Let $M^{n+1}(c)$ be an $(n+1)$-dimensional connected Riemannian space form with constant sectional curvature $c$. According to $c>0$, $c=0$ or $c<0$, it is called sphere space, Euclidean space or hyperbolic space, respectively, and it denoted by $S^{n+1}(c)$, $R^{n+1}$ or $H^{n+1}(c)$. Let $M^n$ be an $n-$dimensional hypersurface in $M^{n+1}(c)$. As it is well known there many rigidity results with constant mean curvature, constant scalar curvature or constant $k$-th mean curvature in $M^{n+1}(c)$, for example, see [1-6] in $S^{n+1}(c)$ or $R^{n+1}$ and [7-9] in hyperbolic space $H^{n+1}(c)$.

In [9], Wu proved the following theorem.

Theorem 1.1  Let $M^n$ $(n\geq 3)$ be a complete hypersurface in $H^{n+1}(-1)$ with constant mean curvature $H$ $(|H|>1)$ and two distinct principal curvatures with multiplicities $n-1$, 1. Set

$ \begin{eqnarray} S_{\pm}=-n+\frac{n^3H^2}{2(n-1)}\mp\frac{n(n-2)}{2(n-1)}\sqrt{n^2H^4-4(n-1)H^2}. \end{eqnarray} $

(1.1)

If the square length of the second fundamental form satisfies $S\leq S_{+}$ or $S\geq S_{-}$, then $S=S_{+}$ or $S=S_{-}$, and $M^n$ is isometric to hyperbolic cylinder $S^{n-1}(\lambda_+^2-1)\times H^1(\frac{1}{\lambda_+^2}-1)$ or $H^{n-1}(\lambda_-^2-1)\times S^1(\frac{1}{\lambda_-^2}-1)$, here

$ \begin{eqnarray} \lambda_{\pm}=\frac{n|H|\pm\sqrt{n^2H^2-4(n-1)}}{2(n-1)}. \end{eqnarray} $

(1.2)

In this note, we shall also investigate $n$-dimensional hypersurfaces with constant curvature $H$ $(|H|>1)$ in $H^{n+1}(-1)$ and obtain the following result:

Theorem 1.2  Let $M^n$ $(n\geq 3)$ be a complete hypersurface in $H^{n+1}(-1)$ with constant mean curvature $H$ $(|H|>1)$ and two distinct principal curvatures with multiplicities $n-1$, 1. If the square length of the second fundamental form satisfies

$ \begin{eqnarray}\label{hfeng13} S_{+}\leq S \leq S_{-}, \end{eqnarray} $

(1.3)

then $S=S_{+}$ or $S=S_{-}$, and $M^n$ is isometric to hyperbolic cylinder $S^{n-1}(\lambda_+^2-1)\times H^1(\frac{1}{\lambda_+^2}-1)$ or $H^{n-1}(\lambda_-^2-1)\times S^1(\frac{1}{\lambda_-^2}-1)$, here

$ \begin{eqnarray} \lambda_{\pm}=\frac{n|H|\pm\sqrt{n^2H^2-4(n-1)}}{2(n-1)}. \end{eqnarray} $

(1.4)

Let $M^{n+1}(c)$ be an $(n+1)$-dimensional connected Riemannian space form with constant sectional curvature $c$. Let $M^n$ be an $n$-dimensional complete connected and oriented hypersurface in $M^{n+1}(c)$. We choose a local orthonormal frame $e_1, \cdots, e_n, e_{n+1}$ in $M^{n+1}(c)$ such that $e_1, \cdots, e_n$ are tangent to $M^n$. Let $\omega_1, \cdots, \omega_{n+1}$ be the dual coframe. We use the following convention on the range of indices:

$ \begin{eqnarray*} 1\leq A, B, \cdots\leq n+1;\quad 1\leq i, j, \cdots \leq n. \end{eqnarray*} $

The structure equations of $M^{n+1}(c)$ are given by

$ \begin{eqnarray} &&d\omega_A=\sum\limits_B \omega_{AB}\wedge \omega_B, \quad \omega_{AB}+\omega_{BA}=0, \end{eqnarray} $

