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ISRN Geometry
VolumeΒ 2012Β (2012), Article IDΒ 367467, 21 pages
http://dx.doi.org/10.5402/2012/367467
Research Article

Some Classes of Pseudosymmetric Contact Metric 3-Manifolds

Department of Mathematics, University of Thessaloniki, Adendron 57007, Greece

Received 13 November 2011; Accepted 22 December 2011

Academic Editors: T.Β Friedrich and O.Β Mokhov

Copyright Β© 2012 Evaggelia Moutafi. 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

We study (a) the class of 3-dimensional pseudosymmetric contact metric manifolds with harmonic curvature and π‘‡π‘Ÿπ‘™ constant along the direction of πœ‰ and (b) the class of (πœ…,πœ‡,𝜈)-contact metric pseudosymmetric 3-manifolds of type constant in the direction of πœ‰.

1. Introduction

A Riemannian manifold (π‘€π‘š,𝑔) is said to be pseudosymmetric according to Deszcz [1] if its curvature tensor 𝑅 satisfies the condition 𝑅(𝑋,π‘Œ)⋅𝑅=𝐿{(π‘‹βˆ§π‘Œ)⋅𝑅}, where the dot means that 𝑅(𝑋,π‘Œ) acts as a derivation on 𝑅, 𝐿 is a smooth function and the endomorphism field π‘‹βˆ§π‘Œ is defined by(π‘‹βˆ§π‘Œ)𝑍=𝑔(π‘Œ,𝑍)π‘‹βˆ’π‘”(𝑍,𝑋)π‘Œ,(1.1) for all vectors fields 𝑋,π‘Œ,𝑍 on 𝑀 and it similarly acts as a derivation on 𝑅.

If 𝐿 is constant, 𝑀 is called a pseudosymmetric manifold of constant type and if particularly 𝐿=0 then 𝑀 is called a semisymmetric manifold first studied by E. Cartan. Semisymmetric spaces [2, 3] are a generalization of locally symmetric spaces (βˆ‡π‘…=0, [4]) while pseudosymmetric spaces are a natural generalization of semisymmetric spaces. There are many details and examples on pseudosymmetric manifolds in [1, 5]. We remark that in dimension three, the pseudosymmetry is equivalent to the condition: the eigenvalues 𝜌1,𝜌2,𝜌3 of the Ricci tensor satisfy 𝜌1=𝜌2 (up to numeration) and the last one is constant [6, 7].

Kowalski and Sekizawa have studied [7–10] 3-dimensional pseudosymmetric spaces of constant type. Hashimoto and Sekizawa classified 3-dimensional conformally flat pseudosymmetric spaces of constant type [11] and finally Calvaruso [12] gave the complete classification of conformally flat pseudosymmetric spaces of constant type for dimensions >2. Cho and Inoguchi [13] studied pseudosymmetric contact homogeneous 3-manifolds while Cho et al. [14] give the conditions so as 3-dimensional trans-Sasakians, quasi-Sasakians, non-Sasakian generalized (πœ…,πœ‡)-spaces to be pseudosymmetric. Belkhelfa et al. [15] studied pseudosymmetric Sasakian space forms of any dimension. Finally Gouli-Andreou and Moutafi in [16, 17] have studied some classes of pseudosymmetric contact metric 3-manifolds.

The aim of this paper is the study of the 3-dimension pseudosymmetric contact metric manifolds. The paper is organized in the following way: in Section 2 we will give some preliminaries on pseudosymmetric manifolds and contact manifolds as well and in the next sections we will study 3-dimensional manifolds which satisfy one of the following conditions.(i)𝑀 is a pseudosymmetric contact metric manifold with harmonic curvature and π‘‡π‘Ÿπ‘™ constant along the direction of πœ‰. (ii)𝑀 is a (πœ…,πœ‡,𝜈)-contact metric pseudosymmetric manifold of type constant in the direction of πœ‰.

2. Preliminaries

Let (π‘€π‘š,𝑔),π‘šβ‰₯3 be a connected Riemannian smooth manifold. We denote by 𝑅 its Riemannian curvature tensor given by the equation 𝑅(𝑋,π‘Œ)𝑍=[βˆ‡π‘‹,βˆ‡π‘Œ]π‘βˆ’βˆ‡[𝑋,π‘Œ]𝑍 for any 𝑋,π‘Œ,π‘βˆˆπ”›(𝑀) and where βˆ‡ is the Levi-Civita connection of π‘€π‘š.

Definition 2.1. A Riemannian manifold (π‘€π‘š,𝑔),π‘šβ‰₯3, is called pseudosymmetric in the sense of Deszcz [1] if at every point of 𝑀 the curvature tensor satisfies the condition:

𝑋(𝑅(𝑋,π‘Œ)⋅𝑅)1,𝑋2,𝑋3𝑋=𝐿((π‘‹βˆ§π‘Œ)⋅𝑅)1,𝑋2,𝑋3ξ€Έξ€Ύ(2.1) or more explixitly:𝑅𝑅𝑋(𝑋,π‘Œ)1,𝑋2𝑋3ξ€Έξ€·π‘…βˆ’π‘…(𝑋,π‘Œ)𝑋1,𝑋2𝑋3ξ€·π‘‹βˆ’π‘…1,𝑅(𝑋,π‘Œ)𝑋2𝑋3ξ€·π‘‹βˆ’π‘…1,𝑋2𝑅(𝑋,π‘Œ)𝑋3ξ€Έξ€½(𝑅𝑋=πΏπ‘‹βˆ§π‘Œ)1,𝑋2𝑋3ξ€Έξ€·(βˆ’π‘…π‘‹βˆ§π‘Œ)𝑋1,𝑋2𝑋3ξ€·π‘‹βˆ’π‘…1,(π‘‹βˆ§π‘Œ)𝑋2𝑋3ξ€·π‘‹βˆ’π‘…1,𝑋2ξ€Έξ€·(π‘‹βˆ§π‘Œ)𝑋3,ξ€Έξ€Ύ(2.2) for any 𝑋,π‘Œ,𝑋1,𝑋2,𝑋3βˆˆπ”›(𝑀), π‘‹βˆ§π‘Œ is given by (1.1) and 𝐿 is a smooth function.

Definition 2.2. A differentiable manifold 𝑀2𝑛+1 endowed with a global 1-form πœ‚ such that πœ‚βˆ§(π‘‘πœ‚)𝑛≠0 everywhere on 𝑀 is called a contact manifold.

Given a contact manifold (𝑀,πœ‚), there is an underlying contact metric structure (πœ‚,πœ‰,πœ™,𝑔) where 𝑔 is a Riemannian metric (the associated metric), πœ™ a global tensor of type (1,1), and πœ‰ a unique global vector field (the characteristic or Reeb vector field). A differentiable (2𝑛+1)-dimensional manifold endowed with a contact metric structure (πœ‚,πœ‰,πœ™,𝑔) is called a contact metric (Riemannian) manifold denoted by 𝑀(πœ‚,πœ‰,πœ™,𝑔). The structure tensors πœ‚, πœ‰, πœ™, and 𝑔 satisfy the equations:πœ™2=βˆ’πΌ+πœ‚βŠ—πœ‰,πœ‚(𝑋)=𝑔(𝑋,πœ‰),πœ‚(πœ‰)=1,π‘‘πœ‚(𝑋,π‘Œ)=𝑔(𝑋,πœ™π‘Œ),𝑔(πœ™π‘‹,πœ™π‘Œ)=𝑔(𝑋,π‘Œ)βˆ’πœ‚(𝑋)πœ‚(π‘Œ).(2.3) The associated metrics can be constructed polarizing π‘‘πœ‚ on the contact subbundle 𝐷 defined by πœ‚=0. Denoting by 𝐿 the Lie differentiation and 𝑅 the curvature tensor, respectively, we define the tensor fields β„Ž, 𝑙, and 𝜏 by1β„Žπ‘‹=2ξ€·πΏπœ‰πœ™ξ€Έξ€·πΏπ‘‹,𝑙𝑋=𝑅(𝑋,πœ‰)πœ‰,𝜏(𝑋,π‘Œ)=πœ‰π‘”ξ€Έ(𝑋,π‘Œ).(2.4) These tensors also satisfy the following formulas:πœ™πœ‰=β„Žπœ‰=π‘™πœ‰=0,πœ‚βˆ˜πœ™=πœ‚βˆ˜β„Ž=0,π‘‘πœ‚(πœ‰,𝑋)=0,(2.5)π‘‡π‘Ÿβ„Ž=π‘‡π‘Ÿβ„Žπœ™=0,βˆ‡π‘‹βˆ‡πœ‰=βˆ’πœ™π‘‹βˆ’πœ™β„Žπ‘‹,β„Žπœ™=βˆ’πœ™β„Ž,(2.6)β„Žπ‘‹=πœ†π‘‹βŸΉβ„Žπœ™π‘‹=βˆ’πœ†πœ™π‘‹,(2.7)πœ‰β„Ž=πœ™βˆ’πœ™π‘™βˆ’πœ™β„Ž2ξ€·πœ™,πœ™π‘™πœ™βˆ’π‘™=22+β„Ž2ξ€Έβˆ‡,(2.8)πœ‰πœ™=0,π‘‡π‘Ÿπ‘™=𝑔(π‘„πœ‰,πœ‰)=2π‘›βˆ’π‘‡π‘Ÿβ„Ž2.(2.9)β„Ž=0 (or equivalently 𝜏=0) if and only if πœ‰ is Killing and 𝑀 is called 𝐾-contact. A contact structure on 𝑀 implies an almost complex structure on the product manifold 𝑀2𝑛+1×ℝ. If this structure is integrable, then the contact metric manifold is said to be Sasakian. A K-contact structure is Sasakian only in dimension 3, and this fails in higher dimensions. More details on contact manifolds we can find in [18, 19].

