CLASSIFICATION OF THREE-DIMENSIONAL CONFORMALLY FLAT QUASI-PARA-SASAKIAN MANIFOLDS

https://doi.org/10.5831/HMJ.2019.41.3.489

CLASSIFICATION OF THREE-DIMENSIONAL CONFORMALLY FLAT QUASI-PARA-SASAKIAN

MANIFOLDS

Irem Kupeli Erken

Abstract. The aim of this paper is to study three-dimensional con-formally flat quasi-para-Sasakian manifolds. First, the necessary and sufficient conditions are provided for three-dimensional quasi-para-Sasakian manifolds to be conformally flat. Next, a character-ization of three-dimensional conformally flat quasi-para-Sasakian manifold is given. Finally, a method for constructing examples of three-dimensional conformally flat quasi-para-Sasakian manifolds is presented.

1. Introduction

Almost paracontact geometry was first introduced and studied by Kaneyuki and Williams in [9] and then many other authors continued to study. Zamkovoy studied almost paracontact metric manifolds in [20]. Because of there are lots of studies on almost contact geometry, it seems there should be new studies about almost paracontact geometry. There-fore, paracontact metric manifolds have been studied in recent years by many authors, emphasizing similarities and differences with respect to the most well known contact case. Interesting papers connecting these fields are, for example, [6], [4], [18], [20], and references therein.

Z. Olszak studied normal almost contact metric manifolds of dimen-sion three [15]. He derived certain necessary and sufficient conditions for an almost contact metric structure on manifold to be normal. He found curvature properties of such structures and he considered normal

Received November 14, 2018. Accepted March 11, 2019. 2010 Mathematics Subject Classification. 53B30, 53D10, 53D15.

Key words and phrases. quasi-para-Sasakian manifold, conformally flat, constant curvature.

almost contact metric manifolds of constant curvature. Curvature and torsion of Frenet-Legendre curves in three-dimensional normal almost paracontact metric manifolds were investigated in [19] and then normal almost paracontact metric manifolds were studied in [1], [10], [11].

The notion of quasi-Sasakian manifolds, introduced by D. E. Blair in [2], unifies Sasakian and cosymplectic manifolds. A quasi-Sasakian manifold is a normal almost contact metric manifold whose fundamental 2-form Φ := g(·, φ·) is closed. Quasi-Sasakian manifolds can be viewed as an odd-dimensional counterpart of Kaehler structures. These manifolds studied by several authors (e.g. [8], [14], [16], [17]).

Quasi-Sasakian manifolds were studied by many different authors and are considered a well-established topic in contact Riemannian geometry. But to the author’s knowledge, there do not exist any comprehensive study about quasi-para-Sasakian manifolds.

Motivated by these considerations, in [13], the author makes the first contribution to investigate basic properties and general curvature iden-tities of quasi-para-Sasakian manifolds.

In this paper, we study three-dimensional conformally flat quasi-para-Sasakian manifolds.

Section 2 is preliminary section, where we recall the definition of almost paracontact metric manifold and quasi-para-Sasakian manifolds. In Section 3, we mainly proved that for a three-dimensional quasi-para-Sasakian manifold M , the followings are equivalent.

i) M is locally symmetric.

ii) M is conformally flat and its scalar curvature τ is const., iii) M is conformally flat and β is const.,

iv)• If β = 0, then M is a paracosymplectic manifold which is locally a product of the real line R and a 2-dimensional para-Kaehlerian manifold or

•If β 6= 0, then M is of constant negative curvature and the quasi-para-Sasakian structure can be obtained by a homothetic deformation of a para-Sasakian structure.

Finally, we gave a theorem which gives a method for constructing ex-amples of three-dimensional conformally flat quasi-para-Sasakian mani-folds.

