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

Abstract

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 an 𝜂-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, Sato [1] introduced the almost paracontact structure (𝜑,𝜉,𝜂) satisfying 𝜑²=𝐼−𝜂⊗𝜉 and 𝜂(𝜉)=1 on a differentiable manifold. Although the structure is an analogue of the almost contact structure [2, 3], it is closely related to almost product structure (in 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 Sato [1] could be even dimensional as well. Takahashi [4] defined 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, Matsumoto [5] replaced the structure vector field 𝜉 by −𝜉 in 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 field 𝜉 is always timelike. These circumstances motivated the authors [6] to 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 (2𝑛+1) and constructed the almost paracomplex structure on 𝑀^2𝑛+1×ℝ. Zamkovoy [8] associated the almost paracontact structure introduced in [7] to a pseudo-Riemannian metric of signature (𝑛+1,𝑛) and showed that any almost paracontact structure admits such a pseudo-Riemannian metric which is called compatible metric.

Tanaka-Webster connection has been introduced by Tanno [9] as 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 (generalized) Tanaka-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 tensor (resp., Ricci tensor) with respect to Levi-Civita connection. In Section 4, conformal curvature tensor of a para-Sasakian manifold with respect to canonical paracontact connection is obtained. Section 5 contains the expression of 𝑊₂-curvature tensor. In Section 6, we study a para-Sasakian manifold satisfying the condition 𝐶(𝜉,𝑋)⋅𝑅=0, where 𝐶(𝜉,𝑋) is considered as a derivation of the tensor algebra at each point of the manifold, 𝑅 is the curvature tensor, and 𝐶 is the conformal curvature tensor with respect to canonical paracontact connection. In Section 7, we obtain some equations in terms of Ricci tensor on a para-Sasakian manifold satisfying 𝑊₂(𝜉,𝑋)⋅𝑅=0 and 𝑊₂(𝜉,𝑋)⋅𝑊₂=0, respectively. In Section 8 it is proved that a concircularly flat para-Sasakian manifold is of constant scalar curvature. Section 9 is devoted to pseudo-projectively flat para-Sasakian manifolds. In Section 10, we show that a para-Sasakian manifold satisfying 𝑍(𝜉,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 condition 𝑍(𝜉,𝑋)⋅𝑆=0 holds on a para-Sasakian manifold with respect to canonical paracontact connection, then the scalar curvature is constant. In Section 11, we give some characterizations for para-Sasakian manifolds with canonical paracontact connection satisfying 𝑃𝑃(𝜉,𝑋)⋅𝑅=0 and 𝑃𝑃(𝜉,𝑋)⋅𝑆=0, respectively. In the last section it is shown that a para-Sasakian manifold on which the condition 𝑃(𝜉,𝑋)⋅𝑃𝑃=0 holds, is either of constant scalar curvature or an 𝜂-Einstein manifold.

2. Preliminaries

A differentiable manifold 𝑀 of dimension (2𝑛+1) is called almost paracontact manifold with the almost paracontact structure (𝜑,𝜉,𝜂) if it admits a tensor field 𝜑 of type (1,1), a vector field 𝜉, a 1-form 𝜂 satisfying the following conditions [7]: 𝜑²𝜂(𝜉)=𝐼−𝜂⊗𝜉,=1,𝜑𝜉=0, (2.1)(2.2) 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 eigendistributions 𝐷^± corresponding to the eigenvalues ±1 of 𝜑 are both 𝑛-dimensional. As an immediate consequence of the conditions (2.2) we have 𝜂∘𝜑=0,rank𝜑=2𝑛. (2.3)

If a (2𝑛+1)-dimensional almost paracontact manifold 𝑀 with an almost paracontact structure (𝜑,𝜉,𝜂) admits a pseudo-Riemannian metric 𝑔 such that [8] 𝑔𝜑𝑋,𝜑𝑌=−𝑔(𝑋,𝑌)+𝜂(𝑋)𝜂(𝑌),𝑋,𝑌∈Γ(𝑇𝑀), (2.4) then we say that 𝑀 is an almost paracontact metric manifold with an almost paracontact metric structure (𝜑,𝜉,𝜂,𝑔), and such metric 𝑔 is called compatible metric. Any compatible metric 𝑔 is necessarily of signature (𝑛+1,𝑛).

From (2.4) it can be easily seen that [8] 𝑔,𝑔()(𝑋),𝑋,𝜑𝑌=−𝑔𝜑𝑋,𝑌𝑋,𝜉=𝜂 (2.5)(2.6) for any 𝑋,𝑌∈Γ(𝑇𝑀). The fundamental 2-form of 𝑀 is defined by Φ(𝑋,𝑌)=𝑔𝑋,𝜑𝑌. (2.7) An almost paracontact metric structure becomes a paracontact metric structure if 𝑔(𝑋,𝜑𝑌)=𝑑𝜂(𝑋,𝑌), for all vector fields 𝑋,𝑌∈Γ(𝑇𝑀), where 𝑑𝜂(𝑋,𝑌)=(1/2){𝑋𝜂(𝑌)−𝑌𝜂(𝑋)−𝜂([𝑋,𝑌])}.

For a (2𝑛+1)-dimensional manifold 𝑀 with an almost paracontact metric structure (𝜑,𝜉,𝜂,𝑔) one can also construct a local orthonormal basis. Let 𝑈 be coordinate neighborhood on 𝑀 and 𝑋₁ any unit vector field on 𝑈 orthogonal to 𝜉. Then 𝜑𝑋₁ is a vector field orthogonal to both 𝑋₁ and 𝜉, and |𝜑𝑋₁|²=−1. Now choose a unit vector field 𝑋₂ orthogonal to 𝜉, 𝑋₁ and 𝜑𝑋₁. Then 𝜑𝑋₂ is also a vector field orthogonal to 𝜉, 𝑋₁, 𝜑𝑋₁, and 𝑋₂ and |𝜑𝑋₂|²=−1. Proceeding in this way we obtain a local orthonormal basis (𝑋_𝑖,𝜑𝑋_𝑖,𝜉),(𝑖=1,2,…,𝑛) called a 𝜑-basis [8].

Remark 2.1. It is also known that a differentiable manifold has an almost paracontact metric structure if it admits a Riemannian metric 𝑔 such that 𝑔(𝜑𝑋,𝜑𝑌)=𝑔(𝑋,𝑌)−𝜂(𝑋)𝜂(𝑌) (see [1]). But in our paper the metric 𝑔 is pseudo-Riemannian and satisfies condition (2.4).

Recall that an almost paracomplex structure [12] on a 2𝑛-dimensional manifold is a tensor field 𝐽 of type (1,1) such that 𝐽²=𝐼 and eigensubbundles 𝑇⁺, 𝑇⁻ corresponding to the eigenvalue 1,−1 of 𝐽, respectively, have equal dimensional 𝑛. The Nijenhuis tensor 𝑁 of 𝐽, given by𝑁_𝐽(𝑋,𝑌)=[𝐽𝑋,𝐽𝑌]+[𝑋,𝑌]−𝐽[𝐽𝑋,𝑌]−𝐽[𝑋,𝐽𝑌],𝑋,𝑌∈Γ(𝑇𝑀), (2.8) is the obstruction for the integrability of the eigensubbundles 𝑇⁺, 𝑇⁻. If 𝑁=0, then the almost paracomplex structure is called paracomplex or integrable [13].

Let 𝑀^2𝑛+1 be an almost paracontact metric manifold with structure (𝜑,𝜉,𝜂) and consider the manifold 𝑀^2𝑛+1×ℝ. We denote a vector field on 𝑀^2𝑛+1×ℝ, by (𝑋,𝑓(𝑑/𝑑𝑡)) where 𝑋 is tangent to 𝑀^2𝑛+1, 𝑡 is the coordinate on ℝ, and 𝑓 is a differentiable function on 𝑀^2𝑛+1×ℝ. An almost paracomplex structure 𝐽 on 𝑀^2𝑛+1×ℝ is defined in [14] by 𝐽𝑑𝑋,𝑓=(𝑋)𝑑𝑑𝑡𝜑𝑋+𝑓𝜉,𝜂𝑑𝑡. (2.9) If 𝐽 is integrable we say that the almost paracontact structure (𝜑,𝜉,𝜂) is normal.

