# Some Classes of Pseudosymmetric Contact Metric 3-Manifolds

Authors: Evaggelia Moutafi
Journal: ISRN Geometry
 (2012)

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## Abstract

We study (a) the class of 3-dimensional pseudosymmetric contact metric manifolds with harmonic curvature andTrl constant along the direction of ξ and (b) the class of (κ,μ,ν)-contact metric pseudosymmetric 3-manifolds of type constant in the direction of ξ.


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## Body

## 1. Introduction

A Riemannian manifold(Mm,g) is said to be pseudosymmetric according to Deszcz [1] if its curvature tensor R satisfies the condition R(X,Y)·R=L{(X∧Y)·R}, where the dot means that R(X,Y) acts as a derivation on R, L is a smooth function and the endomorphism field X∧Y is defined by(1.1)(X∧Y)Z=g(Y,Z)X-g(Z,X)Y,
for all vectors fields X,Y,Z on M and it similarly acts as a derivation on R.IfL is constant, M is called a pseudosymmetric manifold of constant type and if particularly L=0 then M is called a semisymmetric manifold first studied by E. Cartan. Semisymmetric spaces [2, 3] are a generalization of locally symmetric spaces (∇R=0, [4]) while pseudosymmetric spaces are a natural generalization of semisymmetric spaces. There are many details and examples on pseudosymmetric manifolds in [1, 5]. We remark that in dimension three, the pseudosymmetry is equivalent to the condition: the eigenvalues ρ1,ρ2,ρ3 of the Ricci tensor satisfy ρ1=ρ2 (up to numeration) and the last one is constant [6, 7].Kowalski and Sekizawa have studied [7–10] 3-dimensional pseudosymmetric spaces of constant type. Hashimoto and Sekizawa classified 3-dimensional conformally flat pseudosymmetric spaces of constant type [11] and finally Calvaruso [12] gave the complete classification of conformally flat pseudosymmetric spaces of constant type for dimensions >2. Cho and Inoguchi [13] studied pseudosymmetric contact homogeneous 3-manifolds while Cho et al. [14] give the conditions so as 3-dimensional trans-Sasakians, quasi-Sasakians, non-Sasakian generalized (κ,μ)-spaces to be pseudosymmetric. Belkhelfa et al. [15] studied pseudosymmetric Sasakian space forms of any dimension. Finally Gouli-Andreou and Moutafi in [16, 17] have  studied some classes of pseudosymmetric contact metric 3-manifolds.The aim of this paper is the study of the 3-dimension pseudosymmetric contact metric manifolds. The paper is organized in the following way: in Section2 we will give some preliminaries on pseudosymmetric manifolds and contact manifolds as well and in the next sections we will study 3-dimensional manifolds which satisfy one of the following conditions.(i)
M is a pseudosymmetric contact metric manifold with harmonic curvature and Trl constant along the direction of ξ.(ii)
M is a (κ,μ,ν)-contact metric pseudosymmetric manifold of type constant in the direction of ξ.


## 2. Preliminaries

Let(Mm,g),m≥3 be a connected Riemannian smooth manifold. We denote by R its Riemannian curvature tensor given by the equation R(X,Y)Z=[∇X,∇Y]Z-∇[X,Y]Z for any X,Y,Z∈𝔛(M) and where ∇ is the Levi-Civita connection of Mm.Definition 2.1.
A Riemannian manifold(Mm,g),m≥3,is called pseudosymmetric in the sense of Deszcz [1] if at every point of M the curvature tensor satisfies the condition:(2.1)(R(X,Y)⋅R)(X1,X2,X3)=L{((X∧Y)⋅R)(X1,X2,X3)}
or more explixitly:(2.2)R(X,Y)(R(X1,X2)X3)-R(R(X,Y)X1,X2)X3-R(X1,R(X,Y)X2)X3-R(X1,X2)(R(X,Y)X3)=L{(X∧Y)(R(X1,X2)X3)-R((X∧Y)X1,X2)X3-R(X1,(X∧Y)X2)X3-R(X1,X2)((X∧Y)X3)},
for any X,Y,X1,X2,X3∈𝔛(M), X∧Y is given by (1.1) and L is a smooth  function.Definition 2.2.
A differentiable manifoldM2n+1 endowed with a global 1-form η such that η∧(dη)n≠0 everywhere on M is called a contact manifold.Given a contact manifold(M,η), there is an underlying contact metric structure (η,ξ,ϕ,g) where g is a Riemannian metric (the associated metric), ϕ a global tensor of type (1,1), and ξ a unique global vector field (the characteristic or Reeb vector field). A differentiable (2n+1)-dimensional manifold endowed with a contact metric structure (η,ξ,ϕ,g) is called a contact metric (Riemannian) manifold denoted by M(η,ξ,ϕ,g). The structure tensors η, ξ, ϕ, and g satisfy the equations:(2.3)ϕ2=-I+η⊗ξ,η(X)=g(X,ξ),η(ξ)=1,dη(X,Y)=g(X,ϕY),g(ϕX,ϕY)=g(X,Y)-η(X)η(Y).
The associated metrics can be constructed polarizing dη on the contact subbundle D defined by η=0. Denoting by L the Lie differentiation and R the curvature tensor, respectively, we define the tensor fields h, l, and τ by(2.4)hX=12(Lξϕ)X,lX=R(X,ξ)ξ,τ(X,Y)=(Lξg)(X,Y).
