ON α-KENMOTSU MANIFOLDS SATISFYING FLATNESS CONDITIONS

ABSTRACT The main interest of the present paper is to study α-Kenmotsu manifolds that satisfy some certain tensor conditions where α is a smooth function defined by dα⋀η=0 on M n . In particular, the flatness conditions of α-Kenmotsu manifolds are investigated. We conclude the paper with an example on α-Kenmotsu manifolds depending on α. Indexing terms/


INTRODUCTION
Tanno classified connected almost contact metric manifolds whose automorphism groups possess the maximum dimension (see [16]). The sectional curvature of plane sections for such manifolds containing the vector f i e l d ξ is a constant which is called c. The author proved that these manifolds can be divided into three classes: (1) Homogenous normal contact Riemannian manlfolds with c > 0, (2) Global Riemannian products of a line or a circle with Kaehlerian manifold whose constant holomorphic sectional curvature under the condition c = 0, (3) A warped product space R×fC for the case c < 0.
It is well known that the class (1) are characterized by admitting a Sasakian structure. Kenmotsu defined a structure closely related to the warped product which is characterized the differential geometric properties of the manifolds of class (3). The structure is known as Kenmotsu structure and in general, these structures are not Sasakian (see [8]).
It is well known that Kenmotsu manifolds can be characterized through their Levi-Civita connection. He proved that such a manifold M 2n+1 is locally a warped product (-ε,+ε)xfN 2n being a Kaehlerian manifold and f (t) = ce t where c is a positive constant (see [8]).
Weakly symmetric and weakly Ricci-symmetric Riemannian manifolds are generalized locally symmetric manifolds and pseudo symmetric manifold, respectively. These are manifolds in which the covariants derivative ∇ R of the curvature tensor R is a linear expression in R. The appearing coefficients of this expression are called associated 1-forms. They satisfy in the specified types of manifolds gradually weaker conditions.
The notion of special weakly Ricci symmetric manifolds was introduced andstudied by H. Singh, and Q. Khan in 2001(see [13]). The authors considered special weakly symmetric manifolds. Next, Q. Khan studied some geometric properties of conharmonic Sasakian manifolds in 2004 and he also obtained some results on special weakly Ricci-symmetric Sasakian manifolds (see [10]). This paper is devoted to obtain some results on α-Kenmotsu manifolds by choosing a real value-function α instead of any real number α (constant function) with the help of some certain curvature tensor fields. For this reason, we have an α-Kenmotsu structure if there exists a normal almost contact metric structure (φ,ξ,η,g) such that dη=0 and dΦ=2α(η∧Φ) for any vector fields X,Y on Mⁿ, where α is a smooth function defined by dα∧η=0 on Mⁿ .
In this paper, the flatness conditions of α-Kenmotsu manifolds are investigated where α is a smooth function defined by dα∧η=0 on Mⁿ. In particular, we consider φ-conformally flat, φ-conharmonically flat and φ-projectively flat α-cosymplectic manifolds. We prove main results on these manifolds by using the class (3). Moreover, special weakly Ricci-symmetric α-cosymplectic manifolds are examined. Finally, we give an example on α-Kenmotsu manifolds.

PRELIMINARIES
Let (M n ,g) be an n-dimensional Riemannian manifold. We denote by ∇ the covariant differentiation with respect to the Riemannian metric g. Then we have Let (M n ,g), n=dimM , n > 3, be a connected Riemannian manifold of class C ∞ and ∇ be its Riemannian connection. The Weyl conformal curvature tensor C, the conharmonic curvature tensor K and the projective curvature tensor P of (M n ,g) are defined by respectively, where Q is the Ricci operator, S is the Ricci tensor, r=trace(S) is the scalar curvature and X,Y,Z∈χ(Mⁿ), χ(Mⁿ) being the Lie algebra of vector fields of Mⁿ (see [6,17]). It may be natural to consider the following particular cases: (1) C: TP(Mⁿ) x TP(Mⁿ) x TP(Mⁿ)→L(ξp), i.e., the projection of the image of C in φ(TP(Mⁿ) is zero, the projection of the image of in φ(TP(Mⁿ) is zero (see [4]). This condition is equivalent to (2.4) φ²C(φX,φY)φZ=0.
In [12], the author proves that an n-dimensional (n>3) conformally flat Lorentzian para-Sasakian manifold is an η-Einstein manifold, conharmonically flat Lorentzian para-Sasakian manifold is an Einstein manifold with zero scalar curvature. Also the author showed that a projectively flat Lorentzian para-Sasakian manifold is an Einstein manifold with scalar curvature r=n(n-1). D e c e m b e r 29, 2 0 1 5

α-KENMOTSU MANIFOLDS
Let Mⁿ be an n-dimensional differentiable manifold equipped with a triple (φ,ξ,η), where φ is a (1,1)tensor field, ξ is a vector field, η is a 1-form on Mⁿ such that If Mⁿ admits a Riemannian metric g, such that then Mⁿ is said to admit almost contact structure (φ,ξ,η,g) On such a manifold, the fundamental Φ of Mⁿ is defined by for X,Y∈Γ(TM).
An almost contact metric manifold (M,φ,ξ,η,g) is said to be almost cosymplectic if dη=0 and dΦ=0, where d is the exterior differential operator. The products of almost Kaehlerian manifolds and the real line R or the S¹ circle are the simplest examples of almost cosymplectic manifolds. An almost contact manifold (M,φ,ξ,η) is said to be normal if the Nijenhuis torsion vanishes for any vector fields X and Y.
An almost contact metric manifold Mⁿ is said to be almost α-Kenmotsu if dη=0 and dΦ=2αη∧Φ, α being a non-zero real constant. It is worthwhile to note that almost α-Kenmotsu structures are related to some special local conformal deformations of almost cosymplectic structures.
Moreover, an α-Kenmotsu manifold satisfies the following relations where λ is constant and it is called η-Einstein if its Ricci tensor S is of the form for any vector fields X and Y where λ₁ and λ₂ are functions on Mⁿ (see [3,17]).

BASIC CURVATURE PROPERTIES
By using Riemannian curvature tensor properties, the following relations are obtained on α-Kenmotsu manifolds:  for any vector fields X,Y on Mⁿ where α is a smooth function such that dα∧η=0. In these formulas, ∇ is the Levi-Civita connection, R the Riemannian curvature tensor and S is the Ricci tensor of Mⁿ.

CERTAIN TENSOR FIELDS ON α-KENMOTSU MANIFOLDS
In this section, we consider conharmonically flat and special weakly Ricci-symmetric manifold. Thus we give the following results:
Consequently, the manifold Mⁿ has a constant curvature satisfying α² is constant. As a special case, if we choose α=1, we obtain that a conharmonically α-Kenmotsu manifold is locally isometric with a unit sphere which is proved in [10].
Definition 5.1 An n-dimensional Riemannian manifold (Mⁿ,g) is called a special weakly Riccisymmetric manifold if where ζ is a 1-form and is defined by where ρ is the associated vector field.
Then we can give the following results: If a special weakly Ricci-symmetric α-Kenmotsu manifold admits a cyclic Ricci tensor and α is parallel along ξ, then the 1-form ζ must be vanished.

EXAMPLE IN THREE DIMENSIONAL CASE
Let us denote the standart coordinates of R³(x,y,z) and consider 3-dimensional manifold M⊂R³ defined by M={(x,y,z)∈R³ : z≠0}.