On Generalized 𝚽 − Recurrent of Kenmotsu Type Manifolds

: The present paper studies the generalized Φ − recurrent of Kenmotsu type manifolds. This is done to determine the components of the covariant derivative of the Riemannian curvature tensor. Moreover, the conditions which make Kenmotsu type manifolds to be locally symmetric or generalized Φ − recurrent have been established. It is also concluded that the locally symmetric of Kenmotsu type manifolds are generalized Φ − recurrent under suitable condition and vice versa. Furthermore, the study establishes the relationship between the Einstein manifolds and locally symmetric of Kenmotsu type manifolds.


Introduction:
The locally Φ −symmetric property studied by Takahashi 1 for Sasakian manifolds is a weak version of the locally symmetric property. In 1993, Jiménez and Kowalski 2 classified the locally Φ −symmetric Sasakian manifolds. Conversely, the recurrent property studied by Walker 3 is also a weak version of the locally symmetric property, one of its generalization is Φ −recurrent property that was studied by Venkatesha 4 for the generalized Sasakian space forms. There are other generalizations and extensions of recurrent curvature studied by researchers like Tamássy and Binh 5 , Venkatesha et al. 6 and Siddiqi et al. 7 .

Preliminaries:
The notations 2 +1 , ( ), , ∇ and were used to denote the smooth manifold of dimension 2 + 1, the Lie algebra of smooth vector fields of , the exterior differentiation operator, the Riemannian connection and the Riemannian connection form with components respectively, where , = 0,1, . . . ,2 . Definition 1 8 A smooth manifold 2 +1 with the quadruple (Φ, , , ) is called an almost contact metric manifold or briefly ACR −manifold, where Φ is (1,1) −tensor, is a characteristic vector field, is a Riemannian metric and (⋅) = (⋅, ), such that the following conditions hold: ∈ ( ). In the present article, the components of the Riemannian metric of −manifold 2 +1 can be established as follows 8 : where , = 1,2, . . . , , and ̂= + . Moreover, from 3 the components of the endomorphism Φ are given by where ̂̂= and 0 = 0. So, for all , ∈ ( ), the following relations are attained: ( + √−1Φ) , and { 1 , . . . , } is a complex basis of the distribution ker( ). A set of all such −frames given above is called an associated −structure space ( −structure space). For more detail, citations 8 and 9 may be referred to.  10 An ACR −manifold such that the following identity: Suppose that { , 1 , . . . , 2 } are coframe over −structure space of Kenmotsu type manifold such that ̂= = , where is the complex conjugate of and = . Then from 10 , the following theorem is obvious: Theorem 1 Suppose that 2 +1 is the Kenmotsu type manifold, then the Cartan's first structure equations are given by 1.  10 On the AG −structure space, the Kenmotsu type manifold 2 +1 has the following Cartan's second structure equations: where all indexes have range from 1 to , and [⋅ | ⋅ | ⋅] denotes the anti-symmetric operator of the involving indexes except | ⋅ |. Denote and the Riemann curvature tensor with components and Ricci tensor with components of the −manifold respectively, where , = 0,1, . . . ,2 . Theorem 3 10 The components of for the Kenmotsu type manifold over the AG − structure space are given by ( , ) = , and the remaining components of are given by the first Bianchi identity or the conjugate (i.e.

=̂̂̂)
to the above components or identical to zero. Theorem 4 10 The components of for the Kenmotsu type manifold over the AG − structure space are given as follows:

Locally symmetric Kenmotsu type manifolds and its weakened version:
In this section, the Cartan's second structure equations in Theorem 2 are differentiated exteriorly at the beginning. Then on −structure space of the Kenmotsu type manifold 2 +1 , suitable smooth functions exist such that: Now, the components of ∇ on −manifold 2 +1 can be established with respect to the metric from the following identity (12)

Theorem 5
The components of ∇ on AG −structure space of the Kenmotsu type manifold 2 +1 are given by 1.
, ; Proof. The results follow from the equation (6)    Regarding Theorem 5, from item 2 in the above discussion, = is obtained, implying that the 1-forms and must be equal. Moreover, combining the above items again with Theorem 5 leads to deducing the following theorem: