The topological rigidity theorem for submanifolds in space forms

TL;DR

This paper proves a topological sphere theorem for compact submanifolds in space forms under optimal curvature pinching conditions, extending previous results and establishing new topological sphere criteria.

Contribution

It establishes a new topological sphere theorem for submanifolds with optimal curvature pinching conditions, improving previous results and providing new criteria for topological sphere theorems.

Findings

Submanifolds with scalar curvature exceeding a specific bound are homeomorphic to spheres.

The curvature pinching conditions used are proven to be optimal.

New topological sphere theorems are derived for submanifolds with pinched scalar and Ricci curvatures.

Abstract

Let $M$ be an $n(\geq 4)$-dimensional compact submanifold in the simply connected space form $F^{n+p}(c)$ with constant curvature $c\geq 0$, where $H$ is the mean curvature of $M$. We verify that if the scalar curvature of $M$ satisfies $R>n(n-2)(c+H^2)$, and if $Ric_M\geq (n-2-\frac{2\sigma_n}{2n-\sigma_n})(c+H^2)$, then $M$ is homeomorphic to a sphere. Here $\sigma_n=sgn(n-4)((-1)^n+3)$, and $sgn(\cdot)$ is the standard sign function. This improves our previous sphere theorem \cite{XG2}. It should be emphasized that our pinching conditions above are optimal. We also obtain some new topological sphere theorems for submanifolds with pinched scalar curvature and Ricci curvature.

Equations

$$||h(u,u)|{|}^2\le C(n,p)$$

$$C(n,p)=\left\{\begin{array}[]{ll}1,&\mbox{\ for\ }p=1,\mbox{\ and\ }n\mbox{\ is even},\\ \frac{n}{n-1},&\mbox{\ for\ }p=1,\mbox{\ and\ }n\mbox{\ is odd},\\ \frac{1}{3},&\mbox{\ for\ }p\geq 2,\mbox{\ and\ }n\mbox{\ is even},\\ \frac{n}{3n-2},&\mbox{\ for\ }p\geq 2,\mbox{\ and\ }n\mbox{\ is odd}.\end{array}\right.$$

$$Ri{c}_M>(n-2-{ϵ}_n)(c+H^2),$$

$$Ri{c}_M\ge (n-2-\frac{2{\sigma }_n}{2n-{\sigma }_n})(c+H^2),$$

$$Ri{c}_M\le (n-2+\frac{2{\tau }_n}{2n+{\tau }_n})(c+H^2),$$

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

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

$$h=\sum _{\alpha ,i,j}{h}_{ij}^{\alpha }{\omega }_i\otimes {\omega }_j\otimes e_{\alpha },\xi =\frac{1}{n}\sum _{\alpha ,i}{h}_{ii}^{\alpha }{e}_{\alpha },$$

$$R_{ijkl}=c({\delta }_{ik}{\delta }_{jl}-{\delta }_{il}{\delta }_{jk})+\sum _{\alpha }(h_{ik}^{\alpha }{h}_{jl}^{\alpha }-h_{il}^{\alpha }{h}_{jk}^{\alpha }),$$

$$R_{\alpha \beta kl}=\sum _i(h_{ik}^{\alpha }{h}_{il}^{\beta }-h_{il}^{\alpha }{h}_{ik}^{\beta }),$$

$$S=|h{|}^2,H=|\xi |,H_{\alpha }=(h_{ij}^{\alpha }{)}_{n\times n}.$$

$$Ric(e_{i})=(n-1)c+\sum_{\alpha,j}\big{[}h_{ii}^{\alpha}h_{jj}^{\alpha}-(h_{ij}^{\alpha})^{2}\big{]}.$$

$$R=n(n-1)c+n^2{H}^2-S.$$

$$Ri{c}^{[s]}([e_{j_0},\dots ,e_{j_{n-1}}])=\sum _{p=0}^{s-1}\sum _{q=0}^{n-1}{R}_{j_p{j}_q{j}_p{j}_q},$$

$$Ri{c}_{\min }^{[s]}(x)=\underset{\{e_{j_0},\dots ,e_{j_{n-1}}\}\subset T_{x}M}{\min }Ri{c}^{[s]}([e_{j_0},\dots ,e_{j_{n-1}}]),$$

$$Ri{c}_{\max }^{[s]}(x)=\underset{\{e_{j_0},\dots ,e_{j_{n-1}}\}\subset T_{x}M}{\max }Ri{c}^{[s]}([e_{j_0},\dots ,e_{j_{n-1}}]).$$

$$\sum _{k=q+1}^n\sum _{i=1}^q[2|h(e_i,e_k){|}^2-⟨h(e_i,e_i),h(e_k,e_k)⟩]<q(n-q)c$$

