The object of the present paper is to characterize N(k)-contact metric manifolds satisfying certain curvature conditions on the conharmonic curvature tensor. In this paper we study conharmonically symmetric, ξ-conharmonically flat, and ϕ-conharmonically flat N(k)-contact metric manifolds.

1. Introduction

Let M and M¯ be two Riemannian manifolds with g and g¯ being their respective metric tensors related through g¯(X,Y)=e2σg(X,Y),
where σ is a real function. Then M and M¯ are called conformally related manifolds and the correspondence between M and M¯ is known as conformal transformation [1].

It is known that a harmonic function is defined as a function whose Laplacian vanishes. A harmonic function is not invariant, in general. The condition under which a harmonic function remains invariant have been studied by Ishii [2] who introduced the conharmonic transformation as a subgroup of the conformal transformation (1.1) satisfying the condition σ,ii+σ,iσ,i=0,
where comma denotes the covariant differentiation with respect to the metric g.

A rank-four tensor C̃ that remains invariant under conharmonic transformation for a (2n+1)-dimensional Riemannian manifold M is given by C̃(X,Y,Z,W)=R̃(X,Y,Z,W)-12n-1[g(Y,Z)S(X,W)-g(X,Z)S(Y,W)+S(Y,Z)g(X,W)-S(X,Z)g(Y,W)],
where R̃ denotes the Riemannian curvature tensor of type (0,4) defined by R̃(X,Y,Z,W)=g(R(X,Y)Z,W),
where R is the Riemannian curvature tensor of type (1,3) and S denotes Ricci tensor of type (0,2), respectively.

The curvature tensor defined by (1.3) is known as conharmonic curvature tensor. A manifold whose conharmonic curvature vanishes at every point of the manifold is called conharmonically flat manifold. Thus this tensor represents the deviation of the manifold from conharmonic flatness. It satisfies all the symmetric properties of the Riemannian curvature tensor R̃. There are many physical applications of the tensor C̃. For example, in [3], Abdussattar showed that sufficient condition for a space-time to be conharmonic to a flat space-time is that the tensor C̃ vanishes identically. A conharmonically flat space-time is either empty in which case it is flat or filled with a distribution represented by energy momentum tensor T possessing the algebraic structure of an electromagnetic field and conformal to a flat space-time [3]. Also he described the gravitational field due to a distribution of pure radiation in presence of disordered radiation by means of spherically symmetric conharmonically flat space-time. Conharmonic curvature tensor have been studied by Siddiqui and Ahsan [4], Özgür [5], and many others.

Let M be an almost contact metric manifold equipped with an almost contact metric structure (ϕ,ξ,η,g). At each point p∈M, decompose the tangent space TpM into direct sum TpM=ϕ(TpM)⊕{ξp}, where {ξp} is the 1-dimensional linear subspace of TpM generated by {ξp}. Thus the conformal curvature tensor C is a map C:TpM×TpM×TpM⟶ϕ(TpM)⊕{ξp},p∈M.

It may be natural to consider the following particular cases:

C:Tp(M)×Tp(M)×Tp(M)→L(ξp), that is, the projection of the image of C in ϕ(Tp(M)) is zero;

C:Tp(M)×Tp(M)×Tp(M)→ϕ(Tp(M)), that is, the projection of the image of C in L(ξp) is zero;

C:ϕ(Tp(M))×ϕ(Tp(M))×ϕ(Tp(M))→L(ξp), that is, when C is restricted to ϕ(Tp(M))×ϕ(Tp(M))×ϕ(Tp(M)), the projection of the image of C in ϕ(Tp(M)) is zero. This condition is equivalent to

ϕ2C(ϕX,ϕY,ϕZ)=0.

Here cases 1, 2, and 3 are synonymous to conformally symmetric, ξ-conformally flat, and ϕ-conformally flat.

