H-hypersurfaces with three distinct principal curvatures in the Euclidean spaces

Abstract

In this paper, we study hypersurfaces of Euclidean spaces with arbitrary dimension. First, we obtain some results on $\text{ H }$-hypersurfaces. Then, we give the complete classification of $\text{ H }$-hypersurfaces with three distinct curvatures. We also give some explicit examples.

1 Introduction

Let $M$ be an $n$-dimensional submanifold of Euclidean $m$-space $\mathbb E^m$ and $x:M\rightarrow \mathbb E^m$ an isometric immersion. $M$ is said to be biharmonic if $x$ satisfies $\varDelta ^2 x=0$, where $\varDelta$ is the Laplace operator of $M$. In [3, 5], Bang-Yen Chen conjectured that every biharmonic submanifold of a Euclidean space is minimal. Chen’s conjecture is supported by all of the results obtained so far (see for example [8, 9, 13]).

On the other hand, $M$ is said to be null 2-type if $x$ can be expressed as $x=x_0+x_1$ for some non-constant vector valued functions $x_0$ and $x_1$ satisfying $\varDelta x_0=0$ and $\varDelta x_1=\lambda x_1$ for a nonzero constant $\lambda$, [2, 6]. Several works on null 2-type surfaces also have been appeared, [4, 10, 12].

In particular, there are some recent results on biharmonic and null 2-type hypersurfaces, [7, 11, 12]. For example, in [7], authors obtained some results on $\delta (2)$-ideal null 2-type hypersurfaces. Most recently, the complete classification of biharmonic hypersurfaces in $\mathbb E^5$ with three distinct principle curvatures has been obtained by Fu [11].

Now, suppose that $M$ is a hypersurface in Euclidean space $\mathbb E^{n+1}$ and let $N$ be its unit normal vector field. From the definition, one can see that if $M$ is null 2-type or biharmonic, then the equation

$$\begin{aligned} \varDelta ^2x=\lambda \varDelta x \end{aligned}$$

(1.1)

is satisfied for a constant $\lambda$. In addition, Beltrami’s well known formula $\varDelta x=-s_1N$ implies

$$\begin{aligned} \varDelta ^2 x=-\left( \varDelta s_1+s_1(s_1^2-2s_2)\right) N-\left( S(\nabla s_1)+\frac{s_1}{2}\nabla s_1\right) , \end{aligned}$$

where $S$ is the shape operator and $s_1$, $s_2$ denote the first and second mean curvatures of $M$, respectively. Therefore, if a hypersurface $M$ in $\mathbb E^{n+1}$ is biharmonic or null 2-type, then the system of differential equations

$$\begin{aligned} S(\nabla s_1)&= -\frac{s_1}{2}\nabla s_1,\end{aligned}$$

(1.2a)

$$\begin{aligned} \varDelta s_1&= -s_1(s_1^2-2s_2-\lambda ) \end{aligned}$$

(1.2b)

is satisfied for a constant $\lambda$. A hypersurface with non-constant first mean curvature is said to be an $\text{ H }$-hypersurface [13] or biconservative hypersurface [1, 14] if it satisfies (1.2a). Classifying $\text{ H }$-hypersurfaces, or at least understanding their geometry, may play an important role on the theory of hypersurfaces satisfying (1.1).

In this work, we study hypersurfaces with three distinct principal curvatures in the Euclidean space of arbitrary dimension. In Sect. 2, after we describe our notations, we give a summary of the basic facts and formulas that we will use. In Sect. 3, we obtain some geometrical properties of $\text{ H }$-hypersurfaces. In Sect. 4, we give a classification of $\text{ H }$-hypersurfaces with three distinct principal curvatures.

2 Prelimineries

Let $\mathbb E^m$ denote the Euclidean $m$-space with the canonical Euclidean metric tensor given by

$$\begin{aligned} \widetilde{g}=\langle \ ,\ \rangle =\sum \limits _{i=1}^m \mathrm{d}x_i^2, \end{aligned}$$

where $(x_1, x_2, \ldots , x_m)$ is a rectangular coordinate system in $\mathbb E^m$.

Consider an $n$-dimensional Riemannian submanifold $M$ of the space $\mathbb E^m$. We denote Levi-Civita connections of $\mathbb E^m$ and $M$ by $\widetilde{\nabla }$ and $\nabla$, respectively. Then, the Gauss and Weingarten formulas are given, respectively, by

$$\begin{aligned} \widetilde{\nabla }_X Y&= \nabla _X Y + h(X,Y),\end{aligned}$$

(2.1)

$$\begin{aligned} \widetilde{\nabla }_X \rho&= -S_\rho (X)+\nabla ^\perp _X \rho \end{aligned}$$

(2.2)

for all tangent vectors fields $X,\ Y$ and normal vector fields $\rho$, where $h$, $\nabla ^\perp$ and $S$ are the second fundamental form, the normal connection and the shape operator of $M$, respectively. Note that for each $\rho \in T^{\bot }_m M$, the shape operator $S_{\rho }$ along the normal direction $\rho$ is a symmetric endomorphism of the tangent space $T_m M$ at $m \in M$. The shape operator and the second fundamental form are related by $\left\langle h(X, Y), \rho \right\rangle = \left\langle S_{\rho }X, Y \right\rangle .$

The Gauss and Codazzi equations are given, respectively, by

$$\begin{aligned} \langle R(X,Y)Z,W\rangle&= \langle h(Y,Z),h(X,W)\rangle - \langle h(X,Z),h(Y,W)\rangle ,\end{aligned}$$

(2.3)

$$\begin{aligned} (\bar{\nabla }_X h )(Y,Z)&= (\bar{\nabla }_Y h)(X,Z), \end{aligned}$$

(2.4)

where $R$ is the curvature tensor associated with connection $\nabla$ and $\bar{\nabla } h$ is defined by

$$\begin{aligned} (\bar{\nabla }_X h)(Y,Z)=\nabla ^\perp _X h(Y,Z)-h(\nabla _X Y,Z)-h(Y,\nabla _X Z). \end{aligned}$$

