Ricci Solitons on Lorentzian Four-Dimensional Generalized Symmetric Spaces
The existence of non-trivial, i.e., non-Einstein, Ricci solitons on fourdimensional Lorentzian generalized symmetric spaces is proved. Moreover, it is shown that only steady Ricci solitons can be gradient.
Збережено в:
| Дата: | 2018 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2018
|
| Назва видання: | Журнал математической физики, анализа, геометрии |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/145864 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Ricci Solitons on Lorentzian Four-Dimensional Generalized Symmetric Spaces / A. Bouharis, B. Djebbar // Журнал математической физики, анализа, геометрии. — 2018. — Т. 14, № 2. — С. 132-140. — Бібліогр.: 15 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-145864 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1458642019-02-02T01:23:16Z Ricci Solitons on Lorentzian Four-Dimensional Generalized Symmetric Spaces Bouharis, A. Djebbar, B. The existence of non-trivial, i.e., non-Einstein, Ricci solitons on fourdimensional Lorentzian generalized symmetric spaces is proved. Moreover, it is shown that only steady Ricci solitons can be gradient. Доведено iснування нетривiальних (тобто, неейнштейнiвських) солiтонiв Рiччi на чотиривимiрних лоренцевих узагальнених симетричних просторах. Бiльш того, показано, що тiльки стiйкi солiтони Рiччi можуть бути градiєнтними. 2018 Article Ricci Solitons on Lorentzian Four-Dimensional Generalized Symmetric Spaces / A. Bouharis, B. Djebbar // Журнал математической физики, анализа, геометрии. — 2018. — Т. 14, № 2. — С. 132-140. — Бібліогр.: 15 назв. — англ. 1812-9471 DOI: https://doi.org/10.15407/mag14.02.132 Mathematics Subject Classification 2010: 53C20, 53C21 http://dspace.nbuv.gov.ua/handle/123456789/145864 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
The existence of non-trivial, i.e., non-Einstein, Ricci solitons on fourdimensional Lorentzian generalized symmetric spaces is proved. Moreover, it is shown that only steady Ricci solitons can be gradient. |
| format |
Article |
| author |
Bouharis, A. Djebbar, B. |
| spellingShingle |
Bouharis, A. Djebbar, B. Ricci Solitons on Lorentzian Four-Dimensional Generalized Symmetric Spaces Журнал математической физики, анализа, геометрии |
| author_facet |
Bouharis, A. Djebbar, B. |
| author_sort |
Bouharis, A. |
| title |
Ricci Solitons on Lorentzian Four-Dimensional Generalized Symmetric Spaces |
| title_short |
Ricci Solitons on Lorentzian Four-Dimensional Generalized Symmetric Spaces |
| title_full |
Ricci Solitons on Lorentzian Four-Dimensional Generalized Symmetric Spaces |
| title_fullStr |
Ricci Solitons on Lorentzian Four-Dimensional Generalized Symmetric Spaces |
| title_full_unstemmed |
Ricci Solitons on Lorentzian Four-Dimensional Generalized Symmetric Spaces |
| title_sort |
ricci solitons on lorentzian four-dimensional generalized symmetric spaces |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2018 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/145864 |
| citation_txt |
Ricci Solitons on Lorentzian Four-Dimensional Generalized Symmetric Spaces / A. Bouharis, B. Djebbar // Журнал математической физики, анализа, геометрии. — 2018. — Т. 14, № 2. — С. 132-140. — Бібліогр.: 15 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT bouharisa riccisolitonsonlorentzianfourdimensionalgeneralizedsymmetricspaces AT djebbarb riccisolitonsonlorentzianfourdimensionalgeneralizedsymmetricspaces |
| first_indexed |
2025-07-10T22:42:31Z |
| last_indexed |
2025-07-10T22:42:31Z |
| _version_ |
1837301613851574272 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2018, Vol. 14, No. 2, pp. 132–140
doi: https://doi.org/10.15407/mag14.02.132
Ricci Solitons on Lorentzian
Four-Dimensional Generalized Symmetric
Spaces
Amel Bouharis and Bachir Djebbar
The existence of non-trivial, i.e., non-Einstein, Ricci solitons on four-
dimensional Lorentzian generalized symmetric spaces is proved. Moreover,
it is shown that only steady Ricci solitons can be gradient.
Key words: Lorentzian metric, Ricci solitons, gradient Ricci solitons,
generalized symmetric spaces.
Mathematical Subject Classification 2010: 53C20, 53C21.
1. Introduction
A Ricci soliton is a natural generalization of an Einstein metric. It is defined
on a pseudo-Riemannian manifold (M, g) by
LXg + % = λg, (1.1)
where X is a smooth vector field on M, LX denotes the Lie derivative in the
direction of X, % is the Ricci tensor and λ is a real number. Moreover, we say
that the Ricci soliton (M, g) is a gradient Ricci soliton if it admits a vector field
X satisfying X = grad h for some potential function h.
A pseudo-Riemannian manifold (M, g) is said to be Yamabe soliton if it admits
a vector field X such that
LXg = (τ − γ)g, (1.2)
where τ denotes the scalar curvature of (M, g) and γ is a real number.
A Ricci soliton (Yamabe soliton) is said to be shrinking, steady or expanding
according to whether λ > 0, λ = 0 or λ < 0 if γ > 0, γ = 0 or γ < 0, respectively.
In [4], the authors studied Ricci and Yamabe solitons on second-order sym-
metric Lorentzian spaces.
Lorentzian Ricci solitons have been intensively studied showing many essential
differences with respect to the Riemannian case [2,8,14]. In fact, although there
exist three-dimensional Riemannian homogeneous Ricci solitons [1,13], there are
no left-invariant Riemannian Ricci solitons on three-dimensional Lie groups [11]
c© Amel Bouharis and Bachir Djebbar, 2018
https://doi.org/10.15407/mag14.02.132
Ricci Solitons on Lorentzian Four-Dimensional Generalized Symmetric Spaces 133
(see also [15]). Moreover, the Lorentzian case is much richer, allowing the ex-
istence of expanding, steady and shrinking left-invariant Ricci solitons [5]. Re-
cently, Calvaruso and Fino proved, contrarily to the Riemannian case, the ex-
istence of non-compact four-dimensional homogeneous pseudo-Riemannian Ricci
solitons which are not isometric to solvmanifolds [9]. These results make it inter-
esting for further investigation of Ricci solitons on Lorentzian manifolds.
In [12], pseudo-Riemannian four-dimensional generalized symmetric spaces
were classified into four classes, named A, B, C and D, and the pseudo-
Riemannian metrics can have any signature. All these spaces are reductive ho-
mogeneous. Their geometrical properties were studied by Calvaruso and De Leo
in [7].
Four-dimensional generalized symmetric spaces were studied by many au-
thors. In particular, in [3], the authors classified, up to isometry, non-symmetric
simply-connected four-dimensional pseudo-Riemannian generalized symmetric
spaces which are algebraic Ricci solitons. Those of Cerny–Kowalski’s types A, C
and D are algebraic Ricci solitons, whereas those of type B are not.
A complete classification of Ricci solitons of generalized symmetric spaces of
type A, B, D has been recently obtained in [10], leading to new examples. In this
paper, we find the general solution to equation (1.1) for generalized symmetric
spaces of type C.
The paper is organized in the following way. In Section 2, we shall report
the basic description of four-dimensional generalized symmetric spaces. In Sec-
tion 3, the Levi-Civita connection, the curvature tensor and the Ricci tensor of
Lorentzian four-dimensional generalized symmetric spaces will be described in
terms of components with respect to the coordinate vector fields
{
∂i = ∂
∂xi
,
}
.
This provides the needed information for the study, which we do in Section 4.
In Section 4, Ricci solitons on generalized symmetric spaces of type C are char-
acterized via a system of partial differential equations. In particular, we show
that these spaces admit different vector fields resulting in expanding, steady and
shrinking Ricci solitons. Finally, it is proved that those Ricci solitons are gradient
only in the steady case.
2. Four-dimensional generalized symmetric spaces
We start by recalling the definition of the generalized symmetric space. Let
(M, g) be a (pseudo-)Riemannian manifold. A regular s-structure on M is a
family of isometries {sp | p ∈M} of (M, g) such that
• the mapping M ×M →M , (p, q) 7→ sp(q), is smooth,
• p is an isolated fixed point of sp, ∀p ∈M ,
• sp ◦ sq = ssp(q) ◦ sp, ∀p, q ∈M .
The map sp is called the symmetry centered at p. The order of a regular s-
structure is the smallest integer k > 2 such that skp = idM for all p ∈M . If such
an integer does not exist, we say that the regular s-structure has order infinity. A
134 Amel Bouharis and Bachir Djebbar
generalized symmetric space is a connected pseudo-Riemannian manifold carrying
at least one regular s-structure. In particular, a generalized symmetric space is a
pseudo-Riemannian symmetric space if and only if it admits a regular s-structure
of order 2. The order of a generalized symmetric space is the minimum of orders
of all possible s-structures on it. Furthermore, if (M, g) is a generalized symmetric
space, then it is homogeneous, that is, the full isometry group I (M) of M acts
transitively on it, which means that (M, g) can be identified with (G/H, g), where
G ⊂ I (M) is a subgroup of I (M) acting transitively on M , and H is the isotropy
group at a fixed point o ∈M .
Generalized symmetric spaces of low dimension have been completely classi-
fied. The following theorem recalls the classification of non-symmetric simply-
connected 4-dimensional pseudo-Riemannian generalized symmetric spaces.
Theorem 2.1 (Cerny and Kowalski [12]). Non-symmetric, simply-connected
generalized symmetric spaces (M, g) of dimension 4 are of order either 3 or 4, or
infinity. All these spaces are indecomposable and belong, up to isometry, to one
of the following four types.
Type A. The underlying homogeneous space is G/H, where
G =
a b x3
c d x4
0 0 1
, H =
cos t − sin t 0
sin t cos t 0
0 0 1
with ad − bc = 1. (M, g) is the space R4 (x1, x2, x3, x4) with the pseudo-
Riemannian metric
g = λ
[(
1 + x22
)
dx21 +
(
1 + x21
)
dx22 − 2x1x2 dx1 dx2
]
/
(
1 + x21 + x22
)
±
[(
−x1 +
√
1 + x21 + x22
)
dx23
+
(
x1 +
√
1 + x21 + x22
)
dx24 − 2x22dx3 dx4
]
, (2.1)
where λ 6= 0 is a real constant. The order is k = 3, and the possible signatures
are (4, 0), (2, 2) and (0, 4).
Type B. The underlying homogeneous space is G/H, where
G =
e−(x1+x2) 0 0 a
0 ex1 0 b
0 0 ex2 c
0 0 0 1
, H =
1 0 0 −w
0 1 0 −2w
0 0 1 2w
0 0 0 1
.
(M, g) is the space R4 (x1, x2, x3, x4) with the pseudo-Riemannian metric
g = λ
(
dx21 + dx22 + dx1dx2
)
+ e−x2 (2dx1 + dx2) dx4
+ e−x1 (dx1 + 2dx2) dx3, (2.2)
where λ is a real constant. The order is k = 3, and the signature is always (2, 2).
Ricci Solitons on Lorentzian Four-Dimensional Generalized Symmetric Spaces 135
Type C. The underlying homogeneous space G/H is the matrix group
G =
e−x4 0 0 x1
0 ex4 0 x2
0 0 1 x3
0 0 0 1
.
(M, g) is the space R4 (x1, x2, x3, x4) with the Lorentzian metric
g = ε
(
e−2x4dx21 + e2x4dx22
)
+ dx3dx4 with ε = ±1. (2.3)
The order is k = 3, and the possible signatures are (1, 3) , (3, 1).
Type D. The underlying homogeneous space is G/H, where
G =
a b x1
c d x2
0 0 1
, H =
ex4 0 0
0 e−x4 0
0 0 1
with ad − bc = 1. (M, g) is the space R4 (x1, x2, x3, x4) with the pseudo-
Riemannian metric
g = −2 cosh (2x3) cos (2x4) dx1 dx2 + λ
(
dx23 − cosh2 (2x3) dx
2
4
)
+ (sinh (2x3)− cosh (2x3) sin (2x4)) dx
2
1
+ (sinh (2x3) + cosh (2x3) sin (2x4)) dx
2
2, (2.4)
where λ 6= 0 is a real constant. The order is infinite, and the signature is (2, 2).
3. Curvature of four-dimensional generalized symmetric spa-
ce of type C
Let (M, g) be a four-dimensional generalized symmetric space of type C, and
we denote by ∇ and R the Levi-Civita connection and the Riemann curvature
tensor of M , respectively. Throughout this paper, we will always use the sign
convention
R (X,Y ) = ∇[X,Y ] − [∇X ,∇Y ] .
The Ricci tensor of (M, g) is defined by %(X,Y ) = tr{Z → R(X,Z)Y }.
We shall report the nonvanishing Levi-Civita connection, the Riemann cur-
vature tensor, and the corresponding Ricci tensor with respect to the coordinates
vector fields
{
∂1 = ∂
∂x1
, ∂2 = ∂
∂x2
, ∂3 = ∂
∂x3
, ∂4 = ∂
∂x4
}
.
Lemma 3.1. Let M be a four-dimensional generalized symmetric space of
type C. Then the non-vanishing components of the Levi-Civita connection ∇ of
M are given by
∇∂1∂1 = 2εe−2x4∂3, ∇∂1∂4 = ∇∂4∂1 = −∂1,
∇∂2∂2 = −2εe2x4∂3, ∇∂2∂4 = ∇∂4∂2 = ∂2.
136 Amel Bouharis and Bachir Djebbar
The only non-zero components of the Riemann curvature tensor R are
R∂1,∂4∂1 = −2εe−2x4∂3, R∂1,∂4∂4 = ∂1,
R∂2,∂4∂2 = −2εe2x4∂3, R∂2,∂4∂4 = ∂2,
and the ones obtained by them using the symmetries of the curvature tensor. The
non-zero components of the Ricci tensor are given by
%
∂4,∂4
= −2.
4. Ricci solitons on four-dimensional generalized symmetric
space of type C
In this section, we analyze the existence of Ricci solitons on the four-
dimensional generalized symmetric spaces (M, g) of type C. Let X = f1∂1 +
f2∂2 + f3∂3 + f4∂4 be an arbitrary vector field on (M, g), where f1, . . . , f4 are
smooth functions of the variables x1, x2, x3, x4. The Lie derivative of metric (2.3)
with respect to X is given by:
(LXg)∂1,∂1 = 2εe−2x4 (∂1f1 − f4) , (4.1)
(LXg)∂1,∂2 = ε
(
e2x4∂1f2 + e−2x4∂2f1
)
,
(LXg)∂1,∂3 =
1
2
∂1f4 + εe−2x4∂3f1,
(LXg)∂1,∂4 =
1
2
∂1f3 + εe−2x4∂4f1,
(LXg)∂2,∂2 = 2εe2x4 (f4 + ∂2f2) ,
(LXg)∂2,∂3 =
1
2
∂2f4 + εe2x4∂3f2,
(LXg)∂2,∂4 =
1
2
∂2f3 + εe2x4∂4f2,
(LXg)∂3,∂3 = ∂3f4,
(LXg)∂3,∂4 =
1
2
(∂3f3 + ∂4f4) ,
(LXg)∂4,∂4 = ∂4f3.
By using (2.3) and (4.1) in (1.1), a standard calculation gives that a four-
dimensional generalized symmetric space of type C is a Ricci soliton if and only
if the following system holds:
∂1f1 − f4 =
λ
2
, (4.2)
e2x4∂1f2 + e−2x4∂2f1 = 0, (4.3)
∂1f4 + 2εe−2x4∂3f1 = 0, (4.4)
∂1f3 + 2εe−2x4∂4f1 = 0, (4.5)
∂2f2 + f4 =
λ
2
, (4.6)
Ricci Solitons on Lorentzian Four-Dimensional Generalized Symmetric Spaces 137
∂2f4 + 2εe2x4∂3f2 = 0, (4.7)
∂2f3 + 2εe2x4∂4f2 = 0, (4.8)
∂3f3 + ∂4f4 = λ, (4.9)
∂4f3 = 2, (4.10)
∂3f4 = 0. (4.11)
Equation (4.11) yields f4 = f4 (x1,x2, x4) . Deriving equations (4.5) and (4.8) with
respect to x4, we obtain, since ∂4f3 = 2,
∂24f1 = 2∂4f1,
∂24f2 = −2∂4f2.
Hence, integrating, we get
f1 = e2x4h (x1, x2, x3) +H (x1, x2, x3) , (4.12)
f2 = e−2x4k (x1, x2, x3) +K (x1, x2, x3) ,
where h, k, H, and K are smooth functions depending on x1, x2, x3.
Next, from equations (4.2) and (4.6), we have ∂1f1 + ∂2f2 = λ. Then we
replace f1 and f2 to obtain
e2x4∂1h+ e−2x4∂2k + ∂1H + ∂2K = λ.
Deriving with respect to x4, we have
e2x4∂1h− e−2x4∂2k = 0,
which, since x4 is arbitrary, gives ∂1h = ∂2k = 0, and so
∂1H + ∂2K = λ. (4.13)
We replace f1 and f2 in equation (4.3) to find
e2x4∂1K + e−2x4∂2H + ∂1k + ∂2h = 0.
Deriving with respect to x4, we find that ∂1K = ∂2H = 0, which gives
∂1k + ∂2h = 0. (4.14)
Equation in (4.2) yields, since ∂1h = 0, f4 = ∂1H − λ
2 . Then from equation
(4.4), we get
∂21H + 2ε∂3h+ 2εe−2x4∂3H = 0,
which, by derivation with respect to x4, gives ∂3H = 0. Hence, H depends only
on x1 and thus
∂21H + 2ε∂3h = 0. (4.15)
We derive (4.13) with respect to x1, we deduce (since ∂1K = 0) that ∂21H = 0.
Thus H = αx1 + β where α, β ∈ R.
138 Amel Bouharis and Bachir Djebbar
Replacing H in (4.15), we have ∂3h = 0, and we can conclude that h depends
only on x2. Because of (4.13), we then have K = (λ− α)x2 + K̄(x3), where K̄ is
a smooth function.
Next, taking the derivative of (4.14) with respect to x2, we prove, since ∂2k =
0, that h = γx2+δ where γ, δ ∈ R. Hence, k = −γx1+ k̄(x3), where k̄ is a smooth
function. Thus, equation (4.6) yields f4 = α− λ
2 . Moreover, equation (4.7) implies
∂3f2 = 0, that is, k̄′(x3)e
−2x4 +K̄ ′(x3) = 0, which gives k̄′(x3) = K̄ ′(x3) = 0, and
so k̄(x3) = a and K̄(x3) = b, where a, b are arbitrary real constants. Therefore,
(4.12) becomes
f1 = (γx2 + δ) e2x4 + αx1 + β, (4.16)
f2 = (−γx1 + a) e−2x4 + (λ− α)x2 + b.
