The Homology of Warped Product Submanifolds of Spheres and Their Applications

Abstract

The aim of the current article is to formulate sufficient conditions for the Laplacian and a gradient of the warping function of a compact warped product submanifold ${\Sigma }^{{\beta }_1+{\beta }_2}$ in a unit sphere ${\mathbb{S}}^d$ that provides trivial homology and fundamental groups. We also validate the instability of current flows in ${\pi }_1\left({\Sigma }^{{\beta }_1+{\beta }_2}\right)$. The constraints are also applied to the warped function eigenvalues and integral Ricci curvatures.

1. Introduction and Main Results

An algebraic description of a manifold can be found in its homology groups, which are significant topological invariants. This theory has various applications since these groups carry extensive topological information about the connected components, holes, tunnels, and dimensions of the manifold. In fact, homology theory has applications in root construction, protein docking, image segmentation, and gene expression data [1]. It is generally acknowledged that any non-trivial integral homology class in $H_{{\beta }_2}({\Sigma }^n,\mathbb{Z})$ is associated with the topological properties of submanifolds in different ambient spaces. The authors of [2], Federer and Fleming, were the first to demonstrate this idea using the method of variational calculus to represent the idea of geometric measure spaces. Later on, Lawson-Simons [3] constructed an escalation for the second fundamental form, which enforces the vanishing of the homology in a region of intermediate dimensions and the non-existence of stable current flows in the submanifold of the simply connected space form, and discovered the following idea, the main motivation for this work:

Theorem 1 

([3,4]). If the following optimization inequality holds for a compact m-dimensional submanifold in a space form $\tilde{M}\left(c\right)$ such that the curvature $c\ge 0$ and ${\beta }_1$ is an integer satisfying $0<{\beta }_1<m$,

$$\begin{array}{c}\hfill \sum _{b_1={\beta }_1+1}^m\sum _{b_2=1}^{{\beta }_1}(2\left|\right|\mathbf{A}({\mathcal{E}}_{b_1},{\mathcal{E}}_{b_2}){\left|\right|}^2-g\left(\mathbf{A}({\mathcal{E}}_{b_1},{\mathcal{E}}_{b_1}),\mathbf{A}({\mathcal{E}}_{b_2},{\mathcal{E}}_{b_2})\right))<{\beta }_1(m-{\beta }_1)c,\end{array}$$

(1)

then no stable ${\beta }_1$-currents flow in ${\Sigma }^m$ and

$$H_{{\beta }_1}({\Sigma }^m,\mathbb{Z})=H_{{\beta }_2}({\Sigma }^m,\mathbb{Z})=0$$

for any integer ${\beta }_2=m-{\beta }_1$, and $H_i({\Sigma }^m,\mathbb{Z})$ is the i-th homology group of ${\Sigma }^m$ with integer coefficients.

The first study of vanishing homology groups on warped product submanifold theory can be found in [5]. By placing appropriate limits on the Laplacian and the gradient of the warping function, Sahin [5] was able to confirm some conclusions regarding the nonexistence of stable current and vanishing homology groups into the contact CR-warped product that was immersed in a sphere with odd dimensions with the implementation of Theorem 1. In recent years, a significant number of studies have been conducted on the geometric structure and topological characteristics of submanifolds in various ambient spaces. These structures and characteristics have been demonstrated in numerous papers covering numerous applications, such as Euclidean spaces [6], complex projective spaces [7], CR-warped product submanifolds in Sasakian space forms [5], CR-warped product submanifolds in Euclidean spaces [8], CR-warped products in nearly Kaeher manifolds [9,10], CR-warped products in hyperbolic spaces [11], and many others (see [12,13]). There is a closed relationship between the absence of stable currents and the vanishing homology groups of submanifolds in various ambient manifold classes. These conclusions were reached by applying pinching conditions to the second fundamental form. Many writers have examined a variety of topological features in response to the lack of stable currents or stable submanifolds [13,14] motivated by Theorem [3] in the references. In the above literature, we observed that the topological and geometrical approaches have lately become useful ideas in machine learning theory due to the need for deep learning models in curved spaces. The significance of submanifold theory was once again demonstrated by the notion that the data might be viewed as a submanifold of Euclidean space. It is obvious that the theory of submanifolds will continue to be studied in light of this new area of applications.

2. Preliminaries

Let a sphere with a constant sectional curvature, c, be represented by ${\mathbb{S}}^d$, $c=1>0$, and the d-dimension. Given that ${\mathbb{S}}^d$ accepts a canonical isometric immersion in ${\mathbb{R}}^{d+1}$, we use this as our main argument:

$${\mathbb{S}}^d=\{V_2\in {\mathbb{R}}^{d+1}:\left|\right|V_2{\left|\right|}^2=1\}.$$

The Riemannian curvature tensor $\tilde{R}$ of the sphere ${\mathbb{S}}^d$ satisfies

$$\begin{array}{c}\hfill \tilde{R}(V_2,V_3,V_4,V_5)=g(V_2,V_5)g(V_3,V_4)-g(V_3,V_5)g(V_2,V_4),\end{array}$$

(2)

for any $V_3,V_2,V_4,V_5\in \Gamma \left(T\tilde{M}\right)$, where $T\tilde{M}$ is the tangent bundle of ${\mathbb{S}}^d$, and g is the Riemannian metric. In other words, the unit sphere ${\mathbb{S}}^d$ is a manifold with a constant sectional curvature equal to 1.

Let us assume that ${\Sigma }^m$ is an m-dimensional Riemannian submanifold of a Riemannian manifold $\tilde{M}$. Let us denote $\Gamma (T\Sigma )$ for the section of the tangent bundle of $\Sigma$ and $\Gamma \left(T{\Sigma }^{\perp }\right)$ for the set of all normal vector fields of $\Sigma$, respectively. Let us also denote ∇ for the Levi–Civita connection on tangent bundle $T\Sigma$ and ${\nabla }^{\perp }$ for the Levi–Civita connection on the normal bundle $T\Sigma$. If $\tilde{R}$ and R are represented as the Riemannian curvature tensors on the Riemannian manifold $\tilde{M}$ and submanifold ${\Sigma }^m$, respectively, then the Gauss equation is given by

$$\begin{array}{cc}\hfill \tilde{R}(V_2,V_3,V_4,V_5)=& R(V_2,V_3,V_4,V_5)+g\left(\mathbf{A}(V_2,V_4),\mathbf{A}(V_3,V_5)\right)\hfill \\ \hfill & -g\left(\mathbf{A}(V_2,V_5),\mathbf{A}(V_3,V_4)\right),\hfill \end{array}$$

(3)

for any $V_2,V_3,V_4,V_5\in \Gamma \left(T\tilde{M}\right)$, and $\mathbf{A}$ is the second fundamental form of ${\Sigma }^m$. A local orthonormal frame’s $\{{\mathcal{E}}_1,{\mathcal{E}}_2,\cdots ,{\mathcal{E}}_m\}$ and the mean curvature vector $\mathcal{H}$ on $\Sigma$ is defined by

$$\begin{array}{c}\hfill {\left|\right|\mathcal{H}\left|\right|}^2=\frac{1}{m^2}\sum _{r=m+1}^d{\left(\sum _{b_1=1}^m{\mathbf{A}}_{b_1{b}_1}\right)}^2.\end{array}$$

(4)

The scalar curvature of submanifold ${\Sigma }^m$, denoted by $\tau \left(T_x{\Sigma }^m\right)$, is formulated as follows:

$$\begin{array}{c}\hfill \tau \left(T_x{\Sigma }^m\right)=\sum _{1\le b_1<b_2\le m}{K}_{b_1{b}_2}\end{array}$$

(5)

where $K_{b_1{b}_2}=K\left({\mathcal{E}}_{b_1}\wedge {\mathcal{E}}_{b_2}\right)$ is the sectional curvature of ${\Sigma }^m$. The first Equality (5) is proportionate to the following equation, which will be used often in later proofs:

$$\begin{array}{c}\hfill 2\tau \left(T_x{\Sigma }^m\right)=\sum _{1\le b_1\ne b_2\le m}{K}_{b_1{b}_2}.\end{array}$$

(6)

Similarly, the scalar curvature $\tau \left({\prod }_x\right)$ of a $\prod -$ plane is given by

$$\begin{array}{c}\hfill \tau (\prod _x)=\sum _{1\le b_1<b_2\le m}{K}_{b_1{b}_2}.\end{array}$$

(7)

If the plane sections are spanned by ${\mathcal{E}}_{b_1}$ and ${\mathcal{E}}_{b_2}$ at x, then the sectional curvatures of the submanifold ${\Sigma }^m$ and the Riemannian manifold ${\tilde{M}}^d$ are denoted by $K_{b_1{b}_2}$ and ${\tilde{K}}_{b_1{b}_2}$, respectively. If $K_{b_1{b}_2}$ and ${\tilde{K}}_{b_1{b}_2}$ are spanned by $\left\{{\mathcal{E}}_{b_1}and{\mathcal{E}}_{b_2}\right\}$ at x, respectively, then using the Gauss Equations (3) and (5), we have

$$\begin{array}{c}\hfill K\left({\mathcal{E}}_{b_1}\wedge {\mathcal{E}}_{b_2}\right)=\tilde{K}\left({\mathcal{E}}_{b_1}\wedge {\mathcal{E}}_{b_2}\right)+\sum _{r=m+1}^d\left({\mathbf{A}}_{b_1{b}_1}^r{\mathbf{A}}_{b_2{b}_2}^r-{\left({\mathbf{A}}_{b_1{b}_2}^r\right)}^2\right).\end{array}$$

(8)

For more details regarding the above definitions, see [15,16,17].

3. Warped Product Manifolds

A definition of warped product manifolds was given by Bishop and O’Neill [18] by taking the negative curvature of the manifold. The product manifold ${\Sigma }^n=N_1\times N_2$ of two Riemannian manifolds, $N_1$ and $N_2$, with matrices $g_1$ and $g_2$, respectively, is defined as a warped product as ${\Sigma }^n=N_1{\times }_{\mu }{N}_2$ if the metric of ${\Sigma }^n$ satisfies $g=g_1+{\mu }^2{g}_2$, where $\mu$ stands for the warping function defined on the base $N_1$. Of course, in this case, $\mu$ is constant, and ${\Sigma }^n$ is a usual Riemannian product. Some important formulas were given by Bishop and O’Neill [18], including the following equations:

$$\begin{array}{c}\hfill {\nabla }_{V_1}{U}_1={\nabla }_{U_1}{V}_1=\frac{\left(U_1\mu \right)}{\mu }{V}_1\end{array}$$

(9)

$$\begin{array}{c}\hfill \mathcal{R}(U_1,U_2)V_1=\frac{{\mathcal{H}}^{\mu }(U_1,V_1)}{\mu }{U}_2,\end{array}$$

(10)

for any $U_1,U_2\in \Gamma \left(T{N}_1\right)$ and $V_1\in \Gamma \left(T{N}_2\right)$, where ${\mathcal{H}}^{\mu }$ is a Hessian tensor of $\mu$ such that

$$\begin{array}{c}\hfill {\mathcal{H}}^{\mu }(V_2,V_3)=g({\nabla }_{V_2}\nabla \mu ,V_3).\end{array}$$

We also have another interesting relationship regarding the connection ∇ on ${\Sigma }^n$ that will be very useful for our proof in the main results.

$$\begin{array}{c}\hfill g(\nabla lnf,V_1)=V_1(lnf).\end{array}$$

(11)

The following remarks are consequences of the definition of warped products:

Remark 1. 

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

Remark 2. 

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

The ${\left|\right|\nabla \mu \left|\right|}^2$ gradient of the positive differential function $\mu$ for an orthonormal frame $\{{\mathcal{E}}_1,\dots ,{\mathcal{E}}_n\}$ is then defined as

$$\begin{array}{c}\hfill {\left|\right|\nabla \mu \left|\right|}^2=\sum _{i=1}^m{\left({\mathcal{E}}_i\left(\mu \right)\right)}^2.\end{array}$$

(12)

The gradient $\overrightarrow{\nabla }\mu$ in [19] is given by

$$\begin{array}{c}\hfill g(\overrightarrow{\nabla }\mu ,V_2)=V_2\mu ,and\overrightarrow{\nabla }\mu =\sum _{i=1}^m{\mathcal{E}}_i\left(\mu \right){\mathcal{E}}_i,\end{array}$$

(13)

and the Laplacian $\Delta \mu$ of $\mu$ is defined as

$$\begin{array}{cc}\hfill \Delta \mu & =-\sum _{i=1}^m\{\left({\nabla }_{{\mathcal{E}}_i}{\mathcal{E}}_i\right)\mu -{\mathcal{E}}_i\left({\mathcal{E}}_i\left(\mu \right)\right)\}=\sum _{i=1}^{m}g({\nabla }_{{\mathcal{E}}_i}\nabla \mu ,{\mathcal{E}}_i)=trHess\left(\mu \right).\hfill \end{array}$$

(14)

Remark 3. 

It should be emphasized that we take into account Chen’s opposite sign of [19] of the function’s Laplacian μ, that is, $\Delta \mu =div(\nabla \mu )$. The sign convention for the Laplacian Δ adapted by the authors is $\Delta =\frac{{\partial }^2}{\partial t^2}$ on the real line.

In addition, as the vector fields $V_2$ and $V_4$ are tangent to $N_1^{{\beta }_1}$ and $N_2^{{\beta }_2}$, respectively, we obtain

$$\begin{array}{cc}\hfill K(V_2\wedge V_4)& =g(R(V_2,V_4)V_2,V_4)=\left({\nabla }_{V_2}{V}_2\right)lnhg(V_4,V_4)-g\left({\nabla }_{V_2}\left((V_{2}ln\mu )V_5\right),V_4\right)\hfill \\ \hfill & =\left({\nabla }_{V_2}{V}_2\right)lnhg(V_4,V_4)-g\left({\nabla }_{V_2}(V_{2}ln\mu )V_4+(V_{2}ln\mu ){\nabla }_{V_2}{V}_4,V_4\right)\hfill \\ \hfill & =\left({\nabla }_{V_2}{V}_2\right)ln\mu g(V_4,V_4)-{(V_{2}ln\mu )}^2-V_2(V_{2}ln\mu ).\hfill \end{array}$$

(15)

By summing the vector fields with respect to the orthonormal frame’s $\{{\mathcal{E}}_1,\cdots {\mathcal{E}}_n\}$, one obtains

$$\begin{array}{c}\hfill \sum _{i=1}^{{\beta }_1}\sum _{j=1}^{{\beta }_2}K({\mathcal{E}}_i\wedge {\mathcal{E}}_j)=\sum _{i=1}^{{\beta }_1}\sum _{j=1}^{{\beta }_2}\left(\left({\nabla }_{{\mathcal{E}}_i}{\mathcal{E}}_i\right)ln\mu -{\mathcal{E}}_i({\mathcal{E}}_{i}ln\mu )-{({\mathcal{E}}_{i}ln\mu )}^2\right),\end{array}$$

which implies that

$$\begin{array}{c}\hfill \sum _{i=1}^{{\beta }_1}\sum _{j=1}^{{\beta }_2}K({\mathcal{E}}_i\wedge {\mathcal{E}}_j)=-\frac{{\beta }_2\Delta \mu }{\mu }.\end{array}$$

(16)

4. Main Results

We must also use a technique that is an invaluable tool for verifying our results. In the first case, assuming that the warped product submanifold is embedded in ${\mathbb{S}}^d$, and utilizing Theorem 1, we intend to obtain some identical conclusions regarding the warped product submanifold hypothesis, where pinching criteria on the second fundamental form shall be replaced by the warping function.

Using Theorem 1, the first significant outcome of this paper is now provided.

Theorem 2. 

Let $\Psi :{\Sigma }^{{\beta }_1+{\beta }_2}=N_1^{{\beta }_1}{\times }_{\mu }{N}_2^{{\beta }_2}⟶{\mathbb{S}}^d$ be an $N_1^{{\beta }_1}$-minimal isometric embedding from a compact warped product submanifold ${\Sigma }^{{\beta }_1+{\beta }_2}$ into an d-dimensional sphere ${\mathbb{S}}^d$. If the following inequality satisfies

$$\begin{array}{c}\hfill 3\mu \Delta \mu <2({\beta }_2{\parallel \nabla \mu \parallel }^2+{\beta }_1{\mu }^2)\end{array}$$

(17)

where $\Delta \mu$ and $\nabla \mu$ are the Laplacian and gradient of the warping function, respectively, then the following are true:

(a) 

There is no stable integral ${\beta }_1$-current flow in a warped product submanifold ${\Sigma }^{{\beta }_1+{\beta }_2}$.

(b) 

The i-th integral homology groups of ${\Sigma }^{{\beta }_1+{\beta }_2}$ vanish, which means

$$\begin{array}{c}\hfill {\mathbb{H}}_{{\beta }_1}({\Sigma }^{{\beta }_1+{\beta }_2},\mathbb{Z})={\mathbb{H}}_{{\beta }_2}({\Sigma }^{{\beta }_1+{\beta }_2},\mathbb{Z})=0.\end{array}$$

(c) 

If ${\beta }_1=1$, then the fundamental group ${\pi }_1(\Sigma )$ is null, i.e., ${\pi }_1(\Sigma )=0.$ Moreover, ${\Sigma }^{{\beta }_1+{\beta }_2}$ is a simply connected warped product manifold.

Proof. 

Suppose $\dim \left(N_1\right)={\beta }_1$ and $\dim \left(N_2\right)={\beta }_2$. Let $\{{\mathcal{E}}_1,{\mathcal{E}}_2,\cdots ,{\mathcal{E}}_{{\beta }_1}\}$, and $\{{\mathcal{E}}_{{\beta }_1+1}^{*},\cdots ,{\mathcal{E}}_m^{*}\}$ be orthonormal frames of $T{N}_1$ and $T{N}_2$, respectively. From the Gauss Equation (3) for the standard unit sphere ${\mathbb{S}}^d$, we then have

$$\begin{array}{cc}\hfill \sum _{b_1=1}^{{\beta }_1}\sum _{b_2=1}^{{\beta }_2}\left\{2\right||\mathbf{A}({\mathcal{E}}_{b_1},{\mathcal{E}}_{b_2}){\left|\right|}^2-& g\left(\mathbf{A}({\mathcal{E}}_{b_2},{\mathcal{E}}_{b_2}),\mathbf{A}({\mathcal{E}}_{b_1},{\mathcal{E}}_{b_1})\right)\}\hfill \\ \hfill & =\sum _{b_1=1}^{{\beta }_1}\sum _{b_2=1}^{{\beta }_2}g\left(R({\mathcal{E}}_{b_1},{\mathcal{E}}_{b_2}){\mathcal{E}}_{b_1},{\mathcal{E}}_{b_2}\right)\hfill \\ \hfill & -\sum _{b_1=1}^{{\beta }_1}\sum _{b_2=1}^{{\beta }_2}g\left(\tilde{R}({\mathcal{E}}_{b_1},{\mathcal{E}}_{b_2}){\mathcal{E}}_{b_1},{\mathcal{E}}_{b_2}\right)+\sum _{r=1}^d\sum _{b_1=1}^{{\beta }_1}\sum _{b_2=1}^{{\beta }_2}{\left({\mathbf{A}}_{b_1{b}_2}^r\right)}^2.\hfill \end{array}$$

(18)

From $R({\mathcal{E}}_{b_1},{\mathcal{E}}_{b_2}){\mathcal{E}}_{b_1}=\frac{{\mathcal{H}}^{\mu }({\mathcal{E}}_{b_1},{\mathcal{E}}_{b_1})}{\mu }{\mathcal{E}}_{b_2}$ in (10), by taking the trace over $N_1^{{\beta }_1}$ and $N_2^{{\beta }_2}$, we derive

$$\begin{array}{c}\hfill \sum _{b_1=1}^{{\beta }_1}\sum _{b_2=1}^{{\beta }_2}g\left(R({\mathcal{E}}_{b_1},{\mathcal{E}}_{b_2}){\mathcal{E}}_{b_1},{\mathcal{E}}_{b_2}\right)=\frac{{\beta }_2}{\mu }\sum _{b_1=1}^{{\beta }_1}g\left({\nabla }_{{\mathcal{E}}_{b_1}}\nabla \mu ,{\mathcal{E}}_{b_1}\right).\end{array}$$

(19)

Thus, inserting (19) into the first term of the right-hand side of Equation (18), we derive

$$\begin{array}{cc}\hfill \sum _{b_1=1}^{{\beta }_1}\sum _{b_2=1}^{{\beta }_2}\left\{2\right||\mathbf{A}({\mathcal{E}}_{b_1},{\mathcal{E}}_{b_2}){\left|\right|}^2-& g\left(\mathbf{A}({\mathcal{E}}_{b_2},{\mathcal{E}}_{b_2}),\mathbf{A}({\mathcal{E}}_{b_1},{\mathcal{E}}_{b_1})\right)\}\hfill \\ \hfill & =\frac{{\beta }_2}{\mu }\sum _{b_1=1}^{{\beta }_1}g\left({\nabla }_{{\mathcal{E}}_{b_1}}\nabla \mu ,{\mathcal{E}}_{b_1}\right)+\sum _{r=1}^d\sum _{b_1=1}^{{\beta }_1}\sum _{b_2=1}^{{\beta }_2}{\left({\mathbf{A}}_{b_1{b}_2}^r\right)}^2\hfill \\ \hfill & -\sum _{b_1=1}^{{\beta }_1}\sum _{b_2=1}^{{\beta }_2}g\left(\tilde{R}({\mathcal{E}}_{b_1},{\mathcal{E}}_{b_2}){\mathcal{E}}_{b_1},{\mathcal{E}}_{b_2}\right).\hfill \end{array}$$

(20)

By calculating the Laplacian $\Delta \mu$ on ${\Sigma }^{{\beta }_1+{\beta }_2}$, one obtains

$$\begin{array}{c}\hfill \Delta \mu =\sum _{i=1}^{m}g\left({\nabla }_{{\mathcal{E}}_i}\nabla \mu ,{\mathcal{E}}_i\right)=\sum _{b_1=1}^{{\beta }_1}g\left({\nabla }_{{\mathcal{E}}_{b_1}}\nabla \mu ,{\mathcal{E}}_{b_1}\right)+\sum _{b_2=1}^{{\beta }_2}g\left({\nabla }_{{\mathcal{E}}_{b_2}}\nabla \mu ,{\mathcal{E}}_{b_2}\right).\end{array}$$

