The present paper deals with a study of 3-dimensional contact metric generalized (k,μ)-space forms. We obtained necessary and sufficient condition for a 3-dimensional contact metric generalized (k,μ)-space form with Qϕ=ϕQ to be of constant curvature. We also obtained some conditions of such space forms to be pseudosymmetric and ξ-projectively flat, respectively.

1. Introduction

In 1995, Blair et al. [1] introduced the notion of contact metric manifolds with characteristic vector field ξ belonging to the (k,μ)-nullity distribution and such types of manifolds are called (k,μ)-contact metric manifolds. They obtained several results and examples of such a manifold. A full classification of this manifold has been given by Boeckx [2]. A contact metric manifold (M,ϕ,ξ,η,g) is said to be a generalized (k,μ)-space if its curvature tensor satisfies the condition
(1)R(X,Y)ξ=k{η(Y)X-η(X)Y}+μ{η(Y)hX-η(X)hY}
for some smooth functions k and μ on M independent choice of vector fields X and Y. If k and μ are constant, the manifold is called a (k,μ)-space. If a (k,μ)-space M has a constant ϕ-sectional curvature c and a dimension greater than 3, the curvature tensor of this (k,μ)-space form is given by [3]
(2)R=c+34R1+c-14R2+(c+34-k)R3+R4+12R5+(1-μ)R6,
where R1, R2, R3, R4, R5, and R6 are the tensors defined by
(3)R1(X,Y)Z=g(Y,Z)X-g(X,Z)Y,R2(X,Y)Z=g(X,ϕZ)ϕY-g(Y,ϕZ)ϕX+2g(X,ϕY)ϕZ,R3(X,Y)Z=η(X)η(Z)Y-η(Y)η(Z)X+g(X,Z)η(Y)ξ-g(Y,Z)η(X)ξ,R4(X,Y)Z=g(Y,Z)hX-g(X,Z)hY+g(hY,Z)X-g(hX,Z)Y,R5(X,Y)Z=g(hY,Z)hX-g(hX,Z)hY+g(ϕhX,Z)ϕhY-g(ϕhY,Z)ϕhX,R6(X,Y)Z=η(X)η(Z)hY-η(Y)η(Z)hX+g(hX,Z)η(Y)ξ-g(hY,Z)η(X)ξ
for all vector fields X, Y, Z on M, where 2h=Lξϕ and L is the usual Lie derivative.

The notion of generalized Sasakian-space-forms was introduced and studied by Alegre et al. [4] with several examples. A generalized Sasakian-space-form is an almost contact metric manifold (M,ϕ,ξ,η,g) whose curvature tensor is given by
(4)R(X,Y)Z=f1R1+f2R2+R3f3,
where R1, R2, and R3 are the tensors defined above and f1, f2, f3 are differentiable functions on M. In such case we will write the manifold as M(f1,f2,f3). Generalized Sasakian-space-forms have been studied by several authors, namely, [5–11].

By motivating the works on generalized Sasakian-space-forms and (k,μ)-space forms, Carriazo et al. [12] introduced the concept of generalized (k,μ)-space forms. A generalized (k,μ)-space form is an almost contact metric manifold (M,ϕ,ξ,η,g) whose curvature tensor R is given by
(5)R=f1R1+f2R2+f3R3+f4R4+f5R5+f6R6,
where R1, R2, R3, R4, R5, R6 are the tensors defined above and f1, f2, f3, f4, f5, f6 are differentiable functions on M.

The object of the paper is to study 3-dimensional contact metric generalized (k,μ)-space forms. The paper is organized as follows. Section 2 deals with some preliminaries on contact metric manifolds and contact metric generalized (k,μ)-space forms. Section 3 is concerned with 3-dimensional contact metric generalized (k,μ)-space forms. Here it is proved that a 3-dimensional contact metric generalized (k,μ)-space form with Qϕ=ϕQ is of constant curvature if and only if 3f2+f3=0. In Section 4, it is proved that a 3-dimensional contact metric generalized (k,μ)-space form is Ricci-semisymmetric; then either f1=f3 or 3f2+f3=0. In Sections 5 and 6, we obtained that some equivalent conditions for a 3-dimensional contact metric generalized (k,μ)-space form are pseudosymmetric and ξ-projectively flat, respectively.

A contact manifold is a C∞-(2n+1) manifold M2n+1 equipped with a global 1-form η such that η∧(dη)n≠0 everywhere on M2n+1. Given a contact form η it is well known that there exists a unique vector field ξ called the characteristic vector field of η such that η(ξ)=1 and dη(X,ξ)=0 for every vector field X on M2n+1. A Riemannian metric is said to be associated metric if there exists a tensor field ϕ of type (1, 1) such that dη(X,Y)=g(X,ϕY), η(X)=g(X,ξ), ϕ2X=-X+η(X)ξ, ϕξ=0, η(ϕX)=0, and g(ϕX,ϕY)=g(X,Y)-η(X)η(Y) for all vector fields X, Y on M2n+1. Then the structure (ϕ,ξ,η,g) on M2n+1 is called a contact metric structure, and then manifold M2n+1 equipped with such a structure is called a contact metric manifold [13].