Let (𝑀,πœ™,πœ‰,πœ‚,𝑔) be a 3-dimensional contact metric manifold and π‘ˆ the open subset of points π‘βˆˆπ‘€ where β„Žβ‰ 0 in a neighborhood of 𝑝 and π‘ˆ0 the open subset of points π‘βˆˆπ‘€ such that β„Ž=0 in a neighborhood of 𝑝. Because β„Ž is a smooth function on 𝑀 then π‘ˆβˆͺπ‘ˆ0 is an open and dense subset of 𝑀 so if a property is satisfied in π‘ˆ0βˆͺπ‘ˆ then this property will be satisfied in 𝑀. For any point π‘βˆˆπ‘ˆβˆͺπ‘ˆ0 there exists a local orthonormal basis {𝑒,πœ™π‘’,πœ‰} of smooth eigenvectors of β„Ž in a neighborhood of 𝑝 (a πœ™-basis). On π‘ˆ, we put β„Žπ‘’=πœ†π‘’, where πœ† is a non vanishing smooth function which is supposed positive. From (2.7), we have β„Žπœ™π‘’=βˆ’πœ†πœ™π‘’. We recall the following.

Lemma 2.3 (see [20]). On π‘ˆ, one has βˆ‡πœ‰π‘’=π‘Žπœ™π‘’,βˆ‡π‘’π‘’=π‘πœ™π‘’,βˆ‡πœ™π‘’βˆ‡π‘’=βˆ’π‘πœ™π‘’+(πœ†βˆ’1)πœ‰,πœ‰πœ™π‘’=βˆ’π‘Žπ‘’,βˆ‡π‘’πœ™π‘’=βˆ’π‘π‘’+(1+πœ†)πœ‰,βˆ‡πœ™π‘’βˆ‡πœ™π‘’=𝑐𝑒,πœ‰πœ‰=0,βˆ‡π‘’πœ‰=βˆ’(1+πœ†)πœ™π‘’,βˆ‡πœ™π‘’πœ‰=(1βˆ’πœ†)𝑒,(2.10) where π‘Ž is a smooth function and 1𝑏=[]12πœ†(πœ™π‘’β‹…πœ†)+𝐴with𝐴=𝑆(πœ‰,𝑒),𝑐=[]2πœ†(π‘’β‹…πœ†)+𝐡with𝐡=𝑆(πœ‰,πœ™π‘’).(2.11)

From Lemma 2.3 and the formula [𝑋,π‘Œ]=βˆ‡π‘‹π‘Œβˆ’βˆ‡π‘Œπ‘‹ we can prove that[]𝑒,πœ™π‘’=βˆ‡π‘’πœ™π‘’βˆ’βˆ‡πœ™π‘’[]𝑒=βˆ’π‘π‘’+π‘πœ™π‘’+2πœ‰,𝑒,πœ‰=βˆ‡π‘’πœ‰βˆ’βˆ‡πœ‰[]𝑒=βˆ’(π‘Ž+πœ†+1)πœ™π‘’,πœ™π‘’,πœ‰=βˆ‡πœ™π‘’πœ‰βˆ’βˆ‡πœ‰πœ™π‘’=(π‘Žβˆ’πœ†+1)𝑒,(2.12) and from (1.1) we estimate(π‘’βˆ§πœ™π‘’)𝑒=βˆ’πœ™π‘’,(π‘’βˆ§πœ‰)𝑒=βˆ’πœ‰,(πœ™π‘’βˆ§πœ‰)πœ‰=πœ™π‘’,(π‘’βˆ§πœ™π‘’)πœ™π‘’=𝑒,(π‘’βˆ§πœ‰)πœ‰=𝑒,(πœ™π‘’βˆ§πœ‰)πœ™π‘’=βˆ’πœ‰,(2.13) while (π‘‹βˆ§π‘Œ)𝑍=0, whenever π‘‹β‰ π‘Œβ‰ π‘β‰ π‘‹ and 𝑋,π‘Œ,π‘βˆˆ{𝑒,πœ™π‘’,πœ‰}.

By direct computations we calculate the non vanishing independent components of the Riemannian curvature tensor field 𝑅 (1,3):𝑅𝑅(πœ‰,𝑒)πœ‰=βˆ’πΌπ‘’βˆ’π‘πœ™π‘’,𝑅(𝑒,πœ™π‘’)𝑒=βˆ’πΆπœ™π‘’βˆ’π΅πœ‰,(πœ‰,πœ™π‘’)πœ‰=βˆ’π‘π‘’βˆ’π·πœ™π‘’,𝑅(πœ‰,𝑒)πœ™π‘’=βˆ’πΎπ‘’+π‘πœ‰,𝑅(𝑒,πœ™π‘’)πœ‰=π΅π‘’βˆ’π΄πœ™π‘’,𝑅(πœ‰,πœ™π‘’)πœ™π‘’=𝐻𝑒+π·πœ‰,𝑅(πœ‰,𝑒)𝑒=πΎπœ™π‘’+πΌπœ‰,𝑅(𝑒,πœ™π‘’)πœ™π‘’=𝐢𝑒+π΄πœ‰,𝑅(πœ‰,πœ™π‘’)𝑒=βˆ’π»πœ™π‘’+π‘πœ‰,(2.14) where𝐢=βˆ’π‘2βˆ’π‘2+πœ†2βˆ’1+2π‘Ž+(𝑒⋅𝑐)+(πœ™π‘’β‹…π‘),𝐻=𝑏(πœ†βˆ’π‘Žβˆ’1)+(πœ‰β‹…π‘)+(πœ™π‘’β‹…π‘Ž),𝐾=𝑐(πœ†+π‘Ž+1)+(πœ‰β‹…π‘)βˆ’(π‘’β‹…π‘Ž),𝐼=βˆ’2π‘Žπœ†βˆ’πœ†2+1,𝐷=2π‘Žπœ†βˆ’πœ†2+1,𝑍=πœ‰β‹…πœ†.(2.15) Setting 𝑋=𝑒, π‘Œ=πœ™π‘’, 𝑍=πœ‰ in the Jacobi identity [[𝑋,π‘Œ],𝑍]+[[π‘Œ,𝑍],𝑋]+[[𝑍,𝑋],π‘Œ]=0 and using (2.12), we get𝑐𝑏(π‘Ž+πœ†+1)βˆ’(πœ‰β‹…π‘)βˆ’(πœ™π‘’β‹…πœ†)βˆ’(πœ™π‘’β‹…π‘Ž)=0,(π‘Žβˆ’πœ†+1)+(πœ‰β‹…π‘)+(π‘’β‹…πœ†)βˆ’(π‘’β‹…π‘Ž)=0(2.16) or equivalently: 𝐴=𝐻 and 𝐡=𝐾.

We give the components of the Ricci operator 𝑄 with respect to a πœ™-basis:ξ‚€π‘Ÿπ‘„π‘’=2βˆ’1+πœ†2ξ‚ξ‚€π‘Ÿβˆ’2π‘Žπœ†π‘’+π‘πœ™π‘’+π΄πœ‰,π‘„πœ™π‘’=𝑍𝑒+2βˆ’1+πœ†2+2π‘Žπœ†πœ™π‘’+π΅πœ‰,π‘„πœ‰=𝐴𝑒+π΅πœ™π‘’+21βˆ’πœ†2ξ€Έπœ‰,(2.17) whereξ€Ίπ‘Ÿ=π‘‡π‘Ÿπ‘„=21βˆ’πœ†2βˆ’π‘2βˆ’π‘2ξ€»+2π‘Ž+(𝑒⋅𝑐)+(πœ™π‘’β‹…π‘)(2.18) is the scalar curvature. The relations (2.15) and (2.18) yield𝐢=βˆ’π‘2βˆ’π‘2+πœ†2βˆ’1+2π‘Ž+(𝑒⋅𝑐)+(πœ™π‘’β‹…π‘)=2πœ†2π‘Ÿβˆ’2+2,(2.19) and the relation (2.9):ξ€·π‘‡π‘Ÿπ‘™=21βˆ’πœ†2ξ€Έ.(2.20)

Remark 2.4. If 𝑀3=π‘ˆ0 (see [21]), Lemma 2.3 is expressed in a similar form with πœ†=0, 𝑒 is a unit vector field belonging to the contact distribution and for the functions 𝐴,𝐡,𝐷,𝐻,𝐼,𝐾 and 𝑍 we have: 𝐴=𝐡=𝑍=𝐻=𝐾=0, 𝐼=𝐷=1 and 𝐢=π‘Ÿ/2βˆ’2.

