Vanishing Homology of Warped Product Submanifolds in Complex Space Forms and Applications

Abstract

In this paper, we prove the nonexistence of stable integral currents in compact oriented warped product pointwise semi-slant submanifold $M^n$ of a complex space form $\tilde{M}\left(4ϵ\right)$ under extrinsic conditions which involve the Laplacian, the squared norm gradient of the warped function, and pointwise slant functions. We show that i-the homology groups of $M^n$ are vanished. As applications of homology groups, we derive new topological sphere theorems for warped product pointwise semi-slant submanifold $M^n$, in which $M^n$ is homeomorphic to a sphere ${\mathbb{S}}^n$ if $n\ge 4$ and if $n=3$, then $M^3$ is homotopic to a sphere ${\mathbb{S}}^3$ under the assumption of extrinsic conditions. Moreover, the same results are generalized for CR-warped product submanifolds.

1. Introduction and Main Results

A traditional topic in Riemannian geometry is to find the geometrical and topological structures of submanifolds; there has been much progress in this field. For instance, the rigidity theorem was proved by Berger [1] for an even-dimensional complete simply connected manifold M with sectional curvature $\frac{1}{4}\le K_M\le 1$. Further, Gauhmen [2] considered even $n=2m$-dimensional submanifolds minimally immersed in the unit sphere ${\mathbb{S}}^{n+1}$ with a co-dimension equal to one, and showed that if ${\left|\right|h(u,u)\left|\right|}^2<1$ for any unit vector u of $M^n$ where h is the second fundamental form $M^n$, then $M^n$ is totally geodesic in ${\mathbb{S}}^{n+1}$. If $ma{x}_{u\in M}{\left\{\right|\left|h(u,u)\right||}^2\}=1$, then $M^n$ is ${\mathbb{S}}^m\left(\frac{1}{2}\right)\times {\mathbb{S}}^m\left(\frac{1}{2}\right)$ minimally embedded in ${\mathbb{S}}^{2m+1}$, as described. A very famous result in this respect was formulated by Poincare [3], who stated that every simply connected closed 3-manifold is homeomorphic to a 3-sphere. Smale [4] generalized the Poincare conjecture and proved that for a closed $C^{\infty }$-manifold $M^n$ which has the homotopic types of an n-dimensional sphere greater than five, the manifold $M^n$ is homeomorphic to ${\mathbb{S}}^n$. The differentiable sphere theorem was proven by Brendle and Schoen [5] under Ricci flow. In recent years, much attention has been paid to the classification of geometric function theory, topological sphere theorems, and differentiable sphere theorems (see [6,7,8,9,10,11]). In the sequelae, the homology groups of a manifold are important topological invariants that provide algebraic information about the manifold. Federer-Fleming [7] showed that any non-trivial integral homology class in ${\mathbb{H}}_p(M,\mathbb{G}))$ corresponds to a stable current. Motivated by the work of Federer and Fleming [7], Lawson and Simon [9], and Xin [11] proved the nonexistence of stable integral currents in a submanifold $M^n$ and vanishing homology groups of $M^n$ with non-negative sectional curvature according to the following theorem.

Theorem 1

([9,11]). Let $M^n$ be a compact n-dimensional submanifold isometrically immersed in the space form $\tilde{M}\left(c\right)$ of curvature $c\ge 0$ with the second fundamental form h. Let $l_1,l_2$ be any positive integers such that $l_1+l_2=n$ and

$$\begin{array}{c}\hfill \sum _{\alpha =1}^{l_1}\sum _{\beta =l_1+1}^n\left\{2\left|\right|h(e_{\alpha },e_{\beta }){\left|\right|}^2-g\left(h(e_{\alpha },e_{\alpha }),h(e_{\beta },e_{\beta })\right)\right\}<l_1{l}_{2}c,\end{array}$$

(1)

for any $x\in M^n$ and an orthonormal frame ${\left\{e_i\right\}}_{1\le i\le n}$ of the tangent space $T{M}^n$. Then, there do not exist stable $l_1$-currents in $M^n$ and

$${\mathbb{H}}_{l_1}(M^n,\mathbb{G})={\mathbb{H}}_{n-l_1=l_2}(M^n,\mathbb{G})=0,$$

where ${\mathbb{H}}_i(M^n,\mathbb{G})$ stands for i-the homology group of $M^n$ and $\mathbb{G}$ is a finite abelian group with integer coefficients.

Due to these previous studies on large scales, a particular case we consider here is that of warped product pointwise semi-slant submanifolds of complex space form where $4ϵ$ is represented as a constant sectional curvature. In this regard, our motivation comes from the study of Sahin [12], where he discussed the warped product pointwise semi-slant submanifolds in a Kaehler manifold and showed that a warped product pointwise semi-slant submanifold of type $N_T^{l_1}{\times }_f{N}_{\theta }^{l_2}$ is nontrivial when angle $\theta$ is treated as a slant function. Furthermore, it was shown in [12] that the warped product pointwise semi-slant submanifold $N_T^{l_1}{\times }_f{N}_{\theta }^{l_2}$ of a Kaehler manifold is a natural generalization of CR-warped products [13]. Inspired by this notion, we define the extrinsic condition to prove nonexistence-stable integral $l_1$-currents and vanishing homology groups in a warped product pointwise semi-slant submanifold of complex space forms ${\tilde{M}}^m\left(4ϵ\right)$. We use Theorem 1 on this basis to arrive at our first result.

Theorem 2.

Let $M^{l_1+l_2}=N_T^{l_1}{\times }_f{N}_{\theta }^{l_2}$ be a compact warped product pointwise semi-slant submanifold of a complex space form ${\tilde{M}}^m\left(4ϵ\right)$. If the following condition is satisfied

$$\begin{array}{c}\hfill \left({csc}^2\theta +{cot}^2\theta +l_2\right){\left|\right|\nabla f\left|\right|}^2+f\Delta f+\frac{f^2}{l_2}{\parallel h_{\mu }\parallel }^2<3l_1ϵf^2,\end{array}$$

(2)

then there do not exist stable integral $l_1$-currents in $M^{l_1+l_{@}}$ and

$$\begin{array}{c}\hfill {\mathbb{H}}_{l_1}(M^{l_1+l_2},\mathbb{G})={\mathbb{H}}_{l_2}(M^{l_1+l_2},\mathbb{G})=0,\end{array}$$

where ${\mathbb{H}}_i(M^{l_1+l_2},\mathbb{G})$ stands for i-the homology group of $M^{l_1+l_2}$ with integer coefficients, $\nabla f$ and $\Delta f$ are the gradient and the Laplacian of the warped function f, respectively, and $h_{\mu }$ represents the components of the second fundamental form h in an invariant subspace μ.

Our next result is in accordance with Lemma 3.1 in [12], which states that the inner product of the second fundamental form of $N_T^{l_1}$ and F-components of $N_{\theta }^{l_1}$ is equal to zero. To be precise, we have the following result.

Theorem 3.

Let $M^{l_1+l_2}=N_T^{l_1}{\times }_f{N}_{\theta }^{l_2}$ be a compact warped product pointwise semi-slant submanifold of a complex space form ${\tilde{M}}^m\left(4ϵ\right)$. If the inequality

$$\begin{array}{c}\hfill \parallel \nabla f{\parallel }^2<\left\{\frac{\left(4l_1{l}_2ϵ-{\parallel h_{\mu }\parallel }^2\right)f^2}{2l_2\left({csc}^2\theta +{cot}^2\theta \right)}\right\},\end{array}$$

(3)

holds, then there do not exist stable integral $l_1$-currents in $M^{l_1+l_2}$ and

$$\begin{array}{c}\hfill {\mathbb{H}}_{l_1}(M^{l_1+l_2},\mathbb{G})={\mathbb{H}}_{l_2}(M^{l_1+l_2},\mathbb{G})=0.\end{array}$$

The notation is the same as in Theorem 2.

