α-Sasakian 3-Metric as a Ricci Soliton
We prove that if the metric of a 3-dimensional α-Sasakian manifold is a Ricci soliton, then it is either of constant curvature or of constant scalar curvature. We also establish some properties of the potential vector field U of the Ricci soliton. Finally, we give an example of an α-Sasakian 3-metri...
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| author | Kundu, S. Кунду, С. |
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| description | We prove that if the metric of a 3-dimensional α-Sasakian manifold is a Ricci soliton, then it is either of constant curvature or of constant scalar curvature. We also establish some properties of the potential vector field U of the Ricci soliton. Finally, we give an example of an α-Sasakian 3-metric as a nontrivial Ricci soliton. |
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UDC 517.91
S. Kundu (Loreto College, India)
α-SASAKIAN 3-METRIC AS A RICCI SOLITON*
α-САСАКIЄВА 3-МЕТРИКА ЯК СОЛIТОН РIЧЧI
We prove that if the metric of a 3-dimensional α-Sasakian manifold is a Ricci soliton, then it is either of constant curvature
or of constant scalar curvature. We also establish some properties of the potential vector field U of the Ricci soliton. Finally,
we give an example of an α-Sasakian 3-metric as a nontrivial Ricci soliton.
Доведено, що якщо метрика тривимiрного α-сасакiєвого многовиду є солiтоном Рiччi, то вiн має або сталу кривину,
або сталу скалярну кривину. Встановлено деякi властивостi потенцiального векторного поля U солiтона Рiччi.
Наведено приклад α-сасакiєвої 3-метрики як нетривiального солiтона Рiччi.
1. Introduction. Over the last few years, Ricci solitons have been the place of concern for many
geometers and physicists. The whim of Ricci soliton structure was innovated by R. S. Hamilton (for
details we refer to [4]) and there he had excogitated it as a generalisation of an Einstein metric and
specified on a Riemannian manifold (Mn, g) together with a vector field U and a constant λ that
satisfies
£U g + 2S + 2λg = 0, (1.1)
where £ stands for the Lie-derivative operators along the complete vector field U and S as the Ricci
tensor of the manifold. In case, if λ is positive (respectively zero, respectively negative) then the
Ricci soliton is said to be expanding(respectively steady, shrinking). Actually Ricci soliton can be
considered as a fixed point of Hamilton’s Ricci flow:
∂
∂t
gij = −2Sij ; viewed as a dynamical system,
on the space of Riemannian metrics modulo diffeomorphisms and scalings. In particular if U = ∇f,
for some smooth scalar valued function f, then the soliton is said to be a gradient Ricci soliton. In
particular, if U is Killing or U = 0 the soliton is said to be a trivial Ricci soliton (for details refer to
[3, 6]). Also for several classes of these manifolds the existence of nontrivial Ricci solitons is proved.
Recently, Sharma and Ghosh [9], proved that if the metric of a 3-dimensional Sasakian manifold is
a Ricci soliton then it is homothetic to a standard Heisenberg group nil3. Since α-Sasakian is a
generalization of Sasakian manifold, we are interested to study 3-dimensional α-Sasakian manifold
when its metric is a Ricci soliton. We also deduce some properties of the potential vector field U of
the Ricci soliton together with an example of an α-Sasakian 3-metric as a nontrivial Ricci soliton.
2. Preliminaries. An odd-dimensional differentiable manifold (Mn, g) is said to admit an almost
contact metric structure (φ, ξ, η, g) consisting of a Reeb vector field ξ, (1,1)-tensor field φ and a
Riemannian metric g satisfying
φ2 = −I + η ⊗ ξ, η(ξ) = 1, (2.1)
g(φX, φY ) = g(X,Y )− η(X)η(Y ) ∀X,Y ∈ χ(M), (2.2)
where χ(M) represents the collection of all smooth vector fields on M.
* This research was supported by University Grants Commission (India).
c© S. KUNDU, 2013
850 ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
α-SASAKIAN 3-METRIC AS A RICCI SOLITON 851
Moreover, if the relation
dη(X,Y ) = g(φX, Y ),
holds for arbitrary smooth vector fields X and Y, then we call such a structure a contact metric
structure and the manifold with that structure is said to be contact metric manifold. As a consequence
of this, the following relations hold:
φξ = 0, η ◦ φ = 0, dη(ξ,X) = 0 ∀X ∈ χ(M). (2.3)
For details we refer to Blair [1].
An almost contact structure on M is said to be an α-Sasakian manifold, where α is a non-zero
constant, if
(∇Xφ)Y = α
(
g(X,Y )ξ − η(Y )X
)
∀X,Y ∈ χ(M) (2.4)
holds. As a consequence, it follows that:
∇Xξ = −αφX, (2.5)
(∇Xη)Y = −αg(φX, Y ) ∀X,Y ∈ χ(M). (2.6)
If α = 1, then the α-Sasakian structure reduces to Sasakian manifold, thus α-Sasakian structure may
be considered as a generalization of Sasakian one. In other words, Sasakian manifold is a particular
case of α-Sasakian manifold. Also in a 3-dimensional α-Sasakian manifold the following relations
are true:
R(X,Y )ξ = α2{η(Y )X − η(X)Y }, (2.7)
S(X, ξ) = 2α2η(X), (2.8)
Qξ = 2α2ξ ∀X,Y ∈ χ(M), (2.9)
where R is the Riemannian curvature tensor and Q is the Ricci operator associated with the (0, 2)
Ricci tensor S. For details we refer to [5].
Definition 2.1 [1]. In an almost contact Riemannian manifold, if an infinitesimal transformation
U satisfies
(£U η)(X) = ση(X), (2.10)
for a scalar function σ, then we call it an infinitesimal contact transformation. If σ vanishes identi-
cally, then it is called an infinitesimal strict transformation.
3. α-Sasakian 3-metric as a Ricci soliton. Before proceeding towards the main results we state
the following lemma.
Lemma 3.1. In an α-Sasakian 3-metric, the Ricci tensor S is given by
S =
(r
2
− α2
)
g +
(
3α2 − r
2
)
η ⊗ η. (3.1)
Proof. We recall that the Riemannian curvature tensor in a 3-dimensional Riemannian manifold
is given by
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
852 S. KUNDU
R(X,Y )Z = g(Y, Z)QX − g(X,Z)QY + S(Y, Z)X − S(X,Z)Y − r
2
{g(Y,Z)X − g(X,Z)Y },
(3.2)
where r is the Ricci scalar curvature and X,Y, Z ∈ χ(M).
Replacing Z with ξ in (3.2) and recalling (2.8),
η(Y )QX − η(X)QY +
(
α2 − r
2
)
{η(Y )X − η(X)Y } = 0.
Again, replacing Y with ξ and thereby using (2.9), we get the desired result.
Theorem 3.1. If the metric of a 3-dimensional α-Sasakian manifold is a nontrivial Ricci soli-
ton, then it is of constant scalar curvature −2α2 and the soliton is expanding.
Proof. Combining (1.1) and (3.1) yields,
(£U g)(X,Y ) =
(
2α2 − 2λ− r
)
g(X,Y ) +
(
r − 6α2
)
η(X)η(Y ). (3.3)
The identity
(∇X£U g)(Y,Z) = g((£U∇)(X,Y ), Z) + g((£U∇)(X,Z), Y ), (3.4)
can be deduced from the formula [10],
(£U∇Xg −∇X£U g −∇[U,X]
g)(Y,Z) = −g((£U∇)(X,Y ), Z)− g((£U∇)(X,Z), Y ). (3.5)
Differentiating covariantly (1.1) with respect to the vector field Z, we obtain
(∇Z£U g)(X,Y ) + 2(∇ZS)(X,Y ) = 0. (3.6)
Again, differentiating (3.1) covariantly with respect to Z and using (2.6), we have
(∇ZS)(X,Y ) =
1
2
dr(Z){g(X,Y )− η(X)η(Y )}−
−α
(
3α2 − r
2
)
{g(φZ,X)η(Y ) + g(φZ, Y )η(X)}. (3.7)
Combining (3.6) with (3.7), one obtains
(∇Z£U g)(X,Y ) + dr(Z){g(X,Y )− η(X)η(Y )}−
−2α
(
3α2 − r
2
)
{g(φZ,X)η(Y ) + g(φZ, Y )η(X)} = 0. (3.8)
Using (3.4) in (3.8) one obtains
g((£U∇)(Z,X), Y ) + g((£U∇)(Z, Y ), X) + dr(Z){g(X,Y )− η(X)η(Y )}−
−2α
(
3α2 − r
2
)
{g(φZ,X)η(Y ) + g(φZ, Y )η(X)} = 0. (3.9)
Permuting X, Y, Z and then by combinatorial combination we find,
2(£U∇)(Y, Z)+
{
dr(Y )(Z − η(Z)ξ) + dr(Z)(Y − η(Y )ξ)−
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
α-SASAKIAN 3-METRIC AS A RICCI SOLITON 853
−Dr(g(Y,Z)− η(Y )η(Z))
}
−
−2α
(
3α2 − r
2
)
{η(Z)φY + η(Y )φZ} = 0, (3.10)
for all vector field X and D is the gradient operator of g. Now from [10], we have the following
identity:
(£UR)(X,Y )Z = (∇X£U∇)(Y, Z)− (∇Y£U∇)(X,Z). (3.11)
Now the use of (2.5) and (3.10) in the identity (3.11), we obtain on taking Z = ξ
(£UR)(X,Y )ξ =
α
2
[
dr(Y )φX − dr(X)φY
]
+ 2α2(6α2 − r){η(X)Y − η(Y )X}+
+αg(φX, Y )Dr +
α
2
[
dr(φY )(X − η(X)ξ)− dr(φX)(Y − η(Y )ξ)
]
. (3.12)
Taking the Lie-derivative of (2.7) along the direction of U and using (3.3), one obtains
(£UR)(X,Y )ξ = −R(X,Y )£Uξ + 2α2(λ+ 2α2){η(X)Y − η(Y )X}+
+α2{g(Y,£Uξ)X − g(X,£Uξ)Y }. (3.13)
Equating (3.12) and (3.13), it follows that
2α2{g(Y,£Uξ)X − g(X,£Uξ)Y } = 2R(X,Y )£Uξ − 4α2(4α2 − λ− r){η(X)Y − η(Y )X}+
+2αg(φX, Y )Dr + α
[
dr(Y )φX − dr(X)φY
]
+
+α
[
dr(φY )(X − η(X)ξ)− dr(φX)(Y − η(Y )ξ)
]
.
Contracting the above equation over Y and thereby using (3.1), we find
2αdr(φX) = (r − 6α2)[g(£Uξ,X)− η(£Uξ)η(X)] + 8α2(r − 4α2 + λ)η(X). (3.14)
Substituting X = ξ, yields
r = 4α2 − λ, since α 6= 0. (3.15)
Now, the integrability condition of the Ricci soliton (for details refer to [2, 8]) is given by
£U r = −div.Dr + 2λr + 2|S|2.
By using (3.15) and (3.1), one obtains from above
r2 − 4α2r − 12α4 = 0, which implies r = 6α2 or r = −2α2.
Theorem 3.1 is proved.
For r = 6α2 we see that the manifold is Einstein and being of dimension 3 it becomes a space
of constant curvature α2. Hence we have the following corollary.
Corollary 3.1. If the metric of an α-Sasakian manifold is Ricci soliton then it is either a space
of constant curvature α2 or of constant scalar curvature.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
854 S. KUNDU
We now deduce some properties of the potential vector field U related to the underlying contact
structure of the α-Sasakian 3-metric as a Ricci soliton: Putting r = −2α2 and λ = 6α2 in (3.1) and
using the result in (1.1), one obtains
£U g = −8α2
(
g + η ⊗ η),
which implies U is homothetic on the distribution D = ker(η). Replacing X,Y with ξ, one obtains
from (3.10)
(£U∇)(ξ, ξ) = 0, since ξ is killing. (3.16)
Substituting X = Y = ξ in (3.5) provides,
(£U∇)(X,Y ) = ∇X∇Y U −∇∇
X
Y U +R(U,X)Y. (3.17)
Thereby using (3.16) in (3.17) together with X = Y = ξ, we have
∇ξ∇ξU +R(U, ξ)ξ = 0. (3.18)
Hence from (3.18), it is quite evident that U is a Jacobi along geodesics of ξ. Again, using (3.7) in
(3.14), we find
(r − 6α2){£U ξ − η(£U ξ)ξ} = 0.
If g is a nontrivial Ricci soliton, the above equation yields
£U ξ = σξ, where σ = η(£U ξ).
Then the use of (1.1), (3.1) and the above equation, one obtains
£U η = (σ − 16α2)η,
which proves that U is an infinitesimal contact transformation. Setting X = Y = ξ in (1.1) and in
view of (2.8), £U ξ = σξ, we get
σ = 8α2,
from which we see that
σ − 16α2 = −8α2(6= 0).
This implies the infinitesimal contact transformation is nonstrict. Summing up all these results we
can state as follows:
Theorem 3.2. If an α-Sasakian 3-metric admits a nontrivial Ricci soliton together with the
potential vector field U, then the following statements hold:
(1) U is homothetic on the distribution D = ker(η).
(2) U is a Jacobi vector field along geodesics of ξ.
(3) U is an infinitesimal contact transformation.
Thus the φ-sectional curvature of α-Sasakian manifold (of 3-dimension) admitting nontrivial
Ricci soliton is given by α2. Hence, we can state as follows:
Theorem 3.3. For a 3-dimensional α-Sasakian manifold the φ-sectional curvature (sectional
curvature with respect to a plane orthogonal to ξ) is constant and equals to α2.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
α-SASAKIAN 3-METRIC AS A RICCI SOLITON 855
4. Example of an α-Sasakian 3-metric as a Ricci soliton. Let us consider the 3-dimensional
Riemannian manifold M = R3 with a rectangular cartesian coordinate system (xi).
Let us choose the vector fields
{
E1 , E2 , E3
}
as
E1 =
∂
∂x1
, E2 = −2α ∂
∂x2
, E3 = x2
∂
∂x1
+ x3
∂
∂x3
, α being a nonzero constant.
Thus,
{
E1 , E2 , E3
}
forms a basis of χ(M) = χ(R3).
Let g be the Riemannian metric on χ(R3) defined by
g(E1 , E1) = g(E2 , E2) = g(E3 , E3) = 1,
g(E1 , E2) = g(E1 , E3) = g(E2 , E3) = 0.
Let ξ = E1 be the vector field associated with the 1-form η. The (1,1)-tensor field φ be defined by,
φ(E1) = 0, φ(E2) = −E3 , φ(E3) = E2 .
Since,
{
E1 , E2 , E3
}
is a basis, any vector fields X and Y in M can be uniquely expressed as
X = X1E1 +X2E2 +X3E3 and Y = Y 1E1 + Y 2E2 + Y 3E3 ,
where Xi, Y i, i = 1, 2, 3, are smooth functions over M.
Now using the linearity of φ and g, and taking ξ = E1 we have
η(ξ) = 1, φ2X = −X + η(X)ξ, g(φX, φY ) = g(X,Y )− η(X)η(Y )
for any vector fields X and Y in M. Thus (φ, ξ, η, g) defines an almost contact metric structure
on M.
Let ∇ be the Levi – Civita connection with respect to the Riemannian metric g. Then we obtain
[E2 , E3 ] = −2αe1, [E1 , E2 ] = 0, [E1 , E3 ] = 0.
By using Koszul’s formulae (see [7]), we have
∇E1
E3 = −αE2 , ∇E1
E2 = αE3 , ∇E1
E1 = 0,
∇E2
E3 = −αE1 , ∇E2
E1 = αE3 , ∇E2
E2 = 0,
∇E3
E1 = −αE2 , ∇E3
E2 = αE1 , ∇E3
E3 = 0.
Also, the Riemannian curvature tensor R is given by,
R(X,Y )Z = ∇X∇Y Z −∇Y∇XZ −∇[X,Y ]
Z.
Then
R(E1 , E2)E2 = α2E1 , R(E1 , E3)E3 = α2E1 , R(E2 , E1)E1 = α2E2 ,
R(E2 , E3)E3 = −3α2E2 , R(E3 , E1)E1 = α2E3 , R(E3 , E2)E2 = −3α2E3 ,
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
856 S. KUNDU
R(E1 , E2)E3 = 0, R(E2 , E3)E1 = 0, R(E3 , E1)E2 = 0.
Then, the Ricci tensor S is given by
S(E1 , E1) = 2α2, S(E2 , E2) = −2α2, S(E3 , E3) = −2α2,
S(E1 , E2) = 0, S(E1 , E3) = 0, S(E2 , E3) = 0.
Thus the scalar curvature r = −2α2 is constant. The conditions (2.4) to (2.9) hold for any smooth
vector fields X and Y in M. Taking the potential vector field
U = f1E1 + f2E2 + f3E3 ,
where f1, f2 and f3 are smooth functions on M, it can be easily shown that it satisfies the soliton
equation and the soliton is expanding in nature.
The author thankful to Dr. A. Ghosh, Assistant Professor, Krishnanagar Govt. College (Nadia,
India) for his constant and deliberate assistance during the preparation of the paper. The author is
also grateful to Prof. M. Tarafdar (University of Calcutta, India) for her kind help.
1. Blair D. E. Riemannian geometry of contact and symplectic manifolds. – Boston: Birkhäuser, 2002.
2. Chow B., Knopf D. The Ricci flow: An introduction // Math. Surv. and Monogr. – Providence, RI: Amer. Math. Soc.,
2004.
3. Guilfoyle B. S. Einstein metrics adapted to a contact structure on 3-manifolds. – Preprint, http://arXiv.org/abs/
math/0012027, 2000.
4. Hamilton R. S. The Ricci flow on surfaces // Math. and General Relativity (Santa Cruz, CA, 1986). Contemp. Math. –
1988. – 71. – P. 237 – 262.
5. Janssens D., Vanhecke L. Almost contact structures and curvature tensors // Kodai Math. J. – 1981. – 4. – P. 1 – 27.
6. Perelman G. The entropy formula for the Ricci flow and its geometric applications. – Preprint, http://arXiv.org/abs/
math.DG/02111159.
7. Schouten J. A. Ricci calculus. – 2 nd Ed. – Berlin: Springer-Verlag, 1954. – 332 p.
8. Sharma R. Certain results on K-contact and (κ, µ)-contact manifolds // J. Geom. – 2008. – 89. – P. 138 – 147.
9. Sharma R., Ghosh A. Sasakian 3-manifold as a Ricci soliton represents the Heisenberg group // Int. J. Geom. Methods
Modern Phys. – 2011. – 8, № 1. – P. 149 – 154.
10. Yano K. Integral formulas in Riemannian geometry. – New York: Marcel Dekker, 1970.
Received 07.01.12
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
|
| id | umjimathkievua-article-2470 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:24:04Z |
| publishDate | 2013 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/cb/96297bfa8d9cd79218f96a3631f8f8cb.pdf |
| spelling | umjimathkievua-article-24702020-03-18T19:16:10Z α-Sasakian 3-Metric as a Ricci Soliton α-Сасакієва 3-метріка як солітон Річчі Kundu, S. Кунду, С. We prove that if the metric of a 3-dimensional α-Sasakian manifold is a Ricci soliton, then it is either of constant curvature or of constant scalar curvature. We also establish some properties of the potential vector field U of the Ricci soliton. Finally, we give an example of an α-Sasakian 3-metric as a nontrivial Ricci soliton. Доведено, що якщо метрика тривимірного α-сасакієвого многовиду є солітоном Річчі, то він має або сталу кривину, або сталу скалярну кривину. Встановлено деякі властивості потенціального векторного поля U солітона Річчі. Наведено приклад а-сасакієвої 3-метрики як нетривіального солітона Річчі. Institute of Mathematics, NAS of Ukraine 2013-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2470 Ukrains’kyi Matematychnyi Zhurnal; Vol. 65 No. 6 (2013); 850–856 Український математичний журнал; Том 65 № 6 (2013); 850–856 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2470/1706 https://umj.imath.kiev.ua/index.php/umj/article/view/2470/1707 Copyright (c) 2013 Kundu S. |
| spellingShingle | Kundu, S. Кунду, С. α-Sasakian 3-Metric as a Ricci Soliton |
| title | α-Sasakian 3-Metric as a Ricci Soliton |
| title_alt | α-Сасакієва 3-метріка як солітон Річчі |
| title_full | α-Sasakian 3-Metric as a Ricci Soliton |
| title_fullStr | α-Sasakian 3-Metric as a Ricci Soliton |
| title_full_unstemmed | α-Sasakian 3-Metric as a Ricci Soliton |
| title_short | α-Sasakian 3-Metric as a Ricci Soliton |
| title_sort | α-sasakian 3-metric as a ricci soliton |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2470 |
| work_keys_str_mv | AT kundus asasakian3metricasariccisoliton AT kundus asasakian3metricasariccisoliton AT kundus asasakíêva3metríkaâksolítonríččí AT kundus asasakíêva3metríkaâksolítonríččí |