Definition 2.5. An π‘€π‘š Riemannian manifold has harmonic curvature if the Ricci operator 𝑄 satisfies the condition:

ξ€·βˆ‡π‘‹π‘„ξ€Έξ€·βˆ‡π‘Œ=π‘Œπ‘„ξ€Έπ‘‹,βˆ€π‘‹,π‘Œβˆˆπ”›(𝑀).(2.21) From now on we shall work on a (𝑀3,πœ™,πœ‰,πœ‚,𝑔) contact metric 3-manifold concerning a πœ™-basis {𝑒,πœ™π‘’,πœ‰} at any point π‘βˆˆπ‘€. First from the equation (βˆ‡π‘‹π‘„)π‘Œ=βˆ‡π‘‹(π‘„π‘Œ)βˆ’π‘„(βˆ‡π‘‹π‘Œ), Lemma 2.3 and the relations (2.17), we get the equations:ξ€·βˆ‡π‘’π‘„ξ€Έ[]𝑒+ξ‚Έπœ™π‘’=(π‘’β‹…πœ‰β‹…πœ†)βˆ’2π‘πœ†(2π‘Ž+πœ†+1)+(1+πœ†)(πœ™π‘’β‹…πœ†)(π‘’β‹…π‘Ÿ)2ξ‚Ή+ξ‚ƒπ‘Ÿ+2𝑏(πœ‰β‹…πœ†)+2πœ†(π‘’β‹…π‘Ž)+(4πœ†+2π‘Ž+2)(π‘’β‹…πœ†)βˆ’4π‘πœ†(1+πœ†)πœ™π‘’2ξ€·πœ†(πœ†+1)+2πœ†(𝑒⋅𝑐)+2𝑐(π‘’β‹…πœ†)βˆ’(π‘’β‹…π‘’β‹…πœ†)+32ξ€Έξ€·π‘βˆ’1(πœ†+1)βˆ’π‘(πœ™π‘’β‹…πœ†)+2πœ†2ξ€·βˆ‡+π‘Žπœ†+π‘Žξ€Έξ€»πœ‰,𝑒𝑄[]𝑒+ξ‚ƒπ‘Ÿπœ‰=3𝑏(π‘’β‹…πœ†)+2πœ†(𝑒⋅𝑏)βˆ’(π‘’β‹…πœ™π‘’β‹…πœ†)βˆ’2π‘π‘πœ†+(1+πœ†)(πœ‰β‹…πœ†)2ξ€·πœ†(πœ†+1)+2𝑐(π‘’β‹…πœ†)+2πœ†(𝑒⋅𝑐)βˆ’(π‘’β‹…π‘’β‹…πœ†)+32ξ€Έξ€·π‘βˆ’1(πœ†+1)βˆ’π‘(πœ™π‘’β‹…πœ†)+2πœ†2+[]ξ€·βˆ‡+π‘Žπœ†+π‘Žξ€Έξ€»πœ™π‘’4π‘πœ†(1+πœ†)βˆ’(2+6πœ†)(π‘’β‹…πœ†)πœ‰,πœ™π‘’π‘„ξ€Έξ‚Έπ‘’=(πœ™π‘’β‹…π‘Ÿ)2𝑒+[]+ξ‚ƒπ‘Ÿ+(4πœ†βˆ’2π‘Žβˆ’2)(πœ™π‘’β‹…πœ†)βˆ’2πœ†(πœ™π‘’β‹…π‘Ž)+4π‘πœ†(1βˆ’πœ†)+2𝑐(πœ‰β‹…πœ†)4π‘Žπ‘πœ†+(πœ™π‘’β‹…πœ‰β‹…πœ†)+2π‘πœ†(1βˆ’πœ†)+(πœ†βˆ’1)(π‘’β‹…πœ†)πœ™π‘’2ξ€·πœ†(πœ†βˆ’1)+2πœ†(πœ™π‘’β‹…π‘)βˆ’π‘(π‘’β‹…πœ†)βˆ’(πœ™π‘’β‹…πœ™π‘’β‹…πœ†)+32ξ€Έξ€·π‘βˆ’1(πœ†βˆ’1)+2𝑏(πœ™π‘’β‹…πœ†)+2πœ†2ξ€·βˆ‡βˆ’π‘Žπœ†+π‘Žξ€Έξ€»πœ‰,πœ™π‘’π‘„ξ€Έξ‚ƒπ‘Ÿπœ‰=2ξ€·πœ†(πœ†βˆ’1)+2πœ†(πœ™π‘’β‹…π‘)βˆ’(πœ™π‘’β‹…πœ™π‘’β‹…πœ†)βˆ’π‘(π‘’β‹…πœ†)+32ξ€Έξ€·π‘βˆ’1(πœ†βˆ’1)+2𝑏(πœ™π‘’β‹…πœ†)+2πœ†2𝑒+[]+[]ξ€·βˆ‡βˆ’π‘Žπœ†+π‘Žξ€Έξ€»βˆ’2π‘π‘πœ†+3𝑐(πœ™π‘’β‹…πœ†)+2πœ†(πœ™π‘’β‹…π‘)βˆ’(πœ™π‘’β‹…π‘’β‹…πœ†)+(πœ†βˆ’1)(πœ‰β‹…πœ†)πœ™π‘’4π‘πœ†(πœ†βˆ’1)βˆ’(6πœ†βˆ’2)(πœ™π‘’β‹…πœ†)πœ‰,πœ‰π‘„ξ€Έξ‚Έπ‘’=(πœ‰β‹…π‘Ÿ)2𝑒+ξ€Ί+(2πœ†βˆ’4π‘Ž)(πœ‰β‹…πœ†)βˆ’2πœ†(πœ‰β‹…π‘Ž)βˆ’4π‘Ž2ξ€»+[],ξ€·βˆ‡πœ†+(πœ‰β‹…πœ‰β‹…πœ†)πœ™π‘’βˆ’2π‘Žπ‘πœ†+2𝑏(πœ‰β‹…πœ†)+2πœ†(πœ‰β‹…π‘)βˆ’(πœ‰β‹…πœ™π‘’β‹…πœ†)+π‘Ž(π‘’β‹…πœ†)πœ‰π‘„ξ€Έξ€Ίπœ™π‘’=(πœ‰β‹…πœ‰β‹…πœ†)βˆ’4π‘Ž2πœ†ξ€»π‘’+ξ‚Έ(πœ‰β‹…π‘Ÿ)2ξ‚Ή+[]+(2πœ†+4π‘Ž)(πœ‰β‹…πœ†)+2πœ†(πœ‰β‹…π‘Ž)πœ™π‘’2π‘Žπ‘πœ†+2𝑐(πœ‰β‹…πœ†)+2πœ†(πœ‰β‹…π‘)βˆ’(πœ‰β‹…π‘’β‹…πœ†)βˆ’π‘Ž(πœ™π‘’β‹…πœ†)πœ‰.(2.22)

