# Curvature Properties andη-Einstein (k,μ)-Contact Metric Manifolds

Authors: H. G. Nagaraja; C. R. Premalatha
Journal: ISRN Geometry
 (2012)

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## Abstract

We study curvature properties in(k,μ)-contact metric manifolds. We give the characterization of η-Einstein (k,μ)-contact metric manifolds with associated scalars.


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## Body

## 1. Introduction

The class of(k,μ)-contact manifolds [1] is of interest as it contains both the classes of Sasakian and non-Sasakian cases. The contact metric manifolds for which the characteristic vector field ξ belongs to (k,μ)-nullity distribution are called (k,μ) contact metric manifolds. Boeckx [2] gave a classification of (k,μ)-contact metric manifolds. Sharma [3], Papantoniou [4], and many others have made an investigation of (k,μ)-contact metric manifolds. A special class of (k,μ)-contact metric manifolds called N(k)-contact metric manifolds was studied by authors [5, 6] and others. In this paper we study (k,μ)-contact metric manifolds by considering different curvature tensors on it (Table 1). We characterize η-Einstein (k,μ)-contact metric manifolds with associated scalars by considering symmetry, ϕ-symmetry, semisymmetry, ϕ-recurrent, and flat conditions on (k,μ)-contact metric manifolds. The paper is organized as follows: In Section 2, we give some definitions and basic results. In Section 3, we consider conharmonically symmetric, conharmonically semisymmetric, ϕ-conharmonically flat, ξ-conharmonically flat, and ϕ-recurrent (k,μ)-contact metric manifolds and we prove that such manifolds are η-Einstein or η-parallel or cosymplectic depending on the conditions. In Section 4, we prove that ξ-conformally flat (k,μ)-contact metric manifold reduces to N(k)-contact metric manifold if and only if it is an η-Einstein manifold. Further we prove conformally Ricci-symmetric and ϕ-conformally flat (k,μ)-contact metric manifolds are η-Einstein. In Section 5, we prove that pseudoprojectively symmetric and pseudoprojectively Ricci-symmetric (k,μ)-contact metric manifolds are η-Einstein. In Section 6 we consider Ricci-semisymmetric (k,μ)-contact metric manifolds and prove that such manifolds are η-Einstein. In all the cases where (k,μ)-contact metric manifold is an η-Einstein manifold, we obtain associated scalars in terms of k and μ.Table 1
Comparison of the results for different curvature tensors inM(k,μ).
Curvature tensor
Condition
Result
C
~
(
X
,
Y
,
Z
,
W
)
(
∇
W
C
~
)
(
X
,
Y
)
Z
=
0
M
(
k
,
μ
) is cosymplectic ⇔ μ=2(1-n)⇔M(k,μ) is η-Einstein
C
~
(
X
,
Y
,
Z
,
W
)
R
·
C
~
=
0
⇔
η-Einstein with α=-2nk and β=4nkμ(k+μ)
C
~
(
X
,
Y
,
Z
,
W
)
C
~
(
ϕ
X
,
ϕ
Y
,
ϕ
Z
,
ϕ
W
)
=
0
Ricci tensor isη-parallel and μ=2(n-1)n
C
~
(
X
,
Y
,
Z
,
W
)
C
~
(
X
,
Y
)
ξ
=
0
η-Einstein with β=-α-2nk, α=k-(2n-1)μ[2nμ+2(n-1)](k-1)2(n-1)-nμ-k
C
~
(
X
,
Y
,
Z
,
W
)
M
(
k
,
μ
) is ϕ-recurrent
η-Einstein with β=2nk-α, α=-k+μ(2n-1)(k-1)a+k[b-μ(2n-1)]
C
(
X
,
Y
)
Z
C
(
X
,
Y
)
ξ
=
0
M
(
k
,
μ
) reduces to N(k)-c.m.m ⇔M(k,μ) is η-Einstein
C
(
X
,
Y
)
Z
C
·
S
=
0
M
(
1
,
-
2
) is η-Einstein with α=2n(2n-1)5n-2, β=2n(1-n)5n-2
C
(
X
,
Y
)
Z
′C(ϕX,ϕY,ϕZ,ϕW)=0
η-Einstein with α=r(4n-1)+4n2k2n,β=-r(4n-1)2n
P
~
(
X
,
Y
)
Z
R
·
P
~
=
0
η-Einstein
P
~
(
X
,
Y
)
Z
P
~
·
S
=
0
η-Einstein
R
(
X
,
Y
)
Z
R
·
S
=
0
η-Einstein


## 2. Preliminaries

A(2n+1) dimensional C∞-differentiable manifold M is said to admit an almost contact metric structure (ϕ,ξ,η,g) if it satisfies the following relations [7, 8]
(2.1)ϕ2=-I+η⊗ξ,η(ξ)=1,ϕξ=0,η∘ϕ=0,(2.2)g(ϕX,ϕY)=g(X,Y)-η(X)η(Y),g(X,ϕY)=-g(ϕX,Y),g(X,ϕX)=0,g(X,ξ)=η(X),
where ϕ is a tensor field of type (1,1), ξ is a vector field, η is a 1-form, and g is a Riemannian metric on M. A manifold equipped with an almost contact metric structure is called an almost contact metric manifold. An almost contact metric manifold is called a contact metric manifold if it satisfies
(2.3)g(X,ϕY)=dη(X,Y),
for  all  vector  fields  X, Y.The (1,1) tensor fieldh defined by h=(1/2)Lξϕ, where L denotes the Lie differentiation, is a symmetric operator and satisfies hϕ=-ϕh, trh=trϕh=0, and hξ=0. Further we have [1]
(2.4)∇Xξ=-ϕX-ϕhX,(∇Xη)Y=g(X+hX,ϕY),
where ∇ denotes the Riemannian connection of g.The(k,μ)-nullity distribution N(k,μ) of a contact metric manifold M(ϕ,ξ,η,g) is a distribution [1]
(2.5)N(k,μ):p→Np(k,μ)={Z∈Tp(M):R(X,Y)Z=k[g(Y,Z)X-g(X,Z)Y]+μ[g(Y,Z)hX-g(X,Z)hY]},
for any vector fields X and Y on M.Definition 2.1.
A contact metric manifold is said to be(i)
Einstein ifS(X,Y)=λg(X,Y), where λ is a constant and S is the Ricci tensor,(ii)
η-Einstein if S(X,Y)=αg(X,Y)+βη(X)η(Y), where α and β are smooth functions.A contact metric manifold withξ∈N(k,μ) is called a (k,μ)-contact metric manifold. In a (k,μ)-contact metric manifold, we have
(2.6)R(X,Y)ξ=k[η(Y)X-η(X)Y]+μ[η(Y)hX-η(X)hY].
If k=1, μ=0, then the manifold becomes Sasakian [1], and if μ=0, then the notion of (k,μ)-nullity distribution reduces to k-nullity distribution [9]. If k=0, then N(k)-contact metric manifold is locally isometric to the product En+1(0)×Sn(4). In a (2n+1)-dimensional (k,μ)-contact metric manifold, we have the following [1]:
(2.7)h2=(k-1)ϕ2,k≤1,(2.8)(∇Xϕ)(Y)=g(X+hX,Y)ξ-η(Y)(X+hX),(2.9)QX=[2(n-1)-nμ]X+[2(n-1)+μ]hX+[2(1-n)+n(2k+μ)]η(X)ξ,n≥1,(2.10)S(X,Y)=[2(n-1)-nμ]g(X,Y)+[2(n-1)+μ]g(hX,Y)+[2(1-n)+n(2k+μ)]η(X)η(Y),n≥1,(2.11)S(X,ξ)=2nkη(X),(2.12)r=2n(2n-2+k-nμ),(2.13)(∇Xh)(Y)=[(1-k)g(X,ϕY)+g(X,hϕY)]ξ+η(Y)h(ϕX+ϕhX)-μη(X)ϕhY,
where Q is the Ricci operator and r is the scalar curvature of M.Throughout this paperM(k,μ) denotes (2n+1)-dimensional (k,μ)-contact metric manifold.


