Some Curvature Properties of (LCS)_𝑛-Manifolds

Abstract

The object of the present paper is to study (LCS)_𝑛-manifolds with vanishing quasi-conformal curvature tensor. (LCS)_𝑛-manifolds satisfying Ricci-symmetric condition are also characterized.

1. Introduction

Recently, in [1], Shaikh introduced and studied Lorentzian concircular structure manifolds (briefly (LCS)-manifold) which generalizes the notion of LP-Sasakian manifolds, introduced by Matsumoto [2].

Generalizing the notion of LP-Sasakian manifold in 2003 [1], Shaikh introduced the notion of (LCS)_𝑛-manifolds along with their existence and applications to the general theory of relativity and cosmology. Also, Shaikh and his coauthors studied various types of (LCS)_𝑛-manifolds by imposing the curvature restrictions (see [3–6]). In [7, 8], the authors also studied (LCS)_2𝑛+1-manifolds.

The submanifold of an (LCS)_𝑛-manifold is studied by Atceken and Hui [9, 10] and Shukla et al. [11]. In [12], Yano and Sawaki introduced the quasi-conformal curvature tensor, and later it was studied by many authors with curvature restrictions on various structures [13].

After then, the same author studied weakly symmetric (LCS)_𝑛-manifolds by several examples and obtain various results in such manifolds. In [7], authors shown that a pseudo projectively flat and pseudo projectively recurrent (LCS)_𝑛 manifolds are 𝜂-Einstein manifold.

On the other hand, in [5], authors proved the existence of 𝜙-recurrent (LCS)₃ manifold which is neither locally symmetric nor locally 𝜙-symmetric by nontrivial examples. Furthermore, they also give the necessary and sufficient conditions for a (LCS)_𝑛-manifold to be locally 𝜙-recurrent.

In this study, we have investigated the quasi-conformal flat (LCS)_𝑛-manifolds satisfying properties such as Ricci-symmetric, locally symmetric, and 𝜂-Einstein. Finally, we give an example for 𝜂-Einstein manifolds.

2. Preliminaries

An 𝑛-dimensional Lorentzian manifold 𝑀 is a smooth connected paracompact Hausdorff manifold with a Lorentzian metric tensor 𝑔, that is, 𝑀 admits a smooth symmetric tensor field 𝑔 of the type (2,0) such that, for each 𝑝∈𝑀, 𝑔_𝑝∶𝑇_𝑀𝑝×𝑇_𝑀𝑝⟶ℝ (1) is a nondegenerate inner product of signature (−,+,+,…,+). In such a manifold, a nonzero vector 𝑋_𝑝∈𝑇_𝑀(𝑝) is said to be timelike (resp., nonspacelike, null, and spacelike) if it satisfies the condition 𝑔_𝑝(𝑋_𝑝,𝑋_𝑝)<0 (resp., ≤0, =0, >0). These cases are called casual character of the vectors.

Definition 1. In a Lorentzian manifold (𝑀,𝑔), a vector field 𝑃 defined by 𝑔(𝑋,𝑃)=𝐴(𝑋) (2) for any 𝑋∈Γ(𝑇𝑀) is said to be a concircular vector field if ∇_𝑋𝐴𝑌=𝛼𝑔(𝑋,𝑌)+𝑤(𝑋)𝐴(𝑌) (3) for 𝑌∈Γ(𝑇𝑀), where 𝛼 is a nonzero scalar function, 𝐴 is a 1-form, 𝑤 is also closed 1-form, and ∇ denotes the Levi-Civita connection on 𝑀 [7].

Let 𝑀 be a Lorentzian manifold admitting a unit timelike concircular vector field 𝜉, called the characteristic vector field of the manifold. Then we have 𝑔(𝜉,𝜉)=−1. (4) Since 𝜉 is a unit concircular unit vector field, there exists a nonzero 1-form 𝜂 such that 𝑔(𝑋,𝜉)=𝜂(𝑋). (5) The equation of the following form holds: ∇_𝑋𝜂𝑌=𝛼𝑔(𝑋,𝑌)+𝜂(𝑋)𝜂(𝑌),𝛼≠0 (6) for all 𝑋,𝑌∈Γ(𝑇𝑀), where 𝛼 is a nonzero scalar function satisfying ∇_𝑋𝛼=𝑋(𝛼)=𝑑𝛼(𝑋)=𝜌𝜂(𝑋), (7)𝜌 being a certain scalar function given by 𝜌=−𝜉(𝛼).