The mean curvature vector $\zeta$ of $M$ is defined by

$$\begin{aligned} \zeta =\frac{1}{n}\mathrm {tr}\ h. \end{aligned}$$

2.1 Hypersurfaces of Euclidean space

Now, let $M$ be an oriented hypersurface in the Euclidean space $\mathbb E^{n+1}$, $x$ its position vector and $S$ its shape operator along the unit normal vector field $N$ associated with the orientation of $M$. We consider a local orthonormal frame field $\{e_1,e_2,\ldots ,e_n;N\}$ consisting of principal directions of $M$ with corresponding principal curvatures $k_1,\ k_2,\ldots ,\ k_n$. We denote the dual basis of this frame field by $\{\theta _1,\theta _2,\ldots ,\theta _n\}$. Then, the first structural equation of Cartan is

$$\begin{aligned} d\theta _i=\sum \limits _{i=1}^n\theta _j\wedge \omega _{ij},\quad i=1,2,\ldots ,n, \end{aligned}$$

(2.5)

where $\omega _{ij}$ denotes the connection forms corresponding to the chosen frame field, i.e., $\omega _{ij}(e_l)=\langle \nabla _{e_l}e_i,e_j\rangle$.

From the Codazzi equation (2.4), we have

$$\begin{aligned} e_i(k_j)&= \omega _{ij}(e_j)(k_i-k_j),\end{aligned}$$

(2.6a)

$$\begin{aligned} \omega _{ij}(e_l)(k_i-k_j)&= \omega _{il}(e_j)(k_i-k_l) \end{aligned}$$

(2.6b)

for distinct $i,j,l=1,2,\ldots ,n$.

We put $s_1=k_1+k_2+\cdots +k_n$ and, by abuse of terminology, we call this function as the (first) mean curvature of $M$. Note that $M$ is said to be (1-) minimal if $s_1=0$. Throughout this work, we assume $\nabla s_1$ does not vanish at any point of $M$.

3 $\text{ H }$-hypersurfaces

In this section, we give some results on $\text{ H }$-hypersurfaces of Euclidean spaces by extending the results obtained in [13].

3.1 Connection forms of $\text{ H }$-hypersurfaces

Let $M$ be an $\text{ H }$-hypersurface of the Euclidean space $\mathbb E^{n+1}$. Then, (1.2a) is satisfied and $s_1$ is not constant. From (1.2a), we have $\nabla s_1$ is a principal direction of $M$. We consider a frame field $\{e_1,e_2,\ldots ,e_n\}$ consisting of principal directions of $M$ with corresponding principal curvatures $k_1, k_2,\ldots , k_n$ such that $e_1=\nabla s_1/|\nabla s_1|$ and $k_1=-s_1/2$. Therefore, we have

$$\begin{aligned} e_1(k_1)\ne 0,\quad \quad e_x(k_1)=0,\quad x=2,3,\ldots ,n \end{aligned}$$

(3.1)

and

$$\begin{aligned} 3 k_1+k_2+k_3+\cdots +k_n=0. \end{aligned}$$

(3.2)

Remark 1

[13] If $k_1=k_x$ for some $2\le x\le ,n$, then Codazzi equation (2.6a) for $i=1,$ $\ j=x$ implies $e_1(k_1)=e_1(k_x)=\omega _{1x}(e_x)(k_1-k_x)=0$ which contradicts with (3.1). Thus, the dimension of the distribution $D_0$ given by

$$\begin{aligned} D_0(m)=\{X\in T_m|SX=k_1X\} \end{aligned}$$

is 1. Integral curves of $D_0$ are planar and geodesics of $M$. Furthermore, if $\alpha$ and $\beta$ are integral curves of $D_0$ passing through $m$ and $m'$, respectively, then $\alpha$ and $\beta$ are congruent, [13].

By combining (3.1) with Codazzi equation (2.6a) for $i=x,\ j=1$, we get

$$\begin{aligned} \omega _{1x}(e_1)=0, \quad x=2,3,\ldots ,n. \end{aligned}$$

(3.3)

On the other hand, for a tangent vector field $X$ of $M$, $\langle X,e_1\rangle =0$ if and only if $Xk_1=0$. Therefore, $[e_x,e_y](k_1)=0$ implies $\langle [e_x,e_y],e_1\rangle =0$ from which we have

$$\begin{aligned} \omega _{1x}(e_y)=\omega _{1y}(e_x),\quad \quad x,y=2,3,\ldots ,n,\ x\ne y. \end{aligned}$$

From this equation and Codazzi equation (2.6b) for $i=1,\ j=x,\ l=z$, we get

$$\begin{aligned} \omega _{1x}(e_y)=0, \quad x,y=2,3,\ldots ,n\text{, } x\ne y. \end{aligned}$$

(3.4)

Therefore, (2.6b) for $i=x, j=y, l=1$ and (2.6b) for $i=x, j=1, l=y$ imply

$$\begin{aligned} \omega _{xy}(e_1)=0, \quad x,y=2,3,\ldots ,n,\ k_x\ne k_y. \end{aligned}$$

(3.5a)

In fact, we have

$$\begin{aligned} \omega _{xy}(e_z)=0, \quad x,y,z=1,2,3,\ldots ,n,\ x\ne z,\ k_x=k_z\ne k_y \end{aligned}$$

(3.5b)

from the Codazzi equation (2.6b) for $i=x,\ j=y,\ l=z.$

Since (3.3) implies $\langle [e_1,e_x],e_1\rangle =0$, we have $[e_1,e_x](k_1)=0$ from which and (3.1) we obtain

$$\begin{aligned} e_xe_1(k_1)=e_xe_1e_1(k_1)=0,\quad x=2,3,\ldots ,n. \end{aligned}$$

(3.6)

3.2 Some lemmas on $\text{ H }$-hypersurfaces

In this subsection, we obtain some lemmas that we will use on the rest of the paper.

