Biconservative Hypersurfaces in Space Forms $\overline{M}^{{n+1}}({c})$

Abstract

In this paper we study biconservative hypersurfaces M in space forms $\overline{M}^{n+1}(c)$ with four distinct principal curvatures whose second fundamental form has constant norm. We prove that every such hypersurface has constant mean curvature and constant scalar curvature.

1 Introduction

In the last three decades, one of the interesting research topics in Differential Geometry is the study of biharmonic maps, in particular, biharmonic immersions, between Riemannian manifolds. A generalization of harmonic maps, proposed in 1964 by Eells and Sampson [7], these maps are critical points of the bienergy functional, obtained by integrating the squared norm of the tension field, and are characterized by the vanishing of the bitension field. Another interesting research direction, derived from here, is the study of biconservative submanifolds, i.e., those submanifolds for which only the tangent part of the bitension field vanishes.

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. $\mathop {\mathrm {div}}S = 0$ [17]. For the bienergy functional $E_2$, Jiang defined the stress-bienergy tensor $S_2$ and proved that it satisfies $\mathop {\mathrm {div}}S_2 = -\langle \tau _2(\phi ), d\phi \rangle$ [18]. 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 $\mathop {\mathrm {div}}S_2 = 0$.

In a different setting, B. Y. Chen defined biharmonic submanifolds M of the Euclidean space as those with harmonic mean curvature vector field, that is $\Delta \mathbf {H} =0$, where $\Delta$ is the Laplacian operator. If we apply the definition of biharmonic maps to Riemannian immersions into the Euclidean space, we recover Chen’s notion of biharmonic submanifolds. Thus biharmonic Riemannian immersions can also be thought of as a generalization of Chen’s biharmonic submanifolds. The biharmonic submanifolds were studied in [1, 3,4,5,6, 11,12,13, 23] and references therein.

The biconservative submanifolds were studied and classified in $\mathbb {E}^3$ and $\mathbb {E}^4$ by Hasanis and Vlachos [16], in which the biconservative hypersurfaces were called H-hypersurfaces. The terminology “biconservative”was first introduced in [2]. The classification of H-hypersurfaces with three distinct curvatures in Euclidean space of arbitrary dimension were obtained by Turgay in [24]. The classification of biconservative hypersurfaces in $\mathbb {E}^5_2$ with diagonal shape operator having three distinct principal curvatures was obtained by Upadhyay and Turgay in [26]. Also, the first author and Sharfuddin proved that every biconservative Lorentz hypersurface in $\mathbb {E}^{n+1}_1$ with complex eigenvalues has constant mean curvature [15]. For more work on biconservative hypersurfaces in pseudo-Euclidean spaces, please see [15, 26] and references therein.

The constant mean curvature (CMC) biconservative surfaces in $\mathbb {S}^n \times \mathbb {R}$ and $\mathbb {H}^n \times \mathbb {R}$ were studied in [9] by Fetcu et al. A complete classification of CMC biconservative surfaces in a four-dimensional space form was given in [21] by Montaldo et al. Further, the classification of biconservative hypersurfaces in $\mathbb {S}^4$ and $\mathbb {H}^4$ was obtained in [25] by Turgay and Upadhyay. A survey about biharmonic and biconservative hypersurfaces in space forms is provided in [10] by Fetcu et al. In [14] the first author studied the biconservative hypersurfaces in Euclidean 5-space with constant norm of second fundamental form and one of the results states that every such hypersurface has constant mean curvature. Finally, some recent results on complete biconservative surfaces and on closed biconservative surfaces in space forms was obtained by Nistor and Oniciuc [20] and Montaldo and by Pampano [22].

In view of the above developments, in the present paper we study biconservative hypersurfaces in a space form $\overline{M}^{n+1}(c)$ with at most four distinct principal curvatures, whose second fundamental form has constant norm. The main result is the following:

Theorem 1.1

Every biconservative hypersurface M in a space form $\overline{M}^{n+1}(c)$ with at most four distinct principal curvatures, whose second fundamental form has constant norm, is of constant mean curvature and of constant scalar curvature.

Also, the class of biconservative hypersurface in space forms contains some other class of hypersurfaces, that is, the equation (2.5) is fulfilled by a biharmonic submanifold $\triangle \mathbf {H}=0$ and a hypersurface with $\triangle \mathbf {H}=\lambda \mathbf {H}$ [3].

Therefore, from Theorem 1.1, we have following:

Corollary 1.2

Every biharmonic hypersurface M in a space form $\overline{M}^{n+1}(c)$ with at most four distinct principal curvatures, whose second fundamental form has constant norm, is of constant mean curvature and of constant scalar curvature.

Corollary 1.3

Every hypersurface M in a space form $\overline{M}^{n+1}(c)$ satisfying $\triangle \mathbf {H}=\lambda \mathbf {H}$ with at most four distinct principal curvatures, whose second fundamental form has constant norm, is of constant mean curvature and of constant scalar curvature.

The paper is organized as follows: in Sect. 2 we give some preliminaries. In Sect. 3 we discuss biconservative hypersurfaces in space forms with four distinct principal curvatures. We obtain some condition for eigenvalues (Lemma 3.1) and expressions of covariant derivatives of an orthonormal frame in terms of connection forms (Lemma 3.2). In Sect. 4 we use the condition that the second fundamental form has constant norm, and obtain further simplifications of the connection forms (Lemmas 4.1 and 4.2). Sect. 5 is devoted to the proof of Theorem 1.1. In the process, a technical tool about resultant of polynomials is extensively used (Lemma 2.1).

Finally, we would like to note that Lemma 4.1 is crucial to obtain our results in the case of at most four distinct principal curvatures. In case of more than four distinct principal curvatures, it seems difficult to obtain an analogue of this lemma. We may discuss this in a separate paper.

2 Preliminaries

Let (M, g) be a hypersurface isometrically immersed in a $(n+1)$-dimensional space form $(\overline{M}^{n+1}(c), \overline{g})$ and $g = \overline{g}_{|M}$. We denote by $\xi$ a unit normal vector to M where $\overline{g}(\xi , \xi )= 1$, and by h and $\mathcal {A}$ the second fundamental form and the shape operator of M, where $h(X, Y)=g(\mathcal {A}(X), Y)$, for $X, Y \in \Gamma (TM)$.

The mean curvature H of M is given by

$$\begin{aligned} H = \frac{1}{n} \mathop {\mathrm {trace}}\mathcal {A}. \end{aligned}$$

(2.1)

Let $\nabla$ denotes the Levi-Civita connection on M. Then, the Gauss and Codazzi equations are given by

$$\begin{aligned} R(X, Y)Z = c(g(Y, Z) X - g(X, Z) Y)+g(\mathcal {A}Y, Z) \mathcal {A}X - g(\mathcal {A}X, Z) \mathcal {A}Y, \end{aligned}$$

(2.2)

$$\begin{aligned} (\nabla _{X}\mathcal {A})Y = (\nabla _{Y}\mathcal {A})X, \end{aligned}$$

(2.3)

respectively, where R is the curvature tensor and

$$\begin{aligned} (\nabla _{X}\mathcal {A})Y = \nabla _{X}\mathcal {A} Y- \mathcal {A}(\nabla _{X}Y) \end{aligned}$$

(2.4)

for all $X, Y, Z \in \Gamma (TM)$.

A submanifold satisfying $\triangle \mathbf {H} = 0$, is called biharmonic submanifold [3]. The biharmonic equation for $M^n$ in a $(n+1)$-dimensional space form $\overline{M}^{n+1}(c)$ can be decomposed into its normal and tangent part [3, 8], that is

$$\begin{aligned}&2\mathcal {A} (\mathop {\mathrm {grad}}H)+ n H \mathop {\mathrm {grad}}H = 0, \end{aligned}$$

(2.5)

$$\begin{aligned}&\Delta H+H(\mathrm{trace}(\mathcal {A}^2)-n c)=0. \end{aligned}$$

(2.6)

Definition 2.1

A submanifold satisfying equation (2.5) is called biconservative.

In the present work we are concerned with biconservative hypersurfaces $M^n$ in a $(n+1)$-dimensional space form $\overline{M}^{n+1}(c)$.

The following algebraic lemma will be useful to get our result:

Lemma 2.1

([19, Theorem 4.4, pp. 58–59]) Let D be a unique factorization domain, and let $f(X) = a_{0}X^{m} +a_{1}X^{m-1} + \dots + a_{m}, g(X) = b_{0}X^{n} + b_{1}X^{n-1} + \dots + b_{n}$ be two polynomials in D[X]. Assume that the leading coefficients $a_{0}$ and $b_{0}$ of f(X) and g(X) are not both zero. Then f(X) and g(X) have a nonconstant common factor iff the resultant $\Re (f, g)$ of f and g is zero, where

$$\begin{aligned} \Re (f,g)= \begin{vmatrix} a_{0}&\quad a_{1}&\quad a_{2}&\quad \cdots&\quad a_{m}&&\\&\quad a_{0}&\quad a_{1}&\quad \cdots&\quad \cdots&\quad a_{m}&\\&\quad \ddots&\quad \ddots&\quad \ddots&\quad \ddots&\quad \ddots&\\&&\quad a_{0}&\quad a_{1}&\quad a_{2}&\quad \cdots&\quad a_{m} \\ b_{0}&\quad b_{1}&\quad b_{2}&\quad \cdots&\quad b_{n}&&\\&\quad b_{0}&\quad b_{1}&\quad \cdots&\quad \cdots&\quad b_{n}&\\&\quad \ddots&\quad \ddots&\quad \ddots&\quad \ddots&\quad \ddots&\\&&\quad b_{0}&\quad b_{1}&\quad b_{2}&\quad \cdots&\quad b_{n} \\ \end{vmatrix}, \end{aligned}$$

and there are n rows of “a”entries and m rows of “b”entries.