To apply Theorems 2 and 3 in [14], let the slant function $\theta$ become globally constant, setting $\theta =\frac{\pi }{2}$ in Theorems 2 and 3. Then, the pointwise slant submanifold $N_{\theta }^{l_1}$ is turned into a totally real submanifold $N_{\perp }^{l_2}$. Thus, a warped product pointwise semi-slant submanifold $M^{l_1+l_2}=N_T^{l_1}{\times }_f{N}_{\theta }^{l_2}$ becomes CR-warped products in a Kaehler manifold of type $M^n=N_T^{l_1}{\times }_f{N}_{\perp }^{l_2}$. Therefore, following to the motivation of Chen [13], we deduce the following result from Theorem 2 for the nonexistence of stable integral $l_1$-currents and vanishing homology in a CR-warped product submanifold of complex space forms ${\tilde{M}}^m\left(4ϵ\right)$.

Corollary 1.

Let $M^{l_1+l_2}=N_T^{l_1}{\times }_f{N}_{\perp }^{l_2}$ be a compact CR-warped product submanifold of complex space form ${\tilde{M}}^{2m}\left(4ϵ\right)$. If the following condition is satisfied

$$\begin{array}{c}\hfill \left(1+l_2\right){\left|\right|\nabla f\left|\right|}^2+f\Delta f+\frac{f^2}{l_2}{\parallel h_{\mu }\parallel }^2<3l_1ϵf^2,\end{array}$$

(4)

then there do not exist stable integral $l_1$-currents in $M^{l_1+l_2}$ and ${\mathbb{H}}_{l_1}(M^{l_1+l_2},\mathbb{G})={\mathbb{H}}_{l_2}(M^{l_1+l_2},$$\mathbb{G})=0.$

As an immediate consequence of Theorem 3, we have

Corollary 2.

Let $M^{l_1+l_2}=N_T^{l_1}{\times }_f{N}_{\perp }^{l_2}$ be a compact CR-warped product submanifold of complex space form ${\tilde{M}}^{2m}\left(4ϵ\right)$ satisfying the following inequality

$$\begin{array}{c}\hfill \parallel \nabla f{\parallel }^2<\left\{\frac{\left(4l_1{l}_2ϵ-{\parallel h_{\mu }\parallel }^2\right)f^2}{2l_2}\right\}.\end{array}$$

Then, there do not exist stable integral $l_1$-currents in $M^{l_1+l_2}$ and we have the trivial homology groups, i.e.,

$${\mathbb{H}}_{l_1}(M^{l_1+l_2},\mathbb{G})={\mathbb{H}}_{l_1}(M^{l_1+l_2},\mathbb{G})=0.$$

Our next motivation comes from Calin [15] who studied geometric mechanics on Riemannian manifolds and defined a positive differentiable function $\phi$ ( $\phi \in \mathcal{F}\left(M^n\right)$ ) on a compact Riemannian manifold $M^n$. The Dirichlet energy of a function $\phi$ is defined in [15] (see p. 41) as follows:

$$\begin{array}{cc}\hfill \mathbb{E}\left(\phi \right)=& \frac{1}{2}{\int }_{M^n}{\left|\right|\nabla \phi \left|\right|}^{2}dV0<E\left(\phi \right)<\infty .\hfill \end{array}$$

(5)

In view of the kinetic energy formula (5) for a compact oriented manifold without boundary along with Theorem 2, we arrive at the following result.

Theorem 4.

Let $M^{l_1+l_2}=N_T^{l_1}{\times }_f{N}_{\perp }^{l_2}$ be a compact warped product pointwise semi-slant submanifold of a complex space form ${\tilde{M}}^{2m}\left(4ϵ\right)$ without boundary. If the following condition is satisfied

$$\begin{array}{c}\hfill \mathbb{E}\left(f\right)<\left\{\frac{{\int }_{M^n}\left(3l_1{l}_2ϵ-{\parallel h_{\mu }\parallel }^2\right)f^{2}dV}{2l_2(2{csc}^2\theta +l_2)}\right\},\end{array}$$

(6)

where $\mathbb{E}\left(f\right)$ is the Dirichlet energy of the warping function f with respect to the volume element $dV$, then there do not exist stable integral $l_1$-currents in $M^{l_1+l_2}$ and ${\mathbb{H}}_{l_1}(M^{l_1+l_2},\mathbb{G})={\mathbb{H}}_{l_2}(M^{l_1+l_2},\mathbb{G})=0.$

An important concept relates to the geometrical and topological properties on Riemannian manifolds when considering the pinched condition on its metric. It is interesting to investigate the curvature and topology of submanifolds in a Riemannian manifold and the usual sphere theorems in Riemannian geometry. For instance, using the nonexistence of stable currents on compact submanifolds, Lawson and Simon [9] obtained their striking sphere theorem, which proved that for an n-dimensional compact-oriented submanifold $M^n$ in a unit sphere ${\mathbb{S}}^{n+k}$ with the second fundamental form bounded above by a constant which depends on the dimension n, then $M^n$ is homeomorphic to a sphere ${\mathbb{S}}^n$ when $n\ne 3$ and $M^3$ are homotopic to a sphere ${\mathbb{S}}^3$.

Making use of Lawson and Simon [9], Leung [16] proved that for a compact connected oriented submanifold $M^n$ in the unit sphere ${\mathbb{S}}^{n+k}$ such that ${\parallel h(X,X)\parallel }^2<\frac{1}{3}$, when $n\ne 3$ and $M^3$ are homotopic to a sphere ${\mathbb{S}}^3$, then $M^n$ is homeomorphic to a sphere ${\mathbb{S}}^n$. Recently, it has been shown in [17] that if the sectional curvature satisfies some pinching condition $K_M\ge \frac{l_1.sign(l_1-1)}{2(l_1+1)}$ for n-dimensional compact oriented minimal submanifold M in the unit sphere ${\mathbb{S}}^{n+l_1}$ with co-dimension $l_1$, then M is either a totally geodesic sphere, one of the Clifford minimal hyper-surfaces ${\mathbb{S}}^k\left(\frac{k}{n}\right)\times {\mathbb{S}}^{n-k}\left(\frac{n-k}{n}\right)$ in ${\mathbb{S}}^{n+1}$ for $k=1,\cdots ,n-1$, or a Veronese surface in ${\mathbb{S}}^4$. More recently, several results have been derived on topological and differentiable structures of submanifolds when imposing certain conditions on the second fundamental form, Ricci curvatures, and sectional curvatures in a series of articles [4,10,11,18,19,20,21,22,23] by different geometers. For the warped product structure, we refer to [20,24,25,26,27,28,29,30].

The second target of note is to establish topological sphere theorems from the viewpoint of warped product submanifold geometry with positive constant sectional curvature and pinching conditions in terms of the squared norm of the warping function and Laplacian of the warped function as extrinsic invariants. In this sense, we work with conditions on the extrinsic curvature (second fundamental form, warping function), which have the advantage of being invariant under rigid motions. Motivated by Lawson and Simon [9], (p. 441, Theorem 4), we consider a warped product pointwise semi-slant submanifold in a complex space form ${\tilde{M}}^{2m}\left(4ϵ\right)$ such that the constant holomorphic sectional curvature is $4ϵ$, and state our main theorem of this paper.

Theorem 5.

Let $M^{l_1+l_2}=N_T^{l_1}{\times }_f{N}_{\theta }^{l_2}$ be a compact warped product pointwise semi-slant submanifold in a complex space form ${\tilde{M}}^{2m}\left(4ϵ\right)$ satisfying the condition (2). Then, $M^{l_1+l_2}$ is homeomorphic to sphere ${\mathbb{S}}^{l_1+l_2}$ when $l_1+l_2\ge 4,$ while $M^3$ is homotopic to a sphere ${\mathbb{S}}^3$.

Remark 1.

As a consequence of Theorem 5, we obtain the following sphere theorem for a compact CR-warped product submanifold in a complex space form ${\tilde{M}}^{2m}\left(4ϵ\right)$, thanks to Chen [13].

Corollary 3.