These tensors also satisfy the following formulas:(2.5)ϕξ=hξ=lξ=0,η∘ϕ=η∘h=0,dη(ξ,X)=0,(2.6)Trh=Trhϕ=0,∇Xξ=-ϕX-ϕhX,hϕ=-ϕh,(2.7)hX=λX⟹hϕX=-λϕX,(2.8)∇ξh=ϕ-ϕl-ϕh2,ϕlϕ-l=2(ϕ2+h2),(2.9)∇ξϕ=0,Trl=g(Qξ,ξ)=2n-Trh2.h=0 (or equivalently τ=0) if and only if ξ is Killing and M is called K-contact. A contact structure on M implies an almost complex structure on the product manifold M2n+1×ℝ. If this structure is integrable, then the contact metric manifold is said to be Sasakian. A K-contact structure is Sasakian only in dimension 3, and this fails in higher dimensions. More details on contact manifolds we can find in [18, 19].Let(M,ϕ,ξ,η,g) be a 3-dimensional contact metric manifold and U the open subset of points p∈M where h≠0 in a neighborhood of p and U0 the open subset of points p∈M such that h=0 in a neighborhood of p. Because h is a smooth function on M then U∪U0 is an open and dense subset of M so if a property is satisfied in U0∪U then this property will be satisfied in M. For any point p∈U∪U0 there exists a local orthonormal basis {e,ϕe,ξ} of smooth eigenvectors of h in a neighborhood of p (a ϕ-basis). On U, we put he=λe, where λ is a non vanishing smooth function which is supposed positive. From (2.7), we have hϕe=-λϕe. We recall the following.Lemma 2.3 (see [20]).
OnU, one has
(2.10)∇ξe=aϕe,∇ee=bϕe,∇ϕee=-cϕe+(λ-1)ξ,∇ξϕe=-ae,∇eϕe=-be+(1+λ)ξ,∇ϕeϕe=ce,∇ξξ=0,∇eξ=-(1+λ)ϕe,∇ϕeξ=(1-λ)e,
where a is a smooth function and
(2.11)b=12λ[(ϕe⋅λ)+A]withA=S(ξ,e),c=12λ[(e⋅λ)+B]withB=S(ξ,ϕe).From Lemma2.3 and the formula [X,Y]=∇XY-∇YX we can prove that(2.12)[e,ϕe]=∇eϕe-∇ϕee=-be+cϕe+2ξ,[e,ξ]=∇eξ-∇ξe=-(a+λ+1)ϕe,[ϕe,ξ]=∇ϕeξ-∇ξϕe=(a-λ+1)e,
and from (1.1) we estimate(2.13)(e∧ϕe)e=-ϕe,(e∧ξ)e=-ξ,(ϕe∧ξ)ξ=ϕe,(e∧ϕe)ϕe=e,(e∧ξ)ξ=e,(ϕe∧ξ)ϕe=-ξ,
while (X∧Y)Z=0, whenever X≠Y≠Z≠X and X,Y,Z∈{e,ϕe,ξ}.By direct computations we calculate the non vanishing independent components of the Riemannian curvature tensor fieldR (1,3):(2.14)R(ξ,e)ξ=-Ie-Zϕe,R(e,ϕe)e=-Cϕe-Bξ,R(ξ,ϕe)ξ=-Ze-Dϕe,R(ξ,e)ϕe=-Ke+Zξ,R(e,ϕe)ξ=Be-Aϕe,R(ξ,ϕe)ϕe=He+Dξ,R(ξ,e)e=Kϕe+Iξ,R(e,ϕe)ϕe=Ce+Aξ,R(ξ,ϕe)e=-Hϕe+Zξ,
where(2.15)C=-b2-c2+λ2-1+2a+(e⋅c)+(ϕe⋅b),H=b(λ-a-1)+(ξ⋅c)+(ϕe⋅a),K=c(λ+a+1)+(ξ⋅b)-(e⋅a),I=-2aλ-λ2+1,D=2aλ-λ2+1,Z=ξ⋅λ.
Setting X=e, Y=ϕe, Z=ξ in the Jacobi identity [[X,Y],Z]+[[Y,Z],X]+[[Z,X],Y]=0 and using (2.12), we get(2.16)b(a+λ+1)-(ξ⋅c)-(ϕe⋅λ)-(ϕe⋅a)=0,c(a-λ+1)+(ξ⋅b)+(e⋅λ)-(e⋅a)=0
or equivalently: A=H and B=K.We give the components of the Ricci operatorQ with respect to a ϕ-basis:(2.17)Qe=(r2-1+λ2-2aλ)e+Zϕe+Aξ,Qϕe=Ze+(r2-1+λ2+2aλ)ϕe+Bξ,Qξ=Ae+Bϕe+2(1-λ2)ξ,
where(2.18)r=TrQ=2[1-λ2-b2-c2+2a+(e⋅c)+(ϕe⋅b)]
is the scalar curvature. The relations (2.15) and (2.18) yield(2.19)C=-b2-c2+λ2-1+2a+(e⋅c)+(ϕe⋅b)=2λ2-2+r2,
and the relation (2.9):(2.20)Trl=2(1-λ2).Remark 2.4.
IfM3=U0 (see [21]), Lemma 2.3 is expressed in a similar form with λ=0, e is a unit vector field belonging to the contact distribution and for the functions A,B,D,H,I,K and Z we have: A=B=Z=H=K=0, I=D=1 and C=r/2-2.Definition 2.5.
