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abstract: 'In 1972, K. Kenmotsu studied a class of almost contact Riemannian manifolds. Later, such a manifold was called a Kenmotsu manifold. This paper, we studied Kenmotsu manifolds with $(2n+s)$-dimensional $s-$contact metric manifold and this manifold, we have called generalized Kenmotsu manifolds. Necessary and sufficient condition is given for an almost $s-$contact metric manifold to be a generalized Kenmotsu manifold.We show that a generalized Kenmotsu manifold is a locally warped product space. In addition, we study some curvature properties of generalized Kenmotsu manifolds. Moreover, we show that the $\varphi $-sectional curvature of any semi-symmetric and projective semi-symmetric $(2n+s)$-dimensional generalized Kenmotsu manifold is $-s$.'
author:
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    [ ]{}\
    <span style="font-variant:small-caps;">Aysel TURGUT VANLI and Ramazan SARI</span>
---

[^1] [^2] [^3]

Introduction {#introduction .unnumbered}
============

In *[@Y],* K.Yano introduced the notion of a $f-$structure on a differentiable manifold $M$, i.e., a tensor fields $f$ of type $(1,1)$ and $rank$ $2n$ satisfying $f^{3}+f=0$ as a generalization of both (almost) contact (for $s=1$) and (almost) complex structures (for $s=0$). $TM$ splits into two complementary subbundles $\mathcal{L=}$ $Im\varphi $ and $\mathcal{M=}$ $ker\varphi $. The existence of which is equivalent to a reduction of the structural group of the tangent bundle to $\mathit{U(n)\times O(s)}$ *[@BL]*. H. Nakagawa in *[@Nak]* and *[@Nak2]* introduced the notion of globally framed f-manifolds (*called f-manifolds*), later developed and studied by Goldberg and Yano *[@G],* *[@GY],* *[@GY2]*. A wide class of globally frame $f$-manifolds was introduced in *[@BL],* by Blair according to the following definition. A metric $f$-structure is said to be a $K$-structure if the fundamental 2-form $\Phi $, defined usually as $\Phi (X,Y)=g(X,\varphi Y)$, for any vector fields $X$ and $Y$ on $M$, is closed and the normality condition holds, that is; $[\varphi ,\varphi ]+2\sum_{i=1}^{s}d\eta ^{i}\otimes \xi _{i}=0$ where $[\varphi ,\varphi ]$ denotes the Nijenhuis torsion of $\varphi $. Some authors studeid $f$-structure *[@BL2],* *[@CFF],* *[@YK]*. The Riemannian connection $\nabla $ of a metric $f-$manifold satisfies the following formula *[@Bkitap],* $$\begin{aligned}
2g((\nabla _{X}\varphi )Y,Z) &=&3d\Phi (X,\varphi Y,\varphi Z) -3d\Phi(X,Y,Z)+g(N^{1}(Y,Z),\varphi X)  \notag \\
&&+\overset{s}{\underset{i=1}{\sum }}\{N^{2}(Y,Z)\eta ^{i}(X)+2d\eta
^{i}(\varphi Y,X)\eta ^{i}(Z)-2d\eta ^{i}(\varphi Z,X)\eta ^{i}(Y)\},\end{aligned}$$ where the tensor fields $N^{1}$ and $N^{2}$ are defined by $N^{1}=[\varphi
,\varphi ]+2\sum_{i=1}^{s}d\eta ^{i}\otimes \xi _{i},$ $N^{2}(X,Y)=(L_{_{\varphi X}}\eta ^{i})(Y)-(L_{_{\varphi Y}}\eta ^{i})(X)$ respectively, which is by a simple computation can be rewritten as: $N^{2}(X,Y)=2d\eta
^{i}(\varphi X,Y)-2d\eta ^{i}(\varphi Y,X).$

Let $M$ be a $(2n+1)$ dimensional differentiable manifold. $M$ is called an *almost contact metric manifold* if $\varphi $ is $(1,1)$ type tensor field, $\xi $ is vector field, $\eta $ is $1-$ form and $g$ is a compatible Riemannian metric such that $$\varphi ^{2}=-I+\eta \otimes \xi ,\begin{array}{cc}
&
\end{array}\eta (\xi )=1$$$$g(\varphi X,\varphi Y)=g(X,Y)-\eta (X)\eta (Y)$$for all $X,Y\in \Gamma (TM).$

In addition, we have$$\eta (X)=g(X,\xi ),\begin{array}{cc}
&
\end{array}\varphi (\xi )=0,\begin{array}{cc}
&
\end{array}\eta (\varphi )=0$$for all $X,Y\in \Gamma (TM)$ [@Bkitap].

To study manifolds with negative curvature, Bishop and O’Neill introduced the notion of warped product as a generalization of Riemannian product *[@Bis]*. In 1960’s and 1970’s, when almost contact manifolds were studied as an odd dimensional counterpart of almost complex manifolds, the warped product was used to make examples of almost contact manifolds *[@Tan]*. In addition, S. Tanno classified the connected $(2n+1)$ dimensional almost contact manifold $M$ whose automorphism group has maximum dimension $(n+1)^{2}$ in *[@Tan]*$. $ For such a manifold, the sectional curvature of plane sections containing $\xi $ is a constant, say $c$. Then there are three classes.

$i)$ $c>0$, M is homogeneous Sasakian manifold of constant holomorphic sectional curvature.

$ii)$ $c=0$, M is the global Riemannian product of a line or a circle with a K$\ddot{a}$hler manifold of constant holomorphic sectional curvature.

$iii)$ $c<0$, M is warped product space $\mathbb{R}
\times _{f}\mathbb{C}
^{n}.$ Kenmotsu obtained some tensorial equations to characterize manifolds of the third case.

In 1972 , Kenmotsu abstracted the differential geometric properties of the third case. In *[@K],* Kenmotsu studied a class of almost contact Riemannian manifold which satisfy the following two condition,$$\begin{aligned}
(\nabla _{X}\varphi )Y &=&-\eta (Y)\varphi X-g(X,\varphi Y)\xi \\
\nabla _{X}\xi &=&X-\eta (X)\xi  \notag\end{aligned}$$

He showed normal an almost contact Riemannian manifold with $(5)$ but not quasi Sasakian hence not Sasakian. He was to characterize warped product space $L\times _{f}\mathbb{C}
E^{n}$ by an almost contact Riemannian manifold with $(5)$. Moreover, he show that every point of an almost contact Riemannian manifold with $(5)$ has a neighborhood which is a warped $(-\epsilon ,\epsilon )\times _{f}V$ where $f(t)=ce^{t}$ and $V$ is Kähler.

In 1981 *[@Jan],* Janssens and Vanhecke, an almost contact metric manifold satisfiying this $(5)$ is called a Kenmotsu manifold. After this definition, some authors studied Kenmotsu manifold *[@DP], [@JDP], [@P], [@Pr].*

The paper is organized as follows: after a preliminary basic notions of $s-$contact metric manifolds theory, in Section 2, we introduced generalized almost Kenmotsu manifolds and generalized Kenmotsu manifolds. Necessary and sufficient condition is given for a $s-$contact metric manifold to be a generalized Kenmotsu manifold. The warped product $L^{s}\times _{f}V^{2n}$ provides an example. In section 3, some curvature properties are given. In section 4, we studied Ricci curvature tensor. In section 5, we studied semi-symmetric properties of generalized Kenmotsu manifolds. We show that the $\varphi $-sectional curvature of any semi-symmetric and projective semi-symmetric $(2n+s)$-dimensional generalized Kenmotsu manifold is $-s$.

