Some Classes of Pseudosymmetric Contact Metric 3-Manifolds

We study (a) the class of 3-dimensional pseudosymmetric contact metric manifolds with harmonic curvature and 𝑇𝑟𝑙 constant along the direction of 𝜉 and (b) the class of (𝜅,𝜇,𝜈)-contact metric pseudosymmetric 3-manifolds of type constant in the direction of 𝜉.


Introduction
A Riemannian manifold M m , g is said to be pseudosymmetric according to Deszcz 1 if its curvature tensor R satisfies the condition R X, Y • R L{ X ∧ Y • R}, where the dot means that R X, Y acts as a derivation on R, L is a smooth function and the endomorphism field X ∧ Y is defined by for all vectors fields X, Y, 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 ∇R 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 ρ 1 , ρ 2 , ρ 3 of the Ricci tensor satisfy ρ 1 ρ 2 up to numeration and the last one is constant 6, 7 .
Given a contact manifold M, η , there is an underlying contact metric structure η, ξ, φ, g where g is a Riemannian metric the associated metric , φ a global tensor of type 1,1 , and ξ a unique global vector field the characteristic or Reeb vector field .A differentiable 2n 1 -dimensional manifold endowed with a contact metric structure η, ξ, φ, g is called a contact metric Riemannian manifold denoted by M η, ξ, φ, g .The structure tensors η, ξ, φ, and g satisfy the equations: dη X, Y g X, φY , g φX, φY g X, Y − η X η Y .

2.3
The associated metrics can be constructed polarizing dη on the contact subbundle D defined by η 0. Denoting by L the Lie differentiation and R the curvature tensor, respectively, we define the tensor fields h, l, and τ by These tensors also satisfy the following formulas: φξ hξ lξ 0, η• φ η • h 0, dη ξ, X 0, 2.5 Trh Trhφ 0, ∇ X ξ −φX − φhX, hφ −φh, 2.6 hX λX ⇒ hφX −λφX, 2.7 ∇ ξ φ 0, Trl g Qξ, ξ 2n − Trh 2 .2.9 h 0 or equivalently τ 0 if and only if ξ is Killing and M is called K-contact.A contact structure on M implies an almost complex structure on the product manifold M 2n 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, φ, ξ, η, g be a 3-dimensional contact metric manifold and U the open subset of points p ∈ M where h / 0 in a neighborhood of p and U 0 the open subset of points p ∈ M such that h 0 in a neighborhood of p.Because h is a smooth function on M then U ∪ U 0 is an open and dense subset of M so if a property is satisfied in U 0 ∪ U then this property will be satisfied in M. For any point p ∈ U ∪ U 0 there exists a local orthonormal basis {e, φe, ξ} of smooth eigenvectors of h in a neighborhood of p a φ-basis .On U, we put he λe, where λ is a non vanishing smooth function which is supposed positive.From 2.7 , we have hφe −λφe.We recall the following.

2.15
Setting X e, Y φe, Z ξ in the Jacobi identity X, Y , Z Y, Z , X Z, X , Y 0 and using 2.12 , we get where is the scalar curvature.The relations 2.15 and 2.18 yield and the relation 2.9 : From now on we shall work on a M 3 , φ, ξ, η, g contact metric 3-manifold concerning a φbasis {e, φe, ξ} at any point p ∈ M. First from the equation 3 and the relations 2.17 , we get the equations:

2.22
Applying 2.21 to the vectors fields of the φ-basis of the contact metric manifold M 3 we have: We use the previous relations and we get the following nine 9 conditions for a contact metric 3manifold to have harmonic curvature:

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 φe•r 0, b by adding the equations two and six and using similarly 2. 16 we have e • r 0. From the relations φe • r 0, e • r 0, and 2.12 , we can conclude ξ • r 0 and hence we are led to the known result that the scalar curvature r is constant in a contact metric 3-manifold with harmonic curvature.Later for our study, we will use the r as a constant and we will give to these equations a more convenient form.
Definition 2.7 see 22 .Let M 3 be a 3-dimensional contact metric manifold and h λh −λh − the spectral decomposition of h on U.
for all vector fields X on M 3 and all points of an open subset W of U and h 0 on the points of M 3 which do not belong to W, 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 ξ, e 0 and for X φe to ∇ φe φe 0. Hence on a semi-K contact manifold we have a λ 1 c 0. If we apply the deformation e → φe, φe → e, ξ → −ξ, λ → −λ, 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 a − λ 1 b 0. Definition 2.8.A κ, μ, ν -contact metric manifold is defined in 23 as a contact metric manifold M 2n 1 , η, ξ, φ, g on which the curvature tensor satisfies for every X, Y ∈ X M the condition: where κ, μ, ν are smooth functions on M. If ν 0 we have a generalized κ, μ -contact metric manifold 24 and if additionally κ, μ are constants then the manifold is a contact metric κ, μ -space 25, 26 .Moreover, in 23 and 24 it is proved, respectively, that for a κ, μ, ν or a generalized κ, μ -contact metric manifold M 2n 1 of dimension greater than 3 the functions κ, μ 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 Let M, η, g, φ, ξ be a contact metric 3-manifold.In case M U 0 , that is, ξ, η, φ, g is a Sasakian structure, then M is a pseudosymmetric space of constant type 13 .Next, assume that U is not empty and let {e, φe, ξ} be a φ-basis as in Lemma 2.3.We have the following.Lemma 2.10 see 16 .Let M, η, g, φ, ξ be a contact metric three manifold.Then M is pseudosymmetric if and only if

