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.
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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
Let \(\nabla \) denotes the Levi-Civita connection on M. Then, the Gauss and Codazzi equations are given by
respectively, where R is the curvature tensor and
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
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
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
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 \(\{e_{1}, e_{2},\dots , e_{n}\}\),
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
As we have taken \(e_{1}\) parallel to \(\mathop {\mathrm {grad}}H\), it follows that
We express
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
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
Putting the value of \((\nabla _{e_{i}}\mathcal {A})e_{j}\) in (2.3), we find
Using \(i\ne j=k\) and \(i\ne j \ne k\) in the above equation, we obtain
and
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,
Proof
Assume that \(\lambda _{j}= \lambda _{1}\) for some \(j\ne 1\). Then, from (3.6) we get that
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
Here \(p+q+r+1=n\) and \(C_1, C_2\) and \(C_3\) denote the sets
Using (2.1) and (3.1), we obtain that
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:
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
Putting \(i\ne 1, j = 1\) in (3.6) and using (3.5) and (3.3), we find
Putting \(i = 1\) in (3.7), we obtain
Taking \(i \in C_s, \ s\in \{1, 2, 3\}\) in (3.7), we have
Putting \(j = 1\) in (3.7) and using (3.10), we get
Putting \(i = 1\) in (3.7) and using (3.14) and (3.5), we find
Combining (3.14) and (3.12), we obtain
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:
and
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
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
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
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
and
respectively.
Eliminating \(e_{u}(\lambda _{u})\) from (4.2) and (4.3), we find
Putting the value of \(e_{u}(\lambda _{v})\) and \(e_{u}(\lambda _{w})\) from (3.6) in (4.4), we obtain
Differentiating (4.5) with respect to \(e_{1}\) and using (3.6) and (3.18), we have
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
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
Now, we consider two cases.
(i) \(a_1=0.\) Then, from (4.8), we obtain
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
Now, using (3.6) and (3.19) in (4.10), we obtain
where
Differentiating \(a_1\) with respect to \(e_u\) and using (3.6), (3.19) and (4.2), we find
where
Differentiating (3.9) with respect to \(e_1\) and using (3.6), we find
Differentiating (4.13) with respect to \(e_u\) and using (3.3), (3.6), (3.19), (3.20) and (4.2), we find
where
Eliminating \(e_u(\omega _{uu}^{1})\) from (4.12) and (4.14), we get
Eliminating \(e_u(a_1)\) from (4.11) and (4.15), we find
where
Now, we simplify \(f_7\) and \(f_8\). Eliminating \(\omega _{uu}^1\) from \(f_7\) using \(a_1\), we get
Eliminating \(a_1\) from (4.17) using (4.8), we obtain
where
Similarly, eliminating \(\omega _{uu}^1\) from \(f_8\) using \(a_1\), we get
Eliminating \(a_1\) from (4.19) using (4.8), we obtain
where
Eliminating \(f_7\) and \(f_8\) from (4.16) using (4.18) and (4.20), we obtain
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
for \(g_1, g_2\), gives
Now, eliminating \(\lambda _w\) from (4.1) and (4.23) using (3.9), we find
and
where
Eliminating \(\lambda _v\) from (4.25) using (4.24), we obtain
Differentiating (4.26) with respect to \(e_u\) and using (3.3), we get
whereby, we get \(e_u(\lambda _u)=0\).
Therefore, from (4.2) and (3.6), we obtain
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
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
(b) If \(a_1=0\), we have that
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
Putting \(j=w, k=u, i=v\) in (3.7), we get
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
for some smooth function \(\alpha \).
From (4.34), we get
for some smooth function \(\phi \).
Differentiating (4.35) with respect to \(e_1\) and using (3.6), (3.17) and (4.35), we find
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
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
respectively.
Simplifying (5.3), (5.4) and (5.5), we obtain
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
From (5.9), we get
Differentiating (3.9) and (4.1) with respect to \(e_1\) and using (3.6), we find
and
respectively.
Eliminating \(e_1(\lambda _1)\) from (5.12) using (5.11), we obtain
Multiplying (5.13) with \(\omega _{uu}^{1}\) and using (5.9), we find
Eliminating \((\omega _{uu}^{1})^2\) from (5.14) using (5.10), we obtain
Eliminating \(\lambda _w\) from (5.15) using (3.9), we obtain
where
Eliminating \(\lambda _w\) from (4.1) using (3.9), we obtain
where
Equations (5.16) and (5.17) have a common root \(\lambda _v\), so their resultant with respect \(\lambda _v\) vanish. Hence,
which is a polynomial equation
for \(\lambda _1, \lambda _u\).
Differentiating (5.19) with respect to \(e_1\), we get
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
Multiplying (5.21) with \(\omega _{uu}^{1}\) and using (5.9) and (5.10), we obtain
where \(L=\frac{(3G_u-pG_1)(\lambda _u-\lambda _1)}{G_1}\).
Eliminating \(\lambda _w\) from (5.22) using (3.9), we get
where
Equations (5.23) and (5.17) have a common root \(\lambda _v\), so their resultant with respect \(\lambda _v\) vanish. Therefore, we find
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
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
Using (4.31) in (5.26) in Lemma 4.2 we find
Using (4.31) in (5.13), we get
On the other hand, using (3.9), (4.1) and (4.31) in (4.13), we obtain
Eliminating \(\lambda _w\) from (4.1), (5.27) and (5.28) using (3.9), we find
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
and
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
which is a polynomial equation
Differentiating (5.36) with respect to \(e_1\) and using (4.31) and (5.29), we find a polynomial
Differentiating (5.37) with respect to \(e_1\) and using (4.31) and (5.29), we find a polynomial
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
which are polynomial equations
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
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 \(\{e_{1}, e_{2}, \dots ,e_{n}\}\)
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
Eliminating \(\lambda _v\) from (5.44) using (5.43), we obtain
Similarly, eliminating \(\lambda _u\) from (5.44) using (5.43), we get
Differentiating (5.45) with respect to \(e_1\), we find
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
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
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
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
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
Using (3.6), (5.43) and (5.47) in (5.51), we get
Using (3.6) and (5.47) in (5.52), we obtain
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
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
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
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 \(\{e_{1}, e_{2}, \dots ,e_{n}\}\)
Using (5.59) in (3.9) and (4.1), we get
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
which is also constant.
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Acknowledgements
This work is supported by award of grant under FRGS for the year 2017-18, F.No. GGSIPU/DRC/Ph.D./Adm./2017/493. The authors would like to thank the referee for useful suggestions on the paper.
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Gupta, R.S., Arvanitoyeorgos, A. Biconservative Hypersurfaces in Space Forms \(\overline{M}^{{n+1}}({c})\). Mediterr. J. Math. 19, 256 (2022). https://doi.org/10.1007/s00009-022-02134-y
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DOI: https://doi.org/10.1007/s00009-022-02134-y