First, we consider the distribution given by

$$\begin{aligned} D(m)=\{X\in T_mM|SX=k_2 X\}. \end{aligned}$$

(3.7)

Remark 2

Obviously, the dimension of $D$ is equal to multiplicity of $k_2$ as an eigenvalue of the shape operator $S$ of $M$.

We obtain the following lemma.

Lemma 3.1

Let $M$ be an $\text{ H }$-hypersurface in the Euclidean space $\mathbb E^{n+1}$ and $k_2$ one of its principal curvatures. Then, the distribution $D$ given by (3.7) is involutive.

Proof

If the dimension of $D$ is 1, then it is obviously involutive. Thus, we assume $\mathrm {dim}D=p>1$ and, by renaming the indices if necessary,

$$\begin{aligned} k_{2}=k_3=\ldots =k_{p+1}. \end{aligned}$$

(3.8)

Therefore, (3.5b) implies $\langle \nabla _{e_A}e_B,e_i\rangle =\omega _{Bi}(e_A)=0$ for all $i=1,p+2,p+3,\ldots ,n$ and $A,B=2,3,\ldots ,p+1$ with $A\ne B$. Thus, we have $(\nabla _{e_A}e_B)_m\in D(m)$ from which we see that $X_m,Y_m\in D(m)$ implies $[X_m,Y_m]\in D(m)$. Hence, $D$ is involutive. $\square$

Now, we want to construct integral submanifolds of the distribution $D$ given by (3.7).

Lemma 3.2

Let $M$ be an $\text{ H }$-hypersurface in the Euclidean space $\mathbb E^{n+1}$ and $k_2$ one of its principal curvatures. Assume that the distribution $D$ given by (3.7) has dimension greater than $1$. Then, any integral submanifold $H$ of $D$ has parallel mean curvature vector field $\zeta$ in $\mathbb E^{n+1}$. Moreover, for any normal vector field $\rho$ in $H$, we have $\hat{S}_\rho =\tau I$ for a function $\tau$, where $\hat{S}$ denotes the shape operator of $H$ in $\mathbb E^{n+1}$.

Proof

Let the dimension of $D$ is $p>1$. Then, by renaming indices if necessary, we assume (3.8). Using (3.5), we obtain

$$\begin{aligned} \widetilde{\nabla }_{e_A}e_A&=-\omega _{1A}(e_A)e_1+\sum \limits _{C=2}^{p+1}\omega _{AC}(e_A)e_C+\sum \limits _{a=p+2}^{n}\omega _{Aa}(e_A)e_a+k_2N,\nonumber \\ \widetilde{\nabla }_{e_A}e_B&=\sum \limits _{C=2}^{p+1}\omega _{BC}(e_A)e_C \end{aligned}$$

(3.9)

for all $A,B=2,3,\ldots ,p+1$ with $A\ne B$. Note that Codazzi equation (2.6a) for $i=1, j=A$ and $i=a, j=A$ give $\omega _{1A}(e_A)=\frac{e_1(k_A)}{k_1-k_A}$ and $\omega _{Aa}(e_A)=\frac{e_a(k_A)}{k_A-k_a}$, respectively. Thus, (3.8) implies

$$\begin{aligned} \xi&=-\omega _{12}(e_2)=-\omega _{13}(e_3)=\cdots =-\omega _{1(p+1)}(e_{p+1}),\nonumber \\ \eta _a&=\omega _{a2}(e_2)=\omega _{a3}(e_3)=\cdots =\omega _{a(p+1)}(e_{p+1}) \end{aligned}$$

(3.10)

for some functions $\xi$ and $\eta _a$ for $a=p+2,p+3,\ldots ,n$.

Now, let $H$ be an integral submanifold of $D$ and consider the local orthonormal frame field

$$\begin{aligned} \{f_1,f_2,\ldots ,f_p; f_{p+1},f_{p+2},\ldots f_{n+1}\} \end{aligned}$$

on $H$ given by

$$\begin{aligned} f_{A-1}=\left. e_A\right| _{H},\ f_{p+1}=\left. e_1\right| _{H},f_{a}=\left. e_a\right| _{H}, f_{n+1}=\left. N\right| _{H}. \end{aligned}$$

(3.11)

From (3.9) and (3.10), we have

$$\begin{aligned} \widetilde{\nabla }_{f_i}f_i&= \hat{\nabla }_{f_i}f_i+\hat{\xi } f_{p+1}+\sum \limits _{a=p+2}^{n}\hat{\eta }_a f_a+\hat{k}_2f_{n+1}\end{aligned}$$

(3.12a)

$$\begin{aligned} \widetilde{\nabla }_{f_i}f_j&= \hat{\nabla }_{f_i}f_j,\quad i,j=1,2,\ldots ,p,\ i\ne j, \end{aligned}$$

(3.12b)

where $\hat{\nabla }$ denotes the Levi-Civita connection of $H$ and $\hat{\xi }$, $\hat{\eta }_a$, $\hat{k}_2$ are restrictions of $\xi$, $\eta _a$, $k_2$ to $H$, respectively.

Therefore, we have

$$\begin{aligned} \hat{S}_{p+1}=\hat{\xi } I,\ \hat{S}_{a}=\hat{\eta }_{a} I,\ \hat{S}_{n+1}=\hat{k}_2 I \end{aligned}$$

(3.13)

or, equivalently,

$$\begin{aligned} \zeta =\hat{h}(f_1,f_1)=\hat{h}(f_2,f_2)=\cdots =\hat{h}(f_p,f_p)=\hat{\xi } f_{p+1}+\sum \limits _{a=p+2}^{n}\hat{\eta }_a f_a+\hat{k}_2f_{n+1}, \end{aligned}$$

(3.14)

where $\hat{h}$ stands for the second fundamental form of $H$ in $\mathbb E^{n+1}$ and $\hat{S}_\alpha =\hat{S}_{f_\alpha }$ .

