Ricci soliton biharmonic hypersurfaces in the Euclidean space
UDC 515.12 We investigate biharmonic Ricci soliton hypersurfaces $(M^n, g,\xi, \lambda)$ whose potential field $\xi$ satisfies certain conditions. We obtain a result based on the average scalar curvature of the compact Ricci soliton hypersurface $M^n$ where $\xi$ is a general vector field. Then we p...
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2021
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507032970330112 |
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| author | Mosadegh, N. Abedi, E. Ilmakchi, M. Mosadegh, N. Abedi, E. Ilmakchi, Mohammad Mosadegh, N. Abedi, E. Ilmakchi, M. |
| author_facet | Mosadegh, N. Abedi, E. Ilmakchi, M. Mosadegh, N. Abedi, E. Ilmakchi, Mohammad Mosadegh, N. Abedi, E. Ilmakchi, M. |
| author_sort | Mosadegh, N. |
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| datestamp_date | 2025-03-31T08:47:53Z |
| description | UDC 515.12
We investigate biharmonic Ricci soliton hypersurfaces $(M^n, g,\xi, \lambda)$ whose potential field $\xi$ satisfies certain conditions. We obtain a result based on the average scalar curvature of the compact Ricci soliton hypersurface $M^n$ where $\xi$ is a general vector field. Then we prove that there are no proper biharmonic Ricci soliton hypersurfaces in the Euclidean space $E^{n+1}$ provided that the potential field $\xi$ is either a principal vector in grad $H^\perp$ or $\xi=\dfrac{{ \rm{ grad } \,} H}{|{ \rm{ grad } \,} H|}$. |
| doi_str_mv | 10.37863/umzh.v73i7.495 |
| first_indexed | 2026-03-24T02:02:52Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v73i7.495
UDC 515.12
N. Mosadegh, E. Abedi, M. Ilmakchi (Azarbaijan Shahid Madani Univ., Tabriz, Iran)
RICCI SOLITON BIHARMONIC HYPERSURFACES
IN THE EUCLIDEAN SPACE
БIГАРМОНIЧНI ГIПЕРПОВЕРХНI СОЛIТОНIВ РIЧЧI
В ЕВКЛIДОВОМУ ПРОСТОРI
We investigate biharmonic Ricci soliton hypersurfaces (Mn, g, \xi , \lambda ) whose potential field \xi satisfies certain conditions.
We obtain a result based on the average scalar curvature of the compact Ricci soliton hypersurface Mn where \xi is a
general vector field. Then we prove that there are no proper biharmonic Ricci soliton hypersurfaces in the Euclidean space
En+1 provided that the potential field \xi is either a principal vector in grad H\bot or \xi =
\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}H
| \mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}H| .
Вивчаються бiгармонiчнi гiперповерхнi солiтонiв Рiччi (Mn, g, \xi , \lambda ), поле потенцiалу \xi яких задовольняє певнi
умови. Отриманий результат базується на середнiй скалярнiй кривинi гiперповерхнi Mn компактного солiтону Рiччi,
де \xi розглядається як узагальнене векторне поле. Пiсля цього доведено, що не iснує нетривiальних бiгармонiчних
гiперповерхонь солiтонiв Рiччi в евклiдовому просторi En+1, якщо поле потенцiалу \xi є або головним вектором у
grad H\bot , або \xi =
\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}H
| \mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}H| .
1. Introduction. The conception of biharmonic maps was introduced by Eells and Lemair [7] in
1983. It is denoted by C\infty (M,N) the space of smooth maps \varphi : (M, g) \rightarrow (N,h) between two
Riemannian manifolds. A biharmonic map \varphi \in C\infty (M,N) is a critical points of the bienergy
functional
E2 : C\infty (M,N) \rightarrow R, E2(\varphi ) =
1
2
\int
M
| \tau (\varphi )| 2d\nu g,
where \tau (\varphi ) = \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\nabla d\varphi is tension field of \varphi . Actually, the Euler – Lagrange equation correlate to
the bienergy is given by the vanishing of the bitension field
\tau 2(\varphi ) = - \Delta \tau (\varphi ) - \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}RN (d\varphi (.), \tau (\varphi ))d\varphi (.) = 0,
where RN is curvature tensor of N. Infact, bihamonic immersions are special class of biharmonic
maps. An isometric immersion \varphi : (Mn, g) - \rightarrow (Nm, h) is called biharmonic if and only if the
mean curvature vector field
- \rightarrow
H satisfies:
0 = \bigtriangleup
- \rightarrow
H + \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}RN (d\varphi (.),
- \rightarrow
H )d\varphi (.).
Additionally, in Euclidean space biharmonic submanifold and biharmonic immersion are coinsided
with each other. Also, it should be noticed that biharmonic submanifold was introduced by B. Y.
Chen in the middle of 1980s. At first, it was proved the biharmonic surfaces in three dimensions
Euclidean space E3 are minimal in 1985 [2]. Later on, others geometer deal with the result and
extended it. Infact, the result was developed by I. Dimitric [6] and T. Hasanis and T. Vlachos [9].
More precisely, Dimitric got progress on the result where the biharmonic hypersurfaces of Em have
at most two distinct principle curvatures. Also, T. Hasanis and T. Vlachos have extend the result
c\bigcirc N. MOSADEGH, E. ABEDI, M. ILMAKCHI, 2021
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7 931
932 N. MOSADEGH, E. ABEDI, M. ILMAKCHI
when they have proved biharmonic hypersurfaces in E4 are minimal. Consequently, according to the
outcome, a challenging conjecture was made by Chen [3]:
original biharmonic conjecture: ”the only biharmonic submanifolds of Euclidean space are mini-
mal ones”.
Later on, it was proven that biharmonic hypersurfaces in hyperbolic n-space Hn( - 1) with at
most two distinct principle curvatures are minimal [1]. Hence, according to the result, they made the
following generalization of Chen’s conjecture in [1]. Generalized Chen’s conjecture: ”any biharmonic
submanifold of a Reimannian manifold with nonpositive sectional curvature is minimal. Moreover,
Maeta [11] made another generalized Chen’s conjecture: ”the only k-harmonic submanifolds of a
Euclidean space are the minimal ones”.
Recently, authors in [10] have shown that the Hopf biharmonic hypersurfaces in complex
Euclidean space Cn+1 are minimal. Also, they proved that pseudo Hopf biharmonic hypersurface
in unit sphere S2n+1 is either a hypersphere S2n
\biggl(
1\surd
2
\biggr)
or a Clifford hypersurface Sn1
\biggl(
1\surd
2
\biggr)
\times
\times Sn2
\biggl(
1\surd
2
\biggr)
, where n1 + n2 = 2n. Indeed, it was a small progress on the biharmonic conjecture
for hypersurfaces too. In view of the above aspect, studying biharmonic hypersurface with geometric
condition is reasonable. The geometry of Ricci soliton manifolds have been intensively studied by
many geometers, for instance, see the paper was written by Chen – Yen and S. Deshmukh [4]. Indeed,
there was a classification of Ricci solitons on Euclidean hypersurfaces. Furthermore, S. Deshmukh
deal with the geometry of Ricci soliton that to find condition under which it is an Einstein mani-
fold [5].
In this paper, we study about proper biharmonic Ricci soliton hypersurface (Mn, g, \xi , \lambda ) in
Euclidean space En+1, somehow whose potential field \xi has an important role, to obtain the fol-
lowing result. At first, we got a result about compact Ricci soliton hypersurfaces in Euclidean space
with respect to average scaler curvature S, which is defined Av(S) = 1/\mathrm{v}\mathrm{o}\mathrm{l}(M)
\int
M
Sd\nu , where
Ricci soliton vector field \xi is general. Then, it was shown that a nonexisting proper biharmonic
Ricci soliton hypersurface (Mn, g, \xi , \lambda ), in Euclidean space En+1 where either \xi is in gradH\bot or
\xi =
\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}H
| \mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}H|
.
2. Preliminaries. In this section, we recall some fundamental definition for the theorem of Ricci
soliton biharmonic hypersurfaces which are immersed in an Euclidean space En+1.