Let us put ∇_𝑋𝜉=𝛼𝜙𝑋, (8) then from (6) and (8), we can derive 𝜙𝑋=𝑋+𝜂(𝑋)𝜉, (9) which tell us that 𝜙 is a symmetric (1,1)-tensor. Thus the Lorentzian manifold 𝑀 together with the unit timelike concircular vector field 𝜉, its associated 1-form 𝜂, and (1,1)-type tensor field 𝜙 is said to be a Lorentzian concircular structure manifold.

A differentiable manifold 𝑀 of dimension 𝑛 is called (LCS)-manifold if it admits a (1,1)-type tensor field 𝜙, a covariant vector field 𝜂, and a Lorentzian metric 𝑔 which satisfy 𝜂(𝜉)=𝑔(𝜉,𝜉)=−1,𝜙²𝑋=𝑋+𝜂(𝑋)𝜉,𝑔(𝑋,𝜉)=𝜂(𝑋),𝜙𝜉=0,𝜂∘𝜙=0 (10)(11)(12)(13) for all 𝑋∈Γ(𝑇𝑀). Particularly, if we take 𝛼=1, then we can obtain the 𝐿𝑃-Sasakian structure of Matsumoto [2].

Also, in an (LCS)_𝑛-manifold 𝑀, the following relations are satisfied (see [3–6]): 𝜂(𝑅(𝑋,𝑌)𝑍)=𝛼²−𝜌𝑔(𝑌,𝑍)𝜂(𝑋)−𝑔(𝑋,𝑍)𝜂(𝑌),𝑅(𝜉,𝑋)𝑌=𝛼²−𝜌𝑔(𝑋,𝑌)𝜉−𝜂(𝑌)𝑋,𝑅(𝑋,𝑌)𝜉=𝛼²−𝜌𝜂(𝑌)𝑋−𝜂(𝑋)𝑌,∇_𝑋𝜙𝑌=𝛼𝑔(𝑋,𝑌)𝜉+2𝜂(𝑋)𝜂(𝑌)𝜉+𝜂(𝑌)𝑋,𝑆(𝑋,𝜉)=(𝑛−1)𝛼²−𝜌𝜂(𝑋),𝑆𝜙𝑋,𝜙𝑌=𝑆(𝑋,𝑌)+(𝑛−1)𝛼²−𝜌𝜂(𝑋)𝜂(𝑌) (14)(15)(16)(17)(18)(19) for all vector fields 𝑋,𝑌,𝑍 on 𝑀, where 𝑅 and 𝑆 denote the Riemannian curvature tensor and Ricci curvature, respectively, 𝑄 is also the Ricci operator given by 𝑆(𝑋,𝑌)=𝑔(𝑄𝑋,𝑌).

Now let (𝑀,𝑔) be an 𝑛-dimensional Riemannian manifold; then the concircular curvature tensor 𝐶, the Weyl conformal curvature tensor 𝐶, and the pseudo projective curvature tensor 𝑃 are, respectively, defined by 𝐶()()𝑍−𝜏𝑋,𝑌𝑍=𝑅𝑋,𝑌𝑛()𝑔()()𝑌,𝑛−1𝑌,𝑍𝑋−𝑔𝑋,𝑍𝐶()()1𝑋,𝑌𝑍=𝑅𝑋,𝑌𝑍−×𝑆()()𝑌()()+𝜏𝑛−2𝑌,𝑍𝑋−𝑆𝑋,𝑍+𝑔𝑌,𝑍𝑄𝑋−𝑔𝑋,𝑍𝑄𝑌()𝑔()()𝑌,𝑛−1)(𝑛−2𝑌,𝑍𝑋−𝑔𝑋,𝑍𝑃()()𝑍[𝑆()()𝑌]−𝜏𝑋,𝑌𝑍=𝑎𝑅𝑋,𝑌+𝑏𝑌,𝑍𝑋−𝑆𝑋,𝑍𝑛𝑎𝑔()()𝑌,𝑛−1+𝑏𝑌,𝑍𝑋−𝑔𝑋,𝑍 (20)(21)(22) where 𝑎 and 𝑏 are constants such that 𝑎,𝑏≠0, and 𝜏  is  also the scalar curvature of 𝑀  [7].