A normal paracontact metric manifold is a para-Sasakian manifold. An almost paracontact metric structure (𝜑,𝜉,𝜂,𝑔) on a 𝑀^2𝑛+1 is para-Sasakian manifold if and only if [8] ∇_𝑋𝜑𝑌=−𝑔(𝑋,𝑌)𝜉+𝜂(𝑌)𝑋, (2.10) where 𝑋,𝑌∈Γ(𝑇𝑀) and ∇ is Levi-Civita connection of 𝑀.

From (2.10), it can be seen that ∇_𝑋𝜉=−𝜑𝑋. (2.11)

Also in a para-Sasakian manifold, the following relations hold [8]: 𝑔(𝑅())(𝑅()𝑍)()𝜂(𝑌)()𝜂(𝑋),𝑅()(𝑋)(𝑌)𝑅()()(𝑌)𝑅()(𝑋)𝑆()(𝑋),𝑋,𝑌𝑍,𝜉=𝜂𝑋,𝑌=𝑔𝑋,𝑍−𝑔𝑌,𝑍𝑋,𝑌𝜉=𝜂𝑌−𝜂𝑋,𝜉,𝑋𝑌=−𝑔𝑋,𝑌𝜉+𝜂𝑋,𝜉,𝑋𝜉=𝑋−𝜂𝜉,𝑋,𝜉=−2𝑛𝜂 (2.12)(2.13)(2.14)(2.15)(2.16) for any vector fields 𝑋,𝑌,𝑍∈Γ(𝑇𝑀). Here, 𝑅 is Riemannian curvature tensor and 𝑆 is Ricci tensor defined by 𝑆(𝑋,𝑌)=𝑔(𝑄𝑋,𝑌), where 𝑄 is Ricci operator.

In the following we consider the connection ∇ defined by [9] ∇_𝑋𝑌=∇_𝑋𝑌+𝜂(𝑋)𝜑𝑌−𝜂(𝑌)∇_𝑋𝜉+∇_𝑋𝜂𝑌𝜉, (2.17) where 𝑋,𝑌∈Γ(𝑇𝑀). If we use (2.11) in (2.17), then we obtain ∇_𝑋𝑌=∇_𝑋𝑌+𝜂(𝑋)𝜑𝑌+𝜂(𝑌)𝜑𝑋+𝑔𝑋,𝜑𝑌𝜉. (2.18)

Definition 2.2. We call the connection ∇ defined by (2.18) on 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: ∇𝜂=0,∇𝑔=0,∇𝜉=0,∇_𝑋𝜑𝑌=∇_𝑋𝜑𝑌+𝑔(𝑋,𝑌)𝜉−𝜂(𝑌)𝑋, (2.19) for all 𝑋,𝑌∈Γ(𝑇𝑀).

Proof. Calculation is straightforward by using (2.18).

Furthermore, we define homeomorphisms 𝐶(𝑋,𝑌)⋅𝑅=0,𝑊₂(𝑋,𝑌)⋅𝑅=0,𝑊₂(𝑋,𝑌)⋅𝑊₂=0, 𝑍(𝑋,𝑌)⋅𝑅=0, 𝑃𝑃(𝑋,𝑌)⋅𝑅=0 and 𝑃(𝑋,𝑌)⋅𝑃𝑃=0 as follows: 𝐶(𝑋,𝑌)⋅𝑅(𝑈,𝑉)𝑊=𝐶(𝑋,𝑌)𝑅(𝑈,𝑉)𝑊−𝑅𝐶()𝑊−𝑋,𝑌𝑈,𝑉𝑅𝑈,𝐶(𝑋,𝑌)𝑉𝑊−𝑅(𝑈,𝑉)𝐶(𝑋,𝑌)𝑊=0,𝑊₂(𝑋,𝑌)⋅𝑅(𝑈,𝑉)𝑊=𝑊₂(𝑋,𝑌)𝑅()𝑊−𝑈,𝑉𝑅𝑊₂(𝑋,𝑌)𝑈,𝑉𝑊−𝑅𝑈,𝑊₂()𝑉𝑊−𝑋,𝑌𝑅(𝑈,𝑉)𝑊₂(𝑋,𝑌)𝑊=0,𝑊₂(𝑋,𝑌)⋅𝑊₂(𝑈,𝑉)𝑊=𝑊₂(𝑋,𝑌)𝑊₂()𝑊−𝑈,𝑉𝑊₂𝑊₂(𝑋,𝑌)𝑈,𝑉𝑊−𝑊₂𝑈,𝑊₂()𝑉𝑊−𝑋,𝑌𝑊₂(𝑈,𝑉)𝑊₂(𝑋,𝑌)𝑊=0,𝑍(𝑋,𝑌)⋅𝑅(𝑈,𝑉)𝑊=𝑍(𝑋,𝑌)𝑅(𝑈,𝑉)𝑊−𝑅𝑍()𝑊−𝑋,𝑌𝑈,𝑉𝑅𝑈,𝑍(𝑋,𝑌)𝑉𝑊−𝑅(𝑈,𝑉)𝑍(𝑋,𝑌)𝑊=0,𝑃𝑃(𝑋,𝑌)⋅𝑅(𝑈,𝑉)𝑊=𝑃𝑃(𝑋,𝑌)𝑅(𝑈,𝑉)𝑊−𝑅𝑃𝑃()𝑊−𝑋,𝑌𝑈,𝑉𝑅𝑈,𝑃𝑃(𝑋,𝑌)𝑉𝑊−𝑅(𝑈,𝑉)𝑃𝑃(𝑋,𝑌)𝑊=0,𝑃(𝑋,𝑌)⋅𝑃𝑃(𝑈,𝑉)𝑊=𝑃(𝑋,𝑌)𝑃𝑃(𝑈,𝑉)𝑊−𝑃𝑃𝑃()𝑊−𝑋,𝑌𝑈,𝑉𝑃𝑃𝑈,𝑃(𝑋,𝑌)𝑉𝑊−𝑃𝑃(𝑈,𝑉)𝑃(𝑋,𝑌)𝑊=0, (2.20)(2.21)(2.22)(2.23)(2.24)(2.25) for all 𝑋,𝑌,𝑈,𝑉,𝑊∈Γ(𝑇𝑀).

3. Curvature Tensor

Let 𝑀^2𝑛+1 be a para-Sasakian manifold. The curvature tensor 𝑅 of 𝑀 with respect to the canonical paracontact connection ∇ is defined by 𝑅(𝑋,𝑌)𝑍=∇_𝑋∇_𝑌𝑍−∇_𝑌∇_𝑋𝑍−∇_[𝑋,𝑌]𝑍, (3.1) for any 𝑋,𝑌,𝑍∈Γ(𝑇𝑀).

By using (2.18) in (3.1) we obtain 𝑅()()()𝜂(𝑋)()𝜂(𝑌)𝜉(𝑌)𝜂(𝑍)(𝑋)𝜂(𝑍)𝑋,𝑌𝑍=𝑅𝑋,𝑌𝑍+𝑔𝑌,𝑍𝜉−𝑔𝑋,𝑍+𝜂𝑋−𝜂𝑌+2𝑔𝑋,𝜑𝑌𝜑𝑍+𝑔𝑋,𝜑𝑍𝜑𝑌−𝑔𝑌,𝜑𝑍𝜑𝑋, (3.2) where 𝑅(𝑋,𝑌)𝑍=∇_𝑋∇_𝑌𝑍−∇_𝑌∇_𝑋𝑍−∇_[𝑋,𝑌]𝑍 is curvature tensor of 𝑀 with respect to Levi-Civita connection ∇.

Let 𝑇 and 𝑇 be curvature tensors of type (0, 4) with respect to Levi-Civita connection ∇ and the canonical paracontact connection ∇, respectively, given by 𝑇(𝑋,𝑌,𝑍,𝑊)=𝑔(𝑅(𝑋,𝑌)𝑍,𝑊),𝑇(𝑋,𝑌,𝑍,𝑊)=𝑔𝑅(𝑋,𝑌)𝑍,𝑊, (3.3) for all 𝑋,𝑌,𝑍,𝑊∈Γ(𝑇𝑀).

