On the Topology of Warped Product Manifolds Minimally Immersed into a Sphere

Abstract

In this paper, we investigate the geometry and topology of compact warped product minimal submanifolds of arbitrary codimension immersed in a sphere. These submanifolds satisfy a specific pinching condition relating the length and Laplacian of the warping function to the dimensions of the warped product. Our results extend previous work on minimal immersions into the sphere.

1. Introduction and Main Results

A key challenge in differential geometry is to explore the relationship between the geometry and topology of Riemannian manifolds. In submanifold theory, a central question concerns how pinching conditions on intrinsic or extrinsic curvature invariants influence the geometry and topology of submanifolds in space forms. Simons [1] first established a fundamental result on minimal submanifolds of spheres with a sufficiently pinched second fundamental form in his seminal paper. Later, Chern, do Carmo, and Kobayashi [2] proved a celebrated rigidity theorem, which has since inspired numerous significant advances in the study of pinching phenomena.

An important topological invariant that provides useful information about the structure of manifolds is the class of homology groups [3]. It is well known that there are no stable integral currents in the unit sphere ${\mathcal{S}}^n$, nor in a submanifold $N^r$ of ${\mathcal{S}}^n$, provided that the second fundamental form of $N^r$ satisfies a suitable pinching condition [4].

In the Euclidean setting, similar results have been obtained by Leung [5], Xin [6], and Li et al. [7]. Notable contributions in this area can also be found in [8,9,10,11,12,13,14,15,16,17], and the references therein. Understanding the topological invariants of Riemannian space forms and the geometric function theory of Riemannian submanifolds has long been a central objective in differential geometry. In this context, the result of Leung [12] introduced a new perspective for extending such ideas to the setting of warped product manifolds minimally immersed in spheres.

Building upon Leung’s work, Hasanis and Vlachos [11] provided a partial confirmation of his results. They showed that the pinching condition on the second fundamental form is equivalent to an upper bound on the Ricci curvature tensor, under the assumption that the normal connection of the submanifold is flat. Specifically, they proved that a three-dimensional minimal submanifold is homeomorphic to the 3-sphere, and that a higher-dimensional minimal submanifold with positive Ricci curvature is topologically a space form. The result of Hasanis and Vlachos (see Theorem B in [11]) thus provides strong support for Leung’s framework [12].

In the present paper, we further investigate the geometry and topology of compact warped product minimal submanifolds of arbitrary codimension immersed in a sphere, with particular emphasis on those satisfying a condition involving the norm of the second fundamental form, reformulated in terms of the Laplacian of the warping function.

It is of particular interest to examine how constraints on key intrinsic and extrinsic curvature invariants influence the topology of warped product minimal submanifolds. Our work builds upon the foundational results established in [4], contributing to a deeper understanding of the relationship between warped product geometry and topological properties, such as homotopy and homology, within this context.

2. Fundamentals and Terminology

The curvature tensor $\tilde{R}$ of the unit sphere ${\mathcal{S}}^{r+l}\subset {\mathbb{R}}^{r+l+1}$ is given by

$$\begin{array}{c}\hfill \tilde{R}(U_1,U_2,U_3,U_4)=g(U_1,U_4)g(U_2,U_3)-g(U_2,U_4)g(U_1,U_3),\end{array}$$

(1)

where $U_j\in \Gamma \left(T{\mathcal{S}}^{r+l}\right)$, for $1\le j\le 4$.

Let $N^r$ be an r-dimensional Riemannian submanifold of an n-dimensional Riemannian manifold ${\tilde{N}}^n$, equipped with the induced metric g. Let ∇ and ${\nabla }^{\perp }$ denote the induced connections on the tangent bundle $TN$ and the normal bundle $T^{\perp }N$, respectively. Let $\tilde{R}$ and R be the curvature tensors of ${\tilde{N}}^n$ and $N^r$, respectively. Then the Gauss equation is given by

$$\begin{array}{cc}\hfill R(U_1,U_2,U_3,U_4)=& \tilde{R}(U_1,U_2,U_3,U_4)+g\left(h(U_1,U_4),h(U_2,U_3)\right)\hfill \\ \hfill & -g\left(h(U_1,U_3),h(U_2,U_4)\right),\hfill \end{array}$$

(2)

for all vector fields $U_j\in \Gamma \left(TN\right)$, $j=1,\dots ,4$, where h denotes the second fundamental form defined by

$$h(U_1,U_2)={\left({\tilde{\nabla }}_{U_1}{U}_2\right)}^{\perp },$$

where $\tilde{\nabla }$ is the Levi-Civita connection of $\tilde{N}$.

Locally, with respect to orthonormal frames $\{e_1,\dots ,e_r\}$ for $TN$ and $\{e_{r+1},\dots ,e_n\}$ for $T^{\perp }N$, the second fundamental form can be written as

$$\begin{array}{c}\hfill h(e_{a_1},e_{a_2})=\sum _{b=r+1}^n{h}_{a_1{a}_2}^b{e}_b,\end{array}$$

(3)

where the local coefficients $h_{a_1{a}_2}^b$ are given by

$$h_{a_1{a}_2}^b=g\left(h(e_{a_1},e_{a_2}),e_b\right),a_1,a_2=1,\dots ,r.$$

Then, the squared norm of the mean curvature vector $\mathcal{H}$ is given as

$$\begin{array}{c}\hfill {\left|\right|\mathcal{H}\left|\right|}^2={\displaystyle \frac{1}{r^2}}\sum _{b=r+1}^n{\left(\sum _{a_1=1}^r{h}_{a_1{a}_1}^b\right)}^2.\end{array}$$

(4)

Now consider the case where $N^r={\Sigma }_1^t\times {\Sigma }_2^s$ is a Riemannian product manifold with $t+s=r$. Let $\{e_1,\dots ,e_t\}$ and $\{e_{t+1},\dots ,e_r\}$ be local orthonormal frames tangent to ${\Sigma }_1$ and ${\Sigma }_2$, respectively. Then, the squared norms of the mean curvature vectors $H_1$ and $H_2$, corresponding to the factors ${\Sigma }_1$ and ${\Sigma }_2$, respectively, restricted to their tangent bundles, are given by

$$\begin{array}{c}\hfill \left|\right|{\mathcal{H}}_1{\left|\right|}^2={\displaystyle \frac{1}{t^2}}\sum _{b=r+1}^n{\left(\sum _{a_1=1}^t{h}_{a_1{a}_1}^b\right)}^2,\left|\right|{\mathcal{H}}_2{\left|\right|}^2={\displaystyle \frac{1}{s^2}}\sum _{b=r+1}^n{\left(\sum _{a_1=t+1}^r{h}_{a_1{a}_1}^b\right)}^2\end{array}$$

(5)

The squared norm of second fundamental form h is defined by

$$\begin{array}{c}\hfill {\left|\right|h\left|\right|}^2=\sum _{b=r+1}^n\sum _{a_1=1}^r\sum _{a_2=1}^r{\left(h_{a_1{a}_2}^b\right)}^2.\end{array}$$

(6)

We have some classifications for any $U_1,U_2\in \Gamma \left(TN\right)$

(i)

If $h(U_1,U_2)=g(U_1,U_2)\mathcal{H}$, then a submanifold $N^r$ is totally umbilical.

(ii)

If $h(U_1,U_2)=0$, then a submanifold $N^r$ is totally geodesic.

(iii)

If $\mathcal{H}=0$, then $N^r$ is a minimal submanifold.

The scalar curvature $\tau$ of the Riemannian submanifold $N^r$ is locally given by

$$\begin{array}{c}\hfill \tau =\sum _{1\le a_1<a_2\le r}{\mathcal{K}}_{a_1{a}_2},\end{array}$$

(7)

where $\{e_1,\dots ,e_r\}$ is a local orthonormal frame on an open neighborhood of a point $y\in N^r$ and ${\mathcal{K}}_{a_1{a}_2}=\mathcal{K}\left(e_{a_1}\wedge e_{a_2}\right)$ denotes the sectional curvature of the plane spanned by $e_{a_1}$ and $e_{a_2}$ at the point y.

The sectional curvatures of the Riemannian manifold ${\tilde{N}}^n$ and the submanifold $N^r$ are denoted by ${\tilde{\mathcal{K}}}_{a_1{a}_2}$, and ${\mathcal{K}}_{a_1{a}_2}$, respectively, associated with the plane spanned by $e_{a_1}$ and $e_{a_2}$. Thus, ${\tilde{\mathcal{K}}}_{a_1{a}_2}$ and ${\mathcal{K}}_{a_1{a}_2}$ are the sectional curvatures of the span $\{e_{a_1},e_{a_2}\}$. By the Gauss Equation (2) and using (7), we obtain

$$\begin{array}{c}\hfill \tau =\sum _{1\le a_1<a_2\le n}{\tilde{\mathcal{K}}}_{a_1{a}_2}+\sum _{b=r+1}^n\left(h_{a_1{a}_1}^b{h}_{a_2{a}_2}^b-{\left(h_{a_1{a}_2}^b\right)}^2\right).\end{array}$$

(8)

We briefly recall the notion of a warped product manifold (see [18,19]). Let ${\Sigma }_B^t$ and ${\Sigma }_F^s$ be Riemannian manifolds of dimensions t and s, respectively, where ${\Sigma }_B^t$ is called the base manifold and ${\Sigma }_F^s$ the fiber manifold. The warped product manifold $N^r={\Sigma }_B^t{\times }_{\eta }{\Sigma }_F^s$, with $r=t+s$, is equipped with the Riemannian metric $g=g_B+{\eta }^2{g}_F$, where $\eta :{\Sigma }_B^t\to {\mathbb{R}}^{+}$ is a smooth, positive function called the warping function. In this setting, it was shown in ([19], Eq. (3.3)) that

$$\begin{array}{c}\hfill \sum _{a_1=1}^t\sum _{a_2=1}^s\mathcal{K}(e_{a_1}\wedge e_{a_2})={\displaystyle \frac{s\Delta \eta }{\eta }},\end{array}$$

(9)

where $\mathcal{K}$ denotes the sectional curvature of $N^r$ corresponding to the plane spanned by $e_{a_1}\in T{\Sigma }_B$ and $e_{a_2}\in T{\Sigma }_F$, and $\Delta \eta$ is the Laplacian of $\eta$ with respect to the metric $g_B$ on ${\Sigma }_B^t$.

We are now prepared to present the main results of this paper.

3. Proofs of the Main Results

Before proving our main result, we recall the following theorems established in [12] and further developed by Hasanis and Vlachos in [11], which provide confirmation of Leung’s work and build upon the foundational work of Lawson and Simons [4].

Theorem 1

([12]). Let $N^r$ be a compact, connected, minimally immersed submanifold in the sphere ${\mathcal{S}}^{r+t}$ that satisfies the following pinching condition on the second fundamental form

$$\begin{array}{c}\hfill \left|\right|h(U_1,U_1){\left|\right|}^2\le \left({\displaystyle \frac{r}{r-1}}\right)\end{array}$$

(10)

for any unit tangent vector field $U_1$ at any point $u\in N^r$. Then $N^r$ is homeomorphic to the sphere ${\mathcal{S}}^r$ such that the dimension r is odd.

Leung [12] also established the following stronger result by examining the Clifford minimum hypersurfaces in the sphere for the second fundamental form:

Theorem 2

([12]). Let $N^r$ be a compact, connected, minimally immersed submanifold in the sphere ${\mathcal{S}}^{r+l}$. If the following inequality holds

$$\begin{array}{c}\hfill \left|\right|h(U_1,U_1){\left|\right|}^2<\left({\displaystyle \frac{r+1}{r-1}}\right)\end{array}$$

(11)

for any unit tangent vector field $U_1$ at any point u of $N^r$, then $N^r$ is homeomorphic to a sphere ${\mathcal{S}}^r$ such that the dimension r is odd.

In this paper, we aim to extend the rigidity results of Theorems 1 and 2 to the setting of compact, connected, warped product minimal submanifolds. To establish our results, we adopt the approach of Chen [19] for isometric minimal immersions of warped products into ambient manifolds.Motivated by previous studies and Theorem 2, our first main result is stated as follows:

Theorem 3.

Let $\Psi :N^{t+s}={\Sigma }_B^t{\times }_{\eta }{\Sigma }_F^s\to {\mathcal{S}}^{t+s+l}$ be an isometric immersion of a $(t+s)$-dimensional compact, connected, and oriented warped product minimal submanifold into the $(t+s+l)$-dimensional sphere. If the following pinching condition is satisfied

$$\begin{array}{c}\hfill {\displaystyle \frac{\Delta \eta }{\eta }}>\left({\displaystyle \frac{(ts-2)(t+s)-ts}{s\left(t+s-1\right)}}\right)\end{array}$$

(12)

where $\Delta \eta$ denotes the Laplacian of the warping function η on ${\Sigma }_B^t$, then $N^{t+s}$ is homeomorphic to the sphere ${\mathcal{S}}^{t+s}$, provided $t>1$ and $s>2$.

Proof. 

First, we consider that $r=s+t$ is odd, and $\Psi :N^r={\Sigma }_B^t{\times }_{\eta }{\Sigma }_F^s\to {\mathcal{S}}^{r+l}$ is a minimal isometric immersion from $N^r$ into a sphere ${\mathcal{S}}^{r+l}$. Let $\{e_1,\dots ,e_t,e_{t+1},\dots ,e_r\}$ be a local orthonormal frame of the tangent bundle $T{N}^r$ adapted to the product structure, such that the vectors $e_1,\dots ,e_t$ are tangent to ${\Sigma }_B^t$ and $e_{t+1},\dots ,e_r$ are tangent to ${\Sigma }_F^s$.

Thus, by tracing Equations (1) and (2), we find that

$$\begin{array}{c}\hfill {\left|\right|h\left|\right|}^2=r(r-1)-\sum _{1\le a_1\ne a_2\le r}\mathcal{K}(e_{a_1}\wedge e_{a_2}).\end{array}$$

(13)

Now, by referring to Equation (7), we can obtain

$$\begin{array}{cc}\hfill \sum _{1\le a_1\ne a_2\le r}\mathcal{K}(e_{a_1}\wedge e_{a_2})=& \sum _{C=1}^t\sum _{D=t+1}^r\mathcal{K}(e_C\wedge e_D)+\sum _{1\le C\ne E\le t}\mathcal{K}(e_C\wedge e_E)\hfill \\ \hfill & +\sum _{t+1\le D\ne G\le r}\mathcal{K}(e_D\wedge e_G)\hfill \end{array}$$

(14)

Combining Equations (14) and (9), we derive the following:

$$\begin{array}{cc}\hfill \sum _{1\le a_1\ne a_2\le r}\mathcal{K}(e_{a_1}\wedge e_{a_2})=& {\displaystyle \frac{s\Delta \eta }{\eta }}+\sum _{1\le C\ne E\le t}\mathcal{K}(e_C\wedge e_E)\hfill \\ \hfill & +\sum _{t+1\le D\ne G\le r}\mathcal{K}(e_D\wedge e_G).\hfill \end{array}$$

(15)