$$S-n{H}^2=n(n-1)(c+H^2)-R.$$

$$\sum _{k=2}^n[2|h(e_1,e_k){|}^2-⟨h(e_1,e_1),h(e_k,e_k)⟩]$$

$$2Ri{c}_{\min }+R>n(n-1)(c+H^2),$$

$$\sum _{k=2}^n[2|h(e_1,e_k){|}^2-⟨h(e_1,e_1),h(e_k,e_k)⟩]<(n-1)c.$$

$$\sum _{k=2}^n[2|h(e_1,e_k){|}^2-⟨h(e_1,e_1),h(e_k,e_k)⟩]$$

$$3R-2Ri{c}_{\max }^{[n-1]}>n(n-1)(c+H^2),$$

$$\sum _{k=2}^n[2|h(e_1,e_k){|}^2-⟨h(e_1,e_1),h(e_k,e_k)⟩]<(n-1)c.$$

$$Ri{c}_M>\frac{n(n-1)}{n+2}(c+H^2),$$

$$2Ri{c}_{\min }+R>n(n-1)(c+H^2).$$

$$Ri{c}_M>\frac{n(n-1)}{n+2}(c+H^2),$$

$$T_{\alpha }:=\frac{tr{H}_{\alpha }}{n},$$

$$Ric(e_{i})=(n-1)c+\sum_{\alpha}\Big{[}nT_{\alpha}h^{\alpha}_{ii}-(h^{\alpha}_{ii})^{2}-\sum_{j\neq i}(h^{\alpha}_{ij})^{2}\Big{]}.$$

$$\sum _{k=q+1}^n\sum _{i=1}^q[2|h(e_i,e_k){|}^2-⟨h(e_i,e_i),h(e_k,e_k)⟩]$$

$$\displaystyle\sum_{\alpha}\Big{[}aq\sum_{k\neq i}\sum_{i=1}^{q}(h^{\alpha}_{ik})^{2}+(1-a)(n-q)\sum_{i\neq k}\sum_{k=q+1}^{n}(h^{\alpha}_{ik})^{2}\Big{]}$$

$$\sum _{k=q+1}^n\sum _{i=1}^q[2|h(e_i,e_k){|}^2-⟨h(e_i,e_i),h(e_k,e_k)⟩]-q(n-q)c$$

Full text

2010 MSC: 53C20; 53C24; 53C40

**Keywords: **Submanifolds, sphere theorems, Ricci curvature, stable currents

1 Introduction

It plays an important role in global differential geometry to investigate geometrical and topological structures of manifolds. After the pioneering rigidity theorem for closed minimal submanifolds in a sphere due to Simons [19], several striking rigidity results for minimal submanifolds were proved by Chern, do Carmo, Kobayashi, Lawson, Yau and others (see [1, 5, 9, 12, 30], etc.). In 1979, Ejiri [5] obtained the following rigidity theorem for compact minimal submanifold in a unit sphere.

Theorem A. Let $M$ be an $n(\geq 4)$-dimensional oriented compact simply connected minimal submanifold in $S^{n+p}$. If the Ricci curvature of $M$ satisfies $Ric_{M}\geq n-2,$ then $M$ is either the totally geodesic submanifold $S^{n}$, the Clifford torus S^{m}\big{(}\sqrt{\frac{1}{2}}\big{)}\times S^{m}\big{(}\sqrt{\frac{1}{2}}\big{)} in $S^{n+1}$ with $n=2m$, or $\mathbb{C}P^{2}(\frac{4}{3})$ in $S^{7}$. Here $\mathbb{C}P^{2}(\frac{4}{3})$ denotes the $2$-dimensional complex projective space minimally immersed into $S^{7}$ with constant holomorphic sectional curvature $\frac{4}{3}$.

The pinching constant above is the best possible in even dimensions. In 1987, Sun [21] studied the rigidity of compact oriented submanifolds of dimension $n\geq 4$ with parallel mean curvature and pinched Ricci curvature in a sphere, and partially generalized Ejiri’s rigidity theorem. In 1990s, Shen [16] and Li [13] extended Ejiri’s rigidity theorem to the case of 3-dimensional compact minimal submanifolds in a sphere. Moverover, Li [13] discussed $n(\geq 5)$-dimensional compact minimal submanifolds in $S^{n+p}$, and obtained that if $n$ is odd, and $Ric_{M}>n-2-\frac{1}{n-1}$, then $M$ is the totally geodesic. In 2011, Xu and Tian [28] obtained a refined version of the Ejiri rigidity theorem without the assumption that $M$ is simply connected.

Set $UM=\bigcup_{x\in M}U_{x}M$, where $U_{x}M=\{u\in T_{x}M:\|u\|=1\}$, and $T_{x}M$ is the tangent space at $x\in M$. In 1986, Gauchman [6] proved the following rigidity theorem for compact minimal submanifolds in a unit sphere.

Theorem B. Let $M$ be an n-dimensional compact minimal submanifold in $S^{n+p}$. If

[TABLE]

for all $u\in UM$, where $h$ is the second fundamental form of $M$, then either $||h(u,u)||^{2}=0$, and $M$ is totally geodesic, or $n$ is even, and $||h(u,u)||^{2}=C(n,p)$. Here

[TABLE]

When $n$ is even, the above rigidity theorem is optimal. In 1991, Leung [11] proved that if $M^{n}$ is an odd-dimensional compact minimal submanifold in $S^{n+p}$ with flat normal connection, and if $||h(u,u)||^{2}\leq\frac{n}{n-1}$ for all $u\in UM$, then $M$ is totally geodesic. Let M=S^{m}\big{(}\sqrt{\frac{m}{n}}\big{)}\times S^{m+1}\big{(}\sqrt{\frac{m+1}{n}}\big{)} be a Clifford minimal hypersurface in $S^{n+1}$ with $n=2m+1$, then $\frac{n-1}{n+1}\leq||h(u,u)||^{2}\leq\frac{n+1}{n-1}$. Based on these facts, Leung proposed the following conjecture.

Conjecture C. *Let $M^{n}$ be an odd-dimensional compact minimal submanifold in $S^{n+p}$.

(i) If $||h(u,u)||^{2}\leq\frac{n}{n-1}$ for all $u\in UM$, then $M$ is homeomorphic to a sphere.

(ii) If $||h(u,u)||^{2}<\frac{n+1}{n-1}$ for all $u\in UM$, then $M$ is homeomorphic to a sphere.*

In 2001, Hasanis and Vlachos [8] proved if $M^{n}(n\geq 5)$ is an odd-dimensional oriented compact minimal submanifold in $S^{n+p}$, and if $Ric_{M}>n-2-\frac{2}{n-1}$, then $M$ is homeomorphic to a sphere. Moreover, they showed that the condition $Ric_{M}>n-2-\frac{2}{n-1}$ is equivalent to $||h(u,u)||^{2}<\frac{n+1}{n-1}$ when $M$ is a compact minimal submanifold in $S^{n+p}$ with flat normal connection. In 2016, Qian and Tang [15] gave some counter examples to Conjecture C, which are focal submanifolds of isoparametric hypersurfaces in unit spheres with $g=4$.

Since 1970s, the sphere theorems for general submanifolds have been investigated by several authors [10, 17, 18, 22, 23, 24, 26, 29]. Recently, the authors [25] generalized the Ejiri rigidity theorem for minimal submanifolds in a sphere due to Ejiri [5] and Xu-Tian [28] to compact submanifolds with parallel mean curvature in space forms, and proved the following topological sphere theorem for compact submanifolds in a space form with nonnegative constant curvature.

** Theorem D.** Let $M$ be an $n(\geq 4)$-dimensional compact submanifold in $F^{n+p}(c)$ with $c\geq 0$, if $Ric_{M}>(n-2)(c+H^{2}),$ then $M$ is homeomorphic to a sphere.

Moreover, Xu-Leng-Gu [27] investigated the compact submanifolds with odd dimension in space forms, and proved the following topological sphere theorem.

Theorem E. Let $M$ be an $n(\geq 5)$-dimensional compact submanifold in $F^{n+p}(c)$ with $c\geq 0$. If $n$ is odd, and

[TABLE]

then M is homeomorphic to a sphere. Here $\epsilon_{n}=\frac{4}{n^{3}-2n^{2}-n-2}.$

In this paper, we study the compact submanifolds in space forms, and prove the following topological sphere theorems.

Theorem 1.1. Let $M$ be an $n(\geq 4)$-dimensional compact submanifold in $F^{n+p}(c)$ with $c\geq 0$. If the scalar curvature of M satisfies $R>n(n-2)(c+H^{2})$, and

[TABLE]

then $M$ is homeomorphic to a sphere. Here $\sigma_{n}=sgn(n-4)((-1)^{n}+3)$, and $sgn(\cdot)$ is the standard sign function.

Remark. If $Ric_{M}>(n-2)(c+H^{2})$, then $R>n(n-2)(c+H^{2})$. Therefore, Theorem 1.1 is a generalization of Theorem D.

Theorem 1.2. Let $M$ be an $n(\geq 4)$-dimensional compact oriented submanifold in $F^{n+p}(c)$ with $c\geq 0$. If $R>n(n-2)(c+H^{2})$, and

[TABLE]

then $M$ is homeomorphic to a sphere. Here $\tau_{n}=(-1)^{n}+3$.

The following examples show that the pinching conditions in Theorems 1.1 and 1.2 are the best possible.

Example 1.3. Let $S^{n+1}(\frac{1}{\sqrt{c+H^{2}}})$ be the totally umbilic sphere in $F^{n+p}(c)$. Here $H$ is a nonnegative constant.

(i) Let M=S^{m}\big{(}\frac{1}{\sqrt{2(c+H^{2})}}\big{)}\times S^{m}\big{(}\frac{1}{\sqrt{2(c+H^{2})}}\big{)} be a Clifford hypersurface in $S^{n+1}(\frac{1}{\sqrt{c+H^{2}}})$ with $n=2m$ and $c+H^{2}>0$. Then $M$ is a compact submanifold in $F^{n+p}(c)$ with constant mean curvature $H$ and constant Ricci curvature $Ric_{M}\equiv(n-2)(c+H^{2})$. Moreover, $R=n(n-2)(c+H^{2}).$

(ii) Let M=S^{m-1}\big{(}\sqrt{\frac{m-1}{n(c+H^{2})}}\big{)}\times S^{m+1}\big{(}\sqrt{\frac{m+1}{n(c+H^{2})}}\big{)} be a Clifford hypersurface in $S^{n+1}(\frac{1}{\sqrt{c+H^{2}}})$ with $n=2m$ and $c+H^{2}>0$. Then $M$ is a compact submanifold in $F^{n+p}(c)$ with constant mean curvature $H$, $Ric(e_{i})\equiv(n-2-\frac{4}{n-2})(c+H^{2})$ for $1\leq i\leq m-1$, and $Ric(e_{j})\equiv(n-2+\frac{4}{n+2})(c+H^{2})$ for $m\leq j\leq n$. Moreover, $R=n(n-2)(c+H^{2}).$

(iii) When $n=2m+1$, let M=S^{m}\big{(}\sqrt{\frac{m}{n(c+H^{2})}}\big{)}\times S^{m+1}\big{(}\sqrt{\frac{m+1}{n(c+H^{2})}}\big{)} be a Clifford hypersurface in $S^{n+1}(\frac{1}{\sqrt{c+H^{2}}})$ with $c+H^{2}>0$. Then $M$ is a compact submanifold in $F^{n+p}(c)$ with constant mean curvature $H$, $Ric(e_{i})\equiv(n-2-\frac{2}{n-1})(c+H^{2})$ for $1\leq i\leq m$, and $Ric(e_{j})\equiv(n-2+\frac{2}{n+1})(c+H^{2})$ for $m+1\leq j\leq n$. Moreover, $R=n(n-2)(c+H^{2}).$

2 Notation and lemmas

Throughout this paper, let $M$ be an $n$-dimensional compact Riemannian manifold isometrically immersed into an $(n+p)$-dimensional complete and simply connected space form $F^{n+p}(c)$. We shall make use of the following convention on the range of indices:

[TABLE]

We let {$e_{A}$} be local orthonormal frames in $F^{n+p}(c)$ such that, restricted to $M$, the $e_{i}$’s are tangent to M. Let {$\omega_{A}$} and {$\omega_{AB}$} be the dual frame field and the connection 1-forms of $F^{n+p}(c)$ respectively. Restricting these forms to M, we have

[TABLE]

where $h,\xi,R_{ijkl},$ and $R_{\alpha\beta kl}$ are the second fundamental form, the mean curvature vector, the curvature tensor and the normal curvature tensor of $M$, respectively. We define

[TABLE]

Denote by $Ric(u)$ the Ricci curvature of $M$ in direction of $u\in UM$. From the Gauss equation, we have

[TABLE]

Set $Ric_{\min}(x)=\min_{u\in U_{x}M}Ric(u)$. The scalar curvature $R$ of $M$ is given by

[TABLE]

Choose an orthonormal basis $\{e_{i}\}$ in $T_{x}M$ such that $e_{j_{0}}=u,\,\,span\{e_{j_{1}},\ldots,e_{j_{k}}\}=V_{x}^{k}$, where the indices $1\leq j_{0},j_{1},\ldots,j_{k}\leq n$ are distinct with each other. Using the notations of [7], we set

[TABLE]

Definition 2.1([7]). We call $Ric^{[s]}([e_{j_{0}},\ldots,e_{j_{n-1}}])$ the $s$-th weak Ricci curvature of $M$, respectively.

The nonexistence theorem for stable currents in a compact Riemannian manifold $M$ isometrically immersed into $F^{n+p}(c)$ is employed to eliminate the homology groups $H_{q}(M;\mathbb{Z})$ for $0<q<n$, which was initiated by Lawson-Simons [10] and extended by Xin [24].

Theorem 2.1. Let $M^{n}$ be a compact submanifold in $F^{n+p}(c)$ with $c\geq 0$. Assume that

[TABLE]

holds for any orthonormal basis $\{e_{i}\}$ of $T_{x}M$ at any point $x\in M$, where q is an integer satisfying $0<q<n$. Then there do not exist any stable q-currents. Moreover, $H_{q}(M;\mathbb{Z})=H_{n-q}(M;\mathbb{Z})=0,$ and $\pi_{1}(M)=0$ when $q=1$. Here $H_{i}(M;\mathbb{Z})$ is the $i$-th homology group of M with integer coefficients.

To prove the sphere theorems for submanifolds, we need to eliminate the fundamental group $\pi_{1}(M)$ under the s-th weak Ricci curvature and the scalar curvature pinching condition, and get the following lemma.

Lemma 2.2. *Let $M$ be an $n(\geq 4)$-dimensional compact submanifold in $F^{n+p}(c)$ with $c\geq 0$. Assume that one of the following conditions holds:

$(i)$ $R+2Ric_{\min}>n(n-1)(c+H^{2}),$

$(ii)$ $3R-2Ric^{[n-1]}_{\max}>n(n-1)(c+H^{2}),$

then $H_{1}(M;\mathbb{Z})=H_{n-1}(M;\mathbb{Z})=0,$ and $\pi_{1}(M)=0$.*

Proof. From (2.4), we have

[TABLE]

This together with (2.3) implies that

[TABLE]

Since

[TABLE]

we have

[TABLE]

On the other hand, we also get from (2.6) that

[TABLE]

Therefore, if

[TABLE]

then

[TABLE]

Hence, it follows from Theorem 2.1 that $H_{1}(M;\mathbb{Z})=H_{n-1}(M;\mathbb{Z})=0,$ and $\pi_{1}(M)=0$. This proves Lemma 2.2.∎

If

[TABLE]

then we can directly obtain

[TABLE]

Therefore, we have the following corollary from Lemma 2.2.

Corollary 2.3 ([25]). Let $M$ be an $n(\geq 4)$-dimensional compact submanifold in $F^{n+p}(c)$ with $c\geq 0$. If the Ricci curvature of M satisfies

[TABLE]

then $H_{1}(M;\mathbb{Z})=H_{n-1}(M;\mathbb{Z})=0,$ and $\pi_{1}(M)=0$.

3 Proof of the Main Theorem

In this section, we will give the proof of our Main Theorem. we first need to eliminate the homology groups $H_{q}(M;\mathbb{Z})$ for $1<q<n-1$ under the s-th weak Ricci curvature and scalar curvature pinching condition.

Lemma 3.1. *Let $M$ be an $n(\geq 4)$-dimensional compact submanifold in $F^{n+p}(c)$ with $c\geq 0$. Assume that one of the following conditions holds:

$(i)$ $(n-q)(q-1)R+(n-2q)Ric^{[q]}_{\min}>n^{2}(n-q-1)(q-1)(c+H^{2}),$

$(ii)$ $q(n-q-1)R-(n-2q)Ric^{[n-q]}_{\max}>n^{2}(n-q-1)(q-1)(c+H^{2}),$

where q is some integer in $[2,\frac{n}{2}]$. Then $H_{q}(M;Z)=H_{n-q}(M;Z)=0$.*

Proof. Assume that $2\leq q\leq\frac{n}{2}$. Setting

[TABLE]

we have $\sum_{\alpha}T_{\alpha}^{2}=H^{2}.$ Then we get from (2.3)

[TABLE]

Then we get

[TABLE]

for some $a\in(0,1)$. Then it follows from (3.1) and the Cauchy-Schwarz inequality that

[TABLE]

Since $q\geq 2$, and $n-q\geq 2$, we get

[TABLE]

Let $a=\frac{n-q-1}{n-2}$, then we have $a(q-1)=(1-a)(n-q-1).$ This together with (3.3), (3.4) implies

[TABLE]

It follows from the condition (i) and (3.5) that

[TABLE]

We also get from (3.5) and condition (ii) that

[TABLE]

Then the assertion follows from (3.6), (3.7) and Theorem 2.1.∎

** Proof of Theorem 1.1.** If $R>n(n-2)(c+H^{2})$, and $Ric_{M}\geq\frac{n}{2}(c+H^{2})$, then

[TABLE]

It follows from Theorem 2.2 that $H_{1}(M;\mathbb{Z})=H_{n-1}(M;\mathbb{Z})=0,$ and $\pi_{1}(M)=0$.

For $q\in[2,\frac{n}{2}]$, we consider the following three cases:

Case I. $q=\frac{n}{2}$. If $R>n(n-2)(c+H^{2})$, then $(n-q)(q-1)R+(n-2q)Ric^{[q]}_{\min}>n^{2}(n-q-1)(q-1)(c+H^{2})$. Hence we get from Theorem 3.1 that $H_{\frac{n}{2}}(M;Z)=0$.

Case II. $q<\frac{n}{2}$. If $R>n(n-2)(c+H^{2})$ and $Ric^{[q]}_{\min}\geq n(q-1)(c+H^{2})$, then $(n-q)(q-1)R+(n-2q)Ric^{[q]}_{\min}>n^{2}(n-q-1)(q-1)(c+H^{2}),$ for some $q\in[2,\frac{n}{2})$. Therefore, we get from Theorem 3.1 that $H_{q}(M;Z)=H_{n-q}(M;Z)=0$ for some $q\in[2,\frac{n}{2})$.

Since $Ric^{[l]}_{\min}\geq lRic_{\min}$, we have if $n=even$, and

[TABLE]

then $Ric^{[q]}_{\min}\geq n(q-1)(c+H^{2})$ for all $q\in[2,\frac{n}{2})$. If $n=odd$, and

[TABLE]

then $Ric^{[q]}_{\min}\geq n(q-1)(c+H^{2})$ for all $q\in[2,\frac{n}{2})$.

