Curvature Pinching Problems for Compact Pseudo-Umbilical PMC Submanifolds in ${\mathbb{S}}^m\left(c\right)\times \mathbb{R}$

Abstract

Let ${\mathbb{S}}^m\left(c\right)$ denote a sphere with a positive constant curvature c and $M^n(n\ge 3)$ be an n-dimensional compact pseudo-umbilical submanifold in a Riemannian product space ${\mathbb{S}}^m\left(c\right)\times \mathbb{R}$ with a nonzero parallel mean curvature vector (PMC), where $\mathbb{R}$ is a Euclidean line. In this paper, we prove a sequence of pinching theorems with respect to the Ricci, sectional and scalar curvatures of $M^n$, which allow us to generalize some classical curvature pinching results in spheres.

1. Introduction

Curvature pinching problems, as one of the most important topics of differential geometry, have recently attracted significant research interest. Generally speaking, a typical conclusion is that a general manifold M retains particular geometric and topological properties under the given assumptions for a sectional, Ricci or scalar curvature if it is bounded between suitable constants (e.g., [1,2,3,4,5]).

Let $M^n$ be an n-dimensional compact manifold which is minimally immersed in an $(n+p)$ sphere ${\mathbb{S}}^{n+p}\left(c\right)$. The scalar curvature pinching problem of minimal submanifolds of spheres was initiated in 1968 in [6], in which Simons established the famous Simons integral inequality:

$${\int }_{M}S\left\{\left(2-\frac{1}{p}\right)S-nc\right\}dV\ge 0,$$

(1)

where S denotes the square of the second fundamental form and $dV$ is the volume form of $M^n$. It follows from Equation (1) that if $0\le S\le C_1=nc/(2-\frac{1}{p})$, then either $S=0$ and $M^n$ is a totally geodesic submanifold, or $S=C_1$. Soon after, Lawson [7] and Chern, do Carmo and Kobayashi [8] proved that if $S\equiv C_1$, then either $p=1$ and $M^n$ is a Clifford minimal hypersurface $M_{k,n-k}={\mathbb{S}}^k\left(\frac{nc}{k}\right)\times {\mathbb{S}}^{n-k}\left(\frac{nc}{n-k}\right)$ in ${\mathbb{S}}^{n+1}\left(c\right)$ with $1\le k\le n-1$, or $n=p=2$, and $M^n$ is a Veronese surface $M^2$ in ${\mathbb{S}}^4\left(c\right)$. Later, Li and Li [9] improved the first pinching constant $C_1$ to $\frac{2nc}{3}$ for cases where $p\ge 2$ (see also Chen and Xu’s result [10] with another method). Xu [11] obtained a sharp generalization of the above theorems for the compact PMC submanifolds of a sphere.

Instead of the scalar curvature assumption, Yau improved Simons’ inequality under the sectional curvature assumption (see Theorem 15 in [12]). In fact, it was proven in [12] that if $M^n$ is a compact minimal submanifold in ${\mathbb{S}}^{n+p}\left(c\right)$ with the sectional curvature of $M^n$ not being less than $\frac{(p-1)c}{2p-1}$, then $M^n$ is a totally geodesic sphere ${\mathbb{S}}^n\left(c\right)$, a Clifford minimal hypersurface $M_{k,n-k}$ in ${\mathbb{S}}^{n+1}\left(c\right)$ with $1\le k\le n-1$ or a Veronese surface $M^2$ in ${\mathbb{S}}^4\left(c\right)$. Later on, Itoh proved in [13] that if $M^n$ is a compact minimal submanifold in ${\mathbb{S}}^{n+p}\left(c\right)$, and the sectional curvature of $M^n$ is not less than $\frac{nc}{2(n+1)}$, then $M^n$ is either a totally geodesic sphere ${\mathbb{S}}^n\left(c\right)$ or a Veronese manifold of a constant sectional curvature $\frac{nc}{2(n+1)}$ with the co-dimension $p=\frac{1}{2}(n-1)(n+2)$. In 2005, Xu and Han [14] established a geometric rigidity theorem for submanifolds with parallel mean curvatures in a space form, which generalizes Yau and Itoh’s pinching theorems.

In 1979, Ejiri [15] considered the question of classifying all compact minimal n-dimensional submanifolds of a sphere under some restrictions on the Ricci curvature. Actually, the author proved in [15] that if $M^n(n\ge 4)$ is an n-dimensional simply connected compact minimal submanifold immersed in ${\mathbb{S}}^{n+p}\left(c\right)$, the immersion is full, and the Ricci curvature of $M^n$ is not less than $(n-2)c$, then $M^n$ is a totally geodesic sphere ${\mathbb{S}}^n\left(c\right)$, a Clifford minimal hypersurface $M_{k,k}$ in ${\mathbb{S}}^{n+1}\left(c\right)$ with $n=2k$ or a two-dimensional complex projective space $\mathbb{C}{P}^2\left(\frac{4}{3}c\right)$ with a holomorphic sectional curvature $\frac{4}{3}c$ in ${\mathbb{S}}^7\left(c\right)$. Moreover, Shen [16] and Li [17] settled Ejiri’s type of pinching problem for the case where $n=3$. Xu and Tian [18] showed further that Ejiri’s pinching theorem still holds for $n\ge 3$ by removing the simply connected condition. In the case of a nonzero parallel mean curvature vector, Sun [19] initially established an Ejiri-type theorem. He proved in 1987 that if $M^n$ is a compact PMC submanifold in ${\mathbb{S}}^{n+p}\left(c\right)$ with $n\ge 4$, and under the additional assumption that the Ricci curvature of $M^n$ is not less than $n(n-2)(c+H^2)/(n-1)$, then $M^n$ is a totally umbilic sphere. By replacing Sun’s pinching constant with the optimal possible constant $(n-2)(c+H^2)$, He and Luo [20] and Xu et al. [21,22] further generalized Ejiri’s pinching theorem to compact PMC submanifolds in a space form.

Pinching phenomena in submanifolds also arose in the Riemannian product spaces. In 2011, Chen and Cui [23] obtained a pinching theorem concerning the squared length of the second fundamental form or the scalar curvature for compact minimal submanifolds in a kind of product space ${\mathbb{S}}^m\left(1\right)\times \mathbb{R}$. By replacing the real line $\mathbb{R}$ with a general Euclidean space with an arbitrary dimension, the second author [24] further extended Chen and Cui’s rigidity theorem to the compact minimal submanifolds in a generalized cylinder ${\mathbb{S}}^{n_1}\left(c\right)\times {\mathbb{R}}^{n_2}$. After deriving a Simons-type formula on the squared length of the second fundamental form, Chen-Chen-Li [25] investigated the pinching problems of compact minimal submanifolds in ${\mathbb{S}}^m\left(1\right)\times \mathbb{R}$. Actually, they obtained a series of pinching theorems involving the scalar, Ricci and sectional curvatures, which generalize the rigidity theorems due to the works of Li and Li [9], Ejiri, Shen and Li [15,16,17] and Itoh [13]. Later on, the authors of [26] considered a compact minimal submanifold $M^n$ in ${\mathbb{S}}^{n_1}\left(c\right)\times {\mathbb{R}}^{n_2}$ and obtained several rigidity results depending on the Ricci curvature, the squared length and the squared maximum norm of the second fundamental form on $M^n$. Some other types of pinching theorems in general product spaces seem to be of interest and worth further discussion (refer to [27,28,29,30]).