3 Biconservative Hypersurfaces in Space Forms with Four Distinct Principal Curvatures

In this section, we study biconservative hypersurfaces with four distinct principal curvatures in space forms. In view of (2.5), it is easy to see that any CMC hypersurface is biconservative. Therefore, we are interested in the study of non CMC biconservative hypersurfaces in a space form $\overline{M}^{n+1}(c)$.

We assume that the mean curvature is not constant, and we will end up to a contradiction. This implies that $\mathop {\mathrm {grad}}H\ne 0$, hence there exists an open connected subset U of $M^n$ with grad$_{x}H \ne 0$, 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 $-\frac{nH}{2}$.

We denote by A, B, the following sets

$$\begin{aligned} A=\{1, 2,\dots ,n\},\quad B=\{2, 3,\dots , n\}. \end{aligned}$$

Without losing generality, we choose $e_{1}$ in the direction of $\mathop {\mathrm {grad}}H$. Then, the shape operator $\mathcal {A}$ of a hypersurface $M^n$ in a space form $\overline{M}^{n+1}(c)$ takes the following form with respect to a suitable orthonormal frame $\{e_{1}, e_{2},\dots , e_{n}\}$,

$$\begin{aligned} \mathcal {A}e_i=\lambda _ie_i, \quad i \in A, \end{aligned}$$

(3.1)

where $\lambda _i$ is the eigenvalue corresponding to the eigenvector $e_i$ of the shape operator.

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

$$\begin{aligned} \mathop {\mathrm {grad}}H =\sum _{i=1}^{n} e_{i}(H)e_{i}. \end{aligned}$$

(3.2)

As we have taken $e_{1}$ parallel to $\mathop {\mathrm {grad}}H$, it follows that

$$\begin{aligned} e_{1}(H)\ne 0, \quad e_{i}(H)= 0, \quad i \in B. \end{aligned}$$

(3.3)

We express

$$\begin{aligned} \nabla _{e_{i}}e_{j}=\sum _{m=1}^{n}\omega _{ij}^{m}e_{m}, \quad i, j \in A. \end{aligned}$$

(3.4)

From (3.4) and using the compatibility conditions $(\nabla _{e_{k}}g)(e_{i}, e_{i})= 0$ and $(\nabla _{e_{k}}g)(e_{i}, e_{j})= 0$, we obtain

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

(3.5)

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

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

$$\begin{aligned} (\nabla _{e_{i}}\mathcal {A})e_{j}=e_{i}(\lambda _{j})e_{j}+\sum _{k=1}^{n}\omega _{ij}^{k}e_{k} (\lambda _{j}-\lambda _{k}). \end{aligned}$$

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

$$\begin{aligned} e_{i}(\lambda _{j})e_{j}+\sum _{k=1}^{n}\omega _{ij}^{k}e_{k} (\lambda _{j}-\lambda _{k})=e_{j}(\lambda _{i})e_{i}+\sum _{k=1}^{n}\omega _{ji}^{k}e_{k} (\lambda _{i}-\lambda _{k}). \end{aligned}$$

Using $i\ne j=k$ and $i\ne j \ne k$ in the above equation, we obtain

$$\begin{aligned} e_{i}(\lambda _{j})= (\lambda _{i}-\lambda _{j})\omega _{ji}^{j}=(\lambda _{j}-\lambda _{i})\omega _{jj}^{i}, \end{aligned}$$

(3.6)

and

$$\begin{aligned} (\lambda _{i}-\lambda _{j})\omega _{ki}^{j}= (\lambda _{k}-\lambda _{j})\omega _{ik}^{j}, \end{aligned}$$

(3.7)

respectively, for $i, j, k \in A$.

Now, we have

Lemma 3.1

Let $M^n$ be a biconservative hypersurface in a space form $\overline{M}^{n+1}(c)$ having the shape operator given by (3.1) with respect to a suitable orthonormal frame $\{e_{1}, e_{2},\dots , e_{n}\}$. Then,

$$\begin{aligned} \lambda _1\ne \lambda _j,\ \ \text{ for } \text{ all } \ \ j\in B. \end{aligned}$$

(3.8)

Proof

Assume that $\lambda _{j}= \lambda _{1}$ for some $j\ne 1$. Then, from (3.6) we get that

$$\begin{aligned} e_1(\lambda _j)=0, \quad \text{ or }\quad e_1(H)=0, \quad \text{ since }\quad \lambda _1=-\frac{nH}{2}, \end{aligned}$$

which contradicts (3.3), and this completes the proof. $\square$

Therefore, in view of the Lemma 3.1, $\lambda _1= -\frac{n}{2} H$ has multiplicity one. Since M has four distinct principal curvatures, we can assume that $\lambda _1$, $\lambda _u, \lambda _v$ and $\lambda _w$ are the four distinct principal curvatures of the hypersurface M with multiplicities 1, p, q and r respectively, such that

$$\begin{aligned} \lambda _2=\lambda _3=\dots =\lambda _{p+1}=\lambda _u, \ u\in C_1, \end{aligned}$$

$$\begin{aligned} \lambda _{p+2}= & {} \lambda _{p+3}=\dots =\lambda _{p+q+1}=\lambda _v, \ v \in C_2,\\ \lambda _{p+q+2}= & {} \lambda _{p+q+3}=\dots =\lambda _{p+q+r+1}=\lambda _w,\ w\in C_3. \end{aligned}$$

Here $p+q+r+1=n$ and $C_1, C_2$ and $C_3$ denote the sets

$$\begin{aligned} C_1= & {} \{2, 3,\dots , p+1\}, \quad C_2=\{p+2, p+3,\dots , p+q+1\},\\ C_3= & {} \{p+q+2, p+q+3,\dots , n\}. \end{aligned}$$

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

$$\begin{aligned} \sum _{j=2}^n\lambda _j= p\lambda _u+ q\lambda _v+r\lambda _w=\frac{3n}{2}H=-3\lambda _1,\ u\in C_1, \ v\in C_2, \ w\in C_3. \end{aligned}$$

(3.9)

Next, we have

Lemma 3.2

Let $M^n$ be a biconservative hypersurface with four distinct principal curvatures in a space form $\overline{M}^{n+1}(c)$ having the shape operator given by (3.1) with respect to a suitable orthonormal frame $\{e_{1}, e_{2},\dots , e_{n}\}$. Then the following relations are satisfied:

$$\begin{aligned} \nabla _{e_{1}}e_{1}=0, \quad \nabla _{e_{1}}e_{i}=\sum _{ C_{s}}\omega _{1i}^me_{m}\ \ \text{ for } \text{ all }\ \ i \in C_{s}, \ s\in \{1,2,3\}, \quad m\ne i, \end{aligned}$$

$$\begin{aligned} \nabla _{e_{i}}e_{1}= -\omega _{ii}^{1}e_{i} \ \ \text{ for } \text{ all }\ \ i \in B, \quad \nabla _{e_{i}}e_{i}=\sum _{m=1}^n\omega _{ii}^{m}e_{m} \ \ \text{ for } \text{ all }\ \ i \in B, \quad m\ne i, \end{aligned}$$

$$\begin{aligned} \nabla _{e_{i}}e_{j}=\sum _{ C_s}\omega _{ij}^{m}e_{m} \ \ \text{ for } \text{ all }\ \ i, j \in C_s, \ s\in \{1,2,3\},\quad i\ne j, \quad j\ne m, \end{aligned}$$

$$\begin{aligned}&\nabla _{e_{i}}e_{j}=\omega _{ij}^i e_i+\sum _{B\setminus C_s}\omega _{ij}^{m}e_{m} \ \ \text{ for } \text{ all }\ \ i \in C_s, \quad j \in B\setminus C_s, \ s\in \{1,2,3\}, \\&m\ne j, \end{aligned}$$

where $\sum _{C_s}$ and $\sum _{B\setminus C_s}$ denote the summation taken over the corresponding $C_s$ and $B\setminus C_s$, respectively for $s\in \{1, 2, 3\},$ and $\omega _{ij}^{i}$ satisfy (3.5) and (3.6).

Proof

Using (3.3), (3.4) and the fact that $[e_{i} e_{j}](H)=0=\nabla _{e_{i}}e_{j}(H)-\nabla _{e_{j}}e_{i}(H)=\omega _{ij}^{1}e_{1}(H)-\omega _{ji}^{1}e_{1}(H),$ for $i\ne j$, we find that

$$\begin{aligned} \omega _{ij}^{1}=\omega _{ji}^{1}, \quad i, j \in B. \end{aligned}$$

(3.10)

Putting $i\ne 1, j = 1$ in (3.6) and using (3.5) and (3.3), we find

$$\begin{aligned} \omega _{1i}^{1}= 0, \quad i\in A. \end{aligned}$$

(3.11)

Putting $i = 1$ in (3.7), we obtain

$$\begin{aligned} \begin{array}{lcl} \omega _{k1}^{j}= 0, \quad j\ne k \quad \text{ and }\quad j, k \in C_{s}, \ s\in \{1, 2, 3\}. \end{array}\end{aligned}$$

(3.12)

Taking $i \in C_s, \ s\in \{1, 2, 3\}$ in (3.7), we have

$$\begin{aligned} \begin{array}{lcl} \omega _{ki}^{j}= 0, \quad j\ne k\quad \text{ and }\quad j, k \in B\setminus C_s. \end{array}\end{aligned}$$

(3.13)

Putting $j = 1$ in (3.7) and using (3.10), we get

$$\begin{aligned} \begin{array}{lcl} \omega _{ki}^{1}=\omega _{ik}^{1}=0, \quad i \in C_s,\quad k \in B\setminus C_s, \ s\in \{ 1, 2, 3\}. \end{array}\end{aligned}$$

(3.14)

Putting $i = 1$ in (3.7) and using (3.14) and (3.5), we find

$$\begin{aligned} \omega _{1k}^{j}=\omega _{k1}^{j} = 0, \quad j \in C_s, \quad k \in B\setminus C_s, \ s\in \{ 1, 2, 3\}. \end{aligned}$$

(3.15)

Combining (3.14) and (3.12), we obtain

$$\begin{aligned} \begin{array}{lcl} \omega _{ji}^{1}=\omega _{ij}^{1}=0, \quad i\ne j,\quad i, j \in B. \end{array}\end{aligned}$$

(3.16)

Now, using (3.11)$\sim$(3.16) in (3.4), we complete the proof of the lemma. $\square$

We now evaluate $g(R(e_{1},e_{i})e_{1},e_{i}), \quad g(R(e_{1},e_{i})e_{i},e_{j})$ and $g(R(e_{i},e_{j})e_{i},e_{1})$ using Lemma 3.2, (2.2) and (3.1), and find the following relations:

$$\begin{aligned} e_{1}(\omega _{ii}^{1})- (\omega _{ii}^{1})^{2}= c+\lambda _{1} \lambda _{i}, \quad i \in B, \end{aligned}$$

(3.17)

$$\begin{aligned} e_{1}(\omega _{ii}^{j})- \omega _{ii}^{j} \omega _{ii}^{1}= 0, \quad i\in C_s, \quad j \in B\setminus C_{s}, \quad s\in \{1, 2, 3\}, \end{aligned}$$

(3.18)

and

$$\begin{aligned} e_{j}(\omega _{ii}^{1})+ \omega _{ii}^{j} \omega _{jj}^{1}-\omega _{ii}^{j} \omega _{ii}^{1}= 0, \quad i\in C_{s}, \quad j \in B\setminus C_{s}, \quad s\in \{1, 2, 3\}, \end{aligned}$$

(3.19)

respectively. Also, using (3.3), Lemma 3.2, and the fact that $[e_{i}e_{1}](H)=0= \nabla _{e_{i}}e_{1}(H)-\nabla _{e_{1}}e_{i}(H),$ we find that

$$\begin{aligned} e_{i}e_{1}(H)= 0, \ \ \text{ for } \text{ all }\ \ i\in B. \end{aligned}$$

(3.20)

4 Biconservative Hypersurfaces in Space Forms with Constant Norm of Second Fundamental Form

In this section we study biconservative hypersurfaces $M^n$ in space forms $\overline{M}^{n+1}(c)$ with constant norm of second fundamental form. We denote by $\beta$ the squared norm of the second fundamental form h. Then, using (3.1) we find that

$$\begin{aligned} \beta = \lambda _1^{2}+p\lambda _u^{2}+ q\lambda _v^{2}+r \lambda _w^2, \ u\in C_1, \ v\in C_2, \ w\in C_3. \end{aligned}$$

(4.1)

We recall that $B=\{2,3,\dots ,n\}$. Then we have the following:

Lemma 4.1

Let $M^n$ be a biconservative hypersurface with four distinct principal curvatures in a space form $\overline{M}^{n+1}(c)$, having the shape operator given by (3.1) with respect to a suitable orthonormal frame $\{e_{1}, e_{2},\dots , e_{n}\}$. If the second fundamental form is of constant norm, then

$$\begin{aligned} \omega ^u_{vv}=0, \quad \text{ for } \ u \in C_s, \ \text{ and }\ v\in B{\setminus } C_s, \ s\in \{1,2,3\}. \end{aligned}$$

Proof

We will prove the case $u\in C_1$ and $v\in C_2$ and by similar arguments one can prove all other cases. Let $w\in C_3.$ Differentiating (3.9) and (4.1) with respect to $e_{u}$ and using (3.3), we get

$$\begin{aligned} pe_{u}(\lambda _{u})+qe_{u}(\lambda _{v})+re_{u}(\lambda _{w})=0, \end{aligned}$$

(4.2)

and

$$\begin{aligned} p\lambda _{u}e_{u}(\lambda _{u})+q\lambda _{v}e_{u}(\lambda _{v})+r\lambda _{w}e_{u}(\lambda _{w})=0, \end{aligned}$$

(4.3)

respectively.

Eliminating $e_{u}(\lambda _{u})$ from (4.2) and (4.3), we find

$$\begin{aligned} q(\lambda _{v}-\lambda _{u})e_{u}(\lambda _{v})+r(\lambda _{w}-\lambda _{u})e_{u}(\lambda _{w})=0. \end{aligned}$$

(4.4)

Putting the value of $e_{u}(\lambda _{v})$ and $e_{u}(\lambda _{w})$ from (3.6) in (4.4), we obtain

$$\begin{aligned} r(\lambda _{w}-\lambda _{u})^{2}\omega _{ww}^{u}+q(\lambda _{v}-\lambda _{u})^{2}\omega _{vv}^{u}=0. \end{aligned}$$

(4.5)

Differentiating (4.5) with respect to $e_{1}$ and using (3.6) and (3.18), we have

$$\begin{aligned}&r[2(\lambda _{u}-\lambda _{1})\omega _{uu}^{1}+(2\lambda _{1}+\lambda _{u}-3\lambda _{w})\omega _{ww}^{1}](\lambda _w-\lambda _u)\omega _{ww}^{u} \nonumber \\&\quad +q[2(\lambda _{u}-\lambda _{1})\omega _{uu}^{1}+(2\lambda _{1}+\lambda _{u}-3\lambda _{v})\omega _{vv}^{1}](\lambda _v-\lambda _u)\omega _{vv}^{u}=0. \end{aligned}$$

(4.6)

We assume that $\omega _{vv}^{u}\ne 0$ and we will end up to contradiction. Then the value of the determinant formed by the coefficients of $\omega _{vv}^{u}$ and $\omega _{ww}^{u}$ of the system (4.5) and (4.6) will be zero. Therefore, we find

$$\begin{aligned}&2(\lambda _{1}-\lambda _{u})(\lambda _{v}-\lambda _{w})\omega _{uu}^{1}+(2\lambda _{1}+\lambda _{u}-3\lambda _{w})(\lambda _{u}-\lambda _{v})\omega _{ww}^{1} \nonumber \\&\quad -(2\lambda _{1}+\lambda _{u}-3\lambda _{v})(\lambda _{u}-\lambda _{w})\omega _{vv}^{1}=0. \end{aligned}$$

(4.7)

We set $a_{1}=(\lambda _{w}-\lambda _{u})\omega _{vv}^{1}+(\lambda _{u}-\lambda _{v})\omega _{ww}^{1}+(\lambda _{v}-\lambda _{w})\omega _{uu}^{1}$. Eliminating $\omega _{uu}^{1}$ from (4.7) and $a_1$, we get

$$\begin{aligned} \begin{array}{rcl} 2(\lambda _{1}-\lambda _{u})a_1 = 3(\lambda _{u}-\lambda _{v})(\omega _{vv}^{1}-\omega _{ww}^{1}) (\lambda _{u}-\lambda _{w}). \end{array} \end{aligned}$$

(4.8)

Now, we consider two cases.

(i) $a_1=0.$ Then, from (4.8), we obtain

$$\begin{aligned} \begin{array}{rcl} \omega _{ww}^{1}= \omega _{vv}^{1}. \end{array} \end{aligned}$$

(4.9)

Differentiating (4.9) with respect to $e_1$ and using (3.17) and (4.9), we find $\lambda _w=\lambda _v$, a contradiction to four distinct principal curvatures.

(ii) $a_1\ne 0.$ Then, differentiating (4.8) with respect to $e_u$ and using (4.2) and (3.3), we get

$$\begin{aligned}&2p(\lambda _{1}-\lambda _{u})e_u(a_1) +2a_1(qe_u(\lambda _{v})+re_u(\lambda _w))= 3\Big ((-(q+p)e_u(\lambda _{v})-re_u(\lambda _w))\nonumber \\&\quad (\omega _{vv}^{1}-\omega _{ww}^{1}) (\lambda _{u}-\lambda _{w})+p(\lambda _{u}-\lambda _{v})(e_u(\omega _{vv}^{1})-e_u(\omega _{ww}^{1})) (\lambda _{u}-\lambda _{w})\nonumber \\&\qquad +(\lambda _{u}-\lambda _{v})(\omega _{vv}^{1}-\omega _{ww}^{1}) (-qe_u(\lambda _{v})-(p+r)e_u(\lambda _w))\Big ). \end{aligned}$$

(4.10)

Now, using (3.6) and (3.19) in (4.10), we obtain

$$\begin{aligned} \begin{array}{rcl}f_1\omega _{vv}^u +f_2\omega _{ww}^u=2 p(\lambda _{1} - \lambda _{u}) e_u(a_1), \end{array} \end{aligned}$$

(4.11)

where

$$\begin{aligned} f_1= & {} (\lambda _{u} - \lambda _{v}) \Big (3\omega _{ww}^{1}(q\lambda _v-(p+2q)\lambda _u+(p+q)\lambda _w)+3\omega _{vv}^{1}\big (2(p+q)\lambda _u\\&-q\lambda _v-(2p+q)\lambda _w)+p(\lambda _u-\lambda _w)\big )+ 3p\omega _{uu}^{1}(\lambda _w-\lambda _u)+2a_1q\Big ),\\ f_2= & {} (\lambda _{u} - \lambda _{w})\Big (3\omega _{ww}^{1}\big ((2p+r)\lambda _v+r\lambda _w-2(p+r)\lambda _u\big )+3\omega _{vv}^{1}((p+2r)\lambda _u\\&-(p+r)\lambda _v-r\lambda _w)\big )+3p\omega _{uu}^{1}(\lambda _u-\lambda _v)+2a_1r\Big ). \end{aligned}$$

Differentiating $a_1$ with respect to $e_u$ and using (3.6), (3.19) and (4.2), we find

$$\begin{aligned} \begin{array}{rcl} f_3\omega _{vv}^u+f_4\omega _{ww}^u+p(\lambda _{v}-\lambda _{w}) e_u(\omega _{uu}^{1}) = pe_u(a_1), \end{array} \end{aligned}$$