Furthermore, Codazzi equation (2.4) for $X=Z=f_i$ and $Y=f_j$ for $i\ne j$ gives

$$\begin{aligned} \hat{\nabla }^\perp _{f_i}\hat{h}(f_i,f_j)-\hat{h}(\hat{\nabla }_{f_i}f_i,f_j)-\hat{h}(f_i,\hat{\nabla }_{f_i}f_j)=\hat{\nabla }^\perp _{f_j}h(f_i,f_i)-2\hat{h}(\hat{\nabla }_{f_j}f_i,f_i), \end{aligned}$$

where $\hat{\nabla }^\perp$ is the normal connection of $H$ in $\mathbb E^{n+1}$. Using (3.12) in this equation and considering (3.14), we get $\hat{\nabla }^\perp _{f_j}\zeta =0$ for all $j=1,2,\ldots ,p$. Hence, $\zeta$ is parallel. $\square$

Remark 3

Let $M$ be an $\text{ H }$-hypersurface in the Euclidean space $\mathbb E^{n+1}$ and $k_2$ one of its principal curvatures. Assume that the distribution $D$ given by (3.7) has dimension greater than $1$ and $H$ is an (connected) integral submanifold of $D$. From the proof of Lemma 3.2, one can see that $k_2$, $\xi =\omega _{1A}(e_A)$ and $\eta _a=\omega _{Aa}(e_A)$ are constant on $H$.

By the following proposition, we obtain integral submanfiolds of the distribution $D$ given by (3.7).

Proposition 3.3

Let $M$ be an $\text{ H }$-hypersurface in the Euclidean space $\mathbb E^{n+1}$ and $k_2$ one of its principal curvatures. Assume that the distribution $D$ given by (3.7) has dimension $p>1$ and $H$ is an (connected) integral submanifold of $D$ passing through $m\in M$. If $k_2(m)=0$ and $(\nabla k_2)_m=0$ then $H$ is a $p$-plane of $\mathbb E^{n+1}$. Otherwise, $H$ lies on a $(p+1)$-plane of $\mathbb E^{n+1}$ and it is congruent to a hypersphere of $\mathbb E^{p+1}$.

Proof

First, suppose that $k_2$ and $\nabla k_2$ vanish at $m$. Then, we have $\hat{\eta }_a(m)=\hat{\xi }(m)=0$ for $a=p+2,p+3,\ldots ,n$, where $\hat{\eta }_a$ and $\hat{\xi }$ are functions defined in the proof of Lemma 3.2. Remark 3 implies that $\hat{k}_2\equiv 0$, $\hat{\xi }\equiv 0$ and $\hat{\eta }\equiv 0$ on $H$. Thus, from (3.14) we have $\hat{h}=0$, i.e., $M$ is a totally geodesic $p$-dimensional submanifold of $\mathbb E^{n+1}$. Hence, $M$ is a $p$-plane.

Next, assume $k_2(m)\ne 0$. Define $n-p$ normal vector fields $\zeta _1,\zeta _{p+2},\ldots ,\zeta _{n}$ by $\zeta _1= \hat{k}_2 f_p-\hat{\xi } f_{n+1}$ and $\zeta _a=\hat{k}_2f_a-\hat{\eta }_af_{n+1}$. Clearly, $\zeta _1,\zeta _{p+2},\ldots ,\zeta _{n}$ are linearly independent constant vector fields normal to $H$. Thus, $H$ lies in a $(p+1)$-plane $\varPi \cong \mathbb E^{p+1}$ of $\mathbb E^{n+1}$. As its mean curvature vector is parallel, and shape operator is proportional to identity operator $I$, it is a hypersphere of $\varPi$.

If $(\nabla k_2)_m\ne 0$, then we have $\hat{\xi }(m)\ne 0$ or $\hat{\eta }_a(m)\ne 0$ for some $a$ because of Codazzi equation (2.6a). Same proof can be done for both cases. $\square$

4 $\text{ H }$-hypersurfaces with three distinct principal curvatures

Let $M$ be an $\text{ H }$-hypersurfaces in $\mathbb E^{n+1}$ with three distinct principal curvatures $k_1, k_2$ and $k_{p+2}$ and $S$ its shape operator. Because of Remark 1, the multiplicity of $k_1$ is 1. Therefore, the matrix representation of $S$ is

$$\begin{aligned} S=\mathrm {diag}(k_1,\underbrace{k_2,k_2,\ldots ,k_2}_{p \text{ times }},\underbrace{k_{p+2},k_{p+2},\ldots ,k_{p+2}}_{q \text{ times }}),\quad k_2\ne k_{p+2} \end{aligned}$$

(4.1)

corresponding to a local orthonormal frame field $\{e_1,e_2,\ldots ,e_n\}$ consisting of principal directions of $M$, where $p+q+1=n$. We also assume that the functions $k_1-k_2$, $k_1-k_{p+2}$ and $k_2-k_{p+2}$ do not vanish on $M$.

First, we consider the distribution $D^\perp$ given by

$$\begin{aligned} D^\perp (m)=\{X_m\in T_mM| \langle X_m,Y\rangle =0, \text{ for } \text{ all } Y\in D(m)\}. \end{aligned}$$

(4.2)

Lemma 4.1

Let $M$ be an $\text{ H }$-hypersurface in the Euclidean space $\mathbb E^{n+1}$ with the shape operator given by (4.1). Then, the distribution $D^\perp$ given by (4.2) is involutive.

Proof

From the definition, we have $D^\perp (m)=\mathrm {span}\{(e_1)_m,(e_{p+2})_m,(e_{p+3})_m,$ $\ldots ,(e_{n})_m\}$. Moreover, from (3.5) and (4.1) we have $\nabla _{e_a}e_b,\nabla _{e_a}e_1, \nabla _{e_1}e_a\in D^\perp$ for all $a,b=p+2,p+3,\ldots ,n$. Thus, for all $X,Y\in D^\perp$, we have $[X,Y]\in D^\perp$. Hence $D^\perp$ is involutive. $\square$

Remark 4

By combining (3.3) and (3.4) with Cartan’s first structural equation (2.5), we obtained $d\theta _1=0$, i.e., $\theta _1$ is closed. Thus, Poincarè lemma implies that $d\theta _1$ is exact, i.e., there exists a function $s$ such that $\theta _1=ds$. Moreover, since the distributions $D$ and $D^\perp$ given by (3.7) and (4.2) are involutive, there exists a local coordinate system $t_1,t_2,\ldots ,t_n$ on a neighborhood of $m\in M$ such that $t_2,t_3,\ldots ,t_{p+1}$ span $D$ and $t_1=s,t_{p+2},t_{p+3},\ldots ,t_{n}$ span $D^\perp$ because of the local Frobenius theorem. Thus, by redefining the vector fields $e_i,\ i=2,3,\ldots ,n$ properly, we can assume

$$\begin{aligned} e_1=\partial _s,\quad e_i=F_i\partial _{t_i},\ i=2,3,\ldots ,n \end{aligned}$$

(4.3)

for some smooth non-vanishing functions $F_i=F_i(s,t_2,t_3,\ldots ,t_n)$.