Proposition 3.1. In a para-Sasakian manifold one has, 𝑅(𝑋,𝑌)𝑍+𝑅(𝑌,𝑍)𝑋+𝑅(𝑍,𝑋)𝑌=0,𝑇(𝑋,𝑌,𝑍,𝑊)+𝑇(𝑌,𝑋,𝑍,𝑊)=0,𝑇(𝑋,𝑌,𝑍,𝑊)+𝑇(𝑋,𝑌,𝑊,𝑍)=0,𝑇(𝑋,𝑌,𝑍,𝑊)−𝑇(𝑍,𝑊,𝑋,𝑌)=0, (3.4)(3.5)(3.6)(3.7) where 𝑋,𝑌,𝑍,𝑊∈Γ(𝑇𝑀).

Proof. Using (3.2) and first Bianchi identity with respect to Levi-Civita connection ∇ we obtain 𝑅(𝑋,𝑌)𝑍+𝑅(𝑌,𝑍)𝑋+𝑅(𝑍,𝑋)𝑌=2𝑔𝑋,𝜑𝑌𝜑𝑍+𝑔𝑋,𝜑𝑍𝜑𝑌−𝑔𝑌,𝜑𝑍𝜑𝑋+2𝑔𝑌,𝜑𝑍𝜑𝑋+𝑔𝑌,𝜑𝑋𝜑𝑍−𝑔𝑍,𝜑𝑋𝜑𝑌+2𝑔𝑍,𝜑𝑋𝜑𝑌+𝑔𝑍,𝜑𝑌𝜑𝑋−𝑔𝑋,𝜑𝑌𝜑𝑍. (3.8) From (2.5) we get (3.4).
From (3.2) we have 𝑇()()()𝜂(𝑋)𝜂(𝑊)()𝜂(𝑌)𝜂(𝑊)()𝜂(𝑌)𝜂(𝑍)()𝜂(𝑋)𝜂(𝑍)𝑔𝑔𝑔.𝑋,𝑌,𝑍,𝑊=𝑇𝑋,𝑌,𝑍,𝑊+𝑔𝑌,𝑍−𝑔𝑋,𝑍+𝑔𝑋,𝑊−𝑔𝑌,𝑊+2𝑔𝑋,𝜑𝑌𝜑𝑍,𝑊+𝑔𝑋,𝜑𝑍𝜑𝑌,𝑊−𝑔𝑌,𝜑𝑍𝜑𝑋,𝑊 (3.9) It is well known that 𝑇()()()(),𝑇()().𝑋,𝑌,𝑍,𝑊=−𝑇𝑌,𝑋,𝑍,𝑊,𝑇𝑋,𝑌,𝑍,𝑊=𝑇𝑋,𝑌,𝑊,𝑍𝑋,𝑌,𝑍,𝑊=𝑇𝑍,𝑊,𝑋,𝑌 (3.10) By taking into account the previous equations we get (3.5), (3.6), and (3.7), respectively.

Let 𝐸_𝑖={𝑒_𝑖,𝜑𝑒_𝑖,𝜉}(𝑖=1,2,…,𝑛) be a local orthonormal 𝜑-basis of a para-Sasakian manifold 𝑀. Then the Ricci tensor 𝑆 and the scalar curvature 𝜏 of 𝑀 with respect to canonical paracontact connection ∇ are defined by 𝑆(𝑋,𝑌)=^𝑛_𝑖=1𝑔𝑅𝑒_𝑖,𝑋𝑌,𝑒_𝑖−^𝑛_𝑖=1𝑔𝑅𝜑𝑒_𝑖,𝑋𝑌,𝜑𝑒_𝑖+𝑔𝑅(𝜉,𝑋)𝑌,𝜉, (3.11) where 𝑋,𝑌∈Γ(𝑇𝑀), and 𝜏=^𝑛_𝑖=1𝑆𝑒_𝑖,𝑒_𝑖−^𝑛_𝑖=1𝑆𝜑𝑒_𝑖,𝜑𝑒_𝑖+𝑆(𝜉,𝜉), (3.12) respectively.

Theorem 3.2. In a (2𝑛+1)-dimensional para-Sasakian manifold the Ricci tensor 𝑆 and scalar curvature 𝜏 of canonical paracontact connection ∇ are given by 𝑆(𝑋,𝑌)=𝑆(𝑋,𝑌)−2𝑔(𝑋,𝑌)+(2𝑛+2)𝜂(𝑋)𝜂(𝑌),𝜏=𝜏−2𝑛, (3.13)(3.14) where 𝑋,𝑌∈Γ(𝑇𝑀) and 𝑆 and 𝜏 denote the Ricci tensor and scalar curvature of Levi-Civita connection ∇, respectively. Consequently, 𝑆 is symmetric.

Proof. Using (3.2) and (3.9), we have, for any 𝑋,𝑌∈Γ(𝑇𝑀), 𝑆(𝑋,𝑌)=^𝑛_𝑖=1𝑔𝑅𝑒_𝑖,𝑋𝑌,𝑒_𝑖+𝑔𝑒_𝑖,𝑒_𝑖𝜂(𝑋)𝜂(𝑌)+3𝑔𝑒_𝑖,𝜑𝑋𝑔𝜑𝑌,𝑒_𝑖−^𝑛_𝑖=1𝑔𝑅𝜑𝑒_𝑖,𝑋𝑌,𝜑𝑒_𝑖+𝑔𝜑𝑒_𝑖,𝜑𝑒_𝑖𝜂(𝑋)𝜂(𝑌)+3𝑔𝜑𝑒_𝑖,𝜑𝑋𝑔𝜑𝑌,𝜑𝑒_𝑖. (3.15) Since the Ricci tensor of Levi-Civita connection ∇ is given by 𝑆(𝑋,𝑌)=^𝑛_𝑖=1𝑔𝑅𝑒_𝑖,𝑋𝑌,𝑒_𝑖−^𝑛_𝑖=1𝑔𝑅𝜑𝑒_𝑖,𝑋𝑌,𝜑𝑒_𝑖+𝑔(𝑅(𝜉,𝑋)𝑌,𝜉), (3.16) then (3.15) implies (3.13). Equation (3.14) follows from (3.13).
Also from (3.13), it is obvious that 𝑆 is symmetric.

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

Lemma 3.4. Let 𝑀 be a para-Sasakian manifold with canonical paracontact connection ∇. Then 𝑔𝑅(𝑋,𝑌)𝑍,𝜉=𝜂𝑅(𝑋,𝑌)𝑍=0,𝑅(𝑋,𝑌)𝜉=𝑅(𝜉,𝑋)𝑌=𝑅(𝜉,𝑋)𝜉=0,𝑆(𝑋,𝜉)=0, (3.17)(3.18)(3.19) for all 𝑋,𝑌,𝑍∈Γ(𝑇𝑀).

Proof. Calculation is straightforward by using (2.2), (2.6), (2.13), and (2.16) in (3.2).

4. Conformal Curvature Tensor

Let 𝑀 be a (2𝑛+1)-dimensional para-Sasakian manifold. The conformal curvature tensor of 𝑀 with respect to canonical paracontact connection ∇ is defined by 𝐶(𝑋,𝑌)𝑍=𝑅()1𝑋,𝑌𝑍−2𝑛−1𝑆(𝑌,𝑍)𝑋−𝑆()𝑌()𝑋,𝑍+𝑔𝑌,𝑍𝑄𝑋−𝑔(𝑋,𝑍)+𝑄𝑌𝜏()𝑔()()𝑌,2𝑛2𝑛−1𝑌,𝑍𝑋−𝑔𝑋,𝑍 (4.1) where 𝑋,𝑌,𝑍∈Γ(𝑇𝑀). By using (3.2), (3.13), and (3.14) in (4.1) 𝐶()()()𝜂(𝑋)()𝜂(𝑌)𝜉(𝑌)𝜂(𝑍)(𝑋)𝜂(𝑍)−1𝑋,𝑌𝑍=𝑅𝑋,𝑌𝑍+𝑔𝑌,𝑍𝜉−𝑔𝑋,𝑍+𝜂𝑋−𝜂𝑌+2𝑔𝑋,𝜑𝑌𝜑𝑍+𝑔𝑋,𝜑𝑍𝜑𝑌−𝑔𝑌,𝜑𝑍𝜑𝑋𝑆()()𝑋+()𝜂(𝑌)𝜂(𝑍)𝑋()()𝑌−()𝜂(𝑋)𝜂(𝑍)𝑌()()𝑋+()𝑔()𝜂(𝑋)𝜉()()𝑌−()𝑔()𝜂(𝑌)𝜉+2𝑛−1𝑌,𝑍𝑋−2𝑔𝑌,𝑍2𝑛+2−𝑆𝑋,𝑍𝑌+2𝑔𝑋,𝑍2𝑛+2+𝑔𝑌,𝑍𝑄𝑋−2𝑔𝑌,𝑍2𝑛+2𝑌,𝑍−𝑔𝑋,𝑍𝑄𝑌+2𝑔𝑋,𝑍2𝑛+2𝑋,𝑍𝜏−2𝑛()𝑔()()𝑌.2𝑛2𝑛−1𝑌,𝑍𝑋−𝑔𝑋,𝑍 (4.2) Since 𝑅(𝑋,𝑌)𝑍=−𝑅(𝑌,𝑋)𝑍, then 𝐶(𝑋,𝑌)𝑍+𝐶(𝑌,𝑋)𝑍=0. (4.3) Moreover, from the first Bianchi identity we get 𝐶(𝑋,𝑌)𝑍+𝐶(𝑌,𝑍)𝑋+𝐶(𝑍,𝑋)𝑌=0. (4.4)

5. 𝑊₂-Curvature Tensor

In [15] Pokhariyal and Mishra have introduced a new tensor field, called 𝑊₂-curvature tensor field, in a Riemannian manifold and studied its properties.

The 𝑊₂-curvature tensor is defined by 𝑊₂()()+1𝑋,𝑌,𝑍,𝑉=𝑅𝑋,𝑌,𝑍,𝑉𝑔()𝑆()()𝑆(,𝑛−1𝑋,𝑍𝑌,𝑉−𝑔𝑌,𝑍X,𝑉) (5.1) where 𝑋,𝑌,𝑍,𝑉∈Γ(𝑇𝑀) and 𝑆 is a Ricci tensor of type (0,2). The 𝑊₂-curvature tensor of a para-Sasakian manifold 𝑀^2𝑛+1 with respect to canonical paracontact connection ∇ is defined by 𝑊₂(𝑋,𝑌)𝑍=𝑅()1𝑋,𝑌𝑍−𝑔()2𝑛𝑌,𝑍𝑄𝑋−𝑔(𝑋,𝑍)𝑄𝑌. (5.2)

From (3.2) and (3.13) we get 𝑊₂()()()𝜂(𝑋)()𝜂(𝑌)𝜉(𝑌)𝜂(𝑍)(𝑋)𝜂(𝑍)−1𝑋,𝑌𝑍=𝑅𝑋,𝑌𝑍+𝑔𝑌,𝑍𝜉−𝑔𝑋,𝑍+𝜂𝑋−𝜂𝑌+2𝑔𝑋,𝜑𝑌𝜑𝑍+𝑔𝑋,𝜑𝑍𝜑𝑌−𝑔𝑌,𝜑𝑍𝜑𝑋𝑔()()𝑋+()𝑔()𝜂(𝑋)𝜉()()𝑌−()𝑔()𝜂(𝑌)𝜉.2𝑛−1𝑌,𝑍𝑄𝑋−2𝑔𝑌,𝑍2𝑛+2𝑌,𝑍−𝑔𝑋,𝑍𝑄𝑌+2𝑔𝑋,𝑍2𝑛+2𝑋,𝑍 (5.3) Since 𝑅(𝑋,𝑌)𝑍+𝑅(𝑌,𝑋)𝑍=0, we have 𝑊₂(𝑋,𝑌)𝑍+𝑊₂(𝑌,𝑋)𝑍=0. (5.4) Furthermore, by using (5.3) and the first Bianchi identity we obtain 𝑊₂(𝑋,𝑌)𝑍+𝑊₂(𝑌,𝑍)𝑋+𝑊₂(𝑍,𝑋)𝑌=0. (5.5)

6. Para-Sasakian Manifold with Canonical Paracontact Connection Satisfying 𝐶(𝜉,𝑋)⋅𝑅=0

In this section we consider a para-Sasakian manifold 𝑀^2𝑛+1 satisfying the condition 𝐶(𝜉,𝑋)⋅𝑅=0, (6.1) with respect to canonical paracontact connection. From (2.20) we get 𝐶(𝜉,𝑋)⋅𝑅(𝑈,𝑉)𝑊=𝐶(𝜉,𝑋)𝑅(𝑈,𝑉)𝑊−𝑅𝐶()𝑊−𝜉,𝑋𝑈,𝑉𝑅𝑈,𝐶(𝜉,𝑋)𝑉𝑊−𝑅(𝑈,𝑉)𝐶(𝜉,𝑋)𝑊=0, (6.2) for any 𝑋,𝑈,𝑉,𝑊∈Γ(𝑇𝑀). Now putting 𝑈=𝜉 in (6.2), we have 𝐶(𝜉,𝑋)𝑅(𝜉,𝑉)𝑊−𝑅𝐶(𝜉,𝑋)𝜉,𝑉𝑊−𝑅𝜉,𝐶(𝜉,𝑋)𝑉𝑊−𝑅(𝜉,𝑉)𝐶(𝜉,𝑋)𝑊=0. (6.3) Taking inner product with 𝑌 and using (3.18) in (6.3) we get 𝑔𝑅𝐶(𝜉,𝑋)𝜉,𝑉𝑊,𝑌=0. (6.4) Using (4.2) in (6.4), we obtain 1𝑔2𝑛−1𝑅(𝑊,𝑌)𝑄𝑋,𝑉−𝜏()𝑔2𝑛2𝑛−1𝑅(𝑊,𝑌)𝑋,𝑉=0. (6.5) Again using (3.2), (3.13), and (3.14) in (6.5) we obtain 1𝑔(𝑅())()𝜂(𝑉)𝜂(𝑊)()𝜂(𝑌)𝜂(𝑉)()𝜂()𝜂(𝑌)()𝜂()𝜂(𝑊)𝑔𝑔𝑔−2𝑛−1𝑊,𝑌𝑄𝑋,𝑉+𝑔𝑌,𝑄𝑋−𝑔𝑊,𝑄𝑋+𝑔𝑊,𝑉𝑄𝑋−𝑔𝑌,𝑉𝑄𝑋+2𝑔𝑊,𝜑𝑌𝜑𝑄𝑋,𝑉+𝑔𝑊,𝜑𝑄𝑋𝜑𝑌,𝑉−𝑔𝑌,𝜑𝑄𝑋𝜑𝑊,𝑉𝜏−2𝑛()𝑔(𝑅())()𝜂(𝑉)𝜂(𝑊)()𝜂(𝑌)𝜂(𝑉)()𝜂(𝑋)𝜂(𝑌)()𝜂(𝑋)𝜂(𝑊)𝑔𝑔𝑔2𝑛2𝑛−1𝑊,𝑌𝑋,𝑉+𝑔𝑌,𝑋−𝑔𝑊,𝑋+𝑔𝑊,𝑉−𝑔𝑌,𝑉+2𝑔𝑊,𝜑𝑌𝜑𝑋,𝑉+𝑔𝑊,𝜑𝑋𝜑𝑌,𝑉−𝑔𝑌,𝜑𝑋𝜑𝑊,𝑉=0. (6.6)

Let {𝑒_𝑖,𝜑𝑒_𝑖,𝜉}(𝑖=1,…,𝑛) be an orthonormal basis of the tangent space at any point. Hence by suitable contracting of (6.6) we get 1𝑆2𝑛−1²()()()−𝑋,𝑌−4𝑆𝑋,𝑌+4𝑔𝑋,𝑌4𝑛²𝜂(𝑋)𝜂(𝑌)−+8𝑛+4𝜏−2𝑛()𝑆()()2𝑛2𝑛−1+()𝜂(𝑋)𝜂(𝑌)𝑋,𝑌−2𝑔𝑋,𝑌2𝑛+2=0. (6.7)

Theorem 6.1. Let 𝑀^2𝑛+1 be a para-Sasakian manifold satisfying 𝐶(𝜉,𝑋)⋅𝑅=0 with respect to canonical paracontact connection ∇. Then 𝑆()=𝜏𝑄𝑋,𝑌3+𝑆()−𝜏2𝑛𝑋,𝑌2+𝑛𝑔()+𝑋,𝑌4𝑛²𝜏()+6𝑛+2+𝑛+1𝑛𝜂(𝑋)𝜂(𝑌), (6.8) for any 𝑋,𝑌∈Γ(𝑇𝑀).

