# Onϕ-Symmetric N(k)-Paracontact Metric Manifolds

Authors: D. G. Prakasha; K. K. Mirji
Journal: Journal of Mathematics
 (2015)

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## Abstract

The notions ofϕ-symmetric, 3-dimensional locally ϕ-symmetric, ϕ-Ricci symmetric, and 3-dimensional locally ϕ-Ricci symmetric N(k)-paracontact metric manifolds have been introduced and properties of these structures have been discussed.


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## Body

## 1. Introduction

The study of paracontact geometry was initiated by Kaneyuki and Williams [1]. A systematic study of paracontact metric manifolds and their subclasses were started out by Zamkovoy [2]. Since then, several geometers studied paracontact metric manifolds and obtained various important properties of these manifolds ([3–10], etc.). The geometry of paracontact metric manifolds can be related to the theory of Legendre foliations. In [11], the authors introduced the class of paracontact metric manifolds for which the characteristic vector field ξ belongs to the (k,μ)-nullity condition (or distribution) for some real constants k and μ. Such manifolds are known as (k,μ)-paracontact metric manifolds. If μ=0, then the notion of (k,μ)-nullity distribution reduces to k-nullity distribution. A paracontact metric manifold with ξ belonging to k-nullity distribution is called N(k)-paracontact metric manifold.In [12], Takahashi introduced the notion of locally ϕ-symmetric Sasakian manifold as a weaker version of local symmetry of such manifolds. In the context of contact geometry, the notion of ϕ-symmetry was introduced and studied by Boeckx et al. [13] with examples. In ([14, 15]), they studied the notion of ϕ-symmetry and discussed several examples for Kenmotsu manifolds and almost contact metric manifolds of dimension 3. In [16, 17], S. S. Shukla and M. K. Shukla, studied ϕ-Ricci symmetric Kenmotsu manifolds and ϕ-symmetric para-Sasakian manifolds.In the present work, we studyϕ-symmetry and Ricci ϕ-symmetry on N(k)-paracontact metric manifolds. In Section 2, we give a brief account of the N(k)-paracontact metric manifolds. In Section 3, we study the properties of ϕ-symmetric N(k)-paracontact metric manifolds. Section 4 deals with 3-dimensional locally ϕ-symmetric N(k)-paracontact metric manifolds. In this section, we prove that scalar curvature r is constant. Section 5 is devoted to studying the Ricci ϕ-symmetric N(k)-paracontact metric manifolds. Finally, we study the properties of 3-dimensional locally Ricci ϕ-symmetric N(k)-paracontact metric manifolds in Section 6.


## 2. Preliminaries

A(2n+1)-dimensional smooth manifold M2n+1 has an almost paracontact structure (ϕ,ξ,η,g) if it admits a tensor field ϕ of type (1,1), a vector field ξ, a 1-form η, and a Riemannian metric g satisfying the following conditions ([2, 18]):(1)ηX=gX,ξ,ηξ=1,η∘ϕ=0,ϕξ=0,ϕ2X=X-ηXξ,(2)gϕX,ϕY=-gX,Y+ηXηY,dηX,Y=gX,ϕY,for every vector field X,Y on M2n+1.In a paracontact metric manifold(M2n+1,ϕ,ξ,η,g), we define a (1,1) tensor field h by h=(1/2)£ξϕ, where £ denotes the operator of Lie differentiation. Then, h is symmetric and satisfies(3)hξ=0,hϕ=-ϕh,Tr·h=Tr·ϕh=0.If ∇ denotes the Levi-Civita connection of g, then we have the following relation:(4)∇Xξ=-ϕX+ϕhX.A paracontact metric manifold(M2n+1,ϕ,ξ,η,g) is said to be a (k,μ)-space if its curvature tensor R satisfies(5)RX,Yξ=kηYX-ηXY+μηYhX-ηXhY,for all tangent vector fields X,Y, where k,μ are smooth functions on M2n+1.Here, the characteristic vector fieldξ belongs to the (k,μ)-nullity distribution. A paracontact metric manifold with ξ belonging to (k,μ)-nullity distribution is called a (k,μ)-paracontact metric manifold. In particular, if μ=0, then the notion of (k,μ)-nullity distribution reduces to k-nullity distribution. A paracontact metric manifold such that ξ belongs to k-nullity distribution is called N(k)-paracontact metric manifold. Then, curvature tensor R reduces to the following form:(6)RX,Yξ=kηYX-ηXY.ForN(k)-paracontact metric manifold (M2n+1,ϕ,ξ,η,g)(n>1), the following identities hold:(7)h2=1+kϕ2,(8)∇XϕY=-gX-hX,Yξ+ηYX-hX,(9)∇XηY=gX-hX,ϕY,(10)SX,ξ=2nkηX,(11)Qξ=2nkξ,for any vector fields X,Y on M2n+1, where Q and S denote the Ricci operator and Ricci tensor of (M2n+1,g), respectively.N
(
k
)-paracontact metric manifold is called an Einstein manifold if it satisfies (12)SX,Y=λgX,Y,where λ is any scalar.Definition 1.
N
(
k
)-paracontact metric manifold is said to be ϕ-symmetric if(13)ϕ2∇WRX,YZ=0,for arbitrary vector fields X,Y,Z,W.Definition 2.
N
(
k
)-paracontact metric manifold is said to be locally ϕ-symmetric if(14)ϕ2∇WRX,YZ=0,for all vector fields X,Y,Z,W orthogonal to ξ.