In [6], it is proved that a conformally symmetric K-contact manifold is locally isometric to the unit sphere. In [7], it is proved that a K-contact manifold is ξ-conformally flat if and only if it is an η-Einstein Sasakian manifold. In [8], some necessary conditions for a K-contact manifold to be ϕ-conformally flat are proved. In [9], a necessary and sufficient condition for a Sasakian manifold to be ϕ-conformally flat is obtained. In [10], projective curvature tensor in K-contact and Sasakian manifolds has been studied. Moreover, the author [11] considered some conditions on conharmonic curvature tensor C̃, which has many applications in physics and mathematics, on a hypersurface in the semi-Euclidean space Esn+1. He proved that every conharmonically Ricci-symmetric hypersurface M satisfying the condition C̃·R=0 is pseudosymmetric. He also considered the condition C̃·C̃=LC̃Q(g,C̃) on hypersurfaces of the semi-Euclidean space Esn+1.

Motivated by the studies of conformal curvature tensor in (see [6–9]) and the studies of projective curvature tensor in K-contact and Sasakian manifolds in [10] and Lorentzian para-Sasakian manifolds in [5], in this paper we study conharmonic curvature tensor in N(k)-contact metric manifolds.

Analogous to the considerations of conformal curvature tensor, we give following definitions.

Definition 1.1.

A (2n+1)-dimensional N(k)-contact metric manifold is said to be conharmonically symmetric if (∇WC̃)(X,Y)Z=0, where X,Y,Z,W∈TM.

Definition 1.2.

A (2n+1)-dimensional N(k)-contact metric manifold is said to be ξ-conharmonically flat if C̃(X,Y)ξ=0 for X,Y∈TM.

Definition 1.3.

A (2n+1)-dimensional N(k)-contact metric manifold is said to be ϕ-conharmonically flat if C̃(ϕX,ϕY,ϕZ,ϕW)=0, where X,Y,Z,W∈TM.

The paper is organized as follows. After preliminaries in Section 2, in Section 3 we consider conharmonically symmetric N(k)-contact metric manifolds. In this section we prove that if an n-dimensional N(k)-contact metric manifold is conharmonically symmetric, then it is locally isometric to the product E(n+1)(0)×Sn(4). Section 4 deals with ξ-conharmonically flat N(k)-contact metric manifolds and we prove that an n-dimensional N(k)-contact metric manifold is ξ-conharmonically flat if and only if it is an η-Einstein manifold. Besides these some important corollaries are given in this section. Finally, in Section 5, we prove that a ϕ-conharmonically flat N(k)-contact metric manifold is a Sasakian manifold with vanishing scalar curvature.

2. Preliminaries

A (2n+1)-dimensional differentiable manifold M is said to admit an almost contact structure if it admits a tensor field ϕ of type (1,1), a vector field ξ, and a 1-form η satisfying (see [12, 13]) ϕ2X=-X+η(X)ξ,η(ξ)=1,ϕξ=0,η∘ϕ=0.

An almost contact metric structure is said to be normal if the almost induced complex structure J on the product manifold M×ℝ defined by J(X,fddt)=(ϕX-fξ,η(X)ddt)
is integrable, where X is tangent to M, t is the coordinate of ℝ, and f is a smooth function on M×ℝ. Let g be the compatible Riemannian metric with almost contact structure (ϕ,ξ,η), that is, g(ϕX,ϕY)=g(X,Y)-η(X)η(Y).
Then M becomes an almost contact metric manifold equipped with an almost contact metric structure (ϕ,ξ,η,g). From (2.1) it can be easily seen that g(X,ϕY)=-g(ϕX,Y),g(X,ξ)=η(X),
for any vector fields X,Y on the manifold. An almost contact metric structure becomes a contact metric structure if g(X,ϕY)=dη(X,Y), for all vector fields X,Y.

A contact metric manifold is said to be Einstein if S(X,Y)=λg(X,Y), where λ is a constant and η-Einstein if S(X,Y)=αg(X,Y)+βη(X)η(Y), where α and β are smooth functions.

A normal contact metric manifold is a Sasakian manifold. An almost contact metric manifold is Sasakian if and only if(∇Xϕ)Y=g(X,Y)ξ-η(Y)X,X,Y∈TM, where ∇ is the Levi-Civita connection of the Riemannian metric g. A contact metric manifold M2n+1(ϕ,ξ,η,g) for which ξ is a Killing vector field is said to be a K-contact metric manifold. A Sasakian manifold is K-contact but not conversely. However a 3-dimensional K-contact manifold is Sasakian [14].

