JMATH Journal of Mathematics 2314-4785 2314-4629 Hindawi Publishing Corporation 10.1155/2015/728298 728298 Research Article On ϕ-Symmetric N(k)-Paracontact Metric Manifolds Prakasha D. G. Mirji K. K. Nacinovich Mauro Department of Mathematics Karnatak University Dharwad 580 003 India kud.ac.in 2015 29102015 2015 30 07 2015 11 10 2015 13 10 2015 29102015 2015 Copyright © 2015 D. G. Prakasha and K. K. Mirji. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The notions of ϕ-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.

1. Introduction

The study of paracontact geometry was initiated by Kaneyuki and Williams . A systematic study of paracontact metric manifolds and their subclasses were started out by Zamkovoy . Since then, several geometers studied paracontact metric manifolds and obtained various important properties of these manifolds (, etc.). The geometry of paracontact metric manifolds can be related to the theory of Legendre foliations. In , 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 , 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.  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 study ϕ-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,ηϕ=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 field ξ 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.

For N(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)ϕ2WRX,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)ϕ2WRX,YZ=0,for all vector fields X,Y,Z,W orthogonal to ξ.

3. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M98"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow></mml:math></inline-formula>-Symmetric <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M99"><mml:mi>N</mml:mi><mml:mo mathvariant="bold">(</mml:mo><mml:mi>k</mml:mi><mml:mo mathvariant="bold">)</mml:mo></mml:math></inline-formula>-Paracontact Metric Manifolds

Let us consider ϕ-symmetric N(k)-paracontact metric manifold. Then, by virtue of (1) and (13), we have(15)WRX,YZ-ηWRX,YZξ=0.Taking the inner product of (15) by U, we have(16)gWRX,YZ,U-ηWRX,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, 1i(2n+1), we have(17)WSY,Z-i=12n+1ηWRei,YZηei=0.Considering the second term of (17) and setting Z=ξ, we have(18)i=12n+1ηWRei,Yξηei=i=12n+1gWRei,Yξ,ξgei,ξ.Next,(19)gWRei,Yξ,ξ=gWRei,Yξ,ξ-gRWei,Yξ,ξ-gRei,WYξ,ξ-gRei,YWξ,ξ.Since ei is an orthonormal basis, Xei=0. Using (6), we have(20)gRei,WYξ,ξ=gkηWYei-ηeiWY,ξ=kgWY,ξgei,ξ-gei,ξgWY,ξ=0.Using (20) in (19), we have(21)gWRei,Yξ,ξ=gWRei,Yξ,ξ-gRei,YWξ,ξ.Since g(R(ei,Y)ξ,ξ)=-g(R(ξ,ξ)Y,ei)=0, we have(22)gWRei,Yξ,ξ+gRei,Yξ,Wξ=0.Using (22) in (21), we have(23)gWRei,Yξ,ξ=-gRei,Yξ,Wξ-gRei,YWξ,ξ.Using (4) in (23), we have(24)gWRei,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-SWY,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 Locally <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M136"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow></mml:math></inline-formula>-Symmetric <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M137"><mml:mi>N</mml:mi><mml:mo mathvariant="bold">(</mml:mo><mml:mi>k</mml:mi><mml:mo mathvariant="bold">)</mml:mo></mml:math></inline-formula>-Paracontact Metric Manifolds

For a three-dimensional semi-Riemannian manifold, the conformal curvature tensor C 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.

If C=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,ZWηXξ-gY,ZηXϕW+gY,ZηXϕhW-gX,ZWηYξ+gX,ZηYϕW-gX,ZηYϕhW+WηYηZX+ηYWηZX-WηXηZY-ηXWηZY.Applying ϕ2 to both sides of (37), we have(38)ϕ2WRX,YZ=drW2gY,ZX-gX,ZY-gY,ZηXξ+gX,ZηYξ-ηYηZX+ηXηZY+3k-r2gY,ZηXϕhW-gX,ZηYϕhW-gY,ZηXϕW+gX,ZηYϕW+WηYηZX-WηYηZηXξ+ηYWηZX-WηXηZY+WηXηZηYξ-ηXWηZY.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-dimensional N(k)-paracontact metric manifold (M3,ϕ,ξ,η,g) is locally ϕ-symmetric if the scalar curvature tensor r of g is constant.

5. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M164"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow></mml:math></inline-formula>-Ricci Symmetric <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M165"><mml:mi>N</mml:mi><mml:mo mathvariant="bold">(</mml:mo><mml:mi>k</mml:mi><mml:mo mathvariant="bold">)</mml:mo></mml:math></inline-formula>-Paracontact Metric Manifolds Definition 5.

N ( k ) -paracontact metric manifold is said to be ϕ-Ricci symmetric if the Ricci operator Q satisfies(40)ϕ2XQY=0,for all vector fields X,Y on M.

If X,Y are orthogonal to ξ, then manifold is said to be locally ϕ-Ricci symmetric.

Using (1) in (40), we have(41)XQY-ηXQYξ=0.Taking the inner product of (41) with Z, we have(42)gXQY,Z-ηXQYηZ=0.Further simplification of (42) gives the following:(43)gXQY,Z-SXY,Z-ηXQYηZ=0.Putting Y=ξ in (43), we have(44)gXQξ,Z-SXξ,Z-ηXQξηZ=0.Using (4), (10), and (11) in (44), we have(45)SϕX,Z=2nkgϕX,Z-gϕhX,Z+SϕhX,Z+ηXQξηZ.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-Dimensional <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M196"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow></mml:math></inline-formula>-Ricci Symmetric <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M197"><mml:mi>N</mml:mi><mml:mo mathvariant="bold">(</mml:mo><mml:mi>k</mml:mi><mml:mo mathvariant="bold">)</mml:mo></mml:math></inline-formula>-Paracontact Metric Manifolds

On a 3-dimensional N(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)ϕ2WQX=drW2X-ηXξ-3k-r2ηXϕW.Taking X orthogonal to ξ in (51), we get the following form:(52)ϕ2WQX=drW2X.In view of the above equation, we are able to state the following theorem.

Theorem 7.

A 3-dimensional N(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 Locally <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M212"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow></mml:math></inline-formula>-Symmetric <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M213"><mml:mi>N</mml:mi><mml:mo mathvariant="bold">(</mml:mo><mml:mi>k</mml:mi><mml:mo mathvariant="bold">)</mml:mo></mml:math></inline-formula>-Paracontact Metric Manifolds with <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M214"><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula>

We consider the manifold M=R3 with the usual cartesian coordinates (x,y,z). The vector fields(53)e1=x+xz2y-2yz,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 x0.

Let be the Levi-Civita connection. Using the properties of paracontact metric structure and Koszul’s formula(56)2gXY,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=XYZ-YXZ-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 Theorem 4, it follows that the manifold under consideration is locally ϕ-symmetric.

Conflict of Interests

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

Acknowledgments

The authors are thankful to the Referee and the Editorial Board for the valuable comments and suggestions that helped them to improve the paper.

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