## 3.ϕ-Symmetric N(k)-Paracontact Metric Manifolds

Let us considerϕ-symmetric N(k)-paracontact metric manifold. Then, by virtue of (1) and (13), we have(15)∇WRX,YZ-η∇WRX,YZξ=0.Taking the inner product of (15) by U, we have(16)g∇WRX,YZ,U-η∇WRX,YZηU=0.Let ei, i=1,2,…,(2n+1), be an orthonormal basis of the tangent space at any point p of the manifold. Then, putting X=U=ξ in (16) and taking summation over i, 1≤i≤(2n+1), we have(17)∇WSY,Z-∑i=12n+1η∇WRei,YZηei=0.Considering the second term of (17) and setting Z=ξ, we have(18)∑i=12n+1η∇WRei,Yξηei=∑i=12n+1g∇WRei,Yξ,ξgei,ξ.Next,(19)g∇WRei,Yξ,ξ=g∇WRei,Yξ,ξ-gR∇Wei,Yξ,ξ-gRei,∇WYξ,ξ-gRei,Y∇Wξ,ξ.Since ei is an orthonormal basis, ∇Xei=0. Using (6), we have(20)gRei,∇WYξ,ξ=gkη∇WYei-ηei∇WY,ξ=kg∇WY,ξgei,ξ-gei,ξg∇WY,ξ=0.Using (20) in (19), we have(21)g∇WRei,Yξ,ξ=g∇WRei,Yξ,ξ-gRei,Y∇Wξ,ξ.Since g(R(ei,Y)ξ,ξ)=-g(R(ξ,ξ)Y,ei)=0, we have(22)g∇WRei,Yξ,ξ+gRei,Yξ,∇Wξ=0.Using (22) in (21), we have(23)g∇WRei,Yξ,ξ=-gRei,Yξ,∇Wξ-gRei,Y∇Wξ,ξ.Using (4) in (23), we have(24)g∇WRei,Yξ,ξ=-gRei,Yξ,-ϕX+ϕhX-gRei,YϕX+ϕhX,ξ=gRei,Yξ,ϕX-gRei,Yξ,ϕhX+gRei,YϕX,ξ-gRei,YϕhX,ξ=0.Putting Z=ξ in (17) and using (24), it follows that(25)∇WSY,ξ=0.We know that(26)∇WSY,ξ=∇WSY,ξ-S∇WY,ξ-SY,∇Wξ.Using (4), (9), (10), and (25) in (26), we have(27)2nkgW-hW,ϕY-SY,-ϕW+ϕhW=0.Putting W=ϕW in (27) and using (1), (2), (3), and (10), we have(28)SY,W=2nkgY,W+2nkgY,hW-SY,hW.Again, putting W=hW in (28) and using (1) and (7), we obtain(29)2nkgY,hW-SY,hW=1+kSY,W-2nk1+kgY,W.By virtue of (28) and (29), we have(30)SY,W=2nkgY,W.Thus, we can state the following theorem.Theorem 3.
A(2n+1)-dimensional ϕ-symmetric N(k)-paracontact metric manifold (M2n+1,ϕ,ξ,η,g) is an Einstein manifold.


