2- curvature tensor, concircular curvature tensor, projective curvature tensor, and pseudo-projective curvature tensor with respect to canonical paracontact connection on a para-Sasakian manifold.

It is shown that a concircularly flat para-Sasakian manifold with respect to canonical paracontact connection is of constant scalar curvature. We give some characterizations for pseudo-projectively flat para-Sasakian manifolds.

1. Introduction

In 1976, Sato1introduced the almost paracontact structureϕ, ξ, ηsatisfyingϕ²I−η⊗ξ and ηξ 1 on a diﬀerentiable manifold. Although the structure is an analogue of the almost contact structure2,3, it is closely related to almost product structurein contrast to almost contact structure, which is related to almost complex structure. It is well known that an almost contact manifold is always odd-dimensional but an almost paracontact manifold defined by Sato1could be even dimensional as well. Takahashi4defined almost contact manifolds equipped with an associated pseudo-Riemannian metric. In particular he studied Sasakian manifolds equipped with an associated pseudo-Riemannian metric. Also, in 1989, Matsumoto5replaced the structure vector fieldξby−ξin an almost paracontact manifold and associated a Lorentzian metric with the resulting structure and called it a Lorentzian almost paracontact structure. It is obvious that in a Lorentzian almost paracontact manifold, the pseudo-Riemannian metric has only signature 1 and the structure vector fieldξis always

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timelike. These circumstances motivated the authors6to associate a pseudo-Riemannian metric, not necessarily Lorentzian, with an almost paracontact structure.

Kaneyuki and Konzai 7 defined the almost paracontact structure on pseudo- Riemannian manifold M of dimension 2n1 and constructed the almost paracomplex structure onM²ⁿ¹×R. Zamkovoy8associated the almost paracontact structure introduced in7to a pseudo-Riemannian metric of signaturen1, nand showed that any almost paracontact structure admits such a pseudo-Riemannian metric which is called compatible metric.

Tanaka-Webster connection has been introduced by Tanno9as a generalization of the well-known connection defined by Tanaka 10 and, independently, by Webster 11, in context of CR geometry. In a paracontact metric manifold Zamkovoy 8 introduced a canonical connection which plays the same role of the generalized Tanaka-Webster connection 9 in paracontact geometry. In this study we define a canonical paracontact connection on a para-Sasakian manifold which seems to be the paracontact analogue of the generalizedTanaka-Webster connection.

In the present paper we study canonical paracontact connection on a para-Sasakian manifold. Section 2 is devoted to preliminaries. In Section 3, we investigate the relation between curvature tensor resp., Ricci tensor with respect to canonical paracontact connection and curvature tensorresp., Ricci tensorwith respect to Levi-Civita connection.

InSection 4, conformal curvature tensor of a para-Sasakian manifold with respect to canonical paracontact connection is obtained.Section 5contains the expression ofW₂-curvature tensor.

In Section 6, we study a para-Sasakian manifold satisfying the conditionCξ, X·R 0, where Cξ, X is considered as a derivation of the tensor algebra at each point of the manifold,Ris the curvature tensor, andCis the conformal curvature tensor with respect to canonical paracontact connection. InSection 7, we obtain some equations in terms of Ricci tensor on a para-Sasakian manifold satisfying W₂ξ, X ·R 0 and W₂ξ, X ·W₂ 0, respectively. In Section 8 it is proved that a concircularly flat para-Sasakian manifold is of constant scalar curvature.Section 9is devoted to pseudo-projectively flat para-Sasakian manifolds. InSection 10, we show that a para-Sasakian manifold satisfyingZξ, X·R 0 with respect to canonical paracontact connection, is either of constant scalar curvature or an η-Einstein manifold. Also, it is proved that if the conditionZξ, X·S 0 holds on a para- Sasakian manifold with respect to canonical paracontact connection, then the scalar curvature is constant. InSection 11, we give some characterizations for para-Sasakian manifolds with canonical paracontact connection satisfyingP Pξ, X·R0 andP Pξ, X·S0, respectively.

In the last section it is shown that a para-Sasakian manifold on which the conditionPξ, X· P P0 holds, is either of constant scalar curvature or anη-Einstein manifold.