For an 𝑛-dimensional (LCS)_𝑛-manifold the quasi-conformal curvature tensor 𝒞 is given by 𝒞()()𝑍𝑆()()𝑌()()−𝜏𝑋,𝑌𝑍=𝑎𝑅𝑋,𝑌+𝑏𝑌,𝑍𝑋−𝑆𝑋,𝑍+𝑔𝑌,𝑍𝑄𝑋−𝑔𝑋,𝑍𝑄𝑌𝑛𝑎𝑔()()𝑌𝑛−1+2𝑏𝑌,𝑍𝑋−𝑔𝑋,𝑍 (23) for all 𝑋,𝑌,𝑍∈Γ(𝑇𝑀) [14].

The notion of quasi-conformal curvature tensor was defined by Yano and Swaki [12]. If 𝑎=1 and 𝑏=−1/(𝑛−1), then quasi-conformal curvature tensor reduces to conformal curvature tensor.

3. Quasi-Conformally Flat (LCS)_𝑛-Manifolds and Some of Their Properties

For an 𝑛-dimensional quasi-conformally flat (LCS)_𝑛-manifold, we know for 𝑍=𝜉 from (23), ()𝑆()()𝑌()()−𝜏𝑎𝑅𝑋,𝑌𝜉+𝑏𝑌,𝜉𝑋−𝑆𝑋,𝜉+𝑔𝑌,𝜉𝑄𝑋−𝑔𝑋,𝜉𝑄𝑌𝑛𝑎𝑔()()𝑌𝑛−1+2𝑏𝑌,𝜉𝑋−𝑔𝑋,𝜉=0. (24) Here, taking into account of (16), we have𝜂(𝑌)𝑋−𝜂(𝑋)𝑌𝑎𝛼²−𝜌+𝑏(𝑛−1)𝛼²−𝜏−𝜌𝑛𝑎𝜂(𝑌](𝑋)𝑛−1+2𝑏+𝑏𝑄𝑋−𝜂𝑄𝑌=0. (25) Let 𝑌=𝜉 be in (25); then also by using (18) we obtain −𝑋−𝜂(𝑋)𝜉𝑎𝛼²−𝜏−𝜌𝑛𝑎()𝛼𝑛−1+2𝑏+𝑏𝑛−1²(𝑋𝛼−𝜌+𝑏−𝑄𝑋−𝜂)(𝑛−1)²−𝜌𝜉=0. (26) Taking the inner product on both sides of the last equation by 𝑌, we obtain 𝑔(𝑋,𝑌)+𝜂(𝑋)𝜂(𝑌)𝑎𝛼²()×𝛼−𝜌+𝑏𝑛−1₂−𝜏−𝜌𝑛𝑎𝑆()(𝑋)𝜂(𝑌𝛼𝑛−1+2𝑏+𝑏𝑋,𝑌+𝜂)²−𝜌(𝑛−1)=0, (27) that is, 𝑆()()×𝜏𝑋,𝑌=𝑔𝑋,𝑌𝑎𝑛𝑏−𝛼𝑛−1+2𝑏²𝑎−𝜌𝑏+((𝑋)𝜂(𝑌𝜏𝑛−1)+𝜂)𝑎𝑛𝑏−𝛼𝑛−1+2𝑏₂𝑎−𝜌𝑏+2(𝑛−1). (28) Now we are in a proposition to state the following.

Theorem 2. If an 𝑛-dimensional (LCS)_𝑛-manifold 𝑀 is quasi-conformally flat, then 𝑀 is an 𝜂-Einstein manifold.

