Table of Contents
ISRN Geometry
VolumeΒ 2011Β (2011), Article IDΒ 423798, 11 pages
http://dx.doi.org/10.5402/2011/423798
Research Article

Conharmonic Curvature Tensor on 𝑁(𝐾)-Contact Metric Manifolds

1Madanpur K. A. Vidyalaya (H.S.), Vill and PO, Madanpur, Nadia 741245, India
2Department of Pure Mathematics, University of Calcutta, 35 Ballygunge Circular Road, Kol 700019, India
3Department of Mathematics, Faculty of Sciences, Mazandaran University, P.O. Box 47416-1467, Babolsar, Iran

Received 9 April 2011; Accepted 3 June 2011

Academic Editors: A.Β Belhaj, S.Β Hervik, and A. A.Β Ungar

Copyright Β© 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.

Abstract

The object of the present paper is to characterize 𝑁(π‘˜)-contact metric manifolds satisfying certain curvature conditions on the conharmonic curvature tensor. In this paper we study conharmonically symmetric, πœ‰-conharmonically flat, and πœ™-conharmonically flat 𝑁(π‘˜)-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,π‘Œ)=𝑒2πœŽπ‘”(𝑋,π‘Œ),(1.1) 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 πœŽπ‘–,𝑖+𝜎,π‘–πœŽπ‘–,=0,(1.2) where π‘π‘œπ‘šπ‘šπ‘Ž denotes the covariant differentiation with respect to the metric 𝑔.

A rank-four tensor 𝐢 that remains invariant under conharmonic transformation for a (2𝑛+1)-dimensional Riemannian manifold 𝑀 is given by ξ‚ξ‚βˆ’1𝐢(𝑋,π‘Œ,𝑍,π‘Š)=𝑅(𝑋,π‘Œ,𝑍,π‘Š)[],2π‘›βˆ’1𝑔(π‘Œ,𝑍)𝑆(𝑋,π‘Š)βˆ’π‘”(𝑋,𝑍)𝑆(π‘Œ,π‘Š)+𝑆(π‘Œ,𝑍)𝑔(𝑋,π‘Š)βˆ’π‘†(𝑋,𝑍)𝑔(π‘Œ,π‘Š)(1.3) where 𝑅 denotes the Riemannian curvature tensor of type (0,4) defined by 𝑅(𝑋,π‘Œ,𝑍,π‘Š)=𝑔(𝑅(X,π‘Œ)𝑍,π‘Š),(1.4) 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 𝑅. There are many physical applications of the tensor 𝐢. 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 𝐢 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], Γ–zgΓΌr [5], and many others.

Let 𝑀 be an almost contact metric manifold equipped with an almost contact metric structure (πœ™,πœ‰,πœ‚,𝑔). At each point π‘βˆˆπ‘€, decompose the tangent space 𝑇𝑝𝑀 into direct sum 𝑇𝑝𝑀=πœ™(𝑇𝑝𝑀)βŠ•{πœ‰π‘}, where {πœ‰π‘} is the 1-dimensional linear subspace of 𝑇𝑝𝑀 generated by {πœ‰π‘}. Thus the conformal curvature tensor 𝐢 is a map πΆβˆΆπ‘‡π‘π‘€Γ—π‘‡π‘π‘€Γ—π‘‡π‘ξ€·π‘‡π‘€βŸΆπœ™π‘π‘€ξ€ΈβŠ•ξ€½πœ‰π‘ξ€Ύ,π‘βˆˆπ‘€.(1.5)