AnMm Riemannian manifold has harmonic curvature if the Ricci operator Q satisfies the condition:(2.21)(∇XQ)Y=(∇YQ)X,∀X,Y∈X(M).
From now on we shall work on a (M3,ϕ,ξ,η,g) contact metric 3-manifold concerning a ϕ-basis {e,ϕe,ξ} at any point p∈M. First from the equation (∇XQ)Y=∇X(QY)-Q(∇XY), Lemma 2.3 and the relations (2.17), we get the equations:(2.22)(∇eQ)ϕe=[(e⋅ξ⋅λ)-2bλ(2a+λ+1)+(1+λ)(ϕe⋅λ)]e+[(e⋅r)2+2b(ξ⋅λ)+2λ(e⋅a)+(4λ+2a+2)(e⋅λ)-4cλ(1+λ)]ϕe+[(b2+aλ+a)r2(λ+1)+2λ(e⋅c)+2c(e⋅λ)-(e⋅e⋅λ)+3(λ2-1)(λ+1)-b(ϕe⋅λ)+2λ(b2+aλ+a)]ξ,(∇eQ)ξ=[3b(e⋅λ)+2λ(e⋅b)-(e⋅ϕe⋅λ)-2bcλ+(1+λ)(ξ⋅λ)]e+[r2(λ+1)+2c(e⋅λ)+2λ(e⋅c)-(e⋅e⋅λ)+3(λ2-1)(λ+1)-b(ϕe⋅λ)+2λ(b2+aλ+a)]ϕe+[4cλ(1+λ)-(2+6λ)(e⋅λ)]ξ,(∇ϕeQ)e=[(ϕe⋅r)2+(4λ-2a-2)(ϕe⋅λ)-2λ(ϕe⋅a)+4bλ(1-λ)+2c(ξ⋅λ)]e+[4acλ+(ϕe⋅ξ⋅λ)+2cλ(1-λ)+(λ-1)(e⋅λ)]ϕe+[r2(λ-1)+2λ(ϕe⋅b)-c(e⋅λ)-(ϕe⋅ϕe⋅λ)+3(λ2-1)(λ-1)+2b(ϕe⋅λ)+2λ(c2-aλ+a)]ξ,(∇ϕeQ)ξ=[r2(λ-1)+2λ(ϕe⋅b)-(ϕe⋅ϕe⋅λ)-c(e⋅λ)+3(λ2-1)(λ-1)+2b(ϕe⋅λ)+2λ(c2-aλ+a)]e+[-2bcλ+3c(ϕe⋅λ)+2λ(ϕe⋅c)-(ϕe⋅e⋅λ)+(λ-1)(ξ⋅λ)]ϕe+[4bλ(λ-1)-(6λ-2)(ϕe⋅λ)]ξ,(∇ξQ)e=[(ξ⋅r)2+(2λ-4a)(ξ⋅λ)-2λ(ξ⋅a)]e+[-4a2λ+(ξ⋅ξ⋅λ)]ϕe+[-2acλ+2b(ξ⋅λ)+2λ(ξ⋅b)-(ξ⋅ϕe⋅λ)+a(e⋅λ)],(∇ξQ)ϕe=[(ξ⋅ξ⋅λ)-4a2λ]e+[(ξ⋅r)2+(2λ+4a)(ξ⋅λ)+2λ(ξ⋅a)]ϕe+[2abλ+2c(ξ⋅λ)+2λ(ξ⋅c)-(ξ⋅e⋅λ)-a(ϕe⋅λ)]ξ.Applying (2.21) to the vectors fields of the ϕ-basis of the contact metric manifold M3 we have: (∇eQ)ϕe=(∇ϕeQ)e, (∇eQ)ξ=(∇ξQ)e and (∇ϕeQ)ξ=(∇ξQ)ϕe. We use the previous relations and we get the following nine (9) conditions for a contact metric 3-manifold to have harmonic curvature:(2.23)(e⋅ξ⋅λ)+(3-3λ+2a)(ϕe⋅λ)-(ϕe⋅r)2-2c(ξ⋅λ)+2λ(ϕe⋅a)-2bλ(2a+3-λ)=0,2b(ξ⋅λ)+2λ(e⋅a)+(3λ+2a+3)(e⋅λ)+(e⋅r)2-(ϕe⋅ξ⋅λ)-2cλ(3+2a+λ)=0,r+2λ(e⋅c)-2λ(ϕe⋅b)+3c(e⋅λ)-3b(ϕe⋅λ)-(e⋅e⋅λ)+6(λ2-1)+(ϕe⋅ϕe⋅λ)+2λ(b2-c2+2aλ)=0,3b(e⋅λ)+2λ(e⋅b)-(e⋅ϕe⋅λ)+2λ(ξ⋅a)-(ξ⋅r)2-2bcλ+(4a+1-λ)(ξ⋅λ)=0,-(2+6λ+a)(e⋅λ)-2b(ξ⋅λ)-2λ(ξ⋅b)+2λc(a+2+2λ)+(ξ⋅ϕe⋅λ)=0,(a-6λ+2)(ϕe⋅λ)-2c(ξ⋅λ)-2λ(ξ⋅c)+2bλ(2λ-2-a)+(ξ⋅e⋅λ)=0,3c(ϕe⋅λ)+2λ(ϕe⋅c)-(ϕe⋅e⋅λ)-2λ(ξ⋅a)-(ξ⋅r)2-2bcλ-(λ+1+4a)(ξ⋅λ)=0,r2(λ+1)-b(ϕe⋅λ)+2c(e⋅λ)+2λ(e⋅c)-(e⋅e⋅λ)-(ξ⋅ξ⋅λ)+3(λ2-1)(λ+1)+2λ(2a2+b2+aλ+a)=0,r2(λ-1)+2b(ϕe⋅λ)+2λ(ϕe⋅b)-c(e⋅λ)-(ϕe⋅ϕe⋅λ)-(ξ⋅ξ⋅λ)+3(λ2-1)(λ-1)+2λ(2a2+c2-aλ+a)=0.Remark 2.6.
