Generalized inequalities of warped product submanifolds of nearly Kenmotsu $f$-manifolds

TL;DR

This paper establishes sharp inequalities for the squared norm of the second fundamental form of certain warped product submanifolds in nearly Kenmotsu $f$-manifolds, extending and generalizing previous results in the field.

Contribution

It introduces new sharp inequalities for warped product pseudo slant submanifolds in nearly Kenmotsu $f$-manifolds, including equality cases and generalizations of prior results.

Findings

Derived sharp inequalities involving the second fundamental form.

Identified conditions for equality cases.

Extended previous results to more general warped product submanifolds.

Abstract

In the present paper, we discuss the non-trivial warped product pseudo slant submanifolds of type $M_{\bot }\times _{f}M_{\theta }$ and $M_{\theta}\times _{f}M_{\bot }$ of nearly Kenmotsu $f$-manifold $\overline{M}$. Firstly, we get some basic properties of these type warped product submanifolds. Then, we establish the general sharp inequalities for squared norm of second fundamental form for mixed totally geodesic warped product pseudo slant submanifolds of both cases, in terms of the warping function and the slant angle. Also the equality cases are verified. We show that some previous results are trivial from our results.

Equations

$${\phi }^3+\phi =0,$$

$$\begin{array}[]{cc}P_{1}=-\varphi^{2},&P_{2}=\varphi^{2}+I,\end{array}$$

$$\begin{array}[]{ccc}P_{1}+P_{2}=I,&P_{1}^{2}=P_{1},&P_{2}^{2}=P_{2},\\ \varphi P_{1}=P_{1}\varphi=\varphi,&P_{2}\varphi=\varphi P_{2}=0.&\end{array}$$

$${\eta }^j\left({\xi }_i\right)={\delta }_i^j,$$

$${\phi }^2=-I+\sum _{i=1}^s{\eta }^i\otimes {\xi }_i,$$

$$\phi {\xi }_i=0,$$

$${\eta }^i\circ \phi =0.$$

$${\eta }^i\left(X\right)=g\left(X,{\xi }_i\right),$$

$$g\left(\phi X,\phi Y\right)=g\left(X,Y\right)-\sum _{i=1}^s{\eta }^i\left(X\right){\eta }^i\left(Y\right),$$

$$\Phi \left(X,Y\right)=g\left(X,\phi Y\right),$$

$$N_{\phi }+2\sum _{i=1}^{s}d{\eta }^i\otimes {\xi }_i=0,$$

$$\left({\overline{\nabla }}_X\phi \right)Y=\sum _{k=1}^s\left\{g\left(\phi X,Y\right){\xi }_k-{\eta }^k\left(Y\right)\phi X\right\},$$

$$\left({\overline{\nabla }}_X\phi \right)Y+\left({\overline{\nabla }}_Y\phi \right)X=-\sum _{k=1}^s\left\{{\eta }^k\left(X\right)\phi Y+{\eta }^k\left(Y\right)\phi X\right\}$$

$${\overline{\nabla }}_X{\xi }_i=-{\phi }^{2}X,$$

$$R\left({\xi }_i,X\right)Y=\sum _{k=1}^s\left\{-g\left(X,Y\right){\xi }_k+{\eta }^k\left(Y\right)X\right\},$$

$$R\left(X,Y\right){\xi }_i=\sum _{k=1}^s\left\{{\eta }^k\left(X\right)Y-{\eta }^k\left(Y\right)X\right\},$$

$$S\left(\phi X,\phi Y\right)=S\left(X,Y\right)+\left(2n+s-1\right)\sum _{k=1}^s{\eta }^k\left(X\right){\eta }^k\left(Y\right),$$

$$\left({\overline{\nabla }}_X{\eta }^i\right)Y=g\left(X,Y\right)-\sum _{k=1}^s{\eta }^k\left(X\right){\eta }^k\left(Y\right),$$

$$\sum _{k=1}^s{\eta }^k\left(R\left(X,Y\right)Z\right)=\sum _{k=1}^s\left\{g\left(X,Z\right){\eta }^k\left(Y\right)-g\left(Y,Z\right){\eta }^k\left(X\right)\right\},$$

$$\overline{M}=\left\{\left(x_1,\dots ,x_6,y_1,\dots ,y_6,z_1,z_2\right)\in {\mathbb{R}}^{14}:z_1,z_2\ne 0\right\},$$

$$\begin{array}[]{ccc}e_{1}=e^{-\left(z_{1}+z_{2}\right)}\dfrac{\partial}{\partial x_{1}},{}&e_{2}=e^{-\left(z_{1}+z_{2}\right)}\dfrac{\partial}{\partial x_{2}},{}&e_{3}=e^{-\left(z_{1}+z_{2}\right)}\dfrac{\partial}{\partial x_{3}},~{}\vskip 12.0pt plus 4.0pt minus 4.0pt\\ e_{4}=e^{-\left(z_{1}+z_{2}\right)}\dfrac{\partial}{\partial x_{4}},{}&e_{5}=e^{-\left(z_{1}+z_{2}\right)}\dfrac{\partial}{\partial x_{5}},{}&e_{6}=e^{-\left(z_{1}+z_{2}\right)}\dfrac{\partial}{\partial x_{6}},~{}\vskip 12.0pt plus 4.0pt minus 4.0pt\\ e_{7}=e^{-\left(z_{1}+z_{2}\right)}\dfrac{\partial}{\partial y_{1}},{}&e_{8}=e^{-\left(z_{1}+z_{2}\right)}\dfrac{\partial}{\partial y_{2}},{}&e_{9}=e^{-\left(z_{1}+z_{2}\right)}\dfrac{\partial}{\partial y_{3}},~{}\vskip 12.0pt plus 4.0pt minus 4.0pt\\ e_{10}=e^{-\left(z_{1}+z_{2}\right)}\dfrac{\partial}{\partial y_{4}},{}&e_{11}=e^{-\left(z_{1}+z_{2}\right)}\dfrac{\partial}{\partial y_{5}},{}&e_{12}=e^{-\left(z_{1}+z_{2}\right)}\dfrac{\partial}{\partial y_{6}},~{}\vskip 12.0pt plus 4.0pt minus 4.0pt\\ e_{13}=\dfrac{\partial}{\partial z_{1}},{}&e_{14}=\dfrac{\partial}{\partial z_{1}},&\end{array}$$