Inspired by the previous results, the aim of this paper is to investigate the geometric rigidity of compact submanifolds in a kind of Riemannian product space with a parallel mean curvature vector (PMC). We extend some pinching results for the minimal submanifolds to the PMC submanifolds in ${\mathbb{S}}^m\left(c\right)\times \mathbb{R}$. The outline of this paper is as follows. In Section 2, we briefly recall some basic notations, concepts and general properties of compact pseudo-umbilical PMC submanifolds in ${\mathbb{S}}^m\left(c\right)\times \mathbb{R}$. In Section 3, we summarize several relevant inequalities, which is crucial for obtaining our curvature pinching theorems. In Section 4, Section 5 and Section 6, we give the proofs of our pinching theorems (For details see Theorems 1, 2 and 4) for the compact pseudo-umbilical PMC submanifolds in ${\mathbb{S}}^m\left(c\right)\times \mathbb{R}$ in terms of the Ricci, sectional and scalar curvatures of the submanifolds, respectively.

2. Preliminaries

Suppose that $\tilde{M}:={\mathbb{S}}^m\left(c\right)\times \mathbb{R}$ is an $(m+1)$-dimensional Riemannian product space, $M^n$ is a manifold of a dimension $n(\ge 3)$ and $\phi :M^n⟶\tilde{M}$ is an isometric immersion with the co-dimension $p=m+1-n$. We denote by ∇ and $\tilde{\nabla }$ the Levi-Civita connections of $M^n$ and $\tilde{M}$, respectively. Throughout this paper, we agree on the following ranges of indices and the Einstein summation convention to be used unless otherwise stated:

$$1\le A,B,C,\cdots \le m+1;1\le i,j,k,\cdots \le n;n+1\le \alpha ,\beta ,\gamma ,\cdots \le n+p.$$

We can choose $\left\{e_i\right\}$ to be a local orthonormal basis of the tangent bundle $TM$ with a dual basis $\left\{{\omega }_i\right\}$ and $\left\{e_{\alpha }\right\}$ to be a local orthonormal basis of the normal bundle $T^{\perp }M$ with a dual basis $\left\{{\omega }_{\alpha }\right\}$. Let $\left\{{\omega }_{AB}\right\}$ be the connection 1-forms associated with $\left\{{\omega }_A\right\}$. Since ${\omega }_{\alpha }=0$ when restricted on M, it follows immediately that

$$0=d{\omega }_{\alpha }=-{\omega }_{i\alpha }\wedge {\omega }_i.$$

Under Cartan’s lemma, we can write

$${\omega }_{i\alpha }=h_{ij}^{\alpha }{\omega }_j,h_{ij}^{\alpha }=h_{ji}^{\alpha }.$$

(2)

We denote by $A_{\alpha }=\left(h_{ij}^{\alpha }\right)$, $h=h_{ij}^{\alpha }{\omega }_i\otimes {\omega }_j\otimes e_{\alpha }$ and $\nu =H^{\alpha }{e}_{\alpha }=\frac{1}{n}{h}_{ii}^{\alpha }{e}_{\alpha }$ the shape operator with respect to $e_{\alpha }$, the second fundamental form and the mean curvature vector field, respectively. The length of $\nu$, denoted by H, is defined to be the mean curvature of $M^n$. Moreover, $M^n$ is defined to be a submanifold with a parallel mean curvature vector if $\nu$ is parallel on the normal bundle. $M^n$ is said to be a minimal submanifold if $\nu$ vanishes.

The first and second covariant derivatives of $h_{ij}^{\alpha }$ are defined as

$$\begin{array}{cc}\hfill \nabla h_{ij}^{\alpha }=& d{h}_{ij}^{\alpha }+h_{mj}^{\alpha }{\omega }_{mi}+h_{im}^{\alpha }{\omega }_{mj}+h_{ij}^{\beta }{\omega }_{\beta \alpha }:=h_{ijk}^{\alpha }{\omega }_k,\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill \nabla h_{ijk}^{\alpha }=& d{h}_{ijk}^{\alpha }+h_{mjk}^{\alpha }{\omega }_{mi}+h_{imk}^{\alpha }{\omega }_{mj}+h_{ijm}^{\alpha }{\omega }_{mk}+h_{ijk}^{\beta }{\omega }_{\beta \alpha }:=h_{ijkl}^{\alpha }{\omega }_l.\hfill \end{array}$$

(4)

We denote by t the Cartesian coordinate on the Euclidean line $\mathbb{R}$ and ${\partial }_t:=\frac{\partial }{\partial t}$ the unit vector field tangent to $\mathbb{R}$. By decomposing ${\partial }_t$ into its tangential and normal components, we see that

$${\partial }_t=T+\eta =\sum _i{T}_i{e}_i+\sum _{\alpha }{\eta }_{\alpha }{e}_{\alpha },$$

(5)

which satisfies ${\left|T\right|}^2={\sum }_i{T}_i^2,{\left|\eta \right|}^2={\sum }_{\alpha }{\eta }_{\alpha }^2$ and ${\left|T\right|}^2+{\left|\eta \right|}^2=1.$ Moreover, the covariant derivatives of $T_i$ and ${\eta }_{\alpha }$ are defined to be

$$\begin{array}{cc}\hfill & \nabla T_i=d{T}_i+T_j{\omega }_{ji}:=T_{i,j}{\omega }_j,\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill & \nabla {\eta }_{\alpha }=d{\eta }_{\alpha }+{\eta }_{\beta }{\omega }_{\beta \alpha }:={\eta }_{\alpha ,j}{\omega }_j.\hfill \end{array}$$

(7)

Since $\tilde{\nabla }$ is a Euclidean connection when restricted to $\mathbb{R}$, it follows that ${\partial }_t$ is parallel in $\tilde{M}$. Taking the covariant derivative of each side of Equation (5) and using Equations (6) and (7) gives

$$\begin{array}{cc}\hfill & T_{i,j}=\sum _{\beta }{h}_{ij}^{\beta }{\eta }_{\beta },\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill & {\eta }_{\alpha ,i}=-\sum _k{h}_{ik}^{\alpha }{T}_k.\hfill \end{array}$$

(9)

Let $T\tilde{M}$ denote the tangent bundle of $\tilde{M}$. Recall that the curvature tensor of $\tilde{M}$ is the map $\tilde{R}:T\tilde{M}\times T\tilde{M}\times T\tilde{M}⟶T\tilde{M}$, defined as

$$\tilde{R}(X,Y)Z={\tilde{\nabla }}_X{\tilde{\nabla }}_{Y}Z-{\tilde{\nabla }}_Y{\tilde{\nabla }}_{X}Z-{\tilde{\nabla }}_{[X,Y]}Z,$$

(10)

and it is written in terms of the orthonormal basis $\left\{e_A\right\}$ as

$$\tilde{R}(e_C,e_D)e_B=\sum _A{\tilde{R}}_{BCD}^A{e}_A.$$

Moreover, the Riemannian curvature tensor ${\tilde{R}}_{ABCD}$ of $\tilde{M}$ is the covariant 4-tensor field obtained from the (1,3)-type curvature tensor field ${\tilde{R}}_{BCD}^A$ of $\tilde{M}$ by lowering the first index. Here, we continue to use the same symbol $\tilde{R}$ for the Riemannian curvature tensor when there is no risk of confusion. With this definition, we write

$$\tilde{R}(e_A,e_B)e_C=\sum _D{\tilde{R}}_{DCAB}{e}_D,$$

and by utilizing the symmetries of the Riemannian curvature tensor, we obtain

$$⟨\tilde{R}(e_A,e_B)e_C,e_D⟩={\tilde{R}}_{DCAB}={\tilde{R}}_{ABDC}.$$

(11)