Now, let {𝑒₁,𝑒₂,…,𝑒_𝑛−1,𝜉} be an orthonormal basis of the tangent space at any point of the manifold. Then putting 𝑋=𝑌=𝑒_𝑖,𝜉 in (28), and taking summation for 1≤𝑖≤𝑛−1, we have 𝜏=𝑛(𝑛−1)𝛼²−𝜌if𝑎+(𝑛−2)𝑏≠0. (29) In view of (28) and (29), we obtain 𝑆(𝑋,𝑌)=(𝑛−1)𝛼²−𝜌𝑔(𝑋,𝑌), (30) which is equivalent to 𝑄𝑋=(𝑛−1)𝛼²−𝜌𝑋 (31) for any 𝑋∈Γ(𝑇𝑀).

By using (29) and (31) in (23) for a quasi-conformally flat (LCS)_𝑛-manifold 𝑀, we get 𝑅(𝑋,𝑌)𝑍=𝛼²−𝜌𝑔(𝑌,𝑍)𝑋−𝑔(𝑋,𝑍)𝑌, (32) for all 𝑋,𝑌,𝑍∈Γ(𝑇𝑀). If we consider Schur's Theorem, we can give the following the theorem.

Theorem 3. A quasi-conformally flat (LCS)_𝑛-manifold M (𝑛>1) is a manifold of constant curvature (𝛼²−𝜌) provided that 𝑎+𝑏(𝑛−2)≠0.

Now let us consider an (LCS)_𝑛-manifold 𝑀 which is conformally flat. Thus we have from (21) that 𝑅()1𝑋,𝑌𝑍={𝑆()()𝑌()()−𝜏𝑛−2𝑌,𝑍𝑋−𝑆𝑋,𝑍+𝑔𝑌,𝑍𝑄𝑋−𝑔𝑋,𝑍𝑄𝑌()𝑔()()𝑌,𝑛−1)(𝑛−2𝑌,𝑍𝑋−𝑔𝑋,𝑍 (33) for all vector fields 𝑋,𝑌,𝑍 tangent to 𝑀. Setting 𝑍=𝜉 in (33) and using (16), (18) we have 𝜏−𝛼𝑛−1²𝜂(𝑌)(𝑋)𝑌=𝜂(𝑌)(𝑋).−𝜌𝑋−𝜂𝑄𝑋−𝜂𝑄𝑌 (34) If we put 𝑌=𝜉 in (34) and also using (18), we obtain 𝜏𝑄𝑋=−𝛼𝑛−1²𝜏−𝜌𝑋+𝛼𝑛−1−𝑛²−𝜌𝜂(𝑋)𝜉. (35)

Corollary 4. A conformally flat (LCS)_𝑛-manifold is an 𝜂-Einstein manifold.

Generalizing the notion of a manifold of constant curvature, Chen and Yano [15] introduced the notion of a manifold of quasi-constant curvature which can be defined as follows:

Definition 5. A Riemannian manifold is said to be a manifold of quasi-constant curvature if it is conformally flat and its curvature tensor 𝑅 of type (0,4) is of the form 𝑅()𝑔()𝑔()()𝑔(𝑔()𝐴(𝑋)𝐴(𝑊)()𝐴(𝑌)𝐴(𝑊)()𝐴(𝑌)𝐴(𝑍)()𝐴(𝑋)𝐴(𝑍,𝑋,𝑌,𝑍,𝑊=𝑎𝑌,𝑍𝑋,𝑊−𝑔𝑋,𝑍𝑌,𝑊)+𝑏𝑌,𝑍−𝑔𝑋,𝑍+𝑔𝑋,𝑊−𝑔𝑌,𝑊) (36) for all 𝑋,𝑌,𝑍,𝑊∈Γ(𝑇𝑀), where 𝑎,𝑏 are scalars of which 𝑏≠0 and 𝐴 is a nonzero 1-form (for more details, we refer to [13, 16]).

Thus we have the following theorem for (LCS)_𝑛-conformally flat manifolds.

Theorem 6. A conformally flat (LCS)_𝑛-manifold is a manifold of quasi-constant curvature.

Proof. From (33) and (35), we obtain 𝑅()=(𝛼𝑋,𝑌,𝑍,𝑊𝜏−2𝑛−1)²−𝜌()×𝑔()𝑔()()𝑔(+()𝛼𝑛−1)(𝑛−2𝑋,𝑊𝑌,𝑍−𝑔𝑌,𝑊𝑋,𝑍)𝜏−𝑛𝑛−1²−𝜌()×𝑔()𝜂(𝑌)𝜂(𝑍)()𝜂(𝑋)𝜂(𝑍)()𝜂(𝑋)𝜂(𝑊)()𝜂(𝑌)𝜂(𝑊.𝑛−1)(𝑛−2𝑋,𝑊−𝑔𝑌,𝑊+𝑔𝑌,𝑍−𝑔𝑋,𝑍) (37) This implies (36) for (𝛼𝑎=𝜏−2𝑛−1)²−𝜌(),()𝛼𝑛−1)(𝑛−2𝑏=𝜏−𝑛𝑛−1²−𝜌(𝑛−1)(𝑛−2),𝐴=𝜂. (38) This proves our assertion.

Next, differentiating the (19) covariantly with respect to 𝑊, we get ∇_𝑊𝑆𝜙𝑋,𝜙𝑌=∇_𝑊𝑆(𝑋,𝑌)+(𝑛−1)𝑊𝛼²+()𝛼−𝜌𝑛−1²−𝜌𝑊𝜂(𝑋)𝜂(𝑌), (39) for any 𝑋,𝑌∈Γ(𝑇𝑀). Making use of the definition of ∇𝑆 and (8), we have ∇_𝑊𝑆𝜙𝑋,𝜙𝑌+𝑆∇_𝑊𝜙𝑋,𝜙𝑌+𝑆𝜙𝑋,∇_𝑊=∇𝜙𝑌_𝑊𝑆(𝑋,𝑌)+𝑆∇_𝑊𝑋,𝑌+𝑆𝑋,∇_𝑊𝑌+(𝑛−1)𝑊𝛼²𝜂(𝑋)𝜂(𝑌)+()𝛼−𝜌𝑛−1²−𝜌𝜂(𝑌)𝜂∇_𝑊𝑋+()𝛼+𝛼𝑔𝑋,𝜙𝑊𝑛−1²−𝜌𝜂(𝑋)𝜂∇_𝑊𝑌+𝛼𝑔𝑌,𝜙𝑊. (40) Thus we have ∇_𝑊𝑆𝜙𝑋,𝜙𝑌−∇_𝑊𝑆()∇𝑋,𝑌=−𝑆_𝑊𝜙𝑋+𝜙∇_𝑊∇𝑋,𝜙𝑌−𝑆𝜙𝑋,_𝑊𝜙𝑌+𝜙∇_𝑊𝑌+𝑆∇_𝑊𝑋,𝑌+𝑆𝑋,∇_𝑊𝑌+(𝑛−1)𝑊𝛼²𝜂(𝑋)𝜂(𝑌)+()𝛼−𝜌𝑛−1²−𝜌𝜂(𝑌)𝜂∇_𝑊𝑋+()𝛼+𝛼𝑔𝑋,𝜙𝑊𝑛−1²−𝜌𝜂(𝑋)𝜂∇_𝑊𝑌+𝛼𝑔𝑌,𝜙𝑊. (41) Here taking account of (17), we arrive at ∇_𝑊𝑆𝜙𝑋,𝜙𝑌−∇_𝑊𝑆()𝛼𝑔()(𝑋)𝜂(𝑋,𝑌=−𝑆𝑋,𝑊𝜉+2𝜂W)(𝑋)𝑊𝑔()(𝑌)𝜂(𝑊)(𝑌)𝑊𝜉+𝜂,𝜙𝑌−𝑆𝜙𝑋,𝛼𝑌,𝑊𝜉+2𝜂𝜉+𝜂−𝑆𝜙𝑋,𝜙∇_𝑊𝑌+𝑆∇_𝑊𝑋,𝑌+𝑆𝑋,∇_𝑊𝑌+(𝑛−1)𝑊𝛼²−𝜌𝜂(𝑋)𝜂(𝑌)−𝑆𝜙∇_𝑊𝑋,𝜙𝑌+(𝑛−1)𝛼²𝜂(𝑌)×𝜂∇−𝜌_𝑊𝑋+𝛼𝑔𝑋,𝜙𝑊+(𝑛−1)𝛼²𝜂(𝑋)×𝜂∇−𝜌_𝑊𝑌𝑔()𝑆(𝑋)𝜂(𝑊)𝑆(𝑋)𝑆𝑔()𝑆(𝑌)𝜂(𝑊)𝑆(𝑌)𝑆+𝛼𝑔𝑌,𝜙𝑊=−𝛼𝑋,𝑊𝜉,𝜙𝑌+2𝜂𝜉,𝜙𝑌+𝜂𝑊,𝜙𝑌−𝛼𝑌,𝑊𝜙𝑋,𝜉+2𝜂𝜙𝑋,𝜉+𝜂𝜙𝑋,𝑊−𝑆𝜙𝑋,𝜙∇_𝑊𝑌+𝑆∇_𝑊𝑋,𝑌+𝑆𝑋,∇_𝑊𝑌−𝑆𝜙∇_𝑊𝑋,𝜙𝑌+(𝑛−1)𝑊𝛼²𝜂(𝑋)𝜂(𝑌)+()𝛼−𝜌𝑛−1²−𝜌𝜂(𝑌)𝜂∇_𝑊𝑋+()𝛼+𝛼𝑔𝑋,𝜙𝑊𝑛−1²−𝜌𝜂(𝑋)𝜂∇_𝑊𝑌+𝛼𝑔𝑌,𝜙𝑊. (42) Again, by using (13), (18), and (19), we reach ∇_𝑊𝑆𝜙𝑋,𝜙𝑌−∇_𝑊𝑆()(𝑋)𝑆(𝑌)𝑆−()𝛼𝑋,𝑌=−𝛼𝜂𝑊,𝜙𝑌−𝛼𝜂𝜙𝑋,𝑊𝑛−1²−𝜌𝜂(𝑋)𝜂∇_𝑊𝑋−(𝑛−1)𝛼²−𝜌𝜂(𝑌)𝜂∇_𝑊𝑋+(𝑛−1)𝑊𝛼²𝜂(𝑋)𝜂(𝑌)+(𝛼−𝜌𝑛−1)²×𝜂∇−𝜌_𝑊𝑋𝜂(𝑌)(𝑌)𝑔∇+𝛼𝜂𝑋,𝜙𝑊+𝜂_𝑊𝑌𝜂(𝑋)(𝑋)𝑔(𝑋)𝑆(𝑌)𝑆()𝛼+𝛼𝜂𝑌,𝜙𝑊=−𝛼𝜂𝑊,𝜙𝑌−𝛼𝜂𝜙𝑋,𝑊+𝛼𝑛−1²×𝜂(𝑌)𝑔(𝑋)𝑔+()𝑊𝛼−𝜌𝑋,𝜙𝑊+𝜂𝑌,𝜙𝑊𝑛−1²−𝜌𝜂(𝑋)𝜂(𝑌). (43) Thus we have the following theorem.