By equations (4.9) and (4.10), we have
f3 = λx3 + 2x4 + L(x1, x2),
where L is a smooth function.
By equation (4.8) and using the second equation in (4.16), we prove that
∂2L (x1, x2) = 4ε (−γx1 + a) . Then L (x1, x2) = 4ε (−γx1 + a)x2 +L(x1), where
L is a smooth function, and so f3 = 2x4 + λx3 + 4ε (−γx1 + a)x2 + L(x1).
Thus, deriving with respect to x1 and using equation (4.5) and the first equa-
tion in (4.16), we obtain L(x1) = −4εδx1 + η where η ∈ R. The calculations
above proved that the general solution of (4.2)–(4.11) is given by X = f1∂1 +
f2∂2 + f3∂3 + f4∂4, where
f1 = (γx2 + δ) e2x4 + αx1 + β, (4.17)
f2 = (−γx1 + a) e−2x4 + (λ− α)x2 + b,
f3 = 2x4 + λx3 + 4ε (−γx1 + a)x2 − 4εδx1 + η,
f4 = α− λ
2
for arbitrary real constants a, b, α, β, γ, δ, η. Therefore, the four-dimensional
Lorentzian generalized symmetric spaces admit appropriate vector fields for which
(1.1) holds. For any value of λ, we have the following result:
Theorem 4.1. A four-dimensional Lorentzian generalized symmetric space
of type C is an expanding, steady and shrinking Ricci soliton.
Now, let X = grad h be an arbitrary gradient vector field on the four-
dimensional Lorentzian generalized symmetric space (M, g) with potential func-
tion h. Then X is given by
grad h = εe2x4 (∂1h) ∂1 + εe−2x4 (∂2h) ∂2 + 2 (∂4h) ∂3 + 2 (∂3h) ∂4.
By a standard calculation, we prove, using (4.17), that the four-dimensional
generalized symmetric space of type C is a gradient Ricci soliton if and only if
∂1h = εe−2x4
[
(γx2 + δ) e2x4 + αx1 + β
]
, (4.18)
Ricci Solitons on Lorentzian Four-Dimensional Generalized Symmetric Spaces 139
∂2h = εe2x4
[
(−γx1 + a) e−2x4 + (λ− α)x2 + b
]
,
∂3h =
α
2
− λ
4
,
∂4h = x4 +
λ
2
x3 + 2ε (−γx1 + a)x2 − 2εδx1 +
η
2
.
Deriving the last equation in (4.18) with respect to x3 and using the third equation
in (4.18), we prove that λ = 0. Hence the Ricci soliton is necessarily steady. Next,
deriving the first equation in (4.18) with respect to x2 and the second equation
in (4.18) with respect to x1, we get γ = 0. Now, taking the derivative of the last
equation in (4.18) with respect to x1 and using the derivation of the first equation
in (4.18) with respect to x4, we obtain α = β = δ = 0. Then the derivative of the
last equation in (4.18) with respect to x2 gives (since ∂2h = ε
(
a+ be2x4
)
) a =
b = 0. Thus, h depends only on x4. Integrating the last equation in (4.18) with
respect to x4, we deduce that
h =
1
2
(η + x4)x4 + k, k, η ∈ R.
Thus, we have shown the following corollary.
Corollary 4.2. A four-dimensional Lorentzian generalized symmetric space
is a gradient Ricci soliton if and only if it is steady. The potential function h =
h (x4) is given by
h =
1
2
(η + x4)x4 + k, k, η ∈ R.
Following [6], the existence of solutions to the Ricci soliton equation for dif-
ferent values of λ appears to be related to the existence of Ricci and Yamabe
solitons on homogeneous spaces.
Thus, by Theorem 4.1, one can deduce that four-dimensional Lorentzian gen-
eralized symmetric spaces of type C are also Yamabe solitons.
