ON A TYPE OF LORENTZIAN PARA-SASAKIAN MANIFOLDS AHMET YILDIZ, UDAY CHAND DE and ERHAN ATA

AHMET YILDIZ, UDAY CHAND DE and ERHAN ATA

Communicated by the former editorial board

The object of the present paper is to introduce a new concept called generalized η-Einstein manifold in a Lorentzian Para-Sasakian manifold. Some geometric properties have been studied. Finally an example has been constructed to prove the existance of a generalizedη-Einstein Lorentzian Para-Sasakian manifold.

AMS 2010 Subject Classification: 53B20, 53B25, 53B50.

Key words: Lorentzian Para-Sasakian manifolds,η-Einstein manifold, general- izedη-Einstein Lorentzian Para-Sasakian manifold.

1. INTRODUCTION

A Lorentzian Para-Sasakian manifold Mⁿ is said to be an η-Einstein manifold if the following condition

(1.1) S(X, Y) =ag(X, Y) +bη(X)η(Y), holds on Mⁿ,wherea, b are smooth functions.

A Lorentzian Para-Sasakian manifold Mⁿ is said to be a generalized η- Einstein manifold if the following condition

(1.2) S(X, Y) =ag(X, Y) +bη(X)η(Y) +cΩ(X, Y),

holds on M, where a, b, c are smooth functions and Ω(X, Y) = g(φX, Y). If c= 0 then the manifold reduces to anη-Einstein manifold.

In the present paper we have studied generalized η-Einstein manifold in a Lorentzian Para-Sasakian manifold. The paper is organized as follows: After Preliminaries in Section 3 we study some geometric properties of generalizedη- Einstein Lorentzian Para-Sasakian manifolds and conformally flat generalized η-Einstein Lorentzian Para-Sasakian manifolds. In the last section we construct an example of a generalized η-Einstein Lorentzian Para-Sasakian manifold.

MATH. REPORTS16(66),1(2014), 61–67

2.PRELIMINARIES

In [1], K. Matsumoto introduced the notion of Lorentzian Para-Sasakian (brieflyLP-Sasakian) manifold. In [2], the authors defined the same notion in- dependently and they obtained many results about this type manifold (see also [3] and [8]). Several authors have studied Lorentzian para-Sasakian manifolds such as [4, 7, 9] and many others.

LetMⁿbe ann-dimensional differentiable manifold equipped with a triple (φ, ξ, η),whereφis a (1,1)-tensor field,ξ is a vector field,η is a 1-form onMⁿ such that

(2.1) η(ξ) =−1,

(2.2) φ² =I+η⊗ξ,

which implies

(2.3) i) φξ = 0, ii) η(φ) = 0, iii) rank(φ) =n−1.

Then Mⁿ admits a Lorentzian metric g, such that (2.4) g(φX, φY) =g(X, Y) +η(X)η(Y),

and Mⁿ is said to admit a Lorentzian almost paracontact structure (φ,ξ,η,g).

In this case, we have

(2.5) g(X, ξ) =η(X), ∇_Xξ=φX,

Ω(X, Y) =g(X, φY) =g(φX, Y) = Ω(Y, X).

In (2.1) and (2.2) if we replaceξby−ξ, then the triple (φ,ξ,η) is an almost paracontact structure onMⁿdefined by Sato [5]. The Lorentzian metric given by (2.5) stands analogous to the almost paracontact Riemannian metric for any almost paracontact manifold (see [5, 6]).

A Lorentzian almost paracontact manifold Mⁿ equipped with the struc- ture (φ,ξ,η,g) is called Lorentzian paracontact manifold [1] if

Ω(X, Y) = 1

2((∇_Xη)Y + (∇_Yη)X).

A Lorentzian almost paracontact manifold Mⁿ equipped with the struc- ture (φ,ξ,η,g) is calledLorentzian para-Sasakian manifold (brieflyLP-Sasakian manifold) [1] if

(∇_Xφ)Y =g(φX, φY)ξ+η(Y)φ²X.

In aLP-Sasakian manifold the 1-formη is closed. Also in [1], it is proved that if an n-dimensional Lorentzian manifold (Mⁿ, g) admits a timelike unit vector field ξ such that the 1-form η associated to ξ is closed and satisfies

(∇_X∇_Yη)W =g(X, Y)η(W) +g(X, W)η(Y) + 2η(X)η(Y)η(W),

thenMⁿ admits anLP-Sasakian structure.

By definition the Weyl conformal curvature tensor C is given by [10]

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] (2.6)

+ τ

(n−1)(n−2)[g(Y, Z)X−g(X, Z)Y],

where R, S, Q and τ denotes the curvature tensor of type (1,3), Ricci tensor, Ricci operator and scalar curvature of M respectively.

For dimM > 3, if C = 0, then the manifold is called conformally flat manifold.