From these nine conditions, we can derive some useful results: (a) by subtracting the ninth equation from the first and using (2.16), we get ϕe·r=0, (b) by adding the equations two and six and using similarly (2.16) we have e·r=0. From the relations ϕe·r=0, e·r=0, and (2.12), we can conclude ξ·r=0 and hence we are led to the known result that the scalar curvature r is constant in a contact metric 3-manifold with harmonic curvature. Later for our study, we will use the r as a constant and we will give to these equations a more convenient form.Definition 2.7 (see [22]).
LetM3 be a 3-dimensional contact metric manifold and h=λh+-λh- the spectral decomposition of h on U. If(2.24)∇h-Xh-X=[ξ,h+X],
for all vector fields X on M3 and all points of an open subset W of U and h=0 on the points of M3 which do not belong to W, then the manifold is said to be semi-Kcontact manifold. From Lemma 2.3 and the relations (2.12), the above condition for X=e leads to [ξ,e]=0 and for X=ϕe to ∇ϕeϕe=0. Hence on a semi-K contact manifold we have a+λ+1=c=0. If we apply the deformation e→ϕe, ϕe→e, ξ→-ξ, λ→-λ, b→c, and c→b then the contact metric structure remains the same. Hence the condition for a 3-dimensional contact metric manifold to be semi-K contact is equivalent to a-λ+1=b=0.Definition 2.8.
A(κ,μ,ν)-contact metric manifold is defined in [23] as a contact metric manifold (M2n+1,η,ξ,ϕ,g) on which the curvature tensor satisfies for every X,Y∈X(M) the condition:(2.25)R(X,Y)ξ=κ(η(Y)X-η(X)Y)+μ(η(Y)hX-η(X)hY)+ν(η(Y)ϕhX-η(X)ϕhY),
where κ,μ,ν are smooth functions on M. If ν=0 we have a generalized (κ,μ)-contact metric manifold [24] and if additionally κ,μ are constants then the manifold is a contact metric (κ,μ)-space [25, 26]. Moreover, in [23] and [24] it is proved, respectively, that for a (κ,μ,ν) or a generalized (κ,μ)-contact metric manifold M2n+1 of dimension greater than 3 the functions κ,μ are constants and ν is the zero function.Now, we will give some known results concerning contact metric 3-manifolds and pseudosymmetric contact metric 3-manifolds.Proposition 2.9 (see [16]).
In a 3-dimensional contact metric manifold one has(2.26)Qϕ=ϕQ⟺(ξ⋅λ=2bλ-(ϕe⋅λ)=2cλ-(e⋅λ)=aλ=0).Let(M,η,g,ϕ,ξ) be a contact metric 3-manifold. In case M=U0, that is, (ξ,η,ϕ,g) is a Sasakian structure, then M is a pseudosymmetric space of constant type [13]. Next, assume that U is not empty and let {e,ϕe,ξ} be a ϕ-basis as in Lemma 2.3. We have the following.Lemma 2.10 (see [16]).
Let(M,η,g,ϕ,ξ) be a contact metric three manifold. Then M is pseudosymmetric if and only if
(2.27)B(ξ⋅λ)+(-2aλ-λ2+1)A=LA,A(ξ⋅λ)+(2aλ-λ2+1)B=LB,(ξ⋅λ)(r2+2λ2-2)+AB=L(ξ⋅λ),A2-|(ξ⋅λ)|2+(2aλ-λ2+1)(-2aλ-3λ2+3-r2)=L(-2aλ-3λ2+3-r2),B2-|(ξ⋅λ)|2+(-2aλ-λ2+1)(2aλ-3λ2+3-r2)=L(2aλ-3λ2+3-r2),
where L is the function in the pseudosymmetry definition (2.2).Using (2.15), (2.19), the system (2.27) takes a more convenient form:(2.28)ZB+IA=LA,ZA+DB=LB,ZC+AB=LZ,A2-Z2+D(I-C)=L(I-C),B2-Z2+I(D-C)=L(D-C).Remark 2.11.
IfL=0, the manifold is semisymmetric and the above system (2.28) is in accordance with equations (3.1)–(3.5) in [27].Proposition 2.12 (see [16]).
LetM3 be a 3-dimensional contact metric manifold satisfying Qϕ=ϕQ. Then, M3 is a pseudosymmetric space of constant type.


## 3. Pseudosymmetric Contact Metric 3-Manifolds with Harmonic  Curvature andTrl Constant in the Direction of ξ

Theorem 3.1.