Preliminaries
=============

In *[@GY],* a $(2n+s)-$dimensional differentiable manifold $M$ is called metric $f-$manifold if there exist an $(1,1)$ type tensor field $\varphi $, $s$ vector fields $\xi _{1},\dots ,\xi _{s}$, $s$ $1$-forms $\eta
^{1},\dots ,\eta ^{s}$ and a Riemannian metric $g$ on $M$ such that$$\varphi ^{2}=-I+\overset{s}{\underset{i=1}{\sum }}\eta ^{i}\otimes \xi _{i},\begin{array}{cc}
\begin{array}{c}
\end{array}
&
\end{array}\eta ^{i}(\xi _{j})=\delta _{ij}$$$$g(\varphi X,\varphi Y)=g(X,Y)-\overset{s}{\underset{i=1}{\sum }}\eta
^{i}(X)\eta ^{i}(Y),$$for any $X,Y\in \Gamma (TM),$ $i,j\in \{1,\dots ,s\}$. In addition, we have$$\eta ^{i}(X)=g(X,\xi _{i}),\begin{array}{cc}
\begin{array}{c}
\end{array}
&
\end{array}g(X,\varphi Y)=-g(\varphi X,Y),\begin{array}{cc}
&
\end{array}\varphi \xi _{i} =0,\begin{array}{cc}
&
\end{array}\eta ^{i}\circ \varphi =0.$$

Then, a $2$-form $\Phi $ is defined by $\Phi (X,Y)=g(X,\varphi Y)$, for any $X,Y\in \Gamma (TM)$, called the *fundamental* $\mathit{2}$*-form*.

In what follows, we denote by $\mathcal{M}$ the distribution spanned by the structure vector fields $\xi _{1},\dots ,\xi _{s}$ and by $\mathcal{L}$ its orthogonal complementary distribution. Then, $TM=\mathcal{L}\oplus \mathcal{M}$. If $X\in \mathcal{M}$ we have $\varphi X=0$ and if $X\in \mathcal{L}$ we have $\eta ^{i}(X)=0$, for any $i\in \{1,\dots ,s\}$; that is, $\varphi
^{2}X=-X$.

In a metric $f$-manifold, special local orthonormal basis of vector fields can be considered. Let $U$ be a coordinate neighborhood and $E_{1}$ a unit vector field on $U$ orthogonal to the structure vector fields. Then, from $(6)-(8)$, $\varphi E_{1}$ is also a unit vector field on $U$ orthogonal to $E_{1}$ and the structure vector fields. Next, if it is possible, let $E_{2}$ be a unit vector field on $U$ orthogonal to $E_{1}$, $\varphi E_{1}$ and the structure vector fields and so on. The local orthonormal basis $$\{E_{1},\dots ,E_{n},\varphi E_{1},\dots ,\varphi E_{n},\xi _{1},\dots ,\xi
_{s}\},$$so obtained is called an $f$*-basis*. Moreover, a metric $f$-manifold is *normal* if $$\lbrack \varphi ,\varphi ]+2\underset{i=1}{\overset{s}{\sum }}d\eta
^{i}\otimes \xi _{i}=0,$$where $[\varphi ,\varphi ]$ is denoting the Nijenhuis tensor field associated to $\varphi $.

In *[@Van],*  let $M$ a $(2n+s)-$dimensional metric $f-$manifold. If there exists $2$-form $\Phi $ such that $\eta ^{1}\wedge
...\wedge \eta ^{s}\wedge \Phi ^{n}\neq 0$ on $M$ then $M$ is called an *almost s-contact metric manifold.* A normal almost s-contact metric manifold is called an s-contact metric manifold.

Generalized Kenmotsu Manifolds
==============================

As is known in Kenmotsu manifold $dimker\varphi =1$, since $ker\varphi
=sp\{\xi \}$. It was to be $dimker\varphi >1$ open question. Firstly in 2003, L. Bhatt and K. K. Dube introduced* Kenmotsu* $s$*-structure*; that is, an almost $s-$contact metric manifold $M$ is called a Kenmotsu $s-$manifold if * * $$(\nabla _{X}\varphi )Y=g(\varphi X,Y)\overset{s}{\underset{i=1}{\sum }}\xi
_{i}-\varphi X\overset{s}{\underset{i=1}{\sum }}\eta ^{i}(Y)$$for any $X,Y\in \Gamma (TM)$ *[@Bd]*$.$ We will give their definition as a theorem in this paper.

Afterwards in 2006, M. Falcitelli and A.M. Pastore introduced *Kenmotsu f.pk-manifold*. In *[@FP]*, * *a metric $f.pk$-manifold $M$ of dimension $2n+s$, $s\geq 1$, with $f.pk-$structure which is a metrik $f-$structure with parallelizable kernel. $\left( \varphi ,\xi
_{i},\eta ^{i},g\right) $ is said to be a Kenmotsu $f.pk$-manifold if it is normal, the$1$-forms $\eta ^{i}$ are closed and $d\Phi =2\eta ^{1}\wedge
\Phi $ *.* They assume that $d\Phi =2\eta ^{i}\wedge \Phi $ for all $i=1,2,...,s$ in the definition of Kenmotsu $f.pk-$manifold. So, they remark that, since the $1$-forms $\eta ^{i}$ are linearly independent and $\eta
^{i} $ $\wedge \Phi =\eta ^{j}\wedge \Phi $ implies $\eta ^{i}=\eta ^{j}$, then the condition on $d\Phi $ can be satisfied by a unique $\eta ^{i}$ and they can assume that $d\Phi =2\eta ^{1}\wedge \Phi .$ It is clear that authors were equated 1-forms $\eta ^{1},...,\eta ^{s}$, which dual of $\xi
_{1},...,\xi _{s}.$Thus, they studied unique 1-form $\eta ^{1}.$

In this paper, all $\eta ^{1},...,\eta ^{s}$ $1-$forms are unequaled at the definition of generalized Kenmotsu manifolds.

\[Def\] Let $M$ be an almost $s-$contact metric manifold of dimension $(2n+s)$, $s\geq 1$, with $\left( \varphi ,\xi _{i},\eta ^{i},g\right) $ . $M$ is said to be a generalized almost Kenmotsu manifold if for all $1\leq i\leq s,$ $1-$forms $\eta ^{i}$ are closed and $d\Phi =2\underset{i=1}{\overset{s}{\sum }}\eta
^{i}\wedge \Phi .$ A normal generalized almost Kenmotsu manifold $M$ is called a generalized Kenmotsu manifold.

Now, we construct an example of generalized Kenmotsu manifold.

We consider $(2n+s)-$dimensional manifold $$M=\left\{ (x_{1},...,x_{n},y_{1},...,y_{n},z_{1,}...,z_{s})\in \mathbb{R}^{2n+s}:\sum_{\alpha =1}^{s}{z_{\alpha }}\neq 0\right\}$$We choose the vector fields$$X_{i}=e^{-\sum\limits_{\alpha =1}^{s}z_{\alpha }}\frac{\partial }{\partial
x_{i}},Y_{i}=e^{-\sum\limits_{\alpha =1}^{s}z_{\alpha }}\frac{\partial }{\partial
y_{i}},\xi _{\alpha }=\frac{\partial }{\partial z_{\alpha }},1\leq i\leq n,1\leq \alpha \leq s$$which are linearly indepent at each point of $M.$ Let $g$ be the Riemannian metric defined by$$g=e^{2\sum\limits_{\alpha =1}^{s}z_{\alpha }}\left[ \sum\limits_{i=1}^{n}\left( dx_{i}\otimes dx_{i}+dy_{i}\otimes dy_{i}\right) \right]
+\sum\limits_{\alpha =1}^{s}\eta ^{\alpha }\otimes \eta ^{\alpha }.$$Hence, $\left\{ X_{1},...,X_{n},Y_{1},...,Y_{n},\xi _{1},...,\xi
_{s}\right\} $ is an orthonormal basis. Thus, $\eta ^{\alpha }$ be the $1-$form defined by $\eta ^{\alpha }\left( X\right) =g(X,\xi _{\alpha }),\begin{array}{c}
\end{array}\alpha =1,...,s$ for any vector field $X$ on $TM.$ We defined the $(1,1)$ tensor field $\varphi $ as $$\varphi \left( X_{i}\right) =Y_{i},\varphi \left( Y_{i}\right) =-X_{i},\varphi \left( \xi _{\alpha }\right) =0,1\leq i\leq n,1\leq \alpha \leq s.$$The linearity property of $\varphi $ and $g$ yields that