References
[1] P. Baird and L. Danielo, Three-dimensional Ricci solitons which project to surfaces,
J. Reine Angew. Math. 608 (2007), 65–91.
[2] W. Batat, M. Brozos-Vazquez, E. Garćıa-Ŕıo, and S. Gavino-Fernández, Ricci soli-
tons on Lorentzian manifolds with large isometry groups, Bull. Lond. Math. Soc.
43 (2011), 1219–1227.
[3] W. Batat and K. Onda, Four-dimensional pseudo-Riemannian generalized symmet-
ric spaces which are algebraic Ricci solitons, Results Math. 64 (2013), 254–267.
[4] W. Batat and K. Onda, Ricci and Yamabe solitons on second-order symmetric, and
plane wave 4-dimensional Lorentzian manifolds, J. Geom. 105 (2014), 561–575.
[5] M. Brozos-Vázquez, G. Calvaruso, E. Garćıa-Ŕıo, and S. Gavino-Fernández, Three-
dimensional Lorentzian homogeneous Ricci solitons, Israel J. Math. 188 (2012),
385–403.
140 Amel Bouharis and Bachir Djebbar
[6] G. Calvaruso, Oscillator spacetimes are Ricci solitons, Nonlinear Anal. 140 (2016),
254–269.
[7] G. Calvaruso and B. De Leo, Curvature properties of four-dimensional generalized
symmetric spaces, J. Geom. 90 (2008), 30–46.
[8] G. Calvaruso and B. De Leo, Ricci solitons on Lorentzian Walker three-manifolds,
Acta Math. Hungar. 132 (2011), 269–293.
[9] G. Calvaruso and A. Fino, Four-dimensional pseudo-Riemannian homogeneous Ricci
solitons, Int. J. Geom. Methods Mod. Phys. 12 (2015), 1550056, 21 pp.
[10] G. Calvaruso and E. Rosado, Ricci solitons on low-dimensional generalized symmet-
ric spaces, J. Geom. Phys. 112 (2017), 106–117.
[11] L.F. Cerbo, Generic properties of homogeneous Ricci solitons, Adv. Geom. 14
(2014), 225–237.
[12] J. Cerny and O. Kowalski, Classification of generalized symmetric pseudo-Riemann-
ian spaces of dimension n ≤ 4, Tensor (N.S.) 38 (1982), 256–267.
[13] J. Lauret, Ricci soliton solvmanifolds, J. Reine Angew. Math. 650 (2011), 1–21.
[14] K. Onda, Lorentz Ricci solitons on 3-dimensional Lie groups, Geom. Dedicata 147
(2010), 313–322.
[15] T.L. Payne, The existence of soliton metrics for nilpotent Lie groups, Geom. Dedi-
cata 145 (2010), 71–88.
Received April 5, 2017, revised July 5, 2017.
Amel Bouharis,
Université d’Oran 1 Ahmed Ben Bella, BP 1524, ELM Naouer 31000, Oran, Algeria,
E-mail: bouharis@yahoo.fr
Bachir Djebbar,
Université des Sciences et de la Technologie d’Oran “Mohamed Boudiaf”, BP 1505, Bir
El Djir 31000, Oran, Algeria,
E-mail: badj2001@yahoo.fr
Солiтони Рiччi на лоренцевих чотиривимiрних
узагальнених симетричних просторах
Amel Bouharis and Bachir Djebbar
Доведено iснування нетривiальних (тобто, неейнштейнiвських) солi-
тонiв Рiччi на чотиривимiрних лоренцевих узагальнених симетричних
просторах. Бiльш того, показано, що тiльки стiйкi солiтони Рiччi мо-
жуть бути градiєнтними.
Ключовi слова: лоренцева метрика, солiтони Рiччi, градiєнтнi солi-
тони Рiччi, узагальненi симетричнi простори.
mailto:bouharis@yahoo.fr
mailto:badj2001@yahoo.fr
Introduction
Four-dimensional generalized symmetric spaces
Curvature of four-dimensional generalized symmetric space of type C
Ricci solitons on four-dimensional generalized symmetric space of type C
|