Since the study on $\mathbb E^4$ is completed in [13], we focus on the case $n>3$. Therefore, without loss of generality, we may assume $p>1$. Thus, we have (3.8) and Codazzi equation (2.6a) implies

$$\begin{aligned} e_A(k_2)&= 0, \quad A=2,3,\ldots ,p+1. \end{aligned}$$

(4.4)

From (3.1), (3.2), (4.1) and (4.4), we also get

$$\begin{aligned} e_A(k_{p+2})&= 0, \quad a=p+2,p+3,\ldots ,n \end{aligned}$$

(4.5)

from which and Codazzi equation (2.6a) for $i=A,j=a$, we obtain

$$\begin{aligned} \omega _{Aa}(e_a)=0. \end{aligned}$$

(4.6)

4.1 Case $p>1$ and $q>1$

In this case, we have

$$\begin{aligned} e_a(k_{p+2})&= 0, \end{aligned}$$

(4.7a)

$$\begin{aligned} e_a(k_{2})&= 0, \end{aligned}$$

(4.7b)

$$\begin{aligned} \omega _{Aa}(e_A)&= 0, \quad A=2,3,\ldots , p+1,\ a=p+2,p+3,\ldots ,n. \end{aligned}$$

(4.7c)

By combining (3.3), (3.4), (3.5), (4.6) and (4.7c), we get

$$\begin{aligned} \widetilde{\nabla }_{e_1}e_A&= \sum \limits _{C=2}^{p+1}\omega _{AC}(e_1)e_C,\end{aligned}$$

(4.8a)

$$\begin{aligned} \widetilde{\nabla }_{e_A}e_1&= \omega _{12}(e_2)e_A,\end{aligned}$$

(4.8b)

$$\begin{aligned} \widetilde{\nabla }_{e_a}e_1&= \omega _{1{(p+2)}}(e_{p+2})e_a. \end{aligned}$$

(4.8c)

On the other hand, since $[e_1,e_A](k_2)=e_Ae_1(k_2)$, we have

$$\begin{aligned} e_Ae_1(k_2)=\left( \sum \limits _{C=2}^{p+1}\omega _{AC}(e_1)e_C-\omega _{1A}(e_A)e_A\right) (k_2) \end{aligned}$$

from (4.8a) and (4.8b). The right- hand side of this equation is zero because of (4.4). Thus, we obtain

$$\begin{aligned} e_Ae_1(k_2)=0. \end{aligned}$$

(4.9)

Furthermore, from (2.6a) for $i=1, j=2$ we have

$$\begin{aligned} e_A(\omega _{12}(e_2))=e_A\left( \frac{e_1(k_2)}{k_1-k_2}\right) . \end{aligned}$$

By combining this equation with (3.1), (4.4) and (4.9) we get

$$\begin{aligned} e_A(\omega _{12}(e_2))=0. \end{aligned}$$

(4.10a)

By a similar way, we obtain

$$\begin{aligned} e_a(\omega _{12}(e_2))&= 0,\end{aligned}$$

(4.10b)

$$\begin{aligned} e_A(\omega _{1(p+2)}(e_{p+2}))=e_a(\omega _{1(p+2)}(e_{p+2}))&= 0. \end{aligned}$$

(4.10c)

By combining the equations in (4.10), we get

$$\begin{aligned} \omega _{12}(e_2)=\xi (s),\quad \quad \omega _{1(p+2)}(e_{p+2})=\eta (s) \end{aligned}$$

(4.11)

for some functions $\xi ,\eta$, where $s$ is the local coordinate given in Remark 4. Now, we are ready to prove

Theorem 1

Let $M$ be a hypersurface in $\mathbb E^{n+1}$ with the shape operator given by (4.1), $k_2\ne k_{p+2}$ and $p>1, q>1$. Then, $M$ is an $\text{ H }$-hypersurface if and only if it is congruent to one of the following hypersurfaces.

• (i) A generalized rotational hypersurface given by

  $$\begin{aligned} x(s, t_2,\ldots ,t_n)=&\big (\psi (s)\cos t_2,\psi (s)\sin t_2\cos t_3,\ldots ,\psi (s)\sin t_2\ldots \sin t_{p}\cos t_{p+1},\nonumber \\&\psi (s)\sin t_{2}\ldots \sin t_{p}\sin t_{p+1}, \phi (s)\cos t_{p+2}\!,\phi (s)\sin t_{p+2}\cos t_{p+3},\ldots ,\nonumber \\&\phi (s)\sin t_{p+2}\ldots \sin t_{n-1}\cos t_n,\phi (s)\sin t_{p+2}\ldots \sin t_{n-1}\sin t_n\big ) \end{aligned}$$

  (4.12)

  with the profile curve $(\psi ,\phi )$ satisfying $\psi '^2+\phi '^2=1$ and

  $$\begin{aligned} \phi '\psi ''-\phi ''\psi '=\frac{1}{3}\left( p\frac{\phi '}{\psi }-q\frac{\psi '}{\phi }\right) . \end{aligned}$$

  (4.13)