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:

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 M 3 be a 3-dimensional contact metric manifold satisfying Qφ φQ.Then, M 3 is a pseudosymmetric space of constant type.

Pseudosymmetric Contact Metric 3-Manifolds with Harmonic
Curvature and Trl Constant in the Direction of ξ In case M U 0 , M is a pseudosymmetric space of constant type 13 and we get the a case of present Theorem 3.1.Next, assume that U is not empty and let {e, φe, ξ} be a φ-basis.First we note that in the neighborhood U where λ / 0 we have from 2.20 Equations 2.23 because of 3.2 and the fact that r is constant become, respectively, where for the second form of 3.6 , 3.8 , we also used A 2bλ − φe • λ and B 2cλ − e • λ in 3.10 , 3.11 .
In the neighborhood U, the system 2.28 for the pseudosymmetric contact metric 3manifolds of Lemma 2.10 because of 3.2 becomes

3.13
where W ∪ W 3 is open and dense in the closure of U.
In W we have 3.14 hence, we regard the subsets of W:

3.15
where W 1 ∪ W 2 is open and dense in the closure of W and W 1 ∪ W 2 ∪ W 3 is open and dense in the closure of U. We study the initial system at each W i for i 1, 2, 3.
In W 1 the initial system 2.28 becomes 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 W 2 the initial system 2.28 becomes

3.18
Apart from 3.2 , we also have the following equations: while we will also use 3. results and because of 3.34 , we get: a 2λ 2 0. We differentiate this equation with respect to e and because of 3.33 , we get e • a 0, which is a contradiction in V .Hence a − λ 1 0 everywhere in W 2 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 ξ and φe because of 3.23 , 3.24 and 3.25 , 3.30 , 3.31 .
In W 3 the initial system 2.28 becomes

3.35
We have the following equations and 3.2 : .39 while we will also use 3.

3.42
Differentiating 3.39 with respect to ξ and using 3.2 , 3.42 and the fact that r is constant, we get

3.43
The first form of 3.10 and 3.36 give We suppose that there is a point p ∈ W 3 where ξ • a / 0. Because of the continuity of this function, there is a neighborhood of p S ⊂ W 3 ⊂ U: S {q ∈ W 3 : ξ • a / 0}.In S, we have 4aλ − 5λ 2 − 3 r/2 0. Differentiating this equation with respect to ξ and using 3.2 , the constancy of r and the fact that we work in U where λ / 0 we conclude that ξ • a 0 in S, which is a contradiction.Hence ξ • a 0, 3.46 everywhere in W 3 .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 ξ, subtracting and using 2.12 , we get Let's suppose that there is a point p in W 3 where a λ 1 / 0. This function is smooth, then because of its continuity, there is an open neighborhood V of p, V ⊂ W 3 , where a λ 1 / 0 everywhere in V , hence φe • λ 0.

3.49
From 3.37 and 3.49 we have in From 3.3 and 3.49 , we get φe • a b 2a − λ 3 .From the first of 2.16 and because of 3.47 , 3.49 , 3.50 , we get in V : φe • a b a λ 1 / 0. By equalizing these two results and because of 3.50 , we get: a − 2λ 2 0. We differentiate this equation with respect to φe and because of 3.49 , we get φe • a 0, which is a contradiction in V .Hence a λ 1 0 everywhere in W 3 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 ξ 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 W 2 and W 3 , respectively.Remark 3.2.i The conditions of harmonic curvature help us to the systems in the neighborhoods W 2 and W 3 where we had equations of the type A 2 −4aλ −2aλ−3λ 2 3−r/2 and which we could not handle in our previous articles 16, 17 .

ISRN Geometry
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.In case M U 0 , M, ξ, η, φ, g is a Sasakian structure which is a pseudosymmetric space of constant type 13 with κ 1, μ, ν ∈ R and h 0 and we get the a case of present Theorem 4.1.Next, assume that U is not empty and let {e, φe, ξ} be a φ-basis.From 2.25 , we can calculate the following components of the Riemannian curvature tensor: R ξ, e ξ − κ λμ e − λνφe, R e, φe ξ 0, R ξ, φe ξ −λνe − κ − λμ φe.