Theorem 7. If an (LCS)_𝑛-manifold 𝑀 is Ricci-symmetric; then 𝛼²−𝜌 is constant.

Proof. If 𝑛>1-dimensional (LCS)_𝑛-manifold 𝑀 is Ricci-symmetric, then from (43) we conclude that 𝛼(𝑛−1)𝛼²𝜂(𝑌)𝑔(𝑋)𝑔+()𝑊𝛼−𝜌𝑋,𝜙𝑊+𝜂𝑌,𝜙𝑊𝑛−1²𝜂(𝑋)𝜂(𝑌)(𝑋)𝑆(𝑌)𝑆−𝜌−𝛼𝜂𝑊,𝜙𝑌−𝛼𝜂𝜙𝑋,𝑊=0. (44) It follows that 𝛼(𝑛−1)𝛼²𝑔(𝑋)+()𝑊𝛼−𝜌𝑋,𝜙𝑊𝜉−𝜂𝜙𝑊𝑛−1²𝜂(𝑋)𝜉(𝑋)−𝜌−𝛼𝜂𝜙𝑄𝑊−𝛼𝑆𝜙𝑋,𝑊𝜉=0, (45) from which −𝛼(𝑛−1)𝛼²𝑔−()𝑊𝛼−𝜌𝑋,𝜙𝑊𝑛−1²−𝜌𝜂(𝑋)+𝑆𝜙𝑋,𝑊=0, (46) which is equivalent to −𝛼(𝑛−1)𝛼²−𝜌𝜙𝑊−(𝑛−1)𝑊𝛼²−𝜌𝜉+𝛼𝜙𝑄𝑊=0, (47) that is, 𝑊𝛼²−𝜌=0, (48) which proves our assertion.