Let x : Mn - \rightarrow En+1 be an isometric immersion of n-dimensional hypersurface (M, g) into
the Euclidean space En+1. Let \nabla and \nabla stand for Levi – Civita connections on Mn and En+1,
respectively. Let X and Y are tangent vector fields on M also N is considered a locally unit normal
vector field to M in En+1. Then Gauss and Weingarten formulas are
\nabla XY = \nabla XY + h(X,Y ),
\nabla XN = - AX,
where A is Weingarten operator and h is the second fundamental form of M. The mean curvature
vector field
- \rightarrow
H of Mn is defined
- \rightarrow
H =
1
n
(\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}A)N.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
RICCI SOLITON BIHARMONIC HYPERSURFACES IN THE EUCLIDEAN SPACE 933
Assume that
- \rightarrow
H = HN and H implies the mean curvature. One of considerable equation in differen-
tial geometry is \bigtriangleup x = - n
- \rightarrow
H, where \bigtriangleup Laplace – Beltrame operator is defined \bigtriangleup = - \mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}\nabla 2. An
isometric immersion x : Mn - \rightarrow En+1 is called biharmonic if and only if \bigtriangleup
- \rightarrow
H = 0. With respect
to
- \rightarrow
H = HN we have
0 = \bigtriangleup
- \rightarrow
H = 2A(\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}H) + nH\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}H + (\bigtriangleup H +H\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}A2).
So, by identifying the tangent and normal part of above equation, we arrived at necessary and
sufficient condition for Mn to be biharmonic hypersurface in Euclidean space En+1 in following:
\bigtriangleup H +H\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}A2 = 0,
2A(\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}H) + nH\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}H = 0.
(1)
Remark 1. Obviously, any minimal immersion, i.e., H = 0, is biharmonic. The nonharmonic
biharmonic immersions are called proper-biharmonic.
The significant type of smooth vector field on a Riemannian manifold (M, g) is the vector field
that defines a Ricci soliton. A smooth vector field \xi on a Riemannian manifold (M, g) is called to
define a Ricci soliton if it satisfies
1
2
(\$\xi g)(X,Y ) + \mathrm{R}\mathrm{i}\mathrm{c} (X,Y ) = \lambda g(X,Y ), X, Y \in \chi (M), (2)
where \$\xi denotes the Lie derivative in the direction of the vector field \xi , \mathrm{R}\mathrm{i}\mathrm{c} is Ricci tensor of
(M, g) and \lambda is a real number. A Ricci soliton manifold is denoted by (M, g, \xi , \lambda ) and say the
vector field \xi the potential field of the Ricci soliton. Also, the Ricci soliton is named shrinking,
steady or expanding with respect to \lambda > 0, \lambda = 0 or \lambda < 0, respectively. The Ricci soliton is called
trivial when \xi is Killing or zero, so in each case the metric is Einsteinian. If the potential vector
field \xi be the gradient of some smooth function f on M, the (M, g, \xi , \lambda ) is called gradient Ricci
soliton such that is denoted by (M, g, f, \lambda ) and say the smooth function f the potential function.
The gradient Ricci soliton (M, g, f, \lambda ) is named trivial provided that the potential function f be a
constant. Automatically the trivial gradient Ricci solitons are trivial Ricci solitons due to \xi = \nabla f.
In order to show the significant role of Ricci soliton vector field in our principal theorem, we
ended this section with following proposition, where \xi is general.
Proposition 1. Let (Mn, g, \xi , \lambda ) be a compact Ricci soliton hypersurface in Euclidean space
En+1. Then the Ricci soliton hypersurface is expanding, steady or shrinking provided that the ave-
rage scaler curvature Av(S) of Mn be positive, zero or negative, respectively.
Proof. Suppose that \{ ei\} be an appropriate orthogonal frame field on Mn that the Weingarten
operator takes form Aei = \mu iei for 1 \leq i \leq n. Let H be mean curvature vector field of Mn in
En+1 too. Now, by applying the equation (2) we have
g(ei,\nabla ei\xi ) = \lambda g(ei, ei) - \mathrm{R}\mathrm{i}\mathrm{c}\mathrm{c}\mathrm{i} (ei, ei) =
= \lambda + nH\mu i - \mu i
2,
it yields that
n\sum
i
g(ei,\nabla ei\xi ) = n\lambda + n2H2 -
n\sum
i
\mu i
2,
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
934 N. MOSADEGH, E. ABEDI, M. ILMAKCHI
\mathrm{d}\mathrm{i}\mathrm{v} \xi = n\lambda + n2H2 - | h| 2 = n\lambda + S,
where | h| 2 and S = n2H2 - | h| 2 are square of second fundamental form length and scaler curvature
of Mn, respectively. For a Riemannian manifold (M, g) we have that average scaler curvature S
of M as Av(S) = 1/\mathrm{v}\mathrm{o}\mathrm{l} (M)
\int
M
sd\nu . Then, from the last equation and take to account that Mn is
compact, we get
0 =
\int
Mn
\mathrm{d}\mathrm{i}\mathrm{v} \xi d\nu = n\lambda
\int
Mn
d\nu +
\int
Mn
Sd\nu = n\lambda \mathrm{v}\mathrm{o}\mathrm{l} (Mn) +Av(S)\mathrm{v}\mathrm{o}\mathrm{l} (Mn),
and \lambda = - Av(S)
n
. As it was claimed average scaler curvature, determined the type of compact Ricci
soliton hypersurfaces in Euclidean space as an expanding, steady or shrinking one.
