Indefinite almost paracontact metric manifolds

In this paper we introduce the concept of $(\varepsilon)$-almost paracontact manifolds, and in particular, of $(\varepsilon)$-para Sasakian manifolds. Several examples are presented. Some typical identities for curvature tensor and Ricci tensor of $(\varepsilon)$-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 can not admit an $(\varepsilon)$-para Sasakian structure. We show that, for an $(\varepsilon)$-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) $(\varepsilon)$-para Sasakian manifold $M^{n}$ is locally isometric to a pseudohyperbolic space $H_{\nu}^{n}(1)$ (resp. pseudosphere $S_{\nu}^{n}(1)$). In last, it is proved that for an $(\varepsilon)$-para Sasakian manifold, the conditions of being Ricci-semisymmetric, Ricci-symmetric and Einstein are all identical.


Introduction
In 1976, an almost paracontact structure (ϕ, ξ, η) satisfying ϕ 2 = I − η ⊗ ξ and η(ξ) = 1 on a differentiable manifold, was introduced by Sāto [20]. The structure is an analogue of the almost contact structure [17,3] 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, T. Takahashi [23] 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 (ε)-almost contact metric manifolds and (ε)-Sasakian manifolds respectively [2,7,8]. Also, in 1989, K. Matsumoto [9] replaced the structure vector field ξ by − ξ in an almost paracontact manifold and associated a Lorentzian metric with the resulting structure and called it a Lorentzian almost paracontact manifold.
An (ε)-Sasakian manifold is always odd-dimensional. Recently, we have observed that there does not exist a lightlike surface in a 3-dimensional (ε)-Sasakian manifold. On the other hand, in a Lorentzian almost paracontact manifold given by Matsumoto, the semi-Riemannian metric has only index 1 and the structure vector field ξ 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 (ε)-almost paracontact structure, where the structure vector field ξ will be spacelike or timelike according as ε = 1 or ε = −1.
In this paper we initiate study of (ε)-almost paracontact manifolds, and in particular, (ε)-para Sasakian manifolds. The paper is organized as follows. Section 2 contains basic definitions and some examples of (ε)-almost paracontact manifolds. In section 3, some properties of normal almost paracontact structures are discussed. Section 4 contains definitions of an (ε)-paracontact structure and an (ε)-s-paracontact structure. A typical example of an (ε)-s-paracontact structure is also presented. In section 5, we introduce the notion of an (ε)-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 can not admit an (ε)-para Sasakian structure. We show that, for an (ε)-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 (ε)para Sasakian manifold M n is locally isometric to a pseudohyperbolic space H n ν (1) and a symmetric timelike (ε)-para Sasakian manifold M n is locally isometric to a pseudosphere S n ν (1). In last, it is proved that for an (ε)-para Sasakian manifold, the conditions of being Ricci-semisymmetric, Ricci-symmetric and Einstein are all identical. Unlike 3dimensional (ε)-Sasakian manifold, which cannot possess a lightlike surface, the study of lightlike surfaces of 3-dimensional (ε)-para Sasakian manifolds will be presented in a forthcoming paper.
Throughout the paper, by a semi-Riemannian metric [14] on a manifold M, we understand a non-degenerate symmetric tensor field g of type (0, 2). In particular, if its index is 1, it becomes a Lorentzian metric [1]. 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: (1) there exits a Lorentzian metric on M, (2) there exists a non vanishing vector field on M, (3) either M is non compact, or M is compact and has Euler number χ(M) = 0. Also for instance, the only compact surfaces that can be made Lorentzian surfaces are the tori and Klein bottles, and a sphere S n admits a Lorentzian metric if and only if n is odd ≥ 3. Now, we give the following: Definition 2.1 Let M be a manifold equipped with an almost paracontact structure (ϕ, ξ, η). Let g be a semi-Riemannian metric with index(g) = ν such that where ε = ±1. Then we say that M is an (ε)-almost paracontact metric manifold equipped with an (ε)-almost paracontact metric structure (ϕ, ξ, η, g, ε). In particular, if index(g) = 1, then an (ε)-almost paracontact metric manifold will be called a Lorentzian almost paracontact manifold. In particular, if the metric g is positive definite, then an (ε)-almost paracontact metric manifold is the usual almost paracontact metric manifold [20].
The equation (2.7) is equivalent to for all X, Y ∈ T M. From (2.9) it follows that g (ξ, ξ) = ε, (2.10) that is, the structure vector field ξ is never lightlike. Since g is non-degenerate metric on M and ξ is non-null, therefore the paracontact distribution is non-degenerate on M.
Note that a timelike Lorentzian almost paracontact structure is a Lorentzian almost paracontact structure in the sense of Mihai and Rosca [11,10], which differs in the sign of the structure vector field of the Lorentzian almost paracontact structure given by Matsumoto [9]. Example 2.3 Let R 3 be the 3-dimensional real number space with a coordinate system (x, y, z). We define Then the set (ϕ, ξ, η, g 1 ) is a timelike Lorentzian almost paracontact structure, while the set (ϕ, ξ, η, g 2 ) is a spacelike (ε)-almost paracontact metric structure. We note that index(g 1 ) = 1 and index(g 2 ) = 2.
The Nijenhuis tensor [J, J] of a tensor field J of type (1, 1) on a manifold M is a tensor field of type (1, 2) defined by for all X, Y ∈ T M. If M admits a tensor field J of type (1, 1) satisfying then it is said to be an almost product manifold equipped with an almost product structure J. An almost product structure is integrable if its Nijenhuis tensor vanishes. For more details we refer to [24].
Example 2.6 Let (M n , J, G) be a semi-Riemannian almost product manifold, such that Consider the product manifold M n × R. A vector field on M n × R can be represented by Then (ϕ, ξ, η, g, ε) is an (ε)-almost paracontact metric structure on the product manifold M n × R.