It is worth pointing out that we can define the Riemannian curvature tensor R and the normal curvature tensor $R^{\perp }$ of $M^n$. It is convenient to write

$$⟨R(e_i,e_j)e_k,e_l⟩=R_{lkij}=R_{ijlk},⟨R^{\perp }(e_i,e_j)e_{\alpha },e_{\beta }⟩=R_{\beta \alpha ij}^{\perp }.$$

(12)

As we know, the standard Gauss, Codazzi and Ricci equations are given by these respective equations:

$$\begin{array}{cc}\hfill R_{ijkl}=& {\tilde{R}}_{ijkl}+\sum _{\alpha }(h_{ik}^{\alpha }{h}_{jl}^{\alpha }-h_{il}^{\alpha }{h}_{jk}^{\alpha }),\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill -{\tilde{R}}_{\alpha ijk}=& h_{ijk}^{\alpha }-h_{ikj}^{\alpha },\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill R_{\alpha \beta kl}^{\perp }=& {\tilde{R}}_{\alpha \beta kl}+\sum _k(h_{ik}^{\alpha }{h}_{jk}^{\beta }-h_{ik}^{\beta }{h}_{jk}^{\alpha }).\hfill \end{array}$$

(15)

Moreover, the Ricci identity is written as

$$h_{ijkl}^{\alpha }-h_{ijlk}^{\alpha }=h_{mj}^{\alpha }{R}_{mikl}+h_{im}^{\alpha }{R}_{mjkl}+h_{ij}^{\beta }{R}_{\beta \alpha kl}^{\perp }.$$

(16)

Let $\pi :\tilde{M}={\mathbb{S}}^m\left(c\right)\times \mathbb{R}⟶{\mathbb{S}}^m\left(c\right)$ denote the natural projection map. Since $e_A$ projects to the vector field ${\pi }_{*}\left(e_A\right)$ on the tangent bundle $T{\mathbb{S}}^m\left(c\right)$, then

$${\pi }_{*}\left(e_A\right)=e_A-⟨e_A,{\partial }_t⟩{\partial }_t.$$

(17)

By taking into account Equations (10), (11) and (17), it follows that

$$\begin{array}{cc}\hfill {\tilde{R}}_{ABCD}=& c·\{⟨{\pi }_{*}\left(e_A\right),{\pi }_{*}\left(e_C\right)⟩⟨{\pi }_{*}\left(e_B\right),{\pi }_{*}\left(e_D\right)⟩\hfill \\ \hfill & -⟨{\pi }_{*}\left(e_A\right),{\pi }_{*}\left(e_D\right)⟩⟨{\pi }_{*}\left(e_B\right),{\pi }_{*}\left(e_C\right)⟩\}.\hfill \end{array}$$

(18)

Instituting Equation (18) into Equations (13)–(15) leads to the following result.

Proposition 1 

(c.f. [25]). Let $M^n$ be an n-dimensional immersed submanifold in an $(m+1)$-dimensional Riemannian product space $\tilde{M}:={\mathbb{S}}^m\left(c\right)\times \mathbb{R}$. Then, the Gauss, Codazzi and Ricci equations are given by these respective equations:

$$\begin{array}{cc}\hfill R_{ijkl}=& c\left\{{\delta }_{ik}{\delta }_{jl}-{\delta }_{il}{\delta }_{jk}+T_j{T}_k{\delta }_{il}+T_i{T}_l{\delta }_{jk}-T_i{T}_k{\delta }_{jl}-T_j{T}_l{\delta }_{ik}\right\}\hfill \\ & +\sum _{\alpha }(h_{ik}^{\alpha }{h}_{jl}^{\alpha }-h_{il}^{\alpha }{h}_{jk}^{\alpha }),\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill -{\tilde{R}}_{\alpha ijk}=& h_{ijk}^{\alpha }-h_{ikj}^{\alpha }=c{\eta }_{\alpha }(T_j{\delta }_{ik}-T_k{\delta }_{ij}),\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill R_{\alpha \beta ij}^{\perp }=& \sum _k(h_{ik}^{\alpha }{h}_{kj}^{\beta }-h_{ik}^{\beta }{h}_{kj}^{\alpha }).\hfill \end{array}$$

(21)

Using Equation (20) gives the following proposition derived in Proposition 1 of [31]:

Proposition 2 

(c.f. [31]). Let $g_{ijk}^{\alpha }=\frac{1}{3}\left(h_{ijk}^{\alpha }+h_{jki}^{\alpha }+h_{kij}^{\alpha }\right)$. Then, we have

$$\sum _{\alpha ,i,j,k}{\left(h_{ijk}^{\alpha }\right)}^2=\sum _{\alpha ,i,j,k}{\left(g_{ijk}^{\alpha }\right)}^2+\frac{2}{3}(n-1)c^2{\left|T\right|}^2{\left|\eta \right|}^2.$$

(22)

Remark 1. 

If $h_{ijk}^{\alpha }=0$ for any $\alpha ,i,j,k$ in Equation (22), then $\left|T\right|=0$ or $\left|\eta \right|=0$ (i.e., $M^n$ is either contained in a slice ${\mathbb{S}}^m\left(c\right)$, or ${\partial }_t$ is tangent to $\mathbb{R}$ everywhere). In particular, suppose $M^n$ is a compact submanifold of ${\mathbb{S}}^m\left(c\right)\times \mathbb{R}$. It is easy to see that $\left|T\right|=0$, and thus $M^n$ lies in ${\mathbb{S}}^m\left(c\right)$.

The Ricci curvature is a symmetric 2-tensor field, and we find with Equation (19) that the Ricci curvature of $M^n$ is given by

$$R_{ik}=c\left\{{(n-1-|T|}^2){\delta }_{ik}-(n-2)T_i{T}_k\right\}+n\sum _{\alpha }{H}^{\alpha }{h}_{ik}^{\alpha }-\sum _{\alpha ,j}{h}_{ij}^{\alpha }{h}_{jk}^{\alpha }.$$

(23)

In particular, using $k=i$ in Equation (23) gives

$$R_{ii}=c\left\{n-1-{\left|T\right|}^2-(n-2)T_i^2\right\}+n\sum _{\alpha }{H}^{\alpha }{h}_{ii}^{\alpha }-\sum _{\alpha ,j}{\left(h_{ij}^{\alpha }\right)}^2.$$

(24)

In this paper, we assume that $M^n$ has nonzero parallel mean curvature vector (PMC), and thus we can choose $e_{n+1}=\nu /\left|\nu \right|$ such that H is a constant, and for $\alpha \ne n+1$, we have

$$\begin{array}{cc}\hfill & H^{\alpha }=\frac{1}{n}\mathtt{Tr}{A}_{\alpha }=0,\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill & H^{n+1}=\frac{1}{n}\mathtt{Tr}{A}_{n+1}=H,\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill & R_{n+1,\alpha ij}^{\perp }=0.\hfill \end{array}$$

(27)

The detailed proofs are given in [12].

Remark 2. 

Suppose that $M^n$ is a complete pseudo-umbilical PMC hypersurface in a product space ${\mathbb{S}}^n\left(c\right)\times \mathbb{R}$. It follows that $h_{ijk}^{n+1}=0$ for any $1\le i,j,k\le n$. According to Remark 1, it may be concluded that $M^n$ is either a sphere ${\mathbb{S}}^n\left(c\right)$ or a product space ${\mathbb{S}}^{n-1}(c+H^2)\times \mathbb{R}$ of an $(n-1)$-dimensional sphere of a constant sectional curvature $c+H^2$ and a real line $\mathbb{R}$. In the remainder of this paper, we require $p\ge 2$.