Since ∇𝑅=0 implies that ∇𝑆=0, we can give the following corollary.

Corollary 8. If an 𝑛-dimensional (𝐿𝐶𝑆)_𝑛-manifold 𝑀 is locally symmetric, then 𝛼²−𝜌 is constant.

Now, taking the covariant derivation of the both sides of (18) with respect to 𝑌, we have 𝑌𝑆(𝑋,𝜉)=(𝑛−1)𝑊𝛼²−𝜌𝜂(𝑋). (49) From the definition of the covariant derivation of Ricci-tensor, we have ∇_𝑌𝑆(𝑋,𝜉)=∇_𝑌𝑆(𝑋,𝜉)−𝑆∇_𝑌𝑋,𝜉−𝑆𝑋,∇_𝑌𝜉=(𝑛−1)𝑌𝛼²−𝜌𝜂(𝑋)+𝛼²×𝜂∇−𝜌_𝑌𝑋−()𝛼+𝛼𝑔𝑋,𝜙𝑌𝑛−1²−𝜌𝜂∇_𝑌𝑋=()𝑌𝛼−𝛼𝑆𝑋,𝜙𝑌𝑛−1²𝜂(𝑋)()𝛼−𝜌+𝛼𝑛−1²−𝜌𝑔𝑋,𝜙𝑌−𝛼𝑆𝑋,𝜙𝑌. (50) If an (𝐿𝐶𝑆)_𝑛-manifold 𝑀 Ricci symmetric, then Theorem 7 and (43) imply that 𝑆𝑋,𝜙𝑌=(𝑛−1)𝛼²−𝜌𝑔𝜙𝑌,𝑋. (51) This leads us to state the following.

Theorem 9. If an (LCS)_𝑛-manifold 𝑀 is Ricci symmetric, then it is an Einstein manifold.

Corollary 10. If an (LCS)_𝑛-manifold 𝑀 is locally symmetric, then it is an Einstein manifold.

In this section, an example is used to demonstrate that the method presented in this paper is effective. But this example is a special case of Example 6.1 of [6].