$$\begin{aligned}
\eta ^{\alpha }\left( \xi _{\beta }\right) &=&\delta _{\alpha \beta },\varphi ^{2}X=-X+\sum\limits_{\alpha =1}^{s}\eta ^{\alpha }\left( X\right)
\xi _{\alpha }, \\
g\left( \varphi X,\varphi Y\right) &=&g\left( X,Y\right)
-\sum\limits_{\alpha =1}^{s}\eta ^{\alpha }\left( X\right) \eta ^{\alpha
}\left( Y\right) ,\end{aligned}$$

for any vector fields $X$, $Y$ on $M.$ Therefore,$(M,\varphi ,\xi _{\alpha
},\eta ^{\alpha },g)$ defines a metric $f-$manifold. We have $\Phi
(X_{i,},Y_{i})=-1$ and others are zero. Therefore, the essential non-zero component of $\Phi $ is$$\Phi (\frac{\partial }{\partial x_{i}},\frac{\partial }{\partial y_{i}})=g(\frac{\partial }{\partial x_{i}},\varphi \frac{\partial }{\partial y_{i}})=-e^{2\sum\limits_{\alpha =1}^{s}z_{\alpha }}$$and hence, we have $$\Phi =-e^{2\sum\limits_{\alpha =1}^{s}z_{\alpha }}\sum_{i=1}^{n}dx_{i}\wedge
dy_{i}$$Therefore, we get $\eta ^{1}\wedge ...\wedge \eta ^{s}\wedge \Phi ^{n}\neq 0$ on $M$. Thus $(M,\varphi ,\xi _{\alpha },\eta ^{\alpha },g)$ is almost $s-$contact manifold. Consequently, the exterior derivative $d\Phi $ is given by $$d\Phi =2\sum\limits_{\alpha =1}^{s}dz_{\alpha }\wedge
(-e^{2\sum\limits_{\alpha =1}^{s}z_{\alpha }})\sum_{i=1}^{n}dx_{i}\wedge
dy_{i}.$$Therefore, $(M,\varphi ,\xi _{\alpha },\eta ^{\alpha },g)$ is a generalized almost Kenmotsu manifold. It can be seen that $\ (M,\varphi ,\xi _{\alpha
},\eta ^{\alpha },g)$ is normal. So, it is a generalized Kenmotsu manifold. Moreover, we get $$\begin{aligned}
\left[ X_{i},\xi _{\alpha }\right] &=&X_{i},\begin{array}{cc}
&
\end{array}\left[ Y_{i},\xi _{\alpha }\right] =Y_{i}, \\
\left[ X_{i},X_{j}\right] &=&0,\begin{array}{cc}
&
\end{array}\left[ X_{i},Y_{i}\right] =0,\begin{array}{cc}
&
\end{array}\left[ X_{i},Y_{j}\right] =0 \\
\left[ Y_{i},Y_{j}\right] &=&0,\begin{array}{cc}
&
\end{array}1\leq i,j\leq n,1\leq {\alpha }\leq s.\end{aligned}$$The Riemannian connection $\nabla $ of the metric $g$ is given by $$\begin{aligned}
2g(\nabla _{X}Y,Z) &=&Xg(Y,Z)+Yg(Z,X)-Zg(X,Y) \\
&&+g(\left[ X,Y\right] ,Z)-g(\left[ Y,Z\right] ,X)+g(\left[ Z,X\right] ,Y).\end{aligned}$$Using the Koszul’s formula, we obtain $$\begin{aligned}
\nabla _{X_{i}}X_{i} &=&\sum_{\alpha =1}^{s}{\xi _{\alpha }},\begin{array}{c}
\end{array}\nabla _{Y_{i}}Y_{i}=\sum_{\alpha =1}^{s}{\xi _{\alpha }}, \\
\nabla _{X_{i}}X_{j} &=&\nabla _{Y_{i}}Y_{j}=\nabla _{X_{i}}Y_{i}=\nabla
_{X_{i}}Y_{j}=0 \\
\nabla _{X_{i}}\xi _{\alpha } &=&X_{i},\begin{array}{c}
\end{array}\nabla _{Y_{i}}\xi _{\alpha }=Y_{i},1\leq i,j\leq n,1\leq {\alpha }\leq s.\end{aligned}$$

We construct an example of generalized Kenmotsu manifold for $7-$dimensional.

Let $n=2$ and $s=3$. The vector fields$$e_{1}=f_{1}(z_{1},z_{2},z_{3})\frac{\partial }{\partial x_{1}}+f_{2}(z_{1},z_{2},z_{3})\frac{\partial }{\partial y_{1}},$$$$e_{2}=-f_{2}(z_{1},z_{2},z_{3})\frac{\partial }{\partial x_{1}}+f_{1}(z_{1},z_{2},z_{3})\frac{\partial }{\partial y_{1}},$$$$e_{3}=f_{1}(z_{1},z_{2},z_{3})\frac{\partial }{\partial x_{2}}+f_{2}(z_{1},z_{2},z_{3})\frac{\partial }{\partial y_{2}},$$$$e_{4}=-f_{2}(z_{1},z_{2},z_{3})\frac{\partial }{\partial x_{2}}+f_{1}(z_{1},z_{2},z_{3})\frac{\partial }{\partial y_{2}},$$$$e_{5}=\frac{\partial }{\partial z_{1}},e_{6}=\frac{\partial }{\partial z_{2}},e_{7}=\frac{\partial }{\partial z_{3}}$$where $f_{1}$ and $f_{2}$ are given by $$\begin{aligned}
f_{1}(z_{1},z_{2},z_{3}) &=&c_{2}e^{-(z_{1}+z_{2}+z_{3})}\cos
(z_{1}+z_{2}+z_{3})-c_{1}e^{-(z_{1}+z_{2}+z_{3})}\sin (z_{1}+z_{2}+z_{3}), \\
f_{2}(z_{1},z_{2},z_{3}) &=&c_{1}e^{-(z_{1}+z_{2}+z_{3})}\cos
(z_{1}+z_{2}+z_{3})+c_{2}e^{-(z_{1}+z_{2}+z_{3})}\sin (z_{1}+z_{2}+z_{3})\end{aligned}$$for nonzero constant $c_{1},c_{2}.$ It is obvious that $\left\{
e_{1},e_{2},e_{3},e_{4},e_{5},e_{6},e_{7}\right\} $ are linearly independent at each point of $M$. Let $g$ be the Riemannian metric given by $$g=\frac{1}{f_{1}^{2}+f_{2}^{2}}\sum_{i=1}^{2}(dx_{i}\otimes
dx_{i}+dy_{i}\otimes dy_{i})+dz_{1}\otimes dz_{1}+dz_{2}\otimes
dz_{2}+dz_{3}\otimes dz_{3},$$where $\left\{ x_{1},y_{1},x_{2},y_{2},z_{1},z_{2},z_{3}\right\} $ are standard coordinates in $\mathbb{R}
^{7}$. Let $\eta ^{1}$, $\eta ^{2}$ and $\eta ^{3}$ be the $1-$form defined by $\eta ^{1}(X)=g(X,e_{5})$, $\eta ^{2}(X)=g(X,e_{6})$ and $\eta
^{3}(X)=g(X,e_{7})$, respectively, for any vector field $X$ on $M$ and $\phi
$ be the $(1,1)$ tensor field defined by $$\begin{aligned}
\varphi (e_{1}) &=&e_{2},\begin{array}{cc}
\begin{array}{c}
\end{array}
&
\end{array}\varphi (e_{2})=-e_{1},\begin{array}{cc}
&
\end{array}\varphi (e_{3})=e_{4},\begin{array}{cc}
\begin{array}{c}
\end{array}
&
\end{array}\varphi (e_{4})=-e_{3}, \\
\varphi (e_{5} &=&\xi _{1})=0,\begin{array}{cc}
\begin{array}{c}
\end{array}
&
\end{array}\varphi (e_{6}=\xi _{2})=0,\begin{array}{cc}
\begin{array}{c}
\end{array}
&
\end{array}\varphi (e_{7}=\xi _{3})=0.\end{aligned}$$Therefore, the essential non-zero component of $\Phi $ is$$\Phi (\frac{\partial }{\partial x_{i}},\frac{\partial }{\partial y_{i}})=-\frac{1}{f_{1}^{2}+f_{2}^{2}}=-\frac{2e^{2(z_{1}+z_{2}+z_{3})}}{c_{1}^{2}+c_{2}^{2}},\begin{array}{cc}
&
\end{array}i=1,2$$and hence $$\Phi =-\frac{2e^{2(z_{1}+z_{2}+z_{3})}}{c_{1}^{2}+c_{2}^{2}}\sum_{i=1}^{2}dx_{i}\wedge dy_{i}.$$Thus, we have $\eta ^{1}\wedge ...\wedge \eta ^{s}\wedge \Phi ^{n}\neq 0$ on $M$. Consequently, the exterior derivative $d\Phi $ is given by $$d\Phi =-\frac{4e^{2(z_{1}+z_{2}+z_{3})}}{c_{1}^{2}+c_{2}^{2}}(dz_{1}+dz_{2}+dz_{3})\wedge \sum_{i=1}^{2}dx_{i}\wedge dy_{i}.$$Since $\eta ^{1}=dz_{1}$, $\eta ^{2}=dz_{2}$ and $\eta ^{3}=dz_{3},$ we find$$d\Phi =2(\eta ^{1}+\eta ^{2}+\eta ^{3})\wedge \Phi .$$In addition, Nijenhuis tersion of $\varphi $ is equal to zero.

