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ISRN Geometry

VolumeΒ 2012Β (2012), Article IDΒ 367467, 21 pages

http://dx.doi.org/10.5402/2012/367467

## 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 constant along the direction of and (b) the class of ()-contact metric pseudosymmetric 3-manifolds of type constant in the direction of .

#### 1. Introduction

A Riemannian manifold is said to be pseudosymmetric according to Deszcz [1] if its curvature tensor satisfies the condition , where the dot means that acts as a derivation on , is a smooth function and the endomorphism field is defined by for all vectors fields on and it similarly acts as a derivation on .

If is constant, is called a pseudosymmetric manifold of constant type and if particularly then is called a semisymmetric manifold first studied by E. Cartan. Semisymmetric spaces [2, 3] are a generalization of locally symmetric spaces (, [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 of the Ricci tensor satisfy (up to numeration) and the last one is constant [6, 7].

Kowalski and Sekizawa have studied [7β10] 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 -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) is a pseudosymmetric contact metric manifold with harmonic curvature and constant along the direction of . (ii) is a -contact metric pseudosymmetric manifold of type constant in the direction of .

#### 2. Preliminaries

Let be a connected Riemannian smooth manifold. We denote by its Riemannian curvature tensor given by the equation for any and where is the Levi-Civita connection of .

*Definition 2.1. *A Riemannian manifold *, *is called *pseudosymmetric* in the sense of Deszcz [1] if at every point of the curvature tensor satisfies the condition:

or more explixitly: for any , is given by (1.1) and is a smooth function.

*Definition 2.2. *A differentiable manifold endowed with a global 1-form such that everywhere on is called a *contact* manifold.

Given a contact manifold , there is an underlying contact metric structure where is a Riemannian metric (the *associated* metric), a global tensor of type (1,1), and a unique global vector field (the *characteristic* or *Reeb vector field*). A differentiable ()-dimensional manifold endowed with a contact metric structure is called a *contact metric* (Riemannian) manifold denoted by . The structure tensors , , , and satisfy the equations:
The associated metrics can be constructed polarizing on the contact subbundle defined by . Denoting by the Lie differentiation and the curvature tensor, respectively, we define the tensor fields , , and by
These tensors also satisfy the following formulas: (or equivalently ) if and only if is Killing and is called *-contact*. A contact structure on implies an almost complex structure on the product manifold . 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 be a 3-dimensional contact metric manifold and the open subset of points where in a neighborhood of and the open subset of points such that in a neighborhood of . Because is a smooth function on then is an open and dense subset of so if a property is satisfied in then this property will be satisfied in . For any point there exists a local orthonormal basis of smooth eigenvectors of in a neighborhood of (a -basis). On , we put , where is a non vanishing smooth function which is supposed positive. From (2.7), we have . We recall the following.

Lemma 2.3 (see [20]). *On , one has
**
where is a smooth function and
*

From Lemma 2.3 and the formula we can prove that and from (1.1) we estimate while , whenever and .

By direct computations we calculate the non vanishing independent components of the Riemannian curvature tensor field (1,3): where Setting , , in the Jacobi identity and using (2.12), we get or equivalently: and .

We give the components of the Ricci operator with respect to a -basis: where is the scalar curvature. The relations (2.15) and (2.18) yield and the relation (2.9):

*Remark 2.4. *If (see [21]), Lemma 2.3 is expressed in a similar form with , is a unit vector field belonging to the contact distribution and for the functions and we have: , and .

*Definition 2.5. *An Riemannian manifold has harmonic curvature if the Ricci operator satisfies the condition:

From now on we shall work on a contact metric 3-manifold concerning a -basis at any point . First from the equation , Lemma 2.3 and the relations (2.17), we get the equations:

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

*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 , (b) by adding the equations two and six and using similarly (2.16) we have . From the relations , , and (2.12), we can conclude and hence we are led to the known result that the scalar curvature is constant in a contact metric 3-manifold with harmonic curvature. Later for our study, we will use the as a constant and we will give to these equations a more convenient form.