Applying (2.21) to the vectors fields of the πœ™-basis of the contact metric manifold 𝑀3 we have: (βˆ‡π‘’π‘„)πœ™π‘’=(βˆ‡πœ™π‘’π‘„)𝑒, (βˆ‡π‘’π‘„)πœ‰=(βˆ‡πœ‰π‘„)𝑒 and (βˆ‡πœ™π‘’π‘„)πœ‰=(βˆ‡πœ‰π‘„)πœ™π‘’. We use the previous relations and we get the following nine (9) conditions for a contact metric 3-manifold to have harmonic curvature:(π‘’β‹…πœ‰β‹…πœ†)+(3βˆ’3πœ†+2π‘Ž)(πœ™π‘’β‹…πœ†)βˆ’(πœ™π‘’β‹…π‘Ÿ)2βˆ’2𝑐(πœ‰β‹…πœ†)+2πœ†(πœ™π‘’β‹…π‘Ž)βˆ’2π‘πœ†(2π‘Ž+3βˆ’πœ†)=0,2𝑏(πœ‰β‹…πœ†)+2πœ†(π‘’β‹…π‘Ž)+(3πœ†+2π‘Ž+3)(π‘’β‹…πœ†)+(π‘’β‹…π‘Ÿ)2ξ€·πœ†βˆ’(πœ™π‘’β‹…πœ‰β‹…πœ†)βˆ’2π‘πœ†(3+2π‘Ž+πœ†)=0,π‘Ÿ+2πœ†(𝑒⋅𝑐)βˆ’2πœ†(πœ™π‘’β‹…π‘)+3𝑐(π‘’β‹…πœ†)βˆ’3𝑏(πœ™π‘’β‹…πœ†)βˆ’(π‘’β‹…π‘’β‹…πœ†)+62ξ€Έ+ξ€·π‘βˆ’1(πœ™π‘’β‹…πœ™π‘’β‹…πœ†)+2πœ†2βˆ’π‘2ξ€Έ+2π‘Žπœ†=0,3𝑏(π‘’β‹…πœ†)+2πœ†(𝑒⋅𝑏)βˆ’(π‘’β‹…πœ™π‘’β‹…πœ†)+2πœ†(πœ‰β‹…π‘Ž)βˆ’(πœ‰β‹…π‘Ÿ)2βˆ’2π‘π‘πœ†+(4π‘Ž+1βˆ’πœ†)(πœ‰β‹…πœ†)=0,βˆ’(2+6πœ†+π‘Ž)(π‘’β‹…πœ†)βˆ’2𝑏(πœ‰β‹…πœ†)βˆ’2πœ†(πœ‰β‹…π‘)+2πœ†π‘(π‘Ž+2+2πœ†)+(πœ‰β‹…πœ™π‘’β‹…πœ†)=0,(π‘Žβˆ’6πœ†+2)(πœ™π‘’β‹…πœ†)βˆ’2𝑐(πœ‰β‹…πœ†)βˆ’2πœ†(πœ‰β‹…π‘)+2π‘πœ†(2πœ†βˆ’2βˆ’π‘Ž)+(πœ‰β‹…π‘’β‹…πœ†)=0,3𝑐(πœ™π‘’β‹…πœ†)+2πœ†(πœ™π‘’β‹…π‘)βˆ’(πœ™π‘’β‹…π‘’β‹…πœ†)βˆ’2πœ†(πœ‰β‹…π‘Ž)βˆ’(πœ‰β‹…π‘Ÿ)2π‘Ÿβˆ’2π‘π‘πœ†βˆ’(πœ†+1+4π‘Ž)(πœ‰β‹…πœ†)=0,2ξ€·πœ†(πœ†+1)βˆ’π‘(πœ™π‘’β‹…πœ†)+2𝑐(π‘’β‹…πœ†)+2πœ†(𝑒⋅𝑐)βˆ’(π‘’β‹…π‘’β‹…πœ†)βˆ’(πœ‰β‹…πœ‰β‹…πœ†)+32ξ€Έξ€·βˆ’1(πœ†+1)+2πœ†2π‘Ž2+𝑏2ξ€Έπ‘Ÿ+π‘Žπœ†+π‘Ž=0,2ξ€·πœ†(πœ†βˆ’1)+2𝑏(πœ™π‘’β‹…πœ†)+2πœ†(πœ™π‘’β‹…π‘)βˆ’π‘(π‘’β‹…πœ†)βˆ’(πœ™π‘’β‹…πœ™π‘’β‹…πœ†)βˆ’(πœ‰β‹…πœ‰β‹…πœ†)+32ξ€Έξ€·βˆ’1(πœ†βˆ’1)+2πœ†2π‘Ž2+𝑐2ξ€Έβˆ’π‘Žπœ†+π‘Ž=0.(2.23)

Remark 2.6. From these nine conditions, we can derive some useful results: (a) by subtracting the ninth equation from the first and using (2.16), we get πœ™π‘’β‹…π‘Ÿ=0, (b) by adding the equations two and six and using similarly (2.16) we have π‘’β‹…π‘Ÿ=0. From the relations πœ™π‘’β‹…π‘Ÿ=0, π‘’β‹…π‘Ÿ=0, and (2.12), we can conclude πœ‰β‹…π‘Ÿ=0 and hence we are led to the known result that the scalar curvature π‘Ÿ is constant in a contact metric 3-manifold with harmonic curvature. Later for our study, we will use the π‘Ÿ as a constant and we will give to these equations a more convenient form.

Definition 2.7 (see [22]). Let 𝑀3 be a 3-dimensional contact metric manifold and β„Ž=πœ†β„Ž+βˆ’πœ†β„Žβˆ’ the spectral decomposition of β„Ž on π‘ˆ. If

βˆ‡β„Žβˆ’π‘‹β„Žβˆ’ξ€Ίπ‘‹=πœ‰,β„Ž+𝑋,(2.24) for all vector fields 𝑋 on 𝑀3 and all points of an open subset π‘Š of π‘ˆ and β„Ž=0 on the points of 𝑀3 which do not belong to π‘Š, then the manifold is said to be semi-𝐾 contact manifold. From Lemma 2.3 and the relations (2.12), the above condition for 𝑋=𝑒 leads to [πœ‰,𝑒]=0 and for 𝑋=πœ™π‘’ to βˆ‡πœ™π‘’πœ™π‘’=0. Hence on a semi-𝐾 contact manifold we have π‘Ž+πœ†+1=𝑐=0. If we apply the deformation π‘’β†’πœ™π‘’, πœ™π‘’β†’π‘’, πœ‰β†’βˆ’πœ‰, πœ†β†’βˆ’πœ†, 𝑏→𝑐, and 𝑐→𝑏 then the contact metric structure remains the same. Hence the condition for a 3-dimensional contact metric manifold to be semi-𝐾 contact is equivalent to π‘Žβˆ’πœ†+1=𝑏=0.

Definition 2.8. A (πœ…,πœ‡,𝜈)-contact metric manifold is defined in [23] as a contact metric manifold (𝑀2𝑛+1,πœ‚,πœ‰,πœ™,𝑔) on which the curvature tensor satisfies for every 𝑋,π‘Œβˆˆπ‘‹(𝑀) the condition:

𝑅(𝑋,π‘Œ)πœ‰=πœ…(πœ‚(π‘Œ)π‘‹βˆ’πœ‚(𝑋)π‘Œ)+πœ‡(πœ‚(π‘Œ)β„Žπ‘‹βˆ’πœ‚(𝑋)β„Žπ‘Œ)+𝜈(πœ‚(π‘Œ)πœ™β„Žπ‘‹βˆ’πœ‚(𝑋)πœ™β„Žπ‘Œ),(2.25) where πœ…,πœ‡,𝜈 are smooth functions on 𝑀. If 𝜈=0 we have a generalized (πœ…,πœ‡)-contact metric manifold [24] and if additionally πœ…,πœ‡ are constants then the manifold is a contact metric (πœ…,πœ‡)-space [25, 26]. Moreover, in [23] and [24] it is proved, respectively, that for a (πœ…,πœ‡,𝜈) or a generalized (πœ…,πœ‡)-contact metric manifold 𝑀2𝑛+1 of dimension greater than 3 the functions πœ…,πœ‡ are constants and 𝜈 is the zero function.

Now, we will give some known results concerning contact metric 3-manifolds and pseudosymmetric contact metric 3-manifolds.

Proposition 2.9 (see [16]). In a 3-dimensional contact metric manifold one has

π‘„πœ™=πœ™π‘„βŸΊ(πœ‰β‹…πœ†=2π‘πœ†βˆ’(πœ™π‘’β‹…πœ†)=2π‘πœ†βˆ’(π‘’β‹…πœ†)=π‘Žπœ†=0).(2.26)

Let (𝑀,πœ‚,𝑔,πœ™,πœ‰) be a contact metric 3-manifold. In case 𝑀=π‘ˆ0, that is, (πœ‰,πœ‚,πœ™,𝑔) is a Sasakian structure, then 𝑀 is a pseudosymmetric space of constant type [13]. Next, assume that π‘ˆ is not empty and let {𝑒,πœ™π‘’,πœ‰} be a πœ™-basis as in Lemma 2.3. We have the following.

Lemma 2.10 (see [16]). Let (𝑀,πœ‚,𝑔,πœ™,πœ‰) be a contact metric three manifold. Then 𝑀 is pseudosymmetric if and only if 𝐡(πœ‰β‹…πœ†)+βˆ’2π‘Žπœ†βˆ’πœ†2ξ€Έξ€·+1𝐴=𝐿𝐴,𝐴(πœ‰β‹…πœ†)+2π‘Žπœ†βˆ’πœ†2ξ€Έξ‚€π‘Ÿ+1𝐡=𝐿𝐡,(πœ‰β‹…πœ†)2+2πœ†2ξ‚π΄βˆ’2+𝐴𝐡=𝐿(πœ‰β‹…πœ†),2βˆ’||||(πœ‰β‹…πœ†)2+ξ€·2π‘Žπœ†βˆ’πœ†2ξ€Έξ‚€+1βˆ’2π‘Žπœ†βˆ’3πœ†2π‘Ÿ+3βˆ’2=πΏβˆ’2π‘Žπœ†βˆ’3πœ†2π‘Ÿ+3βˆ’2,𝐡2βˆ’||||(πœ‰β‹…πœ†)2+ξ€·βˆ’2π‘Žπœ†βˆ’πœ†2ξ€Έξ‚€+12π‘Žπœ†βˆ’3πœ†2π‘Ÿ+3βˆ’2=𝐿2π‘Žπœ†βˆ’3πœ†2π‘Ÿ+3βˆ’2,(2.27) where 𝐿 is the function in the pseudosymmetry definition (2.2).

Using (2.15), (2.19), the system (2.27) takes a more convenient form:𝐴𝑍𝐡+𝐼𝐴=𝐿𝐴,𝑍𝐴+𝐷𝐡=𝐿𝐡,𝑍𝐢+𝐴𝐡=𝐿𝑍,2βˆ’π‘2𝐡+𝐷(πΌβˆ’πΆ)=𝐿(πΌβˆ’πΆ),2βˆ’π‘2+𝐼(π·βˆ’πΆ)=𝐿(π·βˆ’πΆ).(2.28)

Remark 2.11. If 𝐿=0, the manifold is semisymmetric and the above system (2.28) is in accordance with equations (3.1)–(3.5) in [27].

Proposition 2.12 (see [16]). Let 𝑀3 be a 3-dimensional contact metric manifold satisfying π‘„πœ™=πœ™π‘„. Then, 𝑀3 is a pseudosymmetric space of constant type.

3. Pseudosymmetric Contact Metric 3-Manifolds with Harmonic Curvature and π‘‡π‘Ÿπ‘™ Constant in the Direction of πœ‰

Theorem 3.1. Let 𝑀3 be a 3-dimensional pseudosymmetric contact metric manifold with harmonic curvature and π‘‡π‘Ÿπ‘™ constant in the direction of πœ‰. Then, there are at most eight open subsets of 𝑀3 for which their union is an open and dense subset of 𝑀3 and each of them as an open submanifold of 𝑀3 is either: (a) Sasakian or (b) flat or (c) locally isometric to the Lie groups π‘†π‘ˆ(2), 𝑆𝐿(2,𝑅) equipped with a left invariant metric or (d) pseudosymmetric of constant type and with scalar curvature π‘Ÿ=2(1βˆ’πœ†2+2π‘Ž) or (e) semi-𝐾 contact with 𝐿=βˆ’3a2βˆ’4π‘Ž(π‘Žβ‰ 0) or (f) semi-𝐾 contact with 𝐿=π‘Ž2(π‘Žβ‰ 0) or (g) semi-𝐾 contact of type constant along πœ‰ and πœ™π‘’ or (h) semi-𝐾 contact of type constant along πœ‰ and 𝑒.