We know this from the warped product submanifold. From the hypothesis, $N_1^{{\beta }_1}$ is a geodesic submanifold in ${\Sigma }^m$. This implies that $\nabla \mu \in \mathfrak{X}\left(T{N}_1\right),$ and from the description of the warped product, we obtain

$$\begin{array}{c}\hfill \Delta \mu =\frac{1}{\mu }\sum _{b_2=1}^{{\beta }_2}g({\mathcal{E}}_{b_2},{\mathcal{E}}_{b_2}){\left|\right|\nabla \mu \left|\right|}^2+\sum _{b_1=1}^{{\beta }_1}g\left({\nabla }_{{\mathcal{E}}_{b_1}}\nabla \mu ,{\mathcal{E}}_{b_1}\right).\end{array}$$

By multiplying the above equation by $\frac{1}{\mu }$, we obtain

$$\begin{array}{c}\hfill \frac{\Delta \mu }{\mu }=\frac{1}{\mu }\sum _{b_1=1}^{{\beta }_1}g\left({\nabla }_{{\mathcal{E}}_{b_1}}\nabla \mu ,{\mathcal{E}}_{b_1}\right)+{\beta }_2{\left|\right|\nabla (ln\mu )\left|\right|}^2.\end{array}$$

We rewrite the above equations as follows:

$$\begin{array}{c}\hfill \frac{{\beta }_2}{\mu }\sum _{b_1=1}^{{\beta }_1}g\left({\nabla }_{{\mathcal{E}}_{b_1}}\nabla \mu ,{\mathcal{E}}_{b_1}\right)=\frac{{\beta }_2\Delta \mu }{\mu }-{\beta }_2^2{\left|\right|\nabla ln\mu \left|\right|}^2.\end{array}$$

(21)

Thus, from (20) and (21), one obtains

$$\begin{array}{cc}\hfill \sum _{b_1=1}^{{\beta }_1}\sum _{b_2=1}^{{\beta }_2}\left\{2\right||\mathbf{A}({\mathcal{E}}_{b_1},{\mathcal{E}}_{b_2}){\left|\right|}^2-& g\left(\mathbf{A}({\mathcal{E}}_{b_2},{\mathcal{E}}_{b_2}),\mathbf{A}({\mathcal{E}}_{b_1},{\mathcal{E}}_{b_1})\right)\}\hfill \\ \hfill =& \sum _{r=m+1}^d\sum _{b_1=1}^{{\beta }_1}\sum _{b_2=1}^{{\beta }_2}{\left({\mathbf{A}}_{b_1{b}_2}^r\right)}^2+\frac{{\beta }_2\Delta \mu }{\mu }-{\beta }_2^2{\left|\right|\nabla ln\mu \left|\right|}^2\hfill \\ \hfill & -\sum _{b_1=1}^{{\beta }_1}\sum _{b_2=1}^{{\beta }_2}g\left(\tilde{R}({\mathcal{E}}_{b_1},{\mathcal{E}}_{b_2}){\mathcal{E}}_{b_1},{\mathcal{E}}_{b_2}\right).\hfill \end{array}$$

Next, using the Gauss Equations (3) and (5) for the unit sphere ${\mathbb{S}}^d$, we find that

$$\begin{array}{c}\hfill m^2{\left|\right|\mathcal{H}\left|\right|}^2+m(m-1)={\left|\right|\mathbf{A}\left|\right|}^2+\sum _{1\le C<B\le m}K({\mathcal{E}}_C\wedge {\mathcal{E}}_B).\end{array}$$

(22)

The warped product manifold ${\Sigma }^{{\beta }_1+{\beta }_2}$ can be expressed using the preceding equation, and using (4) for the mean curvature definition and (8), we obtain

$$\begin{array}{cc}\hfill \sum _{r=m+1}^d{\left(\sum _{C=1}^m{\mathbf{A}}_{CC}^r\right)}^2=& \sum _{r=m+1}^d\sum _{i,j=1}^{{\beta }_1}{\left({\mathbf{A}}_{ij}^r\right)}^2+\sum _{r=m+1}^d\sum _{a,b=1}^{{\beta }_2}{\left({\mathbf{A}}_{ab}^r\right)}^2+2\sum _{r=m+1}^d\sum _{b_1=1}^{{\beta }_1}\sum _{b_2=1}^{{\beta }_2}{\left({\mathbf{A}}_{b_1{b}_2}^r\right)}^2\hfill \\ \hfill & +\sum _{b_1=1}^{{\beta }_1}\sum _{b_2=1}^{{\beta }_2}K({\mathcal{E}}_{b_1}\wedge {\mathcal{E}}_{b_2})+\sum _{1\le i<j\le {\beta }_1}K({\mathcal{E}}_i\wedge {\mathcal{E}}_j)+\sum _{1\le a<b\le {\beta }_2}K({\mathcal{E}}_a\wedge {\mathcal{E}}_b).\hfill \end{array}$$

(23)

Using (8) in the fourth term and (16) in the last two terms of the above equation, we derive

$$\begin{array}{cc}\hfill \sum _{r=m+1}^d{\left(\sum _{C=1}^m{\mathbf{A}}_{CC}^r\right)}^2& =\sum _{r=m+1}^d\sum _{i,j=1}^{{\beta }_1}{\left({\mathbf{A}}_{ij}^r\right)}^2+\sum _{r=m+1}^d\sum _{a,b=1}^{{\beta }_2}{\left({\mathbf{A}}_{ab}^r\right)}^2\hfill \\ \hfill & +2\sum _{r=m+1}^d\sum _{b_1=1}^{{\beta }_1}\sum _{b_2=1}^{{\beta }_2}{\left({\mathbf{A}}_{b_1{b}_2}^r\right)}^2-\frac{{\beta }_2\Delta \mu }{\mu }-m(m-1)\hfill \\ \hfill & +\sum _{1\le i<j\le {\beta }_1}\tilde{K}({\mathcal{E}}_i\wedge {\mathcal{E}}_j)+\sum _{1\le a<b\le {\beta }_2}\tilde{K}({\mathcal{E}}_a\wedge {\mathcal{E}}_b)\hfill \\ \hfill & +\sum _{r=m+1}^d\sum _{1\le i<j\le {\beta }_1}\left({\mathbf{A}}_{ii}^r{\mathbf{A}}_{jj}^r-{\left({\mathbf{A}}_{ij}^r\right)}^2\right)+\sum _{r=m+1}^d\sum _{1\le a<b\le {\beta }_2}\left({\mathbf{A}}_{aa}^r{\mathbf{A}}_{bb}^r-{\left({\mathbf{A}}_{ab}^r\right)}^2\right).\hfill \end{array}$$

(24)

