# On Generalized Sasakian-Space-Forms

Authors: H. G. Nagaraja; G. Somashekhara; Savithri Shashidhar
Journal: ISRN Geometry
 (2012)

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## Abstract

The purpose of the present paper is to characterize pseudoprojectively flat and pseudoprojective semisymmetric generalized Sasakian-space-forms.


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## Body

## 1. Introduction

Alegre et al. [1] introduced and studied the generalized Sasakian-space-forms. The authors Alegre and Carriazo [2], Somashekhara and Nagaraja [3, 4], and De and Sarkar [5, 6] studied the generalized Sasakian-space-forms. An almost contact metric manifold (M,ϕ,ξ,η,g) is said to be a generalized Sasakian-space-form if there exist differentiable functions f1,f2,f3 such that curvature tensor R of M is given by
(1.1)R(X,Y)Z=f1R1(X,Y)Z+f2R2(X,Y)Z+f3R3(X,Y)Z,
for any vector fields X, Y, Z on M, where
(1.2)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.
In this paper, we study the curvature properties like flatness, symmetry, and semisymmetry properties in a generalized Sasakian-space-form by considering a pseudoprojective curvature tensor.The paper is organized as follows. Section2 of this paper contains some preliminary results on the generalized Sasakian-space-forms. In Section 3, we study the pseudoprojectively flat generalized Sasakian-space-form and obtain necessary and sufficient conditions for a generalized Sasakian-space-form to be pseudoprojectively flat. In the next section, we deal with pseudoprojectively semisymmetric generalized Sasakian-space-forms, and it is proved that a generalized Sasakian-space-form is pseudoprojectively semisymmetric if and only if the space form is pseudoprojectively flat and f1=f3. The last section is devoted to the study of τ-flat and τ-ϕ-semi symmetric generalized Sasakian-space-forms. In this section, we prove that the associated functions f1,f2,f3 are linearly dependent.In a(2n+1)-dimensional almost contact metric manifold, the pseudoprojective curvature tensor P~ [7] is defined by
(1.3)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 and R, S, and r are the Riemannian curvature tensor of type (0,2), the Ricci tensor, and the scalar curvature of the manifold, respectively. If a=1,  b=-(1/2n), then (1.3) takes the form
(1.4)P~(X,Y)Z=P(X,Y)Z,
where P is the projective curvature tensor. A manifold (M,ϕ,ξ,η,g)  shall be called pseudoprojectively flat if the pseudoprojective curvature tensor P~=0. It is known that the pseudoprojectively flat manifold is either projectively flat (if a≠0) or Einstein (if a=0 and b≠0).


## 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:
(2.1)ϕ2X=-X+η(X)ξ,ϕξ=0,(2.2)η(ξ)=1,g(X,ξ)=η(X),η(ϕX)=0,(2.3)g(ϕX,ϕY)=g(X,Y)-η(X)η(Y),(2.4)g(ϕX,Y)=-g(X,ϕY),(2.5)(∇Xη)Y=g(∇Xξ,Y),
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
(2.6)g(X,ϕY)=dη(X,Y),
for all vector fields X and Y.In a generalized Sasakian-space-form, the following hold:(2.7)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],(2.8)QX=(2nf1+3f2-f3)X-(3f2+(2n-1)f3)η(X)ξ,(2.9)S(X,Y)=(2nf1+3f2-f3)g(X,Y)-(3f2+(2n-1)f3)η(X)η(Y),(2.10)r=2n(2n+1)f1+6nf2-4nf3.