(2.1)

$ \begin{eqnarray} && d\omega_{AB}=\sum\limits_C \omega_{AC}\wedge \omega_{CB}+\Omega_{AB}, \end{eqnarray} $

(2.2)

$ \begin{eqnarray} && \Omega_{AB}=-\frac{1}{2}\sum\limits_{CD}K_{ABCD}\omega_C\wedge\omega_{D}, \end{eqnarray} $

(2.3)

$ \begin{eqnarray} &&K_{ABCD}=c(\delta_{AC}\delta_{BD}-\delta_{AD}\delta_{BC}). \end{eqnarray} $

(2.4)

Restricting to $M^n$ such that

$ \begin{eqnarray} \omega_{n+1}=0, \quad \omega_{n+1i}=\sum\limits_{j}h_{ij}\omega_j, \quad h_{ij}=h_{ji}. \end{eqnarray} $

(2.5)

The structure equations of $M^n$ are

$ \begin{eqnarray} && d\omega_i=\sum\limits_j \omega_{ij}\wedge \omega_j, \quad \omega_{ij}+\omega_{ji}=0, \end{eqnarray} $

(2.6)

$ \begin{eqnarray} && d\omega_{ij}=\sum\limits_k \omega_{ik}\wedge \omega_{kj}-\frac{1}{2}\sum\limits_{k, l}R_{ijkl}\omega_k\wedge\omega_l, \end{eqnarray} $

(2.7)

$ \begin{eqnarray} && R_{ijkl}=c(\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk})+(h_{ik}h_{jl}-h_{il}h_{jk}), \end{eqnarray} $

(2.8)

$ \begin{eqnarray} \label{hfeng1} &&n(n-1)(r-c)=n^2H^2-S, \end{eqnarray} $

(2.9)

where $n(n-1)r$ is the scalar curvature, $H$ is the mean curvature and $S$ is the squared of the second fundamental form of $M^n$.

Let $M^n$ be an $n$ $(n\geq3)$ dimensional complete connected and oriented hypersurface in $M^{n+1}(c)$ with constant mean curvature and with two distinct principal curvatures, one of which is simple. Without loss of generality, we may assume

$ \begin{eqnarray} \lambda_1=\cdots=\lambda_{n_1}=\lambda, \quad \lambda_n=\mu, \end{eqnarray} $

(2.10)

where $\lambda_i$ for $i=1, \cdots, n$ are the principal curvatures of $M^n$. We have

$ \begin{eqnarray}\label{hfeng2} (n-1)\lambda+\mu=nH, \quad S=(n-1)\lambda^2+\mu^2. \end{eqnarray} $

(2.11)

From (2.9) and (2.11), we have, for $c=-1$, that

$ \begin{eqnarray}\label{hfeng10} \lambda\mu=(n-1)(r+1)-(n-2)H^2+(n-2)\sqrt{H^4-(r+1)H^2}, \end{eqnarray} $

(2.12)

on $M^n$, or

$ \begin{eqnarray}\label{hfeng11} \lambda\mu=(n-1)(r+1)-(n-2)H^2-(n-2)\sqrt{H^4-(r+1)H^2}, \end{eqnarray} $

(2.13)

on $M^n$.

Let $M^n$ be a connected hypersurface in $H^{n+1}(-1)$ with constant mean curvature and two distinct principal curvatures $\lambda, \mu$ with multiplicities $n-1, 1$. Since the multiplicites are constant, it is easy to know that their eigenspaces are completely integrable. Let $s$ be the parameter of arc length of the goedesics corresponding to $\mu$, and we may put $\omega_n=ds$. Then $\lambda$ and $\mu$ are locally functions of $s$. Let $\omega=|\lambda-H|^{-\frac{1}{n}}$. In [9], B.Y. Wu got the following equations:

$ \begin{eqnarray}\label{hfeng3} \frac{d^2\omega}{ds^2}+\omega(-1+H^2+(2-n)H\omega^{-n}+(1-n)\omega^{-2n})=0, \end{eqnarray} $