Thus, by modifying the previous equation and applying the sphere’s curvature equation ${\mathbb{S}}^d$, one can obtain

$$\begin{array}{cc}\hfill \sum _{r=m+1}^d{\left(\sum _{A=1}^m{\mathbf{A}}_{AA}^r\right)}^2=& \sum _{r=m+1}^d\sum _{i,j=1}^{{\beta }_1}{\left({\mathbf{A}}_{ij}^r\right)}^2+\sum _{r=m+1}^d\sum _{a,b=1}^{{\beta }_2}{\left({\mathbf{A}}_{ab}^r\right)}^2+2\sum _{r=m+1}^d\sum _{b_1=1}^{{\beta }_1}\sum _{b_2=1}^{{\beta }_2}{\left({\mathbf{A}}_{b_1{b}_2}^r\right)}^2\hfill \\ \hfill & -\frac{{\beta }_2\Delta \mu }{\mu }-\sum _{r=m+1}^d\sum _{1\le i<j\le {\beta }_1}{\left({\mathbf{A}}_{ij}^r\right)}^2+\sum _{r=m+1}^d\sum _{1\le i<j\le {\beta }_1}{\mathbf{A}}_{ii}^r{\mathbf{A}}_{jj}^r\hfill \\ \hfill & +\sum _{r=m+1}^d\left({\left({\mathbf{A}}_{11}^r\right)}^2+\cdots +{\left({\mathbf{A}}_{{\beta }_1{\beta }_2}\right)}^2\right)-\sum _{r=m+1}^d\left({\left({\mathbf{A}}_{11}^r\right)}^2+\cdots +{\left({\mathbf{A}}_{{\beta }_1{\beta }_2}\right)}^2\right)\hfill \\ \hfill & +\sum _{r=m+1}^d\sum _{1\le a<b\le {\beta }_2}{\mathbf{A}}_{aa}^r{\mathbf{A}}_{bb}^r-\sum _{r=m+1}^d\sum _{1\le a<b\le {\beta }_2}{\left({\mathbf{A}}_{ab}^r\right)}^2\hfill \\ \hfill & +\sum _{r=m+1}^d\left({\left({\mathbf{A}}_{{\beta }_1+1{\beta }_2+1}^r\right)}^2+\cdots +{\left({\mathbf{A}}_{mm}\right)}^2\right)\hfill \\ \hfill & -\sum _{r=m+1}^d\left({\left({\mathbf{A}}_{{\beta }_1+1{\beta }_2+1}^r\right)}^2+\cdots +{\left({\mathbf{A}}_{mm}\right)}^2\right)\hfill \\ \hfill & +{\beta }_1({\beta }_1-1)+{\beta }_2({\beta }_2-1)-m(m-1).\hfill \end{array}$$

(25)

The result of rearranging the preceding equation is

$$\begin{array}{cc}\hfill \sum _{r=m+1}^d{\left(\sum _{A=1}^m{\mathbf{A}}_{AA}^r\right)}^2=& \sum _{r=m+1}^d\sum _{i,j=1}^{{\beta }_1}{\left({\mathbf{A}}_{ij}^r\right)}^2+\sum _{r=m+1}^d\sum _{a,b=1}^{{\beta }_2}{\left({\mathbf{A}}_{ab}^r\right)}^2+2\sum _{r=m+1}^d\sum _{b_1=1}^{{\beta }_1}\sum _{b_2=1}^{{\beta }_2}{\left({\mathbf{A}}_{b_1{b}_2}^r\right)}^2\hfill \\ \hfill & +\sum _{r=m+1}^d\left\{\sum _{1\le i<j\le {\beta }_1}{\mathbf{A}}_{ii}^r{\mathbf{A}}_{jj}^r+{\left({\mathbf{A}}_{11}^r\right)}^2+\cdots +{\left({\mathbf{A}}_{{\beta }_1{\beta }_1}\right)}^2\right\}\hfill \\ \hfill & -\sum _{r=m+1}^d\left\{\sum _{1\le i<j\le {\beta }_1}{\left({\mathbf{A}}_{ij}^r\right)}^2+{\left({\mathbf{A}}_{11}^r\right)}^2+\cdots +{\left({\mathbf{A}}_{{\beta }_1{\beta }_1}\right)}^2\right\}\hfill \\ \hfill & +\sum _{r=m+1}^d\left\{\sum _{1\le a<b\le {\beta }_2}{\mathbf{A}}_{aa}^r{\mathbf{A}}_{bb}^r+{\left({\mathbf{A}}_{{\beta }_1+1{\beta }_1+1}^r\right)}^2+\cdots +{\left({\mathbf{A}}_{mm}\right)}^2\right\}\hfill \\ \hfill & -\sum _{r=m+1}^d\left\{\sum _{1\le a<b\le {\beta }_2}{\left({\mathbf{A}}_{ab}^r\right)}^2+{\left({\mathbf{A}}_{{\beta }_1+1{\beta }_1+1}^r\right)}^2+\cdots +{\left({\mathbf{A}}_{mm}\right)}^2\right\}\hfill \\ \hfill & -\frac{{\beta }_2\Delta \mu }{\mu }+2{\beta }_1{\beta }_2.\hfill \end{array}$$

Verifying this using the binomial theorem is straightforward, and it is clear that the base manifold $N_1^{{\beta }_1}$ of a warped product manifold $N_1^{{\beta }_1}{\times }_{\mu }{N}_2^{{\beta }_2}$ is minimal. Therefore, we have

$$\begin{array}{cc}\hfill \sum _{r=m+1}^d{\left(\sum _{A=p+1}^m{\mathbf{A}}_{AA}^r\right)}^2=& 2{\beta }_1{\beta }_2+\sum _{r=m+1}^d\sum _{i,j=1}^{{\beta }_1}{\left({\mathbf{A}}_{ij}^r\right)}^2+\sum _{r=m+1}^d\sum _{a,b=1}^{{\beta }_2}{\left({\mathbf{A}}_{ab}^r\right)}^2+2\sum _{r=m+1}^d\sum _{b_1=1}^{{\beta }_1}\sum _{b_2=1}^{{\beta }_2}{\left({\mathbf{A}}_{b_1{b}_2}^r\right)}^2\hfill \\ \hfill & +\sum _{r=m+1}^d\left({\left({\mathbf{A}}_{11}^r\right)}^2+\cdots +{\left({\mathbf{A}}_{{\beta }_1{\beta }_1}\right)}^2\right)-\sum _{r=m+1}^d\sum _{i,j=1}^{{\beta }_1}{\left({\mathbf{A}}_{ij}^r\right)}^2-\sum _{r=m+1}^d\sum _{a,b=1}^{{\beta }_2}{\left({\mathbf{A}}_{ab}^r\right)}^2\hfill \\ \hfill & +\sum _{r=m+1}^d\left({\left({\mathbf{A}}_{{\beta }_1+1{\beta }_1+1}^r\right)}^2+\cdots +{\left({\mathbf{A}}_{mm}\right)}^2\right)-\frac{{\beta }_2\Delta \mu }{\mu }.\hfill \end{array}$$

(26)

Since the base manifold $N_1^{{\beta }_1}$ of the warped product submanifold $N_1^{{\beta }_1}{\times }_{\mu }{N}_2^{{\beta }_2}$ is known to be minimal according to the theorem’s hypothesis, that is, the partial mean curvature $H_1$ on $N_1^{{\beta }_1}$ vanishes, we can use this knowledge to determine that the $V^{th}$ term on the right-hand side of Equation (26) is equal to zero and that the $VI{I}^{th}$ term is equal to the first term on the left side. Thus,

$$\begin{array}{c}\hfill 2\sum _{r=m+1}^d\sum _{b_1=1}^{{\beta }_1}\sum _{b_2=1}^{{\beta }_2}{\left({\mathbf{A}}_{b_1{b}_2}^r\right)}^2=\frac{{\beta }_2\Delta \mu }{\mu }-2{\beta }_1{\beta }_2.\end{array}$$

(27)