7. Para-Sasakian Manifold with Canonical Paracontact Connection Satisfying 𝑊₂(𝜉,𝑋)⋅𝑅=0 and 𝑊₂(𝜉,𝑋)⋅𝑊₂=0

In this section we consider a para-Sasakian manifold with canonical connection satisfying the condition 𝑊₂(𝜉,𝑋)⋅𝑅=0. (7.1) From (2.21), we get 𝑊₂(𝜉,𝑋)⋅𝑅(𝑈,𝑉)𝑊=𝑊₂(𝜉,𝑋)𝑅(𝑈,𝑉)𝑊−𝑅𝑊₂()𝑊−𝜉,𝑋𝑈,𝑉𝑅𝑈,𝑊₂(𝜉,𝑋)𝑉𝑊−𝑅(𝑈,𝑉)𝑊₂(𝜉,𝑋)𝑊=0, (7.2) where 𝑋,𝑈,𝑉,𝑊∈Γ(𝑇𝑀). Now if we put 𝑈=𝜉 in (7.2), we have 𝑊₂(𝜉,𝑋)𝑅(𝜉,𝑉)𝑊−𝑅𝑊₂()𝑊−𝜉,𝑋𝜉,𝑉𝑅𝜉,𝑊₂(𝜉,𝑋)𝑉𝑊−𝑅(𝜉,𝑉)𝑊₂(𝜉,𝑋)𝑊=0. (7.3) Taking inner product with 𝑌 and from (3.18), we obtain 𝑔𝑅𝑊₂(𝜉,𝑋)𝜉,𝑉𝑊,𝑌=0. (7.4) Using (5.3) in (7.4), we have 1𝑔2𝑛𝑅(𝑊,𝑌)𝑄𝑋,𝑉=0. (7.5) Again using (3.2) and (3.13) in (7.5) we obtain 1𝑔(𝑅())()𝜂(𝑉)𝜂(𝑊)()𝜂(Y)𝜂(𝑉)()𝜂()𝜂(𝑌)()𝜂()𝜂(𝑊)𝑔𝑔𝑔(𝑅())()𝜂(𝑉)𝜂(𝑊)()𝜂(𝑌)𝜂(𝑉)()𝜂(𝑋)𝜂(𝑌)()𝜂(𝑋)𝜂(𝑊)𝑔𝑔𝑔2𝑛𝑊,𝑌𝑄𝑋,𝑉+𝑔𝑌,𝑄𝑋−𝑔𝑊,𝑄𝑋+𝑔𝑊,𝑉𝑄𝑋−𝑔𝑌,𝑉𝑄𝑋+2𝑔𝑊,𝜑𝑌𝜑𝑄𝑋,𝑉+𝑔𝑊,𝜑𝑄𝑋𝜑𝑌,𝑉−𝑔𝑌,𝜑𝑄𝑋𝜑𝑊,𝑉−2𝑔𝑊,𝑌𝑋,𝑉−2𝑔𝑌,𝑋+2𝑔𝑊,𝑋−2𝑔𝑊,𝑉+2𝑔𝑌,𝑉−4𝑔𝑊,𝜑𝑌𝜑𝑋,𝑉−2𝑔𝑊,𝜑𝑋𝜑𝑌,𝑉+2𝑔𝑌,𝜑𝑋𝜑𝑊,𝑉=0. (7.6)

Let {𝑒_𝑖,𝜑𝑒_𝑖,𝜉}(𝑖=1,…,𝑛) be an orthonormal basis of the tangent space at any point. So by a contraction of (7.6) with respect to 𝑉 and 𝑊 we get1𝑆2𝑛²()()()−𝑋,𝑌−4𝑆𝑋,𝑌+4𝑔𝑋,𝑌4𝑛²+8𝑛+4𝜂(𝑋)𝜂(𝑌)=0. (7.7)

Theorem 7.1. Let 𝑀^2𝑛+1 be a para-Sasakian manifold satisfying 𝑊₂(𝜉,𝑋)⋅𝑅=0 with respect to canonical paracontact connection ∇. Then 𝑆(𝑄𝑋,𝑌)=4𝑆(𝑋,𝑌)−4𝑔(𝑋,𝑌)+4𝑛²+8𝑛+4𝜂(𝑋)𝜂(𝑌), (7.8) for any 𝑋,𝑌∈Γ(𝑇𝑀).

Now let us consider a para-Sasakian manifold with canonical paracontact connection satisfying the condition 𝑊₂(𝜉,𝑋)⋅𝑊₂=0. (7.9) From (2.22), we get 𝑊₂(𝜉,𝑋)⋅𝑊₂(𝑈,𝑉)𝑍=𝑊₂(𝜉,𝑋)𝑊₂(𝑈,𝑉)𝑍−𝑊₂𝑊₂()𝑍−𝜉,𝑋𝑈,𝑉𝑊₂𝑈,𝑊₂(𝜉,𝑋)𝑉𝑍−𝑊₂(𝑈,𝑉)𝑊₂(𝜉,𝑋)𝑍=0. (7.10)

Now 𝑈=𝜉 in (7.10), we have 𝑊₂(𝜉,𝑋)𝑊₂(𝜉,V)𝑍−𝑊₂𝑊₂()𝑍−𝜉,𝑋𝜉,𝑉𝑊₂𝜉,𝑊₂(𝜉,𝑋)𝑉𝑍−𝑊₂(𝜉,𝑉)𝑊₂(𝜉,𝑋)𝑍=0. (7.11) Taking inner product with 𝑌 and from (3.18) and (3.19) we obtain 1𝑔2𝑛𝑊₂−1𝑄𝑋,𝑉𝑍,𝑌4𝑛²𝑔𝑄²𝑋,𝑌𝜂(𝑉)𝜂(𝑍)=0. (7.12) Using (5.3) in (7.12), we have 1𝑔2𝑛𝑅−1𝑄𝑋,𝑉𝑍,𝑌4𝑛²𝑔𝑄²𝑋,𝑌𝑔(𝑉,𝑍)−𝑔𝑄𝑋,𝑍𝑔𝑄𝑉,𝑌+𝑔𝑄²𝑋,𝑌𝜂(𝑉)𝜂(𝑍)=0. (7.13)

Let {𝑒_𝑖,𝜑𝑒_𝑖,𝜉}(𝑖=1,…,𝑛) be an orthonormal basis of the tangent space at any point. Hence by suitable contracting (7.13) we get 1𝑆2𝑛²()()()−𝑋,𝑌−4𝑆𝑋,𝑌+4𝑔𝑋,𝑌4𝑛²𝜂(𝑋)𝜂(𝑌)−1+8𝑛+44𝑛²𝑆²()()()−𝑋,𝑌−4𝑆𝑋,𝑌+4g𝑋,𝑌4𝑛²+8𝑛+4𝜂(𝑋)𝜂(𝑌)=0. (7.14)

Theorem 7.2. Let 𝑀^2𝑛+1 be a para-Sasakian manifold satisfying 𝑊₂(𝜉,𝑋)⋅𝑊₂=0 with respect to canonical paracontact connection ∇. Then 𝑆(𝑄𝑋,𝑌)=4𝑆(𝑋,𝑌)−4𝑔(𝑋,𝑌)+4𝑛²+8𝑛+4𝜂(𝑋)𝜂(𝑌), (7.15) for any 𝑋,𝑌∈Γ(𝑇𝑀).