## 3. Pseudoprojectively Flat Generalized Sasakian-Space-Forms

If the generalized Sasakian-space-formM(f1,f2,f3) under consideration is pseudoprojectively flat, then from (1.3) we have
(3.1)′R(X,Y,Z,W)=ba[S(X,Z)g(Y,W)-S(Y,Z)g(X,W)]+r(2n+1)a(a2n+b)[g(Y,Z)g(X,W)-g(X,Z)g(Y,W)],
where a and b are constants and ′R(X,Y,Z,W)=g(R(X,Y)Z,W).Now takingZ=ξ in (3.1) and using (2.1), (2.2), (2.7), and (2.9), we get
(3.2)(f1-f3)[η(Y)g(X,W)-η(X)g(Y,W)]=2nba(f1-f3)(η(Y)g(X,W)-η(X)g(Y,W))+r(2n+1)a(a2n+b)×(η(Y)g(X,W)-η(X)g(Y,W)).
Again putting X=ξ in (3.2), we get
(3.3)[(a+2nba)(2n(2n+1)(f1-f3)-r2n(2n+1))][η(Y)η(W)-g(Y,W)]=0.
The aforementioned equation implies
(3.4)(a+2nba)[2n(2n+1)(f1-f3)-r2n(2n+1)]=0.
That is, either
(3.5)(a+2nb)=0
or
(3.6)r=2n(2n+1)(f1-f3).
If a+2nb=0, a≠0 and b≠0, then, from (1.3), it follows that P~(X,Y)Z=aP(X,Y)Z. Thus in this case pseudoprojective flatness and projective flatness are equivalent.Ifa+2nb≠0, a≠0 and b≠0, then comparing (2.10) and (3.6), we get
(3.7)3f2+(2n-1)f3=0.
Using (3.7) in (2.9), we get
(3.8)S(X,Y)=2n(f1-f3)g(X,Y).
Let {ei} be an orthonormal basis of the tangent space at each point of the manifold. Taking X=Y=ei and summing over 1≤i≤2n+1, we obtain
(3.9)r=2n(2n+1)(f1-f3).
This shows that M(f1,f2,f3) is Einstein with a scalar curvature r=2n(2n+1)(f1-f3). Thus we state the following.Theorem 3.1.
A pseudoprojectively flat generalized Sasakian-space-form is either projectively flat or an Einstein manifold with a scalar curvaturer=[2n(2n+1)(f1-f3)].Suppose that (3.7) holds. Then in view of (2.7) and (2.9), we can write (1.3) as
(3.10)′P~(X,Y,Z,W)=af1(g(Y,Z)g(X,W)-g(X,Z)g(Y,W))+af2[g(X,ϕZ)g(ϕY,W)-g(Y,ϕZ)g(ϕX,W)+2g(X,ϕY)g(ϕZ,W)]+af3[η(X)η(Z)g(Y,W)-η(Y)η(Z)g(X,W)+η(Y)g(X,Z)g(ξ,W)-η(X)g(Y,Z)g(ξ,W)]+b[S(Y,Z)g(X,W)-S(X,Z)g(Y,W)]-(r2n+1)(a2n+b)[g(Y,Z)g(X,W)-g(X,Z)g(Y,W)],
where
(3.11)′P~(X,Y,Z,W)=g(P~(X,Y)Z,W).
Replacing X by ϕX and Y by ϕY,  we get
(3.12)P~(ϕX,ϕY,Z,W)=af3(g(ϕY,Z)g(ϕX,W)-g(ϕX,Z)g(ϕY,W))+af2(g(ϕX,ϕZ)g(ϕ2Y,W)-g(ϕY,ϕZ)g(ϕ2X,W)+2g(ϕX,ϕ2Y)g(ϕZ,W)).
Let {ei} be an orthonormal basis of the tangent space at each point of the manifold.TakingY=W=ei and summation over i,  1≤i≤2n+1, we get
(3.13)∑i=12n+1‍P~(ϕX,ϕei,Z,ei)=af3(g(ϕX,ϕZ))+af2(-g(ϕX,ϕZ)g(ϕei,ϕei)-g(ϕ2X,ϕ2Z)).
Again putting X =Z = ei and taking summation over i,  1≤i≤2n+1, we get f2=0 with a≠0. In view of (3.7), we get f3=0.Now (2.7) reduces to the form
(3.14)R(X,Y)Z=f1[g(Y,Z)X-g(X,Z)Y],
from which we have S(X,Y)=2nf1g(X,Y), and consequently
(3.15)r=2n(2n+1)f1.
By using (3.14) and (3.15) in (1.3), we get P~(X,Y)Z=0. This leads to the following.Theorem 3.2.
A(2n+1)-dimensional generalized Sasakian-space-form M(f1,f2,f3) is pseudoprojectively flat if and only if a+2nb≠0, a≠0, b≠0 and 3f2+(2n-1)f3=0.Alegre and Carriazo [2] proved that any contact metric generalized Sasakian-space-form M(f1,f2,f3) with a dimension greater than or equal to five is a Sasakian manifold and f1, f2, and f3 must be constants.Thus from (3.14), we have the following theorem.Theorem 3.3.
A(2n+1)-dimensional generalized Sasakian-space-form M(f1,f2,f3) with a dimension greater than or equal to 5 is of constant curvature f1 if and only if a+2nb≠0, a≠0, b≠0, and 3f2+(2n-1)f3=0.