2. Preliminaries

A diﬀerentiable manifoldMof dimension2n1is called almost paracontact manifold with the almost paracontact structureϕ, ξ, ηif it admits a tensor fieldϕof type1,1, a vector field ξ, a 1-formηsatisfying the following conditions7:

ϕ²I−η⊗ξ, 2.1

ηξ 1, ϕξ0, 2.2

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where I denotes the identity transformation. Moreover, the tensor field ϕ induces an almost paracomplex structure on the paracontact distribution D kerη; that is, the eigendistributionsD^±corresponding to the eigenvalues±1 ofϕare bothn-dimensional. As an immediate consequence of the conditions2.2we have

η◦ϕ0, rank ϕ

2n. 2.3 If a2n1-dimensional almost paracontact manifoldMwith an almost paracontact structureϕ, ξ, ηadmits a pseudo-Riemannian metricgsuch that8

g

ϕX, ϕY

−gX, Y ηXηY, X, Y∈ΓTM, 2.4

then we say thatMis an almost paracontact metric manifold with an almost paracontact metric structureϕ, ξ, η, g, and such metricgis called compatible metric. Any compatible metricgis necessarily of signaturen1, n.

From2.4it can be easily seen that8 g

X, ϕY −g

ϕX, Y

, 2.5

gX, ξ ηX, 2.6

for anyX, Y ∈ΓTM. The fundamental 2-form ofMis defined by ΦX, Y g

X, ϕY

. 2.7

An almost paracontact metric structure becomes a paracontact metric structure ifgX, ϕY dηX, Y, for all vector fields X, Y ∈ ΓTM, where dηX, Y 1/2{XηY−Y ηX− ηX, Y}.

For a2n1-dimensional manifoldMwith an almost paracontact metric structure ϕ, ξ, η, gone can also construct a local orthonormal basis. LetUbe coordinate neighborhood onMandX1any unit vector field onUorthogonal toξ. ThenϕX1is a vector field orthogonal to both X₁ and ξ, and|ϕX1|² −1. Now choose a unit vector fieldX₂ orthogonal to ξ,X₁ andϕX₁. ThenϕX₂ is also a vector field orthogonal toξ,X₁,ϕX₁, andX₂ and|ϕX2|² −1.

Proceeding in this way we obtain a local orthonormal basisXi, ϕXi, ξ, i1,2, . . . , ncalled aϕ-basis8.

Remark 2.1. It is also known that a diﬀerentiable manifold has an almost paracontact metric structure if it admits a Riemannian metricgsuch thatgϕX, ϕY gX, Y−ηXηY see 1. But in our paper the metricgis pseudo-Riemannian and satisfies condition2.4.

Recall that an almost paracomplex structure12on a 2n-dimensional manifold is a tensor fieldJof type1,1such thatJ² Iand eigensubbundlesT,T⁻corresponding to the eigenvalue 1,−1 ofJ, respectively, have equal dimensionaln. The Nijenhuis tensorN ofJ, given by

NJX, Y JX, JY X, Y−JJX, Y−JX, JY, X, Y ∈ΓTM, 2.8

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is the obstruction for the integrability of the eigensubbundlesT,T⁻. IfN0, then the almost paracomplex structure is called paracomplex or integrable13.

Let M²ⁿ¹ be an almost paracontact metric manifold with structure ϕ, ξ, η and consider the manifold M²ⁿ¹ ×R. We denote a vector field on M²ⁿ¹×R, byX, fd/dt whereX is tangent toM²ⁿ¹,tis the coordinate onR, andf is a diﬀerentiable function on M²ⁿ¹×R. An almost paracomplex structureJonM²ⁿ¹×Ris defined in14by

J

X, f d dt

ϕXfξ, ηXd dt

. 2.9

IfJis integrable we say that the almost paracontact structureϕ, ξ, ηis normal.

A normal paracontact metric manifold is a para-Sasakian manifold. An almost paracontact metric structure ϕ, ξ, η, g on aM²ⁿ¹ is para-Sasakian manifold if and only if 8

∇Xϕ

Y −gX, YξηYX, 2.10

whereX, Y∈ΓTMand∇is Levi-Civita connection ofM.

From2.10, it can be seen that

∇Xξ−ϕX. 2.11

Also in a para-Sasakian manifold, the following relations hold8:

gRX, YZ, ξ ηRX, YZ gX, ZηY−gY, ZηX, 2.12

RX, YξηXY −ηYX, 2.13

Rξ, XY −gX, YξηYX, 2.14

Rξ, XξX−ηXξ, 2.15

SX, ξ −2nηX, 2.16

for any vector fieldsX, Y, Z∈ΓTM. Here,Ris Riemannian curvature tensor andSis Ricci tensor defined bySX, Y gQX, Y, whereQis Ricci operator.

In the following we consider the connection∇defined by9

∇XY ∇XY ηXϕY−ηY∇Xξ

∇Xη Y

ξ, 2.17

whereX, Y∈ΓTM. If we use2.11in2.17, then we obtain

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∇XY ∇XY ηXϕYηYϕXg X, ϕY

ξ. 2.18

Definition 2.2. We call the connection∇defined by 2.18on a para-Sasakian manifold the canonical paracontact connection on a para-Sasakian manifold.