*Definition 2.7 (see [22]). *Let be a 3-dimensional contact metric manifold and the spectral decomposition of on . If

for all vector fields on and all points of an open subset of and on the points of which do not belong to , then the manifold is said to be * contact* manifold. From Lemma 2.3 and the relations (2.12), the above condition for leads to and for to . Hence on a contact manifold we have . If we apply the deformation , , , , , and then the contact metric structure remains the same. Hence the condition for a 3-dimensional contact metric manifold to be semi- contact is equivalent to .

*Definition 2.8. *A *-*contact metric manifold is defined in [23] as a contact metric manifold on which the curvature tensor satisfies for every the condition:

where are smooth functions on . If we have a generalized -contact metric manifold [24] and if additionally are constants then the manifold is a contact metric -space [25, 26]. Moreover, in [23] and [24] it is proved, respectively, that for a or a generalized -contact metric manifold of dimension greater than 3 the functions are constants and 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 be a contact metric 3-manifold. In case , that is, is a Sasakian structure, then is a pseudosymmetric space of constant type [13]. Next, assume that is not empty and let be a -basis as in Lemma 2.3. We have the following.

Lemma 2.10 (see [16]). *Let be a contact metric three manifold. Then is pseudosymmetric if and only if
**
where is the function in the pseudosymmetry definition (2.2).*

Using (2.15), (2.19), the system (2.27) takes a more convenient form:

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

Proposition 2.12 (see [16]). *Let be a 3-dimensional contact metric manifold satisfying . Then, is a pseudosymmetric space of constant type.*

#### 3. Pseudosymmetric Contact Metric 3-Manifolds with Harmonic Curvature and Constant in the Direction of

Theorem 3.1. *Let be a 3-dimensional pseudosymmetric contact metric manifold with harmonic curvature and constant in the direction of . Then, there are at most eight open subsets of for which their union is an open and dense subset of and each of them as an open submanifold of is either: (a) Sasakian or (b) flat or (c) locally isometric to the Lie groups , equipped with a left invariant metric or (d) pseudosymmetric of constant type and with scalar curvature or (e) semi- contact with or (f) semi- contact with or (g) semi- contact of type constant along and or (h) semi- contact of type constant along and .*

*Proof. *We consider the next open subsets of :
where is open and dense subset of .

In case , is a pseudosymmetric space of constant type [13] and we get the (a) case of present Theorem 3.1. Next, assume that is not empty and let be a -basis. First we note that in the neighborhood where we have from (2.20)

Equations (2.23) because of (3.2) and the fact that is constant become, respectively,
where for the second form of (3.6), (3.8), we also used and in (3.10), (3.11).

In the neighborhood , the system (2.28) for the pseudosymmetric contact metric 3-manifolds of Lemma 2.10 because of (3.2) becomes
where are given by (2.15), (2.19).

Studying the third equation, we regard the following open subsets of :
where is open and dense in the closure of .

In we have
hence, we regard the subsets of :
where is open and dense in the closure of and is open and dense in the closure of . We study the initial system at each for .

In the initial system (2.28) becomes
or more explicitly
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 the initial system (2.28) becomes
Apart from (3.2), we also have the following equations:
while we will also use (3.3), (3.4), (3.6), and (3.11).

Differentiating (3.21) with respect to , we get, respectively,
(the derivative can not be estimated any further). Equations (3.3), (3.11) because of (3.19) yield, respectively,
Differentiating (3.22) with respect to and using (3.2), (3.26) and the fact that is constant, we get
The first form of (3.6) and (3.19) give
hence (3.27), (3.28) yield
We suppose that there is a point where . Because of the continuity of the function , there is a neighborhood of this point : . In , we have . Differentiating this equation with respect to and using (3.2), the constancy of and the fact that we work in where we conclude that in , which is a contradiction. Hence
everywhere in . Because of (3.30) the equations (3.20), (3.27) give
Differentiating (3.2) with respect to , (3.19) with respect to , subtracting and using (2.12), we get
Letβs suppose that there is a point in where . The function is continuous, hence there is an open neighborhood of , , where everywhere in , hence
From (3.20) and (3.33) we have in :
Equation (3.4) because of (3.33) gives . From the second of (2.16) and because of (3.31), (3.33), (3.34), we get in : . By equalizing these two results and because of (3.34), we get: . We differentiate this equation with respect to and because of (3.33), we get , which is a contradiction in . Hence everywhere in and because of (3.31), we can conclude according to Definition 2.7 that the structure is semi- contact and pseudosymmetric with constant along the directions of and because of (3.23), (3.24) and (3.25), (3.30), (3.31).