Let $S_I$ be a function given by

$$S_I=\sum _{\alpha \ne n+1}\mathtt{Tr}\left(A_{\alpha }^2\right).$$

We intend to compute the Laplacian of $S_I$. In fact, using the Codazzi equation (Equation (14)) and Ricci identity (Equation (16)), we deduce that

$$\begin{array}{cc}\hfill \frac{1}{2}\Delta S_I=& \frac{1}{2}\Delta \left(\sum _{\alpha \ne n+1}\sum _{i,j}{\left(h_{ij}^{\alpha }\right)}^2\right)\hfill \\ \hfill =& \sum _{\alpha \ne n+1}\sum _{i,j,k}{\left(h_{ijk}^{\alpha }\right)}^2+\sum _{\alpha \ne n+1}\sum _{i,j,k}{h}_{ij}^{\alpha }{h}_{ijkk}^{\alpha }\hfill \\ \hfill =& I_1+I_2+I_3+I_4,\hfill \end{array}$$

(28)

where

$$\begin{array}{cc}\hfill I_1=& \sum _{\alpha \ne n+1}\sum _{i,j,k}{\left(h_{ijk}^{\alpha }\right)}^2,\hfill \\ \hfill I_2=& -\sum _{\alpha \ne n+1}\sum _{i,j,k}{h}_{ij}^{\alpha }({\tilde{R}}_{\alpha kikj}+{\tilde{R}}_{\alpha ijkk}),\hfill \\ \hfill I_3=& \sum _{\alpha \ne n+1}\sum _{i,j,k,\beta }{h}_{ij}^{\alpha }{h}_{ki}^{\beta }{R}_{\beta \alpha jk}^{\perp },\hfill \\ \hfill I_4=& \sum _{\alpha \ne n+1}\sum _{i,j,k,m}{h}_{ij}^{\alpha }(h_{mi}^{\alpha }{R}_{mkjk}+h_{km}^{\alpha }{R}_{mijk}).\hfill \end{array}$$

Recall that $M^n$ is said to be pseudo-umbilical if the shape operator $A_{\nu }$ in the direction of the mean curvature vector field $\nu$ is proportional to the identity map $id$ (i.e., $A_{\nu }=\lambda ·id$ for some function $\lambda$). In particular, $M^n$ is called totally umbilical if its second fundamental form h and its mean curvature vector field $\nu$ satisfy

$$h(X,Y)=⟨X,Y⟩\nu ,X,Y\in T_p{M}^n,p\in M^n.$$

(29)

We now assume $M^n$ to be pseudo-umbilical. Equation (29) means that we can choose a suitable local orthonormal basis $\left\{e_i\right\}$ such that

$$h_{ij}^{n+1}=H{\delta }_{ij},$$

(30)

which gives

$$h_{ijk}^{n+1}=H_{,k}{\delta }_{ij}=0.$$

(31)

When tracing Equation (23) on i and k, the scalar curvature $\mathcal{R}$ of $M^n$ takes the form

$$\mathcal{R}=n(n-1)(c+H^2)-2(n-1)c{\left|T\right|}^2-S_I.$$

(32)

The fundamental $(p\times p)$-matrix $\left(S_{\alpha \beta }\right)$, defined by $S_{\alpha \beta }:=\mathtt{Tr}\left(A_{\alpha }{A}_{\beta }\right)$, is symmetric. We thus find that for each $\alpha \ne n+1$, we have

$$S_{n+1,\alpha }=\mathtt{Tr}\left(A_{n+1}{A}_{\alpha }\right)=\sum _{i,j}H{\delta }_{ij}{h}_{ij}^{\alpha }=H\sum _i{h}_{ii}^{\alpha }=0.$$

(33)

We fix $e_{n+1}=\nu /\left|\nu \right|$ and choose a suitable orthogonal basis ${\left\{e_{\alpha }\right\}}_{\alpha =n+2}^{n+p}$ such that the $(p\times p)$ matrix $\left(S_{\alpha \beta }\right)$ is diagonal; in other words, we have

$$\begin{array}{c}\hfill S_{\alpha \beta }=\left\{\begin{array}{cc}{S}_{\alpha },\hfill & \alpha =\beta ;\hfill \\ 0,\hfill & \alpha \ne \beta .\hfill \end{array}\right.\end{array}$$

By the definition of $S_I$, we conclude that

$$S_I=\sum _{\alpha \ne n+1}{S}_{\alpha },$$

and therefore

$$\sum _{\alpha \ne n+1}{S}_{\alpha }^2\le S_I^2.$$

(34)

According to Equations (19)–(21), (25)–(27) and (33), one computes

$$\begin{array}{cc}\hfill I_2=& nc\sum _{i,j}{\left(\sum _{\alpha \ne n+1}{h}_{ij}^{\alpha }{\eta }_{\alpha }\right)}^2-nc\sum _{\alpha \ne n+1}{|\nabla {\eta }_{\alpha }|}^2,\hfill \end{array}$$

(35)

$$I_3=-\sum _{\alpha ,\beta \ne n+1}\left[\mathtt{Tr}\left(A_{\alpha }^2{A}_{\beta }^2\right)-\mathtt{Tr}{\left(A_{\alpha }{A}_{\beta }\right)}^2\right],$$

(36)

$$\begin{array}{cc}\hfill I_4=& \left[n(c+H^2)-c{\left|T\right|}^2\right]S_I-nc\sum _{\alpha \ne n+1}{|\nabla {\eta }_{\alpha }|}^2\hfill \\ & -\sum _{\alpha ,\beta \ne n+1}\left[\mathtt{Tr}\left(A_{\alpha }^2{A}_{\beta }^2\right)-\mathtt{Tr}{\left(A_{\alpha }{A}_{\beta }\right)}^2\right]-\sum _{\alpha \ne n+1}{S}_{\alpha }^2.\hfill \end{array}$$

(37)

Remark 3. 

It should be emphasized that the formulas mentioned above in this section are closely associated with the similar computations obtained in the proof of Equation (4.3) in [30], where the authors used the method of an invariant operator.

3. Several Key Estimates

For future purposes, in this section, we give several key inequalities which will play crucial roles in the proof of our rigidity theorems. At first, Lemma 1 is immediate from the Cauchy–Schwarz inequality.

Lemma 1. 

The expression ${\displaystyle \sum _{\alpha \ne n+1}}{|\nabla {\eta }_{\alpha }|}^2$ satisfies the following upper bound estimate:

$$\sum _{\alpha \ne n+1}|\nabla {\eta }_{\alpha }{|}^2=\sum _{\alpha \ne n+1}\sum _i{\left(\sum _j{h}_{ij}^{\alpha }{T}_j\right)}^2\le {|T|}^2{S}_I.$$

(38)

Lemma 2 

(c.f. [12]). Let $K_0$ stand for the infimum of the sectional curvature of M at each $x\in M$. Then, we have a lower bound estimate of $I_4$:

$$I_4=\sum _{\alpha \ne n+1}\sum _{i,j,k,m}{h}_{ij}^{\alpha }(h_{mi}^{\alpha }{R}_{mkjk}+h_{km}^{\alpha }{R}_{mijk})\ge n{K}_0{S}_I.$$

(39)

Proof. 

