Biconservative Lorentz hypersurfaces in $\mathbb{E}_{1}^{\lowercase{n}+1}$ with complex eigenvalues

TL;DR

This paper proves that biconservative Lorentz hypersurfaces with complex eigenvalues in Lorentzian space have constant mean curvature, and biharmonic ones are minimal, confirming Chen's conjecture in this setting.

Contribution

It classifies biconservative and biharmonic Lorentz hypersurfaces with complex eigenvalues, establishing their geometric properties and confirming Chen's conjecture.

Findings

Biconservative Lorentz hypersurfaces with complex eigenvalues have constant mean curvature.

Biharmonic Lorentz hypersurfaces with complex eigenvalues are minimal.

The results affirm Chen's conjecture on biharmonic submanifolds.

Abstract

Our paper is an attempt to to verify the Chen's conjecture on biharmonic submanifolds and to classify biconservative submanifolds. In doing so we provide an affirmative answer to Chen's conjecture on biharmonic submanifolds. We prove that every biconservative Lorentz hypersurface $M_{1}^{n}$ in $\mathbb{E}_{1}^{n+1}$ having complex eigenvalues has constant mean curvature. Moreover, every biharmonic Lorentz hypersurface $M_{1}^{n}$ having complex eigenvalues in $\mathbb{E}_{1}^{n+1}$ must be minimal.

Equations

$$\mathcal{A}=\left(\begin{array}[]{ccccccccc}\lambda&\mu&\\ -\mu&\lambda&&\\ &&D_{n-2}&\\ \end{array}\right),\hskip 14.22636pt$$

$$g(e_1,e_1)=-1,g(e_i,e_i)=1,i=2,3,\dots ,n,$$

$$g(e_i,e_j)=0,\forall i,j=1,2,\dots ,n,i\ne j,$$

$$△\vec{H}=0,$$

$$H=\frac{1}{n}\mathrm{trace}\mathcal{A},$$

$$R(X,Y)Z=g(\mathcal{A}Y,Z)\mathcal{A}X-g(\mathcal{A}X,Z)\mathcal{A}Y,$$

$$({\nabla }_X\mathcal{A})Y=({\nabla }_Y\mathcal{A})X,$$

$$({\nabla }_X\mathcal{A})Y={\nabla }_X\mathcal{A}Y-\mathcal{A}({\nabla }_{X}Y)$$

$$\mathcal{A}(\mathrm{grad}H)+\frac{n}{2}H\mathrm{grad}H=0.$$

$${\mathcal{A}}_H{e}_1=\lambda e_1-\mu e_2,{\mathcal{A}}_H{e}_2=\mu e_1+\lambda e_2,{\mathcal{A}}_H{e}_i={\lambda }_i{e}_i,i\in D,$$

$$A=\{1,2,\dots ,n\},B=\{1,2,\dots ,n-1\},C=\{3,4,\dots ,n-1\}.$$

$$\mathrm{grad}H=\sum _{i=1}^n{e}_i(H)e_i.$$

$$e_n(H)\ne 0,e_i(H)=0,i\in B.$$

$${\nabla }_{e_i}{e}_j=\sum _{m=1}^n{\omega }_{ij}^m{e}_m,i,j\in A.$$

$${\omega }_{ki}^i=0,{\omega }_{ki}^j+{\omega }_{kj}^i=0,$$

$$2\lambda +\sum _{j=3}^{n-1}{\lambda }_j=\frac{3nH}{2}=-3{\lambda }_n.$$

$$e_i({\lambda }_j)=({\lambda }_i-{\lambda }_j){\omega }_{ji}^j=({\lambda }_j-{\lambda }_i){\omega }_{jj}^i,$$

$$({\lambda }_i-{\lambda }_j){\omega }_{ki}^j=({\lambda }_k-{\lambda }_j){\omega }_{ik}^j,$$