It may be natural to consider the following particular cases:(1)πΆβˆΆπ‘‡π‘(𝑀)×𝑇𝑝(𝑀)×𝑇𝑝(𝑀)→𝐿(πœ‰π‘), that is, the projection of the image of 𝐢 in πœ™(𝑇𝑝(𝑀)) is zero;(2)πΆβˆΆπ‘‡π‘(𝑀)×𝑇𝑝(𝑀)×𝑇𝑝(𝑀)β†’πœ™(𝑇𝑝(𝑀)), that is, the projection of the image of 𝐢 in 𝐿(πœ‰π‘) is zero;(3)πΆβˆΆπœ™(𝑇𝑝(𝑀))Γ—πœ™(𝑇𝑝(𝑀))Γ—πœ™(𝑇𝑝(𝑀))→𝐿(πœ‰π‘), that is, when 𝐢 is restricted to πœ™(𝑇𝑝(𝑀))Γ—πœ™(𝑇𝑝(𝑀))Γ—πœ™(𝑇𝑝(𝑀)), the projection of the image of 𝐢 in πœ™(𝑇𝑝(𝑀)) is zero. This condition is equivalent toπœ™2𝐢(πœ™π‘‹,πœ™π‘Œ,πœ™π‘)=0.(1.6)

Here cases 1, 2, and 3 are synonymous to conformally symmetric, πœ‰-conformally flat, and πœ™-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 πœ‰-conformally flat if and only if it is an πœ‚-Einstein Sasakian manifold. In [8], some necessary conditions for a 𝐾-contact manifold to be πœ™-conformally flat are proved. In [9], a necessary and sufficient condition for a Sasakian manifold to be πœ™-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 𝐢, 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 𝐢⋅𝑅=0 is pseudosymmetric. He also considered the condition 𝐢⋅𝐢=𝐿𝐢𝑄(𝑔,𝐢) 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(π‘˜)-contact metric manifolds.

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

Definition 1.1. A (2𝑛+1)-dimensional 𝑁(π‘˜)-contact metric manifold is said to be conharmonically symmetric if (βˆ‡π‘Šξ‚πΆ)(𝑋,π‘Œ)𝑍=0, where 𝑋,π‘Œ,𝑍,π‘Šβˆˆπ‘‡π‘€.

Definition 1.2. A (2𝑛+1)-dimensional 𝑁(π‘˜)-contact metric manifold is said to be πœ‰-conharmonically flat if 𝐢(𝑋,π‘Œ)πœ‰=0 for 𝑋,π‘Œβˆˆπ‘‡π‘€.

Definition 1.3. A (2𝑛+1)-dimensional 𝑁(π‘˜)-contact metric manifold is said to be πœ™-conharmonically flat if 𝐢(πœ™π‘‹,πœ™π‘Œ,πœ™π‘,πœ™π‘Š)=0, where 𝑋,π‘Œ,𝑍,π‘Šβˆˆπ‘‡π‘€.

The paper is organized as follows. After preliminaries in Section 2, in Section 3 we consider conharmonically symmetric 𝑁(π‘˜)-contact metric manifolds. In this section we prove that if an 𝑛-dimensional 𝑁(π‘˜)-contact metric manifold is conharmonically symmetric, then it is locally isometric to the product 𝐸(𝑛+1)(0)×𝑆𝑛(4). Section 4 deals with πœ‰-conharmonically flat 𝑁(π‘˜)-contact metric manifolds and we prove that an 𝑛-dimensional 𝑁(π‘˜)-contact metric manifold is πœ‰-conharmonically flat if and only if it is an πœ‚-Einstein manifold. Besides these some important corollaries are given in this section. Finally, in Section 5, we prove that a πœ™-conharmonically flat 𝑁(π‘˜)-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 πœ™ of type (1,1), a vector field πœ‰, and a 1-form πœ‚ satisfying (see [12, 13]) πœ™2𝑋=βˆ’π‘‹+πœ‚(𝑋)πœ‰,πœ‚(πœ‰)=1,πœ™πœ‰=0,πœ‚βˆ˜πœ™=0.(2.1)

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𝑑𝑋,𝑓=ξ‚€π‘‘π‘‘π‘‘πœ™π‘‹βˆ’π‘“πœ‰,πœ‚(𝑋)𝑑𝑑(2.2) is integrable, where 𝑋 is tangent to 𝑀, 𝑑 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 (πœ™,πœ‰,πœ‚), that is, 𝑔(πœ™π‘‹,πœ™π‘Œ)=𝑔(𝑋,π‘Œ)βˆ’πœ‚(𝑋)πœ‚(π‘Œ).(2.3) Then 𝑀 becomes an almost contact metric manifold equipped with an almost contact metric structure (πœ™,πœ‰,πœ‚,𝑔). From (2.1) it can be easily seen that 𝑔(𝑋,πœ™π‘Œ)=βˆ’π‘”(πœ™π‘‹,π‘Œ),𝑔(𝑋,πœ‰)=πœ‚(𝑋),(2.4) for any vector fields 𝑋,π‘Œ on the manifold. An almost contact metric structure becomes a contact metric structure if 𝑔(𝑋,πœ™π‘Œ)=π‘‘πœ‚(𝑋,π‘Œ), for all vector fields 𝑋,π‘Œ.

