Ricci Solitons on α -para Kenmotsu Manifolds with Semi Symmetric Metric Connection

In this paper we introduce notion of Ricci solitons in α-para Kenmotsu manifold with semi -symmetric metric connection. We have found the relations between curvature tensor, Ricci tensors and scalar curvature of α-para Kenmotsu manifold with semi-symmetic metric connection.We have proved that 3-dimensional α-para Kenmotsu manifold with semi -symmetric metric connection is an η-Einstein manifold and the Ricci soliton defined on this manifold is named expanding and steady with respect to the value of λ constant.It is proved that Conharmonically flat α-para Kenmotsu manifold with semi-symmetric metric connection is η-Einstein manifold.


Introduction
In 1972 Kemmotsu studied a class of contact Riemannian manifolds satisfying some special conditions and this manifolds is known as Kenmotsu manifold [1].Sharma and Sinha started to study of the Ricci solitons in contact geometry in 1983 [2]. Later Mukut Mani Tripathi,Cornelia Livia Bejan and Mircea Crasmareanu, and others extensively studied Ricci solitons in contact metric manifolds [3,4]. In 1985, almost paracontact geometry was introduced by kaneyuki and williams and then it was continued by many authors [5]. Nagaraja ve premalatha studied exclusively about Ricci solitons on Kenmotsu manifold in 2012 .Agashe and Chafle ,Liang,pravonovic and Sengupta,Yildiz and Cetinkaya studied semi-symmetric non-metric connection in different ways [6][7][8][9][10][11].
A systematic study of almost paracontact metric manifolds was carried out by Zamkovoy [12]. However such structures were also studied by Buchner and Rosca. Rossca and Venhecke [13]. Further almost Para-Hermitian Structure on the tangent of an almost Para-Co hermitian manifolds was studied by Bejan [3]. A class of α-para kenmotsu manifolds was studied by srivastva and srivastva [14]. We can observe that the concircular curvature tensor on Pseudo-Riemannian manifold to be of constant curvature. Hayden introduced Semi-symmetric linear connection on a Riemannian manifold [8]. Let M be an n-dimensional Riemannian manifold of class C-endowed with the Riemannian metric g and be the Levi-Civita Connection on M^n. A linear connection 2 defined on M^n is said to be semi symmetric if its torsion tensor T is of the form [15].
where ξ is a vector field and η is a 1-form defined by for all vector field X ∈ χ(M n ) where, χ(M n ) is the set of all differentiable vector fields on M n . A relation between the semisymmetric metric connection and the Levi-Civita connection 2 on M n has been obtained by Yano which is given as [16] (1.1)

Preliminaries
A differentiable manifold M n of dimension n is said to have an almost paracontact (ϕ,ξ,η)-structure if it admits an (1,1) tensor field ϕ,a unique vector field ξ,1-form η such that : for any vector field X,Y on M n .The manifold M n equipped with an almost paracontact structure (ϕ,ξ,η) is called almost paracontact manifold . In addition,if an almost paracontact manifold admits a pseudo-Riemannian metric satisfying for any vector field X,Y on M n ,where ϕ is a (1,1) tensor field, ξ is a vector field,η is a 1-form and g is the Riemannian metric.Then M is called an almost contact manifold.For an almost contact manifold M,it follows that [8] (2.6) Let R be Riemann curvature tensor,S Ricci curvature tensor,Q Ricci operator we have and (2.10) for any vector field X,Y on M n ,then (ϕ,ξ,η,g),is called an almost paracontact metric structure and the manifold M equipped with an almost paracontact metric structure is called an almost paracontact metric manifold.Further in addition, if the structure (ϕ,ξ,η,g) satisfies (2.11) for any vector fields X,Y on M n .Then the manifold is called paracontact metric manifold and the corresponding structure (ϕ,ξ,η,g) , is called a paracontact structure with the associated metric g [17]. On an almost paracontact metric manifold,the (1,2) tensor field N ϕ defined as (2.12) Where [ϕ,ϕ] is the nijenhuis tensor of ϕ.If N vanishes identically,then we say that the manifold M n is a normal almost parametric metric manifold. The normality condition implies that the almost paracomplex structure J defined on Mn×R is integrable . Here X is tangent to M n , t is the coordinate of R and λ is a differentiable function on M n ×R.

On 3-Dimensional α-para Kenmotsu Manifold with Semi-Symmetric Metric Connection
In 3-dimensional α-para Kenmotsu manifold, the Ricci tensor S of Levi-Civita connection 2 is given by Let M 3 (ϕ,ξ,η,g) be an α-para Kenmotsu manifold [13],then we have Let M 3 be a 3-dimensional α-para Kenmotsu manifold.The curvature tensor R of M3 with respect to the semi-symmetric metric connection 2 is defined by (3.10) with the help of (3.7) and (3.9), we get

Ricci Solitons in α-para Kenmotsu Manifold with Semi-Symmetric Metric Connection
Let M be a 3-dimensional α-para Kenmotsu manifold with the semi-symmetric metric connection and V be pointwise collinear with ξ (i.e.V =bξ , where b is a function ).Then implies (4.1) using (3.9) in (4.1) , we get Since dη ≠ 0 from, we have (4.7) By using (4.5) and (4.7), we obtain that b is constant. Hence from (4.2) it is verified (4.8) which implies that M is an η-Einstein manifold. This leads to the following Theorem 4.1 If in a 3-dimensional α-para Kenmotsu manifold with the semi symmetric metric connection, the metric g is a Ricci soliton and V is a pointwise collinear with ξ ,then V is a constant multiple of ξ and g is an η-Einstein manifold of the form (4.8) and Ricci soliton is steady and expanding according as λ = 2α(1+α ) is zero and positive , respectively .

Conharmonically Flat α-para Kenmotsu Manifolds with the Semi-Symmetric Metric Connection
We have studied conharmonically flat α-para Kenmotsu manifolds with respect to the semi-symmetric metric connection. In a α-para Kenmotsu manifold the conharmonic curvature tensor with respect to the semi-symmetric metric connection is given by Then using linearity of ϕ and g we have for any Z , W ∈ χ(M) .Now , by direct computations we obtain by using these above equations we get [1] (6.1)

Conclusion
If in a 3-dimensional α-para Kenmotsu manifold with the semisymmetric metric connection , the metric g is a Ricci soliton and In this study , we gave some curvature conditions for 3-dimensional α-para Kenmotsu manifolds with semi-symmetric metric connection.In 3-dimensional α-para Kenmotsu manifolds with semi-symmetric metric connection is also an η-Einstein manifold and Ricci soliton defined steady or expanding on this manifold is named with respect to values of α and λ constant.We also proved that conharmonically flat α-para Kenmotsu manifolds with semi-symmetric metric connection is an η-Einstein manifold.