LetM3 be a 3-dimensional pseudosymmetric contact metric manifold with harmonic curvature and Trl constant in the direction of ξ. Then, there are at most eight open subsets of M3 for which their union is an open and dense subset of M3 and each of them as an open submanifold of M3 is either: (a) Sasakian or (b) flat or (c) locally isometric to the Lie groups SU(2), SL(2,R) equipped with a left invariant metric or (d) pseudosymmetric of constant type and with scalar curvature r=2(1-λ2+2a) or (e) semi-K contact with L=-3a2-4a(a≠0) or (f) semi-K contact with L=a2(a≠0) or (g) semi-K contact of type constant along ξ and ϕe or (h) semi-K contact of type constant along ξ and e.Proof.
We consider the next open subsets ofM:
(3.1)U0={p∈M:λ=0inaneighborhoodofp},U={p∈M:λ≠0inaneighborhoodofp},
where U0∪U is open and dense subset of M.
In caseM=U0, M is a pseudosymmetric space of constant type [13] and we get the (a) case of present Theorem 3.1. Next, assume that U is not empty and let {e,ϕe,ξ} be a ϕ-basis. First we note that in the neighborhood U where λ≠0 we have from (2.20)
(3.2)ξ⋅Trl=0⟺ξ⋅λ=0.
Equations (2.23) because of (3.2) and the fact that r is constant become, respectively,
(3.3)(3-3λ+2a)(ϕe⋅λ)+2λ(ϕe⋅a)-2bλ(2a+3-λ)=0,(3.4)2λ(e⋅a)+(3λ+2a+3)(e⋅λ)-2cλ(3+2a+λ)=0,(3.5)r+2λ(e⋅c)-2λ(ϕe⋅b)+3c(e⋅λ)-3b(ϕe⋅λ)-(e⋅e⋅λ)+6(λ2-1)+(ϕe⋅ϕe⋅λ)+2λ(b2-c2+2aλ)=0,(3.6)3b(e⋅λ)+2λ(e⋅b)-(e⋅ϕe⋅λ)+2λ(ξ⋅a)-2bcλ=0,or3b(e⋅λ)+(e⋅A)-6bcλ+2λ(ξ⋅a)=0,(3.7)r2(λ+1)-b(ϕe⋅λ)+2c(e⋅λ)+2λ(e⋅c)+2λ(2a2+b2+aλ+a)-(e⋅e⋅λ)+3(λ2-1)(λ+1)=0,(3.8)-(2+6λ+a)(e⋅λ)-2λ(ξ⋅b)+2λc(a+2+2λ)+(ξ⋅ϕe⋅λ)=0,or-(2+6λ+a)(e⋅λ)+2λc(a+2+2λ)-(ξ⋅A)=0,(3.9)r2(λ-1)+2b(ϕe⋅λ)+2λ(ϕe⋅b)-c(e⋅λ)-(ϕe⋅ϕe⋅λ)+3(λ2-1)(λ-1)+2λ(2a2+c2-aλ+a)=0,(3.10)3c(ϕe⋅λ)+2λ(ϕe⋅c)-(ϕe⋅e⋅λ)-2λ(ξ⋅a)-2bcλ=0,or3c(ϕe⋅λ)+(ϕe⋅B)-2λ(ξ⋅a)-6bcλ=0,(3.11)(a-6λ+2)(ϕe⋅λ)-2λ(ξ⋅c)+2bλ(2λ-2-a)+(ξ⋅e⋅λ)=0,or(a-6λ+2)(ϕe⋅λ)-(ξ⋅B)+2bλ(2λ-2-a)=0,
where for the second form of (3.6), (3.8), we also used A=2bλ-(ϕe·λ) and B=2cλ-(e·λ) in (3.10), (3.11).
In the neighborhoodU, the system (2.28) for the pseudosymmetric contact metric 3-manifolds of Lemma 2.10 because of (3.2) becomes
(3.12)(I-L)A=0,(D-L)B=0,AB=0,A2+(D-L)(I-C)=0,B2+(I-L)(D-C)=0,
where A,B,C,D,I are given by (2.15), (2.19).
Studying the third equation, we regard the following open subsets ofU:
(3.13)W={p∈U:A=2bλ-(ϕe⋅λ)=0inaneighborhoodofp},W3={p∈U:A=2bλ-(ϕe⋅λ)≠0inaneighborhoodofp},
where W∪W3 is open and dense in the closure of U.
InW we have
(3.14)(D-L)B=0,(D-L)(I-C)=0,B2+(I-L)(D-C)=0,
hence, we regard the subsets of W:
(3.15)W1={p∈W:B=2cλ-(e⋅λ)=0inaneighborhoodofp},W2={p∈W:B=2cλ-(e⋅λ)≠0inaneighborhoodofp},
where W1∪W2 is open and dense in the closure of W and W1∪W2∪W3 is open and dense in the closure of U. We study the initial system at each Wi for i=1,2,3.
InW1 the initial system (2.28) becomes
(3.16)(D-L)(I-C)=0,(I-L)(D-C)=0
or more explicitly
(3.17)(ϕe⋅λ)=2bλ,(e⋅λ)=2cλ,ξ⋅λ=0,[2aλ-2λ2+2+b2+c2-2a-(e⋅c)-(ϕe⋅b)]×(-2aλ-λ2+1-L)=0,[-2aλ-2λ2+2+b2+c2-2a-(e⋅c)-(ϕe⋅b)]×(2aλ-λ2+1-L)=0.
We have studied this system in [17] (Theorem 4.1) and we get the cases (b), (c), (d), (e), and (f) of the present Theorem 3.1.
InW2 the initial system (2.28) becomes
(3.18)D-L=0,B2+(I-L)(D-C)=0.