A contact metric manifold is said to be Einstein if 𝑆(𝑋,π‘Œ)=πœ†π‘”(𝑋,π‘Œ), where πœ† is a constant and πœ‚-Einstein if 𝑆(𝑋,π‘Œ)=𝛼𝑔(𝑋,π‘Œ)+π›½πœ‚(𝑋)πœ‚(π‘Œ), 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 ifξ€·βˆ‡π‘‹πœ™ξ€Έπ‘Œ=𝑔(𝑋,π‘Œ)πœ‰βˆ’πœ‚(π‘Œ)𝑋,(2.5)𝑋,π‘Œβˆˆπ‘‡π‘€, where βˆ‡ is the Levi-Civita connection of the Riemannian metric 𝑔. A contact metric manifold 𝑀2𝑛+1(πœ™,πœ‰,πœ‚,𝑔) for which πœ‰ 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,π‘Œ)πœ‰=0 [15]. Again on a Sasakian manifold [16] we have 𝑅(𝑋,π‘Œ)πœ‰=πœ‚(π‘Œ)π‘‹βˆ’πœ‚(𝑋)π‘Œ.(2.6)

As a generalization of both 𝑅(𝑋,π‘Œ)πœ‰=0 and the Sasakian case, Blair et al. [17] introduced the (π‘˜,πœ‡)-nullity distribution on a contact metric manifold and gave several reasons for studying it. The (π‘˜,πœ‡)-nullity distribution 𝑁(π‘˜,πœ‡) [17] of a contact metric manifold 𝑀 is defined by 𝑁(π‘˜,πœ‡)βˆΆπ‘βŸΆπ‘π‘=ξ€½(π‘˜,πœ‡)π‘Šβˆˆπ‘‡π‘ξ€Ύ,π‘€βˆΆπ‘…(𝑋,π‘Œ)π‘Š=(π‘˜πΌ+πœ‡β„Ž)(𝑔(π‘Œ,π‘Š)π‘‹βˆ’π‘”(𝑋,π‘Š)π‘Œ)(2.7) for all 𝑋,π‘Œβˆˆπ‘‡π‘€, where (π‘˜,πœ‡)βˆˆβ„2. A contact metric manifold 𝑀 with πœ‰βˆˆπ‘(π‘˜,πœ‡) is called a (π‘˜,πœ‡)-contact metric manifold. If πœ‡=0, the (π‘˜,πœ‡)-nullity distribution reduces to π‘˜-nullity distribution [18]. The π‘˜-nullity distribution 𝑁(π‘˜) of a Riemannian manifold is defined by [18] 𝑁(π‘˜)βˆΆπ‘βŸΆπ‘π‘ξ€½(π‘˜)=π‘βˆˆπ‘‡π‘[𝑔]ξ€Ύ,π‘€βˆΆπ‘…(𝑋,π‘Œ)𝑍=π‘˜(π‘Œ,𝑍)π‘‹βˆ’π‘”(𝑋,𝑍)π‘Œ(2.8) with π‘˜ being a constant. If the characteristic vector field πœ‰βˆˆπ‘(π‘˜), then we call a contact metric manifold as 𝑁(π‘˜)-contact metric manifold [19]. If π‘˜=1, then the manifold is Sasakian, and if π‘˜=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 (π‘˜,πœ‡)-contact manifold 𝑀, Boeckx [20] introduced an invariant 𝐼𝑀=1βˆ’πœ‡/2√1βˆ’π‘˜(2.9) and showed that, for two non-Sasakian (π‘˜,πœ‡)-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 (π‘˜,πœ‡)-manifolds of dimension (2𝑛+1) and for every possible value of the invariant 𝐼, one (π‘˜,πœ‡)-manifold 𝑀 can be obtained with 𝐼𝑀=1. For 𝐼>βˆ’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)/|1βˆ’π‘|. Boeckx also gives a Lie algebra construction for any odd dimension and value of 𝐼<βˆ’1.