We fix each $\alpha \ne n+1$ and let ${\lambda }_1^{\alpha },{\lambda }_2^{\alpha },\cdots ,{\lambda }_n^{\alpha }$ be the eigenvalues of $A_{\alpha }$. The matrix $\left(h_{ij}^{\alpha }\right)$ can be diagonalized for a suitable choice of an orthonormal basis $\left\{{\tilde{e}}_i\right\}$ at each point $x\in M$. It can easily be checked that

$$\begin{array}{cc}\hfill & \sum _{i,j,k,m}{h}_{ij}^{\alpha }(h_{mi}^{\alpha }{R}_{mkjk}+h_{km}^{\alpha }{R}_{mijk})\hfill \\ \hfill =& \sum _{i,k}{\lambda }_i^{\alpha }({\lambda }_i^{\alpha }{R}_{ikik}+{\lambda }_k^{\alpha }{R}_{kiik})\hfill \\ \hfill =& \frac{1}{2}\sum _{i,k}{({\lambda }_i^{\alpha }-{\lambda }_k^{\alpha })}^2{R}_{ikik}\hfill \\ \hfill \ge & \frac{1}{2}\sum _{i,k}{({\lambda }_i^{\alpha }-{\lambda }_k^{\alpha })}^2{K}_0\hfill \\ \hfill =& n{K}_0\sum _i{\left({\lambda }_i^{\alpha }\right)}^2-K_0\sum _i{\lambda }_i^{\alpha }\sum _k{\lambda }_k^{\alpha }\hfill \\ \hfill =& n{K}_0\sum _{i,k}{\left(h_{ik}^{\alpha }\right)}^2.\hfill \end{array}$$

(40)

By taking the sum over the index $\alpha$ in Equation (40), not including $n+1$, we can obtain the inequality in Equation (39). In fact, the above inequality (Equation (40)) is originally due to Yau’s estimate (Equation (10.9)) in [12]. □

We end this section by recalling a well-known inequality from the results due to [9].

Lemma 3 

(c.f. [9]). We abbreviate $Tr\left(A_{\alpha }^2{A}_{\beta }^2\right)-Tr{\left(A_{\alpha }{A}_{\beta }\right)}^2$ to $N_{\alpha \beta }$ for $\alpha ,\beta \ne n+1$. Furthermore, we set $N_I={\displaystyle \sum _{\alpha ,\beta \ne n+1}}{N}_{\alpha \beta }$ and $L_I={\displaystyle \sum _{\alpha \ne n+1}}{S}_{\alpha }^2.$ Then, we have

$$2N_I+L_I\le \left[1+\frac{1}{2}\mathrm{sgn}(p-2)\right]S_I^2,$$

(41)

where $\mathrm{sgn}(·)$ stands for the sign function. The equality holds if and only if at most two of the shape operators $\left\{A_{\alpha }\right\}$ are nonzero for $n+2\le \alpha \le n+p$. And if we assume that $A_{n+2}\ne 0,A_{n+3}\ne 0$ and $A_{\alpha }=0$ for any $n+4\le \alpha \le n+p$, then there exists an orthogonal matrix U such that

$$\begin{array}{cc}{U}^{\top }{A}_{n+2}U=\lambda \left(\begin{array}{cc}{J}_{2\times 2}& {\left(0\right)}_{2\times r}\\ {\left(0\right)}_{r\times 2}& {\left(0\right)}_{r\times r}\end{array}\right),& U^{\top }{A}_{n+3}U=\lambda \left(\begin{array}{cc}{K}_{2\times 2}& {\left(0\right)}_{2\times r}\\ {\left(0\right)}_{r\times 2}& {\left(0\right)}_{r\times r}\end{array}\right),\end{array}$$

where $r=n-2,$λ is a nonzero constant and

$$J_{2\times 2}=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),K_{2\times 2}=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right).$$

4. Ricci Curvature Pinching Theorem

In this section, we consider the Ricci curvature pinching problem. Assume that $Q_0$ denotes the infimum of the Ricci curvature of M at every point $x\in M$. From Equations (24)–(26) and (30), one has

$$\sum _{\alpha \ne n+1}\sum _j{\left(h_{ij}^{\alpha }\right)}^2\le (n-1)(c+H^2)-c{\left|T\right|}^2-(n-2)c{T}_i^2-Q_0.$$

(42)

Taking the sum over i in Equation (42) gives

$$S_I\le n(n-1)(c+H^2)-2(n-1)c{\left|T\right|}^2-n{Q}_0.$$

(43)

Since ${({\lambda }_i^{\alpha }-{\lambda }_k^{\alpha })}^2\le 2\left[{\left({\lambda }_i^{\alpha }\right)}^2+{\left({\lambda }_k^{\alpha }\right)}^2\right]$, where the equality holds if and only if ${\lambda }_i^{\alpha }=-{\lambda }_k^{\alpha }$, we obtain

$$\begin{array}{cc}\hfill N_{\alpha \beta }& =\sum _{i,k}({\lambda }_i^{\alpha }{\lambda }_i^{\alpha }{h}_{ik}^{\beta }{h}_{ki}^{\beta }-{\lambda }_i^{\alpha }{h}_{ik}^{\beta }{\lambda }_k^{\alpha }{h}_{ki}^{\beta })\hfill \\ \hfill & =\frac{1}{2}\sum _{\begin{array}{c}i,k\\ i\ne k\end{array}}{({\lambda }_i^{\alpha }-{\lambda }_k^{\alpha })}^2{\left(h_{ik}^{\beta }\right)}^2\hfill \\ \hfill & \le \sum _{\begin{array}{c}i,k\\ i\ne k\end{array}}\left[{\left({\lambda }_i^{\alpha }\right)}^2+{\left({\lambda }_k^{\alpha }\right)}^2\right]{\left(h_{ik}^{\beta }\right)}^2.\hfill \end{array}$$

(44)

This implies in particular that

$$N_{\alpha \beta }\le \sum _{i,k}\left[{\left({\lambda }_i^{\alpha }\right)}^2+{\left({\lambda }_k^{\alpha }\right)}^2\right]{\left(h_{ik}^{\beta }\right)}^2=2\sum _i\left({\left({\lambda }_i^{\alpha }\right)}^2\sum _k{\left(h_{ik}^{\beta }\right)}^2\right).$$

(45)

Combining Equations (45) and (42) gives

$$\begin{array}{cc}\hfill N_I=& \sum _{\alpha ,\beta \ne n+1}{N}_{\alpha \beta }\le 2\sum _{\alpha \ne n+1}\sum _i\left({\left({\lambda }_i^{\alpha }\right)}^2\sum _{\beta \ne n+1}\sum _j{\left(h_{ij}^{\beta }\right)}^2\right)\hfill \\ \hfill \le & 2\sum _{\alpha \ne n+1}\sum _i{\left({\lambda }_i^{\alpha }\right)}^2\left((n-1)(c+H^2)-c{\left|T\right|}^2-(n-2)c{T}_i^2-Q_0\right)\hfill \\ \hfill =& 2\left((n-1)(c+H^2)-c{\left|T\right|}^2-Q_0\right)S_I-2(n-2)c\sum _{\alpha \ne n+1}\sum _i{T}_i^2{\left({\lambda }_i^{\alpha }\right)}^2\hfill \\ \hfill \le & 2\left((n-1)(c+H^2)-c{\left|T\right|}^2-Q_0\right)S_I.\hfill \end{array}$$

(46)

On the other hand, with Equation (44), one obtains

$$\begin{array}{c}\hfill N_{\alpha \beta }\le \sum _{\begin{array}{c}i,k\\ i\ne k\end{array}}\left[{\left({\lambda }_i^{\alpha }\right)}^2+{\left({\lambda }_k^{\alpha }\right)}^2\right]{\left(h_{ik}^{\beta }\right)}^2\le \sum _j{\left({\lambda }_j^{\alpha }\right)}^2\sum _{i,k}{\left(h_{ik}^{\beta }\right)}^2=S_{\alpha }{S}_{\beta },\end{array}$$

from which we have

$$N_I=\sum _{\alpha ,\beta \ne n+1}{N}_{\alpha \beta }\le \sum _{\alpha ,\beta \ne n+1}{S}_{\alpha }{S}_{\beta }=S_I^2-\sum _{\alpha \ne n+1}{S}_{\alpha }^2=S_I^2-L_I.$$

(47)

With these preparations, we shall now prove the following pinching theorem for the Ricci curvature:

Theorem 1. 