Proposition 1 is proved.
In this short note, it will be shown how potential field \xi is used in order to get a little progress
on Chen’s conjecture.
3. Ricci soliton biharmonic hypersurface in Euclidean space \bfitE \bfitn +1 . In this section, we are
going to show that a proper biharmonic Ricci soliton connected hypersurfaces (Mn, g, \xi , \lambda ) in En+1
can not be existing, where the potential vector field \xi is either in gradH\bot or specially \xi =
\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}H
| \mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}H|
.
Now we suppose that the mean curvature is non constant. Taking it to account that, if we have
gradH \equiv 0, so H = constant. Then, according to the first condition of biharmonicity in equation
(1) and due to Mn is a proper biharmonic hypersurface, that is, H \not = 0. Hence, it implies that
Mn is a totally geodesic hypersurface in Euclidean space. Then it is a part of hyperplane in En+1.
Consequently, we obtained that Mn is a steady Ricci soliton biharmonic hypersurface in this case.
Nevertheless, there exists a point p \in M, where gradH \not = 0 at p. So, there is a open subset U of
Mn such that gradH \not = 0 on U. Actually, the second view of biharmonic condition recall that gradH
is an eigenvector corresponding to eigenvalue
- n
2
H. The Weingarten operator A takes following
form in the appropriate local frame field \{ e1, . . . , en\} :
A =
\left[ \lambda 1
. . .
\lambda n
\right] , (3)
where \lambda i is eigenvalue of shape operator A corresponding to eigenvector ei. Without loss of genera-
lity, we suppose that e1 in the direction of \mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}H. Assume that gradH is given by
\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}H =
n\sum
i=1
ei(H)ei.
It is followed that
e1(H) \not = 0, ei(H) = 0, i = 2, . . . , n.
Also, it is written
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
RICCI SOLITON BIHARMONIC HYPERSURFACES IN THE EUCLIDEAN SPACE 935
\nabla eiej =
n\sum
k=1
\omega k
ijek, (4)
where \omega k
ij is called Cartan coefficient. Then computing the compatibility conditions
\nabla ek\langle ei, ej\rangle = 0,
that denotes
\omega i
ki = 0, i = j, (5)
\omega j
ki + \omega i
kj = 0, i \not = j, i, j, k = 1, . . . , n. (6)
Moreover, from the Codazzi equation we have
(\nabla ekA)ei = (\nabla eiA)ek,
\nabla ekAei - A\nabla ekei = \nabla eiAek - A\nabla eiek.
Now take the above equation, (3), (4) and we get
ek(\lambda i)ei + (\lambda i - \lambda j)\omega
j
kiej = ei(\lambda k)ek + (\lambda k - \lambda j)\omega
j
ikej .
We multiply both side of above equation to ej , then we arrived at following equation:
ei(\lambda j) = (\lambda i - \lambda j)\omega
j
ji, (7)
(\lambda i - \lambda j)\omega
j
ki = (\lambda k - \lambda j)\omega
j
ik, (8)
for distinct i, j, k = 1, . . . , n. From \lambda 1 = - n
2
H and (4), we obtain
e1(\lambda 1) \not = 0, ei(\lambda 1) = 0, i = 2, . . . , n, (9)
and
[ei, ej ]\lambda 1 = 0, 2 \leq i, j \leq n, i \not = j,
which implies
\omega 1
ij = \omega 1
ji (10)
for distinct i, j = 2, . . . , n. It is claimed that \lambda j \not = \lambda 1 for j = 2, . . . , n [8]. Since, if \lambda j = \lambda 1 for
j \not = 1, utilize (7) and put i = 1. Then
0 = (\lambda 1 - \lambda j)\omega
j
j1 = e1(\lambda j) = e1(\lambda 1),
which contradicts to (9). For j = 1 and k, i \not = 1, from (8), we get
(\lambda i - \lambda 1)\omega
1
ki = (\lambda k - \lambda 1)\omega
1
ik,
which together with (10) yield
\omega 1
ij = 0, i \not = j, i, j = 2, . . . , n. (11)
Combining (11) with equation (6), we obtain \omega j
i1 = 0, i \not = j , i, j = 2, . . . , n.