Thus from (8), (13), and (15), we derive

$$\begin{array}{cc}\hfill {\left|\right|h\left|\right|}^2=& r(r-1)-{\displaystyle \frac{s\Delta \eta }{\eta }}-\left(\sum _{b=r+1}^{r+l}\sum _{1\le a_1\ne a_2\le t}\left(h_{a_1{a}_1}^b{h}_{a_2{a}_2}^b-{\left(h_{a_1{a}_2}^b\right)}^2\right)\right)\hfill \\ \hfill & -\left(\sum _{b=r+1}^{r+l}\sum _{t+1\le c_1\ne c_2\le r}\left(h_{c_1{c}_1}^b{h}_{c_2{c}_2}^b-{\left(h_{c_1{c}_2}^b\right)}^2\right)\right)\hfill \\ \hfill & -\sum _{1\le a_1\ne a_2\le t}\tilde{\mathcal{K}}(e_{a_1}\wedge e_{a_2})-\sum _{t+1\le c_1\ne c_2\le r}\tilde{\mathcal{K}}(e_{c_1}\wedge e_{c_2}).\hfill \end{array}$$

(16)

As we considered that $N^r$ is a minimal warped product submanifold, then taking the third part of the right-hand side in the above equation,

$$\begin{array}{cc}\hfill \sum _{b=r+1}^{r+l}\sum _{1\le a_1\ne a_2\le t}& \left(h_{a_1{a}_1}^b{h}_{a_2{a}_2}^b-{\left(h_{a_1{a}_2}^b\right)}^2\right)\hfill \\ \hfill & =\sum _{b=r+1}^{r+l}\sum _{1\le a_1\ne a_2\le t}{h}_{a_1{a}_1}^b{h}_{a_2{a}_2}^b-\sum _{b=r+1}^{r+l}\sum _{1\le a_1\ne a_2\le t}{\left(h_{a_1{a}_2}^b\right)}^2.\hfill \end{array}$$

The expression above can be further simplified by strategically adding and subtracting the same term.

$$\begin{array}{cc}\hfill \sum _{b=r+1}^{r+l}\sum _{1\le a_1\ne a_2\le t}& \left(h_{a_1{a}_1}^b{h}_{a_2{a}_2}^b-{\left(h_{a_1{a}_2}^b\right)}^2\right)\hfill \\ \hfill =& \sum _{b=r+1}^{r+l}\sum _{1\le a_1\ne a_2\le t}{h}_{a_1{a}_1}^b{h}_{a_2{a}_2}^b+\sum _{b=r+1}^{r+l}\sum _{1\le m\ne l\le t}{\left(h_{ml}^b\right)}^2\hfill \\ \hfill & -\sum _{b=r+1}^{r+l}\sum _{1\le a_1\ne a_2\le t}{\left(h_{a_1{a}_2}^b\right)}^2-\sum _{b=r+1}^{r+l}\sum _{1\le m\ne l\le t}{\left(h_{ml}^b\right)}^2.\hfill \end{array}$$

Now, by applying the binomial theorem in the right-hand side to the last two terms of the last equation, and computing other terms as well, we can arrive at the following formula:

$$\begin{array}{cc}\hfill \sum _{b=r+1}^{r+l}\sum _{1\le a_1\ne a_2\le t}\left(h_{a_1{a}_1}^b{h}_{a_2{a}_2}^b-{\left(h_{a_1{a}_2}^b\right)}^2\right)=& \sum _{b=r+1}^{r+l}{\left(h_{11}^b+\cdots +h_{tt}^b\right)}^2\hfill \\ \hfill & -\sum _{b=r+1}^{r+l}\sum _{a_1,a_2=1}^t{\left(h_{a_1{a}_2}^b\right)}^2.\hfill \end{array}$$

(17)

We will apply a similar method to the fourth term in (16), and we derive

$$\begin{array}{cc}\hfill \sum _{b=r+1}^{r+l}\sum _{t+1\le c_1\ne c_2\le r}\left(h_{c_1{c}_1}^b{h}_{c_2{c}_2}^b-{\left(h_{c_1{c}_2}^b\right)}^2\right)=& \sum _{b=r+1}^{r+l}{\left(h_{t+1t+1}^b+\cdots +h_{rr}^b\right)}^2\hfill \\ \hfill & -\sum _{b=r+1}^{r+l}\sum _{c_1,c_2=t+1}^r{\left(h_{c_1{c}_2}^b\right)}^2.\hfill \end{array}$$

(18)

In view of (16)–(18), the following conclusion can be drawn:

$$\begin{array}{cc}\hfill {\left|\right|h\left|\right|}^2=& r(r-1)-{\displaystyle \frac{s\Delta \eta }{\eta }}-\sum _{1\le a_1\ne a_2\le t}\tilde{\mathcal{K}}(e_{a_1}\wedge e_{a_2})-\sum _{t+1\le c_1\ne c_2\le r}\tilde{\mathcal{K}}(e_{c_1}\wedge e_{c_2})\hfill \\ \hfill & +\sum _{b=r+1}^{r+l}\sum _{a_1,a_2=1}^t{\left(h_{a_1{a}_2}^b\right)}^2-\sum _{b=r+1}^{r+l}{\left(h_{11}^b+\cdots +h_{tt}^b\right)}^2\hfill \\ \hfill & +\sum _{b=r+1}^{r+l}\sum _{c_1,c_2=t+1}^r{\left(h_{c_1{c}_2}^b\right)}^2-\sum _{b=r+1}^{r+l}{\left(h_{t+1t+1}^b+\cdots +h_{rr}^b\right)}^2.\hfill \end{array}$$

(19)