## 3. Conharmonic Curvature Tensor in(k,μ)-Contact Metric Manifolds

The conharmonic curvature tensor inM(k,μ) is given by [10]
(3.1)C~(X,Y)Z=R(X,Y)Z-12n-1[S(Y,Z)X-S(X,Z)Y+g(Y,Z)QX-g(X,Z)QY].
A (k,μ)-contact metric manifold is said to be (1)
conharmonically symmetric if(3.2)(∇WC~)(X,Y)Z=0,whereX,Y,Z,W∈T(M),(2)
conharmonically semisymmetric if(3.3)(R(U,X)·C~)(Y,Z,W)=R(U,X)C~(Y,Z)W-C~(R(U,X)Y,Z)W-C~(Y,R(U,X)Z)W-C~(Y,Z)R(U,X)W=0.


### 3.1. Conharmonically Symmetric(k,μ)-Contact Metric Manifolds

Differentiating (3.1) covariantly with respect to W, we obtain
(3.4)(∇WC~)(X,Y)Z=(∇WR)(X,Y)Z-12n-1[(∇WS)(Y,Z)X-(∇WS)(X,Z)Y+g(Y,Z)(∇WQ)(X)-g(X,Z)(∇WQ)(Y)].
If M(k,μ) is conharmonically symmetric, then, from (3.4), we obtain
(3.5)(∇WR)(X,Y)Z=12n-1[(∇WS)(Y,Z)X-(∇WS)(X,Z)Y+g(Y,Z)(∇WQ)(X)-g(X,Z)(∇WQ)(Y)].
Differentiating (2.6) covariantly with respect to W and using (2.4), we obtain
(3.6)(∇WR)(X,Y)ξ=k[g(W+hW,ϕY)X-g(W+hW,ϕX)Y]+μ[g(W+hW,ϕY)hX-g(W+hW,ϕX)hY].
Differentiating (2.10) covariantly with respect to W and using (2.11), (2.4), we have
(3.7)(∇WS)(X,Y)=b[(1-k)g(W,ϕX)η(Y)+g(W,hϕX)η(Y)+η(X)g(hϕW,Y)+η(X)g(hϕhW,Y)-μη(W)g(ϕhX,Y)]+c[g(W,ϕX)η(Y)+g(hW,ϕX)η(Y)+η(X)g(W,ϕY)+η(X)g(hW,ϕY)],
where
(3.8)b=2(n-1)+μ,c=2(1-n)+n(2k+μ).
From (3.7), we obtain
(3.9)(∇WQ)(X)=[(1-k+n)μ+2k],[g(W,ϕX)ξ-η(X)(ϕW)]+[μ(1+n)+2nk],[g(W,hϕX)ξ+η(X)(hϕW)]-[2(n-1)+μ]η(W)(ϕhX).
Taking Z=ξ in (3.5) and using (3.6), (3.7),  and (3.9), we obtain
(3.10)(2n-1)[k(g(W+hW,ϕY)X-g(W+hW,ϕX)Y)+μ(g(W+hW,ϕY)hX-g(W+hW,ϕX)hY)]=l[g(W,ϕY)+g(W,hϕY)]X-l[g(W,ϕX)+g(W,hϕX)]Y+η(Y)(mg(W,ϕX)ξ+l[g(W,hϕX)ξ+η(X)(hϕW)]-bη(W)(ϕhX))-η(X)(mg(W,ϕY)ξ+l[g(W,hϕY)ξ+η(Y)(hϕW)]-bη(W)(ϕhY)),
where
(3.11)l=μ(1+n)+2nk,m=(1-k+n)μ+2k.
Contracting (3.10) with ξ and using (2.1), we obtain
(3.12)k[(g(W,ϕY)η(X)-g(W,ϕX)η(Y))(1-μ)+(g(hW,ϕY)η(X)-g(hW,ϕX)η(Y))(2n-1)]=0.
From (3.12), we get either k=0 or
(3.13)[(1-μ)(g(W,ϕY)η(X)-g(W,ϕX)η(Y))+(g(hW,ϕY)η(X)-g(hW,ϕX)η(Y))(2n-1)]=0.
Taking Y=ϕY in (3.13) and using (2.1), we obtain
(3.14)(1-μ)(g(Y,W)-η(Y)η(W))+(2n-1)g(Y,hW)=0.
Taking Y=ϕY in (3.14), we obtain
(3.15)(1-μ)g(ϕY,W)+(2n-1)g(ϕY,hW)=0.
Since μ≠1, from (3.15), it follows that μ=2(1-n) if and only if
(3.16)g(ϕY,W)+g(ϕY,hW)=0.In view of (2.4), the above equation gives that M(k,μ) reduces to a cosymplectic manifold. Thus we have M(k,μ) is cosymplectic if and only if μ=2(1-n).Further from (2.10) and Definition 2.1, we have M(k,μ) is η-Einstein with α=2(n2-1),β=2((1-n2)+nk) if and only if μ=2(1-n). Thus we have the following.Theorem 3.1.
In a conharmonically symmetric(k,μ)-contact metric manifold M(k,μ), the following statements are equivalent. (1)
M(k,μ) is cosymplectic.(2)
M(k,μ) is η-Einstein with α=2(n2-1),β=2((1-n2)+nk).(3)
μ=2(1-n).


### 3.2. Conharmonically Semisymmetric(k,μ)-Contact Metric Manifolds

SupposeR·C~=0. Then from (3.3), we have
(3.17)R(ξ,X)C~(Y,Z)W-C~(R(ξ,X)Y,Z)W-C~(Y,R(ξ,X)Z)W-C~(Y,Z)R(ξ,X)W=0.
Using (2.5), (2.9),  (2.6), (2.10), and (2.11) in (3.17) and taking W=Y=ξ, we get
(3.18)k2[η(Z)η(X)-g(Z,X)]ξ-kμg(hZ,X)ξ-k2n-1(4nkη(Z)η(X)-2nkg(X,Z)-S(Z,X))ξ-kμg(Z,hX)ξ-μ2g(hZ,hX)ξ-μ2n-1[-2nkg(hX,Z)-S(Z,hX)]ξ-k2η(X)[η(Z)ξ-Z]+k2n-1η(X)[4nkη(Z)ξ-2nkZ-QZ]+kR(X,Z)ξ-k2n-1[2nkη(Z)X-2nkη(X)Z+η(Z)QX-η(X)QZ]+μR(hX,Z)ξ-μ2n-1[2nkη(Z)X-2nkη(X)Z+η(Z)QX-QZ]+kη(Z)R(ξ,X)ξ-k2n-1η(Z)[4nkη(X)ξ-2nkX-QX]+μη(Z)R(ξ,hX)ξ-μ2n-1η(Z)[-2nk(hX)-Q(hX)]-k2η(X)[η(Z)ξ-Z]+k2n-1η(X)[-2nkZ-QZ]+kR(ξ,Z)X-k2n-1[S(Z,X)ξ-2nkη(X)Z+2nkg(Z,X)ξ-η(X)QZ]+μR(ξ,Z)hX-μ2n-1[S(Z,hX)ξ-S(hX,ξ)Z+2nkg(Z,hX)ξ]=0.
Taking X=ξ and using (2.9),  (2.6), and (2.11), we obtain
(3.19)QZ=4nkμ(k+μ)η(Z)ξ-2nkZ.
That is, M(k,μ) is an η-Einstein manifold.Conversely, suppose inM(k,μ) the relation (3.19) holds. Then we have
(3.20)R·C~=R(ξ,X)C~(Y,Z)W-C~(R(ξ,X)Y,Z)W-C~(Y,R(ξ,X)Z)W-C~(Y,Z)R(ξ,X)W=η(Z)ξ(-4nk2n-1(k+μ))+μ2n-1(2nkZ+QZ).
Using (3.19) in (3.20), we get R·C~=0 which implies that M(k,μ) is conharmonically semisymmetric. Thus we have the following.Theorem 3.2.
A(k,μ)-contact metric manifold is conharmonically semisymmetric if and only if it is η-Einstein with α=-2nk and β=(4nk/μ)(k+μ).


