# Onϕ-Recurrent Para-Sasakian Manifold Admitting

Authors: K. T. Pradeep Kumar; Venkatesha; C. S. Bagewadi
Journal: ISRN Geometry
 (2012)

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## Abstract

We obtained the relation between the Riemannian connection
and the quarter-symmetric metric connection on a para-Sasakian manifold.
Further, we studyϕ-recurrent and concircular ϕ-recurrent para-Sasakian
manifolds with respect to quarter-symmetric metric connection.


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## Body

## 1. Introduction

The idea of metric connection with torsion in a Riemannian manifold was introduced by Hayden [1]. Further, some properties of semisymmetric metric connection have been studied by Yano [2]. In [3], Golab defined and studied quarter-symmetric connection on a differentiable manifold with affine connection, which generalizes the idea of semisymmetric connection. Various properties of quarter-symmetric metric connection have been studied by many geometers like Rastogi [4, 5], Mishra and Pandey [6], Yano and Imai [7], De et al. [8, 9], Pradeep Kumar et al. [10], and many others.The notion of local symmetry of a Riemannian manifold has been weakened by many authors in several ways to a different extent. As a weaker version of local symmetry, Takahashi [11] introduced the notion of local ϕ-symmetry on a Sasakian manifold. Generalizing the notion of ϕ-symmetry, the authors De et al. [12] introduced the notion of ϕ-recurrent Sasakian manifolds.A linear connection∇̃ on an n-dimensional differentiable manifold is said to be a quarter-symmetric connection [3] if its torsion tensor T is of the form
(1.1)T(X,Y)=∇̃XY-∇̃YX-[X,Y]=η(Y)ϕX-η(X)ϕY,
where η is a 1-form and ϕ is a tensor of type (1,1). In particular, if we replace ϕX by X and ϕY by Y, then the quarter-symmetric connection reduces to the semisymmetric connection [13]. Thus, the notion of quarter-symmetric connection generalizes the idea of the semisymmetric connection. And if quarter-symmetric linear connection ∇̃ satisfies the condition
(1.2)(∇̃Xg)(Y,Z)=0,
for all X,Y,Z∈𝒳(M), where 𝒳(M) is the Lie algebra of vector fields on the manifold M, then ∇̃ is said to be a quarter-symmetric metric connection.


## 2. Preliminaries

Ann-dimensional differentiable manifold M is called an almost paracontact manifold if it admits an almost paracontact structure (ϕ,ξ,η) consisting of a (1,1) tensor field ϕ, a vector field ξ, and a 1-form η satisfying
(2.1)ϕ2X=X-η(X)ξ,(2.2)η(ξ)=1,ϕ∘ξ=0,η∘ϕ=0.Ifg is a compatible Riemannian metric with (ϕ,ξ,η), that is,
(2.3)g(ϕX,ϕY)=g(X,Y)-η(X)η(Y),g(X,ξ)=η(X)(2.4)g(X,ϕY)=g(ϕX,Y),
for all vector fields X and Y on M, then M becomes a almost paracontact Riemannian manifold equipped with an almost paracontact Riemannian structure (ϕ,ξ,η,g).An almost paracontact Riemannian manifold is called a para-Sasakian manifold if it satisfies(2.5)(∇Xϕ)Y=-g(X,Y)ξ-η(Y)X+2η(X)η(Y)ξ,
where ∇ denotes the operator of covariant differentiation. From the above equation it follows that
(2.6)∇Xξ=ϕX,(∇Xη)Y=g(X,ϕY)=(∇Yη)X.In ann-dimensional para-Sasakian manifold M, the following relations hold [14, 15]:
(2.7)η(R(X,Y)Z)=g(X,Z)η(Y)-g(Y,Z)η(X),(2.8)R(X,Y)ξ=η(X)Y-η(Y)X,(2.9)S(X,ξ)=-(n-1)η(X),(2.10)S(ϕX,ϕY)=S(X,Y)+(n-1)η(X)η(Y),
for any vector fields X,Y, and Z, where R and S are the Riemannian curvature tensor and the Ricci tensor of M, respectively.A para-Sasakian manifoldM is said to be η-Einstein if its Ricci tensor S is of the form
(2.11)S(X,Y)=ag(X,Y)+bη(X)η(Y),
for any vector fields X and Y, where a and b are some functions on M.Definition 2.1.
A para-Sasakian manifold is said to be locallyϕ-symmetric if
(2.12)ϕ2((∇WR)(X,Y)Z)=0,
for all vector fields X,Y,Z,W orthogonal to ξ. This notion was introduced for Sasakian manifold by Takahashi [11].Definition 2.2.
A para-Sasakian manifold is said to be locally concircularϕ-symmetric if
(2.13)ϕ2((∇WC¯)(X,Y)Z)=0,
for all vector fields X,Y,Z,W orthogonal to ξ. Where the concircular curvature tensor C¯ is given by [16]
(2.14)C¯(X,Y)Z=R(X,Y)Z-rn(n-1)[g(Y,Z)X-g(X,Z)Y],
where R is the Riemannian curvature tensor and r is the scalar curvature.Definition 2.3.
A para-Sasakian manifold is said to beϕ-recurrent if there exists a nonzero 1-form A such that
(2.15)ϕ2((∇WR)(X,Y)Z)=A(W)R(X,Y)Z,
where A is a 1-form and it is defined by
(2.16)A(W)=g(W,ρ),
and ρ is a vector field associated with the 1-form A.