Let $M^{l_1+l_2}=N_T^{l_1}{\times }_f{N}_{\perp }^{l_2}$ be a compact CR-warped product submanifold in a complex space form ${\tilde{M}}^{2m}\left(4ϵ\right)$ satisfying the pinching condition (4). Then, $M^{l_1+l_2}$ is homeomorphic to a sphere ${\mathbb{S}}^{l_1+l_2}$ when $l_1+l_2\ge 4,$ and $M^3$ is homotopic to a sphere ${\mathbb{S}}^3$.

Using Theorem 4 and 5, we can now obtain an important result.

Corollary 4.

Let $M^{p+q}=N_T^p{\times }_f{N}_{\theta }^q$ be a compact warped product pointwise semi-slant submanifold of complex space form ${\tilde{M}}^{2m}\left(4ϵ\right)$. If (6) is satisfied, then $M^{l_1+l_2}$ is homeomorphic to sphere ${\mathbb{S}}^{l_1+l_2}$ when $l_1+l_2\ge 4$ and $M^3$ is homotopic to a sphere ${\mathbb{S}}^3$.

Remark 2.

The principle behind Cheng’s eigenvalue comparison theorem (see [31]) forms the basis of the following finding. With the help of the first non-zero eigenvalue of the Laplacian operator, Cheng has demonstrated that if M is complete and isometric to the sphere of the standard unit then the following theorem can be inferred using the maximum principle for the first non-zero eigenvalue ${\lambda }_1$, provided that $Ric\left(M\right)geq1$ and $d\left(M\right)=\pi$.

Theorem 6.

Let $M^{l_1+l_2}=N_T^{l_1}{\times }_f{N}_{\theta }^{l_2}$ be a compact warped product pointwise semi-slant submanifold of a complex space form ${\tilde{M}}^{2m}\left(4ϵ\right)$ with f being a non-constant eigenfunction of the first non-zero eigenvalue ${\lambda }_1$ such that the following inequality is satisfied:

$$\begin{array}{c}\hfill {\lambda }_1<\frac{3l_1{l}_2ϵ-\parallel h_{\mu }\parallel }{l_1\left(2{csc}^2\theta +l_2\right)}.\end{array}$$

(7)

Then, $M^{l_1+l_2}$ is homeomorphic to sphere ${\mathbb{S}}^{l_1+l_2}$ when $l_1+l_2\ge 4$ and $M^3$ is homotopic to a sphere ${\mathbb{S}}^3$ when $l_1+l_2=3$.

Motivated by Bochner’s formula [32], we arrive at the following result.

Theorem 7.

Let $M^{l_1+l_2}=N_T^{l_1}{\times }_f{N}_{\theta }^{l_2}$ be a compact warped product pointwise semi-slant submanifold of a complex space form ${\tilde{M}}^{2m}\left(4ϵ\right)$ such that following inequality holds:

$$\begin{array}{c}\hfill \parallel {\nabla }^2{f\parallel }^2+Ric(\nabla f,\nabla f)>\left\{\frac{\left(\parallel h_{\mu }{\parallel }^2-3l_1{l}_2ϵ\right)f\Delta f}{\left(2{csc}^2\theta +l_2\right)}\right\},\end{array}$$

(8)

where $\parallel {\nabla }^2{f\parallel }^2$ denotes the Hessian form of the warping function f and $Ric$ denotes the Ricci curvature along the base manifold $N_T^{l_1}$. Then, $M^{l_1+l_2}$ is homeomorphic to sphere ${\mathbb{S}}^{l_1+l_2}$ when $l_1+l_2\ge 4$ and $M^3$ is homotopic to a sphere ${\mathbb{S}}^3$ when $l_1+l_2=3$.

2. Preliminaries

Let $M^{2m}\left(4ϵ\right)$ be a complex space form with the complex dimension ${dim}_{\mathbb{R}}M=2m$. Then, the curvature tensor R of $M^{2m}\left(4ϵ\right)$ with constant holomorphic sectional curvature $4ϵ$ is expressed as

$$\begin{array}{cc}\hfill R(X_2,Y_2)Z_2=c(& g(X_2,Z_2)Y_2-g(Y_2,Z_2)X_2+g(X_2,J{Z}_2)J{Y}_2\hfill \\ \hfill & -g(Y_2,J{Z}_2)X_2+2g(X_2,J{Y}_2)J{Z}_2).\hfill \end{array}$$

(9)

The Gauss and Weingarten formulas for transforming submanifold $M^n$ into an almost Hermitian manifold ${\tilde{M}}^{2m}$ are provided by

$$\begin{array}{c}\hfill {\tilde{\nabla }}_{X_2}{Y}_2={\nabla }_{X_2}{Y}_2+h(X_2,Y_2),\\ \hfill {\tilde{\nabla }}_{X_2}N=-A_N{X}_2+{\nabla }_{X_2}^{\perp }N,\end{array}$$

for each $X_2,Y_2\in \mathfrak{X}\left(TM\right)$ and $N\in \mathfrak{X}\left(T^{\perp }M\right)$ such that the second fundamental form and the shape operator are denoted by h and $A_N$. They are connected as $g(h(U,V),N)=g(A_{N}U,V)$. Now, for any $X_2\in \mathfrak{X}\left(M\right)$ and $N\in \mathfrak{X}\left(T^{\perp }M\right)$, we have

$$\begin{array}{c}\hfill \left(i\right)J{X}_2=T{X}_2+F{X}_2,\left(ii\right)JN=tN+fN,\end{array}$$

(10)

where $T{X}_2\left(tN\right)$ and $F{X}_2\left(fN\right)$ are the tangential and normal components of $J{X}_2\left(JN\right)$, respectively.

The Gauss equation for a submanifold $M^n$ is defined as

$$\begin{array}{cc}\hfill \tilde{R}(X_2,Y_2,Z_2,W_2)=& R(X_2,Y_2,Z_2,W_2)+g\left(h(X_2,Z_2),h(Y_2,W_2)\right)\hfill \\ \hfill & -g\left(h(X_2,W_2),h(Y_2,Z_2)\right),\hfill \end{array}$$

(11)

for any $X_2,Y_2,Z_2,W_2\in \mathfrak{X}\left(TM\right)$, where $\tilde{R}$ and R are the curvature tensors on ${\tilde{M}}^{2m}$ and $M^n$, respectively.

The norm of second fundamental form h for an orthonormal frame $\{e_1,e_2,\cdots e_n\}$ of the tangent space $TM$ on $M^n$ is defined by

$$\begin{array}{c}\hfill h_{ij}^r=g(h(e_i,e_j),e_r){,\left|\right|h\left|\right|}^2=\sum _{i,j=1}^{n}g(h(e_i,e_j),h(e_i,e_j)).\end{array}$$

(12)

Let $\{e_1,\cdots ,e_n\}$ be an local orthonormal frame of vector field $M^n$. Then, we have

$$\begin{array}{c}\hfill \nabla \phi =\sum _{i=1}^n{e}_i\left(\phi \right)e_i.\\ \hfill and\\ \hfill {\parallel \nabla \phi \parallel }^2=\sum _{i=1}^n{\left(\left(\phi \right)e_i\right)}^2,\end{array}$$

(13)

where $\nabla \phi$ and ${\left|\right|\nabla \phi \left|\right|}^2$ are the gradient of function $\phi$ and its squared norm.

The following classifications can be provided as:

(i)

If $J\left(T_{x}M\right)\subseteq T_{x}M$ for every $x\in M^n$, then $M^n$ is a holomorphic submanifold.

(ii)

If $J\left(T_{x}M\right)\subseteq T^{\perp }M$ for each $x\in M^n$, then $M^n$ is a totally real submanifold.

There are four types of submanifolds of a Kaehler manifold, namely, the CR-submanifold, slant submanifold, semi-slant submanifold, pointwise slant submanifold, and pointwise semi-slant submanifold. The definitions and classifications of such submanifolds are discussed in [12,13]. Moreover, for examples of a pointwise semi-slant submanifold in a Kaehler manifold and related problems, we refer to [12]. It follows from Definition 3.1 in [12] that if we denote as $l_1$ and $l_2$ the dimensions of a complex distribution ${\mathcal{D}}^T$ and pointwise slant distribution ${\mathcal{D}}^{\theta }$ of a pointwise semi-slant submanifold in a Kaehler manifold ${\tilde{M}}^{2m}$, then the following remarks hold:

Remark 3.