$$g=e^{2\left(z_1+z_2\right)}\left(\sum _{i=1}^6\left(d{x}_i\otimes d{x}_i+d{y}_i\otimes d{y}_i\right)+{\eta }_1\otimes {\eta }_1+{\eta }_2\otimes {\eta }_2\right)$$

$$\phi \left(\sum _{i=1}^6\left(x_i\frac{\partial }{\partial x_i}+y_i\frac{\partial }{\partial y_i}\right)+z_1\frac{\partial }{\partial z_1}+z_2\frac{\partial }{\partial z_2}\right)=\phi \left(\sum _{i=1}^6\left(x_i\frac{\partial }{\partial y_i}-y_i\frac{\partial }{\partial x_i}\right)\right).$$

$$\begin{array}[]{ccc}\varphi e_{1}=e_{7},{}&\varphi e_{2}=e_{8},{}&\varphi e_{3}=e_{9}\vskip 12.0pt plus 4.0pt minus 4.0pt\\ \varphi e_{4}=e_{10}&\varphi e_{5}=e_{11}&\varphi e_{6}=e_{12}\vskip 12.0pt plus 4.0pt minus 4.0pt\\ \varphi e_{7}=-e_{1},{}&\varphi e_{8}=-e_{2},{}&\varphi e_{9}=-e_{3}\vskip 12.0pt plus 4.0pt minus 4.0pt\\ \varphi e_{10}=-e_{4}&\varphi e_{11}=-e_{5}&\varphi e_{12}=-e_{6}\vskip 12.0pt plus 4.0pt minus 4.0pt\\ \varphi e_{13}=0&\varphi e_{14}=0.&\end{array}$$

$$\displaystyle\begin{array}[]{ccc}\eta_{1}\left(\xi_{1}\right)=1,{}&\eta_{2}\left(\xi_{2}\right)=1,{}&\varphi^{2}X=-X+\eta_{1}\left(X\right)\xi_{1}+\eta_{2}\left(X\right)\xi_{2}\end{array}$$

$$\displaystyle\begin{array}[]{c}g\left(\varphi X,~{}\varphi Y\right)=g\left(X,~{}Y\right)-\eta_{1}\left(X\right)\eta_{1}\left(X\right)-\eta_{2}\left(X\right)\eta_{2}\left(X\right).\end{array}$$

$$\Phi \left(\sum _{i=1}^6\frac{\partial }{\partial x_i},\sum _{i=1}^6\frac{\partial }{\partial y_i}\right)=$$

$$=$$

$$d\Phi =-2e^{2\left(z_1+z_2\right)}\left(d{z}_1+d{z}_2\right)\wedge \sum _{i=1}^6\left(d{x}_i\wedge d{y}_i\right)=2\left({\eta }_1+{\eta }_2\right)\wedge \Phi .$$

$${\overline{\nabla }}_{X}Y={\nabla }_{X}Y+h\left(X,Y\right)$$

$${\overline{\nabla }}_{X}V=-A_{V}X+{\nabla }_X^{\perp }V$$

Full text

1. Introduction

The warped product which is a natural generalization of Riemannian product was introduced to construct the manifolds with negative curvature by Bishop and O’Neill in 1969 [7]. Then Kenmotsu introduced a remarkable class of almost contact manifolds with negative curvature $-1~{}$by using warped product [16]. In 1978, Bejancu studied a special class of Kähler manifolds and defined the $CR$-submanifolds [6]. Later on, in [10] Chen studied $CR$-submanifolds and introduced $CR$-warped product submanifold in a Kähler manifold by using these two notions. He established a general inequality for the second fundamental form in terms of warping functions for an arbitrary $CR$-warped product in an arbitrary Kähler manifold. After that many authors derived the geometric inequalities of warped product submanifolds in different ambient spaces ([1]-[3], [17], [18], [23]). Recently, Şahin [22] constructed a general inequality for warped product pseudo slant isometrically immersed in a Kähler manifold.

On the other hand, Yano defined and studied the $\left(2n+s\right)$-dimensional globally framed metric $f$-manifold which is a natural generalization complex manifolds and contact manifolds [25]. Then in 1964, Ishihara and Yano investigate the integrability of the structures defined on these manifolds [15]. Blair introduced three classes of globally framed metric $f$-manifold as $K$-manifolds, $S$-manifolds and $C$-manifolds [8]. Moreover Falcitelli and Pastore defined almost Kenmotsu $f$-manifold which is a generalization of an almost Kenmotsu manifold [11]. Öztürk et. al generalized the almost $C$-manifolds and almost Kenmotsu $f$-manifold and they introduced almost $\alpha$-cosymplectic $f$-manifolds [20]. Recently, Balkan studied a globally framed version of nearly Kenmotsu manifolds and obtained the fundamental properties of these type manifolds [5].

The main objective of this paper to consider nearly Kenmotsu $f$-manifolds and compute some geometric sharp inequalities of non-trivial warped product pseudo slant submanifolds. We prove the existence of the warped product pseudo slant submanifolds in nearly Kenmotsu $f$-manifolds by constructing some examples. It well known that the warped product pseudo slant submanifolds are natural extensions of $CR$-warped product submanifold with some geometric condition.

