Chen Inequalities for Warped Product Pointwise Bi-Slant Submanifolds of Complex Space Forms and Its Applications

Abstract

In this paper, by using new-concept pointwise bi-slant immersions, we derive a fundamental inequality theorem for the squared norm of the mean curvature via isometric warped-product pointwise bi-slant immersions into complex space forms, involving the constant holomorphic sectional curvature c, the Laplacian of the well-defined warping function, the squared norm of the warping function, and pointwise slant functions. Some applications are also given.

1. Introduction

In the submanifolds theory, creating a relationship between extrinsic and intrinsic invariants is considered to be one of the most basic problems. Most of these relations play a notable role in submanifolds geometry. The role of immersibility and non-immersibility in studying the submanifolds geometry of a Riemannian manifold was affected by the pioneering work of the Nash embedding theorem [1], where every Riemannian manifold realizes an isometric immersion into a Euclidean space of sufficiently high codimension. This becomes a very useful object for the submanifolds theory, and was taken up by several authors (for instance, see [2,3,4,5,6,7,8,9,10,11,12,13,14,15]). Its main purpose was considered to be how Riemannian manifolds could always be treated as Riemannian submanifolds of Euclidean spaces. Inspired by this fact, Nolker [16] classified the isometric immersions of a warped product decomposition of standard spaces. Motivated by these approaches, Chen started one of his programs of research in order to study the impressibility and non-immersibility of Riemannian warped products into Riemannian manifolds, especially in Riemannian space forms (see [11,17,18,19]). Recently, a lot of solutions have been provided to his problems by many geometers (see [18] and references therein).

The field of study which includes the inequalities for warped products in contact metric manifolds and the Hermitian manifold is gaining importance. In particular, in [17], Chen observed the strong isometrically immersed relationship between the warping function f of a warped product $M_1{\times }_f{M}_2$ and the norm of the mean curvature, which isometrically immersed into a real space form.

Theorem 1.

Let $\tilde{M}(c)$ be a $m-$dimensional real space form and let $\phi :M=M_1{\times }_f{M}_2$ be an isometric immersion of an $n-$dimensional warped product into $\tilde{M}(c)$. Then:

$$\begin{array}{c}\hfill \frac{\Delta f}{f}\le \frac{n^2}{4n_2}{\Vert H\Vert }^2+n_{1}c,\end{array}$$

(1)

where $n_i=dim{M}_i$, $i=1,2$, and Δ is the Laplacian operator of $M_1$ and H is the mean curvature vector of $M^n$. Moreover, the equality holds in (1) if, and only if, φ is mixed and totally geodesic and $n_1{H}_1=n_2{H}_2$ such that $H_1$ and $H_2$ are partially mean curvatures of $M_1$ and $M_2$, respectively.

In [2,5,20,21,22,23,24,25,26,27,28,29,30,31], the authors discuss the study of Einstein, contact metrics, and warped product manifolds for the above-mentioned problems. Furthermore, in regard to the collections of such inequalities, we referred to [12] and references therein. The motivation came from the study of Chen and Uddin [32], which proved the non-triviality of warped-product pointwise bi-slant submanifolds of a Kaehler manifold with supporting examples. If the sectional curvature is constant with a Kaehler metric, then it is called complex space forms. In this paper, we consider the warped-product pointwise bi-slant submanifolds which isometrically immerse into a complex space form, where we then obtain a relationship between the squared norm of the mean curvature, constant sectional curvature, the warping function, and pointwise bi-slant functions. We will announce the main result of this paper in the following.

Theorem 2.

Let ${\tilde{M}}^{2m}(c)$ be the complex space form and let $\phi :M^n=M_1^{n_1}{\times }_f{M}_2^{n_2}\to {\tilde{M}}^{2m}(c)$ be an isometric immersion from warped product pointwise bi-slant submanifolds into ${\tilde{M}}^{2m}(c)$. Then, the following inequality is satisfied:

$$\begin{array}{c}\hfill \Delta (lnf)\le {\Vert \nabla lnf\Vert }^2+\frac{n^2}{4n_2}{\Vert H\Vert }^2+\frac{n_{1}c}{4}-\frac{3c}{4n_2}\left(n_1{cos}^2{\theta }_1+n_2{cos}^2{\theta }_2\right),\end{array}$$

(2)

where ${\theta }_1$ and ${\theta }_2$ are pointwise slant functions along $M_1$ and $M_2$, respectively. Furthermore, ∇ and Δ are the gradient and the Laplacian operator on $M_1^{n_1}$, respectively, and H is the mean curvature vector of $M^n$. The equality case holds in (2) if and only if φ is a mixed totally geodesic isometric immersion and the following satisfies

$$\frac{H_1}{H_2}=\frac{n_2}{n_1}$$

where $H_1$ and $H_2$ are the mean curvature vectors along $M_1^{n_1}$ and $M_2^{n_2}$, respectively.