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

Since the Boeckx invariant for a (1βˆ’1/𝑛,0)-manifold is βˆšπ‘›>βˆ’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 (1βˆ’1/𝑛,0)-manifold. That is, for π‘˜=𝑐(2βˆ’π‘) and πœ‡=βˆ’2𝑐 we solve11βˆ’π‘›=π‘˜+π‘Ž2βˆ’1π‘Ž2,0=πœ‡+2π‘Žβˆ’2π‘Ž(2.10) for π‘Ž and 𝑐. The result is βˆšπ‘=𝑛±1π‘›βˆ’1,π‘Ž=1+𝑐,(2.11) and taking 𝑐 and π‘Ž to be these values we obtain 𝑁(1βˆ’1/𝑛)-contact metric manifold.

However, for a 𝑁(π‘˜)-contact metric manifold 𝑀 of dimension (2𝑛+1), we have [19] ξ€·βˆ‡π‘‹πœ™ξ€Έπ‘Œ=𝑔(𝑋+β„Žπ‘‹,π‘Œ)πœ‰βˆ’πœ‚(π‘Œ)(𝑋+β„Žπ‘‹),(2.12) where β„Ž=(1/2)Β£πœ‰πœ™, β„Ž2=(π‘˜βˆ’1)πœ™2,(2.13)[],𝑅(𝑋,π‘Œ)πœ‰=π‘˜πœ‚(π‘Œ)π‘‹βˆ’πœ‚(𝑋)π‘Œ(2.14)+[]πœ‚π‘†(𝑋,π‘Œ)=2(π‘›βˆ’1)𝑔(𝑋,π‘Œ)+2(π‘›βˆ’1)𝑔(β„Žπ‘‹,π‘Œ)2π‘›π‘˜βˆ’2(π‘›βˆ’1)(𝑋)πœ‚(π‘Œ),𝑛β‰₯1,(2.15)𝑆(π‘Œ,πœ‰)=2π‘›π‘˜πœ‚(𝑋),(2.16)ξ€·βˆ‡π‘‹πœ‚ξ€Έ(π‘Œ)=𝑔(𝑋+β„Žπ‘‹,πœ™π‘Œ),(2.17)ξ€·βˆ‡π‘‹β„Žξ€Έ[],(π‘Œ)={(1βˆ’π‘˜)𝑔(𝑋,πœ™π‘Œ)+𝑔(𝑋,β„Žπœ™π‘Œ)}πœ‰+πœ‚(π‘Œ)β„Ž(πœ™π‘‹+πœ™β„Žπ‘‹)(2.18)

In a (2𝑛+1)-dimensional almost contact metric manifold, if {𝑒1,…,𝑒2𝑛,πœ‰} is a local orthonormal basis of the tangent space of the manifold, then {πœ™π‘’1,…,πœ™π‘’2𝑛,πœ‰} is also a local orthonormal basis. It is easy to verify that 2𝑛𝑖=1𝑔𝑒𝑖,𝑒𝑖=2𝑛𝑖=1π‘”ξ€·πœ™π‘’π‘–,πœ™π‘’π‘–ξ€Έ=2𝑛,(2.19)2𝑛𝑖=1𝑆𝑒𝑖,𝑒𝑖=2𝑛𝑖=1π‘†ξ€·πœ™π‘’π‘–,πœ™π‘’π‘–ξ€Έ=π‘Ÿβˆ’2π‘›π‘˜,(2.20)2𝑛𝑖=1𝑔𝑒𝑖𝑆,π‘π‘Œ,𝑒𝑖=2𝑛𝑖=1π‘”ξ€·πœ™π‘’π‘–ξ€Έπ‘†ξ€·,π‘π‘Œ,πœ™π‘’π‘–ξ€Έ=𝑆(π‘Œ,𝑍)βˆ’2π‘›π‘˜πœ‚(𝑍),(2.21) for π‘Œ,π‘βˆˆπ‘‡(𝑀). In particular in view of πœ‚βˆ˜πœ™=0, we get 2𝑛𝑖=1𝑔𝑒𝑖𝑆,πœ™π‘π‘Œ,𝑒𝑖=2𝑛𝑖=1π‘”ξ€·πœ™π‘’π‘–ξ€Έπ‘†ξ€·,πœ™π‘π‘Œ,πœ™π‘’π‘–ξ€Έ=𝑆(π‘Œ,πœ™π‘).(2.22)