$M^n$ is invariant if $l_1=0$ and pointwise slant if $l_2=0$.

Remark 4.

If we consider the slant function $\theta :M^n\to R$ as globally constant on $M^n$ and $\theta =\frac{\pi }{2}$, then $M^n$ is a CR-submanifold.

Remark 5.

An invariant subspace μ under J of normal bundle $T^{\perp }M$, is defined as $T^{\perp }M=F{\mathcal{D}}^{\theta }\oplus \mu$.

3. Warped Product Submanifolds

A product manifold of the type $M^n=N_1^{l_1}{\times }_f{N}_2^{l_2}$ is a warped product manifold if the metric is defined as $g=g_1+f^2{g}_2$, where $N_1^{l_1}$ and $N_2^{l_2}$ are two Riemannian manifolds and their Riemannian metrics are $g_1$ and $g_2$, respectively. It was discovered by Bishop and O’Neill [33] that the warping function f is a smooth function defined on base $N_1^{l_1}$. The following properties are a direct consequence of the warped product manifold $M^n=N_1^{l_1}{\times }_f{N}_2^{l_2}$:

(i)

${\nabla }_{Z}X={\nabla }_{X}Z=\frac{\left(Xf\right)}{f}Z,$

(ii)

${\nabla }_{Z}W={\nabla }_Z^{{}^{\prime }}W-\frac{g(Z,W)}{f}\nabla f,$

for any $X,Y\in \mathfrak{X}\left(T{N}_1\right)$ and $Z,W\in \mathfrak{X}\left(T{N}_2\right)$, where ∇ and ${\nabla }^{{}^{\prime }}$ denote the Levi-Civita connection on $M^n$ and $N_2$, respectively.

The gradient $\nabla f$ of f is written as

$$\begin{array}{c}\hfill g(\nabla lnf,X_2)=X_2(lnf).\end{array}$$

(14)

The following relation is an interesting property of warped products:

$$\begin{array}{c}\hfill \mathcal{R}(X_2,Z_2)Y_2=\frac{{\mathcal{H}}^f(X_2,Z_2)}{f}{Y}_2,\end{array}$$

(15)

where ${\mathcal{H}}^f$ is a Hessian tensor of f; the remarks below follow as a consequence.

Remark 6.

A warped product manifold $M^n=N_1^{l_1}{\times }_f{N}_2^{l_2}$ is said to be trivial or simply a Riemannian product manifold if the warping function f is a constant function along $N_1^{l_1}$.

Remark 7.

If $M^n=N_1^{l_1}{\times }_f{N}_2^{l_2}$ is a warped product manifold, then $N_1^{l_1}$ is totally geodesic and $N_2^{l_2}$ is a totally umbilical submanifold of $M^n$, respectively.

4. Non-Trivial Warped Product Pointwise Semi-Slant Submanifolds $N_T^{l_1}{\times }_f{N}_{\theta }^{l_2}$

It is well known that warped product submanifolds of types

$$\begin{array}{c}\hfill \left(i\right)N_{\theta }^{l_2}{\times }_f{N}_T^{l_1},and\left(ii\right)N_T^{l_1}{\times }_f{N}_{\theta }^{l_2},\end{array}$$

are called warped product pointwise semi-slant submanifolds, which were discovered in [12]. They contain holomorphic and pointwise slant submanifolds of a Kähler manifold. The first case, with $M^n=N_{\theta }^{l_2}{\times }_f{N}_T^{l_1}$ in a Kähler manifold, is trivial. The second is non-trivial. Before proceeding to the second case, let us recall the following result [12].

Lemma 1.

Let $M^n=N_T^{l_1}{\times }_f{N}_{\theta }^{l_2}$ be a warped product pointwise semi-slant submanifold of a Kähler manifold ${\tilde{M}}^m$. Then,

$$\begin{array}{cc}\hfill g(h(X_2,Z_2),FT{Z}_2)=& -(X_{2}lnf){cos}^2\theta \left|\right|Z_2{\left|\right|}^2,\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill g(h(Z_2,J{X}_2),F{Z}_2)=& (X_{2}lnf)\left|\right|Z_2{\left|\right|}^2,\hfill \end{array}$$

(17)

for any $X_2,Y_2\in \mathfrak{X}\left(T{N}_T\right)$ and $Z_2\in \mathfrak{X}\left(T{N}_{\theta }\right)$.

5. Proof of Main Results

5.1. Proof of Theorem 2

The crucial point of this paper is to derive an upper bound for

$$\sum _{i=1}^{l_1}\sum _{j=l_1+1}^n\left\{2\left|\right|h(e_i,e_j){\left|\right|}^2-g\left(h(e_i,e_i),h(e_j,e_j)\right)\right\}$$

in terms of $\Delta f$ and ${\left|\right|\nabla f\left|\right|}^2$.

Let $M=N_T^{l_1}{\times }_f{N}_{\theta }^{l_2}$ be an $n=l_1+l_2$-dimensional warped product pointwise semi-slant submanifold with $\dim N_T^{l_1}=l_1=2\alpha$ and $\dim N_{\theta }^{l_2}=l_2=2\beta$, where $N_{\theta }^{l_1}$ and $N_T^{l_1}$ are integral manifolds of ${\mathcal{D}}^{\theta }$ and $\mathcal{D},$ respectively. Thus, we consider $\{e_1,e_2,\cdots e_{\alpha },e_{\alpha +1}=J{e}_1,\cdots e_{2\alpha }=J{e}_{\alpha }\}$ and $\{e_{2\alpha +1}=e_1^{\ast },\cdots e_{2\alpha +\beta }=e_{\beta }^{\ast },e_{2\alpha +\beta +1}=e_{\beta +1}^{\ast }=sec\theta P{e}_1^{\ast },\cdots e_{l_1+l_2}=e_{l_2}^{\ast }=sec\theta P{e}_{\beta }^{\ast }\}$ to be orthonormal frames of $T{N}_T$ and $T{N}_{\theta }$, respectively. Thus the orthonormal frames of the normal sub-bundles $F{\mathcal{D}}^{\theta }$ and $\mu$ are $\{e_{n+1}={\overline{e}}_1=csc\theta F{e}_1^{\ast },\cdots e_{n+\beta }={\overline{e}}_{\beta }=csc\theta F{e}_1^{\ast },e_{n+\beta +1}={\overline{e}}_{\beta +1}=csc\theta sec\theta FP{e}_1^{\ast },\cdots e_{n+2\beta }={\overline{e}}_{2\beta }=csc\theta sec\theta FP{e}_{\beta }^{\ast }\}$ and $\{e_{n+2\beta +1},\cdots e_{2m}\}$, respectively. Then, from the Gauss Equation (11), we have

$$\begin{array}{cc}\hfill \sum _{i=1}^{l_1}\sum _{j=1}^{l_2}g\left(R(e_i,e_j)e_i,e_j\right)=& \sum _{i=1}^{l_1}\sum _{j=1}^{l_2}g\left(\tilde{R}(e_i,e_j)e_i,e_j\right)+\left|\right|h(e_i,e_j){\left|\right|}^2\hfill \\ \hfill & -\sum _{i=1}^{l_1}\sum _{j=1}^{l_2}g\left(h(e_i,e_j),h(e_i,e_j)\right).\hfill \end{array}$$

By adding the squared norm of the second fundamental terms in both side of the above equation, we obtain

$$\begin{array}{cc}\hfill \sum _{i=1}^{l_1}\sum _{j=1}^{l_2}g\left(R(e_i,e_j)e_i,e_j\right)+\left|\right|h(e_i,e_j){\left|\right|}^2& =\sum _{i=1}^{l_1}\sum _{j=1}^{l_2}g\left(\tilde{R}(e_i,e_j)e_i,e_j\right)\hfill \\ \hfill & -\sum _{i=1}^{l_1}\sum _{j=1}^{l_2}g\left(h(e_j,e_j),h(e_i,e_i)\right)\hfill \\ \hfill & +2\left|\right|h(e_i,e_j){\left|\right|}^2.\hfill \end{array}$$