2. Preliminaries

A (2n + 1)-dimensional differentiable manifold M has an almost para-contact structure (φ, ξ, η) if it admits a (1, 1) tensor field φ, a vector field ξ and a one-form η satisfying followings

(i) φ2 _{= Id − η ⊗ ξ, η(ξ) = 1,} (ii) distribution

D : p ∈ M → Dp⊂ TpM : Dp = Kerη = {X ∈ TpM : η(X) = 0} is called paracontact distribution generated by η.

The manifold M is said to be an almost paracontact manifold if it is endowed with an almost paracontact structure [20].

If an almost paracontact manifold admits a pseudo-Riemannian met-ric g of a signature (n + 1, n), i.e.

(1) g(φX, φY ) = −g(X, Y ) + η(X)η(Y ),

then the manifold will be called an almost paracontact metric manifold and g is compatible.

For such manifold, we have

(2) η(X) = g(X, ξ), φ(ξ) = 0, η ◦ φ = 0.

Moreover, we can define a skew-symmetric tensor field (a 2-form) Φ by

(3) Φ(X, Y ) = g(X, φY ),

usually called fundamental form.

For an almost paracontact manifold, there exists an orthogonal basis {X₁, . . . , Xn, Y1, . . . , Yn, ξ} such that g(Xi, Xj) = δij, g(Yi, Yj) = −δij and Yi= φXi, for any i, j ∈ {1, . . . , n}. Such basis is called a φ-basis.

On an almost paracontact manifold, one defines the (1, 2)-tensor field N(1) by

(4) N(1)(X, Y ) = [φ, φ] (X, Y ) − 2dη(X, Y )ξ, where [φ, φ] is the Nijenhuis torsion of φ

[φ, φ] (X, Y ) = φ2[X, Y ] + [φX, φY ] − φ [φX, Y ] − φ [X, φY ] . The almost paracontact manifold (structure) is said to be normal when N(1) = 0 [20]. The normality condition implies that the almost paracomplex structure J is integrable which is defined by

J (X, λd

dt) = (φX + λξ, η(X) d dt),

on M × R.

If dη(X, Y ) = g(X, φY ), then η is a paracontact form and the almost paracontact metric manifold (M, φ, ξ, η, g) is said to be paracontact met-ric manifold. In a paracontact metmet-ric manifold one defines a symmetmet-ric, trace-free operator h = 1₂L_ξφ, where Lξ, denotes the Lie derivative. In [20], it is proved that the operator h satisfies the followings: hξ = 0, trh =trhφ = 0 and ∇ξ = −φ + φh, where ∇ is the Levi-Civita connec-tion of the pseudo-Riemannian manifold (M, g). Also h anti-commutes with φ .

Moreover h = 0 if and only if ξ is Killing vector field. In this case (M, φ, ξ, η, g) is said to be a K-paracontact manifold. Similarly as in the class of almost contact metric manifolds [3], a normal almost paracon-tact metric manifold will be called para-Sasakian if Φ = dη [7]. The para-Sasakian condition implies the K-paracontact condition and the converse holds only in dimension three. A paracontact metric manifold will be called paracosymplectic if dΦ = 0, dη = 0 [6].

Now, we will give some results about three-dimensional quasi-para-Sasakian manifolds that we will use next sections.

Proposition 2.1. [19] For a three-dimensional almost paracontact metric manifold M the following three conditions are mutually equivalent

(a) M is normal,

(b) there exist functions α, β on M such that

(5) (∇Xφ)Y = β(g(X, Y )ξ − η(Y )X) + α(g(φX, Y )ξ − η(Y )φX), (c) there exist functions α, β on M such that

(6) ∇Xξ = α(X − η(X)ξ) + βφX.

Corollary 2.2. [10] For a normal almost paracontact metric struc-ture (φ, ξ, η, g) on M , we have ∇ξξ = 0 and dη = −βΦ. The functions α, β realizing ( 5) as well as (6) are given by [19]

(7) 2α = Trace {X −→ ∇Xξ} , 2β = Trace {X −→ φ∇Xξ} . Proposition 2.3. [19] For a three-dimensional almost paracontact metric manifold M , the following three conditions are mutually equiva-lent

(a) M is quasi-para-Sasakian,

(b) there exists a function β on M such that

(c) there exists a function β on M such that

(9) ∇Xξ = βφX.