From (22) and (27), we have

$$\begin{array}{cc}\hfill \sum _{b_1=1}^{{\beta }_1}\sum _{b_2=1}^{{\beta }_2}\left\{2\right||\mathbf{A}({\mathcal{E}}_{b_1},{\mathcal{E}}_{b_2}){\left|\right|}^2-& g\left(\mathbf{A}({\mathcal{E}}_{b_2},{\mathcal{E}}_{b_2}),\mathbf{A}({\mathcal{E}}_{b_1},{\mathcal{E}}_{b_1}\right)\}\hfill \\ \hfill & =\frac{{\beta }_2\Delta \mu }{\mu }-{\beta }_2^2{\left|\right|\nabla ln\mu \left|\right|}^2+\frac{{\beta }_2\Delta \mu }{2\mu }-{\beta }_1{\beta }_2\hfill \\ \hfill & -\sum _{b_1=1}^{{\beta }_1}\sum _{b_2=1}^{{\beta }_2}g\left(\tilde{R}({\mathcal{E}}_{b_1},{\mathcal{E}}_{b_2}){\mathcal{E}}_{b_1},{\mathcal{E}}_{b_2}\right).\hfill \end{array}$$

From Equation (2), one then obtains

$$\begin{array}{c}\hfill \sum _{b_1=1}^{{\beta }_1}\sum _{b_2=1}^{{\beta }_2}g\left(\tilde{R}({\mathcal{E}}_{b_1},{\mathcal{E}}_{b_2}){\mathcal{E}}_{b_1},{\mathcal{E}}_{b_2}\right)=-{\beta }_1{\beta }_2.\end{array}$$

(28)

From this, we obtain

$$\begin{array}{cc}\hfill \sum _{b_1=1}^{{\beta }_1}\sum _{b_2=1}^{{\beta }_2}\left\{2\right||\mathbf{A}({\mathcal{E}}_{b_1},{\mathcal{E}}_{b_2}){\left|\right|}^2-& g\left(\mathbf{A}({\mathcal{E}}_{b_2},{\mathcal{E}}_{b_2}),\mathbf{A}({\mathcal{E}}_{b_1},{\mathcal{E}}_{b_1})\right)\}=\frac{3{\beta }_2\Delta \mu }{2\mu }-\frac{{\beta }_2^2}{{\mu }^2}{\left|\right|\nabla \mu \left|\right|}^2.\hfill \end{array}$$

(29)

Assuming (17) and (29), we obtain

$$\begin{array}{c}\hfill \sum _{b_1=1}^{{\beta }_1}\sum _{b_2=1}^{{\beta }_2}\left\{2\left|\right|\mathbf{A}({\mathcal{E}}_{b_1},{\mathcal{E}}_{b_2}){\left|\right|}^2-g\left(\mathbf{A}({\mathcal{E}}_{b_2},{\mathcal{E}}_{b_2}),\mathbf{A}({\mathcal{E}}_{b_1},{\mathcal{E}}_{b_1})\right)\right\}<{\beta }_1{\beta }_2.\end{array}$$

(30)

By applying Theorem 1 for a constant curvature $c=1$, we find that there are no stable ${\beta }_1$-currents in ${\Sigma }^{{\beta }_1+{\beta }_2}$, and $H_{{\beta }_1}({\Sigma }^{{\beta }_1+{\beta }_2},\mathbb{Z})=H_{{\beta }_2}({\Sigma }^{{\beta }_1+{\beta }_2},\mathbb{Z})=0$, which satisfies Proofs (a) and (b) of the theorem. On the other hand, if in (29) we make the substitution ${\beta }_1=1$, then we obtain

$$\begin{array}{cc}\hfill \sum _{b_2=2}^m\left\{2\right||\mathbf{A}({\mathcal{E}}_1,{\mathcal{E}}_{b_2}){\left|\right|}^2-& g\left(\mathbf{A}({\mathcal{E}}_{b_2},{\mathcal{E}}_{b_2}),\mathbf{A}({\mathcal{E}}_1,{\mathcal{E}}_1)\right)\}=\frac{3{\beta }_2\Delta \mu }{2\mu }-\frac{{\beta }_2}{{\mu }^2}{\left|\right|\nabla \mu \left|\right|}^2\hfill \end{array}$$

(31)

If the pinching condition (17) for ${\beta }_1=1$ and ${\beta }_2=m-1$ holds, then we obtain

$$\begin{array}{cc}\hfill \sum _{b_2=2}^m\left\{2\right||\mathbf{A}({\mathcal{E}}_1,{\mathcal{E}}_{b_2}){\left|\right|}^2-& g\left(\mathbf{A}({\mathcal{E}}_{b_2},{\mathcal{E}}_{b_2}),\mathbf{A}({\mathcal{E}}_1,{\mathcal{E}}_1)\right)\}<(m-1).\hfill \end{array}$$

(32)

There are no stable 1-currents in ${\Sigma }^{1+{\beta }_2}$ and $H_1({\Sigma }^{1+{\beta }_2},\mathbb{Z}=H_{n-1}({\Sigma }^{1+{\beta }_2},\mathbb{Z})=0.$ Let us assume that ${\pi }_1(\Sigma )$ does not equal 0. The traditional theorem, which uses the findings of Cartan and Hadamard, claims that there is a minimal closed geodesic in any non-trivial homotopy class in ${\pi }_1(\Sigma )$, which contradicts itself when applied to the compactness of ${\Sigma }^{1+{\beta }_2}$. Consequently, ${\pi }_1(\Sigma )=0$. The theorem’s third component can be expressed as follows. This Riemannian manifold is simply connected if the finite basic group for any Riemannian manifold is null. Therefore, ${\Sigma }^{{\beta }_1+{\beta }_2}$ is simply connected. □

Inspired by geometric rigidity, the second mission of this study is to show a novel vanishing result for compact warped product submanifolds utilizing the Ricci curvature and the eigenvalue of the warping function’s Laplacian. The following theorem is detailed below.

Theorem 3. 

If the warping function μ is an eigenfunction of the Laplacian of ${\Sigma }^{{\beta }_1+{\beta }_2}$ associated with the first positive eigenvalue ${\lambda }_1$ under the same language of Theorem 2, together they satisfy the following inequality:

$$\begin{array}{c}\hfill \parallel {\nabla }^2{\mu \parallel }^2+Ric(\nabla \mu ,\nabla \mu )+\frac{{\lambda }_1(3{\lambda }_1+2{\beta }_1){\mu }^2}{2{\beta }_2}>0.\end{array}$$

(33)

Thus,

(a) 

There is no stable integral ${\beta }_1$-current flow in a warped product submanifold ${\Sigma }^{{\beta }_1+{\beta }_2}$.

(b) 

The i-th integral homology groups of ${\Sigma }^{{\beta }_1+{\beta }_2}$ with integer coefficients vanish; i.e.,

$$\begin{array}{c}\hfill {\mathbb{H}}_{{\beta }_1}({\Sigma }^{{\beta }_1+{\beta }_2},\mathbb{Z})={\mathbb{H}}_{{\beta }_2}({\Sigma }^{{\beta }_1+{\beta }_2},\mathbb{Z})=0.\end{array}$$

(c) 

The fundamental group ${\pi }_1(\Sigma )$ is null, i.e., ${\pi }_1(\Sigma )=0.$ Furthermore, ${\Sigma }^{{\beta }_1+{\beta }_2}$ is a simply connected warped product submanifold.

Proof. 