In the initial system (2.28) becomes
We have the following equations and (3.2):
while we will also use (3.3), (3.4), (3.6), (3.8).

Differentiating (3.38) with respect to , we get, respectively,
(we neglect the derivative because we can not estimate it). Equations (3.4), (3.8) because of (3.36) yield, respectively,
Differentiating (3.39) with respect to and using (3.2), (3.42) and the fact that is constant, we get
The first form of (3.10) and (3.36) give
hence (3.43), (3.44) yield
We suppose that there is a point where . Because of the continuity of this function, there is a neighborhood of ββ: . In , we have . Differentiating this equation with respect to and using (3.2), the constancy of and the fact that we work in where we conclude that in , which is a contradiction. Hence
everywhere in . Because of (3.46) the equations (3.37), (3.43) give
Differentiating (3.2) with respect to , (3.36) with respect to , subtracting and using (2.12), we get
Letβs suppose that there is a point in where . This function is smooth, then because of its continuity, there is an open neighborhood of , , where everywhere in , hence
From (3.37) and (3.49) we have in :
From (3.3) and (3.49), we get . From the first of (2.16) and because of (3.47), (3.49), (3.50), we get in : . By equalizing these two results and because of (3.50), we get: . We differentiate this equation with respect to and because of (3.49), we get , which is a contradiction in . Hence everywhere in and because of (3.47), we can conclude according to Definition 2.7 that the structure is semi- contact and pseudosymmetric with constant along the directions of and 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 and , respectively.

*Remark 3.2. *(i) The conditions of harmonic curvature help us to the systems in the neighborhoods and where we had equations of the type and which we could not handle in our previous articles [16, 17].

(ii) In case (d) where 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 -Contact Metric 3-Manifolds of Type Constant in the Direction of

Theorem 4.1. *Let be a pseudosymmetric -contact metric 3-manifold of type constant along the direction . Then, there are at most five open subsets of for which their union is an open and dense subset of and each of them as an open submanifold of is either (a) Sasakian or (b) flat or (c) pseudosymmetric of constant type , and of constant scalar curvature or (d) pseudosymmetric generalized -contact metric manifold of type , of scalar curvature and or (e) pseudosymmetric generalized -contact metric manifold of type , of scalar curvature and .*

*Proof. *We study pseudosymmetric -contact metric 3-manifolds with
where is the function in (2.2). We consider the next open subsets of ,
where is open and dense subset of .

In case , is a Sasakian structure which is a pseudosymmetric space of constant type [13] with , and and we get the (a) case of present Theorem 4.1. Next, assume that is not empty and let be a -basis. From (2.25), we can calculate the following components of the Riemannian curvature tensor:
By virtue of (2.14) we can conclude that
and hence the system (2.28) becomes
where are given by (2.19) and (4.4). Substituting from (4.4) (), () in (2.16) we also have
First we will prove that (equivalently as we work in where ). We suppose that there is a point where . By the continuity of this function, we can consider that there is a neighborhood of , where everywhere in . We work in . Then the first equation of (*) becomes or equivalently:
We differentiate this equation with respect to 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 and , respectively, and adding we have
We subtract this last equation from (4.9) and we get
or because of (2.12)
or equivalently: and because we work in , we have , which is a contradiction. Hence, we can deduce everywhere in :
Next we will derive some useful relations. From (4.4) we have:
We differentiate these equations with respect to and , respectively, we subtract, we use the relations (2.12), (4.4) and we get
or because of (4.13)
We differentiate the relations and (4.13) with respect to and , respectively, and subtracting we obtain: or because of (2.12), (4.4), (4.6)
We differentiate the relations and (4.13) with respect to and , respectively, and subtracting we obtain: or because of (2.12), (4.4), (4.5)
Finally, after substituting from (4.4), (4.13) the final form of the system (*) is
In order to study this system we regard the following open subsets of :
where is open and dense in the closure of .

In , we have and hence from (4.4): or or finally and . From (4.17), (4.18) we deduce that . Having also the second equation of (4.19), we regard the open subsets of
where is open and dense in the closure of .