Proof. We consider the next open subsets of 𝑀: π‘ˆ0={π‘βˆˆπ‘€βˆΆπœ†=0inaneighborhoodof𝑝},π‘ˆ={π‘βˆˆπ‘€βˆΆπœ†β‰ 0inaneighborhoodof𝑝},(3.1) where π‘ˆ0βˆͺπ‘ˆ is open and dense subset of 𝑀.
In case 𝑀=π‘ˆ0, 𝑀 is a pseudosymmetric space of constant type [13] and we get the (a) case of present Theorem 3.1. Next, assume that π‘ˆ is not empty and let {𝑒,πœ™e,πœ‰} be a πœ™-basis. First we note that in the neighborhood π‘ˆ where πœ†β‰ 0 we have from (2.20) πœ‰β‹…π‘‡π‘Ÿπ‘™=0βŸΊπœ‰β‹…πœ†=0.(3.2)
Equations (2.23) because of (3.2) and the fact that π‘Ÿ is constant become, respectively, ξ€·πœ†(3βˆ’3πœ†+2π‘Ž)(πœ™π‘’β‹…πœ†)+2πœ†(πœ™π‘’β‹…π‘Ž)βˆ’2π‘πœ†(2π‘Ž+3βˆ’πœ†)=0,(3.3)2πœ†(π‘’β‹…π‘Ž)+(3πœ†+2π‘Ž+3)(π‘’β‹…πœ†)βˆ’2π‘πœ†(3+2π‘Ž+πœ†)=0,(3.4)π‘Ÿ+2πœ†(𝑒⋅𝑐)βˆ’2πœ†(πœ™π‘’β‹…π‘)+3𝑐(π‘’β‹…πœ†)βˆ’3𝑏(πœ™π‘’β‹…πœ†)βˆ’(π‘’β‹…π‘’β‹…πœ†)+62ξ€Έξ€·π‘βˆ’1+(πœ™π‘’β‹…πœ™π‘’β‹…πœ†)+2πœ†2βˆ’π‘2ξ€Έπ‘Ÿ+2π‘Žπœ†=0,(3.5)3𝑏(π‘’β‹…πœ†)+2πœ†(𝑒⋅𝑏)βˆ’(π‘’β‹…πœ™π‘’β‹…πœ†)+2πœ†(πœ‰β‹…π‘Ž)βˆ’2π‘π‘πœ†=0,or3𝑏(π‘’β‹…πœ†)+(𝑒⋅𝐴)βˆ’6π‘π‘πœ†+2πœ†(πœ‰β‹…π‘Ž)=0,(3.6)2ξ€·(πœ†+1)βˆ’π‘(πœ™π‘’β‹…πœ†)+2𝑐(π‘’β‹…πœ†)+2πœ†(𝑒⋅𝑐)+2πœ†2π‘Ž2+𝑏2ξ€Έξ€·πœ†+π‘Žπœ†+π‘Žβˆ’(π‘’β‹…π‘’β‹…πœ†)+32ξ€Έπ‘Ÿβˆ’1(πœ†+1)=0,(3.7)βˆ’(2+6πœ†+π‘Ž)(π‘’β‹…πœ†)βˆ’2πœ†(πœ‰β‹…π‘)+2πœ†π‘(π‘Ž+2+2πœ†)+(πœ‰β‹…πœ™π‘’β‹…πœ†)=0,orβˆ’(2+6πœ†+π‘Ž)(π‘’β‹…πœ†)+2πœ†π‘(π‘Ž+2+2πœ†)βˆ’(πœ‰β‹…π΄)=0,(3.8)2ξ€·πœ†(πœ†βˆ’1)+2𝑏(πœ™π‘’β‹…πœ†)+2πœ†(πœ™π‘’β‹…π‘)βˆ’π‘(π‘’β‹…πœ†)βˆ’(πœ™π‘’β‹…πœ™π‘’β‹…πœ†)+32ξ€Έξ€·βˆ’1(πœ†βˆ’1)+2πœ†2π‘Ž2+𝑐2ξ€Έβˆ’π‘Žπœ†+π‘Ž=0,(3.9)3𝑐(πœ™π‘’β‹…πœ†)+2πœ†(πœ™π‘’β‹…π‘)βˆ’(πœ™π‘’β‹…π‘’β‹…πœ†)βˆ’2πœ†(πœ‰β‹…π‘Ž)βˆ’2π‘π‘πœ†=0,or3𝑐(πœ™eβ‹…πœ†)+(πœ™π‘’β‹…π΅)βˆ’2πœ†(πœ‰β‹…π‘Ž)βˆ’6π‘π‘πœ†=0,(3.10)(π‘Žβˆ’6πœ†+2)(πœ™π‘’β‹…πœ†)βˆ’2πœ†(πœ‰β‹…π‘)+2π‘πœ†(2πœ†βˆ’2βˆ’π‘Ž)+(πœ‰β‹…π‘’β‹…πœ†)=0,or(π‘Žβˆ’6πœ†+2)(πœ™π‘’β‹…πœ†)βˆ’(πœ‰β‹…π΅)+2π‘πœ†(2πœ†βˆ’2βˆ’π‘Ž)=0,(3.11) where for the second form of (3.6), (3.8), we also used 𝐴=2π‘πœ†βˆ’(πœ™π‘’β‹…πœ†) and 𝐡=2π‘πœ†βˆ’(π‘’β‹…πœ†) in (3.10), (3.11).
In the neighborhood π‘ˆ, the system (2.28) for the pseudosymmetric contact metric 3-manifolds of Lemma 2.10 because of (3.2) becomes 𝐴(πΌβˆ’πΏ)𝐴=0,(π·βˆ’πΏ)𝐡=0,𝐴𝐡=0,2𝐡+(π·βˆ’πΏ)(πΌβˆ’πΆ)=0,2+(πΌβˆ’πΏ)(π·βˆ’πΆ)=0,(3.12) where 𝐴,𝐡,𝐢,𝐷,𝐼 are given by (2.15), (2.19).
Studying the third equation, we regard the following open subsets of π‘ˆ: π‘Šπ‘Š={π‘βˆˆπ‘ˆβˆΆπ΄=2π‘πœ†βˆ’(πœ™π‘’β‹…πœ†)=0inaneighborhoodof𝑝},3={π‘βˆˆπ‘ˆβˆΆπ΄=2π‘πœ†βˆ’(πœ™π‘’β‹…πœ†)β‰ 0inaneighborhoodof𝑝},(3.13) where π‘Šβˆͺπ‘Š3 is open and dense in the closure of π‘ˆ.
In π‘Š we have 𝐡(π·βˆ’πΏ)𝐡=0,(π·βˆ’πΏ)(πΌβˆ’πΆ)=0,2+(πΌβˆ’πΏ)(π·βˆ’πΆ)=0,(3.14) hence, we regard the subsets of π‘Š: π‘Š1π‘Š={π‘βˆˆπ‘ŠβˆΆπ΅=2π‘πœ†βˆ’(π‘’β‹…πœ†)=0inaneighborhoodof𝑝},2={π‘βˆˆπ‘ŠβˆΆπ΅=2π‘πœ†βˆ’(π‘’β‹…πœ†)β‰ 0inaneighborhoodof𝑝},(3.15) where π‘Š1βˆͺπ‘Š2 is open and dense in the closure of π‘Š and π‘Š1βˆͺπ‘Š2βˆͺπ‘Š3 is open and dense in the closure of π‘ˆ. We study the initial system at each π‘Šπ‘– for 𝑖=1,2,3.
In π‘Š1 the initial system (2.28) becomes (π·βˆ’πΏ)(πΌβˆ’πΆ)=0,(πΌβˆ’πΏ)(π·βˆ’πΆ)=0(3.16) or more explicitly ξ€Ί(πœ™π‘’β‹…πœ†)=2π‘πœ†,(π‘’β‹…πœ†)=2π‘πœ†,πœ‰β‹…πœ†=0,2π‘Žπœ†βˆ’2πœ†2+2+𝑏2+𝑐2ξ€»Γ—ξ€·βˆ’2π‘Žβˆ’(𝑒⋅𝑐)βˆ’(πœ™π‘’β‹…π‘)βˆ’2π‘Žπœ†βˆ’πœ†2ξ€Έξ€Ί+1βˆ’πΏ=0,βˆ’2π‘Žπœ†βˆ’2πœ†2+2+𝑏2+𝑐2ξ€»Γ—ξ€·βˆ’2π‘Žβˆ’(𝑒⋅𝑐)βˆ’(πœ™π‘’β‹…π‘)2π‘Žπœ†βˆ’πœ†2ξ€Έ+1βˆ’πΏ=0.(3.17) We have studied this system in [17] (Theorem 4.1) and we get the cases (b), (c), (d), (e), and (f) of the present Theorem 3.1.
In π‘Š2 the initial system (2.28) becomes π΅π·βˆ’πΏ=0,2+(πΌβˆ’πΏ)(π·βˆ’πΆ)=0.(3.18) Apart from (3.2), we also have the following equations: (πœ™π‘’β‹…πœ†)=2π‘πœ†,(3.19)𝐡=2π‘πœ†βˆ’(π‘’β‹…πœ†)β‰ 0,(3.20)2π‘Žπœ†βˆ’πœ†2𝐡+1βˆ’πΏ=0,(3.21)2ξ‚€=4π‘Žπœ†2π‘Žπœ†βˆ’3πœ†2π‘Ÿ+3βˆ’2=4π‘Žπœ†πΏβˆ’2πœ†2π‘Ÿ+2βˆ’2(3.22) while we will also use (3.3), (3.4), (3.6), and (3.11).
Differentiating (3.21) with respect to πœ‰, πœ™π‘’ we get, respectively, 2πœ†(πœ‰β‹…π‘Ž)=πœ‰β‹…πΏ,(3.23)2πœ†(πœ™π‘’β‹…π‘Ž)+4π‘Žπ‘πœ†βˆ’4π‘πœ†2=πœ™π‘’β‹…πΏ(3.24) (the derivative 𝑒⋅𝐿=2πœ†(π‘’β‹…π‘Ž)+2(π‘Žβˆ’πœ†)(π‘’β‹…πœ†) can not be estimated any further). Equations (3.3), (3.11) because of (3.19) yield, respectively, πœ™π‘’β‹…π‘Ž=πœ™π‘’β‹…πœ†=2π‘πœ†,(3.25)πœ‰β‹…π΅=βˆ’8π‘πœ†2.(3.26) Differentiating (3.22) with respect to πœ‰ and using (3.2), (3.26) and the fact that π‘Ÿ is constant, we get ξ‚€βˆ’4π‘πœ†π΅=4π‘Žπœ†βˆ’3πœ†2π‘Ÿ+3βˆ’2(πœ‰β‹…π‘Ž).(3.27) The first form of (3.6) and (3.19) give 𝑏𝐡=2πœ†(πœ‰β‹…π‘Ž),(3.28) hence (3.27), (3.28) yield ξ‚€4π‘Žπœ†+5πœ†2π‘Ÿ+3βˆ’2(πœ‰β‹…π‘Ž)=0.(3.29) We suppose that there is a point π‘βˆˆπ‘Š2 where πœ‰β‹…π‘Žβ‰ 0. Because of the continuity of the function πœ‰β‹…π‘Ž, there is a neighborhood of this point π‘†βŠ‚π‘Š2βŠ‚π‘ˆ: 𝑆={π‘žβˆˆπ‘Š2βˆΆπœ‰β‹…π‘Žβ‰ 0}. In 𝑆, we have 4π‘Žπœ†+5πœ†2+3βˆ’π‘Ÿ/2=0. Differentiating this equation with respect to πœ‰ and using (3.2), the constancy of π‘Ÿ and the fact that we work in π‘ˆ where πœ†β‰ 0 we conclude that πœ‰β‹…π‘Ž=0 in 𝑆, which is a contradiction. Hence πœ‰β‹…π‘Ž=0,(3.30) everywhere in π‘Š2. Because of (3.30) the equations (3.20), (3.27) give 𝑏=0.(3.31) Differentiating (3.2) with respect to πœ™π‘’, (3.19) with respect to πœ‰, subtracting and using (2.12), we get (π‘Žβˆ’πœ†+1)(π‘’β‹…πœ†)=0.(3.32) Let’s suppose that there is a point 𝑝 in π‘Š2 where π‘Žβˆ’πœ†+1β‰ 0. The function π‘Žβˆ’πœ†+1 is continuous, hence there is an open neighborhood 𝑉 of 𝑝, π‘‰βŠ‚π‘Š2, where π‘Žβˆ’πœ†+1β‰ 0 everywhere in 𝑉, hence π‘’β‹…πœ†=0.(3.33) From (3.20) and (3.33) we have in π‘‰βŠ‚π‘Š2βŠ‚π‘ˆ: 𝑐≠0.(3.34) Equation (3.4) because of (3.33) gives π‘’β‹…π‘Ž=𝑐(2π‘Ž+πœ†+3). From the second of (2.16) and because of (3.31), (3.33), (3.34), we get in 𝑉: π‘’β‹…π‘Ž=𝑐(π‘Žβˆ’πœ†+1)β‰ 0. By equalizing these two results and because of (3.34), we get: π‘Ž+2πœ†+2=0. We differentiate this equation with respect to 𝑒 and because of (3.33), we get π‘’β‹…π‘Ž=0, which is a contradiction in 𝑉. Hence π‘Žβˆ’πœ†+1=0 everywhere in π‘Š2 and because of (3.31), we can conclude according to Definition 2.7 that the structure is semi-𝐾 contact and pseudosymmetric with 𝐿 constant along the directions of πœ‰ and πœ™π‘’ because of (3.23), (3.24) and (3.25), (3.30), (3.31).
In π‘Š3 the initial system (2.28) becomes π΄πΌβˆ’πΏ=0,2+(π·βˆ’πΏ)(πΌβˆ’πΆ)=0.(3.35) We have the following equations and (3.2): (π‘’β‹…πœ†)=2π‘πœ†,(3.36)𝐴=2π‘πœ†βˆ’(πœ™π‘’β‹…πœ†)β‰ 0,(3.37)βˆ’2π‘Žπœ†βˆ’πœ†2𝐴+1βˆ’πΏ=0,(3.38)2ξ‚€=βˆ’4π‘Žπœ†βˆ’2π‘Žπœ†βˆ’3πœ†2π‘Ÿ+3βˆ’2=βˆ’4π‘Žπœ†πΏβˆ’2πœ†2π‘Ÿ+2βˆ’2(3.39) while we will also use (3.3), (3.4), (3.6), (3.8).
Differentiating (3.38) with respect to πœ‰, 𝑒 we get, respectively, βˆ’2πœ†(πœ‰β‹…π‘Ž)=πœ‰β‹…πΏ,βˆ’2πœ†(π‘’β‹…π‘Ž)βˆ’4π‘Žπ‘πœ†βˆ’4π‘πœ†2,=𝑒⋅𝐿(3.40) (we neglect the derivative πœ™π‘’β‹…πΏ because we can not estimate it). Equations (3.4), (3.8) because of (3.36) yield, respectively, π‘’β‹…π‘Ž=βˆ’π‘’β‹…πœ†=βˆ’2π‘πœ†,(3.41)πœ‰β‹…π΄=βˆ’8π‘πœ†2.(3.42) Differentiating (3.39) with respect to πœ‰ and using (3.2), (3.42) and the fact that π‘Ÿ is constant, we get ξ‚€βˆ’4π‘πœ†π΄=4π‘Žπœ†+3πœ†2π‘Ÿβˆ’3+2(πœ‰β‹…π‘Ž).(3.43) The first form of (3.10) and (3.36) give βˆ’π‘π΄=2πœ†(πœ‰β‹…π‘Ž),(3.44) hence (3.43), (3.44) yield ξ‚€4π‘Žπœ†βˆ’5πœ†2π‘Ÿβˆ’3+2(πœ‰β‹…π‘Ž)=0.(3.45) We suppose that there is a point π‘βˆˆπ‘Š3 where πœ‰β‹…π‘Žβ‰ 0. Because of the continuity of this function, there is a neighborhood of π‘β€‰β€‰π‘†βŠ‚π‘Š3βŠ‚π‘ˆ: 𝑆={π‘žβˆˆπ‘Š3βˆΆπœ‰β‹…π‘Žβ‰ 0}. In 𝑆, we have 4π‘Žπœ†βˆ’5πœ†2βˆ’3+π‘Ÿ/2=0. Differentiating this equation with respect to πœ‰ and using (3.2), the constancy of π‘Ÿ and the fact that we work in π‘ˆ where πœ†β‰ 0 we conclude that πœ‰β‹…π‘Ž=0 in 𝑆, which is a contradiction. Hence πœ‰β‹…π‘Ž=0,(3.46) everywhere in π‘Š3. Because of (3.46) the equations (3.37), (3.43) give 𝑐=0.(3.47) Differentiating (3.2) with respect to 𝑒, (3.36) with respect to πœ‰, subtracting and using (2.12), we get βˆ’(π‘Ž+πœ†+1)(πœ™π‘’β‹…πœ†)=0.(3.48) Let’s suppose that there is a point 𝑝 in π‘Š3 where π‘Ž+πœ†+1β‰ 0. This function is smooth, then because of its continuity, there is an open neighborhood 𝑉 of 𝑝, π‘‰βŠ‚π‘Š3, where π‘Ž+πœ†+1β‰ 0 everywhere in 𝑉, hence πœ™π‘’β‹…πœ†=0.(3.49) From (3.37) and (3.49) we have in π‘‰βŠ‚π‘Š3βŠ‚π‘ˆ: 𝑏≠0.(3.50) From (3.3) and (3.49), we get πœ™π‘’β‹…π‘Ž=𝑏(2π‘Žβˆ’πœ†+3). From the first of (2.16) and because of (3.47), (3.49), (3.50), we get in 𝑉: πœ™π‘’β‹…π‘Ž=𝑏(π‘Ž+πœ†+1)β‰ 0. By equalizing these two results and because of (3.50), we get: π‘Žβˆ’2πœ†+2=0. We differentiate this equation with respect to πœ™π‘’ and because of (3.49), we get πœ™π‘’β‹…π‘Ž=0, which is a contradiction in 𝑉. Hence π‘Ž+πœ†+1=0 everywhere in π‘Š3 and because of (3.47), we can conclude according to Definition 2.7 that the structure is semi-𝐾 contact and pseudosymmetric with 𝐿 constant along the directions of πœ‰ and 𝑒 because of (3.40) and (3.41), (3.46), (3.47).
Finally, we remark that the cases (g) and (h) of the present Theorem 3.1 that result from the structures studied in the sets π‘Š2 and π‘Š3, respectively.