Normal almost paracontact manifolds
Let M be an almost paracontact manifold with almost paracontact structure (ϕ, ξ, η) and consider the product manifold M × R, where R is the real line. A vector field on M × R can be represented by X, f d dt , where X is tangent to M, f a smooth function on M × R and t the coordinates of R. For any two vector fields X, f d dt and Y, h d dt , it is easy to verify the following is integrable, then we say that the almost paracontact structure (ϕ, ξ, η) is normal.
This definition is conformable with the definition of normality given in [5]. As the vanishing of the Nijenhuis tensor [J, J] is a necessary and sufficient condition for the integrability of the almost product structure J, we seek to express the conditions of normality in terms of the Nijenhuis tensor [ϕ, ϕ] of ϕ. In view of (2.11), (3.2), (3.1) and where £ X denotes the Lie derivative with respect to X. Since [J, J] is skew symmetric tensor field of type (1,2), it suffices to compute We are thus led to define four types of tensors N respectively by (see also [20]) Thus the almost paracontact structure (ϕ, ξ, η) will be normal if and only if the tensors defined by (3.3)-(3.6) vanish identically.

(ε)-s-paracontact metric manifolds
The fundamental (0, 2) symmetric tensor of the (ε)-almost paracontact metric structure is defined by for all X, Y ∈ T M. Also, we get for all X, Y, Z ∈ T M.
The condition (4.4) is equivalent to where, £ is the operator of Lie differentiation. For ε = 1 and g Riemannian, M is the usual paracontact metric manifold [21]. A manifold equipped with an (ε)-s-paracontact structure is said to be (ε)-s-paracontact metric manifold.
The equation (4.6) is equivalent to We have

Theorem 4.3 An (ε)-almost paracontact metric manifold is an (ε)-s-paracontact metric manifold if and only if it is an (ε)-paracontact metric manifold such that the structure 1-form η is closed.
Proof. Let M be an (ε)-s-paracontact metric manifold. Then in view of (4.7) we see that η is closed. Consequently, M is an (ε)-paracontact metric manifold.
Conversely, let us suppose that M is an (ε)-paracontact metric manifold and η is closed.
If η is closed, then for any vector X orthogonal to ξ, we get which completes the proof.
Let θ : R p × R q → R be a smooth function. Define a function ψ : R n → R by ψ x 1 , . . . , x n ≡ θ x 1 , . . . , x p+q + x n . Now, define a 1-form η on R n by Next, define a vector field ξ on R n by ξ ≡ ∂ ∂x n (4.10) and a (1, 1) tensor field ϕ on R n by for all vector fields We define a tensor field g of type (0, 2) by Then (ϕ, ξ, η, g) is a timelike Lorentzian almost paracontact structure on R n . Moreover, if the (p + q) smooth functions f i : R n → R are given by f a = F a x 1 , . . . , x p+q e −2x n + (θ a ) 2 , a ∈ {1, . . . , p} , for some smooth functions F i > 0, then we get a timelike Lorentzian s-paracontact manifold.