It is well known that the tangent sphere bundle of a flat Riemannian manifold admits a contact metric structure satisfying R(X,Y)ξ=0 [15]. Again on a Sasakian manifold [16] we have R(X,Y)ξ=η(Y)X-η(X)Y.

As a generalization of both R(X,Y)ξ=0 and the Sasakian case, Blair et al. [17] introduced the (k,μ)-nullity distribution on a contact metric manifold and gave several reasons for studying it. The (k,μ)-nullity distribution N(k,μ) [17] of a contact metric manifold M is defined by N(k,μ):p⟶Np(k,μ)={W∈TpM:R(X,Y)W=(kI+μh)(g(Y,W)X-g(X,W)Y)},
for all X,Y∈TM, where (k,μ)∈ℝ2. A contact metric manifold M with ξ∈N(k,μ) is called a (k,μ)-contact metric manifold. If μ=0, the (k,μ)-nullity distribution reduces to k-nullity distribution [18]. The k-nullity distribution N(k) of a Riemannian manifold is defined by [18] N(k):p⟶Np(k)={Z∈TpM:R(X,Y)Z=k[g(Y,Z)X-g(X,Z)Y]},
with k being a constant. If the characteristic vector field ξ∈N(k), then we call a contact metric manifold as N(k)-contact metric manifold [19]. If k=1, then the manifold is Sasakian, and if k=0, then the manifold is locally isometric to the product En+1(0)×Sn(4) for n>1 and flat for n=1 [15].

Given a non-Sasakian (k,μ)-contact manifold M, Boeckx [20] introduced an invariant
IM=1-μ/21-k
and showed that, for two non-Sasakian (k,μ)-manifoldsM1 andM2, we haveIM1=IM2if and only if, up to a D-homothetic deformation, the two manifolds are locally isometric as contact metric manifolds.

Thus we see that from all non-Sasakian (k,μ)-manifolds of dimension (2n+1) and for every possible value of the invariant I, one (k,μ)-manifold M can be obtained with IM=1. For I>-1 such examples may be found from the standard contact metric structure on the tangent sphere bundle of a manifold of constant curvature c, where we have I=(1+c)/|1-c|. Boeckx also gives a Lie algebra construction for any odd dimension and value of I<-1.

Using this invariant, Blair et al. [19] constructed an example of a (2n+1)-dimensional N(1-1/n)-contact metric manifold, n>1. The example is given in the following.

Since the Boeckx invariant for a (1-1/n,0)-manifold is n>-1, we consider the tangent sphere bundle of an (n+1)-dimensional manifold of constant curvature c so choosing that the resulting D-homothetic deformation will be a (1-1/n,0)-manifold. That is, for k=c(2-c) and μ=-2c we solve1-1n=k+a2-1a2,0=μ+2a-2a
for a and c. The result is c=n±1n-1,a=1+c,
and taking c and a to be these values we obtain N(1-1/n)-contact metric manifold.

However, for a N(k)-contact metric manifold M of dimension (2n+1), we have [19] (∇Xϕ)Y=g(X+hX,Y)ξ-η(Y)(X+hX),
where h=(1/2)£ξϕ, h2=(k-1)ϕ2,R(X,Y)ξ=k[η(Y)X-η(X)Y],S(X,Y)=2(n-1)g(X,Y)+2(n-1)g(hX,Y)+[2nk-2(n-1)]η(X)η(Y),n≥1,S(Y,ξ)=2nkη(X),(∇Xη)(Y)=g(X+hX,ϕY),(∇Xh)(Y)={(1-k)g(X,ϕY)+g(X,hϕY)}ξ+η(Y)[h(ϕX+ϕhX)],

In a (2n+1)-dimensional almost contact metric manifold, if {e1,…,e2n,ξ} is a local orthonormal basis of the tangent space of the manifold, then {ϕe1,…,ϕe2n,ξ} is also a local orthonormal basis. It is easy to verify that ∑i=12ng(ei,ei)=∑i=12ng(ϕei,ϕei)=2n,∑i=12nS(ei,ei)=∑i=12nS(ϕei,ϕei)=r-2nk,∑i=12ng(ei,Z)S(Y,ei)=∑i=12ng(ϕei,Z)S(Y,ϕei)=S(Y,Z)-2nkη(Z),
for Y,Z∈T(M). In particular in view of η∘ϕ=0, we get ∑i=12ng(ei,ϕZ)S(Y,ei)=∑i=12ng(ϕei,ϕZ)S(Y,ϕei)=S(Y,ϕZ).