Let $\mathbb{A}={\left({\sum }_{a_1=1}^t{h}_{a_1{a}_1}^b\right)}^2$ and $\mathbb{B}={\left({\sum }_{a_2=t+1}^r{h}_{a_2{a}_2}^b\right)}^2$. Then, by expanding ${(\mathbb{A}+\mathbb{B})}^2$, we obtain

$$\begin{array}{cc}\hfill {\left|\right|h\left|\right|}^2=& r(r-1)-{\displaystyle \frac{s\Delta \eta }{\eta }}-\sum _{1\le a_1\ne a_2\le t}\tilde{\mathcal{K}}(e_{a_1}\wedge e_{a_2})-\sum _{t+1\le c_1\ne c_2\le r}\tilde{\mathcal{K}}(e_{c_1}\wedge e_{c_2})\hfill \\ \hfill & +\sum _{b=r+1}^{r+l}\sum _{a_1,a_2=1}^t{\left(h_{a_1{a}_2}^b\right)}^2+\sum _{b=r+1}^{r+l}\sum _{c_1,c_2=t+1}^r{\left(h_{c_1{c}_2}^b\right)}^2-r^2{\left|\right|H\left|\right|}^2+\sum _{b=r+1}^{r+l}\sum _{a_1=1}^t{h}_{a_1{a}_1}^b\sum _{a_2=t+1}^r{h}_{a_2{a}_2}^b.\hfill \end{array}$$

By applying Equation (1) and using the minimality assumption, we obtain

$$\begin{array}{cc}\hfill {\left|\right|h\left|\right|}^2\ge & r(r-1)-{\displaystyle \frac{s\Delta \eta }{\eta }}-t(t-1)-s(s-1)\hfill \\ \hfill & +\sum _{b=r+1}^{r+l}\sum _{a_1,a_2=1}^t{\left(h_{a_1{a}_2}^b\right)}^2+\sum _{b=r+1}^{r+l}\sum _{c_1,c_2=t+1}^r{\left(h_{c_1{c}_2}^b\right)}^2.\hfill \end{array}$$

Equality holds if and only if either $\mathbb{A}=0$ or $\mathbb{B}=0$. Now, using the definition of the symmetric bilinear form h and expanding the squared norm on the left-hand side, we have

$$\begin{array}{cc}\hfill \sum _{b=r+1}^{r+l}\sum _{a_1,a_2=1}^t{\left(h_{a_1{a}_2}^b\right)}^2+\sum _{b=r+1}^{r+l}\sum _{c_1,c_2=1}^s{\left(h_{c_1{c}_2}^b\right)}^2& +2\sum _{b=r+1}^{r+l}\sum _{C=1}^t\sum _{D=1}^s{\left(h_{CD}^b\right)}^2\hfill \\ \hfill \ge & 2ts-{\displaystyle \frac{s\Delta \eta }{\eta }}+\sum _{b=r+1}^{r+l}\sum _{a_1,a_2=1}^t{\left(h_{a_1{a}_2}^b\right)}^2\hfill \\ \hfill & +\sum _{b=r+1}^{r+l}\sum _{c_1,c_2=1}^s{\left(h_{c_1{c}_2}^b\right)}^2.\hfill \end{array}$$

Therefore, from (6), we have

$$\begin{array}{c}\hfill {\parallel h\parallel }^2\ge ts-{\displaystyle \frac{s}{2}}\left({\displaystyle \frac{\Delta \eta }{\eta }}\right).\end{array}$$

(20)

To estimate the upper bound of the term ${\parallel h\parallel }^2$, we consider the unit vector fields $U_1$ and $U_2$ defined by:

$$\begin{array}{c}\hfill U_1={\displaystyle \frac{1}{\sqrt{2}}}(e_C+e_D),U_2={\displaystyle \frac{1}{\sqrt{2}}}(e_C-e_D),1\le C\le t,1\le D\le s.\end{array}$$

(21)

Then,

$$\begin{array}{c}\hfill e_C={\displaystyle \frac{1}{\sqrt{2}}}(U_1+U_2),e_D={\displaystyle \frac{1}{\sqrt{2}}}(U_1-U_2).\end{array}$$

(22)

Hence, we obtain:

$$\begin{array}{cc}\hfill {\parallel h\parallel }^2& ={∥h\left({\displaystyle \frac{U_1+U_2}{\sqrt{2}}},{\displaystyle \frac{U_1-U_2}{\sqrt{2}}}\right)∥}^2\hfill \\ \hfill & ={\displaystyle \frac{1}{4}}{∥h(U_1,U_1)-h(U_2,U_2)∥}^2\hfill \\ \hfill & ={\displaystyle \frac{1}{4}}\left\{\parallel h(U_1,U_1){\parallel }^2+{\parallel h(U_2,U_2)\parallel }^2-2g\left(h(U_1,U_1),h(U_2,U_2)\right)\right\}.\hfill \end{array}$$

(23)

By applying the Cauchy–Schwarz inequality, we get

$$\begin{array}{c}\hfill {\parallel h\parallel }^2\le {\displaystyle \frac{1}{4}}\left\{\parallel h(U_1,U_1){\parallel }^2+\parallel h(U_2,U_2){\parallel }^2+2\parallel h(U_1,U_1)\parallel ·\parallel h(U_2,U_2)\parallel \right\}.\end{array}$$