Given a contact metric manifold (M2n+1,ϕ,ξ,η,g) we define a (1, 1) tensor field h by 2h=Lξϕ. Then h is symmetric and satisfies the following relations:
(6)hξ=0,hϕ=-ϕh,tr(h)=tr(ϕh)=0,η·h=0.
Moreover, if ∇ denotes the Riemannian connection of g, then the following relation holds:
(7)∇Xξ=-ϕX-ϕhX.
The vector field ξ is a Killing vector with respect to g if and only if h=0. A contact metric manifold (M2n+1,ϕ,ξ,η,g) for which ξ is a Killing vector is said to be a K-contact manifold. Therefore, a generalized (k,μ)-space form with such a structure is actually a generalized Sasakian-space-form.

A generalized (k,μ)-space-form is an almost contact metric manifold (M,ϕ,ξ,η,g) whose curvature tensor R is given by
(8)R(X,Y)Z=f1{g(Y,Z)X-g(X,Z)Y}+f2{g(X,ϕZ)ϕY-g(Y,ϕZ)ϕX+2g(X,ϕY)ϕZ}+f3{η(X)η(Z)Y-η(Y)η(Z)X+g(X,Z)η(Y)ξ-g(Y,Z)η(X)ξ}+f4{g(Y,Z)hX-g(X,Z)hY+g(hY,Z)X-g(hX,Z)Y}+f5{g(hY,Z)hX-g(hX,Z)hY+g(ϕhX,Z)ϕhY-g(ϕhY,Z)ϕhX}+f6{η(X)η(Z)hY-η(Y)η(Z)hX+g(hX,Z)η(Y)ξ-g(hY,Z)η(X)ξ}
for all vector fields X, Y, Z on M, where f1, f2, f3, f4, f5, f6 are differentiable functions on M, 2h=Lξϕ, and L is the usual Lie derivative. In such case we will denote the manifold as M(f1,…,f6). This kind of manifold appears as natural generalization of the (k,μ)-space forms by taking
(9)f1=c+34,f2=c-14,f3=c+34-k,f4=1,f5=12,f6=1-μ
as constant. Here c denotes constant ϕ-sectional curvature. The ϕ-sectional curvature of generalized (k,μ)-space form M(f1,…,f6) is f1+3f2+f4g((h-ϕhϕ)X,X). Generalized Sasakian-space-forms [4] are also examples with f4=f5=f6=0 and f1, f2, f3 not necessarily constant.

We inferred the following result from [12].

Theorem 1.

Let M(f1,…,f6) be a generalized (k,μ)-space form. If M is a contact metric manifold with f3=f1-1, then it is a Sasakian manifold.

Next, by using the definitions of the tensors R1, R2, R3, R4, R5, R6 and properties (6) of the tensor h in the formula (8), we obtain that the curvature tensor of a generalized (k,μ)-space form satisfies
(10)R(X,Y)ξ=(f1-f3){η(Y)X-η(X)Y}+(f4-f6){η(Y)hX-η(X)hY}
for every X, Y. Thus we have the following Theorem.

Theorem 2 (see [<xref ref-type="bibr" rid="B8">12</xref>]).

If M(f1,…,f6) is a contact metric generalized (k,μ)-space form, then it is a generalized (k,μ)-space, with k=f1-f3 and μ=f4-f6.

We know from Theorem 1 that M(f1,…,f6) is Sasakian if k=f1-f3=1. Under the same hypothesis, we also know that f2=f3=f1-1 and h=0, so we may take f4=f5=f6=0. Therefore, in this paper we will consider non-Sasakian generalized (k,μ)-space forms M(f1,…,f6), that is, those with k=f1-f3≠1.

A contact metric manifold is said to be η-Einstein manifold [13] if it satisfies
(11)S=ag+bη⊗η
for some smooth functions a and b.