2. Preliminaries

Let $\overline{M}$ be $\left(2n+s\right)$-dimensional manifold and $\varphi$ is a non-null $\left(1,~{}1\right)$ tensor field on $M$. If $\varphi$ satisfies

[TABLE]

then $\varphi$ is called an $f$-structure and $\overline{M}$ is called an $f$-manifold [25]. If $rank\varphi=2n$ or $rank\varphi=2n+1$, i.e., $s=0$ or $s=1,~{}$then $\varphi$ is called an almost complex structure or an almost contact structure, respectively [12]. On the other hand, $rank\varphi$ is always constant [21].

On an $f$-manifold $\overline{M}$, $P_{1}$ and $P_{2}$ operators are defined by

[TABLE]

which satisfy

[TABLE]

These properties show that $P_{1}$ and $P_{2}$ are complement projection operators. There are $D$ and $D^{\bot}$ distributions with respect to $P_{1}$ and $P_{2}$ operators, respectively [26]. Moreover $\dim\left(D\right)=2m$ and $\dim\left(D^{\bot}\right)=s.$

Let $\overline{M}$ be $\left(2m+s\right)$-dimensional $f$-manifold and $\varphi$ is a $\left(1,~{}1\right)$ tensor field, $\xi_{i}$ is vector field and $\eta^{i}$ is $1$-form for each $1\leq i\leq s$ on $M,$ respectively. If the following properties are satisfied

[TABLE]

and

[TABLE]

then $\left(\varphi,~{}\xi_{i},~{}\eta^{i}\right)$ is called globally framed $f$-structure or simply framed $f$-structure and $\overline{M}$ is called globally framed $f$-manifold or simply framed $f$-manifold [19]. For a framed $f$-manifold $\overline{M},$ the following properties hold [19]:

[TABLE]

[TABLE]

On a framed $f$-manifold $\overline{M}$ if there exists a Riemannian metric which satisfies

[TABLE]

and

[TABLE]

for all vector fields $X,~{}Y$ on $\overline{M},$ then $\overline{M}~{}$is called a framed metric $f$-manifold [13]. On a framed metric $f$-manifold, the fundamental $2$-form $\Phi$ is defined by

[TABLE]

for all vector fields $X,~{}Y$ on $\overline{M}$ [13]. For a framed metric $f$-manifold, if the following holds

[TABLE]

then $\overline{M}$ is called normal framed metric $f$-manifold, where $N_{\varphi}$ denotes the Nijenhuis torsion tensor of $\varphi$ [14].

A globally framed metric $f$-manifold $\overline{M}$ is called Kenmotsu $f$-manifold if it satisfies

[TABLE]

for all vector fields $X,~{}Y$ on $\overline{M}$ [20]. Furthermore, if a globally framed metric $f$-manifold $\overline{M}$ satisfies

[TABLE]

then it is called a nearly Kenmotsu $f$-manifold. It is easily seen that every Kenmotsu $f$-manifold is a nearly Kenmotsu $f$-manifold, but the converse is not true. When a nearly Kenmotsu $f$-manifold $\overline{M}$ is normal, it turns to a Kenmotsu $f$-manifold [5]. On a nearly Kenmotsu $f$-manifold $\overline{M},$ the following identities hold:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

for any vector fields $X,~{}Y$ on $\overline{M}$ [5]. Now we give the following example for nearly Kenmotsu $f-$manifolds.

Example 2.1**.**

Let us consider a $14$-dimensional manifold

[TABLE]

where $\left(x_{1},~{}\ldots,~{}x_{6},~{}y_{1},~{}\ldots,~{}y_{6},~{}z_{1},~{}z_{2}\right)$ are standard coordinates in $\mathbb{R}^{14}.$ We set the vector fields

[TABLE]

which are linearly independent at any point of $\overline{M}.$ Let $g$ be the Riemannian metric defined by

[TABLE]

where $\eta_{1}$ and $\eta_{2}$ are $1$-forms defined by $\eta_{1}\left(X\right)=g\left(X,~{}e_{13}\right)$ and $\eta_{2}\left(X\right)=g\left(X,~{}e_{14}\right)$, respectively. Thus $\left(e_{1},~{}\ldots,~{}e_{14}\right)$ is an orthornormal basis of $\overline{M}$. Thus we define the $\left(1,1\right)$-tensor field $\varphi$ as follows:

[TABLE]

Then we have

[TABLE]

From the linearity of $g$ and $\varphi$, it yields that

[TABLE]

Hence $\left(\varphi,~{}\xi_{i},~{}\eta^{i},~{}\right)$ defines a globally framed metric $f$-structure on $\overline{M}.$ On the other hand, by the virtue of definition of $1$-forms $\eta_{1}$ and $\eta_{2}$ it is said that they are closed. In other words $d\eta_{1}=0$ and $d\eta_{2}=0.$ Moreover, we get the fundamental $2$-form $\Phi$ on $\overline{M}~{}$as in the follwing form

[TABLE]

Then we have $\Phi=-e^{2\left(z_{1}+z_{2}\right)}\sum_{i=1}^{6}\left(dx_{i}\wedge dy_{i}\right).$ From the exterior derivative of the fundamental $2$-form $\Phi,$ then we derive

[TABLE]

Then the manifold $\overline{M}$ is called an almost Kenmotsu $f$-manifold. After some easy calculation, it is easy to prove that seen that the manifold $\overline{M}$ is normal. Hence by fact that every Kenmotsu $f$-manifold is a nearly Kenmotsu $f$-manifold from [5], then we arrive at $\overline{M}$ is a nearly Kenmotsu $f$-manifold.

Remark 2.1**.**

From (2.13), it is clear that if $s=0$ then $\overline{M}$ is become nearly Kaehler manifold [24]. If $s=1$ then the manifold $\overline{M}$ is called nearly Kenmotsu manifold [3].