Assuming the strongly pinching condition (11), and with $r=t+s$, the above inequality implies:

$$\begin{array}{c}\hfill {\left|\right|h\left|\right|}^2<{\displaystyle \frac{1}{4}}\left({\displaystyle \frac{t+s+1}{t+s-1}}+{\displaystyle \frac{t+s+1}{t+s-1}}+2\left({\displaystyle \frac{t+s+1}{t+s-1}}\right)\right).\end{array}$$

From here, it follows that

$$\begin{array}{c}\hfill {\left|\right|h\left|\right|}^2<{\displaystyle \frac{t+s+1}{t+s-1}}.\end{array}$$

As we noticed that ${\displaystyle \frac{t+s+1}{t+s-1}}<{\displaystyle \frac{t+s+1}{t+s-1}}+1$, then we derive

$$\begin{array}{cc}\hfill {\left|\right|h\left|\right|}^2& <{\displaystyle \frac{t+s+1}{t+s-1}}+1={\displaystyle \frac{t+s+1+t+s-1}{t+s-1}}.\hfill \end{array}$$

Then

$$\begin{array}{c}\hfill {\left|\right|h\left|\right|}^2<{\displaystyle \frac{2(t+s)}{t+s-1}}.\end{array}$$

(24)

Combining (20) and (24), one obtains

$$\begin{array}{c}\hfill ts-\left({\displaystyle \frac{s}{2}}\right){\displaystyle \frac{\Delta \eta }{\eta }}<{\displaystyle \frac{2(t+s)}{t+s-1}}\end{array}$$

which implies that

$$\begin{array}{c}\hfill ts<{\displaystyle \frac{2(t+s)}{t+s-1}}+\left({\displaystyle \frac{s}{2}}\right){\displaystyle \frac{\Delta \eta }{\eta }}.\end{array}$$

It follows that

$$\begin{array}{c}\hfill ts-{\displaystyle \frac{2(t+s)}{t+s-1}}<\left({\displaystyle \frac{s}{2}}\right){\displaystyle \frac{\Delta \eta }{\eta }}.\end{array}$$

(25)

After some computations, we get

$$\begin{array}{c}\hfill \left({\displaystyle \frac{s}{2}}\right){\displaystyle \frac{\Delta \eta }{\eta }}>{\displaystyle \frac{(ts-2)(t+s)-ts}{t+s-1}}.\end{array}$$

(26)

That is,

$$\begin{array}{c}\hfill {\displaystyle \frac{\Delta \eta }{\eta }}>\left({\displaystyle \frac{(ts-2)(t+s)-ts}{s\left(t+s-1\right)}}\right).\end{array}$$

(27)

Therefore, the above inequality holds if and only if inequality (11) is satisfied. To conclude the desired result, it suffices to apply Theorem 2 together with the pinching condition (27). This completes the proof of the theorem. □

By combining the method of proof of Theorem 3 with the idea behind Theorem 1, we are led to the following result:

Theorem 4.

Let $\Psi :N^{t+s}={\Sigma }_B^t{\times }_{\eta }{\Sigma }_F^s⟶{\mathcal{S}}^{t+s+l}$ be an isometric immersion from a compact connected $(t+s)$-dimensional warped product minimal submanifold $N^{t+s}$ into $(t+s+l)$-dimensional sphere ${\mathcal{S}}^{t+s+l}$ satisfying the following pinching condition:

$$\begin{array}{c}\hfill {\displaystyle \frac{\Delta \eta }{\eta }}\ge \left({\displaystyle \frac{(ts-1)(t+s)-ts}{s\left(t+s-1\right)}}\right)\end{array}$$

(28)

where $\Delta \eta$ is the Laplacian of η. Then $N^{t+s}$ is homeomorphic to a sphere ${\mathcal{S}}^{t+s}$ when $t>1$ and $s>2.$

Proof. 

Using Theorem 1, we again consider Equation (23), then we get

$$\begin{array}{cc}\hfill {\left|\right|h\left|\right|}^2=& \left|\right|h\left({\displaystyle \frac{U_1+U_2}{\sqrt{2}}},{\displaystyle \frac{U_1-U_2}{\sqrt{2}}}\right){\left|\right|}^2={\displaystyle \frac{1}{4}}\left|\right|h(U_1,U_1)-h(U_2,U_2){\left|\right|}^2\hfill \\ \hfill =& {\displaystyle \frac{1}{4}}\left\{\left|\right|h(U_1,U_1){\left|\right|}^2+\left|\right|h(U_2,U_2){\left|\right|}^2-2g\left(h(U_1,U_1),h(U_2,U_2)\right)\right\}.\hfill \end{array}$$

Again, for orthogonal vector fields, applying the Cauchy–Schwarz inequality, we conclude that

$$\begin{array}{c}\hfill {\left|\right|h\left|\right|}^2\le {\displaystyle \frac{1}{4}}\left\{\left|\right|h(U_1,U_1){\left|\right|}^2+\left|\right|h(U_2,U_2){\left|\right|}^2+2\left|\right|h(U_1,U_1)\left|\right|\left|\right|h(U_2,U_2)\left|\right|\right\}.\end{array}$$

Now, using the pinching condition (10), we obtain that

$$\begin{array}{c}\hfill {\left|\right|h\left|\right|}^2\le {\displaystyle \frac{1}{4}}\left\{{\displaystyle \frac{r}{r-1}}+{\displaystyle \frac{2r}{r-1}}+{\displaystyle \frac{r}{r-1}}\right\}.\end{array}$$