As an application of Theorem 2 in a compact orientated Riemannian manifold with a free boundary condition, we prove that:

Theorem 3.

Let $M^n=M_1^{n_1}{\times }_f{M}_2^{n_2}$ be a compact, orientate warped product pointwise bi-slant submanifold in a complex space form ${\tilde{M}}^{2m}(c)$ such that $M_1^{n_1}$ is a $n_1$-dimensional and $M_2^{n_2}$ is a $n_2-$dimensional pointwise slant submanifold ${\tilde{M}}^{2m}(c)$. Then, $M^n$ is simply a Riemannian product if, and only if:

$$\begin{array}{c}\hfill {\parallel H\parallel }^2\ge \frac{c}{n^2}\left(3n_1{cos}^2{\theta }_1+3n_2{cos}^2{\theta }_2-n_1{n}_2\right),\end{array}$$

(3)

where H is the mean curvature vector of $M^n$. Moreover, ${\theta }_1$ and ${\theta }_2$ are pointwise slant functions.

By using classifications of pointwise bi-slant submanifolds which were defined in [32], we derived similar inequalities for warped product pointwise pseudo-slant submanifolds [33], warped product pointwise semi-slant submanifolds [34], and CR-warped product submanifolds [17] in a complex space form as well.

2. Preliminaries and Notations

An almost complex structure J and a Riemannian metric g, such that $J^2=-I$ and $g(JX,JY)=g(X,Y)$, for $X,Y\in \mathfrak{X}(\tilde{M})$, where I denotes the identity map and $\mathfrak{X}(\tilde{M})$ is the space containing vector fields tangent to $\tilde{M}$, then $(M,J,g)$ is an almost Hermitian manifold. If the almost complex structure satisfied $({\tilde{\nabla }}_{U}J)V=0,$ for any $U,V\in \mathfrak{X}(\tilde{M})$ and $\tilde{\nabla }$ is a Levi-Cevita connection $\tilde{M}$. In this case, $\tilde{M}$ is called the Kaehler manifold. A complex space form of constant holomorphic sectional curvature c is denoted by ${\tilde{M}}^{2m}(c)$, and its curvature tensor $\tilde{R}$ can be expressed as:

$$\begin{array}{cc}\hfill \tilde{R}\left(U,V,Z,W\right)=\frac{c}{4}& (g(U,Z)g(V,W)-g(V,Z)g(U,W)+g(U,JZ)g(JV,W)\hfill \\ & -g(V,JZ)g(U,JW)+2g(U,JV)g(JZ,W)),\hfill \end{array}$$

(4)

for every $U,V,Z,W\in \mathfrak{X}({\tilde{M}}^{2m}(c))$. A Riemannian manifold ${\tilde{M}}^m$ and its submanifold M, the Gauss and Weingarten formulas are defined by ${\tilde{\nabla }}_{U}V={\nabla }_{U}V+h(U,V),and{\tilde{\nabla }}_U\xi =-A_{\xi }U+{\nabla }_U^{\perp }\xi ,$ respectively for each $U,V\in \mathfrak{X}(M)$ and for the normal vector field $\xi$ of M, where h and $A_{\xi }$ are denoted as the second fundamental form and shape operator. They are related as $g(h(U,V),N)=g(A_{N}U,V)$. Now, for any $U\in \mathfrak{X}(M)$ and for the normal vector field $\xi$ of M, we have:

$$\begin{array}{c}\hfill (i)JU=PU+FU,(ii)J\xi =t\xi +f\xi ,\end{array}$$

(5)

where $PU(t\xi )$ and $FU(f\xi )$ are tangential to M and normal to M, respectively. Similarly, the equations of Gauss are given by:

$$\begin{array}{cc}\hfill R(U,V,Z,W)=\tilde{R}(U,V,Z,W)+& g\left(h(U,W),h(V,Z)\right)-g\left(h(U,Z),h(V,W)\right).\hfill \end{array}$$

(6)

for all $U,V,Z,W$ are tangent M, where R and $\tilde{R}$ are defined as the curvature tensor of ${\tilde{M}}^m$ and $M^n$, respectively.

The mean curvature H of Riemannian submanifold $M^n$ is given by

$$H=\frac{1}{n}trace(h).$$

A submanifold $M^n$ of Riemannian manifold ${\tilde{M}}^m$ is said to be totally umbilical and totally geodesic if $h(U,V)=g(U,V)H$ and $h(U,V)=0$, for any $U,V\in \mathfrak{X}(M)$, respectively, where H is the mean curvature vector of $M^n$. Furthermore, if $H=0$, them $M^n$ is minimal in ${\tilde{M}}^m$.

A new class called a “pointwise slant submanifold” has been studied in almost Hermitian manifolds by Chen-Gray [35]. They provided the following definitions of these submanifolds:

Definition 1.

[35] A submanifold $M^n$ of an almost Hermitian manifold ${\tilde{M}}^{2m}$ is a pointwise slant if, for any non-zero vector $X\in \mathfrak{X}(T_{x}M)$ and each given point $x\in M^n$, the angle $\theta (X)$ between $JX$ and tangent space $T_{x}M$ is free from the choice of the nonzero vector X. In this case, the Wirtinger angle become a real-valued function and it is non-constant along $M^n$, which is defined on $T^{*}M$ such that $\theta :T^{*}M\to \mathbb{R}$.

Chen-Gray in [35] derived a characterization for the pointwise slant submanifold, where $M^n$ is a pointwise slant submanifold if, and only if, there exists a constant $\lambda \in [0,1]$ such that $P^2=-{cos}^2\theta I$, where P is a (1,1) tensor field and I is an identity map. For more classifications, we referred to [35].

Following the above concept, a pointwise bi-slant immersion was defined by Chen-Uddin in [18], where they defined it as follows:

Definition 2.

A submanifold $M^n$ of an almost Hermitian manifold ${\tilde{M}}^{2m}$ is said to be a pointwise bi-slant submanifold if there exists a pair of orthogonal distributions ${\mathcal{D}}_{{\theta }_1}$ and ${\mathcal{D}}_{{\theta }_2}$, such that:

(i) 

$T{M}^n={\mathcal{D}}_{{\theta }_1}\oplus {\mathcal{D}}_{{\theta }_2};$

(ii) 

$J{\mathcal{D}}_{{\theta }_1}\perp {\mathcal{D}}_{{\theta }_2}andJ{\mathcal{D}}_{{\theta }_2}\perp {\mathcal{D}}_{{\theta }_1};$

(iii) 

Each distribution ${\mathcal{D}}_{{\theta }_i}$ is a pointwise slant with a slant function ${\theta }_i:T^{*}M\to \mathbb{R}fori=1,2.$

Remark 1.

A pointwise bi-slant submanifold is a bi-slant submanifold if each slant functions ${\theta }_i:T^{*}M\to \mathbb{R}fori=1,2.$ are constant along $M^n$ (see [13]).

Remark 2.

If ${\theta }_1=\frac{\pi }{2}or{\theta }_2=\frac{\pi }{2}$, then $M^n$ is called a pointwise pseudo-slant submanifold (see [33]).

Remark 3.

If ${\theta }_1=0or{\theta }_2=0$, in this case, $M^n$ is a coinciding pointwise semi-slant submanifold (see [14,34]).

Remark 4.

If ${\theta }_2=\frac{\pi }{2}$ and ${\theta }_1=0$, then $M^n$ is CR-submanifold of the almost Hermitian manifold.

In this context, we shall define another important Riemannian intrinsic invariant called the scalar curvature of ${\tilde{M}}^m$, and denoted at $\tilde{\tau }(T_x{\tilde{M}}^m)$, which, at some x in ${\tilde{M}}^m$, is given:

$$\begin{array}{c}\hfill \tilde{\tau }(T_x{\tilde{M}}^m)=\sum _{1\le \alpha <\beta \le m}{\tilde{K}}_{\alpha \beta },\end{array}$$

(7)

where ${\tilde{K}}_{\alpha \beta }=\tilde{K}\left(e_{\alpha }\wedge e_{\beta }\right)$. It is clear that the first equality (7) is congruent to the following equation, which will be frequently used in subsequent proof:

$$\begin{array}{c}\hfill 2\tilde{\tau }(T_x{\tilde{M}}^m)=\sum _{1\le \alpha <\beta \le m}{\tilde{K}}_{\alpha \beta },1\le \alpha ,\beta \le n.\end{array}$$

(8)

Similarly, scalar curvature $\tilde{\tau }(L_x)$ of $L-$plan is given by:

$$\begin{array}{c}\hfill \tilde{\tau }(L_x)=\sum _{1\le \alpha <\beta \le m}{\tilde{K}}_{\alpha \beta },\end{array}$$

(9)

An orthonormal basis of the tangent space $T_{x}M$ is $\{e_1,\cdots e_n\}$ such that $e_r=(e_{n+1},\cdots e_m)$ belong to the normal space $T^{\perp }M$. Then, we have:

$$\begin{array}{c}\hfill h_{\alpha \beta }^r=g(h(e_{\alpha },e_{\beta }),e_r),\\ \hfill {\Vert h\Vert }^2=\sum _{\alpha ,\beta =1}^{n}g\left(h(e_{\alpha },e_{\beta }),h(e_{\alpha },e_{\beta }\right).\end{array}$$

(10)