## 4.
Three-Dimensional Locallyϕ-Symmetric N(k)-Paracontact Metric Manifolds

For a three-dimensional semi-Riemannian manifold, the conformal curvature tensorC is given by(31)CX,YZ=RX,YZ-SY,ZX-SX,ZY+gY,ZQX-gX,ZQY+r2gY,ZX-gX,ZY,for arbitrary vector fields X,Y,Z.IfC=0, then (31) reduces to the following form:(32)RX,YZ=SY,ZX-SX,ZY+gY,ZQX-gX,ZQY-r2gY,ZX-gX,ZY.Putting Z=ξ in (32) and using (6), we get(33)r2-kηYξ-ηXξ=ηYQX-ηXQY.Again, putting Y=ξ in (33) and using (11), we get(34)QX=r2-kX+3k-r2ηXξ.Taking the inner product of (34) with Y, we obtain(35)SX,Y=r2-kgX,Y+3k-r2ηXηY.Using (34) and (35) in (32), we have(36)RX,YZ=r2-2kgY,ZX-gX,ZY+3k-r2gY,ZηXξ-gX,ZηYξ+ηYηZX-ηXηZY,where R is Riemannian curvature tensor on the 3-dimensional N(k)-paracontact metric manifold.Taking the covariant differentiation of (36) with respect to W, we have(37)∇WRX,YZ=drW2gY,ZX-gX,ZY-gY,ZηXξ+gX,ZηYξ-ηYηZX+ηXηZY+3k-r2gY,Z∇WηXξ-gY,ZηXϕW+gY,ZηXϕhW-gX,Z∇WηYξ+gX,ZηYϕW-gX,ZηYϕhW+∇WηYηZX+ηY∇WηZX-∇WηXηZY-ηX∇WηZY.Applying ϕ2 to both sides of (37), we have(38)ϕ2∇WRX,YZ=drW2gY,ZX-gX,ZY-gY,ZηXξ+gX,ZηYξ-ηYηZX+ηXηZY+3k-r2gY,ZηXϕhW-gX,ZηYϕhW-gY,ZηXϕW+gX,ZηYϕW+∇WηYηZX-∇WηYηZηXξ+ηY∇WηZX-∇WηXηZY+∇WηXηZηYξ-ηX∇WηZY.Now, taking X,Y,Z orthogonal to ξ and using (14), we get(39)drW2gY,ZX-gX,ZY=0.Hence, we can state the following theorem.Theorem 4.
A 3-dimensionalN(k)-paracontact metric manifold (M3,ϕ,ξ,η,g) is locally ϕ-symmetric if the scalar curvature tensor r of g is constant.


## 5.ϕ-Ricci Symmetric N(k)-Paracontact Metric Manifolds

Definition 5.
N
(
k
)-paracontact metric manifold is said to be ϕ-Ricci symmetric if the Ricci operator Q satisfies(40)ϕ2∇XQY=0,for all vector fields X,Y on M.
IfX,Y are orthogonal to ξ, then manifold is said to be locally ϕ-Ricci symmetric.Using (1) in (40), we have(41)∇XQY-η∇XQYξ=0.Taking the inner product of (41) with Z, we have(42)g∇XQY,Z-η∇XQYηZ=0.Further simplification of (42) gives the following:(43)g∇XQY,Z-S∇XY,Z-η∇XQYηZ=0.Putting Y=ξ in (43), we have(44)g∇XQξ,Z-S∇Xξ,Z-η∇XQξηZ=0.Using (4), (10), and (11) in (44), we have(45)SϕX,Z=2nkgϕX,Z-gϕhX,Z+SϕhX,Z+η∇XQξηZ.Putting Z=ϕZ in (45), we have(46)SϕX,ϕZ=2nkgϕX,ϕZ-gϕhX,ϕZ+SϕhX,ϕZ.Again, putting X=ϕX, Z=ϕZ in (46) and then using (1), (3), and (10), we have(47)SX,Z=2nkgX,Z+2nkghX,Z-ShX,Z.Replace Z=hZ in (47), and using (1), (7), and symmetric property of h, we have(48)2nkghX,Z-ShX,Z=k+1SX,Z-2nkgX,Z.By virtue of (47) and (48), we have(49)SY,W=2nkgY,W.Hence, we can state the following theorem.Theorem 6.
A(2n+1)-dimensional N(k)-paracontact metric manifold (M,ϕ,ξ,η,g) is ϕ-Ricci symmetric if g is an Einstein manifold.