$$({\lambda }_j-\lambda ){\omega }_{ij}^1-\mu {\omega }_{ij}^2=({\lambda }_i-\lambda ){\omega }_{ji}^1-\mu {\omega }_{ji}^2$$

$$({\lambda }_j-\lambda ){\omega }_{ij}^2+\mu {\omega }_{ij}^1=({\lambda }_i-\lambda ){\omega }_{ji}^2+\mu {\omega }_{ji}^1,$$

$$({\lambda }_i-{\lambda }_j){\omega }_{1i}^j=-\mu {\omega }_{i2}^j+(\lambda -{\lambda }_j){\omega }_{i1}^j,$$

$$({\lambda }_j-{\lambda }_i){\omega }_{1j}^i=-\mu {\omega }_{j2}^i+(\lambda -{\lambda }_i){\omega }_{j1}^i,$$

$$({\lambda }_i-\lambda ){\omega }_{1i}^1-\mu {\omega }_{1i}^2=e_i(\lambda ),$$

$$({\lambda }_i-\lambda ){\omega }_{1i}^2+\mu {\omega }_{1i}^1=-e_i(\mu ),$$

$$e_1({\lambda }_i)=-\mu {\omega }_{i2}^i+(\lambda -{\lambda }_i){\omega }_{i1}^i,$$

$$({\lambda }_i-{\lambda }_j){\omega }_{2i}^j=\mu {\omega }_{i1}^j+(\lambda -{\lambda }_j){\omega }_{i2}^j,$$

$$({\lambda }_j-{\lambda }_i){\omega }_{2j}^i=\mu {\omega }_{j1}^i+(\lambda -{\lambda }_i){\omega }_{j2}^i,$$

$$({\lambda }_i-\lambda ){\omega }_{2i}^1-\mu {\omega }_{2i}^2=e_i(\mu ),$$

$$({\lambda }_i-\lambda ){\omega }_{2i}^2+\mu {\omega }_{2i}^1=e_i(\lambda ),$$

$$e_2({\lambda }_i)=\mu {\omega }_{i1}^i+(\lambda -{\lambda }_i){\omega }_{i2}^i,$$

Full text

AMS2010 MSC Codes: 53D12, 53C40, 53C42

Key Words: Lorentz hypersurfaces, Biconservative submanifolds, Biharmonic submanifolds, Mean curvature vector.

1. Introduction

The classification of constant mean curvature (CMC) hypersurfaces play an important role in relativity theory [16, 23] and such type of hypersurfaces are associated with the problem of eigenvalues of the shape operator or differential equations arises from Laplacian operator.

In 1964, Eells and Sampson [15] introduced the notion of poly-harmonic maps as a natural generalization of the well-known harmonic maps. Thus, while harmonic maps between Riemannian manifolds $\phi:(M,g)\rightarrow(N,h)$ are critical points of the energy functional $E(\phi)=\frac{1}{2}\int_{M}|d\phi|^{2}v_{g}$, the biharmonic maps are critical points of the bienergy functional $E_{2}(\phi)=\frac{1}{2}\int_{M}|\tau(\phi)|^{2}v_{g}$, where $\tau=\mathop{\mathrm{trace}}\nabla d\phi$ is the tension field of $\phi$.

In 1924, Hilbert pointed that the stress-energy tensor associated to a functional $E$, is a conservative symmetric 2-covariant tensor $S$ at the critical points of $E$, i.e. $\mathop{\mathrm{div}}S=0$ [11]. For the bienergy functional $E_{2}$, Jiang defined the stress-bienergy tensor $S_{2}$ and proved that it satisfies $\mathop{\mathrm{div}}S_{2}=-\langle\tau_{2}(\phi),d\phi\rangle$ [13]. Thus, if $\phi$ is biharmonic, then $\mathop{\mathrm{div}}S_{2}=0$. For biharmonic submanifolds, from the above relation, we see that $\mathop{\mathrm{div}}S_{2}=0$ if and only if the tangent part of the bitension field vanishes. In particular, an isometric immersion $\phi:(M,g)\rightarrow(N,h)$ is called biconservative if $\mathop{\mathrm{div}}S_{2}=0$.