Let $K_{\alpha \beta }$ and ${\tilde{K}}_{\alpha \beta }$ be the sectional curvatures of the plane section spanned by $e_{\alpha }$ and $e_{\beta }$ at x in a submanifold $M^n$ and a Riemannian manifold ${\tilde{M}}^m$, respectively. Thus, $K_{\alpha \beta }$ and ${\tilde{K}}_{\alpha \beta }$ are the intrinsic and extrinsic sectional curvatures of the span $\{e_{\alpha },e_{\beta }\}$ at x. Thus, from the Gauss Equation (6)(i), we have:

$$\begin{array}{cc}\hfill 2\tau (T_x{M}^n)=K_{\alpha \beta }& =2\tilde{\tau }(T_x{M}^n)+\sum _{r=n+1}^m\left(h_{\alpha \alpha }^r{h}_{\beta \beta }^r-{(h_{\alpha \beta }^r)}^2\right)\hfill \\ & ={\tilde{K}}_{\alpha \beta }+\sum _{r=n+1}^m\left(h_{\alpha \alpha }^r{h}_{\beta \beta }^r-{(h_{\alpha \beta }^r)}^2\right).\hfill \end{array}$$

(11)

The following consequences come from (6) and (11), as:

$$\begin{array}{c}\hfill \tau (T_x{M}_1^{n_1})=\sum _{r=n+1}^m\sum _{1\le i<j\le n_1}\left(h_{ii}^r{h}_{jj}^r-{(h_{ij}^r)}^2\right)+\tilde{\tau }(T_x{M}_1^{n_1}).\end{array}$$

(12)

Similarly, we have:

$$\begin{array}{c}\hfill \tau (T_x{M}_2^{n_2})=\sum _{r=n+1}^m\sum _{n_1+1\le a<b\le n}\left(h_{aa}^r{h}_{bb}^r-{(h_{ab}^r)}^2\right)+\tilde{\tau }(T_x{M}_2^{n_2}).\end{array}$$

(13)

Assume that $M_1^{n_1}$ and $M_2^{n_2}$ are two Riemannian manifolds with their Riemannian metrics $g_1$ and $g_2$, respectively. Let f be a smooth function defined on $M_1^{n_1}$. Then, the warped product manifold $M^n=M_1^{n_1}{\times }_f{M}_2^{n_2}$ is the manifold $M_1^{n_1}\times M_2^{n_2}$ furnished by the Riemannian metric $g=g_1+f^2{g}_2$, which defined in [36]. When considering that the $M^n=M_1^{n_1}{\times }_f{M}_2^{n_2}$ is the warped product manifold, then for any $X\in \mathfrak{X}(M_1)$ and $Z\in \mathfrak{X}(M_2)$, we find that:

$$\begin{array}{c}\hfill {\nabla }_{Z}X={\nabla }_{X}Z=(Xlnf)Z.\end{array}$$

(14)

Let $\{e_1,\cdots e_n\}$ be an orthonormal frame for $M^n$; then, summing up the vector fields such that:

$$\begin{array}{c}\hfill \sum _{i=1}^{n_1}\sum _{j=1}^{n_2}K(e_{\alpha }\wedge e_{\beta })=\sum _{\alpha =1}^{n_1}\sum _{\beta =1}^{n_2}\left(\left({\nabla }_{e_{\alpha }}{e}_{\alpha }\right)lnf-e_{\alpha }\left(e_{\beta }lnf\right)-{\left(e_{\alpha }lnf\right)}^2\right).\end{array}$$

From (Equation (3.3) in [11]), the above equation implies that:

$$\begin{array}{c}\hfill \sum _{\alpha =1}^{n_1}\sum _{\beta =1}^{n_2}K(e_{\alpha }\wedge e_{\beta })=n_2\left(\Delta (lnf)-{\Vert \nabla (lnf)\Vert }^2\right)=\frac{n_2\Delta f}{f}.\end{array}$$

(15)

Remark 5.

A warped product manifold $M^n=M_1^{n_1}{\times }_f{M}_2^{n_2}$ is said to be trivial or a simple Riemannian product manifold if the warping function f is constant.

3. Main Inequality for Warped Product Pointwise Bi-Slant Submanifolds

To obtain similar inequalities like Theorem 1, for warped product pointwise bi-slant submanifolds of complex space forms, we need to recall the following lemma.

Lemma 1.

[10] Let $a_1,a_2,\cdots a_n,a_{n+1}$ be $n+1$ be real numbers with

$$\begin{array}{c}\hfill {(\sum _{i=1}^n{a}_i)}^2=(n-1)(\sum _{i=1}^n{a}_i^2+a_{n+1}),n\ge 2.\end{array}$$

Then $2a_1.a_2\ge a_3$ holds if and only if $a_1+a_2=a_3=\cdots =a_k$.

Proof of Theorem 2.

