Some classifications on Kenmotsu manifolds

In this paper, we investigate some curvature problems of   Kenmotsu manifolds satisfying some certain conditions and we reach some classicifications. We consider  -recurrent   Kenmotsu manifolds and we show that  -recurrent   Kenmotsu manifolds are also  -Einstein manifolds. Next, we study  -Ricci symmetric   Kenmotsu manifolds and we find this manifolds are Einstein manifolds too. In addition, we examine locally  -symmetric  -Kenmotsu manifolds. Later we investigate this type manifold with quasi-conformally curvature tensor and concircular curvature tensor. In addition to these, we construct an example of   Kenmotsu manifolds and we see that this example is a locally   symmetric   Kenmotsu manifold.

A. Oubina's sense (Oubina, 1985).Öztürk et al. (2010) study about   Kenmotsu manifolds satisfying some curvature conditions.Dileo (2011)  We generally have interest on conditions about curvature tensor, because curvature tensors play important role in geometry and physics.For example; concircular transformation transforms every geodesic circle of a Riemannian manifold M into a geodesic circle.An interesting invariant of a concircular transformation is the concircular curvature tensor (Yano, 1940).In this paper, proportional to the metric.They are named after Albert Einstein because this condition is equivalent to saying that the metric is a solution of the vacuum Einstein field equations (Besse, 1987).Next, we deal with locally   symmetric  Kenmotsu manifolds and we prove some theorems about the scalar curvature of the manifolds.In addition to these, we consider quasiconformally flat condition on this type manifolds.We find interesting results when we investigate concircularly flat condition on locally   symmetric  -Kenmotsu manifolds.

MATERIALS AND METHODS
Let (M; g) be an (2n + 1)-dimensional Riemannian manifold.We denote by  the covariant differentiation with respect to the Riemannian metric g.The Ricci tensor of M are defined by  n e e e is a locally orthonormal frame and X, Y are vector fields on M. The Ricci operator Q is a tensor field of type (1,1) on M defined by for all vector fields on TM.
where I denotes the identity transformation of the tangent space , is an almost contact metric manifold.Dogan and Karadag 333 Let   is the fundamental 2-form of M. M is called almost  -Kenmotsu manifold, if the 1-form  and the 2form  satisfy the following conditions:  being a non-zero real constant (Janssens and Vanhecke, 1981).We have known that an almost contact metric manifold . Remarking that a normal almost  -Kenmotsu manifold is said to be  -Kenmotsu manifold   0   (Janssens and Vanhecke, 1981).Moreover, if the manifold M satisfies the following relations and (Pitiş, 2007).A Riemannian manifold (M,g) is called a   recurrent Riemannian manifold, if the curvature tensor R satisfies the following condition: where A is 1-form (De et al., 2009;Yıldız et al., 2009).
A Riemannian manifold (M,g) is called Ricci tensor S satisfies the following condition: for all vector fields X and Y in TM (Shukla and Shukla, 2009).A Riemannian manifold M is said to be locally for all vector fields X,Y,Z,W orthogonal to  .This notion was introduced by Takahashi (Binh et al., 2002), for a Sasakian manifold.
A Riemannian manifold (M,g) , where r is the scalar curvature of (M,g).
, where r is the scalar curvature of (M,g).On an   Kenmotsu manifold M, the following relations are held (Janssens and Vanhecke, 1981):  -RECURRENT   KENMOTSU MANIFOLDS Here, we find that a (Dogan, 2014).
In this case; Riemannian curvature tensor of M satisfy the following equation for all X,Y,Z and W in TM: in TM.If we take the inner product of Equation ( 19) with for all X,Y,Z,W,U in TM.Then the sum for From Equations ( 9), ( 14) and ( 16), we get From Equations ( 5) and ( 17), we have Here, we find that a   Ricci symmetric   Kenmotsu manifold is an Einstein manifold.
In this case; Ricci operator of M satisfy the following condition: From this last equation, we have for all vector fields X,Y in TM.If we take the inner product of Equation ( 27) with and we continue the process, we get for all X,Y in TM.From Equations ( 2) and ( 29) for all X in TM.In this case, we have where X,Y,Z and W are orthogonal to  .If we continue the process, we obtain for all X,W,Z orthogonal to  .Then the sum for for all X,W,Z orthogonal to  .So, using Equations ( 9) and ( 15), we obtain, If we continue the process, we get   and we take the sum for for all vector fields W in TM.Then, the proof is complete.

Theorem
for all X,Y,Z in TM.If we write  instead of X and Z and later we take the inner product of Equation ( 33) with If we use Lemma 1 and we consider locally ).In this case, M is Einstein Manifold.

Theorem
Let M be a locally If we consider Lemma 1 and the Equation ( 35), then we complete the proof.

Example
, where   Suppose that  is Levi-Civita connection with respect to the metric g.For all for all vector fields X,Y in TM.Hence  

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Corresponding author.E-mail: saadetdoganmat@gmail.com Author(s) agree that this article remain permanently open access under the terms of the Creative Commons Attribution License 4.0 International License MSC 2010: 53D10, 53C25, 53C21

C
vector fields on M.An almost contact structure on M is defined by (1,1) tensor field  , a vector field  and a 1-form M with metric tensor g and with a triple   M is concircularly flat then M has got constant curvature and its curvature is at each point of M. Let g be Riemannian metric defined by fields X in TM and use the linearity of  and g , then we find TM (Where a,b,c,


Kenmotsu manifold.With the help of above results we can find the following: Einstein manifold.Now, we take X,Y,Z and W orthogonal to  .Then we can write


fields X,Y,Z and W orthogonal to  .In this case, this manifold is a locally Kenmotsu manifold is constant.Actually, if we compute scalar curvature for all vector fields X,Y orthogonal to  , we see that Then the proof is complete.