If $\mu$ is the first eigenfunction of the Laplacian $\Delta \mu =div(\nabla \mu )$ of ${\Sigma }^{{\beta }_1+{\beta }_2}$ associated with the first non-zero eigenvalue ${\lambda }_1$, that is, $\Delta \mu =-{\lambda }_1\mu$, then we recall the Bochner formula (see, e.g., [20]), which declares that the next connection is true for a differentiable function $\mu$ that is defined on a Riemannian manifold:

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

Using the Stokes theorem to integrate the preceding equation, we arrive at

$$\begin{array}{cc}\hfill {\int }_{{\Sigma }^{{\beta }_1+{\beta }_2}}{\parallel {\nabla }^2\mu \parallel }^{2}dV+& {\int }_{{\Sigma }^{{\beta }_1+{\beta }_2}}Ric(\nabla \mu ,\nabla \mu )dV\hfill \\ \hfill & =-{\int }_{{\Sigma }^{{\beta }_1+{\beta }_2}}g\left(\nabla \mu ,\nabla (\Delta \mu )\right)dV\hfill \end{array}$$

(34)

Now, using $\Delta \mu =-{\lambda }_1\mu$ and making a change in Equation (34), we derive

$$\begin{array}{c}\hfill {\int }_{{\Sigma }^{{\beta }_1+{\beta }_2}}{\parallel \nabla \mu \parallel }^{2}dV=\frac{1}{{\lambda }_1}\left({\int }_{{\Sigma }^{{\beta }_1+{\beta }_2}}{\parallel {\nabla }^2\mu \parallel }^{2}dV+{\int }_{{\Sigma }^{{\beta }_1+{\beta }_2}}Ric(\nabla \mu ,\nabla \mu )dV\right).\end{array}$$

(35)

If (33) holds, then one obtains

$$\begin{array}{c}\hfill {\int }_{{\Sigma }^{{\beta }_1+{\beta }_2}}\left\{\parallel {\nabla }^2{\mu \parallel }^2+Ric(\nabla \mu ,\nabla \mu )\right\}dV+\frac{{\lambda }_1(3{\lambda }_1+2{\beta }_1)}{2{\beta }_2}{\int }_{{\Sigma }^{{\beta }_1+{\beta }_2}}{\mu }^{2}dV>0.\end{array}$$

(36)

By substituting Equation (36) in (35), we obtain

$$\begin{array}{c}\hfill -\frac{{\lambda }_1(3{\lambda }_1+2{\beta }_1)}{2{\beta }_2}{\int }_{{\Sigma }^{{\beta }_1+{\beta }_2}}{\mu }^{2}dV<{\lambda }_1{\int }_{{\Sigma }^{{\beta }_1+{\beta }_2}}{\parallel \nabla \mu \parallel }^{2}dV,\end{array}$$

which implies that

$$\begin{array}{c}\hfill -3{\lambda }_1{\int }_{{\Sigma }^{{\beta }_1+{\beta }_2}}{\mu }^{2}dV<2{\beta }_1{\int }_{{\Sigma }^{{\beta }_1+{\beta }_2}}{\mu }^{2}dV+2{\beta }_2{\int }_{{\Sigma }^{{\beta }_1+{\beta }_2}}{\parallel \nabla \mu \parallel }^{2}dV.\end{array}$$

(37)

Now, using $\Delta =-{\lambda }_1\mu$ on the left-hand side of Equation (37), we arrive at

$$\begin{array}{c}\hfill {\int }_{{\Sigma }^{{\beta }_1+{\beta }_2}}\left\{3h\Delta \mu -2{\beta }_2{\parallel \nabla \mu \parallel }^2-2{\beta }_1{\mu }^2\right\}dV<0.\end{array}$$

(38)

One then obtains

$$\begin{array}{c}\hfill 3\mu \Delta \mu <2{\beta }_2{\parallel \nabla \mu \parallel }^2+2{\beta }_1{\mu }^2.\end{array}$$

(39)

Finally, we arrive at the conclusion of our theorem using the preceding equation as well as Theorem 2. The theorem’s proof is now complete. □

Riemannian manifolds with vanishing Ricci curvatures are known as Ricci-flat manifolds. In contrast to Einstein manifolds, Ricci-flat manifolds do not require the cosmological constant to disappear. For Riemannian manifolds of any dimension, with a vanishing cosmological constant, Ricci-flat manifolds are vacuum solutions to the physics equivalents of Einstein’s equations. Hence, we regard the warped product submanifold’s base as Ricci-flat. We give the following result from Theorem 3.

Theorem 4. 

If the warping function μ is an eigenfunction of the Laplacian of ${\Sigma }^{{\beta }_1+{\beta }_2}$ associated with the first positive eigenvalue ${\lambda }_1$ under the same statement of Theorem 2 with assumptions that base manifold is Ricci-flat, then the subsequent stringent inequality holds.

$$\begin{array}{c}\hfill \parallel {\nabla }^2{\mu \parallel }^2+\frac{{\lambda }_1(3{\lambda }_1+2{\beta }_1){\mu }^2}{2{\beta }_2}>0.\end{array}$$

(40)

Then, Statements (a), (b), and (c) in Theorem 2 are satisfied.

Proof. 

As we know that the base manifold $N_1^{{\beta }_1}$ is Ricci-flat, we then have

$$\begin{array}{c}\hfill Ric(\nabla \mu ,\nabla \mu )=0.\end{array}$$

(41)

Thus, by inserting the above-mentioned condition in (33), we obtain the desired result. □

As a quick implementation of Theorem 3, we can provide the following.

Theorem 5. 

Let us assume that $\Psi :{\Sigma }^{{\beta }_1+{\beta }_2}=N_1^{{\beta }_1}{\times }_{\mu }{N}_2^{{\beta }_2}⟶{\mathbb{S}}^{{\beta }_1+{\beta }_2+k_1}$ is an $N_1^{{\beta }_1}$-minimal isometric embedding from a compact warped product submanifold ${\Sigma }^{{\beta }_1+{\beta }_2}$ into an $({\beta }_1+{\beta }_2+k_1)$-dimensional sphere ${\mathbb{S}}^{{\beta }_1+{\beta }_2+k_1}$ that satisfies the following inequality:

$$\begin{array}{cc}\hfill {\int }_{{\Sigma }^{{\beta }_1+{\beta }_2}}{\parallel {\nabla }^2\mu \parallel }^{2}dV& <{\int }_{{\Sigma }^{{\beta }_1+{\beta }_2}}\sum _{i=1}^{{\beta }_1}{\parallel \mathbf{A}\left(\nabla \mu ,{\mathcal{E}}_i\right)\parallel }^{2}dV\hfill \\ \hfill & +\frac{({\beta }_1-1-{\lambda }_1)(3{\lambda }_1+2{\beta }_1)}{2{\beta }_2}{\int }_{{\Sigma }^{{\beta }_1+{\beta }_2}}{\mu }^{2}dV.\hfill \end{array}$$

(42)

Statements (a), (b), and (c) in Theorem 2 are satisfied. Moreover, $\left\{{\mathcal{E}}_i\right\}$ are orthonormal frames for the base $N_1^{{\beta }_1}$.

Proof. 