(4.12)

where

$$\begin{aligned}f_3= & {} p \omega _{uu}^{1} (\lambda _{v} - \lambda _{w}) + \omega _{vv}^{1} (q(\lambda _{v}- \lambda _{u}) +p(\lambda _{w} - \lambda _{u}))+(p+q)(\lambda _{u} - \lambda _{v}) \omega _{ww}^{1},\\ f_4= & {} p\omega _{uu}^{1} (\lambda _{v} - \lambda _{w}) + (p+r)\omega _{vv}^{1} ( \lambda _{w} - \lambda _{u})+\omega _{ww}^{1}(p( \lambda _{u} - \lambda _{v}) -r( \lambda _{w}-\lambda _u)).\end{aligned}$$

Differentiating (3.9) with respect to $e_1$ and using (3.6), we find

$$\begin{aligned} \begin{array}{rcl} p\omega _{uu}^{1} (\lambda _{u} - \lambda _{1}) +q \omega _{vv}^{1} ( \lambda _{v} -\lambda _{1})+r(\lambda _{w} - \lambda _{1}) \omega _{ww}^{1}=\frac{3ne_1(H)}{2}. \end{array} \end{aligned}$$

(4.13)

Differentiating (4.13) with respect to $e_u$ and using (3.3), (3.6), (3.19), (3.20) and (4.2), we find

$$\begin{aligned} \begin{array}{rcl} p(\lambda _{u}-\lambda _{1}) e_u(\omega _{uu}^{1}) =qf_5\omega _{vv}^u+ rf_6\omega _{ww}^u, \end{array} \end{aligned}$$

(4.14)

where

$$\begin{aligned}f_5=(\omega _{vv}^{1}-\omega _{uu}^{1})(\lambda _{1} + \lambda _{u} - 2 \lambda _{v}),\quad f_6=(\omega _{ww}^{1}-\omega _{uu}^{1})(\lambda _{1} + \lambda _{u} - 2 \lambda _{w}). \end{aligned}$$

Eliminating $e_u(\omega _{uu}^{1})$ from (4.12) and (4.14), we get

$$\begin{aligned}&\Big ((\lambda _{u}-\lambda _{1})f_3+(\lambda _{v} - \lambda _{w})qf_5\Big )\omega _{vv}^u+\Big ((\lambda _{u}-\lambda _{1})f_4+(\lambda _{v} - \lambda _{w})rf_6\Big )\omega _{ww}^u\nonumber \\&\quad =(\lambda _{u}-\lambda _{1})p e_u(a_1). \end{aligned}$$

(4.15)

Eliminating $e_u(a_1)$ from (4.11) and (4.15), we find

$$\begin{aligned} \begin{array}{lcl} f_7\omega _{vv}^u+f_8\omega _{ww}^u=0, \end{array} \end{aligned}$$

(4.16)

where

$$\begin{aligned}f_7= & {} f_1+2(\lambda _{u}-\lambda _{1})f_3\\&+2(\lambda _{v} - \lambda _{w})qf_5, \quad f_8=f_2+2(\lambda _{u}-\lambda _{1})f_4+2(\lambda _{v} - \lambda _{w})rf_6. \end{aligned}$$

Now, we simplify $f_7$ and $f_8$. Eliminating $\omega _{uu}^1$ from $f_7$ using $a_1$, we get

$$\begin{aligned} (\lambda _w-\lambda _v)f_7= & {} \Big (3 p \lambda _u^2-q \lambda _v^2+2(3p+2q) \lambda _v \lambda _w-3 (p+q) \lambda _w^2-2\lambda _u ((3 p+ q) \lambda _v\nonumber \\&-q \lambda _w)\Big ) (\omega _{vv}^1-\omega _{ww}^1)(\lambda _u-\lambda _v)+a_1\Big (3 p \lambda _u^2+2 (p+q) \lambda _1( \lambda _v-\lambda _w) \nonumber \\&- p\lambda _u(5\lambda _v+\lambda _w)+\lambda _v(-2q\lambda _v+(3p+2q)\lambda _w)\Big ). \end{aligned}$$

(4.17)

Eliminating $a_1$ from (4.17) using (4.8), we obtain

$$\begin{aligned} \begin{array}{lcl} (\lambda _u-\lambda _v)^2 (\omega _{vv}^1-\omega _{ww}^1)g_1=-2 f_7 (\lambda _1-\lambda _u) (\lambda _v-\lambda _w), \end{array} \end{aligned}$$

(4.18)

where

$$\begin{aligned}g_1= & {} 3 p \lambda _u^2+4 \lambda _u( q\lambda _v-(3p+q)\lambda _w )+2\lambda _1(3p\lambda _u +q \lambda _v-(3 p+ q) \lambda _w)\\&\quad +3\lambda _w ((3 p+2 q) \lambda _w-2 q \lambda _v)\end{aligned}$$

Similarly, eliminating $\omega _{uu}^1$ from $f_8$ using $a_1$, we get

$$\begin{aligned}&a_1\Big ( 2 \lambda _1 (p+r) \left( \lambda _v-\lambda _w\right) +\lambda _w \left( 2 r \lambda _w-(3 p+2 r) \lambda _v\right) -3 p \lambda _u^2\nonumber \\&\qquad +p \lambda _u \left( \lambda _v+5 \lambda _w\right) \Big ) +f_8 (\lambda _v-\lambda _w)=(\lambda _u-\lambda _w) (\omega _{vv}^1-\omega _{ww}^1)\nonumber \\&\quad \Big (2 \lambda _u (r \lambda _v-(3 p+r) \lambda _w)-3 (p+r) \lambda _v^2+2 (3 p+2 r) \lambda _v \lambda _w+3 p \lambda _u^2-r \lambda _w^2\Big ).\nonumber \\ \end{aligned}$$

(4.19)

Eliminating $a_1$ from (4.19) using (4.8), we obtain

$$\begin{aligned} \begin{array}{lcl} 2 f_8 (\lambda _1-\lambda _u) (\lambda _v-\lambda _w)=(\lambda _u-\lambda _w)^2 (\omega _{vv}^1-\omega _{ww}^1)g_2, \end{array} \end{aligned}$$

(4.20)

where

$$\begin{aligned}g_2= & {} -4 \lambda _u ((3 p+r) \lambda _v-r \lambda _w)+2 \lambda _1 (-(3 p+r) \lambda _v+3 p \lambda _u+r \lambda _w)\\&+3 \lambda _v ((3 p+2 r) \lambda _v-2 r \lambda _w)+3 p \lambda _u^2. \end{aligned}$$

Eliminating $f_7$ and $f_8$ from (4.16) using (4.18) and (4.20), we obtain

$$\begin{aligned} \begin{array}{lcl} g_1 \left( \lambda _u-\lambda _v\right) ^2 \omega _{vv}^u-g_2 \left( \lambda _u-\lambda _w\right) ^2 \omega _{ww}^u=0. \end{array} \end{aligned}$$

(4.21)

The value of the determinant formed by the coefficients of $\omega _{vv}^{u}$ and $\omega _{ww}^{u}$ of the system (4.5) and (4.21) will be zero. Therefore, we have

$$\begin{aligned} \begin{array}{lcl} q g_2 +r g_1=0, \end{array} \end{aligned}$$

(4.22)

for $g_1, g_2$, gives

$$\begin{aligned}&p (q+r) \lambda _u^2+2 \lambda _1 p (q+r) \lambda _u+\lambda _w \left( -2 \lambda _1 p r-4 p r \lambda _u-4 q r \lambda _v\right) \nonumber \\&\qquad +\lambda _w^2 (3 p r+2 q r)-4 p q \lambda _u \lambda _v+3 p q \lambda _v^2-2 \lambda _1 p q \lambda _v+2 q r \lambda _v^2=0.\nonumber \\ \end{aligned}$$

(4.23)

Now, eliminating $\lambda _w$ from (4.1) and (4.23) using (3.9), we find

$$\begin{aligned}&-r \beta +(9+r) \lambda _1^2+6 p \lambda _1 \lambda _u+(p^2 +p r) \lambda _u^2+2q(3 \lambda _1+ p \lambda _u) \lambda _v\nonumber \\&\quad +(q^2+q r) \lambda _v^2=0, \end{aligned}$$

(4.24)

and

$$\begin{aligned} \begin{array}{lcl} b_0+b_1 \lambda _v+b_2 \lambda _v^2=0, \end{array} \end{aligned}$$

(4.25)

where

$$\begin{aligned}b_0= & {} p \lambda _u^2 \left( 3 p^2+2 p (q+2 r)+r (q+r)\right) +3 \lambda _1^2 (p (2 r+9)+6 q)\\&+ 2 \lambda _1 p \lambda _u (p (r+9)+(r+6) (q+r)),\\ b_1= & {} 6 \lambda _1 q (3 p+2 (q+r))+2 p q \lambda _u (3 p+2 (q+r)),\\ b_2= & {} q (q+r) (3 p+2 (q+r)). \end{aligned}$$

Eliminating $\lambda _v$ from (4.25) using (4.24), we obtain

$$\begin{aligned}&3 \beta p+\lambda _1^2 (3 p-2 (q+r+9))+2 \lambda _1 p \lambda _u (p+q+r)-p \lambda _u^2 (p+q+r)\nonumber \\&\quad +2 \beta (q+r)=0.\end{aligned}$$

(4.26)

Differentiating (4.26) with respect to $e_u$ and using (3.3), we get

$$\begin{aligned} 2p(p+q+r)( \lambda _1- \lambda _u )e_u(\lambda _u)=0, \end{aligned}$$

(4.27)

whereby, we get $e_u(\lambda _u)=0$.