(3.1)

for $\lambda>H$ or

$ \begin{eqnarray}\label{hfeng4} \frac{d^2\omega}{ds^2}+\omega(-1+H^2+( n-2)H\omega^{-n}+(1-n)\omega^{-2n})=0, \end{eqnarray} $

(3.2)

for $\lambda<H.$ Integrating (3.1) and (3.2), we get

$ \begin{eqnarray}\label{hfeng5}(\frac{d\omega}{ds})^2+(-1+H^2)\omega^2+2H\omega^{2-n}+\omega^{2-2n}=C \end{eqnarray} $

(3.3)

for $\lambda>H$ or

$ \begin{eqnarray}\label{hfeng6}(\frac{d\omega}{ds})^2+(-1+H^2)\omega^2-2H\omega^{2-n}+\omega^{2-2n}=C \end{eqnarray} $

(3.4)

for $\lambda<H$.

We first obtain the following propositions:

Proposition 3.1  Let $M^n$ be an $n$ $(n\geq 3)$-dimensional complete connected hypersurface in $H^{n+1}(-1)$ with constant mean curvature $H$ $(|H|>1)$ and two distinct principal curvatures $\lambda$ and $\mu$ with multiplicities $(n-1)$ and 1, respectively. If $\lambda\mu-1\geq 0$, then $M^n$ is isometric to hyperbolic cylinder $S^{n-1}(\lambda_+^2-1)\times H^1(\frac{1}{\lambda_+^2}-1)$ or $H^{n-1}(\lambda_-^2-1)\times S^1(\frac{1}{\lambda_-^2}-1)$.

Proof  Let $\lambda$ and $\mu$ be the two distinct principal curvatures of $M^n$ with multiplicities $(n-1)$ and 1 respectively. Then, from $nH=(n-1)\lambda+\mu$ and $\omega=|\lambda-H|^{-\frac{1}{n}}$, we have the following :

$ \begin{eqnarray}\lambda\mu-1=-1+H^2+(2-n)H\omega^{-n}+(1-n)\omega^{-2n}, \end{eqnarray} $

(3.5)

for $\lambda>H$ or

$ \begin{eqnarray}\lambda\mu-1=-1+H^2+( n-2)H\omega^{-n}+(1-n)\omega^{-2n}, \end{eqnarray} $

(3.6)

for $\lambda<H$.

Then, if $\lambda\mu-1\geq 0$, we obtain

$ \begin{eqnarray}-1+H^2+(2-n)H\omega^{-n}+(1-n)\omega^{-2n}\geq 0, \end{eqnarray} $

(3.7)

for $\lambda>H$ or

$ \begin{eqnarray}-1+H^2+( n-2)H\omega^{-n}+(1-n)\omega^{-2n}\geq 0, \end{eqnarray} $

(3.8)

for $\lambda<H$. From (3.1) and (3.2), we have $\frac{d^2\omega}{ds^2}\leq 0$. Thus $\frac{d\omega}{ds}$ is a monotonic function of $s\in (-\infty, +\infty)$. Therefore, $\omega(s)$ must monotonic when $s$ tends to infinity. From (3.3) and (3.4), we know that the positive function $\omega(s)$ is bounded from above. Since $\omega(s)$ is bounded and is monotonic when $s$ tends infinity, we find that both $\lim_{s\rightarrow +\infty}\omega(s)$ and $\lim_{s\rightarrow -\infty}\omega(s)$ exist and then we have

$ \begin{eqnarray} \lim\limits_{s\rightarrow+\infty}\frac{d\omega(s)}{ds}=\lim\limits_{s\rightarrow-\infty}\frac{d\omega(s)}{ds}=0. \end{eqnarray} $

(3.9)

By the monotonicity of $\frac{d\omega(s)}{ds}$, we see that $\frac{d\omega(s)}{ds}\equiv 0$ and $\omega(s)$ is constant. Then we have $\lambda$ and $\mu$ are constant, that is, $M^n$ is isoparametric. According to Cartan [10], we know that $M^n$ is isometric to the hyperbolic cylinder $S^{n-1}(\lambda_+^2-1)\times H^1(\frac{1}{\lambda_+^2}-1)$ or $H^{n-1}(\lambda_-^2-1)\times S^1(\frac{1}{\lambda_-^2}-1)$.