Let $\left( M,\varphi ,\xi _{i},\eta ^{i},g\right)$ be an almost $s$-contact metric manifold. $M$ is a generalized Kenmotsu manifold if and only if$$\left( \nabla _{X}\varphi \right) Y=\underset{i=1}{\overset{s}{\sum }}\left\{ g(\varphi X,Y)\xi _{i}-\eta ^{i}(Y)\varphi X\right\}$$for all $X,Y\in \Gamma (TM),$ $i\in \left\{ 1,2,...,s\right\} ,$ where $\nabla $ is Riemannian connection on M.

Let $M$ be a generalized Kenmotsu manifold. From $(1)$, $(6),(7)$ and $(8)$  for all $X,Y\in \Gamma (TM),$ we have$$\begin{aligned}
g\left( \left( \nabla _{X}\varphi \right) Y,Z\right) &=&3\left\{ \underset{i=1}{\overset{s}{\sum }}(\eta ^{i}\wedge \Phi )(X,\varphi Y,\varphi Z)-\underset{i=1}{\overset{s}{\sum }}(\eta ^{i}\wedge \Phi )(X,Y,Z)\right\} \\
&=&\underset{i=1}{\overset{s}{3\sum }}\{\frac{1}{3}(-\eta ^{i}(X)\Phi
(\varphi Y,\varphi Z)+\eta ^{i}(\varphi Y)\Phi (\varphi Z,X)+\eta
^{i}(\varphi Z)\Phi (X,\varphi Y)) \\
&&-\frac{1}{3}(-\eta ^{i}(X)\Phi (Y,Z)+\eta ^{i}(Y)\Phi (Z,X)+\eta
^{i}(Z)\Phi (X,Y))\} \\
&=&\underset{i=1}{\overset{s}{\sum }}\left\{ -\eta ^{i}(X)g(\varphi
Y,\varphi ^{2}Z)+\eta ^{i}(X)g(Y,\varphi Z)-\eta ^{i}(Y)g(Z,\varphi X)-\eta
^{i}(Z)g(X,\varphi Y)\right\} \\
&=&\underset{i=1}{\overset{s}{\sum }}\left\{ -\eta ^{i}(Y)g(Z,\varphi
X)-\eta ^{i}(Z)g(X,\varphi Y)\right\} \\
&=&g(\underset{i=1}{\overset{s}{\sum }}\left\{ g(\varphi X,Y)\xi _{i}-\eta
^{i}(Y)\varphi X\right\} ,Z).\end{aligned}$$

Conversely, firstly, using $(9)$ and $(8)$, we get$$\varphi \nabla _{X}\xi _{j}=\underset{i=1}{\overset{s}{-\sum }}\left\{
g(\varphi X,\xi _{j})\xi _{i}-\eta ^{i}(\xi _{j})\varphi X\right\}$$hence, we get$$\varphi ^{2}\nabla _{X}\xi _{j}=\varphi ^{2}X.$$Therefore, we have$$\nabla _{X}\xi _{j}=-\varphi ^{2}X.$$On the other hand, we get $$d\eta ^{i}(X,Y)=\frac{1}{2}\{g(Y,-\varphi ^{2}X)-g(X,-\varphi ^{2}Y)\}=0$$for all $X,Y\in \Gamma (TM),$ $i\in \left\{ 1,2,...,s\right\} .$ In addition, we know$$\begin{aligned}
3d\Phi (X,Y,Z) &=&Xg(Y,\varphi Z)-Yg(X,\varphi Z)-Zg(X,\varphi
Y)-g([X,Y],\varphi Z)\\&&+g([X,Z],\varphi Y)-g([Y,Z],\varphi X) \\
&=&g(Y,\nabla _{X}\varphi Z-\varphi \nabla _{X}Z)-g(X,\nabla _{Y}\varphi
Z-\varphi \nabla _{Y}Z)+g(X,\nabla _{Z}\varphi Y-\varphi \nabla _{Z}Y).\end{aligned}$$From hypothesis, we have$$\begin{aligned}
3d\Phi (X,Y,Z) &=&\overset{s}{\underset{i=1}{\sum }}\{g(\varphi X,Z)g(Y,\xi
_{i})-\eta ^{i}(Z)g(Y,\varphi X)-g(\varphi Y,Z)g(X,\xi _{i})+\eta
^{i}(Z)g(X,\varphi Y) \\
&&+g(\varphi Z,Y)g(X,\xi _{i})-\eta ^{i}(Y)g(X,\varphi Z)\} \\
&=&2\overset{s}{\underset{i=1}{\sum }}\{\Phi (Z,X)\eta ^{i}(Y)+\Phi
(X,Y)\eta ^{i}(Z)+\Phi (Y,Z)\eta ^{i}(X)\}.\end{aligned}$$Then, we have,$$d\Phi =2\underset{i=1}{\overset{s}{\sum }}\eta ^{i}\wedge \Phi .$$Moreover, the Nijenhuis torsion of $\varphi $ is obtained$$\begin{aligned}
N_{\varphi }(X,Y) &=&\varphi \left( -\overset{s}{\underset{i=1}{\sum }}\{g(\varphi X,Y)\xi _{i}-\eta ^{i}(Y)\varphi X\}+\overset{s}{\underset{i=1}{\sum }}\{g(\varphi Y,X)\xi _{i}-\eta ^{i}(X)\varphi Y\}\right) \\
&&+\overset{s}{\underset{i=1}{\sum }}\{g(\varphi ^{2}X,Y)\xi _{i}-\eta
^{i}(Y)\varphi ^{2}X\}-\overset{s}{\underset{i=1}{\sum }}\{g(\varphi
^{2}Y,X)\xi _{i}-\eta ^{i}(X)\varphi ^{2}Y\} \\
&=&0.\end{aligned}$$Hence, we have$$\lbrack \varphi ,\varphi ]+2\sum_{i=1}^{s}d\eta ^{i}\otimes \xi _{i}=0.$$The proof is completed.