If M3(f1,…,f6) is a contact metric generalized (k,μ)-space form, then its Ricci tensor S and the scalar curvature r can be written as [12]
(12)S(X,Y)=(2f1+3f2-f3)g(X,Y)-(3f2+f3)η(X)η(Y)+(f4-f6)g(hX,Y),(13)r=2(3f1+3f2-2f3).
It is known that in a 3-dimensional Riemannian manifold (M3,g), the curvature tensor R is given by
(14)R(X,Y)Z=S(Y,Z)X-S(X,Z)Y+g(Y,Z)QX-g(X,Z)QY-r2[g(Y,Z)X-g(X,Z)Y]
for any vector fields X, Y, Z on M. By substituting (12) and (13) in (14) we have
(15)R(X,Y)Z=(f1+3f2){g(Y,Z)X-g(X,Z)Y}+(3f2+f3){η(X)η(Z)Y-η(Y)η(Z)X+g(X,Z)η(Y)ξ-g(Y,Z)η(X)ξ}+(f4-f6){g(Y,Z)hX-g(X,Z)hY+g(hY,Z)X-g(hX,Z)Y}.
From (12), we have
(16)S(X,ξ)=2(f1-f3)η(X).
Also, from (15) we obtain
(17)R(ξ,Y)Z=(f1-f3){g(Y,Z)ξ-η(Z)Y}+(f4-f6){g(hY,Z)ξ-η(Z)hY}.

If M3(f1,…,f6) is a contact metric generalized (k,μ)-space form, then Qϕ=ϕQ is true if and only if f4-f6=0. In this connection, the following result appears in [12].

Lemma 3.

If M3(f1,…,f6) is a contact metric generalized (k,μ)-space form, then the following conditions are equivalent:

M3 is an η-Einstein,

Qϕ=ϕQ, where Q denotes the Ricci tensor,

M3 is a (f1-f3,0)-space,

f4-f6=0.

Now we begin with the following.

Lemma 4.

A 3-dimensional contact metric generalized (k,μ)-space form with Qϕ=ϕQ is of constant curvature if and only if 3f2+f3=0.

Proof.

Let a 3-dimensional contact metric generalized (k,μ)-space form M3(f1,…,f6) with Qϕ=ϕQ be a space of constant curvature. Then
(18)R(X,Y)Z=c{g(Y,Z)X-g(X,Z)Y},
where c is constant curvature of the manifold. By using the definition of Ricci curvature and (18) we have
(19)S(X,Y)=2cg(X,Y).
If we use (19) in the definition of the scalar curvature we get
(20)r=6c.
From (19) and (20) one can easily see that
(21)S(X,Y)=r3g(X,Y).
By putting X=Y=ξ in (12) and using (21) we obtain r=6(f1-f3). By comparing this value of r with (13), we have 3f2+f3=0.

Conversely, if 3f2+f3=0, then from (15) we can easily see that
(22)R(X,Y)Z=(f1-f3){g(Y,Z)X-g(X,Z)Y}+(f4-f6){g(Y,Z)hX-g(X,Z)hY+g(hY,Z)X-g(hX,Z)Y}.
If M3(f1,…,f6) satisfies Qϕ=ϕQ, then in view of Lemma 3 we have f4-f6=0. Thus from (22), we have
(23)R(X,Y)Z=(f1-f3){g(Y,Z)X-g(X,Z)Y}.
If M3(f1,…,f6) satisfies f4-f6=0, then f1-f3 is constant. Hence, from (23), we conclude that the M3 is of constant curvature. This completes the proof.

Theorem 5.

Every 3-dimensional contact metric generalized (k,μ)-space form with Qϕ=ϕQ is an N(f1-f3)-η-Einstein manifold.

A contact metric generalized (k,μ)-space form is said to be Ricci-semisymmetric if its Ricci tensor S satisfies the condition
(24)R(X,Y)·S=0,X,Y∈χ(M),
where R(X,Y) acts as a derivation on S. This notion was introduced by Mirzoyan [14] for Riemannian spaces.

From (24) we have
(25)S(R(X,Y)U,V)+S(U,R(X,Y)V)=0.
If we substitute X=ξ in (25) and then using (17), we get
(26)(f1-f3)[g(Y,U)S(ξ,V)-η(U)S(Y,V)+g(Y,V)S(ξ,U)-η(V)S(Y,U)]+(f4-f6)[g(hY,U)S(ξ,V)-η(U)S(hY,V)+g(hY,V)S(ξ,U)-η(V)S(hY,U)]=0.
By using (16) in (26) we obtain
(27)2(f1-f3)[(f1-f3){g(Y,U)η(V)+g(Y,V)η(U)}+(f4-f6){g(hY,U)η(V)+g(hY,V)η(U)}]-(f1-f3)[S(Y,V)η(U)+S(Y,U)η(V)]-(f4-f6)[S(hY,V)η(U)+S(hY,U)η(V)]=0.
Consider that {e1,e2,e3} is an orthonormal basis of the TpM, p∈M. Then by putting Y=U=ei in (27) and taking the summation for 1≤i≤3, we have
(28)(f1-f3)[8(f1-f3)η(V)-S(V,ξ)-rη(V)]=0.
Again by using (16) in (28), we get
(29)(f1-f3)[6(f1-f3)-r]η(V)=0,
which gives either f1=f3 or r=6(f1-f3). For the second case, that is, r=6(f1-f3), we have 3f2+f3=0. Thus we can state the following.