A three-dimensional normal almost paracontact metric manifold is • quasi-para-Sasakian if and only if α = 0 and β is certain function [7], [19], in particular, para-Sasakian if β = −1 [19], [20],

• paracosymplectic if α = β = 0 [6],

• α-para-Kenmotsu if α 6= 0 and α is constant and β = 0 [12]. Namely, the class of para-Sasakian and paracosymplectic manifolds are contained in the class of quasi-para-Sasakian manifolds.

Theorem 2.4. [10]Let (M, φ, ξ, η, g) be a three-dimensional normal almost paracontact metric manifold. Then the following curvature iden-tities hold R(X, Y )Z = (2(ξ(α) + α2+ β2) +1 2τ )(g(Y, Z)X − g(X, Z)Y ) −(ξ(α) + 3(α2+ β2) +1 2τ )((g(Y, Z)η(X)ξ − g(X, Z)η(Y )ξ +η(Y )η(Z)X − η(X)η(Z)Y )+(φZ(β) − Z(α))(η(Y )X − η(X)Y ) +(φY (β) − Y (α))(η(Z)X − g(X, Z)ξ)

(10)

− (φX(β) − X(α)) (η(Z)Y − g(Y, Z)ξ)

+(φgradβ + gradα)(η(Y )g(X, Z) − η(X)g(Y, Z)).

S(Y, Z) = −(ξ(α) + α2_{+ β}2₊ 1

2τ )g(φY, φZ) (11)

+η(Z) (φY (β) − Y (α))

+η(Y ) (φZ(β) − Z(α)) − 2(α2+ β2)η(Y )η(Z), where R, S and τ are resp. Riemannian curvature, Ricci tensor and scalar curvature of M .

Theorem 2.5. Let (M, φ, ξ, η, g) be a three-dimensional quasi-para-Sasakian manifold. Then the following curvature identities hold

R(X, Y )Z = (2β2+1

2τ )(g(Y, Z)X − g(X, Z)Y ) −(3β2+1

2τ )((g(Y, Z)η(X)ξ − g(X, Z)η(Y )ξ

+η(Y )η(Z)X − η(X)η(Z)Y ) + φZ(β)(η(Y )X − η(X)Y ) +φY (β)(η(Z)X − g(X, Z)ξ) −φX(β)(η(Z)Y − g(Y, Z)ξ) +(φgradβ)(η(Y )g(X, Z) − η(X)g(Y, Z)). (12) S(Y, Z) = (β2+1 2τ )g(Y, Z) − (3β 2₊1 2τ )η(Y )η(Z) +η(Y )φZ(β) + η(Z)φY (β). (13)

where R, S and τ are resp. Riemannian curvature, Ricci tensor and scalar curvature of M .

Remark 2.6. In the proof of Theorem 2.4, the author showed that ξ(β) + 2αβ = 0. Namely, for three-dimensional quasi-para-Sasakian manifolds,

(14) ξ(β) = 0.

Theorem 2.7. [13]Let (M2n+1, φ, ξ, η, g) be a quasi-para-Sasakian manifold of constant curvature K. Then K ≤ 0. Furthermore,

•If K = 0, the manifold is paracosymplectic,

•If K < 0, the structure (φ, ξ, η, g) is obtained by a homothetic deformation of a para-Sasakian structure on M2n+1.

3. Three-dimensional conformally flat quasi-Para-Sasakian manifolds

For the conformal flatness of three dimensional semi-Riemannian manifold, we will use linear (1, 1)-tensor field (Weyl-Schouten tensor) L which is defined by

(15) L = Q − τ

4Id, where S(X, Y ) = g(QX, Y )[5].