Let $M$ be a $(2n+s)$-dimensional generalized Kenmotsu manifold with structure $\left( \varphi ,\xi _{i},\eta ^{i},g\right) .$ Then we have$$\nabla _{X}\xi _{j}=-\varphi ^{2}X$$for all $X\in \Gamma (TM),i,$ $j\in \left\{ 1,2,...,s\right\} .$

Let $M$ be a $(2n+s)$-dimensional generalized Kenmotsu manifold with structure $\left( \varphi ,\xi _{i},\eta ^{i},g\right) .$ Then we have$$i)\nabla _{\xi _{j}}\varphi =0,\begin{array}{cc}
\begin{array}{c}
\end{array}
&
\end{array}\nabla _{\xi _{j}}\xi _{i}=0$$$$ii)(L_{\xi _{i}}\varphi )X=0,\begin{array}{cc}
\begin{array}{c}
\end{array}
&
\end{array}(L_{\xi _{i}}\eta ^{j})X=0$$$$iii)(L_{\xi _{i}}g)(X,Y)=2\{g(X,Y)-\sum_{i=1}^{s}\eta ^{i}(X)\eta ^{i}(Y)\}$$for all $X\in \Gamma (TM),$ $i,j\in \left\{ 1,2,...,s\right\} .$

Let $M$ be a $(2n+s)$-dimensional generalized Kenmotsu manifold with structure $\left( \varphi ,\xi _{i},\eta ^{i},g\right) .$ Then we have$$(\nabla _{X}\eta ^{i})Y=g(X,Y)-\underset{j=1}{\overset{s}{\sum }}\eta
^{j}(X)\eta ^{j}(Y)$$

for all $X,Y\in \Gamma (TM),i\in \left\{ 1,2,...,s\right\} .$

Using $(8)$ and $(10)$ we get the desired result.

We have below the corollary in case $s=1$.

Let $\left( M^{2n+1},\varphi ,\xi ,\eta
,g\right) $ be an almost contact metric manifold. $M$ is a Kenmotsu manifold if and only if $$\left( \nabla _{X}\varphi \right) Y=g(\varphi X,Y)\xi -\eta (Y)\varphi X$$for all $X,Y\in \Gamma (TM),$ $i\in \left\{ 1,2,...,s\right\} ,$ where $\nabla $ is Riemannian connection on M *[@K]*.

Let $F$ be  a Kähler manifold $f(t)=ke^{\overset{s}{\underset{i=1}{\sum
}}t_{i}}$ be a function on $
\mathbb{R}^{s},$ and k be a non-zero constant. Then the warped product space $M=\mathbb{R}^{s}\times _{f}F$ have a generalized Kenmotsu manifold.

Let $(F,J,G)$ be a Kähler manifold and consider $M=\mathbb{R}^{s}\times _{f}F,$ with coordinates $(t_{1},...,t_{s},x_{1},...,x_{2n})$. We define $\varphi $ tensor field, $1$-form $\eta ^{i},$ vector field $\xi _{i}$ and Riemannian metric tensor $g$ on $M$ as follows: $$\begin{aligned}
\varphi (\frac{\partial }{\partial t_{i}},U) &=&(0,JU), \\
\eta ^{j}(\frac{\partial }{\partial t_{i}},U) &=&\delta _{ij},
\begin{array}{cc}
&
\end{array}
\xi _{i}=(\frac{\partial }{\partial t_{i}},0) \\
g_{f} &=&\overset{s}{\underset{i=1}{\sum }}dt^{i}\otimes dt^{i}+f^{2}\pi
^{\ast }(G)\end{aligned}$$where $f(t)=ke^{\overset{s}{\underset{i=1}{\sum }}t_{i}},U\in \Gamma (F)$.

Then $(M,\varphi ,\eta ^{i},\xi _{i},g_{f})$ defines $s-$contact metric manifold. Now let us show that this manifold is a generalized Kenmotsu manifold.

It is clear that $\eta ^{i}$ are closed. Thus, we have$$\Phi (X,Y)=g_{f}(X,\varphi Y)=f^{2}\pi ^{\ast }(G(X,JY))$$

or $$\Phi =f^{2}\pi ^{\ast }(\Psi )$$where  is fundamental $2$-form of K$\ddot{a}$hler manifold. Hence, we get

$$d\Phi =2c\overset{s}{\underset{i=1}{\sum }}e^{2\overset{s}{\underset{i=1}{\sum }}t_{i}}dt^{i}\wedge \pi ^{\ast }(\Psi )=2\overset{s}{\underset{i=1}{\sum }}dt^{i}\wedge \Phi .$$

Finally torsion tensor $N_{\varphi }$ of $M$ is vanish, since $\eta ^{i}$ are closed and $N_{J}=0$.

Then $(M=\mathbb{R}
^{s}\times _{f}F,\varphi ,\eta ^{i},\xi _{i},g_{f})$ is a generalized Kenmotsu manifold.

$(\mathbb{R}^{2}\times _{f}V^{4},g_{f}=\overset{2}{\underset{i=1}{\sum }}dt^{i}\otimes
dt^{i}+f^{2}G)$ is warped product with coordinates $(t_{1},t_{2},x_{1},x_{2},x_{3},x_{4})$, where $f^{2}=k^{2}e^{\overset{2}{\underset{i=1}{\sum }}t_{i}}$. Take a orthonormal frame field $\{\overline{E}_{1},\overline{E}_{2},\overline{E}_{3},\overline{E}_{4}\}$ of $V^{4}$ and $\{\overline{e}_{5},\overline{e}_{6}\}$ of $
\mathbb{R}^{2}$ such that $\overline{E}_{2}=J\overline{E}_{1}$, $\overline{E}_{4}=J\overline{E}_{3}.$Then we obtain a local orthonormal field $\{E_{1},E_{2},E_{3},E_{4},E_{5},E_{6}\}$ of $
\mathbb{R}^{2}\times _{f}V^{4}$ by$$\begin{aligned}
E_{1} &=&ke^{-\overset{2}{\underset{i=1}{\sum }}t_{i}}\overline{E}_{1},\begin{array}{cc}
&
\end{array}E_{2}=ke^{-\overset{2}{\underset{i=1}{\sum }}t_{i}}\overline{E}_{2} \\
E_{3} &=&-ke^{-\overset{2}{\underset{i=1}{\sum }}t_{i}}\overline{E}_{3},\begin{array}{cc}
&
\end{array}E_{4}=-ke^{-\overset{2}{\underset{i=1}{\sum }}t_{i}}\overline{E}_{4} \\
E_{5} &=&\xi _{1},\begin{array}{cc}
&
\end{array}E_{6}=\xi _{2}.\end{aligned}$$Then $
\mathbb{R}^{2}\times _{f}V^{4}$ is a generalized Kenmotsu manifold.

Let $(M^{2n+s},\varphi ,\eta ^{i},\xi _{i},g)$ be a generalized Kenmotsu manifold, $V$ and $L$ are Kähler and a flat manifold with locally coordinates $(x_{1},...,x_{2n})$ and $(t_{1},...,t_{s})$  respectively. Then $M$  a locally warped product $L^{s}\times _{f}V^{2n}$  where $f(t)=ke^{\overset{s}{\underset{i=1}{\sum }}t_{i}}$ and $k$ a nonzero positive constant.

We know that $TM=\mathcal{L}\oplus \mathcal{M}$. $ \mathcal{L}\ $ is clearly integrable, since $d\eta ^{i}=0.$ Then $V$ integral manifold of $ \mathcal{L}\ $ is totally umbilical because $\nabla _{X}\xi _{i}=X.$ On the other hand $[\xi _{i},\xi _{j}]=0$ and $\nabla _{\xi _{i}}\xi _{j}=0$, $\mathcal{M}$ is integrable and $L$ integral manifold is totally geodesic.

We select $J=\varphi \mid _{D}$ such that $J^{2}=-I,$ $G=g\mid _{D}.$ Then $(V,J,G)$ is almost Hermitian manifold. Also torsion tensor $N_{J}=N_{\varphi
}=0$ and using $(9),$ we get $(\nabla _{X}J)Y=0.$ Then  $(V,J,G)$ is Kähler manifold.