### 3.3.ϕ-Conharmonically Flat (k,μ)-Contact Metric Manifolds

SupposeM(k,μ) is ϕ-conharmonically flat, that is, C~(ϕX,ϕY,ϕZ,ϕW)=0 for all vector fields X, Y, Z, W. Then from (3.1), we obtain
(3.21)R~(ϕX,ϕY,ϕZ,ϕW)=1(2n-1)[g(ϕY,ϕZ)S(ϕX,ϕW)-g(ϕX,ϕZ)S(ϕY,ϕW)+S(ϕY,ϕZ)g(ϕX,ϕW)-S(ϕX,ϕZ)g(ϕY,ϕW)].
Let {e1,e2,…,e2n,ξ} be a local orthonormal basis of the tangent space TP(M) at each P in M(k,μ). Then in M(k,μ), the following relations hold:
(3.22)∑i=12n‍g(ei,ei)=2n,∑i=12n‍S(ei,ei)=r-2nk,∑i=12n‍g(ei,Z)S(Y,ei)=S(Y,Z)-2nkη(Y)η(Z),(3.23)∑i=12n‍g(ei,ϕZ)S(Y,ei)=S(Y,ϕZ).
Taking X=W=ei in (3.21) and summing up from 1 to 2n, we have
(3.24)∑i=12n‍R~(ϕei,ϕY,ϕZ,ϕei)=1(2n-1)∑i=12n‍[g(ϕY,ϕZ)S(ϕei,ϕei)-g(ϕei,ϕZ)S(ϕY,ϕei)+S(ϕY,ϕZ)g(ϕei,ϕei)-S(ϕei,ϕZ)g(ϕY,ϕei)].
Using (2.13), (3.22), in (3.24), we obtain
(3.25)S(ϕY,ϕZ)=(r-2nk)g(ϕY,ϕZ).
Replacing Y by ϕY and Z by ϕZ in (3.25) and using (2.1), we have
(3.26)S(Y,Z)=(r-2nk)g(Y,Z)+(4nk-r)η(Y)η(Z).
Taking Y=Z=ei in (3.26) and taking summation over i=1 to (2n+1), we obtain r=2nk.Substituting this in (3.26) and taking the covariant derivative with respect to X, we obtain
(3.27)∇XS(ϕY,ϕZ)=0.
That is, S is η-parallel.Further substitutingr=2nk in (2.12), we obtain
(3.28)μ=2n-2n.
Thus from the above discussions we can state the following.Theorem 3.3.
In a(2n+1)-dimensional ϕ-conharmonically flat (k,μ)-contact metric manifold, Ricci tensor is η-parallel and μ=2(n-1)/n.


### 3.4.ξ-Conharmonically Flat (k,μ)-Contact Metric Manifolds

SupposeM(k,μ) is ξ-conharmonically flat, that is, C~(X,Y)ξ=0.Then from (3.1), we obtain
(3.29)R(X,Y)ξ=12n-1(S(Y,ξ)X-S(X,ξ)Y+g(Y,ξ)QX-g(X,ξ)QY).
Using (2.9),  (2.6), and (2.11) in (3.29), we obtain
(3.30)(2n-1)μ[η(Y)hX-η(X)hY]-k[η(Y)X-η(X)Y]-[η(Y)QX-η(X)QY]=0.
Taking Y=ξ in (3.30) and using (2.1), we obtain
(3.31)QX=kX-(k+2nk)η(X)ξ-(2n-1)μhX.
Contracting (3.31) with W, we obtain
(3.32)S(X,W)=kg(X,W)-(k+2nk)η(X)η(W)-(2n-1)μg(hX,W).
Replacing X by hX in (3.32) and using (2.7) and (2.10), we obtain
(3.33)g(hX,W)=[(2n-1)μ+2(n-1)+μ](k-1)[g(X,W)-η(X)η(W)]2(n-1)-nμ-k.
Tha above equation with (3.32) yields
(3.34)S(X,W)=αg(X,W)+βη(X)η(W),
where
(3.35)α=k-(2n-1)μ[2nμ+2(n-1)](k-1)2(n-1)-nμ-k,β=-α-2nk.
Hence M(k,μ) reduces to an η-Einstein manifold.Thus we have the following.Theorem 3.4.
A(2n+1)-dimensional ξ-conharmonically flat (k,μ)-contact metric manifold is an η-Einstein manifold.


### 3.5.ϕ-Recurrent (k,μ)-Contact Metric Manifolds

A(2n+1)-dimensional (k,μ)-contact metric manifold M(k,μ) is said to be ϕ-recurrent if and only if there exists a nonzero 1-form A such that
(3.36)ϕ2((∇WR)(X,Y)Z)=A(W)R(X,Y)Z.
Differentiating (3.1) covariantly with respect to W, we obtain
(3.37)(∇WC~)(X,Y)Z=(∇WR)(X,Y)Z-12n-1((∇WS)(X,Y)Z-(∇WS)(X,Z)Y+g(Y,Z)(∇WQ)(X)-g(X,Z)(∇WQ)(Y)).
Suppose M(k,μ) is ϕ-recurrent. Then from (3.37), we have
(3.38)-(∇WC~)(X,Y)Z+η((∇WC~)(X,Y)Z)ξ=A(W)C~(X,Y)Z.
Contracting (3.38) with ξ, we obtain
(3.39)A(W)η(C~(X,Y)Z)=0.
Since A is a nonzero 1-form, we have η(C~(X,Y)Z)=0.Using (3.1), the above equation yields
(3.40)η(R(X,Y)Z)-12n-1[S(Y,Z)η(X)-S(X,Z)η(Y)+g(Y,Z)η(QX)-g(X,Z)η(QY)]=0.
Using (2.6) and (2.9) in (3.40), we obtain
(3.41)k[g(Y,Z)η(X)-g(X,Z)η(Y)]+μ[g(hY,Z)η(X)-g(hX,Z)η(Y)]=12n-1[S(Y,Z)η(X)-S(X,Z)η(Y)+2nkg(Y,Z)η(X)-2nkg(X,Z)η(Y)].
Taking X=ξ in (3.41), we get
(3.42)S(Y,Z)=-kg(Y,Z)+(2n+1)η(Y)η(Z)+(2n-1)μg(hY,Z).
Replacing Y by hY in (3.42), we obtain
(3.43)S(hY,Z)=-kg(hY,Z)-(2n-1)(k-1)μ(g(Y,Z)-η(Y)η(Z)).
Replacing Y by hY in (2.10) and comparing the resulting equation with (3.43), we obtain
(3.44)g(hY,Z)=(k-1)(b-μ(2n-1))a(g(Y,Z)-η(Y)η(Z)),
where a=2(n-1)-nμ,b=2(n-1)+μ.Using (3.44) in (3.42), we get
(3.45)S(Y,Z)=αg(Y,Z)+βη(Y)η(Z),
where α=-k+(μ(2n-1)(k-1)/(a+k))[b-μ(2n-1)] and β=2nk-α.That is,M(k,μ) is an η-Einstein manifold.Thus we have the following.Theorem 3.5.
Aϕ-recurrent (k,μ)-contact metric manifold is an η-Einstein manifold.