Now we recall some basic facts of submanifold from [9]. Let $M$ be a submanifold immersed in $\overline{M}$. We also denote by $g$ the induced metric on $M$. Let $TM$ be the Lie algebra of vector fields in $M$ and $T^{\bot}M$ the set of all vector fields normal to $M$. Denote by $\nabla$ and $\overline{\nabla}$ the Levi-Civita connections of $M$ and $\overline{M},$ respectively. Then the Gauss and Weingarten formulas are given by

[TABLE]

and

[TABLE]

respectively, for any vector fields $X,~{}Y$ on $\overline{M}$ and any $V\in T^{\bot}M.$ Here, $\nabla^{\bot}$ is normal connection in the normal bundle, $h$ is second fundamental form of $M$ and $A_{V}$ is the Weingarten endomorphism associated with $V$. On the other hand, there is a relation between $A_{V}$ and $h$ such that

[TABLE]

The mean curvature vector $H$ is defined by $H=\dfrac{1}{m}traceh$, where $m$ is the dimension of $M$. $M$ is said to be minimal, totally geodesic and totally umbilical if $H$ vanishes identically and $h=0$,

[TABLE]

respectively. Furthermore, the second fundamental form $h$ satisfies

[TABLE]

3. Submanifolds of Globally Framed Metric $f$-manifolds

In this section, let us recall some basic properties of submanifolds of globally framed metric $f$-manifolds from [4].

Definition 3.1**.**

Let $\overline{M}$ be a globally framed metric $f$-manifold and $M$ is a submanifold of $\overline{M}.$ For any vector field $X$ on $M$ , we can write

[TABLE]

where $TX$ and $NX$ are called tangent and normal component of $\varphi X$, respectively. Similarly, for each $V\in\Gamma\left(T^{\bot}M\right)$, we have

[TABLE]

Here, $tV$ is tangent component and $nV$ is normal component of $\varphi V.$

Corollary 3.1**.**

Let $\overline{M}$ be a globally framed metric $f$-manifold and $M$ is a submanifold of $\overline{M}.$ Then the following identities hold:

[TABLE]

[TABLE]

where $I$ denotes the identity transformation.

Proposition 3.1**.**

Let $\overline{M}$ be a globally framed metric $f$-manifold and $M$ is a submanifold of $\overline{M}.$ Then, $T$ and $n$ are skew-symmetric tensor fields.

Proposition 3.2**.**

Let $\overline{M}$ be a globally framed metric $f$-manifold and $M$ is a submanifold of $\overline{M}.$ Then, for any vector field $X$ on $M$ and $V\in\Gamma\left(T^{\bot}M\right)$, we have

[TABLE]

which gives the relation between $N$ and $t$.

Proposition 3.3**.**

Let $\overline{M}$ be a globally framed metric $f$-manifold and $M$ is a submanifold of $\overline{M}.$ Then, for any vector fields $X,$ $Y~{}$on $M$ and $V\in\Gamma\left(T^{\bot}M\right),$ the following identities hold:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $h$ is the second fundamental form, $\nabla$ is the Levi-Civita connection and $A_{V}$ denotes the shape operator corresponding to the normal vector field $V$.

Definition 3.2**.**

Let $\overline{M}$ be a globally framed metric $f$-manifold and $M$ is a submanifold of $\overline{M}.$ Then the $TM$ tangent bundle of $M$ can be decomposed as

[TABLE]

where for each $1\leq k\leq s$ the $\xi_{k}$ denotes the distributions spanned by the structure vector fields $\xi_{k}$ and $D_{\theta}$ is complementary of distributions $\xi_{k}$ in $TM$, known as the slant distribution on $M.$

Theorem 3.1**.**

Let $\overline{M}$ be a globally framed metric $f$-manifold and $M$ is a submanifold of $\overline{M}.$ Then $M$ is a slant submanifold if and only if there exists a constant $\mu\in\left[0,~{}1\right]$ such that

[TABLE]

Moreover, if $\theta$ is the slant angle of $M$, then $\mu=\cos^{2}\theta.$

Corollary 3.2**.**

Let $M$ be a slant submanifold of a globally framed metric $f$-manifold $\overline{M}$ with slant angle $\theta.$ Then for any vector fields $X,$ $Y~{}$on $M,$ we find

[TABLE]

and

[TABLE]

Definition 3.3**.**

Let $M$ be a submanifold of a globally framed metric $f$-manifold $\overline{M}$ and let $M$ be tangent to the structure vector fields $\xi_{k}$ for each $1\leq k\leq s.$ For each nonzero vector $X$ tangent to $M$ at $p$, we denote by $0\leq\theta\left(X\right)\leq\dfrac{\pi}{2}$, the angle between $\varphi X$ and $T_{p}M$, known as the Wirtinger angle of $X$. If the $\theta\left(X\right)$ is constant, that is, independent of the choice of $p\in M$ and $X\in T_{p}M-\left\{\xi_{k}\right\},$ for each $1\leq k\leq s,$ then $M$ is said to be a slant submanifold and the constant angle $\theta$ is called slant angle of the slant submanifold

Here, if $\theta=0,$ $M$ is invariant submanifold and if $\theta=\dfrac{\pi}{2}$, then $M$ is an anti-invariant submanifold. A slant submanifold is proper slant if it is neither invariant nor anti-invariant submanifold.