Then $M=L\times _{f}V$ is locally a warped product  and metric  is $$g_{f}=\overset{s}{\underset{i=1}{\sum }}dt^{i}\otimes dt^{i}+f^{2}G.$$Its follows that $$(L_{\xi _{i}}g_{f})(X,Y)=\frac{2\xi _{i}(f)}{f}G(X,Y)$$and using $(11)$, we get $$\xi _{i}(f)=f,\begin{array}{cc}
&
\end{array}i=1,...,s.$$Thus,we have $$\frac{\partial f(t_{1},...,t_{s})}{\partial t_{i}}=f(t_{1},...,t_{s}),\begin{array}{cc}
&
\end{array}i=1,...,s.$$Therefore, we obtained $f(t_{1},...,t_{s})=ce^{\overset{s}{\underset{i=1}{\sum }}t_{i}}$ where c is nonzero constant.

Let’s go back to the example 2.3. Let $(\mathbb{R}
^{7},\varphi ,\eta ^{i},\xi _{i},g)$ be a generalized Kenmotsu manifold where $i=1,2,3$. Take a orthonormal frame field $$\left\{ \frac{\partial }{\partial z_{1}}=\xi _{1},\begin{array}{c}
\end{array}\frac{\partial }{\partial z_{2}}=\xi _{2},\begin{array}{c}
\end{array}\frac{\partial }{\partial z_{3}}=\xi _{3}\right\}$$of $\mathbb{R}
^{3}$ and $$\left\{ \frac{e^{2\overset{3}{\underset{i=1}{\sum }}z_{i}}}{c_{1}^{2}+c_{2}^{2}}(f_{1}e_{1}-f_{2}e_{2}),\begin{array}{c}
\end{array}\frac{e^{2\overset{3}{\underset{i=1}{\sum }}z_{i}}}{c_{1}^{2}+c_{2}^{2}}(f_{2}e_{1}+f_{1}e_{2}),\begin{array}{c}
\end{array}\frac{e^{2\overset{3}{\underset{i=1}{\sum }}z_{i}}}{c_{1}^{2}+c_{2}^{2}}(f_{1}e_{3}-f_{2}e_{4}),\begin{array}{c}
\end{array}\frac{e^{2\overset{3}{\underset{i=1}{\sum }}z_{i}}}{c_{1}^{2}+c_{2}^{2}}(f_{2}e_{3}+f_{1}e_{4})\right\}$$of $\mathbb{R}
^{4}.$ Then $\mathbb{R}
^{7}=\mathbb{R}
^{3}\times
\mathbb{R}
^{4}$ is product manifold, the structure by tensor $\varphi $ and metric tensor $g.$ $\mathbb{R}
^{4}$ is the standard Kähler structure $(J,G)$. Here the Riemannian metric $g$ is warped product metric $$g_{0}+cf^{2}G$$where $g_{0}$  is the Euclidean metric of $\mathbb{R}
^{3},$ f is the function defined on $\mathbb{R}
^{3}$ by$$f(z_{1},z_{2},z_{3})=e^{2\overset{3}{\underset{i=1}{\sum }}z_{i}}\text{and}\begin{array}{cc}
&
\end{array}c=\frac{1}{c_{1}^{2}+c_{2}^{2}}.$$

Some Curvature Properties
=========================

Let $M$ be a $(2n+s)$-dimensional generalized Kenmotsu manifold with structure $\left( \varphi ,\xi _{i},\eta ^{i},g\right) .$ Then we have$$R(X,Y)\xi _{i}=\underset{j=1}{\overset{s}{\sum }}\{\eta ^{j}(Y)\varphi
^{2}X-\eta ^{j}(X)\varphi ^{2}Y\}$$for all $X,Y\in \Gamma (TM),i\in \left\{ 1,2,...,s\right\} .$

Firstly, using $(10)$ and $(6)$ we get$$\nabla _{X}\nabla _{Y}\xi _{i}=\nabla _{X}Y+\varphi ^{2}X\underset{j=1}{\overset{s}{\sum }}\eta ^{j}(Y)-\underset{j=1}{\overset{s}{\sum }}\{\eta
^{j}(\nabla _{X}Y)\xi _{j}+g(Y,-\varphi ^{2}X)\xi _{j}\}$$and$$\nabla _{\lbrack X,Y]}\xi _{i}=-\varphi ^{2}\nabla _{X}Y+\varphi ^{2}\nabla
_{Y}X.$$Then,$$\begin{aligned}
R(X,Y)\xi _{i} &=&\nabla _{X}Y+\varphi ^{2}X\underset{j=1}{\overset{s}{\sum }}\eta ^{j}(Y)-\underset{j=1}{\overset{s}{\sum }}\{\eta ^{j}(\nabla _{X}Y)\xi
_{j}+g(Y,-\varphi ^{2}X)\xi _{j}\} \\
&&-\nabla _{Y}X-\varphi ^{2}Y\underset{j=1}{\overset{s}{\sum }}\eta ^{j}(X)+\underset{j=1}{\overset{s}{\sum }}\{\eta ^{j}(\nabla _{Y}X)\xi
_{j}+g(X,-\varphi ^{2}Y)\xi _{j}\} \\
&&+\varphi ^{2}\nabla _{X}Y-\varphi ^{2}\nabla _{Y}X.\end{aligned}$$From $(6)$ desired result.

Let $M$ be a $(2n+s)$-dimensional generalized Kenmotsu manifold with structure $\left( \varphi ,\xi _{i},\eta ^{i},g\right) .$ Then we have$$R(X,\xi _{i})Y=\underset{j=1}{\overset{s}{\sum }}\{\eta ^{j}(Y)\varphi
^{2}X-g(X,\varphi ^{2}Y)\xi _{j}\}$$$$R(X,\xi _{j})\xi _{i}=\varphi ^{2}X,\begin{array}{cc}
\begin{array}{c}
\end{array}
&
\end{array}R(\xi _{k},\xi _{j})\xi _{i}=0$$for all $X,Y\in \Gamma (TM),i,j,k\in \left\{ 1,2,...,s\right\} .$

*[@K] Let* $M$ be a $(2n+1)$-dimensional Kenmotsu manifold with structure $\left( \varphi ,\xi ,\eta ,g\right) .$ Then we have$$\begin{aligned}
R(X,Y)\xi &=&\eta (Y)X-\eta (X)Y \\
R(X,\xi )Y &=&g(X,Y)\xi -\eta (Y)X,\begin{array}{cc}
&
\end{array}R(\xi ,\xi )\xi =0\end{aligned}$$for all $X,Y\in \Gamma (TM).$

Let $M$ be a $(2n+s)$-dimensional generalized Kenmotsu manifold with structure $\left( \varphi ,\xi _{i},\eta ^{i},g\right) .$ Then we have$$\begin{aligned}
(\nabla _{Z}R)(X,Y,\xi _{i}) &=&sg(Z,X)Y-sg(Z,Y)X-R(X,Y)Z \\
&&+s\underset{h=1}{\overset{s}{\sum }}\eta ^{h}(Z)\{\eta ^{h}(Y)X-\eta
^{h}(X)Y\}+\underset{l=1}{\overset{s}{\sum }}\eta ^{l}(Z)R(X,Y)\xi _{l}\end{aligned}$$for all $X,Y\in \Gamma (TM),i\in \left\{ 1,2,...,s\right\} .$

Using $(10)$ and $(13)$, we have$$\begin{aligned}
(\nabla _{Z}R)(X,Y,\xi _{i}) &=&\nabla _{Z}\{\underset{j=1}{\overset{s}{\sum
}}\{\eta ^{j}(X)Y-\eta ^{j}(Y)X\}\}-\underset{j=1}{\overset{s}{\sum }}\{\eta
^{j}(\nabla _{Z}X)Y-\eta ^{j}(Y)\nabla _{Z}X\} \\
&&\text{ \ \ }-\underset{j=1}{\overset{s}{\sum }}\{\eta ^{j}(X)\nabla
_{Z}Y-\eta ^{j}(\nabla _{Z}Y)X\}-R(X,Y)\varphi ^{2}Z.\end{aligned}$$From $(6)$, we get$$(\nabla _{Z}R)(X,Y,\xi _{i})=\underset{j=1}{\overset{s}{\sum }}\{g(X,\nabla
_{Z}\xi _{j})Y-g(Y,\nabla _{Z}\xi _{j})X\}-R(X,Y)Z+\underset{k=1}{\overset{s}{\sum }}\eta ^{k}(Y)R(X,Y)\xi _{k}.$$The proof competes from $(6)$ and $(10)$.