Lemma 3.1. The linear operator L of a three-dimensional quasi-para-Sasakian manifold is given by

(16) LY = τ 4 + β 2_{Y −}_3β2₊ τ 2 

η(Y )ξ − η(Y )φgradβ + dβ(φY )ξ. Proof. By (13), we obtain (17) QY = τ 2 + β 2_{Y −}_3β2₊τ 2 

η(Y )ξ − η(Y )φgradβ + dβ(φY )ξ. The requested equation comes from combining (15) and the above last equation. From (16), we have (∇XL)Y = ∇XLY − L∇XY =  dτ (X) 4 + 2βdβ(X)  Y −  6βdβ(X) +dτ (X) 2  η(Y )ξ −3β2+τ 2  ((∇Xη)(Y )ξ + η(Y )∇Xξ)

−(∇Xη)(Y )φgradβ −η(Y )(∇Xφ)gradβ −η(Y )φ∇Xgradβ +(∇Xdβ)(φY )ξ + dβ((∇Xφ)Y )ξ + dβ(φY )∇Xξ.

If we use (8), (9) and (14) in the last equation, we can state following: Lemma 3.2. For a three-dimensional quasi-para-Sasakian manifold, the following formula is valid for the covariant derivative of the linear operator L (∇XL)Y =  dτ (X) 4 + 2βdβ(X)  Y −  6βdβ(X) + dτ (X) 2  η(Y )ξ −β3β2+τ 2  (g(φX, Y )ξ +η(Y )φX)−βg(φX, Y )φgradβ −βdβ(X)η(Y )ξ − η(Y )φ∇_Xgradβ + (∇Xdβ)(φY )ξ −βη(Y )dβ(X)ξ + βdβ(φY )φX.

(18)

Lemma 3.3. For the the function β of three-dimensional quasi-para-Sasakian manifold, the following equality holds

(19) ∇ξgradβ = βφgradβ.

Proof. By virtue of (14), we have (20)

[ξ, X](β) = ξ(X(β)) − X(ξ(β)) = g(∇ξgradβ, X) + g(gradβ, ∇ξX). By (9), we get

The proof comes from (20) and (21).

There exists a local orthonormal φ-basis {e1= φe2, e2= φe1, e3 = ξ}, such that g(e1, e1) = −g(e2, e2) = g(e3, e3) = 1, for any point p ∈ U ⊂ M .

For the sake of shortness, we will give followings τi = dτ (ei), βi = dβ(ei) βij = (∇eidβ)(ej), Lij = (∇eiL)ej gradβ = β1e1− β2e2+ β3e3, ∇_eigradβ = βi1e1− βi2e2+ βi3e3

for 1 ≤ i, j ≤ 3, where τi, βi, βij are the functions and Lij are the vector fields on U . Also we can write,

(∇eidβ)(ej)−(∇ejdβ)(ei)= ∇eidβ(ej) − dβ∇eiej− ∇ejdβ(ei) + dβ∇ejei

= ei(ej(β))−(∇eiej)(β)−ej(ei(β))+(∇ejei)(β)

= [ei, ej] β −∇eiej− ∇ejei β

= 0. (22)

From (22) we have βij = βji. Moreover, we obtain (∇eidβ)(ej) = ∇eidβ(ej) − dβ(∇eiej) = ∇eihgradβ, eji − hgradβ, ∇eieji = h∇eigradβ, eji (23) and (24) (∇ejdβ)(ei) =∇ejgradβ, ei . Namely (25) h∇eigradβ, eji =∇ejgradβ, ei .

We will use the well known formula for semi-Riemannian manifolds trace {Y → (∇YQ)X} =

1 2∇Xτ.

If we put X = ξ in the above formula and use (17) and (9), we have (26) (∇YQ)ξ = −5βY (β)ξ + (−3β3− β

τ

Using (26), we get 1 2ξ(τ ) = 3 X i=1 εig((∇eiQ)ξ, ei)

= −g(∇e1φgradβ, e1) + g(∇e2φgradβ, e2) − g(∇e3φgradβ, ξ)

where 1 ≤ i ≤ 3.