The biconservative submanifolds were studied and classified in $\mathbb{E}^{4}$ by Hasanis and Vlachos [25] in which the biconservative hypersurfaces were called $H$-hypersurfaces. In [18], the complete classification of H-hypersurfaces with three distinct curvatures in Euclidean space of arbitrary dimension was obtained and some explicit example was given. Upadhyay and Turgay, classified biconservative hypersurfaces in $E^{5}_{2}$ with diagonal shape operator having three distinct principal curvatures [3]. Further, they have constructed the example of biconservative hypersurfaces with four distinct principal curvatures. Recently in [8], it was proved that every biconservative Lorentz hypersurface in $E^{n+1}_{1}$ with complex eigenvalues having at most five distinct principal curvatures has constant mean curvature. Further, it was proved that biconservative Lorentz hypersurface with constant length of second fundamental form and whose shape operator has complex eigenvalues with six distinct principal curvatures has constant mean curvature [8]. For more work on biconservative hypersurfaces in pseudo-Euclidean spaces (please see references in [8, 3]). For work on biharmonic submanifolds (please see [4, 19], and references therein).

In this paper, we study biconservative Lorentz hypersurfaces in $\mathbb{E}_{1}^{n+1}$ whose shape operator has complex eigenvalues. The shape operator of Lorentz hypersurfaces with complex eigenvalues takes the form [2, 17]

[TABLE]

with respect to a suitable orthonormal base field of the tangent bundle $\{e_{1},e_{2},...,e_{n}\}$ of $T_{p}M_{1}^{n}$, which satisfies

[TABLE]

and

[TABLE]

where $D_{n-2}=diag\{\lambda_{3},\lambda_{4},\dots,\lambda_{n}\}$ and $\mu\neq 0$.

We prove

Theorem 1.1**.**

Let $M_{1}^{n}$ be a biconservative Lorentz hypersurface in $\mathbb{E}_{1}^{n+1}$ with complex eigenvalues. Then, it has constant mean curvature.

The submanifolds satisfying

[TABLE]

is called biharmonic submanifold.

The study of biharmonic submanifolds in Euclidean spaces was initiated by Chen in mid 1980s. In particular, he posed the following well-known conjecture in 1991:

The only biharmonic submanifolds of Euclidean spaces are the minimal ones.

The conjecture was later studied by many researchers and so far it is found to be true for hypersurfaces in Euclidean spaces [10, 14, 19, 21, 22, 25]. Chen’s conjecture is not true always for the submanifolds of the semi-Euclidean spaces (see [5, 6, 7]). However, for hypersurfaces in semi-Euclidean spaces, Chen’s conjecture is also right (see [1, 6, 7, 9, 12, 20]).

Since, every biconservative hypersurface is a biharmonic hypersurface, therefore, using Theorem 1.1 and normal part $\triangle H+H\mathop{\mathrm{trace}}(\mathcal{A}^{2})=0$ of the biharmonic equation $\triangle\vec{H}=\vec{0}$, we find

Theorem 1.2**.**

Every biharmonic Lorentz hypersurface $M_{1}^{n}$ in $\mathbb{E}_{1}^{n+1}$ with complex eigenvalues must be minimal.

2. Preliminaries

Let $(M_{1}^{n},g)$ be an $n$-dimensional Lorentz hypersurface isometrically immersed in $(\mathbb{E}^{n+1}_{1},\overline{g})$ and $g=\overline{g}_{|M_{1}^{n}}$. We denote by $\xi$ unit normal vector to $M_{1}^{n}$ where $\overline{g}(\xi,\xi)=1$. A vector $X$ in $\mathbb{E}_{1}^{n+1}$ is called spacelike, timelike or lightlike according as   $\overline{g}(X,X)>0,\hskip 5.69046pt\overline{g}(X,X)<0$ or    $\overline{g}(X,X)=0$, respectively.