Here we state a lemma due to Baikoussis and Koufogiorgos [21] which will be used in this paper.

Lemma 2.1.

Let M2n+1 be an η-Einstein manifold of dimension (2n+1)(n≥1). If ξ belongs to the k-nullity distribution, then k=1 and the structure is Sasakian.

In this section we study conharmonically symmetric N(k)-contact metric manifolds. Differentiating (1.3) covariantly with respect to W, we obtain (∇WC̃)(X,Y)Z=(∇WR)(X,Y)Z-12n-1[g(Y,Z)(∇WQ)X-g(X,Z)(∇WQ)Y+(∇WS)(Y,Z)X-(∇WS)(X,Z)Y].
Therefore for conharmonically symmetric N(k)-contact metric manifolds we have (∇WR)(X,Y)Z=12n-1[g(Y,Z)(∇WQ)X-g(X,Z)(∇WQ)Y+(∇WS)(Y,Z)X-(∇WS)(X,Z)Y].

Differentiating (2.12) covariantly with respect to W and using (2.15) we obtain (∇WR)(X,Y)ξ=k[g(W,ϕY)X+g(hW,ϕY)X-g(W,ϕX)Y-g(hW,ϕX)Y].

Again, differentiating (2.14) covariantly with respect to W and using (2.16) and (2.17) we have (∇WS)(Y,Z)=2(n-1)[(1-k)g(W,ϕY)η(Z)+g(W,hϕY)η(Z)+g(hϕW,Z)η(Y)+g(hϕhW,Z)η(Y)]+{2(1-n)+2nk}[g(W,ϕY)η(Z)+g(hW,ϕY)η(Z)+g(W,ϕZ)η(Y)+g(hW,ϕZ)η(Y)].
Therefore we have (∇WQ)(Y)=2k[g(W,ϕY)ξ-(ϕW)η(Y)]+2nk[g(W,hϕY)+(hϕW)η(Y)].
Putting Z=ξ in (3.2) and using (3.3), (3.4), and (3.5) we obtain(2n-1)k[g(W,ϕY)X+g(hW,ϕY)X-g(W,ϕX)Y-g(hW,ϕX)Y]=2k[g(W,ϕX)ϕ2Y-g(W,ϕY)ϕ2X]+2nk[g(W,hϕX)ϕ2Y-g(W,hϕY)ϕ2X].

Taking inner product of (3.6) with ξ and using (2.1) we obtain (2n-1)k[g(W,ϕY)η(X)+g(hW,ϕY)η(X)-g(W,ϕX)η(Y)-g(hW,ϕX)η(Y)]=0.
From (3.7) we get, either k=0 or g(W,ϕY)η(X)+g(hW,ϕY)η(X)-g(W,ϕX)η(Y)-g(hW,ϕX)η(Y)]=0.
Putting hY instead of Y in (3.8) and using (2.12) we obtain g(W,ϕhY)η(X)=(k-1)g(W,ϕY)η(X).
Using (3.9) in (3.7) yields k[g(W,ϕY)η(X)-g(W,ϕX)η(Y)]=0.
The relation (3.10) gives k=0, since g(W,ϕY)η(X)-g(W,ϕX)η(Y)=0 gives g(W,ϕY)=0 (by putting X=ξ), which is not the case for a N(k)-contact metric manifold, in general.

Therefore in either case we obtain k=0.

Hence we have the following.

Theorem 3.1.

A conharmonically symmetric n-dimensional N(k)-contact metric manifold is locally isometric to the product E(n+1)(0)×Sn(4).

Remark 3.2.

The converse of the above theorem is not true in general. However if k=0, then we get R(X,Y)ξ=0, and hence from the definition of the conharmonic curvature tensor we obtain C̃(X,Y)ξ=0, that is, the manifold under consideration is ξ-conharmonically flat. Thus if an N(k)-contact manifold is locally isometric to E(n+1)(0)×Sn(4), then the manifold is ξ-conharmonically flat.