Proposition 2.3. On a para-Sasakian manifold the connection∇has the following properties:

∇η0, ∇g0, ∇ξ0, ∇Xϕ

Y

∇Xϕ

YgX, Yξ−ηYX, 2.19

for allX, Y ∈ΓTM.

Proof. Calculation is straightforward by using2.18.

Furthermore, we define homeomorphismsCX, Y·R0, W₂X, Y·R0, W₂X, Y·

W20,ZX, Y·R0,P PX, Y·R0 andPX, Y·P P0 as follows:

CX, Y·R

U, VWCX, YRU, VW−R

CX, YU, V W

−R

U, CX, YV

W−RU, VCX, YW 0,

2.20

W₂X, Y·R

U, VWW₂X, YRU, VW

−R

W2X, YU, V

W−R

U, W2X, YV W

−RU, VW2X, YW 0,

2.21

W2X, Y·W2

U, VWW2X, YW2U, VW

−W₂

W₂X, YU, V

W−W₂

U, W₂X, YV W

−W₂U, VW2X, YW 0,

2.22

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ZX, Y·R

U, VWZX, YRU, VW−R

ZX, YU, V W

−R

U, ZX, YV

W−RU, VZX, YW 0,

2.23

P PX, Y·R

U, VWP PX, YRU, VW−R

P PX, YU, V W

−R

U, P PX, YV

W−RU, VP PX, YW 0,

2.24

PX, Y·P P

U, VWPX, YP PU, VW−P P

PX, YU, V W

−P P

U, PX, YV

W−P PU, VPX, YW 0,

2.25

for allX, Y, U, V, W∈ΓTM.

3. Curvature Tensor

Let M²ⁿ¹ be a para-Sasakian manifold. The curvature tensor R of M with respect to the canonical paracontact connection∇is defined by

RX, YZ∇X∇YZ− ∇Y∇XZ− ∇X,YZ, 3.1

for anyX, Y, Z∈ΓTM.

By using2.18in3.1we obtain

RX, YZRX, YZgY, ZηXξ−gX, ZηYξ ηYηZX−ηXηZY2g

X, ϕY ϕZ g

X, ϕZ

ϕY−g Y, ϕZ

ϕX,

3.2

whereRX, YZ∇X∇YZ− ∇Y∇XZ− ∇_X,YZis curvature tensor ofMwith respect to Levi- Civita connection∇.

LetT andT be curvature tensors of type0, 4with respect to Levi-Civita connection

∇and the canonical paracontact connection∇, respectively, given by TX, Y, Z, W gRX, YZ, W, TX, Y, Z, W g

RX, YZ, W ,

3.3

for allX, Y, Z, W∈ΓTM.

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Proposition 3.1. In a para-Sasakian manifold one has,

RX, YZRY, ZXRZ, XY 0, 3.4

TX, Y, Z, W TY, X, Z, W 0, 3.5

TX, Y, Z, W TX, Y, W, Z 0, 3.6

TX, Y, Z, W−TZ, W, X, Y 0, 3.7

whereX, Y, Z, W ∈ΓTM.

Proof. Using 3.2 and first Bianchi identity with respect to Levi-Civita connection ∇ we obtain

RX, YZRY, ZXRZ, XY 2g

X, ϕY

ϕZg X, ϕZ

ϕY−g Y, ϕZ

ϕX2g Y, ϕZ

ϕX g

Y, ϕX

ϕZ−g Z, ϕX

ϕY 2g

Z, ϕX

ϕY g Z, ϕY

ϕX−g X, ϕY

ϕZ.

3.8

From2.5we get3.4.

From3.2we have

TX, Y, Z, W TX, Y, Z, W gY, ZηXηW

−gX, ZηYηW gX, WηYηZ

−gY, WηXηZ 2g X, ϕY

g ϕZ, W g

X, ϕZ g

ϕY, W

−g Y, ϕZ

g ϕX, W

.

3.9

It is well known that

TX, Y, Z, W −TY, X, Z, W, TX, Y, Z, W TX, Y, W, Z,

TX, Y, Z, W TZ, W, X, Y. 3.10 By taking into account the previous equations we get3.5,3.6, and3.7, respectively.

LetE_i {ei, ϕe_i, ξ} i 1,2, . . . , nbe a local orthonormalϕ-basis of a para-Sasakian manifoldM. Then the Ricci tensorSand the scalar curvatureτofMwith respect to canonical paracontact connection∇are defined by

SX, Y ⁿ

i1

g

Rei, XY, ei

−ⁿ

i1

g R

ϕei, X Y, ϕei

g

Rξ, XY, ξ ,

3.11

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whereX, Y∈ΓTM, and

τ ⁿ

i1

Sei, e_i−ⁿ

i1

S

ϕe_i, ϕe_i

Sξ, ξ, 3.12

respectively.