## 6.
Three-Dimensionalϕ-Ricci Symmetric N(k)-Paracontact Metric Manifolds

On a 3-dimensionalN(k)-paracontact metric manifold, the Ricci operator Q is given by (34).Now, taking the covariant differentiation of (34) with respect to W, we have(50)∇WQX=drW2X-ηXξ-3k-r2ηXϕW+3k-r2ηXηWξ+3k-r2gW,ϕXξ+3k-r2ghW,ϕXξ.Applying ϕ2 to both sides of (50), we have(51)ϕ2∇WQX=drW2X-ηXξ-3k-r2ηXϕW.Taking X orthogonal to ξ in (51), we get the following form:(52)ϕ2∇WQX=drW2X.In view of the above equation, we are able to state the following theorem.Theorem 7.
A 3-dimensionalN(k)-paracontact metric manifold (M3,ϕ,ξ,η,g) is locally ϕ-Ricci symmetric if the scalar curvature tensor r of g is constant.


## 7. Example of 3-Dimensional Locallyϕ-Symmetric N(k)-Paracontact Metric Manifolds with k=-1

We consider the manifoldM=R3 with the usual cartesian coordinates (x,y,z). The vector fields(53)e1=∂∂x+xz2∂∂y-2y∂∂z,e2=∂∂y,e3=∂∂zare linearly independent at each point of M. We can compute(54)e1,e2=2e3,e1,e3=2xz3e2,e2,e3=0.We define the semi-Riemannian metric g as the nondegenerate one, whose only nonvanishing components are g(e1,e2)=g(e3,e3)=1, and the 1-form η as η=2ydx+dz, which satisfies η(e1)=η(e2)=0, η(e3)=1. Let ϕ be the (1,1)-tensor field defined by ϕe1=e1, ϕe2=-e2, and ϕξ=0. Then,(55)dηe1,e2=12e1ηe2-e2ηe1-ηe1,e2=-1=-ge1,e2=ge1,ϕe2,dηe1,e3=12e1ηe3-e3ηe1-ηe1,e3=0=ge1,ϕe3,dηe2,e3=12e2ηe3-e3ηe2-ηe2,e3=0=ge2,ϕe3.Therefore, (ϕ,ξ,η,g) is a paracontact metric structure on M.Moreover,he1=-(2x/z3)e2, he2=0, and hξ=0. Hence, h2=0 and, given p=(x,y,z)∈R3, rank(hp)=0 if x=0 and rank(hp)=1 if x≠0.Let∇ be the Levi-Civita connection. Using the properties of paracontact metric structure and Koszul’s formula(56)2g∇XY,Z=XgY,Z+YgZ,X-ZgX,Y-gX,Y,Z-gY,X,Z+gZ,X,Y,we can compute (57)∇e1e3=-e1+2xz3e2,∇e2e3=e2,∇e3e3=0,∇e1e1=-2xz3e3,∇e2e1=-e3,∇e3e1=-e1,∇e1e2=e3,∇e2e2=0,∇e3e2=e2.Hence, (M,ϕ,ξ,η,g) is N(k)-paracontact metric manifold with k=-1.Using the following definition of Riemannian curvature tensor(58)RX,YZ=∇X∇YZ-∇Y∇XZ-∇X,YZ,we obtain (59)Re1,e2e3=0,Re1,e3e3=-e1,Re2,e3e3=-e2,Re1,e2e2=-3e2,Re2,e3e2=0,Re1,e3e2=e3,Re1,e2e1=3e1,Re2,e3e1=e3,Re1,e3e1=4xz3e3.From this, it follows that ϕ2((∇WR)(X,Y)Z)=0 for all vector fields X, Y, and Z are orthogonal to ξ. Thus, the three-dimensional N(k)-paracontact metric manifold with k=-1 is locally ϕ-symmetric.Also from the above expressions for the curvature tensor, we obtain that the scalar curvature tensor is constant. Therefore, from Theorem4, it follows that the manifold under consideration is locally ϕ-symmetric.



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