## 4. Pseudoprojective Semisymmetric Generalized Sasakian-Space-Form

Definition 4.1.
If a generalized Sasakian-space-formM(f1,f2,f3) satisfies
(4.1)R(X,Y)·P~=0,
then the manifold is said to be pseudoprojectively semisymmetric manifold.By using (1.3), (2.1), (2.2), (2.7), and (2.9), we have
(4.2)η(P~(X,Y)Z)=[a(f1-f3)-(r2n+1)(a2n+b)][g(Y,Z)η(X)-g(X,Z)η(Y)]+b[S(Y,Z)η(X)-S(Y,Z)η(Y)].
Taking Z=ξ in (4.2), we get
(4.3)η(P~(X,Y)ξ)=0.
Again putting X=ξ in (4.2), we get
(4.4)(η(P~(ξ,Y)Z))=[a(f1-f3)-(r2n+1)(a2n+b)][g(Y,Z)-η(Y)η(Z)]+b[S(Y,Z)-2n(f1-f3)η(Y)η(Z)].
From (4.1), we have
(4.5)(R(X,Y)P~(U,V)W)-P~(R(X,Y)U,V)W-P~(U,R(X,Y)V)W-P~(U,V)R(X,Y)W=0.
Taking X=ξ and contracting the above with respect to ξ, we get
(4.6)(f1-f3){′P¯(U,V,W,Y)-η(Y)η(P¯(U,V)W)+η(U)η(P¯(Y,V)W)-g(Y,U)η(P¯(ξ,V)W)+η(V)η(P¯(U,Y)W)-g(Y,V)η(P¯(U,ξ)W)+η(W)η(P¯(U,V)Y)-g(Y,W)η(P¯(U,V)ξ)}=0.
Putting U=Y in (4.6) and with the help of (4.2) and (4.3), we get either
(4.7)f1=f3
or
(4.8)′P¯(Y,V,W,Y)-g(Y,Y)η(P¯(ξ,V)W)-g(Y,V)η(P¯(Y,ξ)W)+η(W)η(P¯(Y,V)Y)=0.
Let {ei} be an orthonormal basis of the tangent space at each point of the manifold of the manifold. Putting Y=ei and taking summation over i, 1≤i≤2n+1, and using (4.2) and (4.4), we obtain
(4.9)S(V,W)=Ag(V,W)+Bη(V)η(W),
where
(4.10)A=2nf1+3f2-f3,B=(2n+1)[-3f2-(2n-1)f3].
Now contracting (4.9), we obtain
(4.11)r=(2n+1)A+B.
Using (4.10) in (4.11), we get
(4.12)r=2n(2n+1)(f1-f3).
In view of (2.10), (4.12) yields
(4.13)3f2+(2n-1)f3=0.
From (2.9) and (4.13), we have
(4.14)S(V,W)=2n(f1-f3)g(V,W).
Now using (4.12) and (4.14) in (4.2), we get
(4.15)η(P~(U,V)W)=0.
Plugging (4.15) in (4.6), we obtain
(4.16)P~(U,V,W,Y)=0.
Therefore by taking into account (4.7) and (4.16), we have either f1=f3 or M(f1,f2,f3) is pseudoprojectively flat.Conversely, suppose thatf1=f3. Then, from (2.1), (2.2) and (2.7), we have R(ξ,X)Y=0. Hence R(ξ,U)·P~=0. If the space-form is pseudoprojectively flat then clearly it is pseudoprojectively semisymmetric. Hence we can state the following.Theorem 4.2.
A2n+1-dimensional generalized Sasakian-space-form is pseudoprojectively semisymmetric if and only if the space form is either pseudoprojectively flat or f1=f3.By combining Theorems3.2 and 4.2, we have the following.Corollary 4.3.
A(2n+1)-dimensional generalized Sasakian-space-form M(f1,f2,f3) is pseudoprojectively flat if and only if f1=f3 or a+2nb≠0 and 3f2+(2n-1)f3=0.