Theorem 3.2. In a 2n 1-dimensional para-Sasakian manifold the Ricci tensor S and scalar curvatureτof canonical paracontact connection∇are given by

SX, Y SX, Y−2gX, Y 2n2ηXηY, 3.13

τ τ−2n, 3.14

where X, Y ∈ ΓTM and S and τ denote the Ricci tensor and scalar curvature of Levi-Civita connection∇, respectively. Consequently,Sis symmetric.

Proof. Using3.2and3.9, we have, for anyX, Y ∈ΓTM, SX, Y ⁿ

i1

gRei, XY, ei gei, e_iηXηY 3g

ei, ϕX g

ϕY, ei

−ⁿ

i1

g R

ϕe_i, X Y, ϕe_i

g

ϕe_i, ϕe_i

ηXηY

3g

ϕe_i, ϕX g

ϕY, ϕe_i .

3.15

Since the Ricci tensor of Levi-Civita connection∇is given by SX, Y ⁿ

i1

gRei, XY, ei−ⁿ

i1

g R

ϕe_i, X Y, ϕe_i

gRξ, XY, ξ, 3.16

then3.15implies3.13. Equation3.14follows from3.13.

Also from3.13, it is obvious thatSis symmetric.

Corollary 3.3. If a para-Sasakian manifold is Ricci-flat with respect to canonical paracontact connection, then it is anη-Einstein manifold.

Lemma 3.4. LetMbe a para-Sasakian manifold with canonical paracontact connection∇. Then

g

RX, YZ, ξ η

RX, YZ

0, 3.17

RX, YξRξ, XY Rξ, Xξ0, 3.18

SX, ξ 0, 3.19

for allX, Y, Z∈ΓTM.

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Proof. Calculation is straightforward by using2.2,2.6,2.13, and2.16in3.2.

4. Conformal Curvature Tensor

LetMbe a2n1-dimensional para-Sasakian manifold. The conformal curvature tensor of Mwith respect to canonical paracontact connection∇is defined by

CX, YZRX, YZ− 1 2n−1

SY, ZX−SX, ZY gY, ZQX−gX, ZQY

τ 2n2n−1

gY, ZX−gX, ZY ,

4.1

whereX, Y, Z∈ΓTM. By using3.2,3.13, and3.14in4.1 CX, YZRX, YZgY, ZηXξ−gX, ZηYξ

ηYηZX−ηXηZY2g X, ϕY

ϕZ g

X, ϕZ

ϕY −g Y, ϕZ

ϕX

− 1 2n−1

⎛

⎜⎜

⎜⎝

SY, ZX−2gY, ZX

2n2ηYηZX

−SX, ZY2gX, ZY

−2n2ηXηZY

gY, ZQX−2gY, ZX 2n2gY, ZηXξ

−gX, ZQY 2gX, ZY

−2n2gX, ZηYξ

⎞

⎟⎟

⎟⎠

τ−2n 2n2n−1

gY, ZX−gX, ZY .

4.2

SinceRX, YZ−RY, XZ, then

CX, YZCY, XZ0. 4.3

Moreover, from the first Bianchi identity we get

CX, YZCY, ZXCZ, XY 0. 4.4

5. W

2

-Curvature Tensor

In15Pokhariyal and Mishra have introduced a new tensor field, calledW2-curvature tensor field, in a Riemannian manifold and studied its properties.

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TheW2-curvature tensor is defined by

W2X, Y, Z, V RX, Y, Z, V 1 n−1

gX, ZSY, V−gY, ZSX, V

, 5.1

whereX, Y, Z, V ∈ΓTMandSis a Ricci tensor of type0,2. TheW₂-curvature tensor of a para-Sasakian manifoldM²ⁿ¹with respect to canonical paracontact connection∇is defined by

W₂X, YZRX, YZ− 1 2n

gY, ZQX−gX, ZQY

. 5.2

From3.2and3.13we get

W2X, YZRX, YZgY, ZηXξ−gX, ZηYξ ηYηZX−ηXηZY2g

X, ϕY ϕZ g

X, ϕZ

ϕY−g Y, ϕZ

ϕX

− 1 2n−1

⎛

⎜⎜

⎝

gY, ZQX−2gY, ZX 2n2gY, ZηXξ

−gX, ZQY 2gX, ZY

−2n2gX, ZηYξ

⎞

⎟⎟

⎠.