4.3
By virtue of 2.14 we can conclude that First we will prove that Z ξ • λ 0 equivalently ν 0 as we work in U where λ / 0 .We suppose that there is a point p ∈ U where ξ • λ / 0. By the continuity of this function, we can consider that there is a neighborhood V of p, where ξ • λ / 0 everywhere in V ⊂ U. We work in V .Then the first equation of * becomes C − L 0 or equivalently: We differentiate this equation with respect to ξ and by virtue of 4.1 we get which because of 4.5 , 4.6 becomes Next, we differentiate 4.5 and 4.6 with respect to e and φe, respectively, and adding we have

4.10
We subtract this last equation from 4.
or equivalently: λ ξ • λ 0 and because we work in V ⊂ U, we have ξ • λ 0, which is a contradiction.Hence, we can deduce everywhere in U: Next we will derive some useful relations.From 4.4 we have:

4.14
We differentiate these equations with respect to e and φe, respectively, we subtract, we use the relations 2.12 , 4.

4.19
In order to study this system we regard the following open subsets of U: In Y 1 substituting in κ − C 0, C from 2.15 , a b c 0, we get κ 1 − λ 2 0 and hence the structure is flat with κ μ ν.
In Y 2 from μ 0 we have again I D κ, a 0 and from 4.17

ISRN Geometry
Remark 4.2.The generalized κ, μ -contact metric manifolds in dimension 3 with κ < 1 equivalently λ / 0 and ξ • μ 0 have been studied by Koufogiorgos and Tsichlias 28 .They proved in their Theorem 4.1 of 28 that at any point of P ∈ M, precisely one of the following relations is valid: μ 2 1 √ 1 − κ , or μ 2 1− √ 1 − κ , while there exists a chart U, x, y, z with P ∈ U ⊆ M such that the functions κ, μ depend only on z and the tensors fields η, ξ, φ, g take a suitable form.Each of our submanifolds W 1 and W 2 is such a generalized κ, μ -contact metric 3-manifold.

Theorem 3 . 1 .
Let M 3 be a 3-dimensional pseudosymmetric contact metric manifold with harmonic curvature and Trl constant in the direction of ξ.Then, there are at most eight open subsets of M 3 for which their union is an open and dense subset of M 3 and each of them as an open submanifold of M 3 is either: (a) Sasakian or (b) flat or (c) locally isometric to the Lie groups SU 2 , SL 2, R equipped with a left invariant metric or (d) pseudosymmetric of constant type and with scalar curvature r 2 1 − λ 2 2a or (e) semi-K contact with L −3a 2 − 4a a / 0 or (f) semi-K contact with L a 2 a / 0 or (g) semi-K contact of type constant along ξ and φe or (h) semi-K contact of type constant along ξ and e. ISRN Geometry Proof.We consider the next open subsets of M: U 0 p ∈ M : λ 0 in a neighborhood of p , U p ∈ M : λ / 0 in a neighborhood of p , 3.1 where U 0 ∪ U is open and dense subset of M.

4 .Theorem 4 . 1 . 1 where L is the function in 2 . 2 .
Pseudosymmetric κ, μ, ν -Contact Metric 3-Manifolds of Type Constant in the Direction of ξ Let M 3 be a pseudosymmetric κ, μ, ν -contact metric 3-manifold of type constant along the direction ξ.Then, there are at most five open subsets of M 3 for which their union is an open and dense subset of M 3 and each of them as an open submanifold of M 3 is either (a) Sasakian or (b) flat or (c) pseudosymmetric of constant type L κ 1/2 Trl , μ ν 0 and of constant scalar curvature r 2κ or (d) pseudosymmetric generalized κ, μ -contact metric manifold of type L κ − μλ, of scalar curvature r 2 3κ − μλ and ξ • μ ξ • κ 0 or (e) pseudosymmetric generalized κ, μ -contact metric manifold of type L κ μλ, of scalar curvature r 2 3κ μλ and ξ • μ ξ • κ 0. Proof.We study pseudosymmetric κ, μ, ν -contact metric 3-manifolds with ξ • L 0, 4.We consider the next open subsets of M, U 0 p ∈ M : λ 0 in a neighborhood of p , U p ∈ M : λ / 0 in a neighborhood of p , 4.2 where U 0 ∪ U is open and dense subset of M.
open and dense in the closure of U. In V 1 , we have μ 0 and hence from 4.4 : I D κ or 2aλ − λ 2 1 −2aλ − λ 2 1 or finally a 0 and κ 1 − λ 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 ofV 1 Y 1 p ∈ V 1 : κ − C 0 in a neighborhood of p , Y 2 p ∈ V 1 : κ − C / 0 in a neighborhood of p , 4.21where Y 1 ∪ Y 2 is open and dense in the closure of V 1 .