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 πœ‚-Einstein manifold of dimension (2𝑛+1)(𝑛β‰₯1). If πœ‰ belongs to the π‘˜-nullity distribution, then π‘˜=1 and the structure is Sasakian.

3. Conharmonically Symmetric 𝑁(π‘˜)-Contact Metric Manifolds

In this section we study conharmonically symmetric 𝑁(π‘˜)-contact metric manifolds. Differentiating (1.3) covariantly with respect to π‘Š, we obtain ξ‚€βˆ‡π‘Šξ‚πΆξ‚=ξ€·βˆ‡(𝑋,π‘Œ)π‘π‘Šπ‘…ξ€Έβˆ’1(𝑋,π‘Œ)π‘ξ€Ίξ€·βˆ‡2π‘›βˆ’1𝑔(π‘Œ,𝑍)π‘Šπ‘„ξ€Έξ€·βˆ‡π‘‹βˆ’π‘”(𝑋,𝑍)π‘Šπ‘„ξ€Έξ€·βˆ‡π‘Œ+π‘Šπ‘†ξ€Έξ€·βˆ‡(π‘Œ,𝑍)π‘‹βˆ’π‘Šπ‘†ξ€Έξ€».(𝑋,𝑍)π‘Œ(3.1) Therefore for conharmonically symmetric 𝑁(π‘˜)-contact metric manifolds we have ξ€·βˆ‡π‘Šπ‘…ξ€Έ1(𝑋,π‘Œ)𝑍=ξ€Ίξ€·βˆ‡2π‘›βˆ’1𝑔(π‘Œ,𝑍)π‘Šπ‘„ξ€Έξ€·βˆ‡π‘‹βˆ’π‘”(𝑋,𝑍)π‘Šπ‘„ξ€Έξ€·βˆ‡π‘Œ+π‘Šπ‘†ξ€Έβˆ’ξ€·βˆ‡(π‘Œ,𝑍)π‘‹π‘Šπ‘†ξ€Έξ€».(𝑋,𝑍)π‘Œ(3.2)

Differentiating (2.12) covariantly with respect to π‘Š and using (2.15) we obtain ξ€·βˆ‡π‘Šπ‘…ξ€Έ[𝑔].(𝑋,π‘Œ)πœ‰=π‘˜(π‘Š,πœ™π‘Œ)𝑋+𝑔(β„Žπ‘Š,πœ™π‘Œ)π‘‹βˆ’π‘”(π‘Š,πœ™π‘‹)π‘Œβˆ’π‘”(β„Žπ‘Š,πœ™π‘‹)π‘Œ(3.3)