The mean curvature $H$ of $M_{1}^{n}$ is given by

[TABLE]

where $\mathcal{A}$ is the shape operator of $M_{1}^{n}$.

Let $\nabla$ denotes the Levi-Civita connection on $M_{1}^{n}$. Then, the Gauss and Codazzi equations are given by

[TABLE]

[TABLE]

respectively, where $R$ is the curvature tensor and

[TABLE]

for all $X,Y,Z\in\Gamma(TM_{1}^{n})$.

The submanifolds satisfying $\triangle\vec{H}=0$, is called biharmonic submanifold [4]. The biharmonic equation can be decomposed into its normal and tangent part. Then, the submanifolds satisfying the tangential part of the biharmonic equation is called biconservative. Therefore, the biconservative Lorentz hypersurfaces $M_{1}^{n}$ in $\mathbb{E}^{n+1}_{1}$ characterized by

[TABLE]

3. Biconservative Lorentz hypersurfaces in $\mathbb{E}^{n+1}_{1}$

In this section, we study biconservative Lorentz hypersurface in $\mathbb{E}_{1}^{n+1}$ with complex eigenvalues. Since every hypersurface with constant mean curvature is always biconservative, therefore, we assume that the mean curvature is not constant and $\mathop{\mathrm{grad}}H\neq 0$. Assuming non-constant mean curvature implies the existence of an open connected subset $U$ of $M^{n}_{1}$ with grad${}_{x}H\neq 0$, for all $x\in U$. From (2.5), it is easy to see that $\mathop{\mathrm{grad}}H$ is an eigenvector of the shape operator $\mathcal{A}$ with the corresponding principal curvature $-\frac{n}{2}H$. Therefore, there do not exist biconservative Lorentz hypersurface $M^{n}_{1}$ in $\mathbb{E}^{n+1}_{1}$ with two distinct principal curvatures of non-constant mean curvature with complex eigenvalues. Without losing generality, we choose $e_{n}$ in the direction of $\mathop{\mathrm{grad}}H$. Then, the shape operator $\mathcal{A}$ of hypersurfaces $M_{1}^{n}$ in $\mathbb{E}^{n+1}_{1}$ will take the following form with respect to a suitable orthonormal frame $\{e_{1},e_{2},\dots,e_{n}\}$

[TABLE]

where $D=\{3,4,\dots,n\}$.

Also, we denote the following sets by

[TABLE]

The $\mathop{\mathrm{grad}}H$ can be expressed as

[TABLE]

As we have taken $e_{n}$ parallel to $\mathop{\mathrm{grad}}H$, consequently

[TABLE]

We express

[TABLE]

Differentiating (1.2) and (1.3) with respect to $e_{k}$ and using (3.4), we obtain

[TABLE]

for $i\neq j$ and $i,j,k\in A$.

Using (2.1) and (3.1), we obtain that

[TABLE]

Now, we have

Lemma 3.1**.**

Let $M^{n}_{1}$ be a biconservative Lorentz hypersurface in $\mathbb{E}^{n+1}_{1}$ having the shape operator given by $(\ref{e:b1})$ with respect to a suitable orthonormal frame $\{e_{1},e_{2},\dots,e_{n}\}$. Then,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

for distinct $i,j,k\in D$ such that $\lambda_{k}\neq\lambda_{j}\neq\lambda_{i}$.

Proof. Taking $X=e_{i},Y=e_{j}$ in (2.4) and using (3.1) and (3.4), we get

[TABLE]

Putting the value of $(\nabla_{e_{i}}\mathcal{A})e_{j}$ in (2.3), we find

[TABLE]

whereby for $i\neq j=k$ and $i\neq j\neq k$, we obtain (3.7) and (3.8), respectively. Moreover, using (3.1) in (3.24) and comparing the coefficients of $e_{1}$ and $e_{2}$, we find (3.9) and (3.10), respectively.