## 5.τ-Curvature Tensor in a Generalized Sasakian-Space-Form

In a(2n+1)-dimensional Riemannian manifold M, the τ-curvature tensor is given by [8]
(5.1)τ(X,Y)Z=a0R(X,Y)Z+a1S(Y,Z)X+a2S(X,Z)Y+a3S(X,Y)Z+a4g(Y,Z)QX+a5g(X,Z)QY+a6g(X,Y)QZ+a7r(g(Y,Z)X-g(X,Z)Y),
where a0,…,a7 are some smooth functions on M. For different values of a0,…,a7, the τ-curvature tensor reduces to the curvature tensor, quasiconformal curvature tensor, conformal curvature tensor, conharmonic curvature tensor, concircular curvature tensor, pseudoprojective curvature tensor, projective curvature tensor, M-projective curvature tensor, Wi-curvature tensors (i=0,…9), and Wj*-curvature tensors (j=0,1).Suppose thatM(f1,f2,f3) is τ-flat. Then from (5.1), we have
(5.2)-a0R(X,Y)Z=a1S(Y,Z)X+a2S(X,Z)Y+a3S(X,Y)Z+a4g(Y,Z)QX+a5g(X,Z)QY+a6g(X,Y)QZ+a7r(g(Y,Z)X-g(X,Z)Y).
In view of (2.7), (2.8), and (2.9) in (5.2), we have
(5.3)-a0{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]}=a1[(2nf1+3f2-f3)g(Y,Z)-(3f2+(2n-1)f3)η(Y)η(Z)]X+a2[(2nf1+3f2-f3)g(X,Z)-(3f2+(2n-1)f3)η(X)η(Z)]Y+a3[(2nf1+3f2-f3)g(X,Y)-(3f2+(2n-1)f3)η(X)η(Y)]Z+a4g(Y,Z)[(2nf1+3f2-f3)X-(3f2+(2n-1)f3)η(X)ξ]+a5g(X,Z)[(2nf1+3f2-f3)Y-(3f2+(2n-1)f3)η(Y)ξ]+a6g(X,Y)[(2nf1+3f2-f3)Z-(3f2+(2n-1)f3)η(Z)ξ]+a7r[g(Y,Z)X-g(X,Z)Y].
Putting X=ϕY in (5.3), we get
(5.4)-a0{f1[g(Y,Z)ϕY-g(ϕY,Z)Y]+f2[g(ϕY,ϕZ)ϕY-g(Y,ϕZ,)ϕ2Y+2g(ϕY,ϕY)ϕZ]+f3[-η(Y)η(Z)ϕY+g(ϕY,Z)η(Y)ξ][g(ϕY,ϕZ)ϕY-g(Y,ϕZ,)ϕ2Y+2g(ϕY,ϕY)ϕZ]}=a1[(2nf1+3f2-f3)g(Y,Z)-(3f2+(2n-1)f3)η(Y)η(Z)]ϕY+a2(2nf1+3f2-f3)g(ϕY,Z)Y+a4(2nf1+3f2-f3)g(Y,Z)ϕY+a5g(ϕY,Z)[(2nf1+3f2-f3)Y-(3f2+(2n-1)f3)η(Y)ξ]+a7r[g(Y,Z)ϕY-g(ϕY,Z)Y].
If we choose a unit vector U orthogonal to ξ and taking Y=U, then making use of (2.1) and (2.3) in (5.4), we obtain
(5.5)[-a0f1+(a2+a5)(2nf1+3f2-f3)-a7r+f2]g(ϕU,Z)U+[a0(f1+f2)+(a1+a4)(2nf1+3f2-f3)+a7r]g(U,Z)ϕU+2a0f2g(U,U)ϕZ=0.
Putting Z=U in (5.5), we have
(5.6)λ1f1+λ2f2+λ3f3=0,
where
(5.7)λ1=a0+2n(a1+a4)+2n(2n+1)a7,λ2=3(a0+a1+a4+2na7),λ3=-(a1+a4+4na7).
Thus we have the following.Theorem 5.1.
If a(2n+1)-dimensional generalized Sasakian-space-form M(f1,f2,f3) is τ-flat, then (5.6) holds.From the above theorem, we discuss the following cases.Case
(
i