By the help of (5), (14) and (25) we find that

(27) ξ(τ ) = τ3 = 0.

From (14) and (27), we obtain

(28) β3 = 0.

(19) implies that

(29) β13= β31= −ββ2, β23= β32= −ββ1, β33= 0.

Lemma 3.4. For a three-dimensional quasi-para-Sasakian manifold, the following is valid

Lij− Lji = 0 for 1 ≤ i, j ≤ 3 ⇔ τ1 = −20ββ1, τ2 = −20ββ2, β12= β21= 0, (30) β22 = −β11= β  3β2+ τ 2  .

Proof. By direct computations, using (18), (19), (28) and (29), we derive L12− L21 = − τ₂ 4 + 5ββ2  e1+ τ₁ 4 + 5ββ1  e2 +(β11− β22+ β(τ + 6β2))ξ. L13− L31 = β12e1+  −β₁₁− β3β2+ τ 2  e2 +  −τ1 4 − 5ββ1  ξ. L23− L32 =  β22− β  3β2+τ 2  e1− β12e2 +  −τ2 4 − 5ββ2  ξ. (31)

The proof follows from (31).

We know that a semi-Riemannian manifold is conformally flat⇔ (∇XL)Y − (∇YL)X = 0, for any vector fields X and Y. Hence, we can say that a three-dimensional quasi-para-Sasakian manifold is con-formally flat if and only if (31) holds. By (31), we can give following result.

Theorem 3.5. A three-dimensional quasi-para-Sasakian manifold is conformally flat if and only if the function β satisfies the followings

τ + 10β2 = const., (∇Xdβ)(Y ) = −β  3β2+τ 2  (g(X, Y ) − η(X)η(Y )) −βη(X)dβ(φY ) − βη(Y )dβ(φX). (32)

Theorem 3.6. For a three-dimensional quasi-para-Sasakian mani-fold M , the following assertions are equivalent to each other:

i) M is locally symmetric.

ii) M is conformally flat and its scalar curvature τ is const., iii) M is conformally flat and β is const.,

iv)• If β = 0, then M is a paracosymplectic manifold which is lo-cally a product of the real line R and a 2-dimensional para-Kaehlerian manifold or

• If β 6= 0, then M is of constant negative curvature and the quasi-para-Sasakian structure can be obtained by a homothetic deformation of a para-Sasakian structure.

Proof. First of all, (i) implies (ii) because of the dimM = 3. From (32), one can see (ii) ⇔ (iii). Now, we will show (iii) implies (iv). Using (32), we get β 3β2+τ₂

= 0 and τ is const. Now there are two possibilities. If β = 0, then M is a paracosymplectic manifold which is locally a product of the real line R and a 2-dimensional para-Kaehlerian manifold [6]. If β 6= 0, then τ = −6β2, namely M has constant negative curvature. By using τ = −6β2 in (13), we get M is Einstein since S = τ₃g. Using Theorem 2.7, one can say that the quasi-para-Sasakian structure can be obtained by a homothetic deformation of a para-Sasakian structure. One can easily deduce that (iv) ⇒ (i).

Theorem 3.7. [21](a) The classes of the 3-dimensional normal al-most paracontact metric manifolds are G5, G6 and G5⊕ G6;

(b) The classes of the 3-dimensional paracontact metric manifolds are ¯

G5 and ¯G5⊕ G10;

(c) The class of the 3-dimensional para-Sasakian manifolds is ¯G5 ; (d) The class of the 3-dimensional K-paracontact metric manifolds is ¯

G5;

(e) The class of the 3-dimensional quasi-para-Sasakian manifolds is G5 .