8. Concircularly Flat Para-Sasakian Manifold

The concircular curvature tensor of a para-Sasakian manifold 𝑀^2𝑛+1 with respect to canonical paracontact connection ∇ is defined by 𝑍(𝑋,𝑌)𝑊=𝑅(𝑋,𝑌)𝑊−𝜏()𝑔()()𝑌,2𝑛2𝑛−1𝑌,𝑊𝑋−𝑔𝑋,𝑊 (8.1) for any 𝑋,𝑌,𝑊∈Γ(𝑇𝑀).

By using (3.2) and (3.14) we obtain from (8.1) 𝑍()()()𝜂(𝑋)()𝜂(𝑌)𝜉(𝑌)𝜂(𝑊)(𝑋)𝜂(𝑊)−𝑋,𝑌𝑊=𝑅𝑋,𝑌𝑊+𝑔𝑌,𝑊𝜉−𝑔𝑋,𝑊+𝜂𝑋−𝜂𝑌+2𝑔𝑋,𝜑𝑌𝜑𝑊+𝑔𝑋,𝜑𝑊𝜑𝑌−𝑔𝑌,𝜑𝑊𝜑𝑋𝜏−2𝑛()𝑔()()𝑌.2𝑛2𝑛−1𝑌,𝑊𝑋−𝑔𝑋,𝑊 (8.2)

If 𝑀 is a concircularly flat para-Sasakian manifold with respect to canonical paracontact connection, then we have 𝑔𝑍(𝑋,𝑌)𝑊,𝜑𝑉=0. (8.3) Hence using (2.2) in (8.3) we get 𝑔𝑅()𝜂(𝑌)𝜂(𝑊)𝜂(𝑋)𝜂(𝑊)𝑔𝑔𝑔−𝑋,𝑌𝑊,𝜑𝑉+𝑔𝑋,𝜑𝑉−𝑔𝑌,𝜑𝑉+2𝑔𝑋,𝜑𝑌𝜑𝑊,𝜑𝑉+𝑔𝑋,𝜑𝑊𝜑𝑌,𝜑𝑉−𝑔𝑌,𝜑𝑊𝜑𝑋,𝜑𝑉𝜏−2𝑛()𝑔()𝑔()𝑔2𝑛2𝑛+1𝑌,𝑊𝑋,𝜑𝑉−𝑔𝑋,𝑊𝑌,𝜑𝑉=0. (8.4) Putting 𝑌=𝑊=𝜉 in (8.4) and using (2.2) and (2.14) we have 𝜏−2𝑛()𝑔2𝑛2𝑛+1𝑋,𝜑𝑉=0. (8.5) From (8.5) we 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 dimension 𝑛, which includes projective curvature tensor 𝑃. This tensor field 𝑃 is known as pseudo-projective curvature tensor.

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 with 𝑃𝑃.

Pseudo-projective curvature tensor 𝑃𝑃 of a para-Sasakian manifold 𝑀^2𝑛+1 with respect to canonical paracontact connection ∇ is defined by 𝑃𝑃()()(𝑆()()𝑌)−𝑋,𝑌𝑊=𝑎𝑅𝑋,𝑌𝑊+𝑏𝑌,𝑊𝑋−𝑆𝑋,𝑊𝜏()𝑎2𝑛+1𝑔()()𝑌,2𝑛+𝑏𝑌,𝑊𝑋−𝑔𝑋,𝑊 (9.1) where 𝑋,𝑌,𝑊∈Γ(𝑇𝑀) and 𝑎 and 𝑏 are constants such that 𝑎, 𝑏≠0.

If 𝑎=1 and 𝑏=1/(2𝑛+2), then (9.1) takes the form 𝑃𝑃(𝑋,𝑌)𝑊=𝑅()1𝑋,𝑌𝑊+2𝑛+2𝑆(𝑌,𝑊)𝑋−𝑆()𝑌−𝑋,𝑊𝜏()𝑛𝑔()()𝑌.2𝑛+2𝑌,𝑊𝑋−𝑔𝑋,𝑊 (9.2) Using (3.2), (3.13) and (3.14) in (9.2), we get 𝑃𝑃()()()𝜂(𝑋)()𝜂(𝑌)𝜉(𝑌)𝜂(𝑊)(𝑋)𝜂(𝑊)+1𝑋,𝑌𝑊=𝑅𝑋,𝑌𝑊+𝑔𝑌,𝑊𝜉−𝑔𝑋,𝑊+𝜂𝑋−𝜂𝑌+2𝑔𝑋,𝜑𝑌𝜑𝑊+𝑔𝑋,𝜑𝑊𝜑𝑌−𝑔𝑌,𝜑𝑊𝜑𝑋𝑆()()𝑋+()𝜂(𝑌)𝜂(𝑊)𝑋()()𝑌−()𝜂(𝑋)𝜂(𝑊)𝑌−2𝑛+2𝑌,𝑊𝑋−2𝑔𝑌,𝑊2𝑛+2−𝑆𝑋,𝑊𝑌+2𝑔𝑋,𝑊2𝑛+2𝜏−2𝑛()𝑔()()𝑌.2𝑛+1𝑌,𝑊𝑋−𝑔𝑋,𝑊 (9.3)

If 𝑀 is a pseudo-projectively flat para-Sasakian manifold with respect to canonical paracontact connection, then 𝑔𝑃𝑃(𝑋,𝑌)𝑊,𝜑𝑈=0, (9.4) for any 𝑈∈Γ(𝑇𝑀). Hence using (2.2) in (9.4) we get 𝑔𝑅()𝜂(𝑌)𝜂(𝑊)𝜂(𝑋)𝜂(𝑊)𝑔𝑔𝑔+1𝑋,𝑌𝑊,𝜑𝑈+𝑔𝑋,𝜑𝑈−𝑔𝑌,𝜑𝑈+2𝑔𝑋,𝜑𝑌𝜑𝑊,𝜑𝑈+𝑔𝑋,𝜑𝑊𝜑𝑌,𝜑𝑈−𝑔𝑌,𝜑𝑊𝜑𝑋,𝜑𝑈𝑆()𝑔()𝑔+()𝑔𝜂(𝑌)𝜂(𝑊)()𝑔()𝑔−()𝑔𝜂(𝑋)𝜂(𝑊)−2𝑛+2𝑌,𝑊𝑋,𝜑𝑈−2𝑔𝑌,𝑊𝑋,𝜑𝑈2𝑛+2𝑋,𝜑𝑈−𝑆𝑋,𝑊𝑌,𝜑𝑈+2𝑔𝑋,𝑊𝑌,𝜑𝑈2𝑛+2𝑌,𝜑𝑈𝜏−8𝑛()𝑛𝑔()𝑔()𝑔2𝑛+2𝑌,𝑊𝑋,𝜑𝑈−𝑔𝑋,𝑊𝑌,𝜑𝑈=0. (9.5) Now putting 𝑌=𝑊=𝜉 in (9.5) and using (2.2), (2.14), and (2.16) we have𝜏−2𝑛()𝑛𝑔2𝑛+2𝑋,𝜑𝑈=0. (9.6) From (9.6) we get 𝜏=2𝑛. (9.7) Hence we have the following.

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 𝑍(𝜉,𝑋)⋅𝑅=0 and 𝑍(𝜉,𝑋)⋅𝑆=0

In this section we firstly consider a para-Sasakian manifold 𝑀 satisfying 𝑍(𝜉,𝑋)⋅𝑅=0, (10.1) for any 𝑋∈Γ(𝑇𝑀), with respect to canonical paracontact connection.