## 3.1. Conharmonically Symmetric(k,μ)-Contact Metric Manifolds

Differentiating (3.1) covariantly with respect to W, we obtain
(3.4)(∇WC~)(X,Y)Z=(∇WR)(X,Y)Z-12n-1[(∇WS)(Y,Z)X-(∇WS)(X,Z)Y+g(Y,Z)(∇WQ)(X)-g(X,Z)(∇WQ)(Y)].
If M(k,μ) is conharmonically symmetric, then, from (3.4), we obtain
(3.5)(∇WR)(X,Y)Z=12n-1[(∇WS)(Y,Z)X-(∇WS)(X,Z)Y+g(Y,Z)(∇WQ)(X)-g(X,Z)(∇WQ)(Y)].
Differentiating (2.6) covariantly with respect to W and using (2.4), we obtain
(3.6)(∇WR)(X,Y)ξ=k[g(W+hW,ϕY)X-g(W+hW,ϕX)Y]+μ[g(W+hW,ϕY)hX-g(W+hW,ϕX)hY].
Differentiating (2.10) covariantly with respect to W and using (2.11), (2.4), we have
(3.7)(∇WS)(X,Y)=b[(1-k)g(W,ϕX)η(Y)+g(W,hϕX)η(Y)+η(X)g(hϕW,Y)+η(X)g(hϕhW,Y)-μη(W)g(ϕhX,Y)]+c[g(W,ϕX)η(Y)+g(hW,ϕX)η(Y)+η(X)g(W,ϕY)+η(X)g(hW,ϕY)],
where
(3.8)b=2(n-1)+μ,c=2(1-n)+n(2k+μ).
From (3.7), we obtain
(3.9)(∇WQ)(X)=[(1-k+n)μ+2k],[g(W,ϕX)ξ-η(X)(ϕW)]+[μ(1+n)+2nk],[g(W,hϕX)ξ+η(X)(hϕW)]-[2(n-1)+μ]η(W)(ϕhX).
Taking Z=ξ in (3.5) and using (3.6), (3.7),  and (3.9), we obtain
(3.10)(2n-1)[k(g(W+hW,ϕY)X-g(W+hW,ϕX)Y)+μ(g(W+hW,ϕY)hX-g(W+hW,ϕX)hY)]=l[g(W,ϕY)+g(W,hϕY)]X-l[g(W,ϕX)+g(W,hϕX)]Y+η(Y)(mg(W,ϕX)ξ+l[g(W,hϕX)ξ+η(X)(hϕW)]-bη(W)(ϕhX))-η(X)(mg(W,ϕY)ξ+l[g(W,hϕY)ξ+η(Y)(hϕW)]-bη(W)(ϕhY)),
where
(3.11)l=μ(1+n)+2nk,m=(1-k+n)μ+2k.
Contracting (3.10) with ξ and using (2.1), we obtain
(3.12)k[(g(W,ϕY)η(X)-g(W,ϕX)η(Y))(1-μ)+(g(hW,ϕY)η(X)-g(hW,ϕX)η(Y))(2n-1)]=0.
From (3.12), we get either k=0 or
(3.13)[(1-μ)(g(W,ϕY)η(X)-g(W,ϕX)η(Y))+(g(hW,ϕY)η(X)-g(hW,ϕX)η(Y))(2n-1)]=0.
Taking Y=ϕY in (3.13) and using (2.1), we obtain
(3.14)(1-μ)(g(Y,W)-η(Y)η(W))+(2n-1)g(Y,hW)=0.
Taking Y=ϕY in (3.14), we obtain
(3.15)(1-μ)g(ϕY,W)+(2n-1)g(ϕY,hW)=0.
Since μ≠1, from (3.15), it follows that μ=2(1-n) if and only if
(3.16)g(ϕY,W)+g(ϕY,hW)=0.In view of (2.4), the above equation gives that M(k,μ) reduces to a cosymplectic manifold. Thus we have M(k,μ) is cosymplectic if and only if μ=2(1-n).Further from (2.10) and Definition 2.1, we have M(k,μ) is η-Einstein with α=2(n2-1),β=2((1-n2)+nk) if and only if μ=2(1-n). Thus we have the following.Theorem 3.1.
In a conharmonically symmetric(k,μ)-contact metric manifold M(k,μ), the following statements are equivalent. (1)
M(k,μ) is cosymplectic.(2)
M(k,μ) is η-Einstein with α=2(n2-1),β=2((1-n2)+nk).(3)
μ=2(1-n).


## 3.2. Conharmonically Semisymmetric(k,μ)-Contact Metric Manifolds

SupposeR·C~=0. Then from (3.3), we have
(3.17)R(ξ,X)C~(Y,Z)W-C~(R(ξ,X)Y,Z)W-C~(Y,R(ξ,X)Z)W-C~(Y,Z)R(ξ,X)W=0.
Using (2.5), (2.9),  (2.6), (2.10), and (2.11) in (3.17) and taking W=Y=ξ, we get
(3.18)k2[η(Z)η(X)-g(Z,X)]ξ-kμg(hZ,X)ξ-k2n-1(4nkη(Z)η(X)-2nkg(X,Z)-S(Z,X))ξ-kμg(Z,hX)ξ-μ2g(hZ,hX)ξ-μ2n-1[-2nkg(hX,Z)-S(Z,hX)]ξ-k2η(X)[η(Z)ξ-Z]+k2n-1η(X)[4nkη(Z)ξ-2nkZ-QZ]+kR(X,Z)ξ-k2n-1[2nkη(Z)X-2nkη(X)Z+η(Z)QX-η(X)QZ]+μR(hX,Z)ξ-μ2n-1[2nkη(Z)X-2nkη(X)Z+η(Z)QX-QZ]+kη(Z)R(ξ,X)ξ-k2n-1η(Z)[4nkη(X)ξ-2nkX-QX]+μη(Z)R(ξ,hX)ξ-μ2n-1η(Z)[-2nk(hX)-Q(hX)]-k2η(X)[η(Z)ξ-Z]+k2n-1η(X)[-2nkZ-QZ]+kR(ξ,Z)X-k2n-1[S(Z,X)ξ-2nkη(X)Z+2nkg(Z,X)ξ-η(X)QZ]+μR(ξ,Z)hX-μ2n-1[S(Z,hX)ξ-S(hX,ξ)Z+2nkg(Z,hX)ξ]=0.
Taking X=ξ and using (2.9),  (2.6), and (2.11), we obtain
(3.19)QZ=4nkμ(k+μ)η(Z)ξ-2nkZ.
That is, M(k,μ) is an η-Einstein manifold.Conversely, suppose inM(k,μ) the relation (3.19) holds. Then we have
(3.20)R·C~=R(ξ,X)C~(Y,Z)W-C~(R(ξ,X)Y,Z)W-C~(Y,R(ξ,X)Z)W-C~(Y,Z)R(ξ,X)W=η(Z)ξ(-4nk2n-1(k+μ))+μ2n-1(2nkZ+QZ).
Using (3.19) in (3.20), we get R·C~=0 which implies that M(k,μ) is conharmonically semisymmetric. Thus we have the following.Theorem 3.2.
A(k,μ)-contact metric manifold is conharmonically semisymmetric if and only if it is η-Einstein with α=-2nk and β=(4nk/μ)(k+μ).