As we have seen that $r=t+s$, in this case, r is odd, then we get from the above equation

$$\begin{array}{c}\hfill {\left|\right|h\left|\right|}^2\le {\displaystyle \frac{t+s}{t+s-1}}\end{array}$$

(29)

which, in view of Equation (20), leads to

$$\begin{array}{c}\hfill ts-{\displaystyle \frac{s}{2}}\left({\displaystyle \frac{\Delta \eta }{\eta }}\right)\le {\displaystyle \frac{t+s}{t+s-1}}.\end{array}$$

(30)

Some computation leads to

$$\begin{array}{c}\hfill {\displaystyle \frac{\Delta \eta }{\eta }}\ge {\displaystyle \frac{\left(ts-1\right)\left(t+s\right)-ts}{s\left(t+s-1\right)}}.\end{array}$$

(31)

The above inequality is satisfied if and only if Equation (10) holds. Thus, from the statement of Theorem 1, we get the desired result. □

Consequently, for a positive differentiable function $\Omega$ defined on a compact Riemannian manifold N, the squared norm of its gradient is given by

$$\begin{array}{c}\hfill {\left|\right|\nabla \Omega \left|\right|}^2=\sum _{a=1}^r{\left(e_a(\Omega )\right)}^2,\end{array}$$

(32)

where $\{e_1,\dots ,e_r\}$ is an orthonormal frame tangent to $N^r$.

Let us set

$$\mathcal{E}(\Omega )={\displaystyle \frac{1}{2}}{\int }_N{\parallel \nabla \Omega \parallel }^2\mathrm{dV},$$

(33)

where $\mathcal{E}(\Omega )$ is the Dirichlet energy functional as defined in [20].

Theorem 3 may be restated as follows:

Theorem 5.

Let $\Psi :N^{t+s}={\Sigma }_B^t{\times }_{\eta }{\Sigma }_F^s⟶{\mathcal{S}}^{t+s+l}$ be a minimal isometric immersion from a compact, connected, and oriented $(t+s)$-dimensional warped product submanifold $N^{t+s}$ into the $(t+s+l)$-dimensional sphere ${\mathcal{S}}^{t+s+l}$. Assume that $t>1$ and $s>2$. If the following inequality holds:

$$\begin{array}{c}\hfill \mathcal{E}(\Omega )>\left({\displaystyle \frac{(ts-2)(t+s)-ts}{s(t+s-1)}}\right)Vol\left(N\right),\end{array}$$

(34)

where $Vol\left(N\right)$ is the volume of $N^{t+s}$ and $\mathcal{E}(\Omega )$ denotes the Dirichlet energy of the function $\Omega =ln\eta$, then $N^{t+s}$ is homeomorphic to the sphere ${\mathcal{S}}^{t+s}$.

Proof. 

We start with

$$\begin{array}{c}\hfill {\displaystyle \frac{\Delta \eta }{\eta }}=\Delta (ln\eta )+{\parallel \nabla ln\eta \parallel }^2.\end{array}$$

(35)

From (12) and (35), we have

$$\begin{array}{c}\hfill \Delta \Omega +{\left|\right|\nabla \Omega \left|\right|}^2>\left({\displaystyle \frac{(ts-2)(t+s)-ts}{s\left(t+s-1\right)}}\right).\end{array}$$

(36)

where $\Omega =ln\eta$.

Integration over N in (36) yields, by Green’s lemma (cf. [21]), the following

$$\begin{array}{c}\hfill {\int }_N{\left|\right|\nabla \Omega \left|\right|}^2\mathrm{dV}>\left({\displaystyle \frac{(ts-2)(t+s)-ts}{s\left(t+s-1\right)}}\right)Vol\left(N\right).\end{array}$$

(37)

The conclusion follows from (33), (37), and Theorem 3. □

An important consequence of Theorem 4, derived using Equation (33), is stated as follows

Theorem 6.

Let $\Psi :N^{t+s}={\Sigma }_B^t{\times }_{\eta }{\Sigma }_F^s⟶{\mathcal{S}}^{t+s+l}$ be a minimal isometric immersion from a compact connected manifold $N^{t+s}$ into the sphere ${\mathcal{S}}^{t+s+l}$ satisfying the following

$$\begin{array}{c}\hfill \mathcal{E}(\Omega )\ge \left({\displaystyle \frac{(ts-1)(t+s)-ts}{s\left(t+s-1\right)}}\right)Vol\left(N\right)\end{array}$$

(38)

and assume that $t+s=2a+1$ is odd. Then $N^{t+s}$ is homeomorphic to the sphere ${\mathcal{S}}^{t+s}$ when $t>1$ and $s>2$.

Proof. 

By Green’s lemma and inequality (31), we obtain

$$\begin{array}{c}\hfill {\int }_N{\left|\right|\nabla \Omega \left|\right|}^{2}dV\ge \left({\displaystyle \frac{(ts-1)(t+s)-ts}{s\left(t+s-1\right)}}\right)Vol\left(N\right).\end{array}$$

(39)

Combining Definition (33) with Theorem 4, we obtain the desired conclusion. □

Using the harmonicity of the warping function $\eta$, the results from Theorems 3 and 4, we give the following results.