Next, using (3.1) and (2.4) in $(\nabla_{e_{1}}\mathcal{A})e_{i}=(\nabla_{e_{i}}\mathcal{A})e_{1}$, for $i\in D$, we obtain

[TABLE]

whereby for $i\neq j$, we get (3.11). Further, comparing the coefficients of $e_{1}$, $e_{2}$, and $e_{i}$ and using (3.5), we have (3.13), (3.14) and (3.15), respectively.

Also, using (3.1) and (2.4) in $((\nabla_{e_{1}}\mathcal{A})e_{j},e_{i})=g((\nabla_{e_{j}}\mathcal{A})e_{1},e_{i})$, gives (3.12).

Similarly, using (3.1) and (2.4) in $(\nabla_{e_{2}}\mathcal{A})e_{i}=(\nabla_{e_{i}}\mathcal{A})e_{2}$, for $i\in D$, we get

[TABLE]

whereby for $i\neq j$, we get (3.16). Further, comparing the coefficients of $e_{1}$, $e_{2}$, and $e_{i}$ and using (3.5), we have (3.19), (3.18) and (3.20), respectively.

Also, using (3.1) and (2.4) in $((\nabla_{e_{2}}\mathcal{A})e_{j},e_{i})=g((\nabla_{e_{j}}\mathcal{A})e_{2},e_{i})$, gives (3.17).

Now, using (3.1) and (2.4) in $(\nabla_{e_{1}}\mathcal{A})e_{2}=(\nabla_{e_{2}}\mathcal{A})e_{1}$, for $i\in D$, we obtain

[TABLE]

whereby comparing the coefficients of $e_{1}$, $e_{2}$, and $e_{i}$ and using (3.5), we have (3.21), (3.22) and (3.23), respectively. This completes the proof of the Lemma.

Next, we have

Lemma 3.2**.**

Let $M^{n}_{1}$ be a biconservative Lorentz hypersurface in $\mathbb{E}^{n+1}_{1}$ having the shape operator given by $(\ref{e:b1})$ with respect to a suitable orthonormal frame $\{e_{1},e_{2},\dots,e_{n}\}$. Then,

[TABLE]

Proof. Let $\lambda_{n}=\lambda_{k}$ for $k\in C$, then taking $i=n$ and $j=k$ in (3.7), we get

[TABLE]

which contradicts (3.3). Whereby completing the proof of Lemma.

Using (3.3), (3.4) and the fact that $[e_{i}\hskip 2.84544pte_{j}](H)=0=\nabla_{e_{i}}e_{j}(H)-\nabla_{e_{j}}e_{i}(H)=\omega_{ij}^{n}e_{n}(H)-\omega_{ji}^{n}e_{n}(H),$ for $i\neq j$, we find

[TABLE]

Lemma 3.3**.**

Let $M^{n}_{1}$ be a biconservative Lorentz hypersurface in $\mathbb{E}^{n+1}_{1}$ having the shape operator given by $(\ref{e:b1})$ with respect to a suitable orthonormal frame $\{e_{1},e_{2},\dots,e_{n}\}$. Then,

[TABLE]

Proof. Putting $i\neq n,j=n$ in (3.7) and using (3.3) and (3.5), we find

[TABLE]

Taking $i=n$ in (3.15) and (3.20) and using (3.3) and (3.5), we find

[TABLE]

Combining (3.31) and (3.32), we get (3.30).

Lemma 3.4**.**

Let $M^{n}_{1}$ be a biconservative Lorentz hypersurface in $\mathbb{E}^{n+1}_{1}$ having the shape operator given by $(\ref{e:b1})$ with respect to a suitable orthonormal frame $\{e_{1},e_{2},\dots,e_{n}\}$. Then,

[TABLE]

Proof. Taking $i=n$ in (3.23) and using (3.29), we get

[TABLE]

Taking $i=n$ in (3.13), (3.14), (3.18), (3.19) and using (3.5), (3.29), and (3.34), we find

[TABLE]

Solving (3.35), we get

[TABLE]

Using (3.36), (3.34) and (3.29), we get (3.33).