## 3.3.ϕ-Conharmonically Flat (k,μ)-Contact Metric Manifolds

SupposeM(k,μ) is ϕ-conharmonically flat, that is, C~(ϕX,ϕY,ϕZ,ϕW)=0 for all vector fields X, Y, Z, W. Then from (3.1), we obtain
(3.21)R~(ϕX,ϕY,ϕZ,ϕW)=1(2n-1)[g(ϕY,ϕZ)S(ϕX,ϕW)-g(ϕX,ϕZ)S(ϕY,ϕW)+S(ϕY,ϕZ)g(ϕX,ϕW)-S(ϕX,ϕZ)g(ϕY,ϕW)].
Let {e1,e2,…,e2n,ξ} be a local orthonormal basis of the tangent space TP(M) at each P in M(k,μ). Then in M(k,μ), the following relations hold:
(3.22)∑i=12n‍g(ei,ei)=2n,∑i=12n‍S(ei,ei)=r-2nk,∑i=12n‍g(ei,Z)S(Y,ei)=S(Y,Z)-2nkη(Y)η(Z),(3.23)∑i=12n‍g(ei,ϕZ)S(Y,ei)=S(Y,ϕZ).
Taking X=W=ei in (3.21) and summing up from 1 to 2n, we have
(3.24)∑i=12n‍R~(ϕei,ϕY,ϕZ,ϕei)=1(2n-1)∑i=12n‍[g(ϕY,ϕZ)S(ϕei,ϕei)-g(ϕei,ϕZ)S(ϕY,ϕei)+S(ϕY,ϕZ)g(ϕei,ϕei)-S(ϕei,ϕZ)g(ϕY,ϕei)].
Using (2.13), (3.22), in (3.24), we obtain
(3.25)S(ϕY,ϕZ)=(r-2nk)g(ϕY,ϕZ).
Replacing Y by ϕY and Z by ϕZ in (3.25) and using (2.1), we have
(3.26)S(Y,Z)=(r-2nk)g(Y,Z)+(4nk-r)η(Y)η(Z).
Taking Y=Z=ei in (3.26) and taking summation over i=1 to (2n+1), we obtain r=2nk.Substituting this in (3.26) and taking the covariant derivative with respect to X, we obtain
(3.27)∇XS(ϕY,ϕZ)=0.
That is, S is η-parallel.Further substitutingr=2nk in (2.12), we obtain
(3.28)μ=2n-2n.
Thus from the above discussions we can state the following.Theorem 3.3.
In a(2n+1)-dimensional ϕ-conharmonically flat (k,μ)-contact metric manifold, Ricci tensor is η-parallel and μ=2(n-1)/n.


## 3.4.ξ-Conharmonically Flat (k,μ)-Contact Metric Manifolds

SupposeM(k,μ) is ξ-conharmonically flat, that is, C~(X,Y)ξ=0.Then from (3.1), we obtain
(3.29)R(X,Y)ξ=12n-1(S(Y,ξ)X-S(X,ξ)Y+g(Y,ξ)QX-g(X,ξ)QY).
Using (2.9),  (2.6), and (2.11) in (3.29), we obtain
(3.30)(2n-1)μ[η(Y)hX-η(X)hY]-k[η(Y)X-η(X)Y]-[η(Y)QX-η(X)QY]=0.
Taking Y=ξ in (3.30) and using (2.1), we obtain
(3.31)QX=kX-(k+2nk)η(X)ξ-(2n-1)μhX.
Contracting (3.31) with W, we obtain
(3.32)S(X,W)=kg(X,W)-(k+2nk)η(X)η(W)-(2n-1)μg(hX,W).
Replacing X by hX in (3.32) and using (2.7) and (2.10), we obtain
(3.33)g(hX,W)=[(2n-1)μ+2(n-1)+μ](k-1)[g(X,W)-η(X)η(W)]2(n-1)-nμ-k.
Tha above equation with (3.32) yields
(3.34)S(X,W)=αg(X,W)+βη(X)η(W),
where
(3.35)α=k-(2n-1)μ[2nμ+2(n-1)](k-1)2(n-1)-nμ-k,β=-α-2nk.
Hence M(k,μ) reduces to an η-Einstein manifold.Thus we have the following.Theorem 3.4.
A(2n+1)-dimensional ξ-conharmonically flat (k,μ)-contact metric manifold is an η-Einstein manifold.


## 3.5.ϕ-Recurrent (k,μ)-Contact Metric Manifolds

A(2n+1)-dimensional (k,μ)-contact metric manifold M(k,μ) is said to be ϕ-recurrent if and only if there exists a nonzero 1-form A such that
(3.36)ϕ2((∇WR)(X,Y)Z)=A(W)R(X,Y)Z.
Differentiating (3.1) covariantly with respect to W, we obtain
(3.37)(∇WC~)(X,Y)Z=(∇WR)(X,Y)Z-12n-1((∇WS)(X,Y)Z-(∇WS)(X,Z)Y+g(Y,Z)(∇WQ)(X)-g(X,Z)(∇WQ)(Y)).
Suppose M(k,μ) is ϕ-recurrent. Then from (3.37), we have
(3.38)-(∇WC~)(X,Y)Z+η((∇WC~)(X,Y)Z)ξ=A(W)C~(X,Y)Z.
Contracting (3.38) with ξ, we obtain
(3.39)A(W)η(C~(X,Y)Z)=0.
Since A is a nonzero 1-form, we have η(C~(X,Y)Z)=0.Using (3.1), the above equation yields
(3.40)η(R(X,Y)Z)-12n-1[S(Y,Z)η(X)-S(X,Z)η(Y)+g(Y,Z)η(QX)-g(X,Z)η(QY)]=0.
Using (2.6) and (2.9) in (3.40), we obtain
(3.41)k[g(Y,Z)η(X)-g(X,Z)η(Y)]+μ[g(hY,Z)η(X)-g(hX,Z)η(Y)]=12n-1[S(Y,Z)η(X)-S(X,Z)η(Y)+2nkg(Y,Z)η(X)-2nkg(X,Z)η(Y)].
Taking X=ξ in (3.41), we get
(3.42)S(Y,Z)=-kg(Y,Z)+(2n+1)η(Y)η(Z)+(2n-1)μg(hY,Z).
Replacing Y by hY in (3.42), we obtain
(3.43)S(hY,Z)=-kg(hY,Z)-(2n-1)(k-1)μ(g(Y,Z)-η(Y)η(Z)).
Replacing Y by hY in (2.10) and comparing the resulting equation with (3.43), we obtain
(3.44)g(hY,Z)=(k-1)(b-μ(2n-1))a(g(Y,Z)-η(Y)η(Z)),
where a=2(n-1)-nμ,b=2(n-1)+μ.Using (3.44) in (3.42), we get
(3.45)S(Y,Z)=αg(Y,Z)+βη(Y)η(Z),
where α=-k+(μ(2n-1)(k-1)/(a+k))[b-μ(2n-1)] and β=2nk-α.That is,M(k,μ) is an η-Einstein manifold.Thus we have the following.Theorem 3.5.
Aϕ-recurrent (k,μ)-contact metric manifold is an η-Einstein manifold.


## 4. Conformal Curvature Tensor in(k,μ)-Contact Metric Manifolds

The conformal curvature tensorC in M(k,μ) is defined by [11]
(4.1)C(X,Y)Z=R(X,Y)Z-12n-1[S(Y,Z)X-S(X,Z)Y+g(Y,Z)QX-g(X,Z)QY]+12n(2n-1)[g(Y,Z)X-g(X,Z)Y].Definition 4.1.
A(k,μ)-contact metric manifold M(k,μ) is (1)
ξ-conformally flat if  C(X,Y)ξ=0,(2)
conformally Ricci symmetric ifC·S=0,(3)
ϕ-conformally flat if 'C(ϕX,ϕY,ϕZ,ϕW)=0 for all X, Y, Z, and W∈T(M).