In this section we consider a (2n+1)-dimensional ξ-conharmonically flat N(k)-contact metric manifolds. Then from (1.3) we obtain R(X,Y)ξ=12n-1[g(Y,ξ)QX-g(X,ξ)QY+S(Y,ξ)X-S(X,ξ)Y].

Using (2.1), (2.13), and (2.15) in (4.1) we obtain [η(Y)QX-η(X)QY]+k[η(Y)X-η(X)Y]=0.
Putting Y=ξ in (4.2) and using (2.1) and (2.15) we get QX=-kX+(2n+1)kη(X)ξ.
Taking inner product with W of (4.3) yields S(X,W)=-kg(X,W)+(2n+1)kη(X)η(W).

From relation (4.4), we conclude that the manifold is an η-Einstein manifold.

Conversely, we assume that a (2n+1)-dimensional N(k)-contact manifold satisfies the relation (4.4). Then we easily obtain from (1.3) that C̃(X,Y)ξ=0.

In view of the above discussions we state the following.

Theorem 4.1.

A (2n+1)-dimensional N(k)-contact metric manifold is ξ-conharmonically flat if and only if it is an η-Einstein manifold.

Hence in view of Lemma 2.1 we state the following.

Corollary 4.2.

Let M be a (2n+1)-dimensional ξ-conharmonically flat N(k)-contact metric manifold, then k=1 and the structure is Sasakian.

Let {e1,e2,…,en,en+1,…e2n,e2n+1=ξ} be a local orthonormal basis of the tangent space of the manifold. Putting X=W=ei in (4.4) and summing up from 1 to 2n+1 we obtain in view of (2.18) and (2.19) thatr=0.
Therefore we have the following corollary.

Corollary 4.3.

In a (2n+1)-dimensional ξ-conharmonically flat N(k)-contact metric manifold, the scalar curvature r vanishes.

This section deals with a (2n+1)-dimensional ϕ-conharmonically flat N(k)-contact metric manifold. Then we have from (1.3) that R̃(ϕX,ϕY,ϕZ,ϕW)=12n-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 of the manifold. Then {ϕe1,ϕe2,…,ϕe2n,ξ} is also a local orthonormal basis of the tangent space. Putting X=W=ei in (5.1) and summing up from 1 to 2n we have ∑i=12nR̃(ϕei,ϕY,ϕZ,ϕei)=12n-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 (2.18), (2.19), (2.20), and (2.21) in (5.2) we obtainS(ϕY,ϕZ)=(r-k)g(ϕY,ϕZ).

Replacing Y and Z by ϕY and ϕZ in (5.3) and using (2.1) we have S(Y,Z)=(r-k)g(Y,Z)+[(2n+1)k-r]η(Y)η(Z).

Putting Y=Z=ei in (5.4) and taking summation over i=1 to 2n+1 we get by using (2.18) and (2.19) that r=0.
In view of the above discussions we have the following.

Proposition 5.1.

A (2n+1)-dimensional ϕ-conharmonically flat N(k)-contact metric manifold is an η-Einstein manifold with vanishing scalar curvature.

Therefore in view of the Lemma 2.1 we state the following theorem.

Theorem 5.2.

A (2n+1)-dimensional ϕ-conharmonically flat N(k)-contact metric manifold is a Sasakian manifold with vanishing scalar curvature.

Definition 5.3.

In a (2n+1)-dimensional N(k)-contact metric manifold, if the Ricci tensor S satisfies (∇XS)(ϕY,ϕZ)=0, then the Ricci tensor is said to be η-parallel.

The notion of η-parallel Ricci tensor for Sasakian manifold was introduced by Kon [22].

Putting r=0 in (5.4) we have S(Y,Z)=-kg(Y,Z)+(2n+1)kη(Y)η(Z).

Replacing Y and Z by ϕY and ϕZ in (5.6) and using (2.1) we obtain S(ϕY,ϕZ)=-kg(ϕY,ϕZ).

Relation (5.7) yields (∇XS)(ϕY,ϕZ)=0,
since k is a constant. Therefore we have the following corollary.

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