(18)

Using the orthonormal frames ${\left\{e_i\right\}}_{1\le i\le l_1}$ and ${\left\{e_j\right\}}_{1\le j\le l_2}$ of $N_T^{l_1}$ and $N_{\theta }^{l_2}$, respectively, in (15), we derive

$$\begin{array}{c}\hfill R(e_i,e_j)e_i=\frac{e_j}{f}{\mathcal{H}}^f(e_i,e_i).\end{array}$$

Summing up with an orthonormal frame ${\left\{e_j\right\}}_{1\le j\le l_2}$ (here it should be pointed out that we have adopted the opposite sign from the usual sign convention for the Laplacian), then

$$\begin{array}{c}\hfill \sum _{i=1}^{l_1}\sum _{j=1}^{l_2}g\left(R(e_i,e_j)e_i,e_j\right)=-\frac{l_2}{f}\sum _{i=1}^{l_1}g\left({\nabla }_{e_i}\nabla f,e_i\right).\end{array}$$

(19)

Thus, from Equations (18) and (19), we can derive

$$\begin{array}{cc}\hfill \sum _{i=1}^{l_1}\sum _{j=1}^{l_2}\left\{2\right||h(e_i,e_j){\left|\right|}^2-& g\left(h(e_j,e_j),h(e_i,e_i)\right)\}+\sum _{i=1}^{l_1}\sum _{j=1}^{l_2}g\left(\tilde{R}(e_i,e_j)e_i,e_j\right)\hfill \\ \hfill & =-\frac{l_2}{f}\sum _{i=1}^{l_1}g\left({\nabla }_{e_i}\nabla f,e_i\right)+\sum _{i=1}^{l_1}\sum _{j=1}^{l_2}{\left(h_{ij}^r\right)}^2.\hfill \end{array}$$

(20)

First, we figure out the term $\Delta f$ for $M^n$, which is the Laplacian of f.

$$\begin{array}{cc}\hfill \Delta f=& -\sum _{i=1}^{n}g\left({\nabla }_{e_i}gradf,e_i\right)\hfill \\ \hfill =& -\sum _{\alpha =1}^{l_1}g\left({\nabla }_{e_{\alpha }}gradf,e_{\alpha }\right)-\sum _{\beta =1}^{l_2}g\left({\nabla }_{e_{\beta }}gradf,e_{\beta }\right).\hfill \end{array}$$

The above equation can be expressed as components of $N_{\theta }^q$ from adapted orthonormal framel in this way, we obtain

$$\begin{array}{cc}\hfill \Delta f=& -\sum _{\alpha =1}^{l_1}g\left({\nabla }_{e_{\alpha }}gradf,e_{\alpha }\right)-\sum _{j=1}^{\beta }g\left({\nabla }_{e_j}gradf,e_j\right)\hfill \\ \hfill & -{sec}^2\theta \sum _{j=1}^{\beta }g\left({\nabla }_{T{e}_j}gradf,T{e}_j\right).\hfill \end{array}$$

Benefiting from ∇ being a Levi-Civita connection on $M^n$, we derive

$$\begin{array}{cc}\hfill \Delta f=& -\sum _{\alpha =1}^{l_1}g\left({\nabla }_{e_{\alpha }}gradf,e_{\alpha }\right)-\sum _{j=1}^{\beta }\left(e_{j}g\left(gradf,e_j\right)-g\left({\nabla }_{e_j}{e}_j,gradf\right)\right).\hfill \\ \hfill & -{sec}^2\theta \sum _{j=1}^{\beta }\left(T{e}_{j}g\left(gradf,T{e}_j\right)-g\left({\nabla }_{T{e}_j}T{e}_j,gradf\right)\right).\hfill \end{array}$$

From the property of the gradient of function (14), we obtain

$$\begin{array}{cc}\hfill \Delta f=& -\sum _{\alpha =1}^{l_1}g\left({\nabla }_{e_{\alpha }}gradf,e_{\alpha }\right)-\sum _{j=1}^{\beta }\left(e_j\left(e_{j}f\right)-\left({\nabla }_{e_j}{e}_{j}f\right)\right)\hfill \\ \hfill & -{sec}^2\theta \sum _{j=1}^{\beta }\left(T{e}_j\left(T{e}_j\left(f\right)\right)-\left({\nabla }_{T{e}_j}T{e}_{j}f\right)\right).\hfill \end{array}$$

After computation, we have

$$\begin{array}{cc}\hfill \Delta f=& -\sum _{\alpha =1}^{l_1}g\left({\nabla }_{e_{\alpha }}gradf,e_{\alpha }\right)-\sum _{j=1}^{\beta }\left(e_j\left(g(gradf,e_j)\right)-g({\nabla }_{e_j}{e}_j,gradf)\right)\hfill \\ \hfill & -{sec}^2\theta \sum _{j=1}^{\beta }\left(T{e}_j\left(g(gradf,T{e}_j)\right)-g({\nabla }_{T{e}_j}T{e}_j,gradf)\right).\hfill \end{array}$$

Starting from the hypothesis of a warped product pointwise semi-slant submanifold, $N_T^{l_1}$ is totally geodesic in $M^n$. This implies that $gradf\in \mathfrak{X}\left(T{N}_T\right)$, and from (i)–(ii) in Section 3, we obtain

$$\begin{array}{cc}\hfill \Delta f=& -\frac{1}{f}\sum _{j=1}^{\beta }\left(g(e_j,e_j){\parallel \nabla f\parallel }^2+{sec}^2\theta g(T{e}_j,T{e}_j){\parallel \nabla f\parallel }^2\right)\hfill \\ \hfill & -\sum _{i=1}^{l_1}g\left({\nabla }_{e_i}gradf,e_i\right).\hfill \end{array}$$

By multiplying the above equation by $\frac{1}{f}$, from (3.7) of Corollary 3.1 in [12] we obtain

$$\begin{array}{c}\hfill \frac{\Delta f}{f}=-\frac{1}{f}\sum _{i=1}^{l_1}g\left({\nabla }_{e_i}gradf,e_i\right)-l_2{\parallel \nabla (lnf)\parallel }^2.\end{array}$$

It is not difficult to check that

$$\begin{array}{c}\hfill -\frac{1}{f}\sum _{i=1}^{l_1}g\left({\nabla }_{e_i}gradf,e_i\right)=\frac{\Delta f}{f}+l_2{\left|\right|\nabla lnf\left|\right|}^2.\end{array}$$

This combines with (20) to yield

$$\begin{array}{cc}\hfill l_2^2{\left|\right|\nabla (lnf)\left|\right|}^2+\frac{l_2\Delta f}{f}+& \sum _{\alpha =1}^{l_1}\sum _{\beta =1}^{l_2}{\left(h_{\alpha \beta }^r\right)}^2\hfill \\ \hfill & =\sum _{i=1}^{l_1}\sum _{j=1}^{l_2}\left\{2\left|\right|h(e_i,e_j){\left|\right|}^2-g\left(h(e_j,e_j),h(e_i,e_i)\right)\right\}\hfill \\ \hfill & +\sum _{i=1}^{l_1}\sum _{j=1}^{l_2}g\left(\tilde{R}(e_i,e_j)e_i,e_j\right).\hfill \end{array}$$

(21)

On taking $X=e_i$ and $Z=e_j$ for $1\le i\le l_1$ and $1\le j\le l_2$, respectively, we have

$$\begin{array}{cc}\hfill \sum _{r=n+1}^{2m}\sum _{i=1}^{l_1}\sum _{j=1}^{l_2}{\left(h_{ij}^r\right)}^2=& \sum _{r=n+1}^{n+2\beta }\sum _{i=1}^{l_1}\sum _{j=1}^{l_2}g{\left(h(e_i,e_j^{\ast }),e_r\right)}^2\hfill \\ \hfill & +\sum _{r=n+2\beta +1}^{2m}\sum _{i=1}^{l_1}\sum _{j=1}^{l_2}g{\left(h(e_i,e_j^{\ast }),e_r\right)}^2.\hfill \end{array}$$