). (1) If M(f1,f2,f3) is quasiconformally flat, then a1 = -a2 = a4 = -a5, a3 = a6 = 0, a7 = (-1/(2n+1))(a0/2n+2a1). Putting these in (5.7), we obtain λ1≠0,λ2≠0,λ3≠0.(
2
) If M(f1,f2,f3) is conharmonically flat, then a0=1,a1 = -a2 = a4 = -a5 = -(1/(2n-1)), a3 = a6 = 0, a7 = 0. Putting these in (5.7), we get λ1≠0,λ2≠0,λ3≠0.Similarly forW0*-flat, W1-flat, W3-flat, W9-flat spaces, (5.7) gives λ1≠0,λ2≠0,λ3≠0.Case
(
ii
). If M(f1,f2,f3) is conformally flat, then a0=1, a1 = -a2 = a4 = -a5 = -(1/(2n-1)), a3=a6=0, a7=1/2n(2n-1).Putting these in (5.7), we obtain λ1=0,λ2≠0,λ3=0. Hence f2=0.Case
(
iii
). (a) If M(f1,f2,f3) is pseudoprojectively flat, then a1=-a2,a3=a4=a5=a6=0,a7=-(1/(2n+1))(a0/2n+a1).By putting these values in (5.7), we have λ1=0,λ2≠0,λ3≠0.(
b
) If M(f1,f2,f3) is projectively flat, then a0=1,a1=-a2=-(1/2n),a3=a4=a5=a6=a7=0.Making use of the above functional values in (5.7), we get λ1=0,λ2≠0,λ3≠0.Similarly for concircularly flat,M-projectively flat, W0-flat, W1*-flat, W2-flat, W6-flat, and W8-flat spaces, (5.7) gives λ1=0,λ2≠0,λ3≠0.Case
(
iv
). (a) If M(f1,f2,f3) is W4-flat, then a0=1,a5=-a6=1/2n,a1=a2=a3=a4=a7=0.Putting these in (5.7), we obtain that λ1≠0,λ2≠0,λ3=0.(
b
) If M(f1,f2,f3) is W5-flat, then a0=1,a2=-a5=-(1/2n),a1=a3=a4=a6=a7=0. Putting these in (5.7), we have λ1≠0,λ2≠0,λ3=0.Similarly, for aW7-flat space, (5.7) gives λ1≠0,λ2≠0,λ3=0.Summarizing the above cases, we have the following corollaries.Corollary 5.2.
If a(2n+1)-dimensional generalized Sasakian-space-form M(f1,f2,f3) is either quasiconformally flat, conharmonically flat, W0*-flat, W1-flat, W3-flat, or W9-flat, then f1,f2, and f3 are linearly dependent.Corollary 5.3.
If a(2n+1)-dimensional generalized Sasakian-space-form M(f1,f2,f3) is conformally flat, then f2=0.The above corollary was already proved by Kim [9] and Sarkar and De [10].Corollary 5.4.
If a(2n+1)-dimensional generalized Sasakian-space-form M(f1,f2,f3) is either pseudoprojectively flat, projectively flat, concircularly flat, M-projectively flat, W0-flat, W1*-flat, W2-flat, W6-flat, or W8-flat, then f2 and f3 are linearly dependent.Corollary 5.5.
If a(2n+1)-dimensional generalized Sasakian-space-form M(f1,f2,f3) is either W4-flat, W5-flat, or W7-flat, then f1 and f2 are linearly dependent.