Apart from (3.2), we also have the following equations:
(3.19)(ϕe⋅λ)=2bλ,(3.20)B=2cλ-(e⋅λ)≠0,(3.21)2aλ-λ2+1-L=0,(3.22)B2=4aλ(2aλ-3λ2+3-r2)=4aλ(L-2λ2+2-r2)
while we will also use (3.3), (3.4), (3.6), and (3.11).
Differentiating (3.21) with respect to ξ, ϕe we get, respectively,
(3.23)2λ(ξ⋅a)=ξ⋅L,(3.24)2λ(ϕe⋅a)+4abλ-4bλ2=ϕe⋅L
(the derivative e·L=2λ(e·a)+2(a-λ)(e·λ) can not be estimated any further). Equations (3.3), (3.11) because of (3.19) yield, respectively,
(3.25)ϕe⋅a=ϕe⋅λ=2bλ,(3.26)ξ⋅B=-8bλ2.
Differentiating (3.22) with respect to ξ and using (3.2), (3.26) and the fact that r is constant, we get
(3.27)-4bλB=(4aλ-3λ2+3-r2)(ξ⋅a).
The first form of (3.6) and (3.19) give
(3.28)bB=2λ(ξ⋅a),
hence (3.27), (3.28) yield
(3.29)(4aλ+5λ2+3-r2)(ξ⋅a)=0.
We suppose that there is a point p∈W2 where ξ·a≠0. Because of the continuity of the function ξ·a, there is a neighborhood of this point S⊂W2⊂U: S={q∈W2:ξ·a≠0}. In S, we have 4aλ+5λ2+3-r/2=0. Differentiating this equation with respect to ξ and using (3.2), the constancy of r and the fact that we work in U where λ≠0 we conclude that ξ·a=0 in S, which is a contradiction. Hence
(3.30)ξ⋅a=0,
everywhere in W2. Because of (3.30) the equations (3.20), (3.27) give
(3.31)b=0.
Differentiating (3.2) with respect to ϕe, (3.19) with respect to ξ, subtracting and using (2.12), we get
(3.32)(a-λ+1)(e⋅λ)=0.
Let’s suppose that there is a point p in W2 where a-λ+1≠0. The function a-λ+1 is continuous, hence there is an open neighborhood V of p, V⊂W2, where a-λ+1≠0 everywhere in V, hence
(3.33)e⋅λ=0.
From (3.20) and (3.33) we have in V⊂W2⊂U:
(3.34)c≠0.
Equation (3.4) because of (3.33) gives e·a=c(2a+λ+3). From the second of (2.16) and because of (3.31), (3.33), (3.34), we get in V: e·a=c(a-λ+1)≠0. By equalizing these two results and because of (3.34), we get: a+2λ+2=0. We differentiate this equation with respect to e and because of (3.33), we get e·a=0, which is a contradiction in V. Hence a-λ+1=0 everywhere in W2 and because of (3.31), we can conclude according to Definition 2.7 that the structure is semi-K contact and pseudosymmetric with L constant along the directions of ξ and ϕe because of (3.23), (3.24) and (3.25), (3.30), (3.31).
InW3 the initial system (2.28) becomes
(3.35)I-L=0,A2+(D-L)(I-C)=0.
We have the following equations and (3.2):
(3.36)(e⋅λ)=2cλ,(3.37)A=2bλ-(ϕe⋅λ)≠0,(3.38)-2aλ-λ2+1-L=0,(3.39)A2=-4aλ(-2aλ-3λ2+3-r2)=-4aλ(L-2λ2+2-r2)
while we will also use (3.3), (3.4), (3.6), (3.8).
Differentiating (3.38) with respect to ξ, e we get, respectively,
(3.40)-2λ(ξ⋅a)=ξ⋅L,-2λ(e⋅a)-4acλ-4cλ2=e⋅L,
(we neglect the derivative ϕe·L because we can not estimate it). Equations (3.4), (3.8) because of (3.36) yield, respectively,
(3.41)e⋅a=-e⋅λ=-2cλ,(3.42)ξ⋅A=-8cλ2.
Differentiating (3.39) with respect to ξ and using (3.2), (3.42) and the fact that r is constant, we get
(3.43)-4cλA=(4aλ+3λ2-3+r2)(ξ⋅a).
The first form of (3.10) and (3.36) give
(3.44)-cA=2λ(ξ⋅a),
hence (3.43), (3.44) yield
(3.45)(4aλ-5λ2-3+r2)(ξ⋅a)=0.
We suppose that there is a point p∈W3 where ξ·a≠0. Because of the continuity of this function, there is a neighborhood of p  S⊂W3⊂U: S={q∈W3:ξ·a≠0}. In S, we have 4aλ-5λ2-3+r/2=0. Differentiating this equation with respect to ξ and using (3.2), the constancy of r and the fact that we work in U where λ≠0 we conclude that ξ·a=0 in S, which is a contradiction. Hence
(3.46)ξ⋅a=0,
everywhere in W3. Because of (3.46) the equations (3.37), (3.43) give
(3.47)c=0.
Differentiating (3.2) with respect to e, (3.36) with respect to ξ, subtracting and using (2.12), we get
(3.48)-(a+λ+1)(ϕe⋅λ)=0.
Let’s suppose that there is a point p in W3 where a+λ+1≠0. This function is smooth, then because of its continuity, there is an open neighborhood V of p, V⊂W3, where a+λ+1≠0 everywhere in V, hence
(3.49)ϕe⋅λ=0.