Definition 3.4**.**

Let $M$ be a submanifold of a a globally framed metric $f$-manifold $\overline{M}$  We say that $M$ is a pseudo-slant submanifold if there exist two orthogonal distributions $D_{\theta}$ and $D^{\bot}$ such that

$1)$* The $TM$ tangent bundle of $M$ admits the orthogonal direct decomposition $TM=D^{\bot}\oplus D_{\theta},$ where for each $1\leq k\leq s$ $\xi_{k}\in\Gamma\left(D_{\theta}\right).$*

$2)$* The distribution $D^{\bot}$ is anti-invariant i. e., $\varphi\left(D^{\bot}\right)\subset\left(T^{\bot}M\right).$*

$3)$* The distribution $D_{\theta}$ is slant with angle $\theta\neq\dfrac{\pi}{2},$ that is, the angle between $D_{\theta}$ and $\varphi\left(D_{\theta}\right)$ is a constant.*

A pseudo-slant submanifold of a globally framed metric $f$-manifold is called mixed totally geodesic if $h\left(X,~{}Z\right)=0$ for all $X\in\Gamma\left(D^{\bot}\right)$ and $Z\in\Gamma\left(D_{\theta}\right).$ Now let $\left\{e_{1},~{}\ldots,~{}e_{n}\right\}$ be an orthonormal basis of the tangent space $TM$ and $e_{r}$belongs to the orthonormal basis $\left\{e_{n+1},~{}\ldots,~{}e_{m}\right\}$ of a normal bundle $T^{\bot}M$, then we define

[TABLE]

On the other hand, for a differentiable function $\lambda$ on $M$, we have

[TABLE]

where the gradient $grad\lambda$ is defined by $g\left(\nabla\lambda,~{}X\right)=X\lambda$, for any vector field $X\in\Gamma\left(TM\right).$

4. Warped Product Pseudo-Slant Submanifolds

In this section, we investigate some fundamental properties of warped  product pseudo-slant submanifolds of a nearly Kenmotsu $f$-manifold. First, we recall the definition of warped product manifolds and provide the useful lemma from [7] which will use in the proof of our main results.

Definition 4.1**.**

Let $\left(M_{1},~{}g_{M_{1}}\right)~{}$and $\left(M_{2},~{}g_{M_{2}}\right)$ be two Riemannian manifolds with Riemannian metrics $g_{M_{1}}$ and $g_{M_{2}}$, respectively and $f$ is a positive differentiable function on $M_{1}.$ The warped product $M_{1}\times_{f}M_{2}$ of $M_{1}$ and $M_{2}$ is the Riemannian manifold $\left(M_{1}\times M_{2},~{}g\right)$, where

[TABLE]

More explicitly , if $U$ is tangent to $M=M_{1}\times_{f}M_{2}$ at $\left(p,~{}q\right),$ then

[TABLE]

where for $i=1,~{}2$, $\pi_{i}$ are the canonical projections of $M_{1}\times M_{2}$ on $M_{1}$ and $M_{2}$ respectively.

Lemma 4.1**.**

Let $M=M_{1}\times_{f}M_{2}$ be a warped product manifold. Then we have

$\left(i\right)~{}\nabla_{X}Y\in\Gamma\left(TM_{1}\right),$**

$\left(ii\right)~{}\nabla_{Z}X=\nabla_{X}Z=\left(X\ln f\right)Z,$**

$\left(iii\right)~{}\nabla_{Z}W=\nabla_{Z}^{\circ}W-g\left(Z,~{}W\right)\nabla\ln f,$**

for all $X,~{}Y~{}$on $M_{1}$ and $Z,~{}W~{}$on $M_{2},$where $\nabla$ and $\nabla^{\circ}$ denote the Levi-Civita connections on $M_{1}$ and $M_{2},$respectively. Moreover, $\nabla\ln f,$ the gradient of $\ln f$, is defined by $g\left(\nabla\ln f,~{}U\right)=U\ln f.$ A warped product manifold $M=M_{1}\times_{f}M_{2}$ is trivial if the warping function $f$ is constant. If $M=M_{1}\times_{f}M_{2}$ is a warped product manifold then it is said to be that $M_{1}$ is totally geodesic and $M_{2}$ is totally umbilical submanifold of $M$.

Motivated from the definition of pseudo-slant submanifolds, we see that there are two types of warped product pseudo-slant submanifolds can be constructed. According to them, we have the following cases

[TABLE]

where we consider the the structure vector fields $\xi_{k}^{\prime}s$ are tangent to base manifolds in both cases.

Let us consider the first case of warped product pseudo-slant submanifold of type $M_{\theta}\times_{f}M_{\bot}$ in a nearly Kenmotsu $f-$manifold. The following examples ensure the existence of warped product pseudo-slant submanifold as follows

Example 4.1**.**

It is well-known that $S^{6}$ inherit a nearly Kähler structure, which is not Kählerian. Let us consider $\overline{M}=S^{6}\times_{f}R^{2}$ and let $f\left(t_{1},~{}t_{2}\right)=e^{t_{1}+t_{2}}$. Then it is said that $\overline{M}$ is a nearly Kenmotsu $f$ -manifold which is not Kenmotsu $f$ -manifold with warping function $f.$ On the other hand, $\dim\overline{M}=8$ where $s=2.$ Assume that the coordinates of $M$ are as following

[TABLE]

and for $1\leq i,~{}j\leq 3,~{}k=1,~{}2,$ the following equations are satisfied

[TABLE]

Now, we consider a submanifold of $\overline{M}$ defined by the following immersion $\chi$

[TABLE]

Then the tangent space of $M$ spanned by

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

Then we find

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

It is easy to see that $\varphi W_{3}$ is orthogonal to$~{}TM.$ Thus, the anti-invariant distribution $D_{1}=span\left\{W_{3}\right\}$ and $D_{2}=span\left\{W_{1},~{}W_{2}\right\}$ is a proper slant distribution with slant angle $\theta=arc\cos\left(\dfrac{1}{3}\right)$ such that $\xi_{1}=\dfrac{\partial}{\partial t_{1}}$ and $\xi_{2}=\dfrac{\partial}{\partial t_{2}}$ is tangent to $M$. Thus M is a proper pseudo-slant submanifold. Also, both the distributions are integrable. When we denote the integral manifolds of $D^{\bot}$ and $D^{\theta}$ by $M_{\bot}$ and $M_{\theta},$ respectively. Then the metric tensor $g$ of $M$ is calculated as in the following