If substitute $X=Z=e_{\alpha }$ and $Y=W=e_{\beta }$ for $1\le \alpha ,\beta \le n$ in (4), and (6), taking summing up then

$$\begin{array}{c}\hfill \sum _{\alpha ,\beta =1}^n\tilde{R}(e_{\alpha },e_{\beta },e_{\alpha },e_{\beta })=\frac{c}{4}\left(n(n-1)+3\sum _{\alpha ,\beta =1}^n{g}^2(J{e}_{\alpha },e_{\beta })\right).\end{array}$$

(16)

As $M^n$ is a pointwise bi-slant submanifold, we defined an adapted orthonormal frame as $n=2d_1+2d_2$ follows $\left\{e_1,e_2=sec{\theta }_{1}P{e}_1,\cdots ,e_{2d_1-1},e_{2d_1}=sec{\theta }_{1}P{e}_{2d_1-1},\cdots ,e_{2d_1+1},e_{2d_1+2}=sec{\theta }_{2}P{e}_{2d_1+1},\cdots ,e_{2d_1+2d_2-1},e_{2d_1+2d_2}=sec{\theta }_{2}P{e}_{2d_1+2d_2-1}\right\}$. Thus, we defined it such that $g(e_1,J{e}_2)=-g(J{e}_1,e_2)=g(J{e}_1,sec{\theta }_{1}P{e}_1)$, which implies that $g(e_1,J{e}_2)=-sec{\theta }_{1}g(P{e}_1,P{e}_1)$. □

Following ((2.8) in [32]), we get $g(e_1,J{e}_2)=cos{\theta }_{1}g(e_1,e_2)$. Therefore, we easily obtained the following relation:

$$g^2(e_{\alpha },J{e}_{\beta })=\left\{\begin{array}{c}{cos}^2{\theta }_1,foreach\alpha =1,\cdots ,2d_1-1,\hfill \\ {cos}^2{\theta }_2,foreach\beta =2d_1+1,\cdots ,2d_1+2d_1-1.\hfill \end{array}\right.$$

Hence, we have:

$$\begin{array}{c}\hfill \sum _{\alpha ,\beta =1}^n{g}^2(J{e}_{\alpha },e_{\beta })=\left(n_1{cos}^2\theta +n_2{cos}^2\theta \right).\end{array}$$

(17)

Following from (17), (16), and (6), we find that:

$$\begin{array}{cc}\hfill 2\tau =& \frac{c}{4}n(n-1)+\frac{c}{4}\left(3n_1{cos}^2{\theta }_1+3n_2{cos}^2{\theta }_2\right)+n^2{\Vert H\Vert }^2-{\Vert h\Vert }^2.\hfill \end{array}$$

(18)

Let us assume that:

$$\begin{array}{c}\hfill \delta =2\tau -\frac{c}{4}n(n-1)-\frac{c}{4}\left(3n_1{cos}^2{\theta }_1+3n_2{cos}^2{\theta }_2\right)-\frac{n^2}{2}{\Vert H\Vert }^2.\end{array}$$

(19)

Then, from (19), and (18), we get:

$$\begin{array}{c}\hfill n^2{\Vert H\Vert }^2=2\left(\delta +{\Vert h\Vert }^2\right).\end{array}$$

(20)

Thus, from an orthogonal frame $\{e_1,e_2,\cdots e_n\}$, the proceeding equation takes the new form:

$$\begin{array}{cc}\hfill {\left(\sum _{r=n+1}^{2m}\sum _{i=1}^n{h}_{AA}^r\right)}^2=2(\delta & +\sum _{r=n+1}^{2m}\sum _{i=1}^n{(h_{AA}^r)}^2+\sum _{r=n+1}^{2m+1}\sum _{i<j=1}^n{(h_{AB}^r)}^2\hfill \\ & +\sum _{r=n+1}^{2m}\sum _{A,B=1}^n{(h_{AB}^r)}^2).\hfill \end{array}$$

(21)

This can be expressed in more detail, such as:

$$\begin{array}{cc}\hfill \frac{1}{2}{\left(h_{11}^{n+1}+\sum _{A=2}^{n_1}{h}_{AA}^{n+1}+\sum _{l=n_1+1}^n{h}_{ll}^{n+1}\right)}^2& =\delta +{(h_{11}^{n+1})}^2+\sum _{A=2}^{n_1}{(h_{AA}^{n+1})}^2+\sum _{l=n_1+1}^n{(h_{ll}^{n+1})}^2\hfill \\ & -\sum _{2\le B\ne q\le n_1}{h}_{BB}^{n+1}{h}_{qq}^{n+1}-\sum _{n_1+1\le l\ne s\le n}{h}_{ll}^{n+1}{h}_{ss}^{n+1}\hfill \\ & +\sum _{A<B=1}^n{(h_{AB}^{n+1})}^2+\sum _{r=n+1}^{2m}\sum _{A,B=1}^n{(h_{AB}^r)}^2.\hfill \end{array}$$

(22)