In the above equation, the first term on the right hand side is the $F{\mathcal{D}}^{\theta }$-component and the second term is the $\mu$-component for the orthonormal frame for vector fields of $N_T^{l_1}$ and $N_{\theta }^{l_2}$. Summing over the vector fields of $N_T^{l_1}$ and $N_{\theta }^{l_2}$ and using (16) and () from Lemma 1 in the last equation, we are able to find that

$$\begin{array}{cc}\hfill \sum _{r=n+1}^{2m}\sum _{i=1}^{l_1}\sum _{j=1}^{l_2}{\left(h_{ij}^r\right)}^2=& 2\left({csc}^2\theta +{cot}^2\theta \right)\sum _{i=1}^{\alpha }\sum _{j=1}^{\beta }{\left(e_{i}lnf)\right)}^{2}g{(e_j^{\ast },e_j^{\ast })}^2\hfill \\ \hfill & +2\left({csc}^2\theta +{cot}^2\theta \right)\sum _{i=1}^{\alpha }\sum _{j=1}^{\beta }{\left(J{e}_{i}lnf)\right)}^{2}g{(e_j^{\ast },e_j^{\ast })}^2\hfill \\ \hfill & +\sum _{r=n+2\beta +1}^{2m}\sum _{i=1}^{l_1}\sum _{j=1}^{l_2}g{\left(h(e_i,e_j^{\ast }),e_r\right)}^2.\hfill \end{array}$$

From the adapted orthonormal frame for $N_T$, the last equation can then be expressed as follows:

$$\begin{array}{cc}\hfill \sum _{r=n+1}^{2m}\sum _{i=1}^{l_1}\sum _{j=1}^{l_2}{\left(h_{ij}^r\right)}^2=& 2\left({csc}^2\theta +{cot}^2\theta \right)\sum _{i=1}^{l_1}{\left(e_i(lnf)\right)}^2\sum _{j=1}^{l_2}g{(e_j^{\ast },e_j^{\ast })}^2\hfill \\ \hfill & +\sum _{r=n+2\beta +1}^{2m}\sum _{i=1}^{l_1}\sum _{j=1}^{l_2}g{\left(h(e_i,e_j^{\ast }),e_r\right)}^2.\hfill \end{array}$$

Together with the definition of the squared norm of the gradient function f from (13), the above implies that

$$\begin{array}{c}\hfill \sum _{r=n+1}^{2m}\sum _{i=1}^{l_1}\sum _{j=1}^{l_2}{\left(h_{ij}^r\right)}^2=l_2\left({csc}^2\theta +{cot}^2\theta \right){\left|\right|\nabla lnf\left|\right|}^2+{\parallel h_{\mu }\parallel }^2.\end{array}$$

(22)

Following (21) and (22), we arrive at

$$\begin{array}{cc}\hfill \frac{l_2\Delta f}{f}+& l_2^2{\left|\right|\nabla (lnf)\left|\right|}^2+l_2\left(1+2{cot}^2\theta \right){\left|\right|\nabla (lnf)\left|\right|}^2+{\parallel h_{\mu }\parallel }^2\hfill \\ \hfill & =\sum _{i=1}^{l_1}\sum _{j=1}^{l_2}\left\{2\left|\right|h(e_i,e_j){\left|\right|}^2-g\left(h(e_j,e_j),h(e_i,e_i)\right)\right\}\hfill \\ \hfill & +\sum _{i=1}^{l_1}\sum _{j=1}^{l_2}g\left(\tilde{R}(e_i,e_j)e_i,e_j\right).\hfill \end{array}$$

Because we have the following relation for symmetry of the curvature tensor R,

$$\begin{array}{c}\hfill \sum _{i=1}^{l_1}\sum _{j=1}^{l_2}g\left(\tilde{R}(e_i,e_j)e_i,e_j\right)=\sum _{i=1}^{l_1}\sum _{j=1}^{l_2}\tilde{R}\left(e_i,e_j,e_i,e_j\right).\end{array}$$

(23)

Next, we use the curvature tensor from Formula (9) for the complex space form ${\tilde{M}}^m\left(4ϵ\right)$, which can be simply written as

$$\begin{array}{cc}\hfill \sum _{i=1}^{l_1}\sum _{j=1}^{l_2}\tilde{R}\left(e_i,e_j,e_i,e_j\right)=ϵ\sum _{i=1}^{l_1}\sum _{j=1}^{l_2}\{& g(e_i,e_j)g(e_i,e_j)-g(e_i,e_i)g(e_j,e_j)\hfill \\ \hfill & -g(J{e}_i,e_i)g(J{e}_j,e_j)\hfill \\ \hfill & +3g(J{e}_i,e_j)g(J{e}_j,e_i)\}.\hfill \end{array}$$

(24)

As we know that $e_i\in \mathfrak{X}\left(T{N}_T\right)$ and $e_j\in \mathfrak{X}\left(T{N}_{\theta }\right)$, then $g(e_i,e_j)=0,$ and $g(J{e}_i,e_i)=0(resp,g(J{e}_j,e_i)=0)$ by the fact that for $J{e}_i\perp e_i(J{e}_j\perp e_j)$, respectively. Similarly, from (10)i, we can derive that $g(J{e}_i,e_j)=g(T{e}_i+F{e}_i,e_j)=0$ for $T{e}_i\in \mathfrak{X}\left(T{N}_T\right)$ and $e_j\in \mathfrak{X}\left(T{N}_{\theta }\right)$; thus, (24) implies that

$$\begin{array}{c}\hfill \sum _{i=1}^{l_1}\sum _{j=1}^{l_2}\tilde{R}\left(e_i,e_j,e_i,e_j\right)=-ϵ\sum _{i=1}^{l_1}\sum _{j=1}^{l_2}g(e_i,e_i)g(e_j,e_j).\end{array}$$

After computation using the above equation, we can derive

$$\begin{array}{c}\hfill \sum _{i=1}^{l_1}\sum _{j=1}^{l_2}\tilde{R}\left(e_i,e_j,e_i,e_j\right)=-l_1{l}_2ϵ,\end{array}$$

(25)

Therefore, following (23) and (25), we finally obtain

$$\begin{array}{cc}\hfill \frac{l_2\Delta f}{f}+l_2^2& {\left|\right|\nabla (lnf)\left|\right|}^2+l_2\left({csc}^2\theta +{cot}^2\theta \right){\left|\right|\nabla (lnf)\left|\right|}^2+{\parallel h_{\mu }\parallel }^2+l_1{l}_2ϵ\hfill \\ \hfill & =\sum _{i=1}^{l_1}\sum _{j=1}^{l_2}\left\{2\left|\right|h(e_i,e_j){\left|\right|}^2-g\left(h(e_j,e_j),h(e_i,e_i)\right)\right\}.\hfill \end{array}$$

(26)

If the pinching condition (2) is satisfied, then from (26) we have

$$\begin{array}{c}\hfill \sum _{i=1}^{l_1}\sum _{j=1}^{l_2}\left\{2\left|\right|h(e_i,e_j){\left|\right|}^2-g\left(h(e_j,e_j),h(e_i,e_i)\right)\right\}<4l_1{l}_2ϵ\end{array}$$

By applying Theorem 1 with $c=4ϵ>0$, we obtain the following:

$$\begin{array}{c}\hfill \sum _{i=1}^{l_1}\sum _{j=1}^{l_2}\left\{2\left|\right|h(e_i,e_j){\left|\right|}^2-g\left(h(e_j,e_j),h(e_i,e_i)\right)\right\}<l_1{l}_{2}c.\end{array}$$

This completes the proof of Theorem 2, as the assertion follows from Theorem 1.