### 4.1.ξ-Conformally Flat (k,μ)-Contact Metric Manifolds

Suppose that(k,μ)-contact metric manifold M(k,μ) is ξ-conformally flat. Then from (4.1), we obtain
(4.2)R(X,Y)ξ=12n-1[S(Y,ξ)X-S(X,ξ)Y+g(Y,ξ)QX-g(X,ξ)QY]-12n(2n-1)[g(Y,ξ)X-g(X,ξ)Y].
Using (2.1),  (2.6), and (2.11) in (4.2), we obtain
(4.3)(k-2nk2n-1+12n(2n-1))(η(Y)X-η(X)Y)+μ(η(Y)hX-η(X)hY)-1(2n-1)(η(Y)QX-η(X)QY)=0.
Putting Y=ξ in (4.3) and using (2.1) and (2.10), we obtain
(4.4)QX=(2n-1)(2nk2n-1-(k-2nk2n-1+12n(2n-1)))η(X)ξ+(2n-1)(k-2nk2n-1+12n(2n-1))X+(2n-1)μ(hX).
Contraction of the above with Y yields
(4.5)S(X,Y)=(2nk(2n-1)-12n)η(X)η(Y)+1-2nk2ng(X,Y)+(2n-1)μg(hX,Y).
From (4.5), we have the following.
(4.6)S(X,Y)=αg(X,Y)+βη(X)η(Y),
with α=(1-2nk)/n and β=(2nk(2n+1)-1)/2n  if and only if μ=0.Thus we have the following.Theorem 4.2.
Aξ-conformally flat (k,μ)-contact metric manifold reduces to N(k)-contact metric manifold if and only if it is an η-Einstein manifold.


### 4.2. Conformally Ricci-Symmetric(k,μ)-Contact Metric Manifolds

IfC·S=0, then we have
(4.7)S(C(ξ,X)Y,Z)+S(Y,C(ξ,X)Z)=0.
Taking Z=ξ in (4.7) and using (4.1),  (2.6), (2.9) to (2.12), we obtain
(4.8)(2nk2-4n2k22n-1+2nk(2n-2+k-nμ)2n-1+bμ(k-1)-b2(k-1)2n-1)g(X,Y)+(2nkμ-aμ+ab2n-1)g(hY,X)+((a+c)2nk2n-1+2cnk2n-1-4n2k22n-1-bμ(k-1)+b2(k-1)2n-1)η(Y)η(X)=(k-a2n-1+(2n-2+k-nμ)2n-1)S(X,Y),
where
(4.9)a=2(n-1)-nμ,b=2(n-1)+μ,c=2(1-n)+n(2k+μ).
Taking Y=ξ in (4.8), we obtain μ=-2k.Ifk=1, then h=0, μ=-2.Thus fork=1, (4.8) reduces to
(4.10)S(X,Y)=αg(X,Y)+βη(X)η(Y),
where
(4.11)α=2n(2n-1)5n-2,β=2n(1-n)5n-2,
that is, M(k,μ) reduces to η-Einstein.Thus we have the following.Theorem 4.3.
A conformally Ricci-symmetric(1,-2)-contact metric manifold is an η-Einstein manifold.


### 4.3.ϕ-Conformally Flat (k,μ)-Contact Metric Manifolds

SupposeM(k,μ) is ϕ-conformally flat, that is, 'C(ϕX,ϕY,ϕZ,ϕW)=0 for all vector fields X, Y, Z, and W. Then from (4.1), we obtain
(4.12)′R(ϕX,ϕY,ϕZ,ϕW)=12n-1(S(ϕY,ϕZ)g(ϕX,ϕW)-S(ϕX,ϕZ)g(ϕX,ϕW)+g(ϕY,ϕZ)S(ϕX,ϕW)-g(ϕX,ϕZ)S(ϕY,ϕW))+r2n(2n-1)(g(ϕY,ϕZ)g(ϕX,ϕW)-g(ϕX,ϕZ)g(ϕY,ϕW)).
Let {e1,e2,…e2n,ξ} be a local orthonormal basis of the tangent space TP(M) at P in M(k,μ).TakingX=W=ei in (4.12) and summing up from 1 to 2n, we obtain
(4.13)∑i=12n‍R′(ϕei,ϕY,ϕZ,ϕei)=12n-1∑i=12n‍(S(ϕY,ϕZ)g(ϕei,ϕei)-g(ϕei,ϕZ)S(ϕY,ϕei))-S(ϕei,ϕZ)g(ϕY,ϕei)+g(ϕY,ϕZ)S(ϕei,ϕei)+r2n(2n-1)∑i=12n‍(g(ϕY,ϕZ)g(ϕei,ϕei)-g(ϕei,ϕZ)g(ϕY,ϕei)).
Using (3.22) in (4.13), we obtain
(4.14)S(ϕY,ϕZ)=r(4n-1)+4n2k2ng(ϕY,ϕZ).
Replacing Y by ϕY and Z by ϕZ in (4.14) and using (2.1), we have
(4.15)S(Y,Z)=αg(Y,Z)+βη(Y)η(Z),
where
(4.16)α=r(4n-1)+4n2k2n,β=-r(4n-1)2n.
From the relation (4.15), we conclude that M(k,μ) is an η-Einstein manifold.Hence we can state the following.Theorem 4.4.
Aϕ-conformally flat (k,μ)-contact metric manifold is an η-Einstein manifold with α=(r(4n-1)+4n2k)/2n,β=-r(4n-1)/2n.


## 4.1.ξ-Conformally Flat (k,μ)-Contact Metric Manifolds

Suppose that(k,μ)-contact metric manifold M(k,μ) is ξ-conformally flat. Then from (4.1), we obtain
(4.2)R(X,Y)ξ=12n-1[S(Y,ξ)X-S(X,ξ)Y+g(Y,ξ)QX-g(X,ξ)QY]-12n(2n-1)[g(Y,ξ)X-g(X,ξ)Y].
Using (2.1),  (2.6), and (2.11) in (4.2), we obtain
(4.3)(k-2nk2n-1+12n(2n-1))(η(Y)X-η(X)Y)+μ(η(Y)hX-η(X)hY)-1(2n-1)(η(Y)QX-η(X)QY)=0.
Putting Y=ξ in (4.3) and using (2.1) and (2.10), we obtain
(4.4)QX=(2n-1)(2nk2n-1-(k-2nk2n-1+12n(2n-1)))η(X)ξ+(2n-1)(k-2nk2n-1+12n(2n-1))X+(2n-1)μ(hX).
Contraction of the above with Y yields
(4.5)S(X,Y)=(2nk(2n-1)-12n)η(X)η(Y)+1-2nk2ng(X,Y)+(2n-1)μg(hX,Y).
From (4.5), we have the following.
(4.6)S(X,Y)=αg(X,Y)+βη(X)η(Y),
with α=(1-2nk)/n and β=(2nk(2n+1)-1)/2n  if and only if μ=0.Thus we have the following.Theorem 4.2.
Aξ-conformally flat (k,μ)-contact metric manifold reduces to N(k)-contact metric manifold if and only if it is an η-Einstein manifold.