From (3.37) and (3.49) we have in V⊂W3⊂U:
(3.50)b≠0.
From (3.3) and (3.49), we get ϕe·a=b(2a-λ+3). From the first of (2.16) and because of (3.47), (3.49), (3.50), we get in V: ϕe·a=b(a+λ+1)≠0. By equalizing these two results and because of (3.50), we get: a-2λ+2=0. We differentiate this equation with respect to ϕe and because of (3.49), we get ϕe·a=0, which is a contradiction in V. Hence a+λ+1=0 everywhere in W3 and because of (3.47), we can conclude according to Definition 2.7 that the structure is semi-K contact and pseudosymmetric with L constant along the directions of ξ and e because of (3.40) and (3.41), (3.46), (3.47).
Finally, we remark that the cases (g) and (h) of the present Theorem3.1 that result from the structures studied in the sets W2 and W3, respectively.Remark 3.2.
(i) The conditions of harmonic curvature help us to the systems in the neighborhoodsW2 and W3 where we had equations of the type A2=-4aλ(-2aλ-3λ2+3-r/2) and which we could not handle in our previous articles [16, 17].
(ii) In case (d) whereL is constant, we can also use the classification of [11] to improve our results as the manifolds with harmonic curvature are a special case of conformally flat manifolds in dimension 3.


## 4. Pseudosymmetric(κ,μ,ν)-Contact Metric 3-Manifolds of Type  Constant in the Direction of ξ

Theorem 4.1.
LetM3 be a pseudosymmetric (κ,μ,ν)-contact metric 3-manifold of type constant along the direction ξ. Then, there are at most five open subsets of M3 for which their union is an open and dense subset of M3 and each of them as an open submanifold of M3 is either (a) Sasakian or (b) flat or (c) pseudosymmetric of constant type L=κ=1/2(Trl), μ=ν=0 and of constant scalar curvature r=2κ or (d) pseudosymmetric generalized (κ,μ)-contact metric manifold of type L=κ-μλ, of scalar curvature r=2(3κ-μλ) and ξ·μ=ξ·κ=0 or (e) pseudosymmetric generalized (κ,μ)-contact metric manifold of type L=κ+μλ, of scalar curvature r=2(3κ+μλ) and ξ·μ=ξ·κ=0.Proof.
We study pseudosymmetric(κ,μ,ν)-contact metric 3-manifolds with
(4.1)ξ⋅L=0,
where L is the function in (2.2). We consider the next open subsets of M,
(4.2)U0={p∈M:λ=0inaneighborhoodofp},U={p∈M:λ≠0inaneighborhoodofp},
where U0∪U is open and dense subset of M.
In caseM=U0, (M,ξ,η,ϕ,g) is a Sasakian structure which is a pseudosymmetric space of constant type [13] with κ=1, μ,ν∈ℝ and h=0 and we get the (a) case of present Theorem 4.1. Next, assume that U is not empty and let {e,ϕe,ξ} be a ϕ-basis. From (2.25), we can calculate the following components of the Riemannian curvature tensor:
(4.3)R(ξ,e)ξ=-(κ+λμ)e-λνϕe,R(e,ϕe)ξ=0,R(ξ,ϕe)ξ=-λνe-(κ-λμ)ϕe.
By virtue of (2.14) we can conclude that
(4.4)A=2bλ-(ϕe⋅λ)=0,B=2cλ-(e⋅λ)=0,Z=ξ⋅λ=λν,D=2aλ-λ2+1=κ-λμ,I=-2aλ-λ2+1=κ+λμ,
and hence the system (2.28) becomes
*Z(C-L)=0,-Z2+(D-L)(I-C)=0,-Z2+(I-L)(D-C)=0,
where A,B,C,D,I,Z are given by (2.19) and (4.4). Substituting from (4.4) (ϕe·λ), (e·λ) in (2.16) we also have
(4.5)ξ⋅c=-(ϕe⋅a)+b(a-λ+1),(4.6)ξ⋅b=(e⋅a)-c(λ+a+1).
First we will prove that Z=ξ·λ=0 (equivalently ν=0 as we work in U where λ≠0). We suppose that there is a point p∈U where ξ·λ≠0. By the continuity of this function, we can consider that there is a neighborhood V of p, where ξ·λ≠0 everywhere in V⊂U. We work in V. Then the first equation of (*) becomes C-L=0 or equivalently:
(4.7)(e⋅c)+(ϕe⋅b)=L+b2+c2-λ2+1-2a.
We differentiate this equation with respect to ξ and by virtue of (4.1) we get
(4.8)ξ⋅e⋅c+ξ⋅ϕe⋅b=2b(ξ⋅b)+2c(ξ⋅c)-2λ(ξ⋅λ)-2(ξ⋅a),
which because of (4.5), (4.6) becomes
(4.9)ξ⋅e⋅c+ξ⋅ϕe⋅b=2b(e⋅a)-2c(ϕe⋅a)-2λ(ξ⋅λ)-2(ξ⋅a)-4bcλ.
Next, we differentiate (4.5) and (4.6) with respect to e and ϕe, respectively, and adding we have
(4.10)e⋅ξ⋅c+ϕe⋅ξ⋅b=-[e,ϕe]a-(a+λ+1)(ϕe⋅c)+(a-λ+1)(e⋅b)-c(ϕe⋅a)+b(e⋅a)-4bcλ.