3. MAIN RESULTS

In this section, we give main results of the paper. At first we give the following:

Theorem 3.1. ξ is an eigenvector of the Ricci tensorS corresponding to the eigen value a−b.

Proof. In (1.2) puttingY =ξ, and using (2.1), (2.3) i) and Ω(X, ξ) = 0, we get

(3.1) S(X, ξ) = (a−b)η(X),

which means that ξ is an eigenvector of the Ricci tensor S corresponding to the eigen valuea−b.

Now, puttingX=ξin (3.1) and again using (2.1), we haveS(ξ, ξ) =b−a.

Now we can state the following:

Theorem 3.2. The scalar b−ais the Ricci curvature in the direction of the generator ξ.

Theorem 3.3. If the Ricci tensor S of type (0,2) of an LP-Sasakian manifold is non-vanishing and satisfies the relation

S(Y, Z)S(X, W)−S(X, Z)S(Y, W) = ρ[g(Y, Z)g(X, W)−g(X, Z)g(Y, W)]

+g(φX, W)g(Y, Z), (3.2)

where ρ is non-zero scalar, then the manifold is a generalized η-Einstein man- ifold.

Proof. PuttingY =Z =ξ in (3.2), we obtain

S(ξ, ξ)S(X, W)−S(X, ξ)S(ξ, W) = ρ[g(ξ, ξ)g(X, W)−g(X, ξ)g(ξ, W)]

+g(φX, W)g(ξ, ξ), which can be written as

(n−1)S(X, W)−(n−1)²η(X)η(W) = ρ[g(X, W)−η(X)η(W)]

−g(φX, W).

(3.3)

Now, (3.3) can be written as

S(X, W) =αg(X, W) +βη(X)η(W) +γΩ(X, W), whereα =−_n−1^ρ , β = (n−1)− _n−1^ρ , γ =−_n−1¹ .

Since S 6= 0, ρ is a non-zero scalar, it follows that α, β, γ are non-zero scalars.

This completes the proof.

Now, we consider conformally flat generalized η-Einstein LP-Sasakian manifolds of dim >3.

If a generalizedη-EinsteinLP-Sasakian manifold is conformally flat, then using (1.2) in (2.6), we get

R(X, Y)Z = [ 2α

n−2+ τ

(n−1)(n−2)][g(X, Z)Y −g(Y, Z)X]

+ β

n−2[g(X, Z)η(Y)ξ−g(Y, Z)η(X)ξ +η(X)η(Z)Y −η(Y)η(Z)X]

(3.4)

+ γ

n−2[g(X, Z)φY −g(Y, Z)φX + Ω(X, Z)Y −Ω(Y, Z)X].

Putting Y =ξ in (3.4), we have R(X, ξ)Z = [ 2α

n−2+ τ

(n−1)(n−2)]g(X, Z)ξ (3.5)

− β

n−2g(X, Z)ξ+ γ

n−2Ω(X, Z)ξ, for all X, Y, Z ∈ξ^⊥. Again puttingZ =ξ in (3.5), we have

(3.6) R(X, ξ)ξ= 0,

for all X, Y, Z ∈ξ^⊥. Now we give the following:

Theorem 3.4. In a conformally flat generalizedη-Einstein LP-Sasakian manifold, the curvature tensor of type (1,3) satisfies the following properties (3.4), (3.5) and (3.6 ) for allX, Y, Z ∈ξ^⊥, the(n−1)-dimensional distribution orthogonal to the distribution ξ.

From (3.6), we can state the following:

Corollary3.5. In a conformally flat generalizedη-EinsteinLP-Sasakian manifold the sectional curvature of the plane determined by X, ξ is zero.

4. EXAMPLE

We consider a 3-dimensional manifold M = {(x, y, z) ∈ R³}, where (x, y, z) are the standard coordinates of R³. Let {e₁, e2, e3} be linearly in- dependent global frame onM given by

e1 =e^z ∂

∂x, e2 =e^z−ax ∂

∂y, e3 = ∂

∂z, wherea is non-zero constant.

Let g be the Lorentzian metric defined by

g(e₁, e₃) =g(e₁, e₂) =g(e₂, e₃) = 0, g(e₁, e₁) =g(e₂, e₂) = 1, g(e₃, e₃) =−1.

Let η be the 1-form defined by

η(X) =g(X, e₃),

for any X∈χ(M) . Let φbe the (1,1) tensor field defined by φe1 =−e₁, φe2=−e₂, φe3 = 0.

Then, using the linearily of φand g, we have η(e3) =−1, φ²X=X+η(X)e3

and

g(φX, φY) =g(X, Y) +η(X)η(Y),

for any X, Y ∈ χ(M). Thus, for e3 = ξ, (φ, ξ, η, g) defines a Lorentzian paracontact structure on M.

Let∇be the Levi-Civita connection with respect to the Lorentzian metric g and R be the curvature tensor ofg. Then we have

[e₁, e₂] =−ae^ze₂, [e₁, e₃] =−e₁, [e₂, e₃] =−e₂.