5.2. Proof of Theorem 4

If we consider $M^n$ as the compact-oriented Riemannian manifold without boundary $\partial M^n=\varnothing$, then we are able to prove the strong result in terms of tthe Dirichlet energy and pointwise slant immersion as follows. Taking the integration along the volume element $dV$ in (2), we obtain

$$\begin{array}{cc}\hfill {\int }_{M^n}\left({csc}^2\theta +{cot}^2\theta +l_2\right){\left|\right|\nabla f\left|\right|}^{2}dV+& {\int }_{M^n}f\Delta fdV\hfill \\ \hfill & <{\int }_{M^n}\left(3l_2ϵ-\frac{\parallel h_{\mu }{\parallel }^2}{l_2}\right)f^{2}dV.\hfill \end{array}$$

(27)

From the divergence theorem ${\int }_{M^n}(\Delta f)dV=0$ in [34] without boundary. Using this fact, we can compute the following as

$$\begin{array}{cc}\hfill 0=& {\int }_{M^n}\Delta \left(\frac{f^2}{2}\right)dV=-{\int }_{M^n}div\left(\nabla \left(\frac{f^2}{2}\right)\right)dV\hfill \\ \hfill =& -{\int }_{M^n}div(f\nabla f)dV=-{\int }_{M^n}g(\nabla f,\nabla f)dV+{\int }_{M^n}f\Delta fdV\hfill \end{array}$$

which implies that

$$\begin{array}{c}\hfill {\int }_{M^n}f\Delta fdV={\int }_{M^n}{\parallel \nabla f\parallel }^{2}dV.\end{array}$$

(28)

The inequality (27) takes its new form by virtue of (28), that is,

$$\begin{array}{cc}\hfill 2{\int }_{M^n}\left({csc}^2{\theta \left|\right|\nabla f\left|\right|}^2\right)dV+& l_2{\int }_{M^n}{\left|\right|\nabla f\left|\right|}^{2}dV<{\int }_{M^n}\left(3l_1ϵ-\frac{\parallel h_{\mu }{\parallel }^2}{l_2}\right)f^{2}dV.\hfill \end{array}$$

(29)

Using the Dirichlet energy from Formula (5) in the above equation, we have

$$\begin{array}{c}\hfill 2\left(2{csc}^2\theta +l_2\right)\mathbb{E}\left(f\right)<{\int }_{M^n}\left(3l_1ϵ-\frac{\parallel h_{\mu }{\parallel }^2}{l_2}\right)f^{2}dV.\end{array}$$

Thus, we obtain the required result (6). This completes the proof of the theorem.

5.3. Proof of Corollary 1 and 2

The proof of Corollary 1 and Corollary 3 arises directly from Theorems 2 and 5 by substituting $\theta =\frac{\pi }{2}$ to point out a totally real submanifold from a pointwise slant submanifold, which then provides the promised results.

5.4. Proof of Theorem 5

From Theorem 2, we can find that there do not exist stable integral $l_1$-currents in a warped product pointwise semi-slant submanifold $M^n$ and that the homology groups are zero for all positive integers $l_1,l_2$ such that $n=l_1+l_2\ne 3$; that is, ${\mathbb{H}}_{l_1}(M^n,\mathbb{G})={\mathbb{H}}_{l_2}(M^n,\mathbb{G})=0.$ Therefore, $M^n$ is a homology sphere, and in addition is a homotopic sphere following the same arguments as in [19].

Therefore, applying the generalized Poincarẽ conjecture (Smale $n\ge 5$ [4], Freedman $n=4$ [8]), we know that $M^n$ is homotopic to the sphere ${\mathbb{S}}^n$ as an immediate consequence of Sjerve [10], implying that the fundamental group ${\pi }_1\left(M^n\right)=0$ on $M^n$ when applying the same arguments as above. This implies that $M^{l_1+l_2}$ is homeomorphic to the sphere ${\mathbb{S}}^{l_1+l_2}$. Similarly, it is not hard to check that $M^3$ is homotopic to a sphere ${\mathbb{S}}^3$ when $n=3$ from [9,16]. This completes the proof of Theorem 5.

5.5. Proof of Theorem 6

From the minimum principle on the first eigenvalue ${\lambda }_1$, we can obtain the outcome from [32], p. 186. Let us assume that f is a non-constant warping function

$$\begin{array}{c}\hfill {\lambda }_1{\int }_{M^n}{f}^{2}dV\le {\int }_{M^n}{\parallel \nabla f\parallel }^{2}dV.\end{array}$$

(30)

where the equality holds if and only if $\Delta f={\lambda }_{1}f$. Integrating Equation (29) and Green’s lemma, we have

$$\begin{array}{c}\hfill \left(2{csc}^2\theta +q\right){\int }_{M^n}{\parallel \nabla f\parallel }^{2}dV<{\int }_{M^n}\left(3l_1ϵ-\frac{\parallel h_{\mu }{\parallel }^2}{l_2}\right)f^{2}dV,\end{array}$$

which implies that

$$\begin{array}{c}\hfill {\int }_{M^n}{\parallel \nabla f\parallel }^{2}dV<\frac{1}{\left(2{csc}^2\theta +l_2\right)}{\int }_{M^n}\left(3l_1ϵ-\frac{\parallel h_{\mu }{\parallel }^2}{l_2}\right)f^{2}dV.\end{array}$$

(31)

By virtue of (30) in (31), we can find that

$$\begin{array}{c}\hfill {\int }_{M^n}\left\{{\lambda }_1-\frac{\left(3l_1{l}_2ϵ-\parallel h_{\mu }\parallel \right)}{l_2\left(2{csc}^2\theta +l_2\right)}\right\}{f}^{2}dV<0.\end{array}$$

From this, we arrive at our result (7) by combining Theorems 2 and 5, which completes the proof.

Here, we remember the lemma below.

Lemma 2

([12]). Assume that ${\tilde{M}}^{2m}$ is a Kaehler manifold and $M^{l_1+l_2}=N_T^{l_1}{\times }_f{N}_{\theta }^{l_2}$ is a warped product pointwise semi-slant submanifold of ${\tilde{M}}^{2m}$. Then, we have

$$\begin{array}{c}\hfill g\left(h(X2,Y_2),F{Z}_2\right)=0.\end{array}$$

(32)

for any $X_2,Y_2\in \mathfrak{X}\left(T{N}_T\right)$ and $Z_2,W_2\in \mathfrak{X}\left(T{N}_{\theta }\right)$.

In view of Lemma 2, we can find our next result.

5.6. Proof of Theorem 3

We can write the following from (12) as follows:

$$\begin{array}{cc}\hfill \sum _{i=1}^{l_1}\sum _{j=1}^{l_2}\left\{2\left|\right|h(e_i,e_j){\left|\right|}^2-g\left(h(e_j,e_j),h(e_i,e_i)\right)\right\}=& 2\sum _{i=1}^{l_1}\sum _{j=1}^{l_2}\left|\right|h(e_i,e_j){\left|\right|}^2\hfill \\ \hfill & -\sum _{i=1}^{l_1}\sum _{j=1}^{l_2}g{\left(h(e_i,e_i),e_j\right)}^2,\hfill \end{array}$$

or equivalently as

$$\begin{array}{cc}\hfill \sum _{i=1}^{l_1}\sum _{j=1}^{l_2}\left\{2\right||h(e_i,e_j){\left|\right|}^2-& g\left(h(e_j,e_j),h(e_i,e_i)\right)\}\hfill \\ \hfill & =2\sum _{i=1}^{l_1}\sum _{j=1}^{l_2}\left|\right|h(e_i,e_j){\left|\right|}^2-\sum _{i=1}^{l_1}\sum _{j=1}^{l_2}g{\left(h(e_i,e_i).F{e}_j^{\ast }\right)}^2.\hfill \end{array}$$

By virtue of (32), we have

$$\begin{array}{cc}\hfill \sum _{i=1}^{l_1}\sum _{j=1}^{l_2}\left\{2\right||h(e_i,e_j){\left|\right|}^2-& g\left(h(e_j,e_j),h(e_i,e_i)\right)\}=2\sum _{i=1}^{l_1}\sum _{j=1}^{l_2}\left|\right|h(e_i,e_j){\left|\right|}^2.\hfill \end{array}$$

Using Equation (22) on the right hand side of the above equation, we have

$$\begin{array}{cc}\hfill \sum _{i=1}^{l_1}\sum _{j=1}^{l_2}\left\{2\right||h(e_i,e_j){\left|\right|}^2-& g\left(h(e_j,e_j),h(e_i,e_i)\right)\}=\frac{2l_2}{f^2}\left({csc}^2\theta +{cot}^2\theta \right){\parallel \nabla f\parallel }^2+{\parallel h_{\mu }\parallel }^2.\hfill \end{array}$$

(33)

If assumption (3) is satisfied, then the following inequality is implied by (33):

$$\begin{array}{cc}\hfill \sum _{i=1}^{l_1}\sum _{j=1}^{l_2}\left\{2\right||h(e_i,e_j){\left|\right|}^2-& g\left(h(e_j,e_j),h(e_i,e_i)\right)\}<4l_1{l}_2ϵ.\hfill \end{array}$$

(34)

Thus, the proof is complete from Theorem 1 and from (34).