Assume that $a_1=h_{11}^{n+1}$, $a_2={\sum }_{A=2}^{n_1}{h}_{AA}^{n+1}$, and $a_3={\sum }_{l=n_1+1}^n{h}_{ll}^{n+1}$. Then, applying Lemma 1 in (22), we derive:

$$\begin{array}{cc}\hfill \frac{\delta }{2}+\sum _{A<B=1}^n{(h_{AB}^{n+1})}^2+\frac{1}{2}\sum _{r=n+1}^{2m}\sum _{A,B=1}^n{(h_{AB}^r)}^2\le & \sum _{2\le B\ne q\le n_1}{h}_{BB}^{n+1}{h}_{qq}^{n+1}\hfill \\ & +\sum _{n_1+1\le l\ne s\le n}{h}_{ll}^{n+1}{h}_{ss}^{n+1}.\hfill \end{array}$$

(23)

with equality holds in (23) if and only if

$$\begin{array}{c}\hfill \sum _{A=2}^{n_1}{h}_{AA}^{n+1}=\sum _{B=n_1+1}^n{h}_{BB}^{n+1}.\end{array}$$

(24)

On the other hand, from (15), we have:

$$\begin{array}{c}\hfill \frac{n_2\Delta f}{f}=\tau -\sum _{1\le A<B\le n_1}K(e_A\wedge e_B)-\sum _{n_1+1\le l<q\le n}K(e_l\wedge e_q).\end{array}$$

(25)

Then from (6) and the scalar curvature for the complex space form (11), we get:

$$\begin{array}{cc}\hfill n_2\frac{\Delta f}{f}=\tau & -\frac{n_1(n_1-1)c}{8}-\frac{3n_{1}c}{4}{cos}^2{\theta }_1-\sum _{r=n+1}^{2m}\sum _{1\le A\ne B\le n_1}(h_{AA}^r{h}_{BB}^r-{(h_{AB}^r)}^2)\hfill \\ & -\frac{n_2(n_2-1)c}{8}-\frac{3n_{2}c}{4}{cos}^2{\theta }_2-\sum _{r=n+1}^{2m}\sum _{n_1+1\le l\ne q\le n}(h_{ll}^r{h}_{qq}^r-{(h_{lq}^r)}^2).\hfill \end{array}$$

(26)

Now from (23) and (26), we have:

$$\begin{array}{c}\hfill n_2\frac{\Delta f}{f}\le \rho -\frac{n(n-1)c}{8}+\frac{n_1{n}_{2}c}{4}-\frac{3n_{1}c}{4}{cos}^2{\theta }_1-\frac{\delta }{2}-\frac{3n_{2}c}{4}{cos}^2{\theta }_2.\end{array}$$

(27)

Using (19) in the above equation and relation $\frac{\Delta f}{f}=\Delta (lnf)-{\Vert \nabla lnf\Vert }^2$, we derive:

$$\begin{array}{c}\hfill n_2\left(\Delta (lnf)-{\Vert \nabla lnf\Vert }^2\right)\le \frac{n^2}{4}{\Vert H\Vert }^2+\frac{c}{4}\left(n_1{n}_2+3n_1{cos}^2{\theta }_1+3n_2{cos}^2{\theta }_2\right).\end{array}$$

(28)

which implies inequality. The equality sign holds in (2) if, and only if, the leaving terms in (23) and (24) imply that:

$$\begin{array}{c}\hfill \sum _{r=n+2}^{2m}\sum _{B=1}^{n_1}{h}_{BB}^r=\sum _{r=n+2}^{2m}\sum _{A=n_1+1}^{n_1}{h}_{AA}^r=0,\end{array}$$

(29)

and $n_1{H}_1=n_2{H}_2$, where $H_1$ and $H_2$ are partially mean curvature vectors on $M_1^{n_1}$ and $M_2^{n_2}$, respectively. Moreover, also from (23), we find that

$$\begin{array}{cc}\hfill h_{AB}^r=0,foreach& 1\le A\le n_1\hfill \\ & n_1+1\le B\le n\hfill \\ & n+1\le r\le 2m.\hfill \end{array}$$

(30)

This shows that $\phi$ is a mixed, totally geodesic immersion. The converse part of (30) is true in a warped product pointwise bi-slant into the complex space form. Thus, we reached our promised result.

Consequences of Theorem 2

Inspired by the research in [6,34] and using the Remark 3 in Theorem 2 for pointwise semi-slant warped product submanifolds, we obtained:

Corollary 1.