As we are aware, ${\Sigma }^{{\beta }_1+{\beta }_2}$ is an $N_1^{{\beta }_1}$-minimal compact warped product submanifold. Then, from the Gauss equation, one obtains

$$\begin{array}{c}\hfill R_{jkl}^i={\delta }_{ik}{\delta }_{jl}-{\delta }_{il}{\delta }_{jk}+\sum _{r=1}^k\left({\mathbf{A}}_{ik}^r{\mathbf{A}}_{jl}^r-{\mathbf{A}}_{il}^r{\mathbf{A}}_{jk}^r\right),\end{array}$$

which implies the following:

$$\begin{array}{c}\hfill R_{jij}^i={\delta }_{ii}{\delta }_{jj}-{\delta }_{ij}{\delta }_{ji}+\sum _{r=1}^k\left({\mathbf{A}}_{ii}^r{\mathbf{A}}_{jj}^r-{\mathbf{A}}_{ij}^r{\mathbf{A}}_{ji}^r\right).\end{array}$$

(43)

Taking into account that $N_1^{{\beta }_1}$ is a minimal submanifold and using the argument of the Ricci curvature for a unit sphere, we obtain

$$\begin{array}{c}\hfill Ric({\mathcal{E}}_i,{\mathcal{E}}_j)=({\beta }_1-1){\delta }_{ij}-\sum _{r=1}^k\sum _{l=1}^{{\beta }_1}{\mathbf{A}}_{il}^r{\mathbf{A}}_{jl}^r\end{array}$$

The above equation yields that

$$\begin{array}{c}\hfill Ric({\mu }_i{\mathcal{E}}_i,{\mathcal{E}}_j{\mu }_j)=({\beta }_1-1){\delta }_{ij}{\mu }_i{\mu }_j-\sum _{r=1}^k\sum _{l=1}^{{\beta }_1}{\mathbf{A}}_{il}^r{\mathbf{A}}_{jl}^r{\mu }_i{\mu }_j.\end{array}$$

(44)

Using Equation (44), we obtain

$$\begin{array}{c}\hfill Ric(\nabla \mu ,\nabla \mu )=({\beta }_1-1){\parallel \nabla \mu \parallel }^2-\sum _{i=1}^{{\beta }_1}{\parallel \mathbf{A}\left(\nabla \mu ,{\mathcal{E}}_i\right)\parallel }^2.\end{array}$$

Putting the preceding equation into practice in (35), we obtain

$$\begin{array}{c}\hfill {\int }_{{\Sigma }^{{\beta }_1+{\beta }_2}}\sum _{i=1}^{{\beta }_1}\parallel \mathbf{A}\left(\nabla \mu ,{\mathcal{E}}_i\right){\parallel }^{2}dV={\int }_{{\Sigma }^{{\beta }_1+{\beta }_2}}\parallel {\nabla }^2{\mu \parallel }^{2}dV+({\beta }_1-1-{\lambda }_1){\int }_{{\Sigma }^{{\beta }_1+{\beta }_2}}{\parallel \nabla \mu \parallel }^{2}dV.\end{array}$$

(45)

If our assumption in (42) is satisfied, then

$$\begin{array}{c}\hfill {\int }_{{\Sigma }^{{\beta }_1+{\beta }_2}}\parallel {\nabla }^2{\mu \parallel }^{2}dV<{\int }_{{\Sigma }^{{\beta }_1+{\beta }_2}}\sum _{i=1}^{{\beta }_1}{\parallel \mathbf{A}\left(\nabla \mu ,{\mathcal{E}}_i\right)\parallel }^{2}dV+\frac{({\beta }_1-1-{\lambda }_1)(3{\lambda }_1+2{\beta }_1)}{2{\beta }_2}{\int }_{{\Sigma }^{{\beta }_1+{\beta }_2}}{\mu }^{2}dV.\end{array}$$

The following form can be used to express the previously mentioned equation using $\Delta \mu =-{\lambda }_1\mu$:

$$\begin{array}{cc}\hfill \frac{3({\beta }_1-1-{\lambda }_1)}{2{\beta }_2}{\int }_{{\Sigma }^{{\beta }_1+{\beta }_2}}\mu \Delta \mu dV+& {\int }_{{\Sigma }^{{\beta }_1+{\beta }_2}}{\parallel {\nabla }^2\mu \parallel }^{2}dV\hfill \\ \hfill <& {\int }_{{\Sigma }^{{\beta }_1+{\beta }_2}}\sum _{i=1}^{{\beta }_1}{\parallel \mathbf{A}\left(\nabla \mu ,{\mathcal{E}}_i\right)\parallel }^{2}dV\hfill \\ \hfill & +\frac{{\beta }_1({\beta }_1-1-{\lambda }_1)}{{\beta }_2}{\int }_{{\Sigma }^{{\beta }_1+{\beta }_2}}{\mu }^{2}dV.\hfill \end{array}$$

Including the previous equation in (45), we derive that

$$\begin{array}{cc}\hfill \frac{3({\beta }_1-1-{\lambda }_1)}{2{\beta }_2}{\int }_{{\Sigma }^{{\beta }_1+{\beta }_2}}\mu \Delta \mu dV& <({\beta }_1-1-{\lambda }_1){\int }_{{\Sigma }^{{\beta }_1+{\beta }_2}}{\parallel \nabla \mu \parallel }^{2}dV\hfill \\ \hfill & +\frac{{\beta }_1({\beta }_1-1-{\lambda }_1)}{{\beta }_2}{\int }_{{\Sigma }^{{\beta }_1+{\beta }_2}}{\mu }^{2}dV,\hfill \end{array}$$

which implies the following from the above equation:

$$\begin{array}{c}\hfill 3\mu \Delta \mu <2{\beta }_2{\parallel \nabla \mu \parallel }^2+2{\beta }_1{\mu }^2.\end{array}$$

(46)

Consequently, it is the same as Equation (17), and we achieve the desired outcome, i.e., (42). This completes the proof of the corollary. □

Another intriguing outcome that can be attained as a consequence of Theorem 5 is the following:

Corollary 1. 

Under the same assumption as Theorem 5, if $\nabla \mu \in Ker\mathbf{A}$ holds with

$$\begin{array}{c}\hfill {\int }_{{\Sigma }^{{\beta }_1+{\beta }_2}}{\parallel {\nabla }^2\mu \parallel }^{2}dV<\frac{({\beta }_1-1-{\lambda }_1)(3{\lambda }_1+2{\beta }_1)}{2{\beta }_2}{\int }_{{\Sigma }^{{\beta }_1+{\beta }_2}}{\mu }^{2}dV,\end{array}$$

(47)

then the same statements as (a), (b), and (c) in Theorem 2 are satisfied.

Proof. 

Using the hypothesis of the corollary $\nabla \mu \in Ker\mathbf{A}$, we obtain $\mathbf{A}(\nabla \mu ,{\mathcal{E}}_i)=0$. Using this condition in (42), we can easily obtain the desired outcome. □

5. Conclusions Remarks

In the present paper, we have found sufficient conditions that have given us information regarding vanishing homology groups and fundamental groups. The homology groups were initially defined in algebraic topology and are a general way to associate a sequence of algebraic objects, such as Abelian groups or modules. If the two shapes are distinguished by examining their holes, this idea can force the definition of the homology groups and homology; it was originally a rigorous mathematical method for defining and categorizing holes in a manifold. The most constructive topological invariants for providing the algebraic summary of the manifold are the homology groups of a manifold. These homologies have many applications and are helpful in finding deep topological information regarding the connected components, holes, tunnels, and dimensions of the manifold. Indeed, homology theory has its applications in gene expression data, protein docking, image segmentation, and root architecture; see [21,22,23]. Furthermore, they can provide some significant examples of singularity structures in liquid crystals, systems in low-dimensional statistical mechanics, and physical phase transitions [24]. Moreover, the concept of space–time in general relativity uses warped product manifolds. There are two well-known product spaces with warped products: standard static space–times and the generalization of Robertson–Walker space–times [25]. Particularly in mathematical physics, general relativity relies extensively on differential topological approaches [26]. specifically how quantum gravity uses space–time homology [27,28,29]. The results of this work can be used in physical applications because they are related to the warped product manifold and homotopy–homology theory. We can extend the above work where the curvature is positive or zero to generalized spherical structures.