Taking all the information in to account and summarizing them, we have the following lemma.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
936 N. MOSADEGH, E. ABEDI, M. ILMAKCHI
Lemma 1. Suppose that M be a biharmonic hypersurface in Euclidean space En+1 with non-
constant mean curvature, whose Weingarten operator is given by (3) with respect to an orthogonal
frame \{ e1, . . . , en\} . Then
\nabla e1ei = 0, 1 \leq i \leq n,
\nabla eie1 = - \omega 1
iie1, i = 2, . . . , n,
\nabla eiei =
n\sum
k=1,i \not =k
\omega k
iiek,
\nabla eiej =
n\sum
k=2
\omega k
ijek,
where \nabla denote Livi – Civita connection on M and \omega k
ij satisfies the equation (4).
Generally, a biharmonic hypersurface in Euclidean space En+1 satisfies the equation (1), where
\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}H is an eigenvector of Wiengarten operator.
We are going to prove the following theorem according to the significant point that M is a proper
biharmonic hypersurface in En+1.
Theorem 1. In Euclidean space En+1 does not exist a proper biharmonic Ricci soliton hyper-
surface (Mn, g, \xi , \lambda ) provided that the potential field \xi either be a principal vector in \mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}H\bot , or
specially \xi =
\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}H
| \mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}H|
.
Proof. Let at a point p \in Mn we have gradH \not = 0. So, there exists an open subset U \subset Mn
which gradH \not = 0 there. Suppose that \{ e1, e2, . . . , en\} be an appropriate orthonormal local frame
field at p such that Wiengarten operator A takes form (3). According to equation (1) it can be let
that e1 =
\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}H
| \mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}H|
. Now, by applying the equation (2), we have
(\$\xi g)(ei, ei) = \lambda g(ei, ei) - \mathrm{R}\mathrm{i}\mathrm{c} (ei, ei) = g(\nabla ei\xi , ei).
From above equation we obtain
g(\nabla gradH\xi , \mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}H) = \lambda g(\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}H, \mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}H) - \mathrm{R}\mathrm{i}\mathrm{c} (\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}H, \mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}H).
According to assumption \mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}H = | \mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}H| e1, it yields
g(\nabla e1\xi , e1)| \mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}H| 2 = (\lambda g(e1, e1) - \mathrm{R}\mathrm{i}\mathrm{c} (e1, e1))| \mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}H| 2,
also \mathrm{R}\mathrm{i}\mathrm{c} (e1, e1) = - 3n2H2
4
. Hence, we arrived at
g(e1,\nabla e1\xi ) =
\biggl(
\lambda +
3n2H2
4
\biggr)
. (12)
Now we have two following cases.
Case 1. The potential field \xi =
\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}H
| \mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}H|
.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
RICCI SOLITON BIHARMONIC HYPERSURFACES IN THE EUCLIDEAN SPACE 937
Obviously, g(e1,\nabla e1\xi ) = 0, where \xi = e1 by applying Lemma 1 that \nabla e1\xi = 0. Then,
according to the right-hand side of equation (12), we obtained a contradiction due to gradH \not = 0 and
\lambda is constant. Therefore, the theorem was proven as it was claimed in special case.
Case 2. The potential field \xi is in \mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}H\bot .
Suppose that potential field \xi is in gradH\bot . With respect to the assumption gradH is an
principal vector corresponding to eigenvalue - n
2
H. Then, using equation (2), where we suppose
\xi =
\sum n
t=2
\alpha tet. On the one hand, rewrite the left-hand side of equation (2) and apply Lemma 1, we
arrived at
g
\Biggl(
e1,\nabla e1
n\sum
t=2
\alpha tet
\Biggr)
= g
\Biggl(
e1,
n\sum
t=2
(e1(\alpha t)et + (\alpha t)\nabla e1et)
\Biggr)
=
= g
\Biggl(
e1,
n\sum
t=2
e1(\alpha t)et
\Biggr)
+ g
\Biggl(
e1,
n\sum
t=2
\alpha t\nabla e1et
\Biggr)
= 0.