## 3. Quarter-Symmetric Metric Connection

Let∇̃ be a linear connection and ∇ a Riemannian connection of an almost contact metric manifold M such that
(3.1)∇̃XY=∇XY+U(X,Y),
where U is a tensor of type (1,1). For ∇̃ to be a quarter-symmetric metric connection in M, then we have [3]
(3.2)U(X,Y)=12[T(X,Y)+T′(X,Y)+T′(Y,X)],(3.3)g(T′(X,Y),Z)=g(T(Z,X),Y).
From (1.1) and (3.3), we get
(3.4)T′(X,Y)=η(X)ϕY-g(ϕX,Y)ξ.
Using (1.1) and (3.4) in (3.2), we obtain
(3.5)U(X,Y)=η(Y)ϕX-g(ϕX,Y)ξ.
Thus a quarter-symmetric metric connection ∇̃ in a para-Sasakian manifold is given by
(3.6)∇̃XY=∇XY+η(Y)ϕX-g(ϕX,Y)ξ.
Hence (3.6) is the relation between Riemannian connection and the quarter-symmetric metric connection on a para-Sasakian manifold.A relation between the curvature tensor ofM with respect to the quarter-symmetric metric connection ∇̃ and the Riemannian connection ∇ is given by
(3.7)R̃(X,Y)Z=R(X,Y)Z+3g(ϕX,Z)ϕY-3g(ϕY,Z)ϕX+η(Z)[η(X)Y-η(Y)X]-[η(X)g(Y,Z)-η(Y)g(X,Z)]ξ,
where R̃ and R denote the Riemannian curvatures of the connections ∇̃ and ∇, respectively. From (3.7), it follows that
(3.8)S̃(Y,Z)=S(Y,Z)+2g(Y,Z)-(n+1)η(Y)η(Z),
where S̃ and S are the Ricci tensors of the connections ∇̃ and ∇, respectively.Contracting (3.8), we get
(3.9)r̃=r+(n-1),
where r̃ and r are the scalar curvatures of the connections ∇̃ and ∇, respectively.


## 4.ϕ-Recurrent Para-Sasakian Manifold with respect to Quarter-Symmetric Metric Connection

A para-Sasakian manifold is calledϕ-recurrent with respect to the quarter-symmetric metric connection if its curvature tensor R̃ satisfies the condition
(4.1)ϕ2((∇̃WR̃)(X,Y)Z)=A(W)R̃(X,Y)Z.
By virtue of (2.1) and (4.1), we have
(4.2)(∇̃WR̃)(X,Y)Z-η((∇̃WR̃)(X,Y)Z)ξ=A(W)R̃(X,Y)Z.
From which, it follows that
(4.3)g((∇̃WR̃)(X,Y)Z,U)-η((∇̃WR̃)(X,Y)Z)g(ξ,U)=A(W)g(R̃(X,Y)Z,U).
Let {ei},i=1,2,…,n be an orthonormal basis of the tangent space at any point of the manifold. Then putting X=U=ei in (4.3) and taking summation over i,  1≤i≤n, we get
(4.4)(∇̃WS̃)(Y,Z)-∑i=1nη((∇̃WR̃)(ei,Y)Z)η(ei)=A(W)S̃(Y,Z).
The second term of (4.4) by putting Z=ξ takes the form
(4.5)g((∇̃WR̃)(ei,Y)ξ,ξ)=g(∇̃WR̃(ei,Y)ξ,ξ)-g(R̃(∇̃Wei,Y)ξ,ξ)-g(R̃(ei,∇̃WY)ξ,ξ)-g(R̃(ei,Y)∇̃Wξ,ξ).
On simplification we obtain
(4.6)g((∇̃WR̃)(ei,Y)Z,ξ)=0.
Therefore (4.4) can be written in the form
(4.7)(∇̃WS̃)(Y,Z)=A(W)S̃(Y,Z).
Replacing Z by ξ in the above relation, then using (3.8) and (2.9), we have(4.8)(∇̃WS̃)(Y,ξ)=-2(n-1)A(W)η(Y).
We know that
(4.9)(∇̃WS̃)(Y,ξ)=∇̃WS̃(Y,ξ)-S̃(∇̃WY,ξ)-S̃(Y,∇̃Wξ).
Using (3.8), (2.6) and (2.9) in the above relation, we get
(4.10)(∇̃WS̃)(Y,ξ)=-4(n-1)g(Y,ϕW)-2S(Y,ϕW)+4g(Y,ϕW).
In view of (4.8) and (4.10), we obtain
(4.11)-4(n-1)g(Y,ϕW)-2S(Y,ϕW)+4g(Y,ϕW)=-2(n-1)A(W)η(Y).
Replacing Y by ϕY in (4.11) and then using (2.3) and (2.10), we have
(4.12)S(Y,W)=-2(n-2)g(Y,W)+(n-3)η(Y)η(W).
Hence, we can state the following.Theorem 4.1.
If para-Sasakian manifold isϕ-recurrent with respect to quarter-symmetric metric connection then it is an η-Einstein manifold with respect to Riemannian connection.