• (ii) A generalized cylinder over a rotational hypersurface given by

  $$\begin{aligned} x(s, t_1,\ldots ,t_n)=&\big (\psi (s)\cos t_2,\psi (s)\sin t_2\cos t_3,\ldots ,\psi (s)\sin t_2\ldots \sin t_{p}\cos t_{p+1},\nonumber \\&\psi (s)\sin t_{2}\ldots \sin t_{p}\sin t_{p+1}, \phi (s),t_{p+2},t_{p+3},\ldots ,t_{n}\big ) \end{aligned}$$

  (4.14)

  with the profile curve $(\psi ,\phi )$ satisfying $\psi '^2+\phi '^2=1$ and

  $$\begin{aligned} \phi '\psi ''-\phi ''\psi '=\frac{p}{3}\frac{\phi '}{\psi }. \end{aligned}$$

  (4.15)

Proof

We assume that $M$ is an $\text{ H }$-hypersurface. Then, (3.2) is satisfied. Let $s,t_2,t_3,\ldots ,t_n$ be the local coordinate system given in Remark 4. From (4.8b) and (4.8c) we have

$$\begin{aligned} x_{st_A}&= \omega _{12}(e_2)x_{t_A},\quad A=2,3,\ldots ,p+1 \end{aligned}$$

(4.16a)

$$\begin{aligned} x_{st_a}&= \omega _{1(p+2)}(e_{p+2})x_{t_a},\quad a=p+2,p+3,\ldots ,n. \end{aligned}$$

(4.16b)

By taking into account the (4.11), we integrate (4.16) to obtain

$$\begin{aligned} x_s=\xi (s)x+\tilde{\varTheta }_2(s,t_{p+2},t_{p+3},\ldots ,t_n)=\eta (s)x+\tilde{\varTheta }_1(s,t_2,t_3,\ldots ,t_{p+1}) \end{aligned}$$

for some vector valued functions $\tilde{\varTheta }_1,\tilde{\varTheta }_2$. Therefore, we have

$$\begin{aligned} x(s,t_2,t_3,\ldots ,t_n)=\hat{\varTheta }_1(s,t_2,t_3,\ldots ,t_{p+1})+\hat{\varTheta }_2(s,t_{p+2},t_{p+3},\ldots ,t_n) \end{aligned}$$

(4.17)

for some vector valued functions $\hat{\varTheta }_1$ and $\hat{\varTheta }_2$.

Next, we put (4.17) in (4.16) to get

$$\begin{aligned} \hat{\varTheta }_{1,st_A}= \xi (s)\hat{\varTheta }_{1,t_A},\quad \hat{\varTheta }_{2,st_a}=\eta (s)\hat{\varTheta }_{2,t_a}. \end{aligned}$$

By integrating these equations, we obtain

$$\begin{aligned} x(s,t_2,t_3,\ldots ,t_n)=\psi (s)\varTheta _1(t_2,t_3,\ldots ,t_{p+1})+\phi (s)\varTheta _2(t_{p+2},t_{p+3},\ldots ,t_n)+\varphi (s) \end{aligned}$$

for some functions $\phi ,\psi$ and vector valued functions $\varTheta _1,\varTheta _2,\varphi$. By taking into account Remark 1, we see that $\varphi$ is a constant vector. Thus, we may assume

$$\begin{aligned} x(s,t_2,t_3,\ldots ,t_n)=\psi (s)\varTheta _1(t_2,t_3,\ldots ,t_{p+1})+\phi (s)\varTheta _2(t_{p+2},t_{p+3},\ldots ,t_n). \end{aligned}$$

(4.18)

Because of (4.3), we have

$$\begin{aligned} \left\langle \varTheta _{1,t_A}, \varTheta _{2,t_a}\right\rangle&= 0,\end{aligned}$$

(4.19a)

$$\begin{aligned} \langle x_s,x_s\rangle&= 1. \end{aligned}$$

(4.19b)

Since $k_2\ne k_{p+2}$, without loss of generality, we may assume $k_2\ne 0.$ Now, we consider the slice $H$ of $M$ given by

$$\begin{aligned} y_1(t_2,t_3,\ldots ,t_{p+1})=x(\bar{s},t_2,t_3,\ldots ,t_{p+1},\bar{t}_{p+2},\bar{t}_{p+3},\bar{t}_{n}) \end{aligned}$$

passing through the point $m=x(\bar{s},\bar{t}_2,\bar{t}_3,\ldots ,\bar{t}_n)\in M$. From (4.18) we have

$$\begin{aligned} y_1(t_2,t_3,\ldots ,t_{p+1})=c_0\varTheta _1(t_2,t_3,\ldots ,t_{p+1})+v_0, \end{aligned}$$

(4.20)

where $c_0=\phi (\bar{s})$ is a constant and $v_0=\psi (\bar{s})\varTheta _2(\bar{t}_{p+2},\bar{t}_{p+3},\ldots ,\bar{t}_n)$ is a constant vector.

Since $H$ is an integral submanifold of the distribution $D$ given by (3.7), and $k_2\ne 0$, it is congruent to hypersphere of $\mathbb E^{p+1}$ because of Proposition 3.3. Thus, by choosing suitable coordinates and redefining $\psi$, we may assume

$$\begin{aligned} \varTheta _1(t_2,\ldots ,t_{p+2})&=\big (\cos t_2,\sin t_2\cos t_3,\ldots ,\sin t_2\ldots \sin t_{p}\cos t_{p+1},\nonumber \\&\quad \ \sin t_{2}\ldots \sin t_{p}\sin t_{p+1}, 0,0,\ldots ,0\big ). \end{aligned}$$

(4.21)

Now, consider the submanifold $H'$ given by

$$\begin{aligned} y_2(t_{p+2},t_{p+3},\ldots ,t_{n})=x(\bar{s},\bar{t}_2,\bar{t}_3,\ldots ,\bar{t}_{p+1},t_{p+2},t_{p+3}, t_{n}) \end{aligned}$$

which is an integral submanifold of the distribution $D'$ given by $D'(m')=\{X\in T_{m'}M|SX=k_{p+2}X\}$ passing through the point $m$. Now, we have two cases: $k_{p+2}=0$ and $k_{p+2}\ne 0$.