Let $M$ be a $(2n+s)$-dimensional generalized Kenmotsu manifold Then we have$$\begin{aligned}
(\nabla _{Z}R)(X,Y,\xi _{i}) &=&sg(Z,X)Y-sg(Z,Y)X-R(X,Y)Z,\begin{array}{cc}
&
\end{array}Z\in \mathcal{L} \\
(\nabla _{\xi _{j}}R)(X,Y,\xi _{i}) &=&0\end{aligned}$$for all $X,Y\in \Gamma (TM),i\in \left\{ 1,2,...,s\right\} .$

*[@K] Let* $M$ be a $(2n+1)$-dimensional Kenmotsu manifold with structure $\left( \varphi ,\xi ,\eta ,g\right) .$ Then we have$$(\nabla _{Z}R)(X,Y,\xi )=g(Z,X)Y-g(Z,Y)X-R(X,Y)Z,\begin{array}{cc}
&
\end{array}Z\in \mathcal{L}\text{ and }(\nabla _{\xi }R)(X,Y,\xi )=0.$$

Let $\left( M,\varphi ,\xi _{i},\eta ^{i},g\right) $ be a $(2n+s)$-dimensional locally-symmetric generalized Kenmotsu manifold. Then we have$$R(X,Y)Z=s\{g(Z,X)Y-g(Z,Y)X\}.$$

*[@K] Let* M be a $(2n+1)$-dimensional Kenmotsu manifold with structure $\left( \varphi ,\xi ,\eta ,g\right) .$ If M is a locally symmetric then we have$$R(X,Y)Z=g(Z,X)Y-g(Z,Y)X.$$

The $\varphi -$sectional curvature of any locally symmetric generalized Kenmotsu manifold $\left( M,\varphi ,\xi _{i},\eta ^{i},g\right) $ is equal to $-s$.

In this case $s=1$, we obtain that the $\varphi -$sectional curvature of any locally symmetric Kenmotsu manifold $\left( M,\varphi ,\xi ,\eta ,g\right) $ is equal to $-1$ *[@K].*

Let $M$ be a $(2n+s)$-dimensional generalized Kenmotsu manifold with structure $\left( \varphi ,\xi _{i},\eta ^{i},g\right) .$ Then we have$$R(X,Y)\varphi Z-\varphi R(X,Y)Z=g(Y,Z)\varphi X-g(X,Z)\varphi Y-g(Y,\varphi
Z)X+g(X,\varphi Z)Y$$$$R(\varphi X,\varphi Y)Z=R(X,Y)Z+g(Y,Z)X-g(X,Z)Y+g(Y,\varphi Z)\varphi
X-g(X,\varphi Z)\varphi Y$$for all $X,Y\in \Gamma (TM),i\in \left\{ 1,2,...,s\right\} .$

Ricci Curvature Tensor
======================

Let $M$ be a $(2n+s)$-dimensional generalized Kenmotsu manifold with structure $\left( \varphi ,\xi _{i},\eta ^{i},g\right) .$ Then we have$$S(X,\xi _{i})=-2n\underset{j=1}{\overset{s}{\sum }}\eta ^{j}(X)$$for all $X,Y\in \Gamma (TM),i\in \left\{ 1,2,...,s\right\} .$

If $\{E_{1},E_{2},...,E_{2n+s}\}$ are local orthonormal vector fields, then $S(X,Y)=\underset{k=1}{\overset{2n+s}{\sum }}g(R(E_{k},X)Y,E_{k})$ defines a global tensor field $S$ of type $(0,2)$. Then, we obtain$$\begin{aligned}
S(X,\xi _{i}) &=&\underset{k=1}{\overset{2n}{\sum }}g(\underset{j=1}{\overset{s}{\sum }}\{\eta ^{j}(X)\varphi ^{2}E_{k}-\eta ^{j}(E_{k})\varphi
^{2}X\},E_{k})+\underset{k=1}{\overset{s}{\sum }}g(-\varphi ^{2}X,\xi _{k})
\\
&=&\underset{j=1}{\overset{s}{\sum }}\eta ^{j}(X)\underset{k=1}{\overset{2n}{\sum }}g(\varphi ^{2}E_{k},E_{k}).\end{aligned}$$

In this case $s=1$ we have $S(X,\xi )=-2n\eta (X)$ in [@K].

Let $M$ be a $(2n+s)$-dimensional generalized Kenmotsu manifold with structure $\left( \varphi ,\xi _{i},\eta ^{i},g\right) .$ Then we have$$S(\xi _{k},\xi _{i})=-2n$$for all $X,Y\in TM,i,k\in \left\{ 1,2,...,s\right\} .$

Let $M$ be a $(2n+s)$ dimensional generalized Kenmotsu manifold with structure $\left( \varphi ,\xi _{i},\eta ^{i},g\right) .$ Then we have$$S(\varphi X,\varphi Y)=S(X,Y)+2n\underset{i=1}{\overset{s}{\sum }}\eta
^{i}(X)\eta ^{i}(Y)$$for all $X,Y\in \Gamma (TM),i\in \left\{ 1,2,...,s\right\} .$

We can put $$X=X_{0}+\underset{i=1}{\overset{s}{\sum }}\eta ^{i}(X)\xi _{i}\text{ and }Y=Y_{0}+\underset{i=1}{\overset{s}{\sum }}\eta ^{i}(Y)\xi _{i}$$ where $X_{0},Y_{0}\in \mathcal{L}$. Then from (16) and (17) we have,$$\begin{aligned}
S(X,Y) &=&S(X_{0},Y_{0})+\underset{i=1}{\overset{s}{\sum }}\eta ^{i}(Y)\eta
^{i}(X_{0})+\underset{i=1}{\overset{s}{\sum }}\eta ^{i}(X)\eta ^{i}(Y_{0})+\underset{i=1}{\overset{s}{\sum }}\eta ^{i}(X)\eta ^{i}(Y)S(\xi _{i},\xi
_{i}) \\
&=&S(X_{0},Y_{0})-2n\underset{i=1}{\overset{s}{\sum }}\eta ^{i}(X)\eta
^{i}(Y).\end{aligned}$$ Since $\varphi X,\varphi Y\in \mathcal{L}$ we get $S(X_{0},Y_{0})=S(\varphi
X,\varphi Y)$ which implies the desired result.

Considering $s=1$ in *[@JDP],* we deduce $$S(\varphi X,\varphi Y)=S(X,Y)+2n\eta (X)\eta (Y).$$

Let $M$ be a $(2n+s)$ dimensional generalized Kenmotsu manifold with structure $\left( \varphi ,\xi _{i},\eta ^{i},g\right) .$ Then we have$$\begin{aligned}
(\nabla _{\varphi X}S)(\varphi Y,\varphi Z) &=&(\nabla _{\varphi X}S)(Y,Z)-\underset{i=1}{\overset{s}{\sum }}\eta ^{i}(Y)\{S(X,\varphi Z)+2ng(X,\varphi
Z)\} \\
&&-\underset{i=1}{\overset{s}{\sum }}\eta ^{i}(Z)\{S(X,\varphi
Y)+2ng(X,\varphi Y)\}\end{aligned}$$for all $X,Y\in \Gamma (TM),i\in \left\{ 1,2,...,s\right\} .$