Moreover, we have

[TABLE]

for $n\geq 4$,

[TABLE]

for $n\geq 5$, and

[TABLE]

for $n\geq 6$. Hence it follows from the assumption that $H_{q}(M;Z)=H_{n-q}(M;Z)=0$ for all $q\in[1,\frac{n}{2}]$ and $\pi_{1}(M)=0$. It follows from $H_{i}(M;Z)=0$ for $i=1,2,\cdots,n-1$ that $M$ is a homology sphere. Since $M$ is simply connected, $M$ is a homotopy sphere. This together with the generalized Poincaré conjecture implies that $M$ is a topological sphere. This completes the proof of Theorem 1.1. ∎

Moreover, when $M$ is oriented, we get the following sphere theorem.

Theorem 3.2. Let $M$ be an $n(\geq 4)$-dimensional oriented compact submanifold in $F^{n+p}(c)$ with $c\geq 0$. If $R>n(n-2)(c+H^{2})$, and

[TABLE]

then $M$ is homeomorphic to a sphere. Here $\sigma_{n}=sgn(n-4)((-1)^{n}+3)$, and $sgn(\cdot)$ is the standard sign function.

Proof. We first prove that $Ric_{M}>0$. We consider the following three cases:

$(i)$ If $n=4$, and if there exists a point $x_{0}\in M$ such that $Ric_{\min}(x_{0})\leq 0$, then without loss of generality, we assume that $Ric(e_{1})\leq 0$ at point $x_{0}$. Hence it follows from the assumption that $Ric(e_{i})\geq 2(n-2)(c+H^{2})$ for $i=2,\cdots,n$. Then we get

[TABLE]

On the other hand, we know from the Gauss equation that $R\leq n(n-1)(c+H^{2})$. Therefore $R(x_{0})=n(n-1)(c+H^{2})$, and $S(x_{0})=nH^{2}$, i.e., $x_{0}$ is a totally umbilical point. Then we get from the Gauss equation that $Ric_{M}\equiv(n-1)(c+H^{2})$ at that point, which contradicts with the assumption. Therefore, $Ric_{M}>0$.

$(ii)$ If $n$ is even, $n\geq 6$, and if there exists a point $x_{1}\in M$ such that $Ric_{\min}(x_{1})\leq 0$, then without loss of generality, we assume that $Ric(e_{1})\leq 0$ at point $x_{1}$. Hence it follows from the assumption that $Ric(e_{i})\geq 2(n-2-\frac{4}{n-2})(c+H^{2})$ for $i=2,\cdots,n$. Then we get

[TABLE]

and the equality holds only if $n=6$. This together with the Gauss equation implies that $R(x_{1})=n(n-1)(c+H^{2})$, and $n=6$. But a similar argument as in Case (i) shows that $Ric_{M}\equiv(n-1)(c+H^{2})$ at point $x_{1}$, and it contradicts with the assumption. Therefore, $Ric_{M}>0$.

(iii) If $n$ is odd, and if there exists a point $x_{2}\in M$ such that $Ric_{\min}(x_{2})\leq 0$, then without loss of generality, we assume that $Ric(e_{1})\leq 0$ at point $x_{2}$. Hence it follows from the assumption that $Ric(e_{i})\geq 2(n-2-\frac{2}{n-1})(c+H^{2})$ for $i=2,\cdots,n$. Then we get

[TABLE]

and the equality holds only if $n=5$. This together with the Gauss equation implies that $R(x_{2})=n(n-1)(c+H^{2})$, and $n=5$. But a similar argument as in Case (i) shows that $Ric_{M}\equiv(n-1)(c+H^{2})$ at point $x_{2}$, and it contradicts with the assumption. Therefore, $Ric_{M}>0$.

On the other hand, it follows from the proof of Theorem 1.1 and the assumption that $H_{q}(M;Z)=H_{n-q}(M;Z)=0$ for all $q\in[2,\frac{n}{2}]$. This together with the universal coefficient theorem implies that $H^{n-1}(M;\mathbb{Z})$ has no torsion, and hence neither does $H_{1}(M;\mathbb{Z})$ by the Poincar$\acute{{\rm e}}$ duality. We also get from $Ric_{M}>0$ and the Bonnet-Myers theorem that the fundamental group $\pi_{1}(M)$ is finite. Then we have $H_{1}(M;\mathbb{Z})=0$. Hence, $H_{n-1}(M;\mathbb{Z})=0$. Denote by $\widetilde{M}$ the universal Riemannian covering of $M$. We may consider $\widetilde{M}$ be a Riemannian submanifold of $F^{n+p}(c)$, and hence $\widetilde{M}$ is a homology sphere. Since $\widetilde{M}$ is simply connected, it is a topological sphere, which together with a result of Sjerve [20] implies that $M$ is simply connected. Then $M$ is a homotopy sphere. This together with the generalized Poincaré conjecture implies that $M$ is a topological sphere. ∎

It’s easy to get that if $Ric^{[2]}_{\min}>2(n-2)(c+H^{2})$, then $R>n(n-2)(c+H^{2})$. Hence we have the following corollary.