[TABLE]

Thus $M$ is a warped product pseudo-slant submanifold of the form $M=M_{\theta}\times_{f}M_{\bot}$ with the warping function $f=\sqrt{u_{1}^{2}+u_{2}^{2}}.$

There is another example of warped product pseudo-slant submanifold in a nearly Kenmotsu $f-$manifold as follows

Example 4.2**.**

Now let $M$ be a submanifold of $\overline{M}$ given by the following immersion $\chi$

[TABLE]

Then the tangent space of $M$ spanned by

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

Then we find

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

It is easy to see that $\varphi Z$ is orthogonal to$~{}TM.$ Thus, the anti-invariant distribution $D_{1}=span\left\{Z\right\}$ and $D_{2}=span\left\{X,~{}Y\right\}$ is a proper slant distribution with slant angle $\theta=arc\cos\left(\dfrac{\sqrt{10}}{20}\right)$ such that $\xi_{1}=\dfrac{\partial}{\partial t_{1}}$ and $\xi_{2}=\dfrac{\partial}{\partial t_{2}}$ is tangent to $M$. Thus M is a proper pseudo-slant submanifold. Also, both the distributions are integrable. When we denote the integral manifolds of $D^{\bot}$ and $D^{\theta}$ by $M_{\bot}$ and $M_{\theta},$ respectively. Then the metric tensor $g$ of $M$ is calculated as in the following

[TABLE]

Thus $M$ is a warped product pseudo-slant submanifold of the form $M=M_{\theta}\times_{f}M_{\bot}$ with the warping function $f=\sqrt{\left(w^{4}+v^{4}\right)}.$

Now, we prove some lemmas for the next section. We begin with the following.

Lemma 4.2**.**

Let $M=M_{\theta}\times_{f}M_{\bot}$ be a non-trivial warped product pseudo slant submanifold of a nearly Kenmotsu $f$-manifold $\overline{M}.$ Then we have

[TABLE]

for any X\on $M_{\theta}$ and $Z~{}$on $M_{\bot}$, where the structure vector fields $\xi_{k}^{\prime}s$ are tangent to $M_{\theta}.$

Proof.

By using (2.22) and (3.1), then we get

[TABLE]

From (3.16), it follows

[TABLE]

By virtue of properties, we deduce

[TABLE]

Taking into account of $\sum_{k=1}^{s}\left(\xi_{k}\ln f\right)=s$ and using Lemma 1 (ii), we get $\left(i\right).$ The second property of lemma can be easily gotten by interchanging $X$ by $TX$ in $\left(ii\right)$ of this lemma. This completes proof of lemma ∎

For the second case of warped product pseudo-slant submanifold of type $M_{\bot}\times_{f}M_{\theta}$ where structure vector fields $\xi_{s}$ are tangent to base manifold. We derive some important lemmas which will use in our main results.

Lemma 4.3**.**

Let $M=M_{\bot}\times_{f}M_{\theta}$ be a warped product pseudo slant submanifold of a nearly Kenmotsu $f$-manifold $\overline{M}$ such that the structure vector fields $\xi_{k}^{\prime}s$ are tangent to $M_{\bot},$ for $1\leq k\leq s.$ Then we have

[TABLE]

for any vector field $X~{}$on $M_{\theta}$ and $Z$ on $M_{\bot}.$

Proof.

Let $M=M_{\bot}\times_{f}M_{\theta}$ be a warped product pseudo slant submanifold of a nearly Kenmotsu $f$-manifold $\overline{M}.$ By using (2.22), we find

[TABLE]

By the virtue of the covariant derivative of $\varphi,$we get

[TABLE]

From (2.13) and Theorem 1, we obtain

[TABLE]

Hence by using Lemma 4.1 (ii) and the covariant derivative of $\varphi$, we derive

[TABLE]

Taking into account of (2.23) and (3.1), we deduce

[TABLE]

Now, by using Lemma 4.1 (ii) and (3.17), it follows that

[TABLE]

On the other hand, for any $X~{}$on $M_{\theta}$ and $Z~{}$on $M_{\bot},$ we conclude that

[TABLE]

Since $\xi_{k}~{}$is tangent to $TM_{\bot}~{}$for each $1\leq k\leq s,$ (2.22) and (3.1) imply

[TABLE]

Again by using Lemma 4.1 (ii) and the covariant derivative of $\varphi$, it is said that

[TABLE]

By the virtue of (2.13) and (3.17), the last equation takes the form

[TABLE]

By using Lemma 1 (ii), (2.23) and (3.17), it follows that

[TABLE]

From Lemma 1 (ii) and (2.22), we have

[TABLE]

Hence from (4.4) and (4), we obtain

[TABLE]

which gives us the desired result. ∎

As a consequence of this lemma, we can give the following corollary.

Corollary 4.1**.**

Let $M=M_{\bot}\times_{f}M_{\theta}$ be a totally geodesic warped product pseudo slant submanifold of a nearly Kenmotsu $f$-manifold $\overline{M}.$ Then at least on the following statements is true:

$\left(i\right)~{}M$* is an anti-invariant submanifold,*

$\left(ii\right)~{}M$* is an invariant submanifold or,*

$\left(iii\right)~{}\left(Z\ln f\right)=\sum_{k=1}^{s}\eta^{k}\left(Z\right).$**

Lemma 4.4**.**

Let $M=M_{\bot}\times_{f}M_{\theta}$ be a warped product pseudo slant submanifold of a nearly Kenmotsu $f$-manifold $\overline{M}.$ Then the followings hold.

$\left(i\right)~{}g\left(h\left(X,~{}X\right),~{}\varphi Z\right)=g\left(h\left(Z,~{}X\right),~{}NX\right),$**

$\left(ii\right)~{}g\left(h\left(TX,~{}TX\right),~{}\varphi Z\right)=g\left(h\left(Z,~{}TX\right),~{}NTX\right),$**

for any $X$ on $M_{\theta}$ and $Z~{}$on $M_{\bot}.$

Proof.

By using (2.22), we have

[TABLE]

From the properties we obtain

[TABLE]

By virtue of (2.9) and (2.13), then we derive

[TABLE]

Taking into account of Lemma 4.1 (ii) and (2.22), it follows that

[TABLE]

Since $X$ and $TX$ are orthogonal vector fields and in view of (2.24), we conclude that

[TABLE]

which gives us $\left(i\right).$ By interchanging $X$ by $TX$ in (4.6), we have the last result of this lemma. ∎

5. Inequality for a Warped Product Pseudo Slant Submanifold of the

form $M_{\bot}\times_{f}M_{\theta}$

In this section, we obtain a geometric inequality of warped product pseudo slant submanifold in terms of the second fundamental form such that $\xi_{k}$ is tangent to the anti-invariant submanifold and the mixed totally geodesic submanifold for each $1\leq k\leq s.$

Now let $M=M_{\bot}\times_{f}M_{\theta}$ be $m$-dimensional warped product pseudo slant submanifold of $\left(2m+s\right)$-dimensional nearly Kenmotsu $f$-manifold $\overline{M}$ with $M_{\theta}$ of dimension $d_{1}=2\beta$ and $M_{\bot}~{}$of dimension $d_{2}=\alpha+s,$ where $M_{\theta}$ and $M_{\bot}$ are the integral manifolds of $D_{\theta}$ and $D^{\bot}$, respectively. Then we consider $\{e_{1},~{}\ldots e_{\alpha},~{}e_{d_{2}=\alpha+1}=\xi_{1},~{}\ldots,~{}e_{d_{2}=\alpha+s}=\xi_{s}\}$ and $\{e_{\alpha+s+1}=e_{1}^{\ast},~{}\ldots e_{\alpha+s+\beta+1}=e_{\beta}^{\ast},~{}e_{\alpha+s+\beta+2}=e_{\beta+1}^{\ast}=\sec\theta Te_{1}^{\ast},~{}\ldots,~{}e_{\alpha+s+2\beta}=e_{2\beta}^{\ast}=\sec\theta Te_{2\beta}^{\ast}\}$ are orthonormal basis of $D^{\bot}$and $D_{\theta},$ respectively. Hence the orthonormal basis of the normal subbundles $\varphi D^{\bot},~{}ND_{\theta}$ and $\mu$ are $\{e_{q+1}=\overline{e}_{1}=\varphi e_{1},~{}\ldots,~{}e_{q+\alpha}=\overline{e}_{\alpha}=\varphi e_{\alpha}\},~{}\{e_{q+\alpha+1}=\overline{e}_{\alpha+1}=\widetilde{e}_{1}=\csc\theta Ne_{1}^{\ast},~{}\ldots,~{}e_{q+\alpha+\beta}=\overline{e}_{\alpha+\beta}=\widetilde{e}_{\beta}=\csc\theta Ne_{\beta}^{\ast},~{}e_{q+\alpha+\beta+1}=\overline{e}_{\alpha+\beta+1}=\widetilde{e}_{\beta+1}=\csc\theta NTe_{1}^{\ast},~{}\ldots e_{m+\alpha+2\beta}=\overline{e}_{\alpha+2\beta}=\widetilde{e}_{2\beta}=\csc\theta NTe_{\beta}^{\ast}\}~{}\{e_{2m-1}=\overline{e}_{m},~{}\ldots,~{}e_{2n+s}=\overline{e}_{2(n-m+s)}\}$, respectively.

Theorem 5.1**.**

Let $M=M_{\bot}\times_{f}M_{\theta}$ be $q$-dimensional mixed totally geodesic warped product pseudo slant submanifold of a $\left(2m+s\right)$-dimensional nearly Kenmotsu $f$-manifold $\overline{M}~{}$such that $\xi_{k}\in\Gamma\left(TM_{\bot}\right)$, where $M_{\bot}$ is an anti-invariant submanifold of dimension $d_{2}$, $M_{\theta}$ is a proper slant submanifold of dimension $d_{1}$ of $\overline{M}$ and $\nabla^{\bot}$ denotes the $\nabla^{\bot}$ is normal connection with respect to normal subbunle. Then we have

$\left(i\right)~{}$The squared norm of the second fundamental form of $M$ is given by

[TABLE]

$\left(ii\right)~{}$The equality case holds in (5.1), if $M_{\bot}$ is totally geodesic and $M_{\theta}$ is a totally umbilical submanifold into $\overline{M}.$

Proof.

The squared norm of the second fundamental form is defined by

[TABLE]

Since $M$ is mixed totally geodesic we obtain

[TABLE]

From (3.19), we have

[TABLE]

By rewriting the last equation as in the components of $\varphi D^{\bot},~{}ND_{\theta}~{}$and $v$, then we get

[TABLE]

which implies

[TABLE]

By considering another adapted frame for $D^{\theta},~{}$we derive

[TABLE]

For a mixed totally geodesic submanifold, since the first and last terms of the right hand side in the above equation vanish identically by using Lemma 4.4, then we obtain

[TABLE]

Hence from Lemma 4.3, for a mixed totally geodesic submanifold and by considering the fact that $\eta^{u}\left(e_{i}\right)=0,$ for each $i=1,~{}2,~{}\ldots d_{2}-1$ and $1\leq u\leq s$ for an orthonormal frame, it follows that

[TABLE]

By adding and subtracting the same term $\sum_{u=1}^{s}\xi_{u}\ln f$ in (5.4), it implies that

[TABLE]

We can easily get $\sum_{u=1}^{s}\xi_{u}\ln f=s$ similar to previous studies for a warped product submanifold of a nearly Kenmotsu $f$-manifold. From the last equation, it is said that

[TABLE]

If the equality case holds in the above equation, then from the terms left in (5.2), we arrive at

[TABLE]

which implies that $M_{\bot}$ is totally geodesic in $\overline{M}.$ In a similar way, from the second and third terms in (5.3), we deduce

[TABLE]

which means

[TABLE]

This completes the proof. ∎

Remark 5.1**.**

For globally frame $f-$manifold. If we substitute $s=1$ in Theorem 5.1. Then nearly Kenmotsu $f-$manifold become nearly Kenmotsu manifold. Therefore Theorem 5.1 coincide with Theorem 4.1 in [3]. This means that Theorem 5.1 generalize Theorem 4.1 from [3].