Lemma 3.5**.**

Let $M^{n}_{1}$ be a biconservative Lorentz hypersurface in $\mathbb{E}^{n+1}_{1}$ having the shape operator given by $(\ref{e:b1})$ with respect to a suitable orthonormal frame $\{e_{1},e_{2},\dots,e_{n}\}$. Then,

[TABLE]

Proof. Using (3.11), (3.12) and (3.5), we get

[TABLE]

Similarly, using (3.16), (3.17) and (3.5), we find

[TABLE]

Combining (3.10) and (3.39), we obtain

[TABLE]

Combining (3.9) and (3.38), we find

[TABLE]

Using (3.38), (3.39), (3.40) and (3.41), we get (3.37). Whereby completing the proof of the Lemma.

Now, we find following Lemma for covariant derivative.

Lemma 3.6**.**

Let $M^{n}_{1}$ be a biconservative Lorentz hypersurface in $\mathbb{E}^{n+1}_{1}$ having the shape operator given by $(\ref{e:b1})$ with respect to a suitable orthonormal frame $\{e_{1},e_{2},\dots,e_{n}\}$. Then,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

*for $i\in C$. Moreover,

$(a)$ if $M_{1}^{n}$ has all distinct principal curvatures, then*

[TABLE]

$(b)$* if $M_{1}^{n}$ has ${}^{\prime}q^{\prime}$ distinct principal curvatures $\lambda\pm\sqrt{-1}\mu,\lambda_{3},\dots,\lambda_{q-1},\lambda_{n}$, with multiplicities $p_{3},\dots,p_{q-1}$ of $\lambda_{3},\dots,\lambda_{q-1}$, respectively, such that $p_{3}+p_{4}+\dots+p_{q-2}+p_{q-1}=n-3$. Then*

[TABLE]

[TABLE]

where $i_{1}=3,2,\dots,q-1$, and $C_{3}=\{3,\dots,p_{3}+2\},C_{4}=\{p_{3}+3,\dots,p_{3}+p_{4}+2\},\dots,C_{q-1}=\{p_{3}+p_{4}+\dots+p_{q-2}+3,\dots,n-1\}$, and $\omega_{ij}^{i}$ satisfy $(\ref{e:b5})$ and $(\ref{e:b6})$.

Proof. (a) Let $M_{1}^{n}$ has all distinct principal curvatures. Putting $j=n$ and $k=j$ in (3.8) and using (3.29), we get

[TABLE]

Putting $i=n$ and $k=i$ in (3.8) and using (3.42) and (3.5), we find

[TABLE]

(b) Let $M_{1}^{n}$ has ${}^{\prime}q^{\prime}$ distinct principal curvatures. Putting $i=n$ and $k=i$ in (3.8), we obtain

[TABLE]

Putting $j=n$ and $k=j$ in (3.8) and using (3.29), we get

[TABLE]

Taking $i\in C_{i_{1}}$ in (3.8), we have

[TABLE]

Putting $i=n$ and $k=i$ in (3.8) and using (3.45) and (3.5), we find

[TABLE]

Now, using Lemma 3.3, Lemma 3.4, Lemma 3.5 and (3.42), (3.43), (3.44), (3.45), (3.46) and (3.47) in (3.4), completes the proof of the Lemma.