From (2.23), we have 𝑍(𝜉,𝑋)⋅𝑅(𝑈,𝑉)𝑊=𝑍(𝜉,𝑋)𝑅(𝑈,𝑉)𝑊−𝑅𝑍()𝑊−𝜉,𝑋𝑈,𝑉𝑅𝑈,𝑍(𝜉,𝑋)𝑉𝑊−𝑅(𝑈,𝑉)𝑍(𝜉,𝑋)𝑊=0, (10.2) for all 𝑋,𝑈,𝑉,𝑊∈Γ(𝑇𝑀). Now by putting 𝑈=𝜉 in (10.2), we get 𝑍(𝜉,𝑋)𝑅(𝜉,𝑉)𝑊−𝑅𝑍()𝑊−𝜉,𝑋𝜉,𝑉𝑅𝜉,𝑍(𝜉,𝑋)𝑉𝑊−𝑅(𝜉,𝑉)𝑍(𝜉,𝑋)𝑊=0. (10.3) Using (3.18) in (10.3) we obtain 𝑅𝑍(𝜉,𝑋)𝜉,𝑉𝑊=0. (10.4) Taking inner product with 𝑌 and from (8.2), we have 𝜏−2𝑛()𝑔2𝑛2𝑛−1𝑅(𝑋,𝑉)𝑊,𝑌=0. (10.5) Again using (3.2) in (10.5) we get 𝜏−2𝑛()𝑔(𝑅())()𝜂(𝑋)𝜂(𝑌)()𝜂(𝑌)𝜂(𝑉)()𝜂(𝑉)𝜂(𝑊)()𝜂(𝑋)𝜂(𝑊)𝑔𝑔𝑔2𝑛2𝑛−1𝑋,𝑉𝑊,𝑌+𝑔𝑉,𝑊−𝑔𝑋,𝑊+𝑔𝑋,𝑌−𝑔𝑌,𝑉+2𝑔𝑋,𝜑𝑉𝜑𝑊,𝑌+𝑔𝑋,𝜑𝑊𝑌,𝜑𝑉−𝑔𝑉,𝜑𝑊𝜑𝑋,𝑌=0. (10.6)

Let {𝑒_𝑖,𝜑𝑒_𝑖,𝜉}(𝑖=1,…,𝑛) be an orthonormal basis of the tangent space at any point. So a contraction of (10.6) with respect to 𝑋 and 𝑌 gives 𝜏−2𝑛()𝑆()()2𝑛2𝑛−1+()𝜂(𝑉)𝜂(𝑊)𝑉,𝑊−2𝑔𝑉,𝑊2𝑛+2=0. (10.7) Therefore we have the following.

Theorem 10.1. Let 𝑀^2𝑛+1 be a para-Sasakian manifold satisfying the condition 𝑍(𝜉,𝑋)⋅𝑅=0, for any 𝑋∈Γ(𝑇𝑀). Then either(i)𝜏=2𝑛, that is, the scalar curvature is constant, or(ii)𝑀 is an 𝜂-Einstein manifold with equation 𝑆(𝑉,𝑊)=2𝑔(𝑉,𝑊)−(2𝑛+2)𝜂(𝑉)𝜂(𝑊), (10.8)for all 𝑉,𝑊∈Γ(𝑇𝑀).

Now, let us consider a para-Sasakian manifold 𝑀 satisfying 𝑍(𝜉,𝑋)⋅𝑆=0, (10.9) where 𝑋∈Γ(𝑇𝑀) and 𝑆 is the Ricci tensor of 𝑀 with respect to canonical paracontact connection.

From (10.9), for any 𝑋,𝑈,𝑉∈Γ(𝑇𝑀), we obtain 𝑆𝑍(𝜉,𝑋)𝑈,𝑉+𝑆𝑈,𝑍(𝜉,𝑋)𝑉=0, (10.10) for all 𝑋,𝑈,𝑉∈Γ(𝑇𝑀). Using (3.18) and (3.19) with (8.2) in (10.10) we get 𝐴𝑆(𝑋,𝑉)𝜂(𝑈)+𝑆(𝑋,𝑈)𝜂(𝑉)=0, (10.11) where 𝐴=𝜏/2𝑛(2𝑛−1).

Using (3.13) and (3.14) in (10.11), we get 𝜏−2𝑛()𝑆()𝜂(𝑈)()𝜂(𝑈)+()𝜂(𝑋)𝜂(𝑉)𝜂(𝑈)()𝜂(𝑉)()𝜂(𝑉)−()𝜂(𝑋)𝜂(𝑉)𝜂(𝑈)2𝑛2𝑛−1𝑋,𝑉−2𝑔𝑋,𝑉2𝑛+2−𝑆𝑋,𝑈+2𝑔𝑋,𝑈2𝑛+2=0. (10.12)

Let {𝑒_𝑖,𝜑𝑒_𝑖,𝜉}(𝑖=1,…,𝑛) be an orthonormal basis of the tangent space at any point. By a contraction of (10.12) with respect to 𝑋 and 𝑈 we obtain 𝜏−2𝑛())𝜂(𝑉2𝑛2𝑛−1(𝜏−2𝑛)=0. (10.13) This gives 𝜏=2𝑛 which implies that manifold is of constant scalar curvature.

Theorem 10.2. If the condition 𝑍(𝜉,𝑋)⋅𝑆=0, for all 𝑋∈Γ(𝑇𝑀), holds on a para-Sasakian manifold, then its scalar curvature is constant.

11. Para-Sasakian Manifold with Canonical Paracontact Connection Satisfying 𝑃𝑃(𝜉,𝑋)⋅𝑅=0 and 𝑃𝑃(𝜉,𝑋)⋅𝑆=0

In this section we consider a para-Sasakian manifold 𝑀^2𝑛+1 with canonical paracontact connection satisfying the condition 𝑃𝑃(𝜉,𝑋)⋅𝑅=0, (11.1) for all 𝑋∈Γ(𝑇𝑀).

From (2.24), we get 𝑃𝑃(𝜉,𝑋)⋅𝑅(𝑈,𝑉)𝑊=𝑃𝑃(𝜉,𝑋)𝑅(𝑈,𝑉)𝑊−𝑅𝑃𝑃()𝑊−𝜉,𝑋𝑈,𝑉𝑅𝑈,𝑃𝑃(𝜉,𝑋)𝑉𝑊−𝑅(𝑈,𝑉)𝑃𝑃(𝜉,𝑋)𝑊=0, (11.2) where 𝑋,𝑈,𝑉,𝑊∈Γ(𝑇𝑀). Now we take 𝑈=𝜉 in (11.2), then we have 𝑃𝑃(𝜉,𝑋)𝑅(𝜉,𝑉)𝑊−𝑅𝑃𝑃()𝑊−𝜉,𝑋𝜉,𝑉𝑅𝜉,𝑃𝑃(𝜉,𝑋)𝑉𝑊−𝑅(𝜉,𝑉)𝑃𝑃(𝜉,𝑋)𝑊=0. (11.3) Taking inner product with 𝑌∈Γ(𝑇𝑀) and by using (3.18) and (9.3) we obtain 𝜏−2𝑛()𝑛𝑔2𝑛+2𝑅(𝑋,𝑉)𝑊,𝑌=0. (11.4) Again using (3.2) in (11.4), we have 𝜏−2𝑛()𝑛𝑔(𝑅())()𝜂(𝑋)𝜂(𝑌)()𝜂(𝑌)𝜂(𝑉)()𝜂(𝑉)𝜂(𝑊)()𝜂(𝑋)𝜂(𝑊)𝑔𝑔𝑔2𝑛+2𝑋,𝑉𝑊,𝑌+𝑔𝑉,𝑊−𝑔𝑋,𝑊+𝑔𝑋,𝑌−𝑔𝑌,𝑉+2𝑔𝑋,𝜑𝑉𝜑𝑊,𝑌+𝑔𝑋,𝜑𝑊𝑌,𝜑𝑉−𝑔𝑉,𝜑𝑊𝜑𝑋,𝑌=0. (11.5)

Let {𝑒_𝑖,𝜑𝑒_𝑖,𝜉}(𝑖=1,…,𝑛) be an orthonormal basis of the tangent space at any point. Hence by suitable contracting of (11.5) we get 𝜏−2𝑛()𝑆()()2𝑛2𝑛−1+()𝜂(𝑉)𝜂(𝑊)𝑉,𝑊−2𝑔𝑉,𝑊2𝑛+2=0. (11.6) Therefore we have the following.