6. Inequality for a Warped Product Pseudo Slant Submanifold of the

form $M_{\theta}\times_{f}M_{\bot}$

In this part, we obtain a geometric inequality of warped product pseudo slant submanifold in terms of the second fundamental form such that $\xi_{k}$ is tangent to the slant submanifold for each $1\leq k\leq s.$ By assuming $\xi_{k}$ is tangent to $D_{\theta}$, then we can use the last frame.

Theorem 6.1**.**

Let $M=M_{\theta}\times_{f}M_{\bot}$ be a $q$-dimensional mixed totally geodesic warped product pseudo slant submanifold of a $\left(2m+s\right)$-dimensional nearly Kenmotsu $f$-manifold $\overline{M}~{}$such that $\xi_{k}\in\Gamma\left(TM_{\theta}\right)$, where $M_{\bot}$ is an anti-invariant submanifold of dimension $d_{2}=\alpha$ and $M_{\theta}$ is a proper slant submanifold of dimension $d_{1}=2\beta+s$ of $\overline{M}.$ Then we have

$\left(i\right)~{}$The squared norm of the second fundamental form of $M$ is given by

[TABLE]

$\left(ii\right)~{}$The equality case holds in (6.1), if $M_{\theta}$ is totally geodesic and $M_{\bot}$ is a totally umbilical submanifold of $\overline{M}.$

Proof.

By virtue of definition of second fundamental form, we have

[TABLE]

Since $M$ is mixed totally geodesic we derive

[TABLE]

By using (3.19), then we obtain

[TABLE]

The above equation can be written in the components of $\varphi D^{\bot},~{}ND_{\theta}~{}$and $v$ as

[TABLE]

which gives us

[TABLE]

By taking into account of another adapted frame for $ND^{\theta},~{}$we get

[TABLE]

Thus from Lemma 4.2, for a mixed totally geodesic submanifold and by considering the fact that $\eta^{u}\left(e_{j}\right)=0,$ for each $j=1,~{}2,~{}\ldots d_{1}-1$ and $1\leq u\leq s$ for an orthonormal frame, it implies that

[TABLE]

From the hypothesis, we deduce

[TABLE]

As we seen that $Te_{2\beta+u}^{*}=T\xi_{s}=0$. Then using (3.20), we derive at

[TABLE]

From virtue of (3.17) and the fact that $\sum_{k=1}^{s}\left(\xi_{k}\ln f\right)=s$, we find that

[TABLE]

In view of the trigonometric functions and from (3.17), we conclude

[TABLE]

which implies that

[TABLE]

Thus the above equation yields

[TABLE]

If the equality holds, by using the terms left hand side in (6.2) and (6.3), we get the following conditions

[TABLE]

where $D=D_{\theta}\oplus\overline{\xi},~{}\overline{\xi}=\sum_{u=1}^{s}\xi_{u}.$ This implies that $M_{\theta}$ is totally geodesic in $\overline{M}~{}$and $h\left(D^{\bot},~{}D^{\bot}\right)\subseteq ND_{\theta}.~{}$On the other hand, using Lemma 4 for a mixed totally geodesic submanifold we get

[TABLE]

for all vector fields Z,~{}W\on$~{}M_{\bot}$and $X$ on $M_{\theta}.$ The last equations means that $M_{\bot}$ is a totally umbilical submanifold of $\overline{M}~{}$and so the equality case holds. This completes the proof of theorem ∎

Applications

In this paper, we study warped product pseudo-slant submanifolds of nearly Kenmotsu $f$-manifolds. We generalize some previous results on nearly Kähler manifolds [24] and nearly Kenmotsu manifolds [3]. That is, if we consider $s=0$ in Theorem 6.1, then by the definition of globally frame $f-$manifold. It leads that the nearly Kenmotsu $f-$maniold turn into nearly Kaehler manifiold. So Theorem 6.1 generalize Theorem 4.1 in [24] for warped product pseudo-slant submanifold of nearly Kaehler manifold.

On the other hand, if we choose $s=1$ then we give the following theorem as a consequence of Theorem 6.1 follows

Theorem 6.2**.**

Let $M=M_{\theta}\times_{f}M_{\bot}$ be an $q$-dimensional mixed totally geodesic warped product pseudo slant submanifold of a $\left(2m+1\right)$-dimensional nearly Kenmotsu manifold $\overline{M}~{}$such that $\xi\in\Gamma\left(TM_{\theta}\right)$, where $M_{\bot}$ is an anti-invariant submanifold of dimension $d_{2}=\alpha$ and $M_{\theta}$ is a proper slant submanifold of dimension $d_{1}=2\beta+1$ of $\overline{M}.$ Then we have

$\left(i\right)~{}$The squared norm of the second fundamental form of $M$ is given by

[TABLE]

$\left(ii\right)~{}$The equality case holds in (6.1), if $M_{\theta}$ is totally geodesic and $M_{\bot}$ is a totally umbilical submanifold of $\overline{M}.$

The above result agrees and modified version with the inequality for a warped product pseudo-slant sub-manifold of nearly Kenmotsu manifold obtained in [3]