Let $\phi :M^n=M_1^{n_1}{\times }_f{M}_2^{n_2}\to {\tilde{M}}^{2m}(c)$ be an isometric immersion from the warped product pointwise semi-slant submanifold into a complex space form ${\tilde{M}}^{2m}(c)$, where $M_1^{n_1}$ is the holomorphic and $M_2^{n_2}$ is the pointwise slant submanifolds of ${\tilde{M}}^{2m}(c)$. Then, we have the following inequality:

$$\begin{array}{c}\hfill \Delta (lnf)\le {\Vert \nabla lnf\Vert }^2+\frac{n^2}{4n_2}{\Vert H\Vert }^2+\frac{n_{1}c}{4}-\frac{3c}{4n_2}\left(n_1+n_2{cos}^2\theta \right),\end{array}$$

(31)

where $n_i=dim{M}_i,i=1,2$. Furthermore, ∇ and Δ are the gradient and the Laplacian operator on $M_1^{n_1}$, respectively, and H is the mean curvature vector of $M^n$. The equality sign holds in (31) if, and only if, $n_1{H}_1=n_2{H}_2$, where $H_1$ and $H_2$ are the mean curvature vectors along $M_1^{n_1}$ and $M_2^{n_2}$, respectively, and φ is a mixed, totally geodesic immersion.

From the motivation studied in [14,34], we present the following consequence of Theorem 2 by using the Remark 2 for a nontrivial warped product pointwise pseudo-slant submanifold of a complex space, such that:

Corollary 2.

Let $\phi :M^n=M_1^{n_1}{\times }_f{M}_2^{n_2}\to {\tilde{M}}^{2m}(c)$ be an isometric immersion from a warped product pointwise pseudo-slant submanifold into a complex space form ${\tilde{M}}^{2m}(c)$, such that $M_1^{n_1}$ is a totally real and $M_2^{n_2}$ is a pointwise slant submanifold of ${\tilde{M}}^{2m}(c)$. Then, we have the following inequality:

$$\begin{array}{c}\hfill \Delta (lnf)\le {\Vert \nabla lnf\Vert }^2+\frac{n^2}{4n_2}{\Vert H\Vert }^2+\frac{n_{1}c}{4}-\frac{3c}{4}{cos}^2\theta ,\end{array}$$

(32)

where $n_i=dim{M}_i,i=1,2$. Furthermore, ∇ and Δ are the gradient and the Laplacian operator on $M_1^{n_1}$, respectively, and H is the mean curvature vector of $M^n$. The equality condition holds in (32) if, and only if, the following satisfies

$$\frac{H_1}{H_2}=\frac{n_2}{n_1}$$

: where $H_1$ and $H_2$ are the mean curvature vectors along $M_1^{n_1}$ and $M_2^{n_2}$, respectively, and φ is a mixed, totally geodesic isometric immersion.

Corollary 3.

Let $\phi :M^n=M_1^{n_1}{\times }_f{M}_2^{n_2}\to {\tilde{M}}^{2m}(c)$ be an isometric immersion from a warped product pointwise pseudo-slant submanifold into a complex space form ${\tilde{M}}^{2m}(c)$, such that $M_1^{n_1}$ is a pointwise slant and $M_2^{n_2}$ is a totally real submanifold of ${\tilde{M}}^{2m}(c)$. Then, we have the following:

$$\begin{array}{c}\hfill \Delta (lnf)\le {\Vert \nabla lnf\Vert }^2+\frac{n^2}{4n_2}{\Vert H\Vert }^2+\frac{n_{1}c}{4}-\frac{3n_{1}c}{4n_2}{cos}^2\theta ,\end{array}$$

(33)

where $n_i=dim{M}_i,i=1,2$. Furthermore, ∇ and Δ are the gradient and the Laplacian operator on $M_1^{n_1}$, respectively, and H is the mean curvature vector of $M^n$. This equally holds in (33) if, and only if, φ is a mixed, totally geodesic isometric immersion and the following satisfies

$$\frac{H_1}{H_2}=\frac{n_2}{n_1},$$

, where $H_1$ and $H_2$ are the mean curvature vectors along $M_1^{n_1}$ and $M_2^{n_2}$, respectively.

Similarly, using Remark 4 and from [17], we got the following result from Theorem 2:

Corollary 4.

Let $\phi :M^n=M_1^{n_1}{\times }_f{M}_2^{n_2}\to {\tilde{M}}^{2m}(c)$ be an isometric immersion from a CR-warped product into a complex space form ${\tilde{M}}^{2m}(c)$, such that $M_1^{n_1}$ is a holomorphic submanifold and $M_2^{n_2}$ is a totally real submanifold of ${\tilde{M}}^{2m}(c)$. Then, we get the following:

$$\begin{array}{c}\hfill \Delta (lnf)\le {\Vert \nabla lnf\Vert }^2+\frac{n^2}{4n_2}{\Vert H\Vert }^2+\frac{n_{1}c}{4}-\frac{3n_{1}c}{4n_2},\end{array}$$

(34)

where $n_i=dim{M}_i,i=1,2$. Furthermore, ∇ and Δ are the gradient and the Laplacian operator on $M_1^{n_1}$, respectively, and H is the mean curvature vector of $M^n$. The same holds in (34) if, and only if, φ is mixed and totally geodesic, and $n_1{H}_1=n_2{H}_2$, where $H_1$ and $H_2$ are the mean curvature vectors on $M_1^{n_1}$ and $M_2^{n_2}$, respectively.