5.3

SinceRX, YZRY, XZ0, we have

W₂X, YZW₂Y, XZ0. 5.4

Furthermore, by using5.3and the first Bianchi identity we obtain

W₂X, YZW₂Y, ZXW₂Z, XY 0. 5.5

6. Para-Sasakian Manifold with Canonical Paracontact Connection Satisfying Cξ, X · R 0

In this section we consider a para-Sasakian manifoldM²ⁿ¹satisfying the condition

Cξ, X·R0, 6.1

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with respect to canonical paracontact connection. From2.20we get Cξ, X·R

U, VWCξ, XRU, VW−R

Cξ, XU, V W

−R

U, Cξ, XV

W−RU, VCξ, XW 0,

6.2

for anyX, U, V, W∈ΓTM. Now puttingUξin6.2, we have Cξ, XRξ, VW−R

Cξ, Xξ, V

W−R

ξ, Cξ, XV

W−Rξ, VCξ, XW 0.

6.3

Taking inner product withY and using3.18in6.3we get g

R

Cξ, Xξ, V W, Y

0. 6.4

Using4.2in6.4, we obtain 1

2n−1g

RW, YQX, V

− τ

2n2n−1g

RW, YX, V

0. 6.5

Again using3.2,3.13, and3.14in6.5we obtain

1 2n−1

⎡

⎢⎢

⎣

gRW, YQX, V gY, QXηVηW

−gW, QXηYηV gW, VηQXηY

−gY, VηQXηW 2g W, ϕY

g

ϕQX, V g

W, ϕQX g

ϕY, V

−g

Y, ϕQX g

ϕW, V

⎤

⎥⎥

⎦

− τ−2n 2n2n−1

⎡

⎢⎢

⎣

gRW, YX, V gY, XηVηW

−gW, XηYηV gW, VηXηY

−gY, VηXηW 2g W, ϕY

g ϕX, V g

W, ϕX g

ϕY, V

−g Y, ϕX

g ϕW, V

⎤

⎥⎥

⎦0.

6.6

Let{ei, ϕe_i, ξ}i1, . . . , nbe an orthonormal basis of the tangent space at any point.

Hence by suitable contracting of6.6we get 1

2n−1

S²X, Y−4SX, Y 4gX, Y

−

4n²8n4

ηXηY

− τ−2n 2n2n−1

SX, Y−2gX, Y

2n2ηXηY

0. 6.7

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Theorem 6.1. Let M²ⁿ¹ be a para-Sasakian manifold satisfyingCξ, X·R 0 with respect to canonical paracontact connection∇. Then

SQX, Y

3 τ 2n

SX, Y−

2 τ n

gX, Y

4n²6n2τn1 n

ηXηY,

6.8

for anyX, Y ∈ΓTM.

7. Para-Sasakian Manifold with Canonical Paracontact Connection Satisfying W

2

ξ, X · R 0 and W

2

ξ, X · W

2

0

In this section we consider a para-Sasakian manifold with canonical connection satisfying the condition

W2ξ, X·R0. 7.1

From2.21, we get

W2ξ, X·R

U, VW W2ξ, XRU, VW−R

W2ξ, XU, V W

−R

U, W₂ξ, XV

W−RU, VW2ξ, XW 0,

7.2

whereX, U, V, W∈ΓTM. Now if we putUξin7.2, we have

W₂ξ, XRξ, VW−R

W₂ξ, Xξ, V W

−R

ξ, W₂ξ, XV

W−Rξ, VW2ξ, XW 0.

7.3

Taking inner product withY and from3.18, we obtain

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g R

W₂ξ, Xξ, V W, Y

0. 7.4

Using5.3in7.4, we have 1 2ng

RW, YQX, V

0. 7.5

Again using3.2and3.13in7.5we obtain

1 2n

⎡

⎢⎢

⎢⎣

gRW, YQX, V gY, QXηVηW

−gW, QXηYηV gW, VηQXηY

−gY, VηQXηW 2g W, ϕY

g

ϕQX, V g

W, ϕQX g

ϕY, V

−g

Y, ϕQX g

ϕW, V

−2gRW, YX, V−2gY, XηVηW 2gW, XηYηV−2gW, VηXηY 2gY, VηXηW−4g

W, ϕY g

ϕX, V

−2g W, ϕX

g ϕY, V

2g Y, ϕX

g ϕW, V

⎤

⎥⎥

⎥⎦

0. 7.6

So by a contraction of7.6with respect toV andWwe get 1

2n

−

4n²8n4

ηXηY

0. 7.7

Theorem 7.1. LetM²ⁿ¹ be a para-Sasakian manifold satisfyingW₂ξ, X·R 0 with respect to canonical paracontact connection∇. Then

SQX, Y 4SX, Y−4gX, Y

4n²8n4

ηXηY, 7.8

Now let us consider a para-Sasakian manifold with canonical paracontact connection satisfying the condition

W2ξ, X·W20. 7.9

From2.22, we get

W₂ξ, X·W₂

U, VZW₂ξ, XW2U, VZ−W₂

W₂ξ, XU, V Z

−W₂

U, W₂ξ, XV

Z−W₂U, VW2ξ, XZ 0.