In substituting in , from (2.15), , we get and hence the structure is flat with .

In from we have again , and from (4.17), (4.18) while we must also have . Hence, and from (2.18) of constant scalar curvature .

In having , the second equation of (4.19) becomes . Hence, we regard the open subsets of
where is open and dense in the closure of .

In we must have while in we have . We differentiate these equations with respect to and because of (4.13) we get

By virtue of and in (4.4) we deduce and hence
In we differentiate with respect to and similarly we also obtain (4.24). Each of and is a generalized -contact metric 3-manifold with and scalar curvature or respectively and from (4.1), (4.13) and (4.23) or (4.24) and .

Concluding: the structure in gives the Sasakian case, the structures in and give the (b) and (c) cases of the present Theorem 4.1 and the structures in and give (d) and (e) respectively.

*Remark 4.2. *The generalized -contact metric manifolds in dimension 3 with (equivalently ) and have been studied by Koufogiorgos and Tsichlias [28]. They proved in their Theorem 4.1 of [28] that at any point of , precisely one of the following relations is valid: , or , while there exists a chart with such that the functions , depend only on and the tensors fields , , , take a suitable form. Each of our submanifolds and is such a generalized -contact metric 3-manifold.

#### Acknowledgments

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

#### References

- R. Deszcz, βOn pseudo-symmetric spaces,β
*Bulletin de la Société Mathématique de Belgique. Série A*, vol. 44, no. 1, pp. 1β34, 1992. View at Google Scholar Β· View at Zentralblatt MATH - Z. I. Szabó, βStructure theorems on Riemannian spaces satisfying $R(X,Y)\xb7R=0$. I. The local version,β
*Journal of Differential Geometry*, vol. 17, no. 4, pp. 531β582, 1982. View at Google Scholar - Z. I. Szabó, βStructure theorems on Riemannian spaces satisfying $R(X,Y)\xb7R=0$ II. Global versions,β
*Geometriae Dedicata*, vol. 19, no. 1, pp. 65β108, 1985. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - H. Takagi, βAn example of Riemannian manifolds satisfying $R(X,Y)\xb7R=0$ but not $\nabla R=0$,β
*The Tohoku Mathematical Journal*, vol. 24, pp. 105β108, 1972. View at Google Scholar Β· View at Zentralblatt MATH - M. Belkhelfa, R. Deszcz, M. Głogowska, M. Hotloś, D. Kowalczyk, and L. Verstraelen, βOn some type of curvature conditions,β in
*PDEs, Submanifolds and Affine Differential Geometry*, vol. 57 of*Banach Center Publications*, pp. 179β194, Institute of Mathematics of the. Polish Academy of Sciences, Warsaw, Poland, 2002. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - J. Deprez, R. Deszcz, and L. Verstraelen, βExamples of pseudo-symmetric conformally flat warped products,β
*Chinese Journal of Mathematics*, vol. 17, no. 1, pp. 51β65, 1989. View at Google Scholar Β· View at Zentralblatt MATH - O. Kowalski and M. Sekizawa,
*Three—Dimensional Riemannian Manifolds of c—Conullity Two*, Riemannian Manifolds of Conullity Two, chapter 11, World Scientific, Singapore, 1996. - O. Kowalski and M. Sekizawa, βLocal isometry classes of Riemannian 3-manifolds with constant Ricci eigenvalues ${\rho}_{1}=\rho 2\ne \rho 3$,β
*Archivum Mathematicum*, vol. 32, no. 2, pp. 137β145, 1996. View at Google Scholar - O. Kowalski and M. Sekizawa, βPseudo-symmetric spaces of constant type in dimension three—elliptic spaces,β
*Rendiconti di Matematica e Delle sue Applicazioni. Serie VII*, vol. 17, no. 3, pp. 477β512, 1997. View at Google Scholar Β· View at Zentralblatt MATH - O. Kowalski and M. Sekizawa, βPseudo-symmetric spaces of constant type in dimension three—non-elliptic spaces,β
*Bulletin of Tokyo Gakugei University*, vol. 50, pp. 1β28, 1998. View at Google Scholar Β· View at Zentralblatt MATH - N. Hashimoto and M. Sekizawa, βThree-dimensional conformally flat pseudo-symmetric spaces of constant type,β
*Archivum Mathematicum*, vol. 36, no. 4, pp. 279β286, 2000. View at Google Scholar Β· View at Zentralblatt MATH - G. Calvaruso, βConformally flat pseudo-symmetric spaces of constant type,β
*Czechoslovak Mathematical Journal*, vol. 56, no. 2, pp. 649β657, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - J. T. Cho and J.-I. Inoguchi, βPseudo-symmetric contact 3-manifolds,β
*Journal of the Korean Mathematical Society*, vol. 42, no. 5, pp. 913β932, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - J. T. Cho, J.-I. Inoguchi, and J.-E. Lee, βPseudo-symmetric contact 3-manifolds. III,β
*Colloquium Mathematicum*, vol. 114, no. 1, pp. 77β98, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - M. Belkhelfa, R. Deszcz, and L. Verstraelen, βSymmetry properties of Sasakian space forms,β
*Soochow Journal of Mathematics*, vol. 31, no. 4, pp. 611β616, 2005. View at Google Scholar Β· View at Zentralblatt MATH - F. Gouli-Andreou and E. Moutafi, βTwo classes of pseudosymmetric contact metric 3-manifolds,β
*Pacific Journal of Mathematics*, vol. 239, no. 1, pp. 17β37, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - F. Gouli-Andreou and E. Moutafi, βThree classes of pseudosymmetric contact metric 3-manifolds,β
*Pacific Journal of Mathematics*, vol. 245, no. 1, pp. 57β77, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - D. E. Blair,
*Contact Manifolds in Riemannian geometry*, vol. 509 of*Lecture Notes in Mathematics*, Springer, Berlin, Germany, 1976. - D. E. Blair,
*Riemannian Geometry of Contact and Symplectic Manifolds*, vol. 203 of*Progress in Mathematics*, Birkhäuser, Boston, Mass, USA, 2002. - F. Gouli-Andreou and Ph. J. Xenos, βOn 3-dimensional contact metric manifolds with ${\nabla}_{\xi}\tau =0$,β
*Journal of Geometry*, vol. 62, no. 1-2, pp. 154β165, 1998. View at Publisher Β· View at Google Scholar - F. Gouli-Andreou and Ph. J. Xenos, βOn a class of 3-dimensional contact metric manifolds,β
*Journal of Geometry*, vol. 63, no. 1-2, pp. 64β75, 1998. View at Publisher Β· View at Google Scholar - F. Gouli-Andreou, J. Karatsobanis, and Ph. Xenos, βConformally flat $3-\tau -a$ manifolds,β
*Differential Geometry—Dynamical Systems*, vol. 10, pp. 107β131, 2008. View at Google Scholar - T. Koufogiorgos, M. Markellos, and V. J. Papantoniou, βThe harmonicity of the Reeb vector field on contact metric 3-manifolds,β
*Pacific Journal of Mathematics*, vol. 234, no. 2, pp. 325β344, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - T. Koufogiorgos and C. Tsichlias, βOn the existence of a new class of contact metric manifolds,β
*Canadian Mathematical Bulletin*, vol. 43, no. 4, pp. 440β447, 2000. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - D. E. Blair, T. Koufogiorgos, and B. J. Papantoniou, βContact metric manifolds satisfying a nullity condition,β
*Israel Journal of Mathematics*, vol. 91, no. 1–3, pp. 189β214, 1995. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - E. Boeckx, βA full classification of contact metric ($\kappa ,\mu $)-spaces,β
*Illinois Journal of Mathematics*, vol. 44, no. 1, pp. 212β219, 2000. View at Google Scholar - G. Calvaruso and D. Perrone, βSemi-symmetric contact metric three-manifolds,β
*Yokohama Mathematical Journal*, vol. 49, no. 2, pp. 149β161, 2002. View at Google Scholar Β· View at Zentralblatt MATH - T. Koufogiorgos and C. Tsichlias, βGeneralized $(\kappa ,\mu )$-contact metric manifolds with $\xi \mu =0$,β
*Tokyo Journal of Mathematics*, vol. 31, no. 1, pp. 39β57, 2008. View at Google Scholar Β· View at Zentralblatt MATH