Let $M^n$ be a compact pseudo-umbilical PMC submanifold immersed in a product space ${\mathbb{S}}^m\left(c\right)\times \mathbb{R}$ $(n\ge 3,p\ge 2)$. If the infimum of the Ricci curvature of $M^n$ satisfies $Q_0\ge D_1$, where $D_1$ is defined as Equation (51), then $\left|T\right|=0$, and $M^n$ is a totally umbilical sphere ${\mathbb{S}}^n(c+H^2)$ in a slice ${\mathbb{S}}^m\left(c\right)$.

Proof. 

According to Equations (35)–(37), a direct computation shows that

$$\begin{array}{cc}\hfill \frac{1}{2}\Delta S_I=& I_1+I_2+I_3+I_4\hfill \\ \hfill =& I_1+\left[n(c+H^2)-c{\left|T\right|}^2\right]S_I+nc\sum _{i,j}{\left(\sum _{\alpha \ne n+1}{h}_{ij}^{\alpha }{\eta }_{\alpha }\right)}^2\hfill \\ \hfill & -2nc\sum _{\alpha \ne n+1}{|\nabla {\eta }_{\alpha }|}^2-2N_I-L_I\hfill \\ \hfill \ge & I_1+\left[n(c+H^2)-c{\left|T\right|}^2\right]S_I-2nc\sum _{\alpha \ne n+1}{|\nabla {\eta }_{\alpha }|}^2-N_I-(N_I+L_I).\hfill \end{array}$$

(48)

By applying Equations (38) and (43)–(48), it is straightforward to show that

$$\begin{array}{cc}\hfill \frac{1}{2}\Delta S_I\ge & I_1+\left[n(c+H^2)-c{\left|T\right|}^2\right]S_I-2nc{\left|T\right|}^2{S}_I\hfill \\ \hfill & -2\left((n-1)(c+H^2)-c{\left|T\right|}^2-Q_0\right)S_I-S_I^2\hfill \\ \hfill \ge & I_1+\left[n(c+H^2)-c{\left|T\right|}^2\right]S_I-2nc{\left|T\right|}^2{S}_I\hfill \\ \hfill & -2\left((n-1)(c+H^2)-c{\left|T\right|}^2-Q_0\right)S_I\hfill \\ \hfill & -\left(n(n-1)(c+H^2)-2(n-1)c{\left|T\right|}^2-n{Q}_0\right)S_I\hfill \\ \hfill =& I_1+(n+2)\left(Q_0-\frac{n^2-2}{n+2}(c+H^2)-\frac{1}{n+2}c{\left|T\right|}^2\right)S_I.\hfill \end{array}$$

(49)

On the other hand, applying Equations (38), (41) and (43) to Equation (48) gives rise to

$$\begin{array}{cc}\hfill \frac{1}{2}\Delta S_I\ge & I_1+\left[n(c+H^2)-c{\left|T\right|}^2\right]S_I-2nc{\left|T\right|}^2{S}_I-\left[1+\frac{1}{2}\mathrm{sgn}(p-2)\right]S_I^2\hfill \\ \hfill \ge & I_1+\left[n(c+H^2)-(2n+1)c{\left|T\right|}^2\right]S_I\hfill \\ \hfill & -\left[1+\frac{1}{2}\mathrm{sgn}(p-2)\right]\left(n(n-1)(c+H^2)-2(n-1)c{\left|T\right|}^2-n{Q}_0\right)S_I\hfill \\ \hfill =& I_1+n\left[1+\frac{1}{2}\mathrm{sgn}(p-2)\right][Q_0-\frac{2(n-2)+(n-1)\mathrm{sgn}(p-2)}{2+\mathrm{sgn}(p-2)}(c+H^2)\hfill \\ \hfill & -\frac{6-2(n-1)\mathrm{sgn}(p-2)}{n\left[2+\mathrm{sgn}(p-2)\right]}c{\left|T\right|}^2]S_I.\hfill \end{array}$$

(50)

We set

$$\begin{array}{cc}\hfill D_1=& min\{\frac{n^2-2}{n+2}(c+H^2)+\frac{1}{n+2}c{\left|T\right|}^2,\hfill \\ \hfill & \frac{2(n-2)+(n-1)\mathrm{sgn}(p-2)}{2+\mathrm{sgn}(p-2)}(c+H^2)\hfill \\ \hfill & +\frac{6-2(n-1)\mathrm{sgn}(p-2)}{n\left[2+\mathrm{sgn}(p-2)\right]}c{\left|T\right|}^2\}.\hfill \end{array}$$

(51)

Provided that $Q_0\ge D_1$, taking integration over $M^n$ on Equations (49) and (50), and using the compactness of $M^n$, we are led to the conclusion that $\Delta S_I=0$ and

$$I_1=\sum _{\alpha \ne n+1}\sum _{i,j,k}{\left(h_{ijk}^{\alpha }\right)}^2=0.$$

(52)

Taking into consideration Equations (31) and (52), we find $h_{ijk}^{\alpha }=0$ for any $\alpha ,i,j,k$. It follows from Remark 1 that $\left|T\right|=0$, and $M^n$ is a compact pseudo-umbilical PMC submanifold in ${\mathbb{S}}^m\left(c\right)$. Under Theorem 1 in [32] (see also [33]), we find that $M^n$ is either a sphere ${\mathbb{S}}^{m-1}(c+H^2)$ or a minimal submanifold immersed in ${\mathbb{S}}^{m-1}(c+H^2)$ with the co-dimension $\tilde{p}=p-2.$ Moreover, we see that

$$S_I=\sum _{\alpha \ne n+1}\sum _{i,j}{\left(h_{ij}^{\alpha }\right)}^2=0,$$

or

$$Q_0=D_1=min\left\{\frac{n^2-2}{n+2}(c+H^2),\frac{2(n-2)+(n-1)\mathrm{sgn}(p-2)}{2+\mathrm{sgn}(p-2)}(c+H^2)\right\}.$$

If $S_I=0$ or $p=2$, then $M^n$ is a totally umbilical sphere $S^n(c+H^2)$ in ${\mathbb{S}}^m\left(c\right)$.

If $S_I>0$ and $p\ge 3$, then the Ricci curvature of $M^n$ is identically equal to $D_1$. Now, we also observe that

$$\begin{array}{cc}\hfill D_1=& min\left\{\frac{n^2-2}{n+2}(c+H^2),\frac{3n-5}{3}(c+H^2)\right\}\hfill \\ \hfill =& \left\{\begin{array}{cc}\frac{4}{3}(c+H^2),\hfill & \mathrm{if}n=3;\hfill \\ \frac{n^2-2}{n+2}(c+H^2),\hfill & \mathrm{if}n\ge 4.\hfill \end{array}\right.\hfill \end{array}$$

It can easily be seen that the Ricci curvature satisfies

$$R_{ik}=D_1>(n-2)(c+H^2)$$

for any $n\ge 3$. This, combined with the results of Ejiri [15] and Li [17], implies that $M^n$ must be a totally geodesic sphere $S^n(c+H^2)$ in ${\mathbb{S}}^{m-1}(c+H^2)$, which contradicts the assumption that $S_I>0$. We have thus proven Theorem 1. □

5. Sectional Curvature Pinching Theorem

Now, we shall state and prove the following sectional curvature pinching theorem:

Theorem 2. 