## 4.2. Conformally Ricci-Symmetric(k,μ)-Contact Metric Manifolds

IfC·S=0, then we have
(4.7)S(C(ξ,X)Y,Z)+S(Y,C(ξ,X)Z)=0.
Taking Z=ξ in (4.7) and using (4.1),  (2.6), (2.9) to (2.12), we obtain
(4.8)(2nk2-4n2k22n-1+2nk(2n-2+k-nμ)2n-1+bμ(k-1)-b2(k-1)2n-1)g(X,Y)+(2nkμ-aμ+ab2n-1)g(hY,X)+((a+c)2nk2n-1+2cnk2n-1-4n2k22n-1-bμ(k-1)+b2(k-1)2n-1)η(Y)η(X)=(k-a2n-1+(2n-2+k-nμ)2n-1)S(X,Y),
where
(4.9)a=2(n-1)-nμ,b=2(n-1)+μ,c=2(1-n)+n(2k+μ).
Taking Y=ξ in (4.8), we obtain μ=-2k.Ifk=1, then h=0, μ=-2.Thus fork=1, (4.8) reduces to
(4.10)S(X,Y)=αg(X,Y)+βη(X)η(Y),
where
(4.11)α=2n(2n-1)5n-2,β=2n(1-n)5n-2,
that is, M(k,μ) reduces to η-Einstein.Thus we have the following.Theorem 4.3.
A conformally Ricci-symmetric(1,-2)-contact metric manifold is an η-Einstein manifold.


## 4.3.ϕ-Conformally Flat (k,μ)-Contact Metric Manifolds

SupposeM(k,μ) is ϕ-conformally flat, that is, 'C(ϕX,ϕY,ϕZ,ϕW)=0 for all vector fields X, Y, Z, and W. Then from (4.1), we obtain
(4.12)′R(ϕX,ϕY,ϕZ,ϕW)=12n-1(S(ϕY,ϕZ)g(ϕX,ϕW)-S(ϕX,ϕZ)g(ϕX,ϕW)+g(ϕY,ϕZ)S(ϕX,ϕW)-g(ϕX,ϕZ)S(ϕY,ϕW))+r2n(2n-1)(g(ϕY,ϕZ)g(ϕX,ϕW)-g(ϕX,ϕZ)g(ϕY,ϕW)).
Let {e1,e2,…e2n,ξ} be a local orthonormal basis of the tangent space TP(M) at P in M(k,μ).TakingX=W=ei in (4.12) and summing up from 1 to 2n, we obtain
(4.13)∑i=12n‍R′(ϕei,ϕY,ϕZ,ϕei)=12n-1∑i=12n‍(S(ϕY,ϕZ)g(ϕei,ϕei)-g(ϕei,ϕZ)S(ϕY,ϕei))-S(ϕei,ϕZ)g(ϕY,ϕei)+g(ϕY,ϕZ)S(ϕei,ϕei)+r2n(2n-1)∑i=12n‍(g(ϕY,ϕZ)g(ϕei,ϕei)-g(ϕei,ϕZ)g(ϕY,ϕei)).
Using (3.22) in (4.13), we obtain
(4.14)S(ϕY,ϕZ)=r(4n-1)+4n2k2ng(ϕY,ϕZ).
Replacing Y by ϕY and Z by ϕZ in (4.14) and using (2.1), we have
(4.15)S(Y,Z)=αg(Y,Z)+βη(Y)η(Z),
where
(4.16)α=r(4n-1)+4n2k2n,β=-r(4n-1)2n.
From the relation (4.15), we conclude that M(k,μ) is an η-Einstein manifold.Hence we can state the following.Theorem 4.4.
Aϕ-conformally flat (k,μ)-contact metric manifold is an η-Einstein manifold with α=(r(4n-1)+4n2k)/2n,β=-r(4n-1)/2n.


## 5. Pseudoprojective Curvature Tensor in(k,μ)-Contact Metric Manifolds

InM(k,μ), the pseudoprojective curvature tensor P~ is given by [11]
(5.1)P~(X,Y)Z=aR(X,Y)Z+b[S(Y,Z)X-S(X,Z)Y]-r2n+1(a2n+b)[g(Y,Z)X-g(X,Z)Y],
where a and b are constants such that a, b≠0, R is the curvature tensor, S is the Ricci tensor, r is the scalar curvature.


### 5.1. Pseudoprojectively Symmetric(k,μ)-Contact Metric Manifolds

SupposeR·P~=0 holds in M(k,μ). Then we have
(5.2)R(ξ,X)P~(Y,Z)W-P~(R(ξ,X)Y,Z)W-P~(Y,R(ξ,X)Z)W-P~(Y,Z)R(ξ,X)W=0.
Taking Y=W=ξ in (5.2) and using (5.1),  (2.5),  (2.10), and (2.11), we have
(5.3)(ak2+2bnk2-kr2n+1(a2n+b)+bμ(2(n-1)+μ)(k-1))[η(Z)η(X)-g(Z,X)]ξ-[akμ+2nkbμ-μ-bμ(2(n-1)-nμ)]g(Z,hX)ξ+(2kr2n+1(a2n+b)2ak2-4nk2b)η(X)(η(Z)ξ-Z)+(ak2+2bnk2-kr2n+1(a2n+b)-aμ2(k-1))η(Z)(η(X)ξ-X)+η(Z)hX(-akμ+akμη(Y))+η(X)hZ(-2akμ-μ-akμη(Y))+(ak2+2bnk2-kr2n+1(a2n+b))η(Y)[η(Z)X-η(X)Z]+μ2(k-1)[η(Z)η(X)ξ-η(Z)X]+ak2[g(X,Z)ξ-η(X)Z]+kbS(Z,X)ξ-k(2nkb+r2n+1(a2n+b))η(X)Z-kr2n+1(a2n+b)g(Z,X)ξ=0.
Contracting the above with ξ, we obtain
(5.4)S(Z,X)=1kb([2nk2b+bμ((2n-1)+μ)(k-1)]g(Z,X)+[2kr2n+1(a2n+b)-bμ((2n-1)+μ)(k-1)]η(X)η(Z)+[2nkbμ-bμ((2n-1)-nμ)(k-1)]g(Z,hX)[2nk2b+bμ((2n-1)+μ)(k-1)]).
Replacing X by hX in (5.4) and using (2.7) and (2.10), we obtain
(5.5)g(Z,hX)=2nkbμ-bμl-kbmkbl-(2nk2b+bμm(k-1))(k-1)[η(X)η(Z)-g(X,Z)],
where
(5.6)l=2(n-1)-nμ,m=2(n-1)+μ.
Substituting for g(Z,hX) in (5.4), we obtain
(5.7)S(Z,X)=αg(Z,X)+βη(X)η(Z),
where
(5.8)α=1kb(2nk2b+bμm(k-1)-(2nkbμ-bμl-kbm)(2nkbμ-bμl)kbl-(2nk2b+bμm(k-1))(k-1)),β=1kb(2kr2n+1(a2n+b)-bμm(k-1)+(2nkbμ-bμl-kbm)(2nkbμ-bμl)kbl-(2nk2b+bμm(k-1))(k-1)).
From relation (5.7), we conclude that M(k,μ) is an η-Einstein manifold.Hence we can state the following.Theorem 5.1.
A pseudoprojective symmetric(k,μ)-contact metric manifold is an η-Einstein manifold.


