Some characterizations of three-dimensional trans-Sasakian manifolds admitting η- Ricci solitons and trans-Sasakian manifolds as Kagan subprojective space
UDC 514.7 The object of the present paper is to study three-dimensional trans-Sasakian manifolds admitting $\eta$-Ricci soliton. Actually, we study such manifolds whose Ricci tensor satisfy some special conditions like cyclic parallelity, Ricci semisymmetry, $\phi$-Ricci semisymmetry, after reviewin...
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| author | Sarkar, A. Sil, A. Paul, A. K. Саркар, А. Сіл, А. Пауль, А. К. |
| author_facet | Sarkar, A. Sil, A. Paul, A. K. Саркар, А. Сіл, А. Пауль, А. К. |
| author_sort | Sarkar, A. |
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| description | UDC 514.7
The object of the present paper is to study three-dimensional trans-Sasakian manifolds admitting $\eta$-Ricci soliton. Actually, we study such manifolds whose Ricci tensor satisfy some special conditions like cyclic parallelity, Ricci semisymmetry, $\phi$-Ricci semisymmetry, after reviewing the properties of second order parallel tensors on such manifolds. We determine the form of Riemann curvature tensor of trans-Sasakian manifolds of dimension greater than three as Kagan subprojective spaces. We also give some classification results of trans-Sasakian manifolds of dimension greater than three as Kagan subprojective spaces. |
| doi_str_mv | 10.37863/umzh.v72i3.6047 |
| first_indexed | 2026-03-24T03:25:50Z |
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UDC 514.7
A. Sarkar, A. Sil, A. K. Paul (Univ. Kalyani, West Bengal, India)
SOME CHARACTERIZATIONS OF THREE-DIMENSIONAL
TRANS-SASAKIAN MANIFOLDS ADMITTING \bfiteta -RICCI SOLITONS
AND TRANS-SASAKIAN MANIFOLDS AS KAGAN SUBPROJECTIVE SPACES
ДЕЯКI ХАРАКТЕРИСТИКИ ТРИВИМIРНИХ ТРАНС-МНОГОВИДIВ
САСАКЯНА, ЩО ДОПУСКАЮТЬ \bfiteta -СОЛIТОНИ РIЧЧI,
ТА ТРАНС-МНОГОВИДИ САСАКЯНА ЯК СУБПРОЕКТИВНI
ПРОСТОРИ КАГАНА
The object of the present paper is to study three-dimensional trans-Sasakian manifolds admitting \eta -Ricci soliton. Actually,
we study such manifolds whose Ricci tensor satisfy some special conditions like cyclic parallelity, Ricci semisymmetry,
\phi -Ricci semisymmetry, after reviewing the properties of second order parallel tensors on such manifolds. We determine
the form of Riemann curvature tensor of trans-Sasakian manifolds of dimension greater than three as Kagan subprojective
spaces. We also give some classification results of trans-Sasakian manifolds of dimension greater than three as Kagan
subprojective spaces.
Вивчаються тривимiрнi транс-многовиди Сасакяна, якi допускають \eta -солiтони Рiччi. Власне, пiсля огляду власти-
востей паралельних тензорiв другого порядку на таких многовидах ми вивчаємо многовиди, тензор Рiччi яких
задовольняє деякi спецiальнi умови, такi як циклiчна паралельнiсть, напiвсиметрiя Рiччi, \phi -напiвсиметрiя Рiччi.
Визначено форму тензора кривини Рiмана для транс-многовидiв Сасакяна, розмiрнiсть яких бiльша нiж 3, як суб-
проективних просторiв Кагана. Також наведено деякi класифiкацiйнi результати для транс-многовидiв Сасакяна,
розмiрнiсть яких бiльша нiж 3, як субпроективних просторiв Кагана.
1. Introduction. In [18], R. S. Hamilton introduced the revolutionary concept of Ricci flow on
surfaces. The concepts of Ricci flow in physics was introduced by Friedan [14] almost around in the
same time but with different motivations. Now a days such geometric flows have become popular,
largely, because of Perelman’s [22] work which lead to the proof of well known Poincaré conjecture.