Corollary 3.3. Let $M$ be an $n(\geq 4)$-dimensional oriented compact submanifold in $F^{n+p}(c)$ with $c\geq 0$. If $Ric^{[2]}_{\min}>2(n-2)(c+H^{2})$, then $M$ is homeomorphic to a sphere.

For $n=4$, we also get the following differentiable theorem.

Theorem 3.4. Let $M$ be a $4$-dimensional compact oriented submanifold in $F^{n+p}(c)$ with $c\geq 0$. If $R>8(c+H^{2})$, and $Ric_{M}>0$, then $M$ is diffeomorphic to a sphere.

Proof. Since $R>8(c+H^{2})$ is equal to $S<4c+8H^{2}$, it follows from a theorem due to Xu-Zhao [29] that $M$ is diffeomorphic to a spherical space form. Moreover, we know from Lemma 3.1 that $H_{2}(M;\mathbb{Z})=0$. Using the same argument as in the proof of Theorem 3.2 implies that $H_{1}(M;Z)=H_{n-1}(M;Z)=0$, and $M$ is simply connected. Therefore, $M$ is diffeomorphic to a sphere.∎

Theorem 3.5. Let $M$ be an $n(\geq 5)$-dimensional oriented compact submanifold in $F^{n+p}(c)$ with $c\geq 0$. If $n$ is odd, and

[TABLE]

then M is homeomorphic to a sphere. Here $\epsilon_{n}=\frac{4}{n^{3}-2n^{2}-n-2}.$

Proof. Since $R\geq\frac{n}{l}Ric^{[l]}_{\min}$, we get from Lemma 3.1 that if

[TABLE]

where $q$ is some integer in $[2,\frac{n}{2}]$, then $H_{q}(M;Z)=H_{n-q}(M;Z)=0$. On the other hand, we know that $\frac{Ric^{[l]}_{\min}}{l}\geq\frac{Ric^{[2]}_{\min}}{2}$ for $l\geq 2$, and the function

[TABLE]

is strictly monotone decreasing for $2\leq s\leq\frac{n}{2}$. Hence, if

[TABLE]

for some integer $s\in[2,\frac{n}{2}]$, then $H_{q}(M;Z)=H_{n-q}(M;Z)=0$ for all $q\in[2,s].$ Let $s=\frac{n-1}{2}$. Then we get from the assumption that $H_{q}(M;Z)=H_{n-q}(M;Z)=0$ for $q\in[2,\frac{n}{2}]$. Moreover, it follows from the proof of Theorem 3.2 that $Ric_{M}>0$. Using a same argument as in the proof of Theorem 3.2, we get that $H_{1}(M;Z)=H_{n-1}(M;Z)=0$, and $M$ is simply connected. Then $M$ is a homotopy sphere. This together with the generalized Poincaré conjecture implies that $M$ is a topological sphere. This proves Theorem 3.5.∎

Theorem 3.6. *Let $M$ be an $n(\geq 5)$-dimensional compact submanifold in $F^{n+p}(c)$ with $c\geq 0$. Assume that the scalar curvature of M satisfies $R>n(n-2)(c+H^{2})$. Then we have

$(i)$ if $n=5,6,8$, and $\frac{Ric^{[s]}_{\max}}{s}\leq[n-2+\frac{n-4}{2(n-1)}](c+H^{2}),$ then $M$ is homeomorphic to a sphere.

$(ii)$ if $n=7$, $n\geq 9$, and $\frac{Ric^{[s]}_{\max}}{s}\leq(n-2+\frac{2\tau_{n}}{2n+\tau_{n}})(c+H^{2})$, then $M$ is homeomorphic to a sphere.

Here $s$ is some integer in $[1,\frac{n+2}{2}]$, and $\tau_{n}=(-1)^{n}+3$.*

Remark. It follows from (ii) of Example 1.3 that $\frac{Ric^{[s]}_{\max}}{s}=(n-2+\frac{4}{n+2})(c+H^{2})$, for the submanifold S^{m-1}\big{(}\sqrt{\frac{m-1}{n(c+H^{2})}}\big{)}\times S^{m+1}\big{(}\sqrt{\frac{m+1}{n(c+H^{2})}}\big{)} with $n=2m$. It follows from (iii) of Example 1.3 that $\frac{Ric^{[s]}_{\max}}{s}=(n-2+\frac{2}{n+1})(c+H^{2}),$ for the submanifold S^{m}\big{(}\sqrt{\frac{m}{n(c+H^{2})}}\big{)}\times S^{m+1}\big{(}\sqrt{\frac{m+1}{n(c+H^{2})}}\big{)} with $n=2m+1$.

** Proof.** If $R>n(n-2)(c+H^{2})$, and $Ric^{[n-1]}_{\max}\leq(n^{2}-2.5n)(c+H^{2})$, then

[TABLE]

It follows from Theorem 2.2 that $H_{1}(M;\mathbb{Z})=H_{n-1}(M;\mathbb{Z})=0,$ and $\pi_{1}(M)=0$.