(ε)-para Sasakian manifolds
We begin with the following: Definition 5.1 An (ε)-almost contact metric structure is called an (ε)-para Sasakian structure if where ∇ is the Levi-Civita connection with respect to g. A manifold endowed with an (ε)-para Sasakian structure is called an (ε)-para Sasakian manifold.
Proof. Let M be an (ε)-para Sasakian manifold. Then from (5.1) we get Operating by ϕ to the above equation we get (4.6).
The converse of the above Theorem is not true. Indeed, the (ε)-s-paracontact structure in the Example 4.5 need not be (ε)-para Sasakian.
Proof. In an almost paracontact manifold M, we have for all vector fields X, Y in M. Now, let M be an (ε)-para Sasakian manifold. Then it is (ε)-s-paracontact and therefore using (5.1) and (4.6) in (5.2), we get 1 N = 0.
Lemma 5.6 Let M be an (ε)-para Sasakian manifold. Then the curvature tensor R satisfies Consequently, for all vector fields X, Y, Z in M.
Proof. Writing the equation (5.1) equivalently as and differentiating covariantly with respect to X we get for all X, Y, Z, W ∈ T M. Now using (5.13) in the Ricci identity we obtain (5.9). The equation ( The equation (5.3) may also be obtained by (5.12). The equations (5.9)-(5.12) are generalizations of the equations (3.2) and (3.3) in [13]. Now, we prove the following: A non-flat semi-Riemannian manifold M is said to be recurrent [16] if its Ricci tensor R satisfies the recurrence condition where α is a 1-form. If α = 0 in the above equation, then the manifold becomes symmetric in the sense of Cartan [6]. We say that M is proper recurrent, if α = 0.
2. The pseudohyperbolic space of radius r > 0 in R n+1 ν+1 is the hyperquadric with dimension n and index ν.
Theorem 5.10 An (ε)-para Sasakian manifold is symmetric if and only if it is of constant curvature − ε. Consequently, a symmetric spacelike (ε)-para Sasakian manifold is locally isometric to a pseudohyperbolic space H n ν (1) and a symmetric timelike (ε)-para Sasakian manifold is locally isometric to a pseudosphere S n ν (1).
Proof. Let M be a symmetric (ε)-para Sasakian manifold. Then putting α = 0 in (5.15) we obtain for all X, Y, Z, W ∈ T M. Writing ϕW in place of W in the above equation and using (2.7) and (5.4), we get 16) which shows that M is a space of constant curvature − ε. The converse is trivial.

Corollary 5.11
If an (ε)-para Sasakian manifold is of constant curvature, then for all X, Y, Z, W ∈ T M.
Apart from recurrent spaces, semi-symmetric spaces are another well-known and important natural generalization of symmetric spaces. A semi-Riemannian manifold (M, g) is a semi-symmetric space if its curvature tensor R satisfies the condition R(X, Y ) · R = 0 for all vector fields X, Y on M, where R (X, Y ) acts as a derivation on R. 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 [4].
Given a class of semi-Riemannian manifolds, it is always interesting to know that whether, inside that class, semi-symmetry implies symmetry or not. Here, we prove the following: Theorem 5. 12 In an (ε)-para Sasakian manifold, the condition of semi-symmetry implies the condition of symmetry.
Proof. Let M be a symmetric (ε)-para Sasakian manifold. Let the condition of being semi-symmetric be true, that is, In particular, from the condition R(ξ, U) · R = 0, we get which in view of (5.8) gives Therefore M is of constant curvature − ε, and hence symmetric.
A semi-Riemannian manifold M is said to be Ricci-recurrent [15] if its Ricci tensor S satisfies the condition (∇ X S) (Y, Z) = α (X) S (Y, Z) , X, Y, Z ∈ T M, (5.21) where α is a 1-form. If α = 0 in the above equation, then the manifold becomes Riccisymmetric. We say that M is proper Ricci-recurrent, if α = 0. Putting Y = ξ in the above equation, we get α (X) = 0, a contradiction.
A semi-Riemannian manifold M is said to be Ricci-semi-symmetric [12] if its Ricci tensor S satisfies the condition R(X, Y ) · S = 0 for all vector fields X, Y on M, where R (X, Y ) acts as a derivation on S.