Case 1. $k_{p+2}=0$. In this case, $H'$ is a $q$-plane because of Proposition 3.3. Thus, $\varTheta _2$ is the position vector of a $q$-plane. Because of (4.19a) without loss of generality, we may assume

$$\begin{aligned} \varTheta _2(t_{p+2},t_{p+3},\ldots ,t_{n})=&\big (0,0,\ldots , 1,t_{p+2},t_{p+3},\ldots ,t_{n}\big ). \end{aligned}$$

By redefining $t_{p+2},\ldots ,t_{n}$, we obtain (4.14). Because of (4.19b), we have $\psi '^2+\phi '^2=1$.

Moreover, the shape operator of this hypersurface is

$$\begin{aligned} S=\mathrm {diag}(k_1,\underbrace{\frac{\phi '}{\psi },\frac{\phi '}{\psi },\ldots ,\frac{\phi '}{\psi }}_{p \text{ times }},\underbrace{0,0,\ldots ,0}_{q \text{ times }}). \end{aligned}$$

(4.22)

From (3.2) and (4.22), we get (4.15). Hence, we have the case (ii) of theorem.

Case 2. $k_{p+2}\ne 0$. In this case, $H'$ is congruent to a hypersphere of $\mathbb E^{q+1}$ because of Proposition 3.3. Because of (4.19a), without loss of generality, we choose

$$\begin{aligned} \varTheta _2(t_{p+2},t_{p+3},\ldots ,t_{n})=&\big (\underbrace{0,0,\ldots ,0}_{p+1 \text{ times }},\cos t_{p+2},\sin t_{p+2}\cos t_{p+3},\ldots ,\\&\sin t_{p+2}\ldots \sin t_{n-1}\cos t_{n},\psi \sin t_{p+2}\ldots \sin t_{n}\big ). \end{aligned}$$

Therefore, we obtain (4.12). Because of (4.19b), we have $\psi '^2+\phi '^2=1$.

Moreover, the shape operator of this hypersurface is

$$\begin{aligned} S=\mathrm {diag}(k_1,\underbrace{\frac{\phi '}{\psi },\frac{\phi '}{\psi },\ldots ,\frac{\phi '}{\psi }}_{p \text{ times }},\underbrace{-\frac{\psi '}{\phi },-\frac{\psi '}{\phi },\ldots ,-\frac{\psi '}{\phi }}_{q \text{ times }}). \end{aligned}$$

(4.23)

From (3.2) and (4.23) we get (4.13). Hence, we have the case (i) of theorem. $\square$

Remark 5

In [14], Montaldo et al. proved that a curve satisfying (4.13) is of catenary type. The authors also proved that none of these type of hypersurfaces are biharmonic. Recently, in [11], Yu Fu remarked that he extended this result by proving that there is no non-minimal biharmonic hypersurface in $\mathbb E^{n+1}$ with three distinct principal curvature. However, classifying null 2-type hypersurfaces with three distinct principal curvature is an open problem.

4.2 Case $p>1$ and $q=1$

In the remaining part, we consider the case $p>1$ and $q=1$ to obtain a necessary condition for null 2-type hypersurfaces with 3 principal curvatures. In this case (4.1) becomes

$$\begin{aligned} S=\mathrm {diag}(k_1,\underbrace{k_2,k_2,\ldots ,k_2}_{n-2 \text{ times }},k_n). \end{aligned}$$

(4.24)

Since $p=n-2>1$, the equations (4.4)-(4.6) are still satisfied. Moreover, the distribution $D$ given in (3.7) is involutive and its integral submanifold are congruent to hyperspheres or hyperplanes of $\mathbb E^{n-1}$ because of Lemma 3.1 and Lemma 3.2. From [13, Lemma 2.2], we also know that integral curves of $e_1=\partial _s$ are some planar curves and congruent to each other. Therefore, we first want to focus on the remaining part, integral curves of $e_n$.

Let $M$ be a hypersurface with the shape operator given in (4.24). We also suppose that the functions $k_1-k_2,$ $k_1-k_n$ and $k_2-k_n$ do not vanish on $M$. Now, assume that $M$ is a null 2-type hypersurface. Then, $M$ is an $\text{ H }$-surface satisfying (1.2b). Moreover, from (3.2) and (4.24), we have

$$\begin{aligned} 3k_1+(n-2)k_2+k_n=0 \end{aligned}$$

(4.25)

because $M$ is an $\text{ H }$-hypersurface.

By combining (4.4) and (4.5) with Codazzi equation (2.6a), we have $\omega _{An}(e_n)=0$. Therefore, we have

$$\begin{aligned} \widetilde{\nabla }_{e_n}e_1=\omega _{1n}(e_n)e_n,&\quad&\widetilde{\nabla }_{e_A}e_1=\omega _{1A}(e_A)e_A\end{aligned}$$

(4.26a)

$$\begin{aligned} \widetilde{\nabla }_{e_A}e_n=-\omega _{An}(e_A)e_A,&\quad&\widetilde{\nabla }_{e_n}e_A=\omega _{AB}(e_n)e_B \end{aligned}$$

(4.26b)

Now, we want to show $e_n(k_2)=0$ using a method similar with [11].