By using the same method in Proposition 3.1, we can obtain the following proposition:

Proposition 3.2  Let $M^n$ be an $n$ $(n\geq 3)$-dimensional complete connected hypersurface in $H^{n+1}(-1)$ with constant mean curvature $H$ $(|H|>1)$ and two distinct principal curvatures $\lambda$ and $\mu$ with multiplicities $(n-1)$ and 1, respectively. If $\lambda\mu-1\leq0$, then $M^n$ is isometric to hyperbolic cylinder $S^{n-1}(\lambda_+^2-1)\times H^1(\frac{1}{\lambda_+^2}-1)$ or $H^{n-1}(\lambda_-^2-1)\times S^1(\frac{1}{\lambda_-^2}-1)$.

Proof of Theorem 1.2  Since $M^n$ is oriented and the mean curvature $H$ is constant, we can choose an orientation for $M^n$ such that $H>0$. From (2.9), we know that the inequality $S_{+}\leq S \leq S_{-}$ is equivalent to

$ \begin{eqnarray}\label{hfeng9} &&\frac{1}{2(n-1)^2}[n^2H^2-n\sqrt{n^2H^4-4(n-1)H^2}-2(n-1)]\\ &\leq& \frac{n(r+1)-2}{n-2}\leq\frac{1}{2(n-1)^2}[n^2H^2+n\sqrt{n^2H^4-4(n-1)H^2}-2(n-1)], \nonumber \end{eqnarray} $

(3.10)

where $n(n-1)r$ is the scalar curvature of $M^n$.

We define the function

$ \begin{eqnarray}\label{hfeng50} f(x)=(n-1)^2x^2-[n^2H^2-2(n-1)]x+1. \end{eqnarray} $

(3.11)

Since $f(0)=1$ and $|H|=H>1$, we know that function (3.11) has two positive real roots.

$ \begin{eqnarray} x_{1, 2}=\frac{1}{2(n-1)^2}[n^2H^2\pm n\sqrt{n^2H^4-4(n-1)H^2}-2(n-1)]. \end{eqnarray} $

(3.12)

It can be easily checked that $x_1\leq x_2$ and if $x_1\leq x\leq x_2$, then $f(x)\leq 0$.

Now we set $x= \frac{n(r+1)-2}{n-2}$, from (3.10), we have

$ \begin{eqnarray}\label{hfeng12} f(\frac{n(r+1)-2}{n-2})\leq 0.\end{eqnarray} $

(3.13)

If there exists a point $p$ on $M^n$ such that (2.12) and (2.13) hold at $p$, we have $H^2=r+1$ at $p$, from (2.9), we have $S=nH^2$ at $p$, that is, $p$ is a umbilical point on $M^n$, that is contradiction to $M^n$ has no umbilical points. Therefore, we only consider two cases:

Case Ⅰ  If (2.12) holds on $M^n$, we shall prove that $-1+\lambda\mu\geq 0$ on $M^n$. We consider three subcases:

(a) If $-1+(n-1)(r+1)-(n-2)H^2\geq0$, then from (2.12), we have $-1+\lambda\mu\geq0$ on $M^n$.

(b) If $-1+(n-1)(r+1)-(n-2)H^2<0$, suppose $-1+\lambda\mu<0$ on $M^n$, from (2.12), we have

$ \begin{eqnarray} (n-2)\sqrt{H^4-(r+1)H^2}<-[-1+(n-1)(r+1)-(n-2)H^2]. \end{eqnarray} $

(3.14)

Therefore, we have

$ \begin{eqnarray} \frac{(n-2)^2}{n^2}\{(n-1)^2[\frac{n(r+1)-2}{n-2}]^2-[n^2H^2-2(n-1)]\frac{n(r+1)-2}{n-2}+1\}>0, \end{eqnarray} $

(3.15)

that is, $f(\frac{n(r+1)-2}{n-2})>0$. This is a contradiction to (3.13), we deduce that $-1+\lambda\mu\geq0$ on $M^n$.