Therefore, from (4.2) and (3.6), we obtain

$$\begin{aligned} \begin{array}{lcl} q(\lambda _{v} - \lambda _{u})\omega _{vv}^u+r(\lambda _{w} -\lambda _{u})\omega _{ww}^u=0. \end{array} \end{aligned}$$

(4.28)

The value of the determinant formed by the coefficients of $\omega _{vv}^{u}$ and $\omega _{ww}^{u}$ of the system (4.5) and (4.28) will be zero. Therefore, we get

$$\begin{aligned} \begin{array}{lcl} (\lambda _{v} - \lambda _{u})(\lambda _{w} - \lambda _{u})(\lambda _{v} - \lambda _{w})=0, \end{array} \end{aligned}$$

(4.29)

which gives a contradiction to four distinct principal curvatures. Hence, we obtain that $\omega _{vv}^u=0$ and the proof of the lemma is completed. $\square$

Next, we have:

Lemma 4.2

Under the assumtions of Lemma 4.1, let $a_{1}=(\lambda _{w}-\lambda _{u})\omega _{vv}^{1}+(\lambda _{u}-\lambda _{v})\omega _{ww}^{1}+(\lambda _{v}-\lambda _{w})\omega _{uu}^{1}.$ Then,

(a) If $a_1\ne 0$, it is

$$\begin{aligned} \omega _{wv}^{u}=\omega _{vw}^{u}= \omega _{wu}^{v}=\omega _{uw}^{v}=\omega _{vu}^{w}=\omega _{uv}^{w}=0, \quad \text{ where }\ u\in C_1, v\in C_2, w\in C_3. \end{aligned}$$

(4.30)

(b) If $a_1=0$, we have that

$$\begin{aligned} \omega _{ii}^{1}=\alpha \lambda _i+\phi ,\quad e_1(\alpha )=\alpha \phi +\lambda _{1}(1+\alpha ^2),\quad e_1(\phi )=\phi ^2+\alpha \lambda _1\phi +c,\nonumber \\ \end{aligned}$$

(4.31)

for some smooth functions $\alpha$ and $\phi$, and for $i\in B$.

Proof

(a) Let $a_1\ne 0.$ Evaluating $g(R(e_{v},e_{u})e_{w},e_{1})$ using (2.2), (3.1) and Lemma 3.2 and Lemma 4.1, we find

$$\begin{aligned} \omega _{vu}^{w}(\omega _{uu}^{1}-\omega _{ww}^{1})=\omega _{uv}^{w}(\omega _{vv}^{1}-\omega _{ww}^{1}). \end{aligned}$$

(4.32)

Putting $j=w, k=u, i=v$ in (3.7), we get

$$\begin{aligned} (\lambda _{u}-\lambda _{w})\omega _{vu}^{w}=(\lambda _{v}-\lambda _{w})\omega _{uv}^{w}. \end{aligned}$$

(4.33)

The value of the determinant formed by the coefficients of $\omega _{vu}^{w}$ and $\omega _{uv}^{w}$ in (4.32) and (4.33) is $a_1\ne 0$, hence $\omega _{vu}^{w}=0=\omega _{uv}^{w}$. Also, from (3.5), we get $\omega _{vw}^{u}=-\omega _{vu}^{w}$ and $\omega _{uv}^{w}=-\omega _{uw}^{v}$. Consequently, we obtain $\omega _{vw}^{u}=0$, and $\omega _{uw}^{v}=0$, which together with (3.7) gives $\omega _{wv}^{u}=0$, and $\omega _{wu}^{v}=0$.

(b) Let $a_1=0.$ Then, we have

$$\begin{aligned} \frac{\omega _{uu}^{1}-\omega _{vv}^{1}}{\lambda _{u}-\lambda _{v}}=\frac{\omega _{ww}^{1}-\omega _{vv}^{1}}{\lambda _{w}-\lambda _{v}} =\frac{\omega _{uu}^{1}-\omega _{ww}^{1}}{\lambda _{u}-\lambda _{w}}=\alpha , \end{aligned}$$

(4.34)

for some smooth function $\alpha$.

From (4.34), we get

$$\begin{aligned} \omega _{ii}^{1}=\alpha \lambda _{i}+\phi , \end{aligned}$$

(4.35)

for some smooth function $\phi$.

Differentiating (4.35) with respect to $e_1$ and using (3.6), (3.17) and (4.35), we find

$$\begin{aligned} e_1(\alpha )=\alpha \phi +\lambda _{1}(1+\alpha ^2),\quad e_1(\phi )=\phi ^2+\alpha \lambda _1\phi +c, \end{aligned}$$

(4.36)

whereby completing the proof of the lemma.$\square$

5 Proof of Theorem 1.1

Depending upon principal curvatures, we consider the following cases.

Case 1. The case of four distinct principal curvatures

From (3.7) and (3.5), we obtain

$$\begin{aligned} (\lambda _{u}-\lambda _{v})\omega _{wu}^{v}=(\lambda _{w}-\lambda _{v})\omega _{uw}^{v}=(\lambda _{u}-\lambda _{w})\omega _{vu}^{w}. \end{aligned}$$

(5.1)

From (5.1) and (3.5), we find

$$\begin{aligned} \omega _{vw}^{u}\omega _{wv}^{u}+\omega _{wu}^{v}\omega _{uw}^{v}+\omega _{vu}^{w}\omega _{uv}^{w}=0. \end{aligned}$$

(5.2)

Evaluating $g(R(e_{u},e_{v})e_{u},e_{v})$, $g(R(e_{u},e_{w})e_{u},e_{w})$ and $g(R(e_{v},e_{w})e_{v},e_{w})$, using (2.2), (3.1), (5.2) and Lemmas 3.2 and 4.1, we find that

$$\begin{aligned} -\omega _{uu}^{1}\omega _{vv}^{1} +\sum \nolimits _{k\in B\setminus \{C_1,C_2\}}2\omega _{uv}^{k}\omega _{vu}^{k}=c+ \lambda _{u} \lambda _{v}, \end{aligned}$$

(5.3)

$$\begin{aligned} -\omega _{uu}^{1}\omega _{ww}^{1} +\sum \nolimits _{k\in B\setminus \{C_1,C_3\}}2\omega _{uw}^{k}\omega _{wu}^{k}=c+ \lambda _{u} \lambda _{w}, \end{aligned}$$

(5.4)

$$\begin{aligned} -\omega _{vv}^{1}\omega _{ww}^{1} +\sum \nolimits _{k\in B\setminus \{C_2,C_3\}}2\omega _{vw}^{k}\omega _{wv}^{k}=c+ \lambda _{v} \lambda _{w}, \end{aligned}$$

(5.5)

respectively.

Simplifying (5.3), (5.4) and (5.5), we obtain

$$\begin{aligned} -\omega _{uu}^{1}\omega _{vv}^{1} +2 r\omega _{uv}^{w}\omega _{vu}^{w}=c+ \lambda _{u} \lambda _{v}, \end{aligned}$$

(5.6)

$$\begin{aligned} -\omega _{uu}^{1}\omega _{ww}^{1} +2 q\omega _{uw}^{v}\omega _{wu}^{v}=c+ \lambda _{u} \lambda _{w}, \end{aligned}$$

(5.7)

$$\begin{aligned} -\omega _{vv}^{1}\omega _{ww}^{1} +2 p\omega _{vw}^{u}\omega _{wv}^{u}=c+ \lambda _{v} \lambda _{w}, \end{aligned}$$

(5.8)

respectively.

Depending upon $a_1$, we consider the following cases.

Subcase A. Assume that $a_1\ne 0$. Using (4.30) in (5.3), (5.4) and (5.5), we obtain

$$\begin{aligned} -\omega _{uu}^{1}\omega _{vv}^{1}=c+\lambda _{u} \lambda _{v},\quad -\omega _{uu}^{1}\omega _{ww}^{1}=c+\lambda _{u} \lambda _{w},\quad -\omega _{vv}^{1}\omega _{ww}^{1}=c+\lambda _{v} \lambda _{w}. \end{aligned}$$

(5.9)

From (5.9), we get

$$\begin{aligned} -(c+\lambda _{w} \lambda _{v})(\omega _{uu}^{1})^{2}= (c+\lambda _{u} \lambda _{v})(c+\lambda _{u} \lambda _{w}). \end{aligned}$$

(5.10)

Differentiating (3.9) and (4.1) with respect to $e_1$ and using (3.6), we find

$$\begin{aligned} p(\lambda _{u}- \lambda _{1})\omega _{uu}^{1}+q(\lambda _{v}- \lambda _{1})\omega _{vv}^{1}+r(\lambda _{w}- \lambda _{1})\omega _{ww}^{1}= -3e_1(\lambda _{1}), \end{aligned}$$

(5.11)

and

$$\begin{aligned} p\lambda _{u}(\lambda _{u}- \lambda _{1})\omega _{uu}^{1}+q\lambda _{v}(\lambda _{v}- \lambda _{1})\omega _{vv}^{1}+r\lambda _{w}(\lambda _{w}- \lambda _{1})\omega _{ww}^{1}= -\lambda _{1}e_1(\lambda _{1}), \end{aligned}$$

(5.12)

respectively.

Eliminating $e_1(\lambda _1)$ from (5.12) using (5.11), we obtain

$$\begin{aligned} p(3\lambda _{u}- & {} \lambda _{1})(\lambda _{u}- \lambda _{1})\omega _{uu}^{1}+q(3\lambda _{v}- \lambda _{1})(\lambda _{v}- \lambda _{1})\omega _{vv}^{1}\nonumber \\&\quad +r(3\lambda _{w}- \lambda _{1})(\lambda _{w}- \lambda _{1})\omega _{ww}^{1}=0. \end{aligned}$$

(5.13)

Multiplying (5.13) with $\omega _{uu}^{1}$ and using (5.9), we find

$$\begin{aligned} p(3\lambda _{u}- \lambda _{1})(\lambda _{u}- \lambda _{1})(\omega _{uu}^{1})^2= & {} q(3\lambda _{v}- \lambda _{1})(\lambda _{v}- \lambda _{1})(c+\lambda _u\lambda _v)\nonumber \\&+r(3\lambda _{w}- \lambda _{1})(\lambda _{w}- \lambda _{1})(c+\lambda _u\lambda _w).\nonumber \\ \end{aligned}$$

(5.14)

Eliminating $(\omega _{uu}^{1})^2$ from (5.14) using (5.10), we obtain

$$\begin{aligned}&-p(3\lambda _{u}- \lambda _{1})(\lambda _{u}- \lambda _{1})(c+\lambda _u\lambda _w)(c+\lambda _u\lambda _v)=\big ( q(3\lambda _{v}- \lambda _{1})(\lambda _{v}- \lambda _{1})\nonumber \\&\quad (c+\lambda _u\lambda _v)+r(3\lambda _{w}- \lambda _{1})(\lambda _{w}- \lambda _{1})(c+\lambda _u\lambda _w)\big )(c+\lambda _v\lambda _w). \end{aligned}$$

(5.15)

Eliminating $\lambda _w$ from (5.15) using (3.9), we obtain

$$\begin{aligned} \begin{array}{rcl} v_0+v_1\lambda _v+v_2\lambda _v^2+v_3\lambda _v^3+v_4\lambda _v^4+v_5\lambda _v^5=0,\end{array}\end{aligned}$$

(5.16)

where

$$\begin{aligned}v_0= & {} c r (\lambda _1^2 (c r (r (p+q+12)+r^2+27)-p ((p+12) r+r^2+81) \lambda _u^2)+9 \lambda _1 (2 c p r \lambda _u\\&-p (3 p+r) \lambda _u^3)+3 p (p+r) \lambda _u^2 (c r-p \lambda _u^2)-3 \lambda _1^3 ((p+12) r+r^2+27) \lambda _u), \\ v_1= & {} \lambda _1 (18 c^2 q r^2-c p r \lambda _u^2 (p (4 r+27)+4 q r+54 q+4 r^2)\\&+p (4 p^2 (r+9)+4 p r^2-9 r^2) \lambda _u^4) +3 p \lambda _u (2 c^2 q r^2-c r \lambda _u^2 (p^2+3 p q+r (q-r))\\&+(p^3-p r^2) \lambda _u^4-3 \lambda _1^3 (c r ((q+12) r+r^2+27)-p (r^2+36 r+108) \lambda _u^2)\\&-\lambda _1^2 (c r \lambda _u (p (2 (q+12) r+81) +3 q (8 r+27))-6 p (3 p (2 r+9)+2 r^2) \lambda _u^3)\\&+9 \lambda _1^4 (r^2+12 r+27) \lambda _u, \\ v_2= & {} q (3 (c^2 r^2 (q+r)-3 c p r (p+q) \lambda _u^2+(4 p^3-p r^2) \lambda _u^4)-\lambda _1^2 (c r ((q+12) r+r^2+81)\\&- 36 p (2 r+9) \lambda _u^2)-\lambda _1 (c r \lambda _u (p (4 r+54)+4 q r+27 q+4 r^2)-4 p (3 p (r+9)+r^2) \lambda _u^3)\\&+ 3 \lambda _1^3 (r^2+36 r+108) \lambda _u), \\ v_3= & {} q (-3 c r \lambda _u (3 p q+p r+q^2-r^2)+\lambda _1 (4 p (3 q (r+9)+r^2) \lambda _u^2-9 c r (3 q+r))+18 p^2 q \lambda _u^3\\&+6 \lambda _1^2 (3 q (2 r+9)+2 r^2) \lambda _u), \\ v_4= & {} q (\lambda _1 (4 q^2 (r+9)+4 q r^2-9 r^2) \lambda _u-3 (c q r (q+r)+p (r^2-4 q^2) \lambda _u^2)), \\ v_5= & {} 3 q^2(q^2 - r^2 )\lambda _u.\end{aligned}$$

Eliminating $\lambda _w$ from (4.1) using (3.9), we obtain

$$\begin{aligned} \begin{array}{rcl} v_6+v_7\lambda _v+v_8\lambda _v^2=0,\end{array}\end{aligned}$$

(5.17)

where

$$\begin{aligned}v_6= & {} -r \beta +9 \lambda _1^2+r \lambda _1^2+6 p \lambda _1 \lambda _u+p^2 \lambda _u^2+p r \lambda _u^2,\quad v_7=6 q \lambda _1+2 p q \lambda _u, \quad \\ v_8= & {} q^2+q r.\end{aligned}$$

Equations (5.16) and (5.17) have a common root $\lambda _v$, so their resultant with respect $\lambda _v$ vanish. Hence,

$$\begin{aligned}&v_5^2 v_6^5-v_4 v_5 v_6^4 v_7+v_3 v_5 v_6^3 v_7^2-v_2 v_5 v_6^2 v_7^3+v_1 v_5 v_6 v_7^4-v_0 v_5 v_7^5+v_4^2 v_6^4 v_8\nonumber \\&\quad -2 v_3 v_5 v_6^4 v_8-v_3 v_4 v_6^3 v_7 v_8+3 v_2 v_5 v_6^3 v_7 v_8+v_2 v_4 v_6^2 v_7^2 v_8-4 v_1 v_5 v_6^2 v_7^2 v_8\nonumber \\&\quad -v_1 v_4 v_6 v_7^3 v_8+5 v_0 v_5 v_6 v_7^3 v_8+v_0 v_4 v_7^4 v_8+v_3^2 v_6^3 v_8^2-2 v_2 v_4 v_6^3 v_8^2\nonumber \\&\quad +2 v_1 v_5 v_6^3 v_8^2 -v_2 v_3 v_6^2 v_7 v_8^2+3 v_1 v_4 v_6^2 v_7 v_8^2-5 v_0 v_5 v_6^2 v_7 v_8^2+v_1 v_3 v_6 v_7^2 v_8^2\nonumber \\&\quad -4 v_0 v_4 v_6 v_7^2 v_8^2 -v_0 v_3 v_7^3 v_8^2+v_2^2 v_6^2 v_8^3-2 v_1 v_3 v_6^2 v_8^3+2 v_0 v_4 v_6^2 v_8^3-v_1 v_2 v_6 v_7 v_8^3\nonumber \\&\quad +3 v_0 v_3 v_6 v_7 v_8^3 +v_0 v_2 v_7^2 v_8^3+v_1^2 v_6 v_8^4-2 v_0 v_2 v_6 v_8^4-v_0 v_1 v_7 v_8^4+v_0^2 v_8^5=0\nonumber \\ \end{aligned}$$

(5.18)

which is a polynomial equation

$$\begin{aligned} G(\lambda _1, \lambda _u)=0, \end{aligned}$$

(5.19)

for $\lambda _1, \lambda _u$.

Differentiating (5.19) with respect to $e_1$, we get

$$\begin{aligned} \begin{array}{rcl} G_1e_1(\lambda _1)+G_ue_1(\lambda _u)=0,\end{array}\end{aligned}$$

(5.20)

where $G_1=\frac{\partial G}{\partial \lambda _1}, G_u=\frac{\partial G}{\partial \lambda _u}$.

Eliminating $e_1(\lambda _1)$ from (5.20) using (5.11) and using (3.6), we find

$$\begin{aligned} \begin{array}{rcl} (3G_u-pG_1)(\lambda _u-\lambda _1)\omega _{uu}^1=\Big (q(\lambda _v-\lambda _1)\omega _{vv}^1+r(\lambda _w-\lambda _1)\omega _{ww}^1\Big )G_1.\end{array}\end{aligned}$$

(5.21)

Multiplying (5.21) with $\omega _{uu}^{1}$ and using (5.9) and (5.10), we obtain

$$\begin{aligned} L(c+\lambda _u\lambda _v)(c+\lambda _u\lambda _w)= & {} \Big (q(\lambda _v-\lambda _1)(c+\lambda _u\lambda _v)+r(\lambda _w-\lambda _1)\nonumber \\&\times (c+\lambda _u\lambda _w)\Big )(c+\lambda _w\lambda _v),\end{aligned}$$

(5.22)

where $L=\frac{(3G_u-pG_1)(\lambda _u-\lambda _1)}{G_1}$.

Eliminating $\lambda _w$ from (5.22) using (3.9), we get

$$\begin{aligned} \begin{array}{rcl}v_9+v_{10}\lambda _v+v_{11}\lambda _v^2+v_{12}\lambda _v^3+v_{13}\lambda _v^4=0,\end{array}\end{aligned}$$

(5.23)

where

$$\begin{aligned} v_9= & {} c^2 L r^2+(3 c^2 r^2 +c^2 q r^2 +c^2 r^3) \lambda _1+c^2 p r^2 \lambda _u-3 c L r \lambda _1 \lambda _u\nonumber \\&+(-9 c r -3 c r^2) \lambda _1^2 \lambda _u - c L p r \lambda _u^2 +(-6 c p r-c p r^2 )\lambda _1 \lambda _u^2-c p^2 r \lambda _u^3,\nonumber \\ v_{10}= & {} (-3 -q-r)3cr\lambda _1^2+(r-q)cLr \lambda _u-(6 p +6 q + p q +p r)cr \lambda _1 \lambda _u\nonumber \\&+(27+9 r )\lambda _1^3 \lambda _u-( p+2 q) cpr\lambda _u^2-3 L r \lambda _1 \lambda _u^2 +(27 p +6 p r )\lambda _1^2 \lambda _u^2\nonumber \\&-L p r \lambda _u^3+(9 + r )p^2\lambda _1 \lambda _u^3+p^3 \lambda _u^4,\nonumber \\ v_{11}= & {} [(-3-cq-r )\lambda _1 -(p+q+r)\lambda _u]cqr+[(9+r )3\lambda _1\nonumber \\&+(18+r )p \lambda _u]q\lambda _1 \lambda _u+(3 p^2 \lambda _u-Lr)q \lambda _u^2,\nonumber \\ v_{12}= & {} 9 q^2 \lambda _1 \lambda _u +3 q r \lambda _1 \lambda _u+3 p q^2 \lambda _u^2+p q r \lambda _u^2,\quad \quad v_{13}=q^3 \lambda _u +q^2 r \lambda _u.\end{aligned}$$

Equations (5.23) and (5.17) have a common root $\lambda _v$, so their resultant with respect $\lambda _v$ vanish. Therefore, we find

$$\begin{aligned} \begin{array}{lcl} \mathcal {G}(\lambda _1, \lambda _u)=0, \end{array}\end{aligned}$$

(5.24)

which is a polynomial equation in terms of $\lambda _1, \lambda _u$.

Rewrite (5.19) and (5.24) as polynomials $G_{\lambda _1}(\lambda _u), \mathcal {G}_{\lambda _1}(\lambda _u)$ of $\lambda _u$ with coefficients in the polynomial ring $\mathbb {R}[\lambda _1]$ over real field. According to Lemma 2.1, equations $G_{\lambda _1}(\lambda _u)=0$ and $\mathcal {G}_{\lambda _1}(\lambda _u)=0$ have a common root if and only if

$$\begin{aligned} \begin{array}{lcl} \mathfrak {R}(G_{\lambda _1}(\lambda _u),\mathcal {G}_{\lambda _1}(\lambda _u))=0, \end{array}\end{aligned}$$

(5.25)

which is a polynomial of $\lambda _1$ with real coefficients. Then (5.25) shows that $\lambda _1$ must be a constant, a contradiction.

Subcase B. Assume that $a_1= 0$.

Adding (5.6), (5.7) and (5.8) and using (5.2), we obtain

$$\begin{aligned} pq(\omega _{uu}^{1}\omega _{vv}^{1}+c+\lambda _{u} \lambda _{v})+pr(\omega _{uu}^{1}\omega _{ww}^{1}+c+\lambda _{u} \lambda _{w})+qr(\omega _{vv}^{1}\omega _{ww}^{1}+c+\lambda _{v} \lambda _{w})=0. \end{aligned}$$

(5.26)

Using (4.31) in (5.26) in Lemma 4.2 we find

$$\begin{aligned}&\left( c+\phi ^2\right) (p q+p r+q r)+\left( \alpha ^2+1\right) \left( p q \lambda _u \lambda _v+p r \lambda _u \lambda _w+q r \lambda _v \lambda _w\right) \nonumber \\&\quad +\alpha \phi \left( p q \lambda _u+p q \lambda _v+p r \lambda _u+p r \lambda _w+q r \lambda _v+q r \lambda _w\right) =0. \end{aligned}$$

(5.27)

Using (4.31) in (5.13), we get

$$\begin{aligned}&p(3\lambda _u-\lambda _1)(\lambda _u-\lambda _1)(\alpha \lambda _u+\phi )+q\left( 3\lambda _v-\lambda _1\right) \left( \lambda _v-\lambda _1\right) \left( \alpha \lambda _v+\phi \right) \nonumber \\&\quad +r\left( 3\lambda _w-\lambda _1\right) \left( \lambda _w-\lambda _1\right) \left( \alpha \lambda _w+\phi \right) =0. \end{aligned}$$

(5.28)

On the other hand, using (3.9), (4.1) and (4.31) in (4.13), we obtain

$$\begin{aligned} (n+2)\lambda _1\phi -\alpha (\beta +2\lambda _1^2)=3e_1(\lambda _1). \end{aligned}$$

(5.29)

Eliminating $\lambda _w$ from (4.1), (5.27) and (5.28) using (3.9), we find

$$\begin{aligned}&9 \lambda _1^2+r \lambda _1^2+6 p \lambda _1 \lambda _u+p^2 \lambda _u^2+p r \lambda _u^2+6 q \lambda _1 \lambda _v+2 p q \lambda _u \lambda _v\nonumber \\&\quad +q^2 \lambda _v^2+q r \lambda _v^2-r \beta =0,\end{aligned}$$

(5.30)

$$\begin{aligned}&{\left( c+\phi ^2\right) (p q+p r+q r)-3 \alpha \lambda _1 \phi (p+q)-(\alpha \phi (p-r)+3 (\alpha ^2+1) \lambda _1) p \lambda _u}\nonumber \\&\quad {-(\alpha ^2+1) p^2 \lambda _u^2+\lambda _v q(\alpha \phi (r-q)-(\alpha ^2+1) (p \lambda _u+3 \lambda _1 ))-\left( \alpha ^2+1\right) q^2 \lambda _v^2=0,}\nonumber \\ \end{aligned}$$

(5.31)

(5.32)

respectively.

Rewrite (5.30), (5.31) and (5.32) as polynomials ${\mathcal {F}_{2}}_{(\lambda _1,\lambda _u,\alpha ,\phi )}(\lambda _v), {G_{2}}_{(\lambda _1,\lambda _u,\alpha ,\phi )}(\lambda _v)$, and ${\mathcal {G}_1}_{(\lambda _1,\lambda _u,\alpha ,\phi )}(\lambda _v)$ of $\lambda _v$ with coefficients in polynomial ring $\mathbb {R}[\lambda _1,\lambda _u,\alpha ,\phi ]$. According to Lemma 2.1, the equations ${\mathcal {F}_{2}}_{(\lambda _1,\lambda _u,\alpha ,\phi )}(\lambda _v)=0, {G_{2}}_{(\lambda _1,\lambda _u,\alpha ,\phi )}(\lambda _v)=0$, and ${\mathcal {F}_{2}}_{(\lambda _1,\lambda _u,\alpha ,\phi )}(\lambda _v)=0, {\mathcal {G}_1}_{(\lambda _1,\lambda _u,\alpha ,\phi )}(\lambda _v)=0$ have a common root if and only if

$$\begin{aligned} \begin{array}{lcl} \mathfrak {R}({\mathcal {F}_{2}}_{(\lambda _1,\lambda _u,\alpha ,\phi )}(\lambda _v), {G_{2}}_{(\lambda _1,\lambda _u,\alpha ,\phi )}(\lambda _v))=0, \end{array}\end{aligned}$$

(5.33)

and

$$\begin{aligned} \begin{array}{lcl} \mathfrak {R}({\mathcal {F}_{2}}_{(\lambda _1,\lambda _u,\alpha ,\phi )}(\lambda _v), {\mathcal {G}_1}_{(\lambda _1,\lambda _u,\alpha ,\phi )}(\lambda _v))=0, \end{array}\end{aligned}$$

(5.34)

which is polynomials of $\lambda _1,\lambda _u,\alpha ,\phi$ with real coefficients, in degree 4 and 6 respectively. Rewrite, equations (5.33) and (5.34) as polynomials ${\mathcal {F}_3}_{(\lambda _1,\alpha ,\phi )}(\lambda _u), {\mathcal {G}_2}_{(\lambda _1,\alpha ,\phi )}(\lambda _u)$ with coefficients in polynomial ring $\mathbb {R}[\lambda _1,\alpha ,\phi ]$ over real field. According to Lemma 2.1, the equations ${\mathcal {F}_3}_{(\lambda _1,\alpha ,\phi )}(\lambda _u)=0, {\mathcal {G}_2}_{(\lambda _1,\alpha ,\phi )}(\lambda _u)=0$, have a common root if and only if

$$\begin{aligned} \begin{array}{lcl} \mathfrak {R}({\mathcal {F}_{3}}_{(\lambda _1,\alpha ,\phi )}(\lambda _u), {\mathcal {G}_2}_{(\lambda _1,\alpha ,\phi )}(\lambda _u))=0, \end{array}\end{aligned}$$

(5.35)

which is a polynomial equation

$$\begin{aligned} \begin{array}{rcl} {\mathcal {F}_{4}}(\alpha ,\phi ,\lambda _1)=0. \end{array} \end{aligned}$$

(5.36)

Differentiating (5.36) with respect to $e_1$ and using (4.31) and (5.29), we find a polynomial

$$\begin{aligned} \begin{array}{rcl} {\mathcal {F}_{5}}(\alpha ,\phi ,\lambda _1)=0. \end{array} \end{aligned}$$

(5.37)

Differentiating (5.37) with respect to $e_1$ and using (4.31) and (5.29), we find a polynomial

$$\begin{aligned} \begin{array}{rcl} {\mathcal {F}_{6}}(\alpha ,\phi ,\lambda _1)=0. \end{array} \end{aligned}$$

(5.38)

Rewrite (5.36), (5.37) and (5.38) as polynomials ${\mathcal {F}_{4}}_{(\lambda _1,\phi )}(\alpha ), {\mathcal {F}_{5}}_{(\lambda _1,\phi )}(\alpha )$, and ${\mathcal {F}_{6}}_{(\lambda _1,\phi )}(\alpha )$ of $\alpha$ with coefficients in polynomial ring $\mathbb {R}[\lambda _1,\phi ]$. According to Lemma 2.1, the equations ${\mathcal {F}_{4}}_{(\lambda _1,\phi )}(\alpha )=0, {\mathcal {F}_{5}}_{(\lambda _1,\phi )}(\alpha )=0$, and ${\mathcal {F}_{4}}_{(\lambda _1,\phi )}(\alpha )=0, {\mathcal {F}_{6}}_{(\lambda _1,\phi )}(\alpha )=0$ have a common root if and only if

$$\begin{aligned} \begin{array}{lcl} \mathfrak {R}({\mathcal {F}_{4}}_{(\lambda _1,\phi )}(\alpha ), {\mathcal {F}_{5}}_{(\lambda _1,\phi )}(\alpha ))=0, \mathfrak {R}({\mathcal {F}_{4}}_{(\lambda _1,\phi )}(\alpha ), {\mathcal {F}_{6}}_{(\lambda _1,\phi )}(\alpha ))=0, \end{array}\end{aligned}$$

(5.39)

which are polynomial equations

$$\begin{aligned} \begin{array}{rcl} h_1(\phi , \lambda _1)=0, \quad \text{ and } \quad h_2(\phi , \lambda _1)=0, \end{array} \end{aligned}$$

(5.40)

respectively. Finally, rewrite ${h_1}_{\lambda _1}(\phi )=0$ and ${h_2}_{\lambda _1}(\phi )=0$, as a polynomial equations $\phi$ with coefficients in polynomial ring $\mathbb {R}[\lambda _1]$ over real field. According to Lemma 2.1, the equations ${h_1}_{\lambda _1}(\phi )=0$ and ${h_2}_{\lambda _1}(\phi )=0$ have a common root if and only if

$$\begin{aligned} \begin{array}{lcl} \mathfrak {R}({h_1}_{\lambda _1}(\phi ), {h_2}_{\lambda _1}(\phi ))=0, \end{array}\end{aligned}$$

(5.41)

which is a polynomial equation ${h_3}(\lambda _1)=0$ of $\lambda _1$ with constant coefficients. Thus, the real function $\lambda _1$ satisfies a polynomial equation ${h_3}(\lambda _1)=0$ with constant coefficients, and, therefore, $\lambda _1$ must be a constant, a contradiction.