On the other hand, it is observed that the left-hand side of above equation is equal to
\biggl(
\lambda +
+
3n2H2
4
\biggr)
= 0 according to equation (12), which yields it is impossible, where the mean curvature
H is not constant. Consequently, we obtained that proper biharmonic Ricci soliton hypersurfaces
(Mn, g, \xi , \lambda ) in Euclidean space are not existing whenever the potential field \xi is in gradH\bot .
Theorem 1 is proved.
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Received 22.11.18
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
|
| id | umjimathkievua-article-495 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:02:52Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/67/1ac283176aa66c870079500cafe9a367.pdf |
| spelling | umjimathkievua-article-4952025-03-31T08:47:53Z Ricci soliton biharmonic hypersurfaces in the Euclidean space Ricci soliton biharmonic hypersurfaces in the Euclidean space Ricci soliton biharmonic hypersurfaces in the Euclidean space Mosadegh, N. Abedi, E. Ilmakchi, M. Mosadegh, N. Abedi, E. Ilmakchi, Mohammad Mosadegh, N. Abedi, E. Ilmakchi, M. Biharmonic Hypersurfaces Ricci Soliton Biharmonic Hypersurfaces Ricci Soliton UDC 515.12 We investigate biharmonic Ricci soliton hypersurfaces $(M^n, g,\xi, \lambda)$ whose potential field $\xi$ satisfies certain conditions. We obtain a result based on the average scalar curvature of the compact Ricci soliton hypersurface $M^n$ where $\xi$ is a general vector field. Then we prove that there are no proper biharmonic Ricci soliton hypersurfaces in the Euclidean space $E^{n+1}$ provided that the potential field $\xi$ is either a principal vector in grad $H^\perp$ or $\xi=\dfrac{{ \rm{ grad } \,} H}{|{ \rm{ grad } \,} H|}$. UDC 515.12 Бiгармонiчнi гiперповерхнi солiтонiв Рiчi у евклiдовому просторi Вивчаються бігармонічні гіперповерхні солітонів Річчі $(M^n, g,\xi, \lambda),$ поле потенціалу $\xi$ яких задовольняє певні умови. Отриманий результат базується на середній скалярній кривині гіперповерхні $M^n$ компактного солітону Річчі, де $\xi$ розглядається як узагальнене векторне поле. Після цього доведено, що не існує нетривіальних бігармонічних гіперповерхонь солітонів Річчі в евклідовому просторі $E^{n+1},$ якщо поле потенціалу $\xi$ є або головним вектором у grad $H^\perp,$ або $\xi=\dfrac{{ \rm{ grad } \,} H}{|{ \rm{ grad } \,} H|}.$ Institute of Mathematics, NAS of Ukraine 2021-07-20 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/495 10.37863/umzh.v73i7.495 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 7 (2021); 931 - 937 Український математичний журнал; Том 73 № 7 (2021); 931 - 937 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/495/9067 Copyright (c) 2021 Esmaiel Abedi, Mohammad Ilmakchi |
| spellingShingle | Mosadegh, N. Abedi, E. Ilmakchi, M. Mosadegh, N. Abedi, E. Ilmakchi, Mohammad Mosadegh, N. Abedi, E. Ilmakchi, M. Ricci soliton biharmonic hypersurfaces in the Euclidean space |
| title | Ricci soliton biharmonic hypersurfaces in the Euclidean space |
| title_alt | Ricci soliton biharmonic hypersurfaces in the Euclidean space Ricci soliton biharmonic hypersurfaces in the Euclidean space |
| title_full | Ricci soliton biharmonic hypersurfaces in the Euclidean space |
| title_fullStr | Ricci soliton biharmonic hypersurfaces in the Euclidean space |
| title_full_unstemmed | Ricci soliton biharmonic hypersurfaces in the Euclidean space |
| title_short | Ricci soliton biharmonic hypersurfaces in the Euclidean space |
| title_sort | ricci soliton biharmonic hypersurfaces in the euclidean space |
| topic_facet | Biharmonic Hypersurfaces Ricci Soliton Biharmonic Hypersurfaces Ricci Soliton |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/495 |
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