(c) If $-1+(n-1)(r+1)-(n-2)H^2\geq0$ at a point $p$ of $M^n$ and $-1+(n-1)(r+1)-(n-2)H^2<0$ at other points of $M^n$, in this case, from (a) and (b), we have $-1+\lambda\mu\geq 0$ on $M^n$.

Therefore, we know that if (2.12) holds on $M^n$, then $-1+\lambda\mu\geq 0$ on $M^n$. By Proposition 3.1, we obtain that $M^n$ is isometric to hyperbolic cylinder $S^{n-1}(\lambda_+^2-1)\times H^1(\frac{1}{\lambda_+^2}-1)$ or $H^{n-1}(\lambda_-^2-1)\times S^1(\frac{1}{\lambda_-^2}-1)$, and $\lambda_{\pm}=\frac{n|H|\pm\sqrt{n^2H^2-4(n-1)}}{2(n-1)}$.

Case Ⅱ  If (2.13) holds on $M^n$, we consider three subcases:

(d) If $-1+(n-1)(r+1)-(n-2)H^2\leq0$, then from (2.13), we have $-1+\lambda\mu\leq0$ on $M^n$.

(e) If $-1+(n-1)(r+1)-(n-2)H^2>0$ on $M^n$, suppose $-1+\lambda\mu>0$ on $M^n$, from (2.13), we have

$ \begin{eqnarray} (n-2)\sqrt{H^4-(r+1)H^2}<-1+(n-1)(r+1)-(n-2)H^2. \end{eqnarray} $

(3.16)

Therefore, we have

$ \begin{eqnarray} \frac{(n-2)^2}{n^2}\{(n-1)^2[\frac{n(r+1)-2}{n-2}]^2-[n^2H^2-2(n-1)]\frac{n(r+1)-2}{n-2}+1\}>0, \end{eqnarray} $

(3.17)

that is, $f(\frac{n(r+1)-2}{n-2})>0$. This is a contradiction to (3.13), we deduce that $-1+\lambda\mu\leq0$ on $M^n$.

(f) If $-1+(n-1)(r+1)-(n-2)H^2\leq0$ at a point $p$ of $M^n$ and $-1+(n-1)(r+1)-(n-2)H^2>0$ at other points of $M^n$, in this case, from (d) and (e), we have $-1+\lambda\mu\leq 0$ on $M^n$.

Therefore, we know that if (2.13) holds on $M^n$, then $-1+\lambda\mu\leq 0$ on $M^n$. By Proposition 3.2, we obtain that $M^n$ is isometric to hyperbolic cylinder $S^{n-1}(\lambda_+^2-1)\times H^1(\frac{1}{\lambda_+^2}-1)$ or $H^{n-1}(\lambda_-^2-1)\times S^1(\frac{1}{\lambda_-^2}-1)$, and $\lambda_{\pm}=\frac{n|H|\pm\sqrt{n^2H^2-4(n-1)}}{2(n-1)}$.

This proves Theorem 1.2.

Remark  Wu in Theorem 1.1 (Theorem 5.2 in [9]) considered the complete hypersurfaces in $H^{n+1}(-1)$ which satisfied the condition: $S\geq S_-$ or $S\leq S_+$. He obtained the existence of the global solutions of (3.3) or (3.4) under some conditions for $C$. From the existence of global solutions, he proved the Theorem 1.1. On the other hand, we in Theorem 1.2 consider the complete hypersurfaces in $H^{n+1}(-1)$ which satisfy the condition: $S_+\leq S\leq S_-$. We obtain that the sectional curvature $\lambda\mu-1$ of $M^n$ satisfies that $\lambda\mu-1\geq 0$ or $\lambda\mu-1\leq 0$. From Proposition 3.1 or Proposition 3.2, we can prove Theorem 1.2.