7.10

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NowUξin7.10, we have

W2ξ, XW2ξ,VZ−W2

W2ξ, Xξ, V Z

−W2

ξ, W2ξ, XV

Z−W2ξ, VW2ξ, XZ 0.

7.11

Taking inner product withY and from3.18and3.19we obtain 1

2ng W2

QX, V

Z, Y

− 1 4n²

g

Q²X, Y

ηVηZ

0. 7.12

Using5.3in7.12, we have

1 2ng

R QX, V

Z, Y

− 1 4n²

⎡

⎣g

Q²X, Y

gV, Z−g QX, Z

g QV, Y g

Q²X, Y

ηVηZ

⎤

⎦0. 7.13

Hence by suitable contracting7.13we get 1

2n

−

4n²8n4

ηXηY

− 1 4n²

−

4n²8n4

ηXηY

0. 7.14

Theorem 7.2. LetM²ⁿ¹be a para-Sasakian manifold satisfyingW₂ξ, X·W₂ 0 with respect to canonical paracontact connection∇. Then

SQX, Y 4SX, Y−4gX, Y

4n²8n4

ηXηY, 7.15

8. Concircularly Flat Para-Sasakian Manifold

The concircular curvature tensor of a para-Sasakian manifoldM²ⁿ¹with respect to canonical paracontact connection∇is defined by

ZX, YW RX, YW− τ 2n2n−1

gY, WX−gX, WY

, 8.1

for anyX, Y, W ∈ΓTM.

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By using3.2and3.14we obtain from8.1

ZX, YW RX, YWgY, WηXξ−gX, WηYξ ηYηWX−ηXηWY2g

X, ϕY ϕW g

X, ϕW

ϕY −g Y, ϕW

ϕX

− τ−2n 2n2n−1

gY, WX−gX, WY .

8.2

If M is a concircularly flat para-Sasakian manifold with respect to canonical paracontact connection, then we have

g

ZX, YW, ϕV

0. 8.3

Hence using2.2in8.3we get g

RX, YW, ϕV g

X, ϕV

ηYηW

−g Y, ϕV

ηXηW 2g X, ϕY

g

ϕW, ϕV g

X, ϕW g

ϕY, ϕV

−g Y, ϕW

g

ϕX, ϕV

− τ−2n 2n2n1

gY, Wg X, ϕV

−gX, Wg

Y, ϕV 0.

8.4

PuttingY W ξin8.4and using2.2and2.14we have τ−2n

2n2n1g X, ϕV

0. 8.5

From8.5we obtain

τ 2n. 8.6

Hence we have the following.

Theorem 8.1. If a para-Sasakian manifold M is concircularly flat with respect to canonical paracontact connection, then it is of constant scalar curvature.

9. Pseudo-projectively Flat Para-Sasakian Manifold

In 2002, Prasad 16 defined and studied a tensor field P on a Riemannian manifold of dimensionn, which includes projective curvature tensorP. This tensor fieldP is known as pseudo-projective curvature tensor.

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In this section, we study pseudo-projective curvature tensor with respect to canonical paracontact connection∇in a para-Sasakian manifold and we denote this curvature tensor withP P.

Pseudo-projective curvature tensor P P of a para-Sasakian manifold M²ⁿ¹ with respect to canonical paracontact connection∇is defined by

P PX, YW aRX, YWbSY, WX−SX, WY

− τ 2n1

a 2nb

gY, WX−gX, WY ,

9.1

whereX, Y, W ∈ΓTMandaandbare constants such thata,b /0.

Ifa1 andb1/2n2, then9.1takes the form

P PX, YWRX, YW 1 2n2

SY, WX−SX, WY

− τ

2n2n

9.2

Using3.2,3.13and3.14in9.2, we get

P PX, YW RX, YWgY, WηXξ−gX, WηYξ ηYηWX−ηXηWY2g

X, ϕY ϕW g

X, ϕW

ϕY −g Y, ϕW

ϕX

1 2n2

⎛

⎜⎜

⎝

SY, WX−2gY, WX 2n2ηYηWX

−SX, WY2gX, WY

−2n2ηXηWY

⎞

⎟⎟

⎠

− τ−2n 2n1

9.3

IfM is a pseudo-projectively flat para-Sasakian manifold with respect to canonical paracontact connection, then

g

P PX, YW, ϕU

0, 9.4

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for anyU∈ΓTM. Hence using2.2in9.4we get g

RX, YW, ϕU g

X, ϕU

ηYηW

−g Y, ϕU

ηXηW 2g X, ϕY

g

ϕW, ϕU g

X, ϕW g

ϕY, ϕU

−g Y, ϕW

g

ϕX, ϕU

1 2n2

⎛

⎜⎜

⎝

SY, Wg X, ϕU

−2gY, Wg X, ϕU 2n2g

X, ϕU

ηYηW

−SX, Wg Y, ϕU

2gX, Wg Y, ϕU

−2n2g Y, ϕU

ηXηW

⎞

⎟⎟

⎠

− τ−8n 2n2n

gY, Wg X, ϕU

−gX, Wg

Y, ϕU 0.