Case 2. The case of three distinct principal curvatures

Suppose that M is a biconservative hypersurface with three distinct principal curvatures $\lambda _1=-\frac{nH}{2}, \lambda _u$, $\lambda _v$, with multiplicities 1, p and $n-p-1$ respectively. Further, suppose that M has constant norm of second fundamental forms. Without losing generality, we choose $e_{1}$ in the direction of $\mathop {\mathrm {grad}}H$ and therefore shape operator $\mathcal {A}$ of the hypersurface will take the following form with respect to a suitable frame $\{e_{1}, e_{2}, \dots ,e_{n}\}$

$$\begin{aligned} \mathcal {A}e_1=-\frac{nH}{2}e_1, \quad \mathcal {A}e_i=\lambda _u e_i,\quad \mathcal {A}e_j=\lambda _v e_j, \end{aligned}$$

(5.42)

where $i=2,3,\dots , p+1$, and $j=p+2, p+3,\dots , n.$

Using (5.42) in (3.9) and (4.1), we get

$$\begin{aligned} p\lambda _u+(n-p-1)\lambda _{v}=-3\lambda _1. \end{aligned}$$

(5.43)

$$\begin{aligned} p\lambda _u^{2}+(n-p-1)\lambda _{v}^{2}=\beta -\lambda _1^2. \end{aligned}$$

(5.44)

Eliminating $\lambda _v$ from (5.44) using (5.43), we obtain

$$\begin{aligned} 8 \lambda _1^2+n \lambda _1^2-p \lambda _1^2+6 p \lambda _1 \lambda _u-p \lambda _u^2+n p \lambda _u^2=(-1+n-p) \beta . \end{aligned}$$

(5.45)

Similarly, eliminating $\lambda _u$ from (5.44) using (5.43), we get

$$\begin{aligned} 9 \lambda _1^2+p \lambda _1^2+\left( -6 \lambda _1+6 n \lambda _1-6 p \lambda _1\right) \lambda _v+\left( 1-2 n+n^2+p-n p\right) \lambda _v^2=p \beta . \end{aligned}$$

(5.46)

Differentiating (5.45) with respect to $e_1$, we find

$$\begin{aligned} e_1(\lambda _u) =\mu e_1(\lambda _1),\quad e_1(\mu )=\frac{-8+p \left( 1-6 \mu +\mu ^2\right) -n \left( 1+p \mu ^2\right) }{p \left( 3 \lambda _1+(-1+n) \lambda _u\right) }e_1(\lambda _1), \end{aligned}$$

(5.47)

where $\mu =-\frac{(8+n-p) \lambda _1+3 p \lambda _u}{p \left( 3 \lambda _1+(-1+n) \lambda _u\right) }$ and to find $e_1(\mu )$ we have used the first expression of (5.47).

Differentiating (5.45) with respect to $e_j$ and using (3.6), we find

$$\begin{aligned} e_{j}(\lambda _u)=0, \omega ^j_{ii}=0, \end{aligned}$$

(5.48)

where $i=2,3,\dots , p+1$, and $j=p+2, p+3,\dots , n.$

Differentiating (5.46) with respect to $e_i$ and using (3.6), we have

$$\begin{aligned} e_{i}(\lambda _v)=0, \omega ^i_{jj}=0, \end{aligned}$$

(5.49)

where $i=2,3,\dots , p+1$, and $j=p+2, p+3,\dots , n.$

Also, putting $k\in \{p+2, p+3,\dots , n\}$, and $j, i \in \{2,3,\dots , p+1\}$ in (3.7) and using (3.5) and (3.7), we get

$$\begin{aligned} \omega ^j_{ik}=\omega ^k_{ij}=\omega ^k_{ji}=\omega ^i_{jk}=0, \quad i\ne j. \end{aligned}$$

(5.50)

Evaluating $g(R(e_i,e_j)e_i,e_j)$ using (2.2), (5.42), (5.48), (5.49) and (5.50), we obtain

$$\begin{aligned} \omega _{ii}^1\omega _{jj}^1=-c-\lambda _u\lambda _v,\quad \text{ for }\quad j\in \{p+2, p+3,\dots , n\}, i \in \{2,3,\dots , p+1\}, \end{aligned}$$

(5.51)

where $\omega _{ii}^1=\frac{e_{1}(\lambda _u)}{\lambda _u-\lambda _1}, \omega _{jj}^1=\frac{e_{1}(\lambda _v)}{\lambda _v-\lambda _1}$.

Also, equation (3.17) is valid for three distinct principal curvatures. Therefore, from (3.17), we find

$$\begin{aligned} e_{1}\left( \frac{e_{1}(\lambda _u)}{\lambda _u-\lambda _1}\right) =\left( \frac{e_{1}(\lambda _u)}{\lambda _u-\lambda _1}\right) ^{2}+\lambda _1\lambda _u+c, \end{aligned}$$

(5.52)

$$\begin{aligned} e_{1}\left( \frac{e_{1}(\lambda _v)}{\lambda _v-\lambda _1}\right) =\left( \frac{e_{1}(\lambda _v)}{\lambda _v-\lambda _1}\right) ^{2}+\lambda _1\lambda _v+c. \end{aligned}$$

(5.53)

Using (3.6), (5.43) and (5.47) in (5.51), we get

$$\begin{aligned} \begin{array}{lcl} \frac{\lambda _u \left( -\lambda _1+\lambda _u\right) \left( 3 \lambda _1+p \lambda _u\right) \left( (2+n-p) \lambda _1 +p \lambda _u\right) }{-1+n-p}=e_1^2(\lambda _1) \mu (3+p \mu ).\end{array} \end{aligned}$$

(5.54)

Using (3.6) and (5.47) in (5.52), we obtain

$$\begin{aligned} \begin{array}{lcl} \mu (\lambda _u-\lambda _1)e_1e_1(\lambda _1)= \lambda _1 \lambda _u \left( -\lambda _1 +\lambda _u\right) {}^2+A e_1^2(\lambda _1),\end{array} \end{aligned}$$

(5.55)

where $A=\frac{-\left( 8+n+n p \mu ^2-p \left( 1-6 \mu +\mu ^2\right) \right) \left( \lambda _1-\lambda _u\right) +p \mu (-1+2 \mu ) \left( 3 \lambda _1+(-1+n) \lambda _u\right) }{p \left( 3 \lambda _1+(-1+n) \lambda _u\right) }.$

Using (3.6) and (5.47) in (5.53), we find

$$\begin{aligned} \begin{array}{rcl} (3+2\mu )(2\lambda +4\lambda _1)e_1e_1(\lambda _1)=Be_1^2(\lambda _1)-\lambda _1(3\lambda _1+2\lambda )(2\lambda +4\lambda _1)^2,\end{array} \end{aligned}$$

(5.56)

where $B=\frac{[\lambda (36\mu ^2+102\mu +83)+\lambda _1(48\mu ^2+126\mu +103)]}{3(\lambda _1+\lambda )}.$

Eliminating $e_1e_1(\lambda _1)$ from (5.55) using (5.56), we obtain

$$\begin{aligned} \begin{array}{rcl} C e_1^2(\lambda _1)=D,\end{array} \end{aligned}$$

(5.57)

where $C=(3+2\mu )(2\lambda +4\lambda _1)A-\mu (\lambda -\lambda _1)B$ and $D=-\lambda _1(\lambda -\lambda _1)(2\lambda +4\lambda _1)[\mu (3\lambda _1+2\lambda )(4\lambda _1+2\lambda )+\lambda (3+2\mu )(\lambda -\lambda _1)].$

Eliminating $e_1^2(\lambda _1)$ from (5.54) using (5.57), we find

$$\begin{aligned} \begin{array}{lcl} (3\mu +2\mu ^2)D-C \lambda (\lambda -\lambda _1)(3\lambda _1+2\lambda )(4\lambda _1+2\lambda )=0.\end{array} \end{aligned}$$

(5.58)

Finally, eliminating $\lambda$ from (5.45) and (5.58), we get a polynomial equation $\varphi (H)=0$ in H with constant coefficients. Thus, the real function H satisfies a polynomial equation $\varphi (H) = 0$ with constant coefficients, and, therefore, it must be a constant.

Case 3. The case of two distinct principal curvatures

Suppose that M is a biconservative hypersurface with two distinct principal curvatures $\lambda _1=-\frac{nH}{2}$ and $\lambda$, with multiplicities 1 and $n-1$ respectively. Without losing generality, we choose $e_{1}$ in the direction of $\mathop {\mathrm {grad}}H$ and therefore shape operator $\mathcal {A}$ of the hypersurface will take the following form with respect to a suitable frame $\{e_{1}, e_{2}, \dots ,e_{n}\}$

$$\begin{aligned} \mathcal {A}e_1=-\frac{nH}{2}e_1, \quad \mathcal {A}e_i=\lambda e_i, \quad i=2,\dots ,n-1. \end{aligned}$$

(5.59)

Using (5.59) in (3.9) and (4.1), we get

$$\begin{aligned} \lambda =-\frac{3\lambda _1}{n-1}, \end{aligned}$$

(5.60)

$$\begin{aligned} (n-1)\lambda ^{2}=\beta -\lambda _1^2, \end{aligned}$$

(5.61)

respectively.

Further, if M has constant norm of second fundamental forms, then, from (5.60) and (5.61), we get $\lambda _1$ is a constant, which gives that H constant, a contradiction.

Combining Cases 1, 2, and 3 it follows that M has constant mean curvature.

Now, using (2.2) we get that the scalar curvature $\rho$ is given by

$$\begin{aligned} \rho = n(n-1)c+n^2H^2+\beta , \end{aligned}$$

(5.62)

which is also constant.