Taking e3 =ξ and using Koszul formula for the Lorentzian metric g, we have

∇_e1e1 =−e₃, ∇_e1e2 = 0, ∇_e1e3=−e₁

∇_e2e1 =ae^ze2, ∇_e2e2=−ae^ze1−e3, ∇_e2e3=−e₂

∇_e3e₁ = 0, ∇_e3e₂= 0, ∇_e3e₃= 0.

From the above it can be easily seen that (φ, ξ, η, g) is an LP-Sasakian structure on M . So, M³(φ, ξ, η, g) is an LP-Sasakian manifold. Using the

above relations, we can easily calculate the non-vanishing components of the curvature tensor R as follows:

R(e₂, e₃)e₃ =−e₂, R(e₁, e₃)e₃ =−e₁, R(e₁, e₂)e₂ = (1−a²e^2z)e₁ R(e2, e3)e2 =−ae^ze1−e3, R(e1, e3)e1=−e₃, R(e1, e2)e1 =−(1−a²e^2z)e2

and the components which can be obtained from these by the symmetry prop- erties. From the above, we can easily calculate the non-vanishing components of the Ricci tensorS as follows:

S(e₁, e₁) =−a²e^2z, S(e₂, e₂) =−a²e^2z, S(e₃, e₃) =−2.

Since {e₁, e2, e3} forms a basis, any vector field X, Y ∈ χ(M) can be written as

X =a₁e₁+b₁e₂+c₁e₃ and

Y =a2e1+b2e2+c2e3, wherea1, a2, b1, b2, c1, c2∈R⁺.Hence,

g(X, Y) =a1a2+b1b2−c1c2

and

S(X, Y) =−

(a₁a₂+b₁b₂)a²e^2z+ 2c₁c₂ . Let Ω(X, Y) be a (0,2) tensor defined by

Ω(X, Y) =g(φX, Y) . Then

Ω(e1, e1) =g(φe1, e1) =g(−e₁, e1) =−1 Ω(e₂, e₂) =g(φe₂, e₂) =g(−e₂, e₂) =−1 and

Ω(e₃, e₃) =g(φe₃, e₃) =g(0, e₃) = 0.

We take the scalars a´,band f respectively as follows:

a´= 1−2a²e^2z, b=−(1 + 2a²e^2z),

f = 1−a²e^2z. Then, we have

S(e₁, e₁) =a´g(e₁, e₁) +bη(e₁)η(e₁) +f Ω(e₁, e₁), S(e2, e2) =a´g(e2, e2) +bη(e2)η(e2) +f Ω(e2, e2), S(e₃, e₃) =a´g(e₃, e₃) +bη(e₃)η(e₃) +f Ω(e₃, e₃).

Thus, the manifold under consideration is a generalized η-Einstein LP- Sasakian manifold.

REFERENCES

[1] K. Matsumoto, On Lorentzian paracontact manifolds. Bull. Yamagata Univ. Natur.

Sci. 12(2) (1989), 151–156.

[2] I. Mihai and R. Rosca,On Lorentzian P-Sasakian manifolds. Classical Analysis, World Scientific Publi., Singapore, (1992), 155–169.

[3] K. Matsumoto and I. Mihai,On a certain transformation in a Lorentzian para-Sasakian manifold. Tensor (N.S.)47(1988), 189–197.

[4] C. ¨Ozg¨ur,On φ-conformally flat Lorentzian para-Sasakian manifolds. Radovi Matem- aticki12(2003), 99–106.

[5] I. Sato, On a structure similar to almost contact structure. Tensor (N.S.) 30 (1976), 219–224.

[6] I. Sato,On a structure similar to almost contact structure II. Tensor (N.S.)31(1977), 199–205.

[7] A.A. Shaikh and K.K. Baishya, On φ-symmetric Lorentzian para-Sasakian manifolds.

Yokohama Math. J.52(2005), 97–112.

[8] M.M. Tripathi and U.C. De,Lorentzian almost paracontact manifolds and their subman- ifolds. J. Korea Soc. Math. Educ. Ser. B, Pure Appl. Math.8(2001), 101–105.

[9] A. Taleshian and N. Asghari, On LP-Sasakian manifolds satisfying certain conditions on the concircular curvature tensor. Differ. Geom. Dyn. Syst. 12(2010), 228–232.

[10] K. Yano and M. Kon,Structures on manifolds. Series in Pure Math., Vol. 3, World Sci, (1984).

Received 25 October 2011 Inonu University,

Education Faculty, Malatya 44280, Turkey

a.yildiz@inonu.edu.tr ayildiz44@yahoo.com Calcutta University, Department of Pure Mathematics,

35, Ballygaunge Circular Road, Kolkata 700019, West Bengal, India

uc de@yahoo.com Dumlupınar University, Department of Mathematics,

K¨utahya, Turkey erhan.ata@dumlupinar.edu.tr