On (k,μ)-paracontact metric spaces satisfying some conditions on the w0-curvature tensor

The object of the present paper is to study (k,μ)-Paracontact metric manifold. We introduce the curvature tensors of (k,μ)Paracontact manifold satisfying the conditions W ? 0 (X ,Y ) ·P= 0, W ? 0 (X ,Y ) ·R= 0, W ? 0 (X ,Y ) · Z̃ = 0, W ? 0 (X ,Y ) ·S= 0 and W ? 0 (X ,Y ) ·C̃ = 0. According these cases, (k,μ)-Paracontact manifolds have been characterized. In my opinion some exciting results on a (k,μ)Paracontact metric manifold are obtained.


Introduction
In the modern geometry, the geometry of paracontact manifolds has turn into a subject of growing interest for its substantial applications in applied mathematics and physics. Paracontact manifolds are smooth manifolds of dimension (2n + 1) equipped with a (1, 1)−tensor φ , a vector field ξ , and a 1-form η satisfying η(ξ ) = 1, φ 2 = I − η ⊗ ξ and φ induces an almost paracomplex structure on each fibre of D = ker(η) [1]. Moreover if the manifold is equipped with a pseudo-Riemannian metric g so that weyl curvature tensor, R is the Riemannian curvature tensor, Z is the concircular curvature tensor, S is the Ricci tensor, C is the quasi-conformal curvature tensor and W 0 is the W 0 − curvature tensor.

Preliminaries
A contact manifold is a C ∞ −(2n+1) dimensional manifold M 2n+1 equipped with a global 1-form η such that η ∧(dη) n = 0 everywhere on M 2n+1 . Given such a form η, it is well known that there exists a unique vector field ξ , called the characteristic vector field, such that η(ξ ) = 1 and dη(X, ξ ) = 0 for every vector field X on M 2n+1 . A Riemannian metric g is said to be associated metric if there exists a tensor field φ of type (1, 1) such that g(φ X, φY ) = −g(X,Y ) + η(X)η(Y ), g(X, ξ ) = η(X) (2) for all vector fields X,Y on M. Then the structure (φ , ξ , η, g) on M is called a paracontact metric structure and the manifold equipped with such a structure is called a almost paracontact metric manifold [7].
We now define a (1, 1) tensor field h by h = 1 2 L ξ φ , where L denotes the Lie derivative. Then h is symmetric and satisfies the conditions If ∇ denotes the Levi-Civita connection of g, then we have the following relation for any X ∈ χ(M) [15]. For a paracontact metric manifold M 2n+1 (φ , ξ , η, g), if ξ is a killing vector field or equivalently, h = 0, then it is called a K-paracontact manifold.
A paracontact metric structure (φ , ξ , η, g) is normal, that is, satisfies [φ , φ ] + 2dη ⊗ ξ = 0, which is equivalent to for all X,Y ∈ χ(M) [15]. If an almost paracontact metric manifold is normal, then it called paracontact metric manifold. Any para-Sasakian manifold is K-paracontact, and the converse holds when n = 1, that is, for 3-dimensional spaces. Any para-Sasakian manifold satisfies for all X,Y ∈ χ(M), but this is not a sufficient condition for a paracontact manifold to be para-Sasakian. It is clear that every para-Sasakian manifold is K-paracontact. But the converse is not always true [4].

Definition 1.
A paracontact manifold M is said to be η-Einstein if its Ricci tensor S of type (0, 2) is of the from S(X,Y ) = ag(X,Y ) + bη(X)η(Y ),where a, b are smooth functions on M. If b = 0, then the manifold is also called Einstein [23].

Definition 2.
A paracontact metric manifold is said to be a (k, µ)−paracontact manifold if the curvature tensor R satisfies for all X,Y ∈ χ(M), where k and µ are real constants.
This class is very wide containing the para-Sasakian manifolds as well as the paracontact metric manifolds satisfying R(X,Y )ξ = 0 [16].
In particular, if µ = 0, then the paracontact metric (k, µ)−manifold is called paracontact metric N(k)-manifold . Thus for a paracontact metric N(k)-manifold the curvature tensor satisfies the following relation for all X,Y ∈ χ(M). Though the geometric behavior of paracontact metric (k, µ)−spaces is different according as k < −1, or k > −1, but there are also some common results for k < −1 and k > −1.

Lemma 1.
There does not exist any paracontact (k, µ)−manifold of dimension greater than 3 with k > −1 which is Einstein whereas there exits such manifolds for k < −1 [5].
The concept of quasi-conformal curvature tensor was defined by K. Yano and S. Sawaki [11]. Quasi-conformal curvature tensor of a (2n + 1)-dimensional Riemannian manifold is defined as where a and b are arbitrary scalars, and r is the scalar curvature of the manifold, Q, S and r denote the Ricci operator, Ricci tensor and scalar curvature of manifold, respectively.
Let (M, g) be an (2n + 1)-dimensional Riemannian manifold. Then the concircular curvature tensor Z is defined by for all X,Y, Z ∈ χ(M) [10]. Then the projective curvature tensor P is defined by for all X,Y, Z ∈ χ(M), where r is the scalar curvature of M and Q is the Ricci operator given by g(QX,Y ) = S(X,Y ) [10].
Then the curvature tensor W 0 is defined by for all X,Y, Z ∈ χ(M) [8].
3 A (k, µ)− paracontact manifold satisfying certain conditions on the W 0 -curvature tensor In this section, we will give the main results for this paper.
So, M is an η−Einstein manifold. The converse is obvious. This completes of the proof.