Let L be a three-dimensional real connected Lie group and g be its Lie algebra with a basis {E1, E2, E3} of left invariant vector fields. An

almost paracontact structure (φ, ξ, η) and a pseudo-Riemannian metric g defined by

φE1 = E2, φE2 = E1, φE3= 0,

ξ = E3, η(E3) = 1, η(E1) = η(E2) = 0, g(E1, E1) = g(E3, E3) = −g(E2, E2) = 1,

g(Ei, Ej) = 0, i 6= j ∈ {1, 2, 3} . (33)

Then (L, φ, ξ, η, g) is a three-dimensional almost paracontact metric man-ifold. Because of the metric g is left invariant, one can write Koszul equality by following

(34) 2g(∇xy, z) = g([x, y] , z) + g([z, x] , y) + g([z, y] , x), where ∇ is the Levi-Civita connection of g.

Let the commutators of g be defined by [Ei, Ej] = CijkEk, where the structure constants C_ijk are real numbers and C_ijk = −C_jik.

Theorem 3.8. [21]The manifold (L, φ, ξ, η, g) belongs to the class Gi(i ∈ {5, 6, 10, 12}) if and only if the corresponding Lie algebra g is determined by the following commutators:

G5 : [E1, E2] = C121 E1+ C122 E2+ C123 E3, (35) : [E1, E3] = C132 E2, [E2, E3] = C132 E1 : C123 6= 0, C121 C132 = 0, C122 C132 = 0; G6 : [E1, E2] = C121 E1+ C122 E2, (36) : [E1, E3] = C131 E1+ C132 E2, [E2, E3] = C132 E1+ C131 E2: −2C131 6= 0, C₁₂2 C₁₃2 − C₁₃1 C₁₂1 = 0, C₁₂1 C₁₃2 − C₁₃1 C₁₂2 = 0; G10 : [E1, E2] = C121 E1+ C122 E2, (37) : [E1, E3] = C131 E1+ C132 E2, [E2, E3] = C231 E1− C131 E2: C132 6= C231 or C131 6= 0, C₁₂1 C₁₃1 + C₁₂2 C₂₃1 = 0, C₁₂1 C₁₃2 − C₁₂2 C₁₃1 = 0; G12 : [E1, E2] = C121 E1+ C122 E2, (38) : [E1, E3] = C132 E2+ C133 E3, [E2, E3] = C132 E1+ C233 E3: C133 6= 0 or C233 6= 0, (C₁₂1 − C3 23)C132 = 0, (C122 + C133 )C132 = 0, (C₁₂1 − C₂₃3 )C₁₃3 + (C₁₂2 + C₁₃3 )C₂₃3 = 0

Theorem 3.9. A three-dimensional quasi-para-Sasakian manifold is conformally flat if and only if the corresponding Lie algebra g is deter-mined by the following commutator G5

(39) G5: [E1, E2] = C123 E3, [E1, E3] = C123 E2, [E2, E3] = C123 E1. Proof. Assume that the three-dimensional quasi-para-Sasakian man-ifold is conformally flat. Using (33), (34) and (36) we have

∇_E1E1 = C121 E2, ∇E2E1 = −C 2 12E2− 1 2C 3 12E3, ∇_E3E1 = (−C132 + 1 2C 3 12)E2, ∇_E1E2 = C121 E1+ 1 2C 3 12E3, ∇E2E2 = −C 1 12E1, ∇E3E2 = (−C 2 13+ 1 2C 3 12)E1, ∇E1E3 = 1 2C 3 12E2, ∇E2E3 = 1 2C 3 12E1, ∇E3E3 = 0.

From (9), we get β = 1₂C₁₂3 is a constant function. Using the above covariant derivatives, we obtain

R(E1, E2)E3 = ( 1 2C 3 12C121 − 1 2C 2 12C123 )E1+ ( 1 2C 3 12C121 − 1 2C 1 12C123 )E2, R(E1, E2)E2 = (− 3 4(C 3 12)2− (C121 )2+ C122 C121 + C123 C132 )E1 +(−(C₁₂1 )2+ C₁₂1 C₁₂2 )E2, R(E1, E2)E1 = (−C122 C121 + (C121 )2)E1 +(−3 4(C 3 12)2− (C121 )2+ (C122 )2+ C123 C132 )E2, R(E2, E3)E3 = (− 1 4(C 3 12)2)E2, R(E2, E3)E2 = (−C132 C121 )E1 +(C₁₃2 C₁₂2 −1 2C 3 12C122 − C121 C132 + 1 2C 1 12C123 )E2 +  −1 4(C 3 12)2E3  , R(E2, E3)E1 = C132 (C121 − C122 )E1+ 1 2C 3 12(C122 − C121 )E1− C132 C121 E2,