Next, we have

Lemma 3.7**.**

Let $M^{n}_{1}$ be a biconservative Lorentz hypersurface in $\mathbb{E}^{n+1}_{1}$ having the shape operator given by $(\ref{e:b1})$ with respect to a suitable orthonormal frame $\{e_{1},e_{2},\dots,e_{n}\}$. Then,

[TABLE]

Proof. Evaluating $g(R(e_{n},e_{1})e_{n},e_{1})$, using (2.2), (3.1) and Lemma 3.6, we have

[TABLE]

which gives

[TABLE]

Since $\lambda_{n}\neq 0$, therefore, from (3.50), we find (3.48). Thus completing the proof of the Lemma.

Now, using Lemma 3.7, we find following Theorem.

Theorem 3.8**.**

There do not exist biconservative Lorentz hypersurface $M^{n}_{1}$ in $\mathbb{E}^{n+1}_{1}$ with three distinct principal curvatures of non-constant mean curvature with complex eigenvalues.

Proof. Let $M_{1}^{n}$ has three distinct principal curvatures. Then, from (3.6) and (3.48), we get $H=0$, a contradiction. Which completes the proof of the theorem.

Next, we have

Lemma 3.9**.**

Let $M^{n}_{1}$ be a biconservative Lorentz hypersurface in $\mathbb{E}^{n+1}_{1}$ having the shape operator given by $(\ref{e:b1})$ with respect to a suitable orthonormal frame $\{e_{1},e_{2},\dots,e_{n}\}$. Then,

[TABLE]

and

[TABLE]

Proof. Using (3.48) and (3.5) in (3.13) and (3.19), we find

[TABLE]

respectively.

On the other hand, adding (3.14) and (3.18), and therein using (3.48), (3.53) and (3.5), we obtain

[TABLE]

which together with (3.53) gives

[TABLE]

Using (3.48), (3.54) and (3.55) in (3.23), we get

[TABLE]

Therefore, from (3.53) and (3.56), we obtain

[TABLE]

We claim that $\omega_{11}^{i}=0$. In fact, if $\omega_{11}^{i}\neq 0$, then $\mu^{2}-\lambda_{i}^{2}=0$. Which gives $\lambda_{i}=\pm\mu$ for all $i\in C$. In view of Theorem 3.8, we consider the following cases:

Case I. Let $M_{1}^{n}$ has four distinct principal curvatures. Then, using (3.48) and Lemma 3.2 in (3.6), we obtain $(n-3)\lambda_{i}=\frac{3nH}{2}$ or $\pm(n-3)\mu=\frac{3nH}{2}$. Which on differentiating with respect to $e_{n}$ gives $\pm(n-3)e_{n}(\mu)=\frac{3ne_{n}(H)}{2}$. Also, using (3.33) in (3.14), we find $e_{n}(\mu)=0$. Therefore, we obtain $e_{n}(H)=0$, a contradiction.

Case II. Let $M_{1}^{n}$ has five distinct principal curvatures $\lambda\pm\sqrt{-1}\mu,\lambda_{3}=\mu,\lambda_{4}=-\mu,\lambda_{n}$. Then, using (3.48) and Lemma 3.2 in (3.6), we get $(p_{3}-p_{4})\mu=\frac{3nH}{2}$, where $p_{3}$ and $p_{4}$ are the multiplicities of $\lambda_{3}$ and $\lambda_{4}$, respectively. Now, proceeding as in Case I, we get a contradiction.

Case III. Let $M_{1}^{n}$ has more than five distinct principal curvatures. Then, $\lambda_{i}=\pm\mu$ for all $i\in C$ gives a contradiction to more than five distinct principal curvatures.

Hence $\omega_{11}^{i}=0$. Using this in (3.54), (3.55) and (3.56), we find (3.51).

Using (3.33), (3.48) and (3.51) in (3.14), (3.21) and (3.22), we get

[TABLE]

Hence $\mu$ is constant in all direction. This completes the proof of the Lemma.