### 5.1.τ-ϕ-Semisymmetric Generalized Sasakian-Space-Form

Definition 5.6.
M
(
f
1
,
f
2
,
f
3
) is τ-ϕ-semisymmetric if
(5.8)τ(X,Y)·ϕ=0
holds   in   M(f1,f2,f3).We know that(5.9)(τ(X,Y)·ϕ)Z=τ(X,Y)ϕZ-ϕ(τ(X,Y)Z).
From (5.8) and (5.9), we have
(5.10)τ(X,Y)ϕZ-ϕ(τ(X,Y)Z)=0.
By using (5.1) in (5.10), we have
(5.11)a0R(X,Y)ϕZ+a1S(Y,ϕZ)X+a2S(X,ϕZ)Y+a3S(X,Y)ϕZ+a4g(Y,ϕZ)QX+a5g(X,ϕZ)QY+a6g(X,Y)Q(ϕZ)+a7r[g(Y,ϕZ)X-g(X,ϕZ)Y]-a7r[g(Y,Z)ϕX-g(X,Z)ϕY]-{a0ϕ(R(X,Y)ϕZ)+a1S(Y,Z)ϕX+a2S(X,Z)ϕY+a3S(X,Y)ϕZ=+a4g(Y,Z)ϕ(QX)+a5g(X,Z)ϕ(QY)+a6g(X,Y)ϕ(QZ)}=0.
Let {ei} be an orthonormal basis of the tangent space at each point of the manifold. Contracting (5.11) with respect to W and putting Y=W=ei, also taking summation over i, 1≤i≤2n+1, and making use of (2.1), (2.4), (2.7), (2.9), and (2.8), we have
(5.12)[2a1+(2n+1)a2]S(X,ϕZ)=Ag(X,ϕZ),
where
(5.13)A=[-(2n-1)a0+4na4+2n(2n+1)a5-2n(4n2-1)a7]f1+[2(n-1)a0+6a4+6na5-6n(2n-1)a7]f2+[-2a4-4na5+4n(2n-1)a7]f3.
Changing Z to ϕZ in (5.12) and also in view of (2.1) and (2.2), (2.9) yields
(5.14)S(X,Z)=[A(2a1+(2n+1)a2)]g(X,Z)+[2n(2a1+(2n+1)a2)(f1-f3)-A(2a1+(2n+1)a2)]η(X)η(Z).
Thus we can state the following.Theorem 5.7.
Aτ-ϕ-semisymmetric generalized Sasakian-space-form is η-Einstein provided (2a1+(2n+1)a2)≠0.


## 5.1.τ-ϕ-Semisymmetric Generalized Sasakian-Space-Form

Definition 5.6.
M
(
f
1
,
f
2
,
f
3
) is τ-ϕ-semisymmetric if
(5.8)τ(X,Y)·ϕ=0
holds   in   M(f1,f2,f3).We know that(5.9)(τ(X,Y)·ϕ)Z=τ(X,Y)ϕZ-ϕ(τ(X,Y)Z).
From (5.8) and (5.9), we have
(5.10)τ(X,Y)ϕZ-ϕ(τ(X,Y)Z)=0.
By using (5.1) in (5.10), we have
(5.11)a0R(X,Y)ϕZ+a1S(Y,ϕZ)X+a2S(X,ϕZ)Y+a3S(X,Y)ϕZ+a4g(Y,ϕZ)QX+a5g(X,ϕZ)QY+a6g(X,Y)Q(ϕZ)+a7r[g(Y,ϕZ)X-g(X,ϕZ)Y]-a7r[g(Y,Z)ϕX-g(X,Z)ϕY]-{a0ϕ(R(X,Y)ϕZ)+a1S(Y,Z)ϕX+a2S(X,Z)ϕY+a3S(X,Y)ϕZ=+a4g(Y,Z)ϕ(QX)+a5g(X,Z)ϕ(QY)+a6g(X,Y)ϕ(QZ)}=0.
Let {ei} be an orthonormal basis of the tangent space at each point of the manifold. Contracting (5.11) with respect to W and putting Y=W=ei, also taking summation over i, 1≤i≤2n+1, and making use of (2.1), (2.4), (2.7), (2.9), and (2.8), we have
(5.12)[2a1+(2n+1)a2]S(X,ϕZ)=Ag(X,ϕZ),
where
(5.13)A=[-(2n-1)a0+4na4+2n(2n+1)a5-2n(4n2-1)a7]f1+[2(n-1)a0+6a4+6na5-6n(2n-1)a7]f2+[-2a4-4na5+4n(2n-1)a7]f3.
Changing Z to ϕZ in (5.12) and also in view of (2.1) and (2.2), (2.9) yields
(5.14)S(X,Z)=[A(2a1+(2n+1)a2)]g(X,Z)+[2n(2a1+(2n+1)a2)(f1-f3)-A(2a1+(2n+1)a2)]η(X)η(Z).
Thus we can state the following.Theorem 5.7.
Aτ-ϕ-semisymmetric generalized Sasakian-space-form is η-Einstein provided (2a1+(2n+1)a2)≠0.



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*Source: 309486-2012-12-27.xml*