Theorem 11.1. Let 𝑀^2𝑛+1 be a para-Sasakian manifold. If 𝑃𝑃(𝜉,𝑋)⋅𝑅=0 holds on 𝑀, then either(i) scalar curvature is constant with equation 𝜏=2𝑛 or(ii)𝑀 is an 𝜂-Einstein manifold with equation 𝑆(𝑉,𝑊)=2𝑔(𝑉,𝑊)−(2𝑛+2)𝜂(𝑉)𝜂(𝑊), (11.7)for all 𝑉,𝑊∈Γ(𝑇𝑀).

Now let us consider a para-Sasakian manifold satisfying 𝑃𝑃(𝜉,𝑋)⋅𝑆=0, (11.8) where 𝑋∈Γ(𝑇𝑀) and 𝑆 is the Ricci tensor of 𝑀 with respect to canonical paracontact connection.

From (11.8) we obtain 𝑆𝑃𝑃(𝜉,𝑋)𝑌,𝑍+𝑆𝑌,𝑃𝑃(𝜉,𝑋)𝑍=0. (11.9) Using (3.18) and (3.19) with (9.3) in (11.9) we have 𝐵𝑆(𝑋,𝑍)𝜂(𝑌)+𝑆(𝑋,𝑌)𝜂(𝑍)=0, (11.10) where 𝐵=𝜏/(2𝑛+2)𝑛.

Using (3.13) and (3.14) in (11.10) we get 𝜏−2𝑛()𝑛𝑆()𝜂(𝑌)()𝜂(𝑌)+()𝜂(𝑋)𝜂(𝑌)𝜂(𝑍)()𝜂(𝑍)()𝜂(𝑍)−()𝜂(𝑋)𝜂(𝑌)𝜂(𝑍)2𝑛+2𝑋,𝑍−2𝑔𝑋,𝑍2𝑛+2−𝑆𝑋,𝑌+2𝑔𝑋,𝑌2𝑛+2=0. (11.11)

Let {𝑒_𝑖,𝜑𝑒_𝑖,𝜉}(𝑖=1,…,𝑛) be an orthonormal basis of the tangent space at any point. So a contraction of (11.11) with respect to 𝑋 and 𝑌 gives 𝜏−2𝑛()𝑛)𝜂(𝑍2𝑛+2(𝜏−2𝑛)=0, (11.12) which implies that 𝜏=2𝑛.

So we have the following.

Theorem 11.2. If the condition 𝑃𝑃(𝜉,𝑋)⋅𝑆=0, for all 𝑋∈Γ(𝑇𝑀), 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)⋅𝑃𝑃=0

The projective curvature tensor 𝑃 of a para-Sasakian manifold 𝑀^2𝑛+1 with respect to canonical paracontact connection ∇ is defined by 𝑃(𝑋,𝑌)𝑍=𝑅()1𝑋,𝑌𝑍−2𝑛𝑆(𝑌,𝑍)𝑋−𝑆(𝑋,𝑍)𝑌, (12.1) where 𝑋,𝑌,Z∈Γ(𝑇𝑀). By using (3.2) and (3.13), from (12.1) we obtain 𝑃()()()𝜂(𝑋)()𝜂(𝑌)𝜉(𝑌)𝜂(𝑍)(𝑋)𝜂(𝑍)−1𝑋,𝑌𝑍=𝑅𝑋,𝑌𝑍+𝑔𝑌,𝑍𝜉−𝑔𝑋,𝑍+𝜂𝑋−𝜂𝑌+2𝑔𝑋,𝜑𝑌𝜑𝑍+𝑔𝑋,𝜑𝑍𝜑𝑌−𝑔𝑌,𝜑𝑍𝜑𝑋𝑆()()𝑋+()𝜂(𝑌)𝜂(𝑍)𝑋()()𝑌−()𝜂(𝑋)𝜂(𝑍)𝑌.2𝑛𝑌,𝑍𝑋−2𝑔𝑌,𝑍2𝑛+2−𝑆𝑋,𝑍𝑌+2𝑔𝑋,𝑍2𝑛+2 (12.2) Let us consider the equation 𝑃(𝜉,𝑋)⋅𝑃𝑃=0 (12.3) holds on a para-Sasakian manifold. From (2.25), we get 𝑃(𝜉,𝑋)⋅𝑃𝑃(𝑈,𝑉)𝑊=𝑃(𝜉,𝑋)𝑃𝑃(𝑈,𝑉)𝑊−𝑃𝑃𝑃()𝑊−𝜉,𝑋𝑈,𝑉𝑃𝑃𝑈,𝑃(𝜉,𝑋)𝑉𝑊−𝑃𝑃(𝑈,𝑉)𝑃(𝜉,𝑋)𝑊=0, (12.4) for all 𝑋,𝑈,𝑉,𝑊∈Γ(𝑇𝑀). Using (12.2) with (3.18) and (3.19) we obtain 12𝑛𝑆𝑋,𝑃𝑃(𝑈,𝑉)𝑊𝜉−𝑆(𝑋,𝑈)𝑃𝑃()𝑊−𝜉,𝑉𝑆(𝑋,𝑉)𝑃𝑃(𝑈,𝜉)𝑊−𝑆(𝑋,𝑊)𝑃𝑃(𝑈,𝑉)𝜉=0. (12.5) Taking inner product with 𝑌 and put 𝑈=𝜉, by using (3.19) we get 12𝑛𝑆𝑋,𝑃𝑃(𝜉,𝑉)𝑊𝜂(𝑌)−𝑆(𝑋,𝑊)𝑔𝑃𝑃(𝜉,𝑉)𝜉,𝑌=0. (12.6) Using (9.3) in (12.6), we obtain 𝐶𝑆(𝑋,𝑉)𝜂(𝑌)𝜂(𝑊)+𝑆()𝜂(𝑌)𝜂(𝑉)−𝑋,𝑊𝑆(𝑋,𝑊)𝑔(𝑌,𝑉)=0, (12.7) where 𝐶=𝜏/(2𝑛+2)2𝑛².

From the last equation, with (3.13) we get 𝐶𝑆()𝜂(𝑌)𝜂(𝑊)()𝜂(𝑌)𝜂(𝑊)+()𝜂(𝑋)𝜂(𝑌)𝜂(𝑉)𝜂(𝑊)()𝜂(𝑌)𝜂(𝑉)()𝜂(𝑌)𝜂(𝑉)+()𝜂(𝑋)𝜂(𝑌)𝜂(𝑉)𝜂(𝑊)()𝑔()()𝑔()−()𝜂(𝑋)𝜂(𝑌)𝜂(𝑉)𝜂(𝑊)𝑋,𝑉−2𝑔𝑋,𝑉2𝑛+2+𝑆𝑋,𝑊−2𝑔𝑋,𝑊2𝑛+2−𝑆𝑋,𝑊𝑌,𝑉+2𝑔𝑋,𝑊𝑌,𝑉2𝑛+2=0. (12.8) Hence by suitable contracting of (12.8), we obtain 𝜏−2𝑛(2𝑛+2)2𝑛²𝑆()()+()𝜂(𝑉)𝜂(𝑊)𝑉,𝑊−2𝑔𝑉,𝑊2𝑛+2=0. (12.9) Hence we have the following.

Theorem 12.1. Let 𝑀 be a (2𝑛+1)-dimensional para-Sasakian manifold. If the condition 𝑃(𝜉,𝑋)⋅𝑃𝑃=0 holds on 𝑀, then either(i) scalar curvature is constant with equation 𝜏=2𝑛 or(ii)𝑀 is an 𝜂-Einstein manifold with equation 𝑆(𝑉,𝑊)=2𝑔(𝑉,𝑊)−(2𝑛+2)𝜂(𝑉)𝜂(𝑊), (12.10)for all 𝑉,𝑊∈Γ(𝑇𝑀).

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.