A Ricci soliton is a special solution of Ricci flow. This is considered as a natural generalization of
Einstein metric and is defined on a Riemannian manifold (M, g) by
(\$V g)(X,Y ) + 2S(X,Y ) + 2\lambda g(X,Y ) = 0, (1.1)
where \$V denotes the Lie derivative operator along a complete vector field V. V is known as
potential vector field. \lambda is a constant, called soliton constant. S is the Ricci tensor and g is the
metric. X, Y are the arbitrary vector fields on M. The Ricci soliton is said to be shrinking, steady
or expanding as \lambda is negative, zero or positive, respectively [7]. The study of Ricci solitons on
contact manifolds was initiated by R. Sharma [24]. Later several authors have studied Ricci soliton
on almost contact manifolds. For example, we may refer the papers [12, 15, 16, 27]. In [5], it has
been proved that a real hypersurface in a non-flat complex space form does not admit a Ricci soliton
with \xi as soliton vector field and then the author adopted the notion of \eta -Ricci soliton. The \eta -Ricci
soliton (g, \xi , \lambda , \mu ) on a Riemannian manifold is defined by
\$\xi g + 2S + 2\lambda - 2\mu n\otimes n = 0,
where \xi is the Reeb vector field, \mu is a constant and the other objects are as described in equa-
tion (1.1). For details see also [2, 6, 8, 23]. Since 1923 [13], second order parallel tensors are studied
c\bigcirc A. SARKAR, A. SIL, A. K. PAUL, 2020
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3 427
428 A. SARKAR, A. SIL, A. K. PAUL
by several authors [20]. Second order parallel tensors were studied in the frame work of contact
manifolds by R. Sharma [25].
If geodesics are represented by n - 2 homogeneous linear equations for a suitable coordinate
system in an affine space An, then An is called a subprojective space by B. Kagan [19]. T. Adati [1]
have studied such spaces intensively and proved that Kagan subprojective spaces are conformally
flat [1, p. 160]. In this paper, we would like to find form of Riemann curvature tensors of trans-
Sasakian manifolds of dimension greater than three as Kagan subprojective spaces. We give some
classification results of such spaces.
2. Preliminaries. Let M be a differentiable manifold of dimension 2n + 1. M is said to have
almost contact structure [3] if there is a (1,1) tensor field \phi , a vector field \xi and a 1-form \eta on M
such that
\phi 2X = - X + \eta (X)\xi , \eta (\xi ) = 1,
where X \in \chi (M), the set of all differentiable vector fields on M. On such manifolds it can be also
proved that
\phi \xi = 0, \eta (\phi X) = 0, g(X,\phi Y ) = - g(\phi X, Y ), g(X, \xi ) = \eta (X)
for X,Y \in \chi (M). An almost contact structure is called almost contact metric structure if there exists
a Riemannian metric g on M satisfying
g(\phi X, \phi Y ) = g(X,Y ) - \eta (X)\eta (Y ).
The (0,2) tensor field \Phi defined by \Phi (X,Y ) = g(X,\phi Y ) is known as fundamental 2-form of the
manifold. If \Phi is closed, an almost contact metric structure reduces to contact metric structure [3]. An
almost contact metric structure (\phi , \xi , \eta , g) on a differentiable manifold M is called a trans-Sasakian
structure [21] if (M \times R, J,G) belongs to the class W4 in the Gray – Hervella classification [17].
Here J is the almost complex structure on M\times R defined by J
\biggl(
X, f
d
dt
\biggr)
=
\biggl(
\phi X - f\xi , \eta (X)
d
dt
\biggr)
,
for all vector fields X on M and smooth functions f on M \times R and G is the product metric on
M \times R. This fact may be formulated by the following equation [4]:
(\nabla X\phi )Y = \alpha
\bigl(
g(X,Y )\xi - \eta (Y )X
\bigr)
+ \beta
\bigl(
g(\phi X, Y )\xi - \eta (Y )\phi X
\bigr)
,
where \alpha and \beta are smooth functions on M. The above formula implies
\nabla X\xi = \beta (X - \eta (X)\xi ) - \alpha \phi X, (2.1)
(\nabla X\eta )Y = - \alpha g(\phi X, Y ) + \beta g(\phi X, \phi Y ). (2.2)