We subtract this last equation from (4.9) and we get
(4.11)[ξ,e]c+[ξ,ϕe]b=b(e⋅a)-c(ϕe⋅a)-2(ξ⋅a)-2λ(ξ⋅λ)+[e,ϕe]a+(a+λ+1)(ϕe⋅c)-(a-λ+1)(e⋅b)
or because of (2.12)
(4.12)(a+λ+1)(ϕe⋅c)+(λ-a-1)(e⋅b)=b(e⋅a)-c(ϕe⋅a)-2(ξ⋅a)-2λ(ξ⋅λ)-b(e⋅a)+c(ϕe⋅a)+2(ξ⋅a)+(λ+a+1)(ϕe⋅c)+(λ-a-1)(e⋅b)
or equivalently: λ(ξ·λ)=0 and because we work in V⊂U, we have ξ·λ=0, which is a contradiction. Hence, we can deduce everywhere in U:
(4.13)ξ⋅λ=0⟺ν=0.
Next we will derive some useful relations. From (4.4) we have:
(4.14)ϕe⋅λ=2bλ,e⋅λ=2cλ.
We differentiate these equations with respect to e and ϕe, respectively, we subtract, we use the relations (2.12), (4.4) and we get
(4.15)ξ⋅λ=λ[(e⋅b)-(ϕe⋅c)]
or because of (4.13)
(4.16)e⋅b=ϕe⋅c.
We differentiate the relations ϕe·λ=2bλ and (4.13) with respect to ξ and ϕe, respectively, and subtracting we obtain: [ξ,ϕe]λ=2λ(ξ·b) or because of (2.12), (4.4), (4.6)
(4.17)e⋅a=2cλ.
We differentiate the relations e·λ=2cλ and (4.13) with respect to ξ and e, respectively, and subtracting we obtain: [ξ,e]λ=2λ(ξ·c) or because of (2.12), (4.4), (4.5)
(4.18)ϕe⋅a=-2bλ.
Finally, after substituting D,I,Z from (4.4), (4.13) the final form of the system (*) is
(4.19)μ(C-L)=0,(κ-λμ-L)(κ+λμ-C)=0.
In order to study this system we regard the following open subsets of U:
(4.20)V1={p∈U:C-L≠0inaneighborhoodofp},V2={p∈U:C-L=0inaneighborhoodofp},
where V1∪V2 is open and dense in the closure of U.
InV1, we have μ=0 and hence from (4.4): I=D=κ or 2aλ-λ2+1=-2aλ-λ2+1 or finally a=0 and κ=1-λ2. From (4.17), (4.18) we deduce that b=c=0. Having also the second equation of (4.19), we regard the open subsets of V1(4.21)Y1={p∈V1:κ-C=0inaneighborhoodofp},Y2={p∈V1:κ-C≠0inaneighborhoodofp},
where Y1∪Y2 is open and dense in the closure of V1.
InY1 substituting in κ-C=0, C from (2.15), a=b=c=0, we get κ=1-λ2=0 and hence the structure is flat with κ=μ=ν.
InY2 from μ=0 we have again I=D=κ, a=0 and from (4.17), (4.18) b=c=0 while we must also have κ=L. Hence, L=κ=(1/2)Trl and from (2.18) of constant scalar curvature r=2(1-λ2).
InV2 having C=L, the second equation of (4.19) becomes (2aλ-λ2+1-L)(-2aλ-λ2+1-L)=0. Hence, we regard the open subsets of V2(4.22)W1={p∈V2:-2aλ-λ2+1-L≠0inaneighborhoodofp},W2={p∈V2:-2aλ-λ2+1-L=0inaneighborhoodofp},
where W1∪W2 is open and dense in the closure of V2.
InW1 we must have 2aλ-λ2+1-L=0 while in W2 we have -2aλ-λ2+1-L=0. We differentiate these equations with respect to ξ and because of (4.13) we get
(4.23)ξ⋅a=0.
By virtue ofI and D in (4.4) we deduce μ=-2a and hence
(4.24)ξ⋅μ=0.
In W2 we differentiate -2aλ-λ2+1-L=0 with respect to ξ and similarly we also obtain (4.24). Each of W1 and W2 is a generalized (κ,μ)-contact metric 3-manifold with ξ·μ=0 and scalar curvature r=2(2aλ-3λ2+3)=2(3κ-μλ) or r=2(-2aλ-3λ2+3)=2(3κ+μλ) respectively and from (4.1), (4.13) and (4.23) or (4.24) ξ·κ=0 and ξ·r=0.
Concluding: the structure inU0 gives the Sasakian case, the structures in Y1 and Y2 give the (b) and (c) cases of the present Theorem 4.1 and the structures in W1 and W2 give (d) and (e) respectively.Remark 4.2.
The generalized(κ,μ)-contact metric manifolds in dimension 3 with κ<1 (equivalently λ≠0) and ξ·μ=0 have been studied by Koufogiorgos and Tsichlias [28]. They proved in their Theorem 4.1 of [28] that at any point of P∈M, precisely one of the following relations is valid: μ=2(1+1-κ), or μ=2(1-1-κ), while there exists a chart (U,(x,y,z)) with P∈U⊆M such that the functions κ, μ depend only on z and the tensors fields η, ξ, ϕ, g take a suitable form. Each of our submanifolds W1 and W2 is such a generalized (κ,μ)-contact metric 3-manifold.



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*Source: 367467-2012-04-24.xml*