9.5

Now puttingY Wξin9.5and using2.2,2.14, and2.16we have τ−2n

2n2ng X, ϕU

0. 9.6

From9.6we get

τ 2n. 9.7

Theorem 9.1. If a para-Sasakian manifold is pseudo-projectively flat with respect to canonical paracontact connection, then its scalar curvature is constant.

10. Para-Sasakian Manifold with Canonical Paracontact Connection Satisfying Zξ, X · R 0 and Zξ, X · S 0

In this section we firstly consider a para-Sasakian manifoldMsatisfying

Zξ, X·R0, 10.1

for anyX∈ΓTM, with respect to canonical paracontact connection.

From2.23, we have Zξ, X·R

U, VWZξ, XRU, VW−R

Zξ, XU, V W

−R

U, Zξ, XV

W−RU, VZξ, XW 0,

10.2

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for allX, U, V, W∈ΓTM. Now by puttingUξin10.2, we get Zξ, XRξ, VW−R

Zξ, Xξ, V W

−R

ξ, Zξ, XV

W−Rξ, VZξ, XW0.

10.3

Using3.18in10.3we obtain R

Zξ, Xξ, V

W 0. 10.4

Taking inner product withY and from8.2, we have τ−2n

2n2n−1g

RX, VW, Y

0. 10.5

Again using3.2in10.5we get

τ−2n 2n2n−1

⎡

⎢⎢

⎣

gRX, VW, Y gV, WηXηY

−gX, WηYηV gX, YηVηW

−gY, VηXηW 2g X, ϕV

g ϕW, Y g

X, ϕW g

Y, ϕV

−g V, ϕW

g ϕX, Y

⎤

⎥⎥

⎦0. 10.6

Let{ei, ϕei, ξ}i1, . . . , nbe an orthonormal basis of the tangent space at any point.

So a contraction of10.6with respect toXandY gives τ−2n

2n2n−1

SV, W−2gV, W 2n2ηVηW

0. 10.7

Therefore we have the following.

Theorem 10.1. LetM²ⁿ¹be a para-Sasakian manifold satisfying the conditionZξ, X·R0, for anyX∈ΓTM. Then either

iτ 2n, that is, the scalar curvature is constant, or

iiMis anη-Einstein manifold with equation

SV, W 2gV, W−2n2ηVηW, 10.8

for allV, W∈ΓTM.

Now, let us consider a para-Sasakian manifoldMsatisfying

Zξ, X·S0, 10.9

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where X ∈ ΓTM and S is the Ricci tensor of M with respect to canonical paracontact connection.

From10.9, for anyX, U, V ∈ΓTM, we obtain S

Zξ, XU, V S

U, Zξ, XV

0, 10.10

for allX, U, V ∈ΓTM. Using3.18and3.19with8.2in10.10we get A

SX, VηU SX, UηV

0, 10.11 whereAτ/2n2n−1.

Using3.13and3.14in10.11, we get

τ−2n 2n2n−1

⎡

⎢⎢

⎣

SX, VηU−2gX, VηU

2n2ηXηVηU

−SX, UηV 2gX, UηV

−2n2ηXηVηU

⎤

⎥⎥

⎦0. 10.12

By a contraction of10.12with respect toXandUwe obtain τ−2n

2n2n−1

τ−2nηV

0. 10.13

This givesτ2nwhich implies that manifold is of constant scalar curvature.

Theorem 10.2. If the conditionZξ, X·S 0, for all X ∈ ΓTM, holds on a para-Sasakian manifold, then its scalar curvature is constant.

11. Para-Sasakian Manifold with Canonical Paracontact Connection Satisfying P P ξ, X · R 0 and P P ξ, X · S 0

In this section we consider a para-Sasakian manifold M²ⁿ¹ with canonical paracontact connection satisfying the condition

P Pξ, X·R0, 11.1

for allX∈ΓTM.

From2.24, we get

P Pξ, X·R

U, VWP Pξ, XRU, VW−R

P Pξ, XU, V W

−R

U, P Pξ, XV

W−RU, VP Pξ, XW 0,

11.2

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whereX, U, V, W∈ΓTM. Now we takeUξin11.2, then we have P Pξ, XRξ, VW−R

P Pξ, Xξ, V W

−R

ξ, P Pξ, XV

W−Rξ, VP Pξ, XW0.