Theorem 6.

If a 3-dimensional contact metric generalized (k,μ)-space form is Ricci-semisymmetric then either f1=f3 or 3f2+f3=0.

Let (M,g) be a Riemannian manifold with the Riemannian metric ∇. A tensor field F:χ(M)×χ(M)×χ(M)→χ(M) of type (1, 3) is said to be curvature-like if it has the properties of R. For example, the tensor R given by
(30)R(X,Y)Z=(X∧Y)Z=g(Y,Z)X-g(X,Z)Y,0000000000000000000000000000X,Y,Z∈χ(M)
defines a curvature-like tensor field on M. For example, if R(X,Y)Z=k((X∧Y)Z)=k(g(Y,Z)X-g(X,Z)Y), then the manifold is of constant curvature k.

It is well known that every curvature-like tensor field F acts on the algebra 𝔗s1(M) of all tensor fields on M of type (1,s) as a derivation [15]
(31)(F·P)(X1,…,Xs;Y,X)=F(X,Y){P(X1,…,Xs)}-∑j=1sP(X1,…,F(X,Y)Xj,…,Xs)
for all X1,…,Xs∈χ(M) and P∈𝔗s1(M). The derivation F·P of P by F is a tensor field of type (1,s+2). For a tensor field P of type (1,s), we define the derivative of P with respect to the curvature-like tensor defined by (30) as
(32)Q(g,P)(X1,…,Xs;Y,X)=(X∧Y)P(X1,…,Xs)-∑j=1sP(X1,…,(X∧Y)Xj,…,Xs).
As a generalization of locally symmetric (∇R=0) manifolds, the notion of a semisymmetric (R·R=0) manifold is introduced by Szabó [16]. As a proper generalization of a semisymmetric manifold, the notion of pseudosymmetric manifold is introduced by Deszcz [17]. A Riemannian manifold (M,g) is said to be pseudosymmetric if there exists a real valued smooth function L:M→ℝ such that R·R=L(g,R). In particular, if L is constant, then M is said to be a pseudosymmetric manifold of constant type [17]. A pseudosymmetric manifold is said to be proper if it is not semisymmetric.

For 3-dimensional Riemannian spaces, the following characterization of pseudosymmetry is known (cf. [18, 19]).

Theorem 7.

A 3-dimensional Riemannian space is pseudosymmetric if and only if it is η-Einstein.

We also know that a 3-dimensional contact metric generalized (k,μ)-space form is η-Einstein if and only if f4-f6=0. Hence, in view of Theorem 7, we can state the following.

Theorem 8.

A 3-dimensional contact metric generalized (k,μ)-space form is pseudosymmetric if and only if f4-f6=0.

From Lemma 3 and Theorem 8, we have following.

Lemma 9.

A 3-dimensional contact metric generalized (k,μ)-space form with Qϕ=ϕQ is pseudosymmetric.

In a 3-dimensional contact metric generalized (k,μ)-space form M3(f1,…,f6) the projective curvature tensor 𝒫 is defined as
(33)𝒫(X,Y)Z=R(X,Y)Z-12[S(Y,Z)X-S(X,Z)Y]
for all X,Y,Z∈TM.

By plugging Z=ξ in (33) and using (16) and (10), we get
(34)𝒫(X,Y)ξ=(f4-f6)[η(Y)hX-η(X)hY].
If f4-f6=0, then from (34) one can get
(35)𝒫(X,Y)ξ=0;
that is, M3(f1,…,f6) is ξ-projectively flat.

Conversely, suppose that M3(f1,…,f6) is ξ-projectively flat. Then from (34) we obtain
(36)(f4-f6)[η(Y)hX-η(X)hY]=0.
This implies that f4-f6=0. Thus we can state the following.

Theorem 10.

A 3-dimensional contact metric generalized (k,μ)-space form is ξ-projectively flat if and only if f4-f6=0.

From Lemma 3 and Theorem 10, we obtain the following.

Lemma 11.

A 3-dimensional contact metric generalized (k,μ)-space form with Qϕ=ϕQ is ξ-projectively flat.

By combining Lemma 3, Theorem 5, Theorem 8, and Theorem 10 the following result is obtained.

Theorem 12.

If M3(f1,…,f6) is a contact metric generalized (k,μ)-space form, then the following conditions are equivalent to each other:

M3 is N(f1-f3)-η-Einstein,

Qϕ=ϕQ, where Q denotes the Ricci operator,

f4-f6=0,

M3 is pseudosymmetric,

M3 is ξ-projectively flat.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The first author is thankful to University Grants Commission, New Delhi, India, for financial support in the form of Major Research Project F. no. 39-30/2010 (SR), dated 23-12-2010.

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