In particular, if both pointwise slant functions ${\theta }_1={\theta }_2=\frac{\pi }{2}$, then $M^n$ is becomes a totally real warped product submanifold—thus, we obtain:

Corollary 5.

Let $\phi :M^n=M_1^{n_1}{\times }_f{M}_2^{n_2}\to {\tilde{M}}^{2m}(c)$ be an isometric immersion from an n-dimensional, totally real warped product submanifold into a $2m$-dimensional complex space form ${\tilde{M}}^{2m}(c)$, where $M_1^{n_1}$ and $M_2^{n_2}$ are totally real submanifolds of ${\tilde{M}}^{2m}(c)$. Then, we have the following:

$$\begin{array}{c}\hfill \Delta (lnf)\le {\Vert \nabla lnf\Vert }^2+\frac{n^2}{4n_2}{\Vert H\Vert }^2+\frac{n_{1}c}{4},\end{array}$$

(35)

where $n_i=dim{M}_i,i=1,2$ and Δ is the Laplacian operator on $M_1^{n_1}$. The same holds in (35) if, and only if, φ is mixed and totally geodesic, and the following satisfies

$$\frac{H_1}{H_2}=\frac{n_2}{n_1},$$

where $H_1$ and $H_2$ are the mean curvature vectors on $M_1^{n_1}$ and $M_2^{n_2}$, respectively.

Proof of Theorem 3.

In this direction, we consider the warped product pointwise bi-slant submanifolds as a compact oriented Riemannian manifold without boundary. If the inequality (2) holds:

$$\begin{array}{c}\hfill \Delta (lnf)-{\Vert \nabla lnf\Vert }^2\le \frac{n^2}{4n_2}{\Vert H\Vert }^2+\frac{c}{4n_2}\left(n_1{n}_2-3n_1{cos}^2{\theta }_1-3n_2{cos}^2{\theta }_2\right).\end{array}$$

(36)

Since $M^n$ is a compact oriented Riemannian submanifold without boundary, then we have following formula with respect to the volume element:

$$\begin{array}{c}\hfill {\int }_{M^n}\Delta fdV=0.\end{array}$$

(37)

From the hypothesis of the theorem, $M^n$ is a compact warped product submanifold; then from (37), we derive:

$$\begin{array}{c}\hfill {\int }_M\left(\frac{c}{4n_2}\left(3n_1{cos}^2{\theta }_1+3n_2{cos}^2{\theta }_2-n_1{n}_2\right)-\frac{1}{4n_2}\sum _{i=1}^n{(h_{ii}^{n+1})}^2\right)dV\le {\int }_M{(\Vert \nabla lnf\Vert }^2)dV.\end{array}$$

(38)

Now, we assume that $M^n$ is a Riemannian product, and the warping function f must be constant on $M^n$. Then, from (38), we get the inequality (3). □

Conversely, let the inequality (3) hold; then from (38), we derive:

$$\begin{array}{c}\hfill 0\le {\int }_{M^n}{(\Vert \nabla lnf\Vert }^2)\le 0.\end{array}$$

The above condition implies that ${\Vert \nabla lnf\Vert }^2=0$, where this means that f is a constant function on $M^n$. Hence, $M^n$ is simply a Riemannian product of $M_1^{n_1}$ and $M_2^{n_2}$, respectively. Thus, the theorem is proved. We give some other important corollaries as consequences of Theorem 2, as follows:

Corollary 6.

Let $M^n=M_1^{n_1}{\times }_f{M}_2^{n_2}$ be a warped product pointwise bi-slant submanifold of a complex space form ${\tilde{M}}^{2m}(c)$ with warping function f, such that $n_1=dim{M}_1$ and $n_2=dim{M}_2$. If φ is an isometrically minimal immersion from warped product $M^n$ into ${\tilde{M}}^{2m}(c)$, then we obtain:

$$\begin{array}{c}\hfill \Delta (lnf)\le {\Vert \nabla lnf\Vert }^2+\frac{c}{4n_2}\left(n_1{n}_2-3n_1{cos}^2{\theta }_1-3n_2{cos}^2{\theta }_2\right).\end{array}$$

(39)

Corollary 7.

Let $M^n=M_1^{n_1}{\times }_f{M}_2^{n_2}$ be a warped product pointwise bi-slant submanifold of a complex space form ${\tilde{M}}^{2m}(c)$ with warping function f, such that $n_1=dim{M}_1$ and $n_2=dim{M}_{\theta }$. Then, there is no existing minimal isometric immersion φ from warped product $M^n$ into ${\tilde{M}}^{2m}(c)$ with:

$$\begin{array}{c}\hfill {\Delta (lnf)>\Vert \nabla lnf\Vert }^2+\frac{c}{4n_2}\left(n_1{n}_2-3n_1{cos}^2{\theta }_1-3n_2{cos}^2{\theta }_2\right).\end{array}$$

(40)