Let $M^n$ be a compact pseudo-umbilical PMC submanifold immersed in a product space ${\mathbb{S}}^m\left(c\right)\times \mathbb{R}$ $(n\ge 3,p\ge 2)$. If the infimum of the sectional curvature of $M^n$ satisfies $K_0\ge D_2$, where $D_2$ is defined as Equation (60), then $\left|T\right|=0$ (i.e., $M^n$ lies in a slice ${\mathbb{S}}^m\left(c\right)$). Furthermore, $M^n$ is locally either a totally umbilical sphere ${\mathbb{S}}^n(c+H^2)$ in ${\mathbb{S}}^m\left(c\right)$ or a Veronese manifold with a constant sectional curvature $\frac{n}{2(n+1)}(c+H^2)$.

Proof. 

By substituting Equations (35)–(37) into Equation (28), a direct computation shows that

$$\begin{array}{cc}\hfill \frac{1}{2}\Delta S_I=& I_1+I_2+I_3+(1+\frac{n}{n+2})I_4-\frac{n}{n+2}{I}_4\hfill \\ \hfill =& I_1+\frac{2(n+1)}{n+2}{I}_4-\frac{n}{n+2}\left[n(c+H^2)-c{\left|T\right|}^2\right]S_I+nc\sum _{i,j}{\left(\sum _{\alpha \ne n+1}{h}_{ij}^{\alpha }{\eta }_{\alpha }\right)}^2\hfill \\ \hfill & -\frac{2nc}{n+2}\sum _{\alpha \ne n+1}{|\nabla {\eta }_{\alpha }|}^2+\frac{1}{n+2}(n{L}_I-2N_I)\hfill \\ \hfill \ge & I_1+\frac{2(n+1)}{n+2}{I}_4-\frac{n}{n+2}\left[n(c+H^2)-c{\left|T\right|}^2\right]S_I\hfill \\ \hfill & -\frac{2nc}{n+2}\sum _{\alpha \ne n+1}{|\nabla {\eta }_{\alpha }|}^2+\frac{1}{n+2}(n{L}_I-2N_I).\hfill \end{array}$$

(53)

In light of Equations (38) and (39), we can derive through Equation (53) that

$$\begin{array}{cc}\hfill \frac{1}{2}\Delta S_I\ge & I_1+\frac{2n(n+1)}{n+2}{K}_0{S}_I-\frac{n}{n+2}\left[n(c+H^2)-c{\left|T\right|}^2\right]S_I\hfill \\ \hfill & -\frac{2nc}{n+2}{\left|T\right|}^2{S}_I+\frac{1}{n+2}(n{L}_I-2N_I)\hfill \\ \hfill \ge & I_1+\frac{2n(n+1)}{n+2}\left(K_0-\frac{n}{2(n+1)}(c+H^2)-\frac{{c\left|T\right|}^2}{2(n+1)}\right)S_I,\hfill \end{array}$$

(54)

where the last inequality is followed by

$$N_I\le \frac{n}{2}{L}_I,$$

which is taken from T. Itoh (see [34], Proposition 1).

Likewise, by substituting Equations (35)–(37) into Equation (28) again, we have

$$\begin{array}{cc}\hfill \frac{1}{2}\Delta S_I=& I_1+I_2+I_3+(1+\frac{p-2}{p-1})I_4-\frac{p-2}{p-1}{I}_4\hfill \\ \hfill =& I_1+\frac{2p-3}{p-1}{I}_4-\frac{p-2}{p-1}\left[n(c+H^2)-c{\left|T\right|}^2\right]S_I+nc\sum _{i,j}{\left(\sum _{\alpha \ne n+1}{h}_{ij}^{\alpha }{\eta }_{\alpha }\right)}^2\hfill \\ \hfill & -\frac{nc}{p-1}\sum _{\alpha \ne n+1}{|\nabla {\eta }_{\alpha }|}^2+\frac{1}{p-1}\left((p-2)L_I-N_I\right)\hfill \\ \hfill \ge & I_1+\frac{2p-3}{p-1}{I}_4-\frac{p-2}{p-1}\left[n(c+H^2)-c{\left|T\right|}^2\right]S_I\hfill \\ \hfill & -\frac{nc}{p-1}\sum _{\alpha \ne n+1}{|\nabla {\eta }_{\alpha }|}^2+\frac{1}{p-1}\left((p-2)L_I-N_I\right).\hfill \end{array}$$

(55)

Applying Equations (38) and (39) to Equation (55) implies that

$$\begin{array}{cc}\hfill \frac{1}{2}\Delta S_I\ge & I_1+\frac{2p-3}{p-1}n{K}_0{S}_I-\frac{p-2}{p-1}\left[n(c+H^2)-c{\left|T\right|}^2\right]S_I\hfill \\ \hfill & -\frac{nc}{p-1}{\left|T\right|}^2{S}_I+\frac{1}{p-1}\left((p-2)L_I-N_I\right)\hfill \\ \hfill =& I_1+\frac{2p-3}{p-1}n\left(K_0-\frac{p-2}{2p-3}(c+H^2)-\frac{n+2-p}{n(2p-3)}c{\left|T\right|}^2\right)S_I\hfill \\ \hfill & +\frac{1}{p-1}\left((p-2)L_I-N_I\right).\hfill \end{array}$$

(56)

With the Cauchy–Schwarz inequality, we obtain

$$S_I^2={\left(\sum _{\alpha \ne n+1}1·S_{\alpha }\right)}^2\le \sum _{\alpha \ne n+1}{1}^2·\sum _{\alpha \ne n+1}{S}_{\alpha }^2=(p-1)\sum _{\alpha \ne n+1}{S}_{\alpha }^2=(p-1)L_I,$$

(57)

where the equality holds if and only if one of ${\left\{S_{\alpha }\right\}}_{\alpha =n+2}^{n+p}$ is nonzero or

$$S_{n+2}=S_{n+3}=\cdots =S_{n+p}.$$

Also, it follows from Equations (47) and (57) that

$$(p-2)L_I-N_I\ge (p-2)L_I-(S_I^2-L_I)=(p-1)L_I-S_I^2\ge 0.$$

(58)

Applying Equation (58) to Equation (56) shows that

$$\begin{array}{c}\hfill \frac{1}{2}\Delta S_I\ge I_1+\frac{2p-3}{p-1}n\left(K_0-\frac{p-2}{2p-3}(c+H^2)-\frac{n+2-p}{n(2p-3)}c{\left|T\right|}^2\right)S_I.\end{array}$$

(59)

We set

$$\begin{array}{cc}\hfill D_2=& min\{\frac{n}{2(n+1)}(c+H^2)+\frac{{c\left|T\right|}^2}{2(n+1)},\hfill \\ \hfill & \frac{p-2}{2p-3}(c+H^2)+\frac{n+2-p}{n(2p-3)}c{\left|T\right|}^2\}.\hfill \end{array}$$

(60)

Provided that $K_0\ge D_2$, by integrating Equations (54) and (59) and using the compactness of $M^n$, we therefore conclude that $\Delta S_I=0$ and

$$I_1=\sum _{\alpha \ne n+1}\sum _{i,j,k}{\left(h_{ijk}^{\alpha }\right)}^2=0.$$

An argument similar to that used in proof of Theorem 1 shows that $\left|T\right|=0$, and $M^n$ is either a sphere ${\mathbb{S}}^{m-1}(c+H^2)$ or a minimal submanifold immersed in ${\mathbb{S}}^{m-1}(c+H^2)$ with the co-dimension $\tilde{p}=p-2.$ Moreover, we see that

$$S_I=\sum _{\alpha \ne n+1}\sum _{i,j}{\left(h_{ij}^{\alpha }\right)}^2=0,$$

or

$$K_0=D_2=min\left\{\frac{n}{2(n+1)}(c+H^2),\frac{\tilde{p}}{2\tilde{p}+1}(c+H^2)\right\}.$$

For the former case, $M^n$ is a totally umbilical sphere $S^n(c+H^2)$ in ${\mathbb{S}}^m\left(c\right)$. Hereafter, we consider the latter case, where the sectional curvature $R_{ikik}$ is identically equal to $D_2$.