Lemma 3.10**.**

Let $M^{n}_{1}$ be a biconservative Lorentz hypersurface in $\mathbb{E}^{n+1}_{1}$ having the shape operator given by $(\ref{e:b1})$ with respect to a suitable orthonormal frame $\{e_{1},e_{2},\dots,e_{n}\}$. Then, $g(R(e_{n},e_{i})e_{n},e_{i})$, $g(R(e_{n},e_{i})e_{i},e_{1})$, $g(R(e_{n},e_{i})e_{i},e_{2})$, $g(R(e_{i},e_{1})e_{i},e_{n})$, $g(R(e_{i},e_{2})e_{i},e_{n})$, $g(R(e_{i},e_{2})e_{i},e_{1})$ and $g(R(e_{i},e_{1})e_{i},e_{2})$ give the following:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

respectively, for all $i\in C$.

Proof: Here, we give the proof of the first two relations (3.59) and (3.60). The proof of the other relations can be obtained in a similar way.

Using (2.2) and (3.1), we have

[TABLE]

[TABLE]

for all $i\in C$.

(i) Let $M_{1}^{n}$ has all the distinct principal curvatures. Then, using Lemma 3.6, we get

[TABLE]

for all $i\in C$.

Therefore, from (3.66) and (3.68), we get (3.59).

Next, we know that

[TABLE]

for all $i\in C$.

Now, using Lemma 3.6 and Lemma 3.9, we have

[TABLE]

[TABLE]

[TABLE]

Hence, using above in (3.69), we get

[TABLE]

for all $i\in C$.

Therefore, from (3.67) and (3.70), we get (3.60) for $i\in C$.

(ii) Let $M_{1}^{n}$ has ’q’ distinct principal curvatures. Then, using Lemma 3.6, we find

\begin{array}[]{lcl}g(R(e_{n},e_{i})e_{n},e_{i})=g(\nabla_{e_{n}}\nabla_{e_{i}}e_{n}-\nabla_{e_{i}}\nabla_{e_{n}}e_{n}-\nabla_{[e_{n}\hskip 2.84544pte_{i}]}e_{n},e_{i})\\ =g(\nabla_{e_{n}}(-\omega_{ii}^{n}e_{i})-\sum_{m\neq i}^{C_{3}}\omega_{ni}^{m}(\nabla_{e_{m}}e_{n})-\omega_{ii}^{n}\nabla_{e_{i}}e_{n},e_{i})\\ =g(-e_{n}(\omega_{ii}^{n})e_{i}-\sum_{m\neq i}^{C_{3}}\omega_{ni}^{m}(\sum_{l\neq n}^{C_{3}}\omega_{mn}^{l}e_{l})+(\omega_{ii}^{n})^{2}e_{i},e_{i}),\end{array}

wherein using (3.44), gives

[TABLE]

for all $i\in C_{3}$.

Therefore, from (3.66) and (3.71), we get (3.59) for $i\in C_{3}$. Similarly, for all $i\in C$, we find (3.59).

Next, using Lemma 3.6 and Lemma 3.9 for $i\in C_{3}$, we have

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Hence, using above in (3.69), we get

[TABLE]

for all $i\in C_{3}$.

Therefore, from (3.67) and (3.72), we get (3.60) for $i\in C_{3}$. Similarly, for all $i\in C$, we find (3.60).

4. Proof of the theorem

Using Lemma 3.6, we get

[TABLE]

Operating $\omega_{ii}^{n}$ on both sides in (4.1), we find

[TABLE]

Using (3.59), (3.60), (3.62), (3.63), (3.15), (3.3) and Lemma 3.7 in (4.2), we obtain

[TABLE]

whereby, we find

[TABLE]

Now, using Lemma 3.6, we get

[TABLE]

Using (3.59), (3.61), (3.62), (3.63), (3.20), (3.3) and Lemma 3.7 in (4.5), we obtain

[TABLE]

Using (4.4) and (4.6) in (3.64), we find $\lambda_{i}=0$ for all $i\in C$. Using this in (3.6), we get $H=0$, a contradiction. Whereby proof of Theorem 1.1 is complete.