η- ℝicci solitons in α- para kenmotsu manifolds

The moto of the paper is to examine “η-Ricci solitons” in “α-para Kenmotsu manifolds” with second order parallel tensor. Also study “η-Ricci solitons” are considered manifolds fulfilling some particular curvature conditions: S(ζ, X 1) · R = 0, R · S = 0. Further we show that Pseudo quasi conformally flat is an Einstein with scalar curvature is constant.


Introduction
In 1972 Kenmotsu have introduced Kenmotsu manifolds [12]. and "almost Kenmotsu manifolds" to investigated in many angles.Maximum outcomes carried in [9], [10] would be simply generalized to the "class of almost α-Kenmotsu manifolds" [8]. The properties of "α-para Kenmotsu manifolds" have studied by Srivastava and Srivastava [20]. In the examination of Ricci flow, "Ricci Solitons" are the major things, since they are self-similar solutions of the flow. A "η-Ricci solitons" is a triple (g, ϑ, λ) with g a "Riemannian metric", ϑ a "vector field" generated by ϕ t ∈ R and "λ a real scalar" such that "L V g + 2S + 2λg = 0". "Ricci soliton" is a generalization of Einstein metric. Where S is a Ricci tensor of M ,L V indicates the "Lie derivative" operator with the "vector field" V . The "Ricci soliton" is noted "shrinking, steady and expanding when λ is negative, zero and positive" respectively. Motivated by the above results we studied, in section 3 the η-Ricci Solitons in (α − pkm) 3 , in the section 4 we discuss η-Ricci Solitons in (α − pkm) 3 Satisfying S(ζ, B 1 ) · R = 0. Section 5 is devoted to the study of η-Ricci Solitons on (α − pkm) 3 satisfying R · S = 0. In section 6, we establish η-Ricci Solitons in Pseudo Quasi Conformally flat (α − pkm) 3 . In section 7, we discuss the η-Ricci Solitons in Partially Ricci Pseudo Symmetric (α − pkm) 3 .

η-Ricci
Solitons in (α − pkm) 3 Let (M, ϕ, ζ, η, g) be an "almost paracontact metric manifold",By examine the equation where L ζ is the "Lie derivative operator" with the "vector field" ζ, S is the Ricci curvature tensor field of the metric g, "λ and µ" are real constants.Creating L ζ g in terms of the "Levi-Civita connection D", we get: for any B 1 ,B 2 ∈ χ(M ).
The content (g, ζ, λ, µ) it delivers the equation (15) is called as "η-Ricci solitons" on M [5]; in some specific, it is labeled "Ricci soliton" when µ = 0 and it is noted to be a shrinking, steady or expanding based on λ is negative, zero or positive,respectively [6].
Definition 1 A tensor ϑ of second order is said to be a parallel tensor if Dϑ = 0, where D indicates the operator of covariant differentiation with respect to the metric tensor g. Let ϑ be a (0, 2)-symmetric tensor field on a (α − pkm) 3 M ,such that Dϑ = 0. Using Ricci identity [18]we get for arbitrary vector field Since ϑ is symmetric. By using the expression (8) for (α−pkm) 3 and (11) in the above equation, we get Definition 2 If (α 2 = 0) then M 2n+1 (ζ) its known as regular .
In the sense of getting a characterisation of such manifolds we consider: Definition 3 [16] ζ is known as semi-torse forming "vector field for (M, g)", for all "vector fields" B 1 : and therefore, if B 1 η = ζ ⊥ ,then R(B 1 , ζ)ζ = −α 2 B 1 and we obtain: ,ζ is non-degenetate with respect to S, iv) Q(ζ) = 0 i.e.,ζ does not belong to the kernel of Q. In particular, if ζ is parallel (Dζ = 0) then M is not regular.
Regards to the above we restrict to the regular case. Returning to (21), with B 4 = ζ then we obtain: By differentiating (24) covariantly along B 2 , we get From (25) and (26), we get Replace B 1 by ϕB 2 in (24), we have Again replace B 2 by ϕB 2 in (29) and using (2) and (24),we get By differentiating (30) invariantly along any "vector field on M ", it can be easily seen that ϑ(ζ, ζ) is constant when α 3 = 0. Hence we can state the following theorem: Theorem 1 Let M be a (α − pkm) 3 with non vanishing ζ sectional curvature and admit with a tensor field ϑ which is symmetric. If ϑ is parallel with respect to D then it is a constant multiple of metric tensor g when α 3 = 0.
Proposition 2 Under this hypothesis if α is positive or negative then Ricci soliton is expanding.
6. η-Ricci Solitons in Pseudo Quasi Conformally flat (α − pkm) 3 Definition 4 The Pseudo quasi conformal curvature tensor [17] L on a (α − pkm) 3 is defined by for all vector fields B 1 , B 2 , B 3 where p, q, d are arbitrary constants not simultaneously zero, S is the "Ricci tensor", Q is the Ricci operator, R is the Reimannian curvature tensor and r is the scalar curvaur tensor of the manifold M .
We consider (α − pkm) 3 M which is Pseudo-Quasi conformally flat.Then from definition (2) and (43) we have Contracting with B 1 and B 2 in (44), we obtain Hence we get Lemma 3 A pseudo quasi conformally flat(α − pkm) 3 is an Einstein with contant scalar curvature.
In view of (17) and (46), we state that hold on the set A = {B ∈ M : Q(g, S) = 0 at B}. where j ∈ C ∞ (M ) for p ∈ A. R · S, Q(g, S) and (B 1 ∧ g B 2 ) are respectively defined as for all B 1 , B 2 , U and V ∈ T M n .