### 5.2. Pseudoprojective Ricci-Symmetric(k,μ)-Contact Metric Manifolds

IfP~·S=0, then we have
(5.9)S(P~(ξ,X)Y,Z)+S(Y,P~(ξ,X)Z)=0.
Taking Y=ξ in (5.9) and using (5.1),  (2.1), (2.5),  (2.10), and (2.11), we obtain
(5.10)S(X,Z)=1m-ak([2mnk-aμ(k-1)q-a2nk2]g(X,Z)+[aμl-aμ2nk]g(hX,Z)+aμ(k-1)qη(X)η(Z)[2mnk-aμ(k-1)q-a2nk2]),
where
(5.11)l=2(n-1)-nμ,q=2(n-1)+μ,m=r2n+1(a2n+b).
Replacing X by hX in (5.10) and using (2.7) and (2.10), we obtain
(5.12)g(hX,Z)=[aμl-aμ2nk-q(m-ak)](k-1)l(m-ak)-[2mnk-aμ(k-1)q-a2nk2][η(X)η(Z)-g(X,Z)].
Now substituting for g(hX,Z) in (5.10), we obtain
(5.13)S(X,Z)=αg(X,Z)+βη(X)η(Y),
where
(5.14)α=1m-ak(2mnk-aμ(k-1)q-a2nk2-[aμl-aμ2nk-q(m-ak)](k-1)l(m-ak)-[2mnk-aμ(k-1)q-a2nk2]),β=1m-ak(aμ(k-1)q+[aμl-aμ2nk-q(m-ak)](k-1)l(m-ak)-[2mnk-aμ(k-1)q-a2nk2]).
From (5.13), we have that M(k,μ) is an η-Einstein manifold.Hence we can state the following.Theorem 5.2.
A(k,μ)-contact metric manifold is an η-Einstein manifold if P~·S=0.


## 5.1. Pseudoprojectively Symmetric(k,μ)-Contact Metric Manifolds

SupposeR·P~=0 holds in M(k,μ). Then we have
(5.2)R(ξ,X)P~(Y,Z)W-P~(R(ξ,X)Y,Z)W-P~(Y,R(ξ,X)Z)W-P~(Y,Z)R(ξ,X)W=0.
Taking Y=W=ξ in (5.2) and using (5.1),  (2.5),  (2.10), and (2.11), we have
(5.3)(ak2+2bnk2-kr2n+1(a2n+b)+bμ(2(n-1)+μ)(k-1))[η(Z)η(X)-g(Z,X)]ξ-[akμ+2nkbμ-μ-bμ(2(n-1)-nμ)]g(Z,hX)ξ+(2kr2n+1(a2n+b)2ak2-4nk2b)η(X)(η(Z)ξ-Z)+(ak2+2bnk2-kr2n+1(a2n+b)-aμ2(k-1))η(Z)(η(X)ξ-X)+η(Z)hX(-akμ+akμη(Y))+η(X)hZ(-2akμ-μ-akμη(Y))+(ak2+2bnk2-kr2n+1(a2n+b))η(Y)[η(Z)X-η(X)Z]+μ2(k-1)[η(Z)η(X)ξ-η(Z)X]+ak2[g(X,Z)ξ-η(X)Z]+kbS(Z,X)ξ-k(2nkb+r2n+1(a2n+b))η(X)Z-kr2n+1(a2n+b)g(Z,X)ξ=0.
Contracting the above with ξ, we obtain
(5.4)S(Z,X)=1kb([2nk2b+bμ((2n-1)+μ)(k-1)]g(Z,X)+[2kr2n+1(a2n+b)-bμ((2n-1)+μ)(k-1)]η(X)η(Z)+[2nkbμ-bμ((2n-1)-nμ)(k-1)]g(Z,hX)[2nk2b+bμ((2n-1)+μ)(k-1)]).
Replacing X by hX in (5.4) and using (2.7) and (2.10), we obtain
(5.5)g(Z,hX)=2nkbμ-bμl-kbmkbl-(2nk2b+bμm(k-1))(k-1)[η(X)η(Z)-g(X,Z)],
where
(5.6)l=2(n-1)-nμ,m=2(n-1)+μ.
Substituting for g(Z,hX) in (5.4), we obtain
(5.7)S(Z,X)=αg(Z,X)+βη(X)η(Z),
where
(5.8)α=1kb(2nk2b+bμm(k-1)-(2nkbμ-bμl-kbm)(2nkbμ-bμl)kbl-(2nk2b+bμm(k-1))(k-1)),β=1kb(2kr2n+1(a2n+b)-bμm(k-1)+(2nkbμ-bμl-kbm)(2nkbμ-bμl)kbl-(2nk2b+bμm(k-1))(k-1)).
From relation (5.7), we conclude that M(k,μ) is an η-Einstein manifold.Hence we can state the following.Theorem 5.1.
A pseudoprojective symmetric(k,μ)-contact metric manifold is an η-Einstein manifold.


## 5.2. Pseudoprojective Ricci-Symmetric(k,μ)-Contact Metric Manifolds

IfP~·S=0, then we have
(5.9)S(P~(ξ,X)Y,Z)+S(Y,P~(ξ,X)Z)=0.
Taking Y=ξ in (5.9) and using (5.1),  (2.1), (2.5),  (2.10), and (2.11), we obtain
(5.10)S(X,Z)=1m-ak([2mnk-aμ(k-1)q-a2nk2]g(X,Z)+[aμl-aμ2nk]g(hX,Z)+aμ(k-1)qη(X)η(Z)[2mnk-aμ(k-1)q-a2nk2]),
where
(5.11)l=2(n-1)-nμ,q=2(n-1)+μ,m=r2n+1(a2n+b).
Replacing X by hX in (5.10) and using (2.7) and (2.10), we obtain
(5.12)g(hX,Z)=[aμl-aμ2nk-q(m-ak)](k-1)l(m-ak)-[2mnk-aμ(k-1)q-a2nk2][η(X)η(Z)-g(X,Z)].
Now substituting for g(hX,Z) in (5.10), we obtain
(5.13)S(X,Z)=αg(X,Z)+βη(X)η(Y),
where
(5.14)α=1m-ak(2mnk-aμ(k-1)q-a2nk2-[aμl-aμ2nk-q(m-ak)](k-1)l(m-ak)-[2mnk-aμ(k-1)q-a2nk2]),β=1m-ak(aμ(k-1)q+[aμl-aμ2nk-q(m-ak)](k-1)l(m-ak)-[2mnk-aμ(k-1)q-a2nk2]).
From (5.13), we have that M(k,μ) is an η-Einstein manifold.Hence we can state the following.Theorem 5.2.
A(k,μ)-contact metric manifold is an η-Einstein manifold if P~·S=0.


## 6. Ricci Semisymmetric(k,μ)-Contact Metric Manifolds

If a(2n+1)-dimensional (k,μ)-contact metric manifold is Ricci semisymmetric, then R·S=0. That is,
(6.1)S(R(W,X)Y,Z)+S(Y,R(W,X)Z)=0.
Taking W=Y=ξ in (6.1) and using (2.5), (2.7), and  (2.11), we obtain
(6.2)S(X,Z)=1k[(2nk2+bμ(k-1))g(X,Z)+(2nkμ-aμ)g(hX,Z)-bμ(k-1)η(X)η(Z)(2nk2+bμ(k-1))],
where
(6.3)a=2(n-1)-nμ,b=2(n-1)+μ.
Replacing Z by hZ in (6.2) and using (2.7) and (2.10), we obtain
(6.4)g(X,hZ)=(2nkμ-aμ-bk)(k-1)ak-2nk2-bμ(k-1)(η(X)η(Z)-g(X,Z)).
Then (6.2) reduces to
(6.5)S(X,Z)=αg(X,Z)+βη(X)η(Z),
where
(6.6)a=2(n-1)-nμ,b=2(n-1)+μ,α=1k(2nk2+bμ(k-1)-(2nkμ-aμ)(2nkμ-aμ-bk)(k-1)ak-2nk2-bμ(k-1)),β=1k((2nkμ-aμ)(2nkμ-aμ-bk)(k-1)ak-2nk2-bμ(k-1)-bμ(k-1)).
From relation (6.5), we conclude that the manifold is an η-Einstein manifold.Hence we can state the following.Theorem 6.1.
A Ricci semisymmetric(k,μ)-contact metric manifold is an η-Einstein manifold.



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*Source: 970682-2012-11-05.xml*