Case A. If $2\tilde{p}\ge n$, then

$$D_2=\frac{n}{2(n+1)}(c+H^2),$$

Therefore, $M^n$ is a Veronese manifold with a constant sectional curvature $\frac{n}{2(n+1)}(c+H^2)$ and $\tilde{p}=\frac{1}{2}(n-1)(n+2)$ (see Example 2 in [8] and Main Theorem in [13]). Here, we assume that the immersion

$${\phi }_1:M^n⟶{\mathbb{S}}^{m-1}(c+H^2)$$

is full. In particular, for $n=\tilde{p}=2$, we recover the Veronese surface.

Case B. If $2\tilde{p}<n$, then

$$D_2=\frac{\tilde{p}}{2\tilde{p}+1}(c+H^2),$$

Therefore, the equality holds in Equation (58), and so does Equation (57).

Suppose that $S_{n+2}=S_{n+3}=\cdots =S_{n+p}=0$. It is easy to see that $M^n$ is a totally geodesic sphere $S^n(c+H^2)$ in ${\mathbb{S}}^{m-1}(c+H^2)$.

Suppose that $S_{n+2}=S_{n+3}=\cdots =S_{n+p}\ne 0$. Note that $M^n$ is contained in a slice ${\mathbb{S}}^m\left(c\right)$, and we thus find that ${\partial }_t=\eta$ is perpendicular to the mean curvature vector field $\nu$. We can certainly assume that $e_{n+2}=\eta$. Since the equality holds in Equation (53), we obtain ${\left(h_{ij}^{n+2}\right)}^2=0$ for arbitrary $1\le i,j\le n$, namely $S_{n+2}=0$, which is a contradiction.

We now turn our attention to the case where $S_{n+2}\ne 0$ and $S_{n+3}=\cdots =S_{n+p}=0$. Therefore, we assume that $M^n$ is a minimal immersed hypersurface of ${\mathbb{S}}^{n+1}(c+H^2)$ with the shape operator $\tilde{A}=diag({\tilde{\lambda }}_1,{\tilde{\lambda }}_2,\cdots ,{\tilde{\lambda }}_n)$. According to the Gauss equation, it follows that

$$\frac{\tilde{p}}{2\tilde{p}+1}(c+H^2)=(c+H^2)+{\tilde{\lambda }}_i{\tilde{\lambda }}_k$$

(61)

for any $i\ne k$. Hence, ${\tilde{\lambda }}_1={\tilde{\lambda }}_2=\cdots ={\tilde{\lambda }}_n$, which is due to the fact that $n\ge 3$. From Equation (61), one easily finds that for any i,

$${\tilde{\lambda }}_i^2=-\frac{\tilde{p}+1}{2\tilde{p}+1}(c+H^2)<0,$$

which is impossible. This finishes the proof of Theorem 2. □

6. Scalar Curvature Pinching Theorem

Upon substituting Equations (38) and (41) into Equation (48), one sees immediately that

$$\begin{array}{cc}\hfill \frac{1}{2}\Delta S_I\ge & I_1+\left[n(c+H^2)-c{\left|T\right|}^2\right]S_I-2nc{\left|T\right|}^2{S}_I-\left[1+\frac{1}{2}\mathrm{sgn}(p-2)\right]S_I^2\hfill \\ \hfill =& I_1+\left\{n(c+H^2)-(2n+1)c{\left|T\right|}^2-\left[1+\frac{1}{2}\mathrm{sgn}(p-2)\right]S_I\right\}{S}_I.\hfill \end{array}$$

(62)

By the same argument as that in [8,9], we can now establish the desired pinching theorem for $S_I$.

Theorem 3. 

Let $M^n$ be a compact pseudo-umbilical PMC submanifold immersed in a product space ${\mathbb{S}}^m\left(c\right)\times \mathbb{R}$ $(n\ge 3,p\ge 2)$. If $S_I\le D_3$ everywhere on $M^n$, then $\left|T\right|=0$ (i.e., $M^n$ lies in a slice ${\mathbb{S}}^m\left(c\right)$). Here, $D_3$ is given by

$$D_3=\frac{1}{1+\frac{1}{2}\mathrm{sgn}(p-2)}\{n(c+H^2)-(2n+1){c\left|T\right|}^2\}.$$

Furthermore, one of the following assertions holds:

(i) 

$M^n$ is a sphere ${\mathbb{S}}^n(c+H^2)$;

(ii) 

$M^n$ is a Clifford minimal hypersurface $M_{k,m-2-k}$ contained in ${\mathbb{S}}^{m-1}(c+H^2)$ with $1\le k\le m-3$, where the Clifford minimal hypersurface $M_{k,m-2-k}$ is given by

$$M_{k,m-2-k}=S^k\left(\frac{(m-2)(c+H^2)}{k}\right)\times S^{m-2-k}\left(\frac{(m-2)(c+H^2)}{m-2-k}\right);$$

(iii) 

$M^n$ is a Veronese

surface $M^2$ in ${\mathbb{S}}^4(c+H^2)$.

In light of Equation (32), we point out that if $S_I\le D_3$, then $\mathcal{R}\ge D_4$, where

$$\begin{array}{cc}\hfill D_4=& n(n-1)(c+H^2)-2(n-1)c{\left|T\right|}^2-D_3\hfill \\ \hfill =& \frac{2(n-2)+(n-1)\mathrm{sgn}(p-2)}{2+\mathrm{sgn}(p-2)}n(c+H^2)\hfill \\ \hfill & +\frac{6-2(n-1)\mathrm{sgn}(p-2)}{2+\mathrm{sgn}(p-2)}c{\left|T\right|}^2.\hfill \end{array}$$

Consequently, Theorem 3 can be formulated equivalently in terms of the scalar curvature $\mathcal{R}$ of $M^n$ as follows:

Theorem 4. 

Let $M^n$ be a compact pseudo-umbilical PMC submanifold immersed in a product space ${\mathbb{S}}^m\left(c\right)\times \mathbb{R}$ $(n\ge 3,p\ge 2)$. If $\mathcal{R}\ge D_4$ everywhere on $M^n$, then $\left|T\right|=0$, and $M^n$ is a sphere ${\mathbb{S}}^n(c+H^2)$, a Clifford minimal hypersurface $M_{k,m-2-k}$ contained in ${\mathbb{S}}^{m-1}(c+H^2)$ or a Veronese surface $M^2$ in ${\mathbb{S}}^4(c+H^2)$.