Again, differentiating (2.14) covariantly with respect to π‘Š and using (2.16) and (2.17) we have ξ€·βˆ‡π‘Šπ‘†ξ€Έ[][].(π‘Œ,𝑍)=2(π‘›βˆ’1)(1βˆ’π‘˜)𝑔(π‘Š,πœ™π‘Œ)πœ‚(𝑍)+𝑔(π‘Š,β„Žπœ™π‘Œ)πœ‚(𝑍)+𝑔(β„Žπœ™π‘Š,𝑍)πœ‚(π‘Œ)+𝑔(β„Žπœ™β„Žπ‘Š,𝑍)πœ‚(π‘Œ)+{2(1βˆ’π‘›)+2π‘›π‘˜}𝑔(π‘Š,πœ™π‘Œ)πœ‚(𝑍)+𝑔(β„Žπ‘Š,πœ™π‘Œ)πœ‚(𝑍)+𝑔(π‘Š,πœ™π‘)πœ‚(π‘Œ)+𝑔(β„Žπ‘Š,πœ™π‘)πœ‚(π‘Œ)(3.4) Therefore we have ξ€·βˆ‡π‘Šπ‘„ξ€Έ[𝑔][𝑔].(π‘Œ)=2π‘˜(π‘Š,πœ™π‘Œ)πœ‰βˆ’(πœ™π‘Š)πœ‚(π‘Œ)+2π‘›π‘˜(π‘Š,β„Žπœ™π‘Œ)+(β„Žπœ™π‘Š)πœ‚(π‘Œ)(3.5) Putting 𝑍=πœ‰ in (3.2) and using (3.3), (3.4), and (3.5) we obtain[]𝑔(2π‘›βˆ’1)π‘˜π‘”(π‘Š,πœ™π‘Œ)𝑋+𝑔(β„Žπ‘Š,πœ™π‘Œ)π‘‹βˆ’π‘”(π‘Š,πœ™π‘‹)π‘Œβˆ’π‘”(β„Žπ‘Š,πœ™π‘‹)π‘Œ=2π‘˜(π‘Š,πœ™π‘‹)πœ™2π‘Œβˆ’π‘”(π‘Š,πœ™π‘Œ)πœ™2𝑋+2π‘›π‘˜π‘”(π‘Š,β„Žπœ™π‘‹)πœ™2π‘Œβˆ’π‘”(π‘Š,β„Žπœ™π‘Œ)πœ™2𝑋.(3.6)

Taking inner product of (3.6) with πœ‰ and using (2.1) we obtain [](2π‘›βˆ’1)π‘˜π‘”(π‘Š,πœ™π‘Œ)πœ‚(𝑋)+𝑔(β„Žπ‘Š,πœ™π‘Œ)πœ‚(𝑋)βˆ’π‘”(π‘Š,πœ™π‘‹)πœ‚(π‘Œ)βˆ’π‘”(β„Žπ‘Š,πœ™π‘‹)πœ‚(π‘Œ)=0.(3.7) From (3.7) we get, either π‘˜=0 or ]𝑔(π‘Š,πœ™π‘Œ)πœ‚(𝑋)+𝑔(β„Žπ‘Š,πœ™π‘Œ)πœ‚(𝑋)βˆ’π‘”(π‘Š,πœ™π‘‹)πœ‚(π‘Œ)βˆ’π‘”(β„Žπ‘Š,πœ™π‘‹)πœ‚(π‘Œ)=0.(3.8) Putting β„Žπ‘Œ instead of π‘Œ in (3.8) and using (2.12) we obtain 𝑔(π‘Š,πœ™β„Žπ‘Œ)πœ‚(𝑋)=(π‘˜βˆ’1)𝑔(π‘Š,πœ™π‘Œ)πœ‚(𝑋).(3.9) Using (3.9) in (3.7) yields π‘˜[]𝑔(π‘Š,πœ™π‘Œ)πœ‚(𝑋)βˆ’π‘”(π‘Š,πœ™π‘‹)πœ‚(π‘Œ)=0.(3.10) The relation (3.10) gives π‘˜=0, since 𝑔(π‘Š,πœ™π‘Œ)πœ‚(𝑋)βˆ’π‘”(π‘Š,πœ™π‘‹)πœ‚(π‘Œ)=0 gives 𝑔(π‘Š,πœ™π‘Œ)=0 (by putting 𝑋=πœ‰), which is not the case for a 𝑁(π‘˜)-contact metric manifold, in general.

Therefore in either case we obtain π‘˜=0.

Hence we have the following.

Theorem 3.1. A conharmonically symmetric 𝑛-dimensional 𝑁(π‘˜)-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 π‘˜=0, then we get 𝑅(𝑋,π‘Œ)πœ‰=0, and hence from the definition of the conharmonic curvature tensor we obtain 𝐢(𝑋,π‘Œ)πœ‰=0, that is, the manifold under consideration is πœ‰-conharmonically flat. Thus if an 𝑁(π‘˜)-contact manifold is locally isometric to 𝐸(𝑛+1)(0)×𝑆𝑛(4), then the manifold is πœ‰-conharmonically flat.