Example 11. Now, we consider the 3-dimensional manifold 𝑀=𝑥,𝑦,𝑧∈ℝ³,𝑧≠0, (52) where (𝑥,𝑦,𝑧) denote the standard coordinates in ℝ³. The vector fields 𝑒₁=𝑒^𝑧𝑥𝜕𝜕𝜕𝑥+𝑦𝜕𝑦,𝑒₂=𝑒^𝑧𝜕,𝑒𝜕𝑦₃=𝜕𝜕𝑧 (53) are linearly independent of each point of 𝑀. Let 𝑔 be the Lorentzian metric tensor defined by 𝑔𝑒₁,𝑒₁=𝑔𝑒₂,𝑒₂=−𝑔𝑒₃,𝑒₃𝑔𝑒=1,_𝑖,𝑒_𝑗=0,𝑖≠𝑗, (54) for 𝑖,𝑗=1,2,3. Let 𝜂 be the 1-form defined by 𝜂(𝑍)=𝑔(𝑍,𝑒₃) for any 𝑍∈Γ(𝑇𝑀). Let 𝜙 be the (1,1)-tensor field defined by 𝜙𝑒₁=𝑒₁,𝜙𝑒₂=𝑒₂,𝜙𝑒₃=0. (55) Then using the linearity of 𝜙 and 𝑔, we have 𝜂(𝑒₃)=−1, 𝜙²𝑍=𝑍+𝜂(𝑍)𝑒₃,𝑔𝜙𝑍,𝜙𝑊=𝑔(𝑍,𝑊)+𝜂(𝑍)𝜂(𝑊), (56) for all 𝑍,𝑊∈Γ(𝑇𝑀). Thus for 𝜉=𝑒₃, (𝜙,𝜉,𝜂,𝑔) defines a Lorentzian paracontact structure on 𝑀.
Now, let ∇ be the Levi-Civita connection with respect to the Lorentzian metric 𝑔, and let 𝑅 be the Riemannian curvature tensor of 𝑔. Then we have 𝑒₁,𝑒₂=−𝑒^𝑧𝑒₂,𝑒₁,𝑒₃=−𝑒₁,𝑒₂,𝑒₃=−𝑒₂. (57) Making use of the Koszul formulae for the Lorentzian metric tensor 𝑔, we can easily calculate the covariant derivations as follows: ∇_𝑒1𝑒₁=−𝑒₃,∇_𝑒2𝑒₁=𝑒^𝑧𝑒₂,∇_𝑒1𝑒₃=−𝑒₁,∇_𝑒2𝑒₃=−𝑒₂,∇_𝑒2𝑒₂=−𝑒^𝑧𝑒₁−𝑒₃,∇_𝑒1𝑒₂=∇_𝑒3𝑒₁=∇_𝑒3𝑒₂=∇_𝑒3𝑒₃=0. (58) From the previously mentioned, it can be easily seen that (𝜙,𝜉,𝜂,𝑔) is an (LCS)₃-structure on 𝑀, that is, 𝑀 is an (LCS)₃-manifold with 𝛼=−1 and 𝜌=0. Using the previous relations, we can easily calculate the components of the Riemannian curvature tensor as follows: 𝑅𝑒₁,𝑒₂𝑒₁=𝑒^2𝑧−1𝑒₂,𝑅𝑒₁,𝑒₂𝑒₂=1−𝑒^2𝑧𝑒₁,𝑅𝑒₁,𝑒₃𝑒₁=−𝑒₃,𝑅𝑒₁,𝑒₃𝑒₃=−𝑒₁,𝑅𝑒₂,𝑒₃𝑒₂=−𝑒₃,𝑅𝑒₂,𝑒₃𝑒₃=−𝑒₂,𝑅𝑒₁,𝑒₂𝑒₃=𝑅𝑒₁,𝑒₃𝑒₂=𝑅𝑒₂,𝑒₃𝑒₁=0. (59) By using the properties of 𝑅 and definition of the Ricci tensor, we obtain 𝑆𝑒₁,𝑒₁=𝑆𝑒₂,𝑒₂=−𝑒^2𝑧,𝑆𝑒₃,𝑒₃𝑆𝑒=−2,₁,𝑒₂=𝑆𝑒₁,𝑒₃=𝑆𝑒₂,𝑒₃=0. (60) Thus the scalar curvature 𝜏 of 𝑀 is given by 𝜏=³_𝑖=1𝑔𝑒_𝑖,𝑒_𝑖𝑆𝑒_𝑖,𝑒_𝑖=21−𝑒^2𝑧. (61) On the other hand, for any 𝑍,𝑊∈Γ(𝑇𝑀), 𝑍 and 𝑊 can be written as 𝑍=∑³_𝑖=1𝑓_𝑖𝑒_𝑖 and 𝑊=∑³_𝑗=1𝑔_𝑗𝑒_𝑗, where 𝑓_𝑖 and 𝑔_𝑖 are smooth functions on 𝑀. By direct calculations, we have 𝑆(𝑍,𝑊)=−𝑒^2𝑧𝑓₁𝑔₁+𝑓₂𝑔₂−2𝑓₃𝑔₃=−𝑒^2𝑧𝑓₁𝑔₁+𝑓₂𝑔₂−𝑓₃𝑔₃−𝑓₃𝑔₃𝑒^2𝑧+2. (62) Since 𝜂(𝑍)=−𝑓₃ and 𝜂(𝑊)=−𝑔₃ and 𝑔(𝑍,𝑊)=𝑓₁𝑔₁+𝑓₂𝑔₂−𝑓₃𝑔₃, we have 𝑆(𝑍,𝑊)=−𝑒^2𝑧𝑔(𝑍,𝑊)−𝑒^2𝑧+2𝜂(𝑍)𝜂(𝑊). (63) This tell us that 𝑀 is an 𝜂-Einstein manifold.

Acknowledgment

The authors would like to thank the reviewers for the extremely carefully reading and for many important comments, which improved the paper considerably.