## 5. Concircularϕ-Recurrent Para-Sasakian Manifold with respect to Quarter-Symmetric Metric Connection

A concircularϕ-recurrent para-Sasakian manifold with respect to the quarter-symmetric metric connection is defined by
(5.1)ϕ2((∇̃WC̅̃)(X,Y)Z)=A(W)C̅̃(X,Y)Z,
where C̅̃ is a concircular curvature tensor with respect to the quarter-symmetric metric connection given by
(5.2)C̅̃(X,Y)Z=R̃(X,Y)Z-r̃n(n-1)[g(Y,Z)X-g(X,Z)Y].
By virtue of (2.1) and (5.1), we have
(5.3)(∇̃WC̅̃)(X,Y)Z-η((∇̃WC̅̃)(X,Y)Z)ξ=A(W)C̅̃(X,Y)Z,
from which it follows that
(5.4)g((∇̃WC̅̃)(X,Y)Z,U)-η((∇̃WC̅̃)(X,Y)Z)g(ξ,U)=A(W)g(C̅̃(X,Y)Z,U),
where
(5.5)(∇̃WC̅̃)(X,Y)Z=((∇WR)(X,Y)Z)+6[g(ϕY,Z)g(W,X)-g(ϕX,Z)g(W,Y)]ξ+6[η(Y)g(W,Z)+η(Z)g(W,Y)]ϕX-6[η(X)g(W,Z)+η(Z)g(W,X)]ϕY+2[η(Y)g(X,Z)-η(X)g(Y,Z)]ϕW+6[η(X)g(ϕY,Z)-η(Y)g(ϕX,Z)]W+12η(W)η(Z)[η(X)ϕY-η(Y)ϕX]+η(Z)[g(W,Y)X-g(W,X)Y]+2η(W)η(Z)[η(X)Y-η(Y)X]+12η(W)[η(Y)g(ϕX,Z)-η(X)g(ϕY,Z)]ξ+η(W)[η(Y)g(X,Z)-η(X)g(Y,Z)]ξ+η(Z)[g(ϕW,X)Y-g(ϕW,Y)X]+g(W,Z)[η(Y)X-η(X)Y]+[g(W,X)g(Y,Z)-g(W,Y)g(X,Z)]ξ-[g(ϕW,X)g(Y,Z)-g(ϕW,Y)g(X,Z)]ξ+g(ϕW,Z)[η(X)Y-η(Y)X]-∇Wrn(n-1)[g(Y,Z)X-g(X,Z)Y].
Let {ei},i=1,2,…,n be an orthonormal basis of the tangent space at any point of the manifold. Then putting X=U=ei in (5.4) and taking summation over i,1≤i≤n, we get
(5.6)(∇WS)(Y,Z)=∇Wrng(Y,Z)+(n+4)η(Z)g(ϕW,Y)+(n+3)η(Y)g(ϕW,Z)+(2n-3)η(W)η(Y)η(Z)-(n-1)η(Y)g(W,Z)-∇Wrn(n-1)[g(Y,Z)-η(Y)η(Z)]+A(W)S(Y,Z)-A(W){(n+1)η(Y)η(Z)+r-(n+1)ng(Y,Z)}.
Replacing Z by ξ in (5.6) and using (2.9), we have
(5.7)(∇WS)(Y,ξ)=∇Wrnη(Y)+(n+4)g(ϕW,Y)+(n-2)η(W)η(Y)-A(W)η(Y)[2n+r-(n+1)n].
We know that
(5.8)(∇WS)(Y,ξ)=∇WS(Y,ξ)-S(∇WY,ξ)-S(Y,∇Wξ).
Using (2.6) and (2.9) in the above relation, it follows that
(5.9)(∇WS)(Y,ξ)=-(n-1)[g(ϕW,Y)]-S(Y,ϕW).
In view of (5.7) and (5.9), we obtain
(5.10)S(Y,ϕW)=-(n-1)g(ϕW,Y)-∇Wrnη(Y)-(n+4)g(ϕW,Y)-(n-2)η(W)η(Y)+A(W)η(Y)[2n+r-(n+1)n].
Replacing Y by ϕY in (5.10) and then using (2.3) and (2.10), we obtain
(5.11)S(Y,W)=-(2n+3)g(W,Y)+(n+4)η(W)η(Y).
This leads to the following theorem.Theorem 5.1.