4. πœ‰-Conharmonically Flat 𝑁(π‘˜)-Contact Metric Manifolds

In this section we consider a (2𝑛+1)-dimensional πœ‰-conharmonically flat 𝑁(π‘˜)-contact metric manifolds. Then from (1.3) we obtain 1𝑅(𝑋,π‘Œ)πœ‰=[].2π‘›βˆ’1𝑔(π‘Œ,πœ‰)π‘„π‘‹βˆ’π‘”(𝑋,πœ‰)π‘„π‘Œ+𝑆(π‘Œ,πœ‰)π‘‹βˆ’π‘†(𝑋,πœ‰)π‘Œ(4.1)

Using (2.1), (2.13), and (2.15) in (4.1) we obtain [][]πœ‚(π‘Œ)π‘„π‘‹βˆ’πœ‚(𝑋)π‘„π‘Œ+π‘˜πœ‚(π‘Œ)π‘‹βˆ’πœ‚(𝑋)π‘Œ=0.(4.2) Putting π‘Œ=πœ‰ in (4.2) and using (2.1) and (2.15) we get 𝑄𝑋=βˆ’π‘˜π‘‹+(2𝑛+1)π‘˜πœ‚(𝑋)πœ‰.(4.3) Taking inner product with π‘Š of (4.3) yields 𝑆(𝑋,π‘Š)=βˆ’π‘˜π‘”(𝑋,π‘Š)+(2𝑛+1)π‘˜πœ‚(𝑋)πœ‚(π‘Š).(4.4)

From relation (4.4), we conclude that the manifold is an πœ‚-Einstein manifold.

Conversely, we assume that a (2𝑛+1)-dimensional 𝑁(π‘˜)-contact manifold satisfies the relation (4.4). Then we easily obtain from (1.3) that 𝐢(𝑋,π‘Œ)πœ‰=0.

In view of the above discussions we state the following.

Theorem 4.1. A (2𝑛+1)-dimensional 𝑁(π‘˜)-contact metric manifold is πœ‰-conharmonically flat if and only if it is an πœ‚-Einstein manifold.

Hence in view of Lemma 2.1 we state the following.

Corollary 4.2. Let 𝑀 be a (2n+1)-dimensional πœ‰-conharmonically flat 𝑁(π‘˜)-contact metric manifold, then π‘˜=1 and the structure is Sasakian.

Let {𝑒1,𝑒2,…,𝑒𝑛,𝑒𝑛+1,…e2𝑛,𝑒2𝑛+1=πœ‰} be a local orthonormal basis of the tangent space of the manifold. Putting 𝑋=π‘Š=𝑒𝑖 in (4.4) and summing up from 1 to 2𝑛+1 we obtain in view of (2.18) and (2.19) thatπ‘Ÿ=0.(4.5) Therefore we have the following corollary.

Corollary 4.3. In a (2𝑛+1)-dimensional πœ‰-conharmonically flat 𝑁(π‘˜)-contact metric manifold, the scalar curvature π‘Ÿ vanishes.

5. πœ™-Conharmonically Flat 𝑁(π‘˜)-Contact Metric Manifolds

This section deals with a (2𝑛+1)-dimensional πœ™-conharmonically flat 𝑁(π‘˜)-contact metric manifold. Then we have from (1.3) that 𝑅=1(πœ™π‘‹,πœ™π‘Œ,πœ™π‘,πœ™π‘Š)[].2π‘›βˆ’1𝑔(πœ™π‘Œ,πœ™π‘)𝑆(πœ™π‘‹,πœ™π‘Š)βˆ’π‘”(πœ™π‘‹,πœ™π‘)𝑆(πœ™π‘Œ,πœ™π‘Š)+𝑆(πœ™π‘Œ,πœ™π‘)𝑔(πœ™π‘‹,πœ™π‘Š)βˆ’π‘†(πœ™π‘‹,πœ™π‘)𝑔(πœ™π‘Œ,πœ™π‘Š)(5.1)