Corollary 1.

Let $\Psi :N^{t+s}={\Sigma }_B^t{\times }_{\eta }{\Sigma }_F^s⟶{\mathcal{S}}^{t+s+l}$ be an isometric immersion of a compact, connected, and oriented warped product minimal submanifold into the unit sphere ${\mathcal{S}}^{t+s+l}$. Assume that the warping function η is harmonic and that the pinching condition $\left(ts-2\right)(t+s)<ts$ holds, with $t>1$ and $s>2$. Then $N^{t+s}$ is a Riemannian product and is homeomorphic to the sphere ${\mathcal{S}}^{t+s}$.

Proof. 

Let the warping function $\eta$ be harmonic, then $\Delta \eta =0$. Hence, from the pinching condition and the inequality in (12), we get the required result. □

As an application of Theorem 4, we can prove

Corollary 2.

Let $\Psi :N^{t+s}={\Sigma }_B^t{\times }_{\eta }{\Sigma }_F^s⟶{\mathcal{S}}^{t+s+l}$ be an isometric immersion of a compact, connected, and oriented warped product minimal submanifold into the unit sphere ${\mathcal{S}}^{t+s+l}$. Assume that $r=t+s$ is odd, that the warping function η is harmonic, and that the pinching condition $\left(ts-1\right)(t+s)\le ts$ holds, with $t>1$ and $s>2$. Then $N^{t+s}$ is a Riemannian product and is homeomorphic to the sphere ${\mathcal{S}}^{t+s}$.

Proof. 

The conclusion follows directly from the proof of Corollary 1, together with Theorem 4. □

The following conclusion is derived from Cheng’s eigenvalue comparison theorem [22], which demonstrates that N is complete and isometric to the standard unit sphere with the assumptions $Ric\left(N\right)\ge 1$ and $d\left(N\right)=\pi$ by using the first non-zero eigenvalue of the Laplacian operator. According to [23] and Theorem 3, the following can be determined as an application of the maximum principle for the first non-zero eigenvalue ${\lambda }_1$:

Theorem 7.

Let $\Psi :N^{t+s}={\Sigma }_B^t{\times }_{\eta }{\Sigma }_F^s⟶{\mathcal{S}}^{t+s+l}$ be an isometric immersion of a compact, connected, and oriented warped product minimal submanifold into the unit sphere ${\mathcal{S}}^{t+s+l}$. Assume that the warping function η is non-constant, is an eigenfunction corresponding to the first non-zero eigenvalue ${\lambda }_1$, and that the pinching condition

$$\begin{array}{c}\hfill {\lambda }_1<\left\{{\displaystyle \frac{\left((ts-2)(t+s)-ts\right)Vol\left(N\right)}{s(t+s-1){\int }_N{(ln\eta )}^{2}dV}}\right\}\end{array}$$

(40)

holds, with $t>1$ and $s>2$. Then $N^{t+s}$ is homeomorphic to the sphere ${\mathcal{S}}^{t+s}$.

Proof. 

The minimum principle on ${\lambda }_1$ yields (see, for instance, [22,23]) for $\eta$ being a non-constant warping function

$$\begin{array}{c}\hfill {\lambda }_1{\int }_N{(ln\eta )}^{2}dV\le {\int }_N{\left|\right|\nabla (ln\eta )\left|\right|}^{2}dV\end{array}$$

(41)

where equality holds if and only if one has $\Delta (ln\eta )={\lambda }_{1}ln\eta$. So, by combining the above inequality in (41) and the inequality (37), we obtain the pinching inequality (40). This completes the proof of the theorem. □

The following significant result is a direct consequence of Theorem 4:

Corollary 3.

Let $\Psi :N^{t+s}={\Sigma }_B^t{\times }_{\eta }{\Sigma }_F^s⟶{\mathcal{S}}^{t+s+l}$ be an isometric immersion of a compact, connected warped product minimal submanifold into the unit sphere ${\mathcal{S}}^{t+s+l}$. Assume that the warping function η is non-constant, is an eigenfunction corresponding to the first non-zero eigenvalue ${\lambda }_1$, and that the pinching condition

$$\begin{array}{c}\hfill {\lambda }_1\le \left\{{\displaystyle \frac{\left((ts-1)(t+s)-ts\right)Vol\left(N\right)}{s(t+s-1){\int }_N{(ln\eta )}^{2}dV}}\right\}\end{array}$$

(42)

holds, with $t>1$ and $s>2$. Then $N^{t+s}$ is homeomorphic to the sphere ${\mathcal{S}}^{t+s}$.

Proof. 

By applying similar arguments to those used in Equations (39) and (41), we obtain the desired result. □

The following example motivates our study.

Example 1.

Let $N^r=[0,\mathbb{R})\times {\mathcal{S}}^{r-1}$ be a smooth Riemannian manifold equipped with the metric $g=d{r}^2+{\eta }^2\left(r\right)g_{{\mathcal{S}}^{r-1}}$, where $r\ge 2$. It is evident that $N^r$ is a warped product manifold of the form $N^r=[0,\mathbb{R}){\times }_{\eta }{\mathcal{S}}^{r-1}$. Moreover, $N^r$ is diffeomorphic to the Euclidean ball (for further details, see [24]).