GEOMETRYISRN Geometry2090-63152090-6307International Scholarly Research Network42379810.5402/2011/423798423798Research ArticleConharmonic Curvature Tensor on N(K)-Contact Metric ManifoldsGhoshSujit1DeU. C.2TaleshianA.3BelhajA.HervikS.UngarA. A.1Madanpur K. A. Vidyalaya (H.S.)Vill and PO, MadanpurNadia 741245India2Department of Pure MathematicsUniversity of Calcutta35 Ballygunge Circular RoadKol 700019Indiacaluniv.ac.in3Department of Mathematics, Faculty of Sciences, Mazandaran University, P.O. Box 47416-1467, BabolsarIranumz.ac.ir20112772011201109042011030620112011Copyright © 2011 Sujit Ghosh et al.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 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 .

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  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 , 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 . 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 , Özgür , 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 pM, 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×TpMϕ(TpM){ξp},pM.

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 , it is proved that a conformally symmetric K-contact manifold is locally isometric to the unit sphere. In , it is proved that a K-contact manifold is ξ-conformally flat if and only if it is an η-Einstein Sasakian manifold. In , some necessary conditions for a K-contact manifold to be ϕ-conformally flat are proved. In , a necessary and sufficient condition for a Sasakian manifold to be ϕ-conformally flat is obtained. In , projective curvature tensor in K-contact and Sasakian manifolds has been studied. Moreover, the author  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 ) and the studies of projective curvature tensor in K-contact and Sasakian manifolds in  and Lorentzian para-Sasakian manifolds in , 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,WTM.

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,YTM.

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,WTM.

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,  ηϕ=0.

An almost contact metric structure is said to be normal if the almost induced complex structure J on the product manifold M× defined by J(X,fddt)=(ϕX-fξ,η(X)ddt) is integrable, where X is tangent to M, t is the coordinate of , and f is a smooth function on M  ×  . 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,YTM, 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 .

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 . Again on a Sasakian manifold  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.  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)  of a contact metric manifold M is defined by N(k,μ):pNp(k,μ)={WTpM:R(X,Y)W=(kI+μh)(g(Y,W)X-g(X,W)Y)}, for all X,YTM, where (k,μ)2. A contact metric manifold M with ξN(k,μ) is called a (k,μ)-contact metric manifold. If μ=0, the (k,μ)-nullity distribution reduces to k-nullity distribution . The k-nullity distribution N(k) of a Riemannian manifold is defined by  N(k):pNp(k)={ZTpM:R(X,Y)Z=k[g(Y,Z)X-g(X,Z)Y]}, with k being a constant. If the characteristic vector field ξN(k), then we call a contact metric manifold as N(k)-contact metric manifold . 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 .

Given a non-Sasakian (k,μ)-contact manifold M, Boeckx  introduced an invariant IM=  1-μ/21-k   and showed that, for two non-Sasakian (k,μ)-manifolds  M1 and  M2, we have  IM1=IM2  if 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.  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  (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),n1,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,ZT(M). In particular in view of ηϕ=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  which will be used in this paper.

Lemma 2.1.

Let M2n+1 be an η-Einstein manifold of dimension (2n+1)(n1). If ξ belongs to the k-nullity distribution, then k=1 and the structure is Sasakian.

3. Conharmonically Symmetric <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M230"><mml:mi>N</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula>-Contact Metric Manifolds

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.

4. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M267"><mml:mrow><mml:mi>ξ</mml:mi></mml:mrow></mml:math></inline-formula>-Conharmonically Flat <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M268"><mml:mi>N</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula>-Contact Metric Manifolds

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.

5. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M299"><mml:mrow><mml:mi>ϕ</mml:mi></mml:mrow></mml:math></inline-formula>-Conharmonically Flat <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M300"><mml:mi>N</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula>-Contact Metric Manifolds

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-1i=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 .

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.

Corollary 5.4.

A (2n+1)-dimensional ϕ-conharmonically flat N(k)-contact metric manifold satisfies η-parallel Ricci tensor.

AhsanZ.Tensor Analysis with Applications2008New Delhi, IndiaAnamaya PublishersIshiiY.On conharmonic transformationsThe Tensor Society. Tensor. New Series1957773800102837ZBL0079.15702AbdussattarD. B.On conharmonic transformations in general relativityBulletin of the Calcutta Mathematical Society196641409416SiddiquiS. A.AhsanZ.Conharmonic curvature tensor and the space-time of general relativityDifferential Geometry - Dynamical Systems201012213220ÖzgürC.On ϕ-conformally flat Lorentzian para-Sasakian manifoldsRadovi Matematički2003121991062022248ZBL1074.53057GuoZ.Conformally symmetric K-contact manifoldsChinese Quarterly Journal of Mathematics1992715101166659ZhenG.CabrerizoJ. L.FernándezL. M.FernándezM.On ξ-conformally flat contact metric manifoldsIndian Journal of Pure and Applied Mathematics19972867257341461184CabrerizoJ. L.FernándezL. M.FernándezM.GuoZ.The structure of a class of K-contact manifoldsActa Mathematica Hungarica1999824331340167562110.1023/A:1006696410826ZBL0924.53024DwibediM. K.FernándezL. M.TripathiM. M.The structure of some classes of contact metric manifoldsThe Georgian Mathematical Journal2009162295304TripathiM. M.DwivediM. K.The structure of some classes of K-contact manifoldsIndian Academy of Sciences. Proceedings. Mathematical Sciences20081183371379245024110.1007/s12044-008-0029-1ÖzgürC.Hypersurfaces satisfying some curvature conditions in the semi-Euclidean spaceChaos, Solitons and Fractals200939524572464251944210.1016/j.chaos.2007.07.018ZBL1197.53088BlairD. E.Contact Manifolds in Riemannian Geometry1976Berlin, GermanySpringervi+146Lecture Notes in Mathematics, Vol. 5090467588BlairD. E.Riemannian geometry of contact and symplectic manifolds2002203Boston, MABirkhäuser Boston Inc.xii+260Progress in Mathematics1874240JunJ.-B.KimI. B.KimU. K.On 3—dimensional almost contact metric manifoldsKyungpook Mathematical Journal19943422933011325608BlairD. E.Two remarks on contact metric structuresThe Tohoku Mathematical Journal. Second Series19772933193240464108ZBL0376.53021SasakiS.Lecture Notes on Almost Contact Manifolds1965Tokyo, JapanTohoku UniversityBlairD. E.KoufogiorgosT.PapantoniouB. J.Contact metric manifolds satisfying a nullity conditionIsrael Journal of Mathematics1995911–3189214134831210.1007/BF02761646ZBL0837.53038TannoS.Ricci curvatures of contact Riemannian manifoldsThe Tohoku Mathematical Journal. Second Series198840344144810.2748/tmj/1178227985957055ZBL0655.53035BlairD. E.KimJ.-S.TripathiM. M.On the concircular curvature tensor of a contact metric manifoldJournal of the Korean Mathematical Society2005425883892215735010.4134/JKMS.2005.42.5.883ZBL1084.53039BoeckxE.A full classification of contact metric (k, μ)-spacesIllinois Journal of Mathematics20004412122191731388BaikoussisC.KoufogiorgosT.On a type of contact manifoldsJournal of Geometry1993461-219120569210.1007/BF01230994ZBL0780.53036KonM.Invariant submanifolds in Sasakian manifoldsMathematische Annalen19762193277290042584410.1007/BF01354288ZBL0301.53031