Remark 3.2. (i) The conditions of harmonic curvature help us to the systems in the neighborhoods π‘Š2 and π‘Š3 where we had equations of the type 𝐴2=βˆ’4π‘Žπœ†(βˆ’2π‘Žπœ†βˆ’3πœ†2+3βˆ’π‘Ÿ/2) and which we could not handle in our previous articles [16, 17].
(ii) In case (d) where 𝐿 is constant, we can also use the classification of [11] to improve our results as the manifolds with harmonic curvature are a special case of conformally flat manifolds in dimension 3.

4. Pseudosymmetric (πœ…,πœ‡,𝜈)-Contact Metric 3-Manifolds of Type Constant in the Direction of πœ‰

Theorem 4.1. Let 𝑀3 be a pseudosymmetric (πœ…,πœ‡,𝜈)-contact metric 3-manifold of type constant along the direction πœ‰. Then, there are at most five open subsets of 𝑀3 for which their union is an open and dense subset of 𝑀3 and each of them as an open submanifold of 𝑀3 is either (a) Sasakian or (b) flat or (c) pseudosymmetric of constant type 𝐿=πœ…=1/2(π‘‡π‘Ÿπ‘™), πœ‡=𝜈=0 and of constant scalar curvature π‘Ÿ=2πœ… or (d) pseudosymmetric generalized (πœ…,πœ‡)-contact metric manifold of type 𝐿=πœ…βˆ’πœ‡πœ†, of scalar curvature π‘Ÿ=2(3πœ…βˆ’πœ‡πœ†) and πœ‰β‹…πœ‡=πœ‰β‹…πœ…=0 or (e) pseudosymmetric generalized (πœ…,πœ‡)-contact metric manifold of type 𝐿=πœ…+πœ‡πœ†, of scalar curvature π‘Ÿ=2(3πœ…+πœ‡πœ†) and πœ‰β‹…πœ‡=πœ‰β‹…πœ…=0.