11.3

Taking inner product withY ∈ΓTMand by using3.18and9.3we obtain τ−2n

2n2ng

RX, VW, Y

0. 11.4

Again using3.2in11.4, we have

τ−2n 2n2n

⎡

⎢⎢

⎣

gRX, VW, Y gV, WηXηY

−gX, WηYηV gX, YηVηW

−gY, VηXηW 2g X, ϕV

g ϕW, Y g

X, ϕW g

Y, ϕV

−g V, ϕW

g ϕX, Y

⎤

⎥⎥

⎦0. 11.5

Hence by suitable contracting of11.5we get τ−2n

2n2n−1

0. 11.6

Therefore we have the following.

Theorem 11.1. LetM²ⁿ¹be a para-Sasakian manifold. IfP Pξ, X·R0 holds onM, then either iscalar curvature is constant with equationτ2n

or

SV, W 2gV, W−2n2ηVηW, 11.7

Now let us consider a para-Sasakian manifold satisfying

P Pξ, X·S0, 11.8

where X ∈ ΓTM and S is the Ricci tensor of M with respect to canonical paracontact connection.

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From11.8we obtain

S

P Pξ, XY, Z S

Y, P Pξ, XZ

0. 11.9

Using3.18and3.19with9.3in11.9we have

B

SX, ZηY SX, YηZ

0, 11.10

whereBτ/2n2n.

Using3.13and3.14in11.10we get

τ−2n 2n2n

⎡

⎢⎢

⎣

SX, ZηY−2gX, ZηY 2n2ηXηYηZ

−SX, YηZ 2gX, YηZ

−2n2ηXηYηZ

⎤

⎥⎥

⎦0. 11.11

So a contraction of11.11with respect toXandYgives

τ−2n 2n2n

τ−2nηZ

0, 11.12

which implies thatτ2n.

So we have the following.

Theorem 11.2. If the conditionP Pξ, X·S 0, for allX ∈ ΓTM, holds on a para-Sasakian manifold, then the scalar curvature of the manifold is constant.

12. Para-Sasakian Manifold with Canonical Paracontact Connection Satisfying P ξ, X · P P 0

The projective curvature tensorPof a para-Sasakian manifoldM²ⁿ¹with respect to canonical paracontact connection∇is defined by

PX, YZRX, YZ− 1 2n

SY, ZX−SX, ZY

, 12.1

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whereX, Y,Z∈ΓTM. By using3.2and3.13, from12.1we obtain PX, YZRX, YZgY, ZηXξ−gX, ZηYξ

ηYηZX−ηXηZY2g X, ϕY

ϕZ g

X, ϕZ

ϕY−g Y, ϕZ

ϕX

− 1 2n

⎛

⎜⎜

⎝

SY, ZX−2gY, ZX 2n2ηYηZX

−SX, ZY2gX, ZY

−2n2ηXηZY

⎞

⎟⎟

⎠.

12.2

Let us consider the equation

Pξ, X·P P0 12.3

holds on a para-Sasakian manifold. From2.25, we get Pξ, X·P P

U, VWPξ, XP PU, VW−P P

Pξ, XU, V W

−P P

U, Pξ, XV

W−P PU, VPξ, XW 0,

12.4

for allX, U, V, W∈ΓTM. Using12.2with3.18and3.19we obtain

1 2n

S

X, P PU, VW

ξ−SX, UP Pξ, VW

−SX, VP PU, ξW−SX, WP PU, Vξ

0. 12.5

Taking inner product withY and putUξ, by using3.19we get 1

2n S

X, P Pξ, VW

ηY−SX, Wg

P Pξ, Vξ, Y

0. 12.6

Using9.3in12.6, we obtain

C

SX, VηYηW SX, WηYηV

−SX, WgY, V

0, 12.7

whereCτ/2n22n².

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From the last equation, with3.13we get

C

⎡

⎢⎢

⎢⎣

SX, VηYηW−2gX, VηYηW

2n2ηXηYηVηW

SX, WηYηV−2gX, WηYηV

2n2ηXηYηVηW

−SX, WgY, V 2gX, WgY, V

−2n2ηXηYηVηW

⎤

⎥⎥

⎥⎦

0. 12.8

Hence by suitable contracting of12.8, we obtain τ−2n

2n22n²

0. 12.9

Theorem 12.1. LetMbe a2n1-dimensional para-Sasakian manifold. If the conditionPξ, X· P P0 holds onM, then either

iscalar curvature is constant with equationτ2n or

SV, W 2gV, W−2n2ηVηW, 12.10

Acknowledgments

This paper was supported by Adıyaman University, under Scientific Research Project no.

FEFBAP/2012-005. The authors thank the referee for useful suggestions and remarks for the revised version.

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