Let {𝑒1,𝑒2,…,𝑒2𝑛,πœ‰} be a local orthonormal basis of the tangent space of the manifold. Then {πœ™π‘’1,πœ™π‘’2,…,πœ™π‘’2𝑛,πœ‰} is also a local orthonormal basis of the tangent space. Putting 𝑋=π‘Š=𝑒𝑖 in (5.1) and summing up from 1 to 2𝑛 we have 2𝑛𝑖=1ξ‚π‘…ξ€·πœ™π‘’π‘–,πœ™π‘Œ,πœ™π‘,πœ™π‘’π‘–ξ€Έ=12π‘›βˆ’12𝑛𝑖=1𝑔(πœ™π‘Œ,πœ™π‘)π‘†πœ™π‘’π‘–,πœ™π‘’π‘–ξ€Έξ€·βˆ’π‘”πœ™π‘’π‘–ξ€Έπ‘†ξ€·,πœ™π‘πœ™π‘Œ,πœ™π‘’π‘–ξ€Έξ€·+𝑆(πœ™π‘Œ,πœ™π‘)π‘”πœ™π‘’π‘–,πœ™π‘’π‘–ξ€Έξ€·βˆ’π‘†πœ™π‘’π‘–ξ€Έπ‘”ξ€·,πœ™π‘πœ™π‘Œ,πœ™π‘’π‘–.ξ€Έξ€»(5.2)

Using (2.18), (2.19), (2.20), and (2.21) in (5.2) we obtain𝑆(πœ™π‘Œ,πœ™π‘)=(π‘Ÿβˆ’π‘˜)𝑔(πœ™π‘Œ,πœ™π‘).(5.3)

Replacing π‘Œ and 𝑍 by πœ™π‘Œ and πœ™π‘ in (5.3) and using (2.1) we have []𝑆(π‘Œ,𝑍)=(π‘Ÿβˆ’π‘˜)𝑔(π‘Œ,𝑍)+(2𝑛+1)π‘˜βˆ’π‘Ÿπœ‚(π‘Œ)πœ‚(𝑍).(5.4)

Putting π‘Œ=𝑍=𝑒𝑖 in (5.4) and taking summation over 𝑖=1 to 2𝑛+1 we get by using (2.18) and (2.19) that π‘Ÿ=0.(5.5) In view of the above discussions we have the following.

Proposition 5.1. A (2𝑛+1)-dimensional πœ™-conharmonically flat 𝑁(π‘˜)-contact metric manifold is an πœ‚-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 πœ™-conharmonically flat 𝑁(π‘˜)-contact metric manifold is a Sasakian manifold with vanishing scalar curvature.

Definition 5.3. In a (2𝑛+1)-dimensional 𝑁(π‘˜)-contact metric manifold, if the Ricci tensor 𝑆 satisfies (βˆ‡π‘‹π‘†)(πœ™π‘Œ,πœ™π‘)=0, then the Ricci tensor is said to be πœ‚-parallel.

The notion of πœ‚-parallel Ricci tensor for Sasakian manifold was introduced by Kon [22].

Putting π‘Ÿ=0 in (5.4) we have 𝑆(π‘Œ,𝑍)=βˆ’π‘˜π‘”(π‘Œ,𝑍)+(2𝑛+1)π‘˜πœ‚(π‘Œ)πœ‚(𝑍).(5.6)

Replacing π‘Œ and 𝑍 by πœ™π‘Œ and πœ™π‘ in (5.6) and using (2.1) we obtain 𝑆(πœ™π‘Œ,πœ™π‘)=βˆ’π‘˜π‘”(πœ™π‘Œ,πœ™π‘).(5.7)

Relation (5.7) yields ξ€·βˆ‡π‘‹π‘†ξ€Έ(πœ™π‘Œ,πœ™π‘)=0,(5.8) since π‘˜ is a constant. Therefore we have the following corollary.

Corollary 5.4. A (2𝑛+1)-dimensional πœ™-conharmonically flat 𝑁(π‘˜)-contact metric manifold satisfies πœ‚-parallel Ricci tensor.

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