Proof. We study pseudosymmetric (πœ…,πœ‡,𝜈)-contact metric 3-manifolds with πœ‰β‹…πΏ=0,(4.1) where 𝐿 is the function in (2.2). We consider the next open subsets of 𝑀, π‘ˆ0={π‘βˆˆπ‘€βˆΆπœ†=0inaneighborhoodof𝑝},π‘ˆ={π‘βˆˆπ‘€βˆΆπœ†β‰ 0inaneighborhoodof𝑝},(4.2) where π‘ˆ0βˆͺπ‘ˆ is open and dense subset of 𝑀.
In case 𝑀=π‘ˆ0, (𝑀,πœ‰,πœ‚,πœ™,𝑔) is a Sasakian structure which is a pseudosymmetric space of constant type [13] with πœ…=1, πœ‡,πœˆβˆˆβ„ and β„Ž=0 and we get the (a) case of present Theorem 4.1. Next, assume that π‘ˆ is not empty and let {𝑒,πœ™π‘’,πœ‰} be a πœ™-basis. From (2.25), we can calculate the following components of the Riemannian curvature tensor: 𝑅𝑅(πœ‰,𝑒)πœ‰=βˆ’(πœ…+πœ†πœ‡)π‘’βˆ’πœ†πœˆπœ™π‘’,𝑅(𝑒,πœ™π‘’)πœ‰=0,(πœ‰,πœ™π‘’)πœ‰=βˆ’πœ†πœˆπ‘’βˆ’(πœ…βˆ’πœ†πœ‡)πœ™π‘’.(4.3) By virtue of (2.14) we can conclude that 𝐴=2π‘πœ†βˆ’(πœ™π‘’β‹…πœ†)=0,𝐡=2π‘πœ†βˆ’(π‘’β‹…πœ†)=0,𝑍=πœ‰β‹…πœ†=πœ†πœˆ,𝐷=2π‘Žπœ†βˆ’πœ†2+1=πœ…βˆ’πœ†πœ‡,𝐼=βˆ’2π‘Žπœ†βˆ’πœ†2+1=πœ…+πœ†πœ‡,(4.4) and hence the system (2.28) becomes 𝑍(πΆβˆ’πΏ)=0,βˆ’π‘2+(π·βˆ’πΏ)(πΌβˆ’πΆ)=0,βˆ’π‘2+(πΌβˆ’πΏ)(π·βˆ’πΆ)=0,(βˆ—) where 𝐴,𝐡,𝐢,𝐷,𝐼,𝑍 are given by (2.19) and (4.4). Substituting from (4.4) (πœ™π‘’β‹…πœ†), (π‘’β‹…πœ†) in (2.16) we also have πœ‰β‹…π‘=βˆ’(πœ™π‘’β‹…π‘Ž)+𝑏(π‘Žβˆ’πœ†+1),(4.5)πœ‰β‹…π‘=(π‘’β‹…π‘Ž)βˆ’π‘(πœ†+π‘Ž+1).(4.6) First we will prove that 𝑍=πœ‰β‹…πœ†=0 (equivalently 𝜈=0 as we work in π‘ˆ where πœ†β‰ 0). We suppose that there is a point π‘βˆˆπ‘ˆ where πœ‰β‹…πœ†β‰ 0. By the continuity of this function, we can consider that there is a neighborhood 𝑉 of 𝑝, where πœ‰β‹…πœ†β‰ 0 everywhere in π‘‰βŠ‚π‘ˆ. We work in 𝑉. Then the first equation of (*) becomes πΆβˆ’πΏ=0 or equivalently: (𝑒⋅𝑐)+(πœ™π‘’β‹…π‘)=𝐿+𝑏2+𝑐2βˆ’πœ†2+1βˆ’2π‘Ž.(4.7) We differentiate this equation with respect to πœ‰ and by virtue of (4.1) we get πœ‰β‹…π‘’β‹…π‘+πœ‰β‹…πœ™π‘’β‹…π‘=2𝑏(πœ‰β‹…π‘)+2𝑐(πœ‰β‹…π‘)βˆ’2πœ†(πœ‰β‹…πœ†)βˆ’2(πœ‰β‹…π‘Ž),(4.8) which because of (4.5), (4.6) becomes πœ‰β‹…π‘’β‹…π‘+πœ‰β‹…πœ™π‘’β‹…π‘=2𝑏(π‘’β‹…π‘Ž)βˆ’2𝑐(πœ™π‘’β‹…π‘Ž)βˆ’2πœ†(πœ‰β‹…πœ†)βˆ’2(πœ‰β‹…π‘Ž)βˆ’4π‘π‘πœ†.(4.9) Next, we differentiate (4.5) and (4.6) with respect to 𝑒 and πœ™π‘’, respectively, and adding we have []π‘’β‹…πœ‰β‹…π‘+πœ™π‘’β‹…πœ‰β‹…π‘=βˆ’π‘’,πœ™π‘’π‘Žβˆ’(π‘Ž+πœ†+1)(πœ™π‘’β‹…π‘)+(π‘Žβˆ’πœ†+1)(𝑒⋅𝑏)βˆ’π‘(πœ™π‘’β‹…π‘Ž)+𝑏(π‘’β‹…π‘Ž)βˆ’4π‘π‘πœ†.(4.10) We subtract this last equation from (4.9) and we get [][][]π‘Ž+πœ‰,𝑒𝑐+πœ‰,πœ™π‘’π‘=𝑏(π‘’β‹…π‘Ž)βˆ’π‘(πœ™π‘’β‹…π‘Ž)βˆ’2(πœ‰β‹…π‘Ž)βˆ’2πœ†(πœ‰β‹…πœ†)+𝑒,πœ™π‘’(π‘Ž+πœ†+1)(πœ™π‘’β‹…π‘)βˆ’(π‘Žβˆ’πœ†+1)(𝑒⋅𝑏)(4.11) or because of (2.12) (π‘Ž+πœ†+1)(πœ™π‘’β‹…π‘)+(πœ†βˆ’π‘Žβˆ’1)(𝑒⋅𝑏)=𝑏(π‘’β‹…π‘Ž)βˆ’π‘(πœ™π‘’β‹…π‘Ž)βˆ’2(πœ‰β‹…π‘Ž)βˆ’2πœ†(πœ‰β‹…πœ†)βˆ’π‘(π‘’β‹…π‘Ž)+𝑐(πœ™π‘’β‹…π‘Ž)+2(πœ‰β‹…π‘Ž)+(πœ†+π‘Ž+1)(πœ™π‘’β‹…π‘)+(πœ†βˆ’π‘Žβˆ’1)(𝑒⋅𝑏)(4.12) or equivalently: πœ†(πœ‰β‹…πœ†)=0 and because we work in π‘‰βŠ‚π‘ˆ, we have πœ‰β‹…πœ†=0, which is a contradiction. Hence, we can deduce everywhere in π‘ˆ: πœ‰β‹…πœ†=0⟺𝜈=0.(4.13) Next we will derive some useful relations. From (4.4) we have: πœ™π‘’β‹…πœ†=2π‘πœ†,π‘’β‹…πœ†=2π‘πœ†.(4.14) We differentiate these equations with respect to 𝑒 and πœ™π‘’, respectively, we subtract, we use the relations (2.12), (4.4) and we get []πœ‰β‹…πœ†=πœ†(𝑒⋅𝑏)βˆ’(πœ™π‘’β‹…π‘)(4.15) or because of (4.13) 𝑒⋅𝑏=πœ™π‘’β‹…π‘.(4.16) We differentiate the relations πœ™π‘’β‹…πœ†=2π‘πœ† and (4.13) with respect to πœ‰ and πœ™π‘’, respectively, and subtracting we obtain: [πœ‰,πœ™π‘’]πœ†=2πœ†(πœ‰β‹…π‘) or because of (2.12), (4.4), (4.6) π‘’β‹…π‘Ž=2π‘πœ†.(4.17) We differentiate the relations π‘’β‹…πœ†=2π‘πœ† and (4.13) with respect to πœ‰ and 𝑒, respectively, and subtracting we obtain: [πœ‰,𝑒]πœ†=2πœ†(πœ‰β‹…π‘) or because of (2.12), (4.4), (4.5) πœ™π‘’β‹…π‘Ž=βˆ’2π‘πœ†.(4.18) Finally, after substituting 𝐷,𝐼,𝑍 from (4.4), (4.13) the final form of the system (*) is πœ‡(πΆβˆ’πΏ)=0,(πœ…βˆ’πœ†πœ‡βˆ’πΏ)(πœ…+πœ†πœ‡βˆ’πΆ)=0.(4.19) In order to study this system we regard the following open subsets of π‘ˆ: 𝑉1𝑉={π‘βˆˆπ‘ˆβˆΆπΆβˆ’πΏβ‰ 0inaneighborhoodof𝑝},2={π‘βˆˆπ‘ˆβˆΆπΆβˆ’πΏ=0inaneighborhoodof𝑝},(4.20) where 𝑉1βˆͺ𝑉2 is open and dense in the closure of π‘ˆ.
In 𝑉1, we have πœ‡=0 and hence from (4.4): 𝐼=𝐷=πœ… or 2π‘Žπœ†βˆ’πœ†2+1=βˆ’2π‘Žπœ†βˆ’πœ†2+1 or finally π‘Ž=0 and πœ…=1βˆ’πœ†2. From (4.17), (4.18) we deduce that 𝑏=𝑐=0. Having also the second equation of (4.19), we regard the open subsets of 𝑉1π‘Œ1=ξ€½π‘βˆˆπ‘‰1ξ€Ύ,π‘ŒβˆΆπœ…βˆ’πΆ=0inaneighborhoodof𝑝2=ξ€½π‘βˆˆπ‘‰1ξ€Ύ,βˆΆπœ…βˆ’πΆβ‰ 0inaneighborhoodof𝑝(4.21) where π‘Œ1βˆͺπ‘Œ2 is open and dense in the closure of 𝑉1.
In π‘Œ1 substituting in πœ…βˆ’πΆ=0, 𝐢 from (2.15), π‘Ž=𝑏=𝑐=0, we get πœ…=1βˆ’πœ†2=0 and hence the structure is flat with πœ…=πœ‡=𝜈.
In π‘Œ2 from πœ‡=0 we have again 𝐼=𝐷=πœ…, π‘Ž=0 and from (4.17), (4.18) 𝑏=𝑐=0 while we must also have πœ…=𝐿. Hence, 𝐿=πœ…=(1/2)π‘‡π‘Ÿπ‘™ and from (2.18) of constant scalar curvature π‘Ÿ=2(1βˆ’πœ†2).
In 𝑉2 having 𝐢=𝐿, the second equation of (4.19) becomes (2π‘Žπœ†βˆ’πœ†2+1βˆ’πΏ)(βˆ’2π‘Žπœ†βˆ’πœ†2+1βˆ’πΏ)=0. Hence, we regard the open subsets of 𝑉2π‘Š1=ξ€½π‘βˆˆπ‘‰2βˆΆβˆ’2π‘Žπœ†βˆ’πœ†2ξ€Ύ,π‘Š+1βˆ’πΏβ‰ 0inaneighborhoodof𝑝2=ξ€½π‘βˆˆπ‘‰2βˆΆβˆ’2π‘Žπœ†βˆ’πœ†2ξ€Ύ,+1βˆ’πΏ=0inaneighborhoodof𝑝(4.22) where π‘Š1βˆͺπ‘Š2 is open and dense in the closure of 𝑉2.
In π‘Š1 we must have 2π‘Žπœ†βˆ’πœ†2+1βˆ’πΏ=0 while in π‘Š2 we have βˆ’2π‘Žπœ†βˆ’πœ†2+1βˆ’πΏ=0. We differentiate these equations with respect to πœ‰ and because of (4.13) we get πœ‰β‹…π‘Ž=0.(4.23)
By virtue of 𝐼 and 𝐷 in (4.4) we deduce πœ‡=βˆ’2π‘Ž and hence πœ‰β‹…πœ‡=0.(4.24) In π‘Š2 we differentiate βˆ’2π‘Žπœ†βˆ’πœ†2+1βˆ’πΏ=0 with respect to πœ‰ and similarly we also obtain (4.24). Each of π‘Š1 and π‘Š2 is a generalized (πœ…,πœ‡)-contact metric 3-manifold with πœ‰β‹…πœ‡=0 and scalar curvature π‘Ÿ=2(2π‘Žπœ†βˆ’3πœ†2+3)=2(3πœ…βˆ’πœ‡πœ†) or π‘Ÿ=2(βˆ’2π‘Žπœ†βˆ’3πœ†2+3)=2(3πœ…+πœ‡πœ†) respectively and from (4.1), (4.13) and (4.23) or (4.24) πœ‰β‹…πœ…=0 and πœ‰β‹…π‘Ÿ=0.
Concluding: the structure in π‘ˆ0 gives the Sasakian case, the structures in π‘Œ1 and π‘Œ2 give the (b) and (c) cases of the present Theorem 4.1 and the structures in π‘Š1 and π‘Š2 give (d) and (e) respectively.

Remark 4.2. The generalized (πœ…,πœ‡)-contact metric manifolds in dimension 3 with πœ…<1 (equivalently πœ†β‰ 0) and πœ‰β‹…πœ‡=0 have been studied by Koufogiorgos and Tsichlias [28]. They proved in their Theorem 4.1 of [28] that at any point of π‘ƒβˆˆπ‘€, precisely one of the following relations is valid: βˆšπœ‡=2(1+1βˆ’πœ…), or βˆšπœ‡=2(1βˆ’1βˆ’πœ…), while there exists a chart (π‘ˆ,(π‘₯,𝑦,𝑧)) with π‘ƒβˆˆπ‘ˆβŠ†π‘€ such that the functions πœ…, πœ‡ depend only on 𝑧 and the tensors fields πœ‚, πœ‰, πœ™, 𝑔 take a suitable form. Each of our submanifolds π‘Š1 and π‘Š2 is such a generalized (πœ…,πœ‡)-contact metric 3-manifold.

Acknowledgments

The author thanks Professors F. Gouli-Andreou, Ph. J. Xenos, R. Deszcz, J. Inoguchi, and C. Γ–zgΓΌr for useful information on pseudosymmetric manifolds.

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