R(E1, E3)E3 = − 1 4(C 3 12)2E1, R(E1, E3)E2 = C132 C121 E1, R(E1, E3)E1 = (C132 C122 )E2+ 1 4(C 3 12)2E3.

Using above equations, we have constant scalar curvature as follows, S(E1, E1) = 1 2(C 3 12)2+ (C121 )2− C122 C121 − C123 C132 , (40) S(E2, E2) = − 1 2(C 3 12)2− (C121 )2+ (C122 )2+ C123 C132 , (41) S(E3, E3) = − 1 2(C 3 12)2, (42)

τ = S(E1, E1) − S(E2, E2) + S(E3, E3), (43) τ = 1 2(C 3 12)2+ 2(C121 )2− (C122 )2− 2C123 C132 − C122 C121 . (44)

Using the fact that our manifold is conformally flat, if we use Theorem 3.5 in (13) and by the equations (40), (41), we have

−1 2(C 3 12)2 = 1 2(C 3 12)2+ (C121 )2− C122 C121 − C123 C132 . (45) 1 2(C 3 12)2 = − 1 2(C 3 12)2− (C121 )2+ (C122 )2+ C123 C132 . (46)

If we sum (45) and (46), we have C₁₂2 (C₁₂2 − C1

12) = 0.

By virtue of the last equation, following cases occurs.

Case I: Accept C₁₂2 = 0. If we subtract (45) from (46) and use C₁₂2 = 0, we obtain

(47) 2(C₁₂3 )2= −2(C₁₂1 )2+ 2C₁₂3 C₁₃2 . Taking into account τ = −6β2 = −3₂(C₁₂3 )2 in (44), we get

(48) C₁₂3 (C₁₃2 − C3

12) = (C121 )2.

If we act C₁₃2 on both sides of the last equation and using the fact that C₁₂3 6= 0 and C1

12C132 = 0 in G5, we obtain C₁₃2 = 0 or C₁₃2 = C₁₂3 .

If we take C₁₃2 = 0, by (48) we get −(C₁₂3 )2 = (C₁₂1 )2. So C₁₃2 shold be different from zero. If we take C₁₃2 = C₁₂3 in (47), we get C₁₂1 = 0.

Case II: Assume C₁₂2 = C₁₂1 . By virtue of (45), we get (49) C₁₂3 (C₁₃2 − C₁₂3 ) = 0.

From (49), there are two possibilities. The first one is C₁₂3 = 0. But this contradicts with the C₁₂3 6= 0 in G₅. So the second one is C₁₃2 = C₁₂3 . If we use C₁₃2 = C₁₂3 in the fact that C₁₂3 6= 0, C1

12C132 = 0, C122 C132 = 0 in G5, we obtain C121 = C122 = 0.

Namely, by Case I and Case II, we get C₁₂1 = C₁₂2 = 0, C₁₃2 = C₁₂3 which gives us (39). The proof of converse side is obvious.

Remark 3.10. Using Theorem 3.9, one can construct several ex-amples of three-dimensional conformally flat quasi-para-Sasakian mani-folds. For example,

•For β = 2, using the commutators [E1, E2] = 4E3, [E1, E3] = 4E2, [E2, E3] = 4E1, one can get a 3-dimensional conformally flat proper quasi-Sasakian manifold with τ = −24 which is neither the para-cosymplectic manifold nor the para-Sasakian manifold.

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Irem Kupeli Erken

Department of Mathematics,

Faculty of Engineering and Natural Sciences, Bursa Technical University,

16330, Bursa-Turkey.