On 3-Dimensional Contact Metric Generalized -Space Forms

Abstract

The present paper deals with a study of 3-dimensional contact metric generalized -space forms. We obtained necessary and sufficient condition for a 3-dimensional contact metric generalized -space form with to be of constant curvature. We also obtained some conditions of such space forms to be pseudosymmetric and -projectively flat, respectively.

1. Introduction

In 1995, Blair et al. [1] introduced the notion of contact metric manifolds with characteristic vector field belonging to the -nullity distribution and such types of manifolds are called -contact metric manifolds. They obtained several results and examples of such a manifold. A full classification of this manifold has been given by Boeckx [2]. A contact metric manifold is said to be a generalized -space if its curvature tensor satisfies the condition for some smooth functions and on independent choice of vector fields and . If and are constant, the manifold is called a -space. If a -space has a constant -sectional curvature and a dimension greater than 3, the curvature tensor of this -space form is given by [3] where , , , , , and are the tensors defined by for all vector fields , , on , where and is the usual Lie derivative.

The notion of generalized Sasakian-space-forms was introduced and studied by Alegre et al. [4] with several examples. A generalized Sasakian-space-form is an almost contact metric manifold whose curvature tensor is given by where , , and are the tensors defined above and , , are differentiable functions on . In such case we will write the manifold as . Generalized Sasakian-space-forms have been studied by several authors, namely, [5–11].

By motivating the works on generalized Sasakian-space-forms and -space forms, Carriazo et al. [12] introduced the concept of generalized -space forms. A generalized -space form is an almost contact metric manifold whose curvature tensor is given by where , , , , , are the tensors defined above and , , , , , are differentiable functions on .

The object of the paper is to study 3-dimensional contact metric generalized -space forms. The paper is organized as follows. Section 2 deals with some preliminaries on contact metric manifolds and contact metric generalized -space forms. Section 3 is concerned with 3-dimensional contact metric generalized -space forms. Here it is proved that a 3-dimensional contact metric generalized -space form with is of constant curvature if and only if . In Section 4, it is proved that a 3-dimensional contact metric generalized -space form is Ricci-semisymmetric; then either or . In Sections 5 and 6, we obtained that some equivalent conditions for a 3-dimensional contact metric generalized -space form are pseudosymmetric and -projectively flat, respectively.

2. Contact Metric Generalized -Space Forms

A contact manifold is a -() manifold equipped with a global 1-form such that everywhere on . Given a contact form it is well known that there exists a unique vector field called the characteristic vector field of such that and for every vector field on . A Riemannian metric is said to be associated metric if there exists a tensor field of type (1, 1) such that , , , , , and for all vector fields , on . Then the structure on is called a contact metric structure, and then manifold equipped with such a structure is called a contact metric manifold [13].

Given a contact metric manifold we define a (1, 1) tensor field by . Then is symmetric and satisfies the following relations: Moreover, if denotes the Riemannian connection of , then the following relation holds: The vector field is a Killing vector with respect to if and only if . A contact metric manifold for which is a Killing vector is said to be a -contact manifold. Therefore, a generalized -space form with such a structure is actually a generalized Sasakian-space-form.

A generalized -space-form is an almost contact metric manifold whose curvature tensor is given by for all vector fields , , on , where , , , , , are differentiable functions on , , and is the usual Lie derivative. In such case we will denote the manifold as . This kind of manifold appears as natural generalization of the -space forms by taking as constant. Here denotes constant -sectional curvature. The -sectional curvature of generalized -space form is . Generalized Sasakian-space-forms [4] are also examples with and , , not necessarily constant.

We inferred the following result from [12].

Theorem 1. Let be a generalized -space form. If is a contact metric manifold with , then it is a Sasakian manifold.

Next, by using the definitions of the tensors , , , , , and properties (6) of the tensor in the formula (8), we obtain that the curvature tensor of a generalized -space form satisfies for every , . Thus we have the following Theorem.

Theorem 2 (see [12]). If is a contact metric generalized -space form, then it is a generalized -space, with and .

We know from Theorem 1 that is Sasakian if . Under the same hypothesis, we also know that and , so we may take . Therefore, in this paper we will consider non-Sasakian generalized -space forms , that is, those with .

A contact metric manifold is said to be -Einstein manifold [13] if it satisfies for some smooth functions and .

3. On 3-Dimensional Contact Metric Generalized -Space Forms

If is a contact metric generalized -space form, then its Ricci tensor and the scalar curvature can be written as [12] It is known that in a 3-dimensional Riemannian manifold , the curvature tensor is given by for any vector fields , , on . By substituting (12) and (13) in (14) we have From (12), we have Also, from (15) we obtain

If is a contact metric generalized -space form, then is true if and only if . In this connection, the following result appears in [12].

Lemma 3. If is a contact metric generalized -space form, then the following conditions are equivalent:(i) is an -Einstein,(ii), where denotes the Ricci tensor,(iii) is a -space,(iv).

Now we begin with the following.

Lemma 4. A 3-dimensional contact metric generalized -space form with is of constant curvature if and only if .

Proof. Let a 3-dimensional contact metric generalized -space form with be a space of constant curvature. Then where is constant curvature of the manifold. By using the definition of Ricci curvature and (18) we have If we use (19) in the definition of the scalar curvature we get From (19) and (20) one can easily see that By putting in (12) and using (21) we obtain . By comparing this value of with (13), we have .
Conversely, if , then from (15) we can easily see that If satisfies , then in view of Lemma 3 we have . Thus from (22), we have If satisfies , then is constant. Hence, from (23), we conclude that the is of constant curvature. This completes the proof.

Theorem 5. Every 3-dimensional contact metric generalized -space form with is an --Einstein manifold.

Proof. The proof follows from (12), (10), and Lemma 3.

4. Ricci-Semisymmetric 3-Dimensional Contact Metric Generalized -Space Forms

A contact metric generalized -space form is said to be Ricci-semisymmetric if its Ricci tensor satisfies the condition where acts as a derivation on . This notion was introduced by Mirzoyan [14] for Riemannian spaces.

From (24) we have If we substitute in (25) and then using (17), we get By using (16) in (26) we obtain Consider that is an orthonormal basis of the , . Then by putting in (27) and taking the summation for , we have Again by using (16) in (28), we get which gives either or . For the second case, that is, , we have . Thus we can state the following.

Theorem 6. If a 3-dimensional contact metric generalized -space form is Ricci-semisymmetric then either or .

5. Pseudosymmetric 3-Dimensional Contact Metric Generalized -Space Forms

Let be a Riemannian manifold with the Riemannian metric . A tensor field of type (1, 3) is said to be curvature-like if it has the properties of . For example, the tensor given by defines a curvature-like tensor field on . For example, if , then the manifold is of constant curvature .

It is well known that every curvature-like tensor field acts on the algebra of all tensor fields on of type as a derivation [15] for all and . The derivation of by is a tensor field of type . For a tensor field of type , we define the derivative of with respect to the curvature-like tensor defined by (30) as As a generalization of locally symmetric manifolds, the notion of a semisymmetric manifold is introduced by Szabó [16]. As a proper generalization of a semisymmetric manifold, the notion of pseudosymmetric manifold is introduced by Deszcz [17]. A Riemannian manifold is said to be pseudosymmetric if there exists a real valued smooth function such that . In particular, if is constant, then is said to be a pseudosymmetric manifold of constant type [17]. A pseudosymmetric manifold is said to be proper if it is not semisymmetric.

For 3-dimensional Riemannian spaces, the following characterization of pseudosymmetry is known (cf. [18, 19]).

Theorem 7. A 3-dimensional Riemannian space is pseudosymmetric if and only if it is -Einstein.

We also know that a 3-dimensional contact metric generalized -space form is -Einstein if and only if . Hence, in view of Theorem 7, we can state the following.

Theorem 8. A 3-dimensional contact metric generalized -space form is pseudosymmetric if and only if .

From Lemma 3 and Theorem 8, we have following.

Lemma 9. A 3-dimensional contact metric generalized -space form with is pseudosymmetric.

6. -Projectively Flat 3-Dimensional Contact Metric Generalized -Space Forms

In a 3-dimensional contact metric generalized -space form the projective curvature tensor is defined as for all .

By plugging in (33) and using (16) and (10), we get If , then from (34) one can get that is, is -projectively flat.

Conversely, suppose that is -projectively flat. Then from (34) we obtain This implies that . Thus we can state the following.

Theorem 10. A 3-dimensional contact metric generalized -space form is -projectively flat if and only if .

From Lemma 3 and Theorem 10, we obtain the following.

Lemma 11. A 3-dimensional contact metric generalized -space form with is -projectively flat.

By combining Lemma 3, Theorem 5, Theorem 8, and Theorem 10 the following result is obtained.

Theorem 12. If is a contact metric generalized -space form, then the following conditions are equivalent to each other:(i) is --Einstein,(ii), where denotes the Ricci operator,(iii),(iv) is pseudosymmetric,(v) is -projectively flat.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The first author is thankful to University Grants Commission, New Delhi, India, for financial support in the form of Major Research Project F. no. 39-30/2010 (SR), dated 23-12-2010.