Based on Theorem 3 and the similar proof of Theorem 5, we reach the following result.

Corollary 5.

Assume that $M^{l_1+l_2}=N_T^{l_1}{\times }_f{N}_{\theta }^{l_2}$ is a compact warped product pointwise semi-slant submanifold of a complex space form ${\tilde{M}}^{2m}\left(4ϵ\right)$ satisfying the following:

$$\begin{array}{c}\hfill \parallel \nabla f{\parallel }^2<\left\{\frac{\left(4l_1{l}_2ϵ-{\parallel h_{\mu }\parallel }^2\right)f^2}{2l_2\left({csc}^2\theta +{cot}^2\theta \right)}\right\}.\end{array}$$

Then, $M^{p+q}$ is homeomorphic to a sphere ${\mathbb{S}}^{p+q}$ when $p+q\ne 3$, while $M^3$ is homotopic to a sphere ${\mathbb{S}}^3$.

5.7. Proof of Theorem 7

In this theorem, we replace our pinching condition (2) with the Hessian of the warping function and Ricci curvature by using the concept of the eigenvalue of the warped function. If f is a first eigenfunction of the Laplacian of $M^n$ associated with the first eigenvalue ${\lambda }_1$, that is, $\Delta f={\lambda }_{1}f$, then we an recall Bochner’s formula (see, e.g., [32]), which states that for a differentiable function f defined on a Riemannian manifold, the following relation holds:

$$\begin{array}{c}\hfill \frac{1}{2}{\Delta \parallel \nabla f\parallel }^2={\parallel {\nabla }^{2}f\parallel }^2+Ric(\nabla f,\nabla f)+g\left(\nabla f,\nabla (\Delta f)\right).\end{array}$$

By integrating the above equation with the aid of Stokes’ theorem, we obtain

$$\begin{array}{c}\hfill \int \parallel {\nabla }^2{f\parallel }^{2}dV+\int Ric(\nabla f,\nabla f)dV+\int g\left(\nabla f,\nabla (\Delta f)\right)dV=0.\end{array}$$

Now, by using $\Delta f={\lambda }_{1}f$ and slightly rearranging the above equation, we derive

$$\begin{array}{c}\hfill {\int \parallel \nabla f\parallel }^{2}dV=-\frac{1}{{\lambda }_1}\left(\int \parallel {\nabla }^2{f\parallel }^{2}dV+\int Ric(\nabla f,\nabla f)dV\right).\end{array}$$

(35)

On combing Equations (28) and (27), we obtain

$$\begin{array}{c}\hfill \left(2{csc}^2\theta +l_2\right){\int }_{M^n}{\parallel \nabla f\parallel }^{2}dV+{\int }_{M^n}\frac{f^2{\parallel h_{\mu }\parallel }^2}{l_2}dV<3l_1ϵ{\int }_{M^n}{f}^{2}dV.\end{array}$$

(36)

Following from (35) and (36), we find that

$$\begin{array}{c}\hfill {\int }_{M^n}\left(\frac{\parallel h_{\mu }{\parallel }^2}{l_2}-3l_1ϵ\right)f^{2}dV<\frac{\left(2{csc}^2\theta +l_2\right)}{{\lambda }_1}{\int }_{M^n}\left(\parallel {\nabla }^2{f\parallel }^2+Ric(\nabla f,\nabla f)\right)dV,\end{array}$$

which implies that

$$\begin{array}{c}\hfill \parallel {\nabla }^2{f\parallel }^2+Ric(\nabla f,\nabla f)>\left\{\frac{\left(\parallel h_{\mu }{\parallel }^2-3l_1{l}_2ϵ\right){\lambda }_1{f}^2}{\left(2{csc}^2\theta +l_2\right)}\right\}.\end{array}$$

(37)

The proof follows from the above Equation (37) along with Theorem 2.

6. Consequences

It is well known that a complete simply-connected complex space form ${\tilde{M}}^{2m}\left(4ϵ\right)$ is holomorphicaly isometric to the complex Euclidean space ${\mathbb{C}}^m$, the complex projective m-space $\mathbb{C}{P}^m\left(4\right)$, and a complex hyperbolic m-space $\mathbb{C}{H}^m(-4)$ with $ϵ=0,1\&ϵ=-1$. Therefore, we define the following corollaries in consequence of our Theorem 2 and Theorem 5.

Corollary 6.

Let $M^{l_1+l_2}=N_T^{l_1}{\times }_f{N}_{\theta }^{l_2}$ be a compact warped product pointwise semi-slant submanifold in a complex Euclidean space ${\mathbb{C}}^m$ satisfying the condition

$$\begin{array}{c}\hfill \left({csc}^2\theta +{cot}^2\theta +l_2\right){\left|\right|\nabla f\left|\right|}^2+f\Delta f+\frac{f^2}{l_2}{\parallel h_{\mu }\parallel }^2<0.\end{array}$$

Then, there do not exist stable integral $l_1$-currents in $M^{l_1+l_2}$ and ${\mathbb{H}}_{l_1}(M^{l_1+l_2},\mathbb{G})={\mathbb{H}}_{l_2}(M^{l_1+l_2},$$\mathbb{G})=0$. Furthermore, $M^{l_1+l_2}$ is homeomorphic to a sphere ${\mathbb{S}}^{l_1+l_2}$ when $l_1+l_2\ge 4$, while $M^3$ is homotopic to a sphere ${\mathbb{S}}^3$.

Similarly, for the complex projective m-space $\mathbb{C}{P}^m\left(4\right)$ we have the following.

Corollary 7.

Let $M^{l_1+l_2}=N_T^{l_1}{\times }_f{N}_{\theta }^{l_2}$ be a compact warped product pointwise semi-slant submanifold in a complex projective m-space $\mathbb{C}{P}^{2m}\left(4\right)$ satisfying the condition

$$\begin{array}{c}\hfill \left({csc}^2\theta +{cot}^2\theta +l_2\right){\left|\right|\nabla f\left|\right|}^2+f\Delta f<\frac{f^2}{l_2}\left(3l_1{l}_2-{\parallel h_{\mu }\parallel }^2\right).\end{array}$$

Then, there do not exist stable integral $l_1$-currents in $M^{l_1+l_2}$ and ${\mathbb{H}}_{l_1}(M^{l_1+l_2},$$\mathbb{G})={\mathbb{H}}_{l_2}(M^{l_1+l_2},$$\mathbb{G})=0$. In addition, $M^{l_1+l_2}$ is homeomorphic to a sphere ${\mathbb{S}}^{l_1+l_2}$ when $l_1+l_2\ge 4$, while $M^3$ is homotopic to a sphere ${\mathbb{S}}^3$.

7. Conclusions

The presented study is significant in light of the extant literature thanks to the new pinching conditions presented in terms of pointwise slant functions and the Laplacian of the warped function. We have discussed the rigidity results and investigated several topological classifications. In addition, we have derived a number of extrinsic conditions involving relevant geometric quantities by analyzing the extent to which the topology of warped product submanifolds is affected by the conditions on the main intrinsic and main extrinsic curvature invariants. A number of topological sphere theorems have been investigated in refeence to the connection between warped product submanifolds and homotopic–homologic theory. The contents of the present paper can be expected to attract researchers to the prospect of finding possible applications in various research areas of physics.