If para-Sasakian manifold is concircularϕ-recurrent with respect to quarter-symmetric metric connection then it is an η-Einstein manifold with respect to Riemannian connection.Now from (5.3), we have
(5.12)(∇̃WC̅̃)(X,Y)Z=η((∇̃WC̅̃)(X,Y)Z)ξ+A(W)C̅̃(X,Y)Z.
This gives
(5.13)((∇WR)(X,Y)Z)=η((∇WR)(X,Y)Z)ξ+6[η(Y)g(W,Z)-η(Z)g(W,Y)]ϕX+6[η(X)g(W,Z)+η(Z)g(W,X)]ϕY+2[η(X)g(Y,Z)-η(Y)g(X,Z)]ϕW-6[η(X)g(ϕY,Z)-η(Y)g(ϕX,Z)]W-2η(W)η(Z)[η(X)Y-η(Y)X]+12η(W)η(Z)[η(Y)ϕX-η(X)ϕY]-η(Z)[g(W,Y)X-g(W,X)Y]+η(Z)[g(ϕW,Y)X-g(ϕW,X)Y]+η(Z)[η(X)g(W,Y)-η(Y)g(W,X)]ξ+η(Z)[η(Y)g(ϕW,X)-η(X)g(ϕW,Y)]ξ-g(W,Z)[η(Y)X-η(X)Y]+g(ϕW,Z)[η(Y)X-η(X)Y]+6η(W)[η(X)g(ϕY,Z)-η(Y)g(ϕX,Z)]ξ+∇Wrn(n-1)[g(Y,Z)X-g(X,Z)Y-η(X)g(Y,Z)ξ+η(Y)g(X,Z)ξ]+A(W)R(X,Y)Z+3A(W)[g(ϕX,Z)ϕY-g(ϕY,Z)ϕX]+A(W)η(Z)[η(X)Y-η(Y)X]-A(W)[η(X)g(Y,Z)-η(Y)g(X,Z)]ξ-r+(n-1)n(n-1)A(W)[g(Y,Z)X-g(X,Z)Y].
Now from (5.13) and Bianchi’s second identity, we have
(5.14)A(W)η(R(X,Y)Z)+A(X)η(R(Y,W)Z)+A(Y)η(R(W,X)Z)=(n+1)(n-1)+rn(n-1)A(W)[η(X)g(Y,Z)-η(Y)g(X,Z)]+(n+1)(n-1)+rn(n-1)A(X)[η(Y)g(W,Z)-η(W)g(Y,Z)]+(n+1)(n-1)+rn(n-1)A(Y)[η(W)g(X,Z)-η(X)g(W,Z)].
By virtue of (2.7), we obtain from (5.14) that
(5.15)A(W)[g(X,Z)η(Y)-g(Y,Z)η(X)]+A(X)[g(Y,Z)η(W)-g(W,Z)η(Y)]+A(Y)[g(W,Z)η(X)-g(X,Z)η(W)]=(n+1)(n-1)+rn(n-1)A(W)[η(X)g(Y,Z)-η(Y)g(X,Z)]+(n+1)(n-1)+rn(n-1)A(X)[η(Y)g(W,Z)-η(W)g(Y,Z)]+(n+1)(n-1)+rn(n-1)A(Y)[η(W)g(X,Z)-η(X)g(W,Z)].
Putting Y=Z=ei in (5.15) and taking summation over i, 1≤i≤n, we get
(5.16)A(W)η(X)=A(X)η(W),
for all vector fields X,W. Replacing X by ξ in (5.16), we get
(5.17)A(W)=η(W)η(ρ),
for any vector field W.Hence from (5.16) and (5.17), we can state the following.Theorem 5.2.
In a concircularϕ-recurrent para-Sasakian manifold with respect to quarter-symmetric metric connection, the characteristic vector field ξ and the vector field ρ associated to the 1-form A are in codirectional and the 1-form A is given by (5.17).



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*Source: 317253-2012-02-16.xml*