Using $(9)$, we get$$\begin{aligned}
(\nabla _{\varphi X}S)(\varphi Y,\varphi Z) &=&\nabla _{\varphi X}S(Y,Z)+2n\underset{i=1}{\overset{s}{\sum }}\{\eta ^{i}(Y)\nabla _{\varphi X}\eta
^{i}(Z)+\eta ^{i}(Z)\nabla _{\varphi X}\eta ^{i}(Y)\} \\
&&+\underset{i=1}{\overset{s}{\sum }}\{-S(g(\varphi ^{2}X,Y)\xi _{i}-\eta
^{i}(Y)\varphi ^{2}X,\varphi Z)-S(\nabla _{\varphi X}Y,Z)-2n\eta ^{i}(\nabla
_{\varphi X}Y)\eta ^{i}(Z) \\
&&-S(\varphi Y,g(\varphi ^{2}X,Z)\xi _{i}-\eta ^{i}(Z)\varphi
^{2}X)-S(Y,\nabla _{\varphi X}Z)-2n\eta ^{i}(Y)\eta ^{i}(\nabla _{\varphi
X}Z)\}.\end{aligned}$$From $(6)$, $(16)$ and $(17)$ we have$$\begin{aligned}
(\nabla _{\varphi X}S)(\varphi Y,\varphi Z) &=&(\nabla _{\varphi X}S)(Y,Z)+\underset{i=1}{\overset{s}{\sum }}\{2n\eta ^{i}(Y)(\nabla _{\varphi X}\eta
^{i})Z+2n\eta ^{i}(Z)(\nabla _{\varphi X}\eta ^{i})Y \\
&&-\eta ^{i}(Y)S(X,\varphi Z)-\eta ^{i}(Z)S(\varphi Y,X)\}.\end{aligned}$$

Let $M$ be a $(2n+s)$ dimensional generalized Kenmotsu manifold with structure $\left( \varphi ,\xi _{i},\eta ^{i},g\right) .$ Then we have$$\begin{aligned}
(\nabla _{X}S)(\varphi Y,\varphi Z) &=&(\nabla _{X}S)(Y,Z)+2n\underset{i=1}{\overset{s}{\sum }}\{g(X,Y)\eta ^{i}(Z)+g(X,Z)\eta ^{i}(Y)\} \\
&&+\underset{i=1}{\overset{s}{\sum }}\{\eta ^{i}(Y)S(X,Z)+\eta
^{i}(Z)S(X,Y)\}\end{aligned}$$for all $X,Y\in \Gamma (TM),i\in \left\{ 1,2,...,s\right\} .$

The Ricci tensor $S$ of a $(2n+s)$-dimensional generalized Kenmotsu manifold $M$ is called $\eta -parallel$, if it satisfies$$(\nabla _{X}S)(\varphi Y,\varphi Z)=0$$for all vector fields $X,Y$ and $Z$ on $M$.

Let $\left( M,\varphi ,\xi _{i},\eta ^{i},g\right) $ a $(2n+s)$-dimensional generalized Kenmotsu manifold. $M$ has $\eta -parallel$ if and only if$$\begin{aligned}
(\nabla _{X}S)(Y,Z) &=&-2n\underset{i=1}{\overset{s}{\sum }}\{g(X,Y)\eta
^{i}(Z)+g(X,Z)\eta ^{i}(Y)\} \\
&&-\underset{i=1}{\overset{s}{\sum }}\{\eta ^{i}(Y)S(X,Z)+\eta
^{i}(Z)S(X,Y)\}\end{aligned}$$for all $X,Y,Z\in \Gamma (TM),i\in \left\{ 1,2,...,s\right\} .$

[@K] Let $\left( M,\varphi ,\xi ,\eta ,g\right) $ a $(2n+1)$-dimensional Kenmotsu manifold. $M$ has $\eta -parallel$ if and only if$$\begin{aligned}
(\nabla _{X}S)(Y,Z) &=&-2n\{g(X,Y)\eta (Z)+g(X,Z)\eta (Y)\} \\
&&\text{ \ \ \ }-\eta (Y)S(X,Z)-\eta (Z)S(X,Y)\end{aligned}$$for all $X,Y,Z\in \Gamma (TM).$

Semi-Symmetric Properties of Generalized Kenmotsu Manifolds
===========================================================

With respect to the Riemannian connection $\nabla $ of a generalized Kenmotsu manifold $\left( M,\varphi ,\xi _{i},\eta ^{i},g\right) $, we can prove:

The $\varphi $- sectional curvature of any semi-symmetric $(2n+s)$-dimensional generalized Kenmotsu manifold $\left( M,\varphi ,\xi _{i},\eta
^{i},g\right) $ is equal to $-s$.

Let $X$ be a unit vector field. Since $\left( M,\varphi ,\xi _{i},\eta
^{i},g\right) $ is semi-symmetric, then $$(R.R)(X,\xi _{i},X,\varphi X,\varphi X,\xi _{i})=0,$$for any $i,j\in \left\{ 1,2,...,s\right\} .$ Expanding this formula from $(7)$ and taking into account $(14)$, we get $$R(X,\varphi X,\varphi X,X)=-s,$$which completes the proof.

Observe that, in the case $s=1$, by using the *Theorem 11* we obtain that a semi-symmetric Kenmotsu manifolds is constant curvature equal to $-1$ [@Bin].

Let $\left( M,\varphi ,\xi _{i},\eta ^{i},g\right) $ be a $(2n+s)$ dimensional Ricci semi-symmetric generalized Kenmotsu manifold. Then its Ricci tensor field $S$ respect the Riemannian connection satisfies$$S(X,Y)=-2n\{sg(\varphi X,\varphi Y)+\underset{i,j=1}{\overset{s}{\sum }}\eta
^{i}(X)\eta ^{j}(Y)\}$$for any $X,Y\in \Gamma (TM).$

Since $\left( M,\varphi ,\xi _{i},\eta ^{i},g\right) $ is Ricci semi-symmetric, then $$S(R(X,\xi _{i})\xi _{j},Y)+S(\xi _{j},R(X,\xi _{i})Y)=0,$$for any $X,Y\in \Gamma (TM)$ and $i,j\in \left\{ 1,2,...,s\right\} .$ Now, from $(14),$ $(15)$ and $(16)$ we get the desired result.

In this case $s=1$ we have following the corollary.

Any Ricci semi-symmetric $(2n+1)-$dimensional Kenmotsu manifold is an Einstein manifold.

Considering $s=1$ in $(19)$, we deduce $$S(X,Y)=-2ng(X,Y)$$for any $X,Y\in \Gamma (TM).$

For the Weyl projective curvature tensor field $P$, the weyl projective curvature tensor $P$ of a $(2n+s)$-dimensional generalized Kenmotsu manifold $M$ is given by$$P(X,Y)Z=R(X,Y)Z-\frac{1}{2n+s-1}\{S(Y,Z)X-S(X,Z)Y\}$$where $R$ is curvature tensor and $S$ is the ricci curvature tensor of $M$, we have the following theorem.

The $\varphi $- sectional curvature of any projectively semi-symmetric generalized Kenmotsu manifold $\left( M,\varphi ,\xi _{i},\eta ^{i},g\right)
$ is equal to $-s$.

Let $X$ be a unit vector field. Then, from $(6)$ and taking into account $(14)$ and $(16)$ we have $$(R.P)(X,\xi _{i},X,\varphi X,\varphi X,\xi _{j})=(R.R)(X,\xi _{i},X,\varphi
X,\varphi X,\xi _{j}).$$This completes the proof from the *Theorem 5.1*.

Let $\left( M,\varphi ,\xi ,\eta ,g\right) $ a $(2n+1)$-dimensional Kenmotsu manifold. The $\varphi $- sectional curvature of any projectively semi-symmetric Kenmotsu manifold if and only if $M$ is an Einstein manifold.

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[^1]: 2010 *Mathematics Subject Classification*. Primary 53C15 ; Secondary 53C25, 53D10.

[^2]: *Key words and phrases*. Kenmotsu manifolds, metric $f$-manifolds, s-contact metric manifolds, generalized Kenmotsu manifolds, semi-symmetric, ricci semi-symmetric, projective semi-symmetric.

[^3]:
