#### Abstract

We introduce the concept of ()-almost paracontact manifolds, and in particular, of ()-para-Sasakian manifolds. Several examples are presented. Some typical identities for curvature tensor and Ricci tensor of ()-para Sasakian manifolds are obtained. We prove that if a semi-Riemannian manifold is one of flat, proper recurrent or proper Ricci-recurrent, then it cannot admit an ()-para Sasakian structure. We show that, for an ()-para Sasakian manifold, the conditions of being symmetric, semi-symmetric, or of constant sectional curvature are all identical. It is shown that a symmetric spacelike (resp., timelike) ()-para Sasakian manifold is locally isometric to a pseudohyperbolic space (resp., pseudosphere ). At last, it is proved that for an ()-para Sasakian manifold the conditions of being Ricci-semi-symmetric, Ricci-symmetric, and Einstein are all identical.

#### 1. Introduction

In 1976, an almost paracontact structure satisfying and on a differentiable manifold was introduced by Satō [1]. The structure is an analogue of the almost contact structure [2, 3] and is closely related to almost product structure (in contrast to almost contact structure, which is related to almost complex structure). An almost contact manifold is always odd dimensional but an almost paracontact manifold could be even dimensional as well. In 1969, Takahashi [4] introduced almost contact manifolds equipped with associated pseudo-Riemannian metrics. In particular, he studied Sasakian manifolds equipped with an associated pseudo-Riemannian metric. These indefinite almost contact metric manifolds and indefinite Sasakian manifolds are also known as -almost contact metric manifolds and -Sasakian manifolds, respectively [5–7]. Also, in 1989, Matsumoto [8] replaced the structure vector field by in an almost paracontact manifold and associated a Lorentzian metric with the resulting structure and called it a Lorentzian almost paracontact manifold.

An -Sasakian manifold is always odd dimensional. Recently, we have observed that there does not exist a lightlike surface in a -dimensional -Sasakian manifold. On the other hand, in a Lorentzian almost paracontact manifold given by Matsumoto, the semi-Riemannian metric has only index and the structure vector field is always timelike. These circumstances motivate us to associate a semi-Riemannian metric, not necessarily Lorentzian, with an almost paracontact structure, and we shall call this indefinite almost paracontact metric structure an -almost paracontact structure, where the structure vector field will be spacelike or timelike according as or .

In this paper we initiate study of -almost paracontact manifolds, and in particular, -para Sasakian manifolds. The paper is organized as follows. Section 2 contains basic definitions and some examples of -almost paracontact manifolds. In Section 3, some properties of normal almost paracontact structures are discussed. Section 4 contains definitions of an -paracontact structure and an --paracontact structure. A typical example of an --paracontact structure is also presented. In Section 5, we introduce the notion of an -para Sasakian structure and study some of its basic properties. We find some typical identities for curvature tensor and Ricci tensor. We prove that if a semi-Riemannian manifold is one of flat, proper recurrent, or proper Ricci-recurrent, then it cannot admit an -para Sasakian structure. We show that, for an -para Sasakian manifold, the conditions of being symmetric, semi-symmetric, or of constant sectional curvature are all identical. More specifically, it is shown that a symmetric spacelike -para Sasakian manifold is locally isometric to a pseudohyperbolic space , and a symmetric timelike -para Sasakian manifold is locally isometric to a pseudosphere . At last, it is proved that for an -para Sasakian manifold, the conditions of being Ricci-semi-symmetric, Ricci-symmetric, and Einstein are all identical. Unlike -dimensional -Sasakian manifold, which cannot possess a lightlike surface, the study of lightlike surfaces of -dimensional -para Sasakian manifolds will be presented in a forthcoming paper.

#### 2. -Almost Paracontact Metric Manifolds

Let be an almost paracontact manifold [1] equipped with an almost paracontact structure consisting of a tensor field of type , a vector field , and a -form satisfying
It is easy to show that the relation (2.1) and one of the three relations (2.2), (2.3), and (2.4) imply the remaining two relations of (2.2), (2.3), and (2.4). On an -dimensional almost paracontact manifold, one can easily obtain
Equation (2.5) gives an *-structure* [9].

Throughout the paper, by a semi-Riemannian metric [10] on a manifold , we understand a non-degenerate symmetric tensor field of type . In particular, if its index is , it becomes a Lorentzian metric [11]. A sufficient condition for the existence of a Riemannian metric on a differentiable manifold is paracompactness. The existence of Lorentzian or other semi-Riemannian metrics depends upon other topological properties. For example, on a differentiable manifold, the following statements are equivalent: there exits a Lorentzian metric on , there exists a non-vanishing vector field on , and either is non-compact, or is compact and has Euler number . Also, for instance, the only compact surfaces that can be made Lorentzian surfaces are the tori and Klein bottles, and a sphere admits a Lorentzian metric if and only if is odd .

Now, we give the following.

*Definition 2.1. *Let be a manifold equipped with an almost paracontact structure . Let be a semi-Riemannian metric with index such that
where . Then we say that is an -*almost paracontact metric manifold* equipped with an *-almost paracontact metric structure *. In particular, if index, then an -almost paracontact metric manifold will be called a* Lorentzian almost paracontact manifold*. In particular, if the metric is positive definite, then an -almost paracontact metric manifold is the usual* almost paracontact metric manifold* [1].

Equation (2.7) is equivalent to
along with
for all . From (2.9) it follows that
that is, the structure vector field is never lightlike. Since is non-degenerate metric on and is non-null, therefore the paracontact distribution
is non-degenerate on .

*Definition 2.2. *Let be an -almost paracontact metric manifold (resp., a Lorentzian almost paracontact manifold). If , then will be said to be a* spacelike *-*almost paracontact metric manifold* (resp., a *spacelike Lorentzian almost paracontact manifold*). Similarly, if , then will be said to be a* timelike *-*almost paracontact metric manifold* (resp., a *timelike Lorentzian almost paracontact manifold*).

Note that a timelike Lorentzian almost paracontact structure is a Lorentzian almost paracontact structure in the sense of Mihai and Roşca [12], Matsumoto [13], which differs in the sign of the structure vector field of the Lorentzian almost paracontact structure given by Matsumoto [8].

*Example 2.3. * Let be the -dimensional real number space with a coordinate system . We define
Then the set is a timelike Lorentzian almost paracontact structure, while the set is a spacelike -almost paracontact metric structure. We note that index and index.

*Example 2.4. *Let be the -dimensional real number space with a coordinate system . We define
Then, the set is an almost paracontact structure in . The set is a timelike Lorentzian almost paracontact structure. Moreover, the trajectories of the timelike structure vector are geodesics. The set is a spacelike Lorentzian almost paracontact structure. The set is a spacelike -almost paracontact metric structure with index.

*Example 2.5. *Let be the -dimensional real number space with a coordinate system . Defining
the set becomes a timelike Lorentzian almost paracontact structure in , while the set is a spacelike -almost paracontact structure. Note that index.

The Nijenhuis tensor of a tensor field of type on a manifold is a tensor field of type defined by
for all . If admits a tensor field of type satisfying
then it is said to be an* almost product manifold* equipped with an* almost product structure *. An almost product structure is* integrable *if its Nijenhuis tensor vanishes. For more details we refer to [14].

*Example 2.6. *Let be a semi-Riemannian almost product manifold such that
Consider the product manifold . A vector field on can be represented by , where is tangent to , a smooth function on , and the coordinates of . On we define
Then is an -almost paracontact metric structure on the product manifold .

*Example 2.7. *Let be an -almost contact metric manifold. If we put , then is an -almost paracontact metric manifold.

#### 3. Normal Almost Paracontact Manifolds

Let be an almost paracontact manifold with almost paracontact structure and consider the product manifold , where is the real line. A vector field on can be represented by , where is tangent to , a smooth function on , and the coordinates of . For any two vector fields and , it is easy to verify the following:

*Definition 3.1. *If the induced almost product structure on defined by
is integrable, then we say that the almost paracontact structure is *normal*.

This definition is conformable with the definition of normality given in [15]. As the vanishing of the Nijenhuis tensor is a necessary and sufficient condition for the integrability of the almost product structure , we seek to express the conditions of normality in terms of the Nijenhuis tensor of . In view of (2.15), (3.2), (3.1), and (2.1)–(2.4) we have
where denotes the Lie derivative with respect to . Since is skew symmetric tensor field of type , it suffices to compute and . Thus we have

We are thus led to define four types of tensors , , , and , respectively, by (see also [1])
Thus the almost paracontact structure will be normal if and only if the tensors defined by (3.5)–(3.8) vanish identically.

Taking account of (2.1)–(2.5) and (3.5)–(3.8), it is easy to obtain the following.

Lemma 3.2. *Let be an almost paracontact manifold with an almost paracontact structure . Then
**
Consequently,
*

From (3.14), it follows that if or vanishes then vanishes. In view of (3.14), (3.16), and (3.17), we can state the following.

Theorem 3.3. * If, in an almost paracontact manifold , vanishes, then , , and vanish identically. Hence, the almost paracontact structure is normal if and only if . *

Some equations given in Lemma 3.2 are also in [1]. First part of Theorem 3.3 is given as Theorem of [1]. Now, we find a necessary and sufficient condition for the vanishing of in the following.

Proposition 3.4. *The tensor vanishes if and only if
*

*Proof. *The necessary part follows from (3.13). Conversely, from (3.18) and (2.3), we have
which along with (2.1), when used in (3.18), yields
which in view of (3.10) proves that .

From the definition of and , it follows that [1, Theorem ] the tensor (resp., ) vanishes identically if and only if (resp., ) is invariant under the transformation generated by infinitesimal transformations . Consequently, in a normal almost paracontact manifold, and are invariant under the transformation generated by infinitesimal transformations .

The tangent sphere bundle over a Riemannian manifold has naturally an almost paracontact structure in which and [16]. Also an almost paracontact structure is said to be* weak normal* [15] if the almost product structures and are integrable. Then an almost paracontact structure is normal if and only if it is weak normal and .

#### 4. --Paracontact Metric Manifolds

The fundamental symmetric tensor of the -almost paracontact metric structure is defined by for all . Also, we get for all .

*Definition 4.1. *We say that is an *-paracontact metric structure* if
In this case is an *-paracontact metric manifold*.

The condition (4.3) is equivalent to
where is the operator of Lie differentiation. For and Riemannian, is the usual paracontact metric manifold [17].

*Definition 4.2. *An-almost paracontact metric structure is called an -*-paracontact metric structure* if
A manifold equipped with an --paracontact structure is said to be -*-paracontact metric manifold*.

Equation (4.5) is equivalent to

We have the following.

Theorem 4.3. *An -almost paracontact metric manifold is an -s-paracontact metric manifold if and only if it is an -paracontact metric manifold such that the structure -form is closed.*

*Proof. *Let be an --paracontact metric manifold. Then in view of (4.6) we see that is closed. Consequently, is an -paracontact metric manifold.

Conversely, let us suppose that is an -paracontact metric manifold and is closed. Then
which implies (4.6).

Proposition 4.4. *If in an -almost paracontact metric manifold the structure -form is closed, then
*

*Proof. *First we note that and in particular
If is closed, then for any vector orthogonal to we get
which completes the proof.

Using techniques similar to those introduced in [18, Section ], we give the following.

*Example 4.5. *Let us assume the following:
Let be a smooth function. Define a function by
Now, define a -form on by
Next, define a vector field on by
and a tensor field on by
for all vector fields
Let be smooth functions. We define a tensor field of type by
where are smooth functions such that
Then is a timelike Lorentzian almost paracontact structure on . Moreover, if the smooth functions are given by
for some smooth functions , then we get a timelike Lorentzian -paracontact manifold.

#### 5. -Para Sasakian Manifolds

We begin with the following.

*Definition 5.1. *An -almost paracontact metric structure is called an *-para Sasakian structure* if
where is the Levi-Civita connection with respect to . A manifold endowed with an -para Sasakian structure is called an *-para Sasakian manifold*.

For and Riemannian, is the usual para Sasakian manifold [17, 18]. For , Lorentzian, and replaced by , becomes a Lorentzian para Sasakian manifold [8].

*Example 5.2. * Let be the -dimensional real number space with a coordinate system . We define
Then is an -para Sasakian structure.

Theorem 5.3. *An -para Sasakian structure is always an - -paracontact metric structure, and hence an -paracontact metric structure. *

*Proof. * Let be an -para Sasakian manifold. Then from (5.1) we get
Operating by to the above equation, we get (4.5).

The converse of the above theorem is not true. Indeed, the --paracontact structure in the Example 4.5 need not be -para Sasakian.

Theorem 5.4. *An -para Sasakian structure is always normal. *

*Proof. *In an almost paracontact manifold , we have
for all vector fields in . Now, let be an -para Sasakian manifold. Then it is --paracontact, and therefore using (5.1) and (4.5) in (5.4), we get .

*Problem 1. *Whether a normal -paracontact structure is -para Sasakian or not, consider the following.

Lemma 5.5. *Let be an -para Sasakian manifold. Then the curvature tensor satisfies
**
Consequently,
**
for all vector fields in .*

*Proof. * Using (4.5), (5.1), and (2.1) in
we obtain (5.5).

If we put then in an -para Sasakian manifold (5.5) and (5.8) can be rewritten as respectively.

Lemma 5.6. *In an -para Sasakian manifold , the curvature tensor satisfies
**
for all vector fields in .*

*Proof. * Writing (5.1) equivalently as
and differentiating covariantly with respect to , we get
for all . Now using (5.18) in the Ricci identity
we obtain (5.13). Equation (5.14) follows from (5.13) and (5.6). Equation (5.15) follows from (5.14). Finally, equation (5.16) follows from (5.14) and (5.15).

Equation (5.5) may also be obtained by (5.16). Equations (5.13)–(5.16) are generalizations of the and in [19]. Now, we prove the following:

Theorem 5.7. *An -para Sasakian manifold cannot be flat. *

*Proof. *Let be a flat -para Sasakian manifold. Then from (5.6) we get
from which we obtain
for all , a contradiction.

A non-flat semi-Riemannian manifold is said to be* recurrent* [20] if its Ricci tensor satisfies the recurrence condition
where is a -form. If in the above equation, then the manifold becomes *symmetric* in the sense of Cartan [21]. We say that is proper recurrent if .

Theorem 5.8. *An -para Sasakian manifold cannot be proper recurrent.*

*Proof. * Let be a recurrent -para Sasakian manifold. Then from (5.22), (5.6), and (4.5) we obtain
for all . Putting in the above equation, we get
a contradiction.

Let and . Then [10, Definition , page 110] the following are given.

(1) The*pseudosphere*of radius in is the hyperquadric with dimension and index .(2) The

*pseudohyperbolic space*of radius in is the hyperquadric with dimension and index .

Theorem 5.9. *An -para Sasakian manifold is symmetric if and only if it is of constant curvature . Consequently, a symmetric spacelike -para Sasakian manifold is locally isometric to a pseudohyperbolic space and a symmetric timelike -para Sasakian manifold is locally isometric to a pseudosphere .*

*Proof. * Let be a symmetric -para Sasakian manifold. Then putting in (5.23), we obtain
for all . Writing in place of in the above equation and using (2.7) and (5.6), we get
which shows that is a space of constant curvature . The converse is trivial.

Corollary 5.10. *If an -para Sasakian manifold is of constant curvature, then
**
for all .*

*Proof. * Obviously, if an -para Sasakian manifold is of constant curvature , then . Therefore, using (5.28) in (5.14), we get (5.29).

Apart from recurrent spaces, semi-symmetric spaces are another well-known and important natural generalization of symmetric spaces. A semi-Riemannian manifold is a *semi-symmetric space* if its curvature tensor satisfies the condition
for all vector fields on , where acts as a derivation on . Symmetric spaces are obviously semi-symmetric, but the converse need not be true. In fact, in dimension greater than two there always exist examples of semi-symmetric spaces which are not symmetric. For more details we refer to [22].

Given a class of semi-Riemannian manifolds, it is always interesting to know that whether, inside that class, semisymmetry implies symmetry or not. Here, we prove the following.

Theorem 5.11. *In an -para Sasakian manifold, the condition of semi-symmetry implies the condition of symmetry.*

*Proof. * Let be a symmetric -para Sasakian manifold. Let the condition of being semi-symmetric be true, that is,
In particular, from the condition , we get
which in view of (5.12) gives
Equation (5.11) then gives
Therefore is of constant curvature , and hence symmetric.

In view of Theorems 5.9 and 5.11, we have the following.

Corollary 5.12. *Let be an -para Sasakian manifold. Then the following statements are equivalent.*(i)* is symmetric.*(ii)* is of constant curvature .*(iii)* is semi-symmetric.*(iv)* satisfies . *

Now, we need the following.

Lemma 5.13. *In an -dimensional -para Sasakian manifold the Ricci tensor satisfies
**
for all . Consequently,
*

*Proof. * Contracting (5.16), we get (5.35). Replacing by in (5.35), we get (5.36). Putting in (5.35), we get (5.37).

A semi-Riemannian manifold is said to be* Ricci-recurrent* [23] if its Ricci tensor satisfies the condition
where is a -form. If in the above equation, then the manifold becomes* Ricci-symmetric*. We say that is proper Ricci-recurrent, if .

Theorem 5.14. *An -para Sasakian manifold cannot be proper Ricci-recurrent.*

*Proof. *Let be an -dimensional -para Sasakian manifold. If possible, let be proper Ricci-recurrent. Then
But we have
Using (5.40) in (5.39), we get
Putting in the above equation, we get , a contradiction.

A semi-Riemannian manifold is said to be* Ricci-semi-symmetric* [24] if its Ricci tensor satisfies the condition
for all vector fields on , where acts as a derivation on .

at last, we prove the following.

Theorem 5.15. *For an -dimensional -para Sasakian manifold , the following three statements are equivalen. *(a)* is an Einstein manifold.*(b)* is Ricci-symmetric.*(c)* is Ricci-semi-symmetric. *

*Proof. * Obviously, the statement (a) implies each of the statements (b) and (c). Let (b) be true. Then putting in (5.41), we get
Replacing by in the above equation, we get
which shows that the statement is true. At last, let be true. In particular,
implies that
which in view of (5.8) and (5.37) again gives (5.44). This completes the proof.

#### Acknowledgments

The authors are very much thankful to the referees for their valuable suggestions. This paper was prepared during the first visit of the first author to nönü University, Turkey, in July 2009. The first author was supported by the Scientific and Technical Research Council of Turkey (TÜBTAK) for Advanced Fellowships Programme.