Since $e_A(k_2)=0$ and $e_A(k_n)=0$, we have $[e_A,e_1](k_2)=e_Ae_1(k_2)$ and $[e_A,e_1](k_n)=e_Ae_1(k_n)$. By computing the left-hand side of each of these equations using (4.26b), we get

$$\begin{aligned} e_Ae_1(k_2)=e_Ae_1(k_n)=0,\quad A=2,3,\ldots ,n-1. \end{aligned}$$

(4.27)

Furthermore, from the Gauss equation (2.3) for $X=e_A,$ $Y=e_n,$ $Z=e_1,$ and $W=e_A$ we obtained

$$\begin{aligned} e_n(\omega _{12}(e_2))=\frac{e_n(k_2)}{k_2-k_n}(\omega _{12}(e_2)-\omega _{1n}(e_n)). \end{aligned}$$

(4.28)

By a direct calculation using Codazzi equation (2.6a), (3.1), (4.25) and (4.28), we also obtain

$$\begin{aligned} e_n(\omega _{1n}(e_n))=(n-2)\frac{(2k_2-k_1-k_n)e_n(k_2)}{(k_1-k_n)(k_2-k_n)}(\omega _{12}(e_2)-\omega _{1n}(e_n)). \end{aligned}$$

(4.29)

On the other hand, from (4.24) and (1.2b) we have

$$\begin{aligned} e_1e_1(k_1){+}(n-2)\omega _{12}(e_2)e_1(k_1){+}\omega _{1n}(e_n)e_1(k_1){=}k_1(k_1^2+(n-2)k_2^2+k_n^2-\lambda ).\qquad \end{aligned}$$

(4.30)

By applying $e_n$ to both hand side of this equation and using (3.6), (4.28) and (4.29) we obtain

$$\begin{aligned} e_n(k_n)\Big (e_1(k_1)(\omega _{12}(e_2)-\omega _{1n}(e_n))-k_1(k_2-k_n)(k_1-k_n))\Big )=0. \end{aligned}$$

(4.31)

From the assumptions, we have the functions $\omega _{12}(e_2)-\omega _{1n}(e_n)$ and $k_1$ do not vanish. Thus, if $e_n(k_n)\ne 0$, then we have

$$\begin{aligned} \frac{e_1(k_1)}{k_1}=\frac{(k_2-k_n)(k_1-k_n)}{\omega _{12}(e_2)-\omega _{1n}(e_n)} \end{aligned}$$

because of (4.31). By applying $e_n$ to this equation we obtain

$$\begin{aligned} e_n\left( \frac{(k_2-k_n)(k_1-k_n)}{\omega _{12}(e_2)-\omega _{1n}(e_n)}\right) =0. \end{aligned}$$

Next, we compute the left-hand side of this equation using (3.1), (4.25), (4.28) and (4.29) to get $k_n=a_0k_2$ for a constant $a_0$. However, this equation, (3.1) and (4.25) give us $e_n(k_2)=0$ which is a contradiction. Therefore, we have

$$\begin{aligned} e_n(k_2)=0 \end{aligned}$$

(4.32)

and (3.1), (4.25) imply

$$\begin{aligned} e_n(k_n)=0. \end{aligned}$$

(4.33)

Moreover, from Codazzi equation (2.6a) and (4.32) we have

$$\begin{aligned} \omega _{An}(e_A)=0. \end{aligned}$$

(4.34)

On the other hand, from (4.28), (4.29) and (4.32) we get

$$\begin{aligned} e_n(\omega _{1n}(e_n))= e_n(\omega _{1A}(e_A))=0 \end{aligned}$$

(4.35a)

and by taking into account (4.26) and using Gauss equation (2.3) for $X=e_A,$ $Y=e_n,$ $Z=e_1,$ $W=e_n$ we obtain

$$\begin{aligned} e_A(\omega _{1n}(e_n))=0. \end{aligned}$$

(4.35b)

From Codazzi equation (2.6a) for $i=1,\ j=A$ we have $\omega _{1A}(e_A)=e_1(k_A)/(k_1-k_A)$. Thus, we have

$$\begin{aligned} e_A(\omega _{1A}(e_A))=0. \end{aligned}$$

(4.35c)

Next, we want to give a geometric interpretation of these results.

Proposition 4.2

Let $M$ be a null 2-type hypersurface in $\mathbb E^{n+1}$ with the shape operator given by (4.24) and non-constant first mean curvature. Then, an integral curve of $e_n$ is either a circle or line.

Proof

Using (4.34), we get

$$\begin{aligned} \widetilde{\nabla }_{e_n}e_n&=-\omega _{1n}(e_n)e_1+k_nN, \quad \widetilde{\nabla }_{e_n}e_1=\omega _{1n}(e_n)e_n,\quad \widetilde{\nabla }_{e_n}N=-k_ne_n, \end{aligned}$$

(4.36)

Moreover, (4.35a) and (4.33) imply that $\omega _{1n}(e_n)$ and $k_n$ are constant on any (connected) integral curve $\alpha$ of $e_n$. Let $t,n$ be tangent and normal vector fields of $\alpha$. Note that we have $t=\left. e_n\right| _\alpha .$ If

$$\begin{aligned} \Vert \hat{\nabla }_t t\Vert =a=0 \end{aligned}$$

then $\alpha$ is a line and proof is completed, where $\hat{\nabla }$ is the Levi-Civita connection of $\alpha$ and $a$ is the constant given by $\left. \big (\omega _{1n}(e_n)^2+k_n^2\big )^{1/2}\right| _\alpha$.

We assume $\hat{\nabla }_{t} t\ne 0$. Then, we have $n=\hat{\nabla }_t t/\Vert \hat{\nabla }_t t\Vert$. From (4.36) we have $\hat{\nabla }_tt=an,\quad \hat{\nabla }_t n=-an$. Thus, $\alpha$ is planar and its curvature $a>0$. $\square$

By summing up (4.35), we see that (4.11) is satisfied for $q=1.$ Thus, by taking into account Proposition 4.2, we obtain a necessary condition for being null 2-type of hypersurfaces that we are considering. The following proposition can be proved like Theorem 1.

Proposition 4.3

Let $M$ be an hypersurface with non-constant first mean curvature and the shape operator given by (4.24). If $M$ is a null 2-type hypersurface then it must be congruent to one of the following hypersurfaces.

• (i) A generalized rotational hypersurfaces given by (4.12) with $p=n-2$, $q=1$ for some functions satisfying $\psi '^2+\phi '^2=1$ and (4.13),

• (ii) A generalized cylinder over a rotational hypersurface, given by (4.14) with $p=n-2$, $q=1$ for some functions satisfying $\psi '^2+\phi '^2=1$ and (4.15),

• (iii) A generalized cylinder over a rotational surface, given by (4.14) with $p=1$, $q=n-2$, for some functions satisfying $\psi '^2+\phi '^2=1$ and (4.15).