For $q\in[2,\frac{n}{2}]$, we consider the following three cases:

Case I. $q=\frac{n}{2}$. If $R>n(n-2)(c+H^{2})$, then $q(n-q-1)R-(n-2q)Ric^{[n-q]}_{\max}>n^{2}(n-q-1)(q-1)(c+H^{2}).$ Hence we get from Lemma 3.1 that $H_{\frac{n}{2}}(M;Z)=0$.

Case II. $q<\frac{n}{2}$. If $R>n(n-2)(c+H^{2})$ and $Ric^{[n-q]}_{\max}\leq n(n-q-1)(c+H^{2})$, then $q(n-q-1)R-(n-2q)Ric^{[n-q]}_{\max}>n^{2}(n-q-1)(q-1)(c+H^{2}),$ for some $q\in[2,\frac{n}{2})$. Therefore, we get from Lemma 3.1 that $H_{q}(M;Z)=H_{n-q}(M;Z)=0$ for some $q\in[2,\frac{n}{2})$.

Since $\frac{Ric^{[l_{1}]}_{\max}}{l_{1}}\leq\frac{Ric^{[l_{2}]}_{\max}}{l_{2}}$ for $l_{1}\geq l_{2}$, we have if $n=even$, and

[TABLE]

then $Ric^{[n-q]}_{\max}\leq n(n-q-1)(c+H^{2})$ for all $q\in[2,\frac{n}{2})$. If $n=odd$, and

[TABLE]

then $Ric^{[n-q]}_{\max}\leq n(n-q-1)(c+H^{2})$ for all $q\in[2,\frac{n}{2})$.

Moreover, we have

[TABLE]

for $n=5$, and

[TABLE]

for $n\geq 7$;

[TABLE]

for $n=6,8$, and

[TABLE]

for $n\geq 10$. Hence it follows from the assumption that $H_{q}(M;Z)=H_{n-q}(M;Z)=0$ for all $q\in[1,\frac{n}{2}]$ and $\pi_{1}(M)=0$. It follows from $H_{i}(M;Z)=0$ for $i=1,2,\cdots,n-1$ that $M$ is a homology sphere. Since $M$ is simply connected, $M$ is a homotopy sphere. This together with the generalized Poincaré conjecture implies that $M$ is a topological sphere. This completes the proof of Theorem 3.6.∎

Theorem 3.7. *Let $M$ be an $n$-dimensional compact oriented submanifold in $F^{n+p}(c)$ with $c\geq 0$. Assume that the scalar curvature of M satisfies $R>n(n-2)(c+H^{2})$. We have that

$(i)$ if $n=5$, and $\frac{Ric^{[s]}_{\max}}{s}\leq(n-2+\frac{2}{n+1})(c+H^{2}),$ then $M$ is homeomorphic to a sphere.

$(ii)$ if $n=4,6,8$, and $\frac{Ric^{[s]}_{\max}}{s}\leq(n-2+\frac{4}{n+2})(c+H^{2})$, then $M$ is homeomorphic to a sphere.

Here $s$ is some integer in $[1,\frac{n+2}{2}]$.*

Proof. It follows from the assumption and the proof of Theorem 3.6 that $H_{2}(M;Z)=H_{3}(M;Z)=0$ for $n=5$, and

[TABLE]

for $n=4$, $6$, $8$, where $q\in[2,\frac{n}{2}]$. Moreover, we have

[TABLE]

Then a similar argument as in the proof of Theorem 3.2 shows that

[TABLE]

and $M$ is simply connected. Therefore, $M$ is homeomorphic to a sphere. This proves Theorem 3.7.∎

Proof of Theorem 1.2. Choosing $s=1$, we get the conclusion from Theorems 3.6 and 3.7.∎

Motivated by Theorem 1.1, Example 1.3, the convergence results of Brendle and Schoen for Ricci flow [2, 3, 4], and the differentiable sphere theorems for submanifolds with positive Ricci curvature proved in [25, 28], we propose the following conjecture.

Conjecture 3.8. Let $M$ be an $n(\geq 5)$-dimensional compact submanifold in the space form $F^{n+p}(c)$ with $c+H^{2}>0$. If $n$ is odd, and

[TABLE]

then $M$ is diffeomorphic to $S^{n}$.

To verify Conjecture 3.8, we hope to prove the following conjecture on the normalized Ricci flow.

Conjecture 3.9. Let $(M,g_{0})$ be an $n(\geq 5)$-dimensional compact submanifold in an $(n+p)$-dimensional space form $F^{n+p}(c)$ with $c+H^{2}>0$. If $n$ is odd, and the Ricci curvature of $M$ satisfies

[TABLE]

then the normalized Ricci flow with initial metric $g_{0}$

[TABLE]

exists for all time and converges to a constant curvature metric as $t\rightarrow\infty$. Moreover, $M$ is diffeomorphic to a spherical space form.