The Ricci tensor [11] of a three-dimensional trans-Sasakian manifold is given by
S(X,Y ) =
\Bigl( r
2
+ \xi \beta -
\bigl(
\alpha 2 - \beta 2
\bigr) \Bigr)
g(X,Y ) -
\Bigl( r
2
+ \xi \beta - 3
\bigl(
\alpha 2 - \beta 2
\bigr) \Bigr)
\eta (X)\eta (Y ) -
-
\bigl(
Y \beta + (\phi Y )\alpha
\bigr)
\eta (X) -
\bigl(
X\beta + (\phi X)\alpha
\bigr)
\eta (Y ), (2.3)
where r is the scalar curvature of the manifold. Again from [9], we known that the Riemann
curvature of a three-dimensional trans-Sasakian manifold is given by
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
SOME CHARACTERIZATIONS OF THREE-DIMENSIONAL TRANS-SASAKIAN MANIFOLDS . . . 429
R(X,Y )Z =
\Bigl( r
2
+ 2\xi \beta - 2
\bigl(
\alpha 2 - \beta 2
\bigr) \Bigr) \bigl(
g(Y,Z)X - g(X,Z)Y
\bigr)
-
- g(Y, Z)
\Bigl[ \Bigl( r
2
+ \xi \beta - 3
\bigl(
\alpha 2 - \beta 2
\bigr) \Bigr)
\eta (X)\xi - \eta (X)(\phi \mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}\alpha - \mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}\beta )+
+ (X\beta + (\phi X)\alpha )\xi
\Bigr]
+ g(X,Z)
\Bigl[ \Bigl( r
2
+ \xi \beta - 3
\bigl(
\alpha 2 - \beta 2
\bigr) \Bigr)
\eta (Y )\xi - \eta (Y )(\phi \mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}\alpha - \mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}\beta )+
+ (Y \beta + (\phi Y )\alpha )\xi
\Bigr]
-
\Bigl[ \bigl(
Z\beta + (\phi Z)\alpha
\bigr)
\eta (Y ) + (Y \beta + (\phi Y )\alpha )\eta (Z) +
+
\Bigl( r
2
+ \xi \beta - 3
\bigl(
\alpha 2 - \beta 2
\bigr) \Bigr)
\eta (Y )\eta (Z)
\Bigr]
X +
\Bigl[
(Z\beta + (\phi Z)\alpha )\eta (X) +
+
\bigl(
X\beta + (\phi X)\alpha
\bigr)
\eta (Z) +
\Bigl( r
2
+ \xi \beta - 3
\bigl(
\alpha 2 - \beta 2
\bigr) \Bigr)
\eta (X)\eta (Z)
\Bigr]
Y. (2.4)
Again
2\alpha \beta + \xi \alpha = 0.
3. Existence criteria of \bfiteta -Ricci soliton on three-dimensional trans-Sasakian manifolds.
Theorem 3.1. A three-dimensional trans-Sasakian manifold with constant \xi -sectional curvature
admits \eta -Ricci soliton if and only if \$\xi g + 2S + 2\mu n\otimes n is parallel.
Theorem 3.2. A three-dimensional proper trans-Sasakian manifold with cyclic parallel Ricci
tensor does not admit \eta -Ricci soliton. It reduces to Einstein manifold.
Proof. Let (M, g, \xi , \lambda , \mu ) be a three-dimensional trans-Sasakian \eta -Ricci soliton. Then we have
(\$\xi g)(X,Y ) + 2S(X,Y ) + 2\lambda g(X,Y ) + 2\mu \eta (X)\eta (Y ) = 0.
Expressing the Lie derivative in terms of covariant derivative and using (2.1), we obtain
S(X,Y ) = - 2\lambda + \beta
2
g(X,Y ) +
2\mu - \beta
2
\eta (X)\eta (Y ). (3.1)
In [12], it was proved that if T is a symmetric parallel tensor on a trans-Sasakian manifold of
dimension three with non-zero \xi -sectional curvature, then
T (X,Y ) = T (\xi , \xi )g(X,Y ).
We see that (\$\xi g)(X,Y ) + 2S(X,Y ) + 2\mu \eta (X)\eta (Y ) is a symmetric (0, 2) tensor. Hence, by using
its property, we obtain Theorem 3.1. Consider the manifold has cyclic parallel Ricci tensor [9]. By
virtue of (3.1) and (2.3), after simplification we have S = - 2(\lambda + \beta )
2
g.
Theorem 3.2 is proved.
4. Ricci semisymmetric three-dimensional trans-Sasakian manifold admitting \bfiteta -Ricci soli-
ton.
Theorem 4.1. If a three-dimensional trans-Sasakian manifold of type (\alpha , \beta ), where \alpha \not = \pm \beta
and \beta = \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}, admitting \eta -Ricci soliton is Ricci semisymmetric, then \mu =
\beta
2
.
Corollary 4.1. A Ricci semisymmetric three-dimensional \alpha -Sasakian manifold does not admit
proper \eta -Ricci soliton.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
430 A. SARKAR, A. SIL, A. K. PAUL
Proof. It is well known that a Riemannian manifold is called Ricci semisymmetric if
(R(X,Y )S)(U, V ) = 0.
The above condition implies
S
\bigl(
R(X,Y )U, V
\bigr)
+ S
\bigl(
U,R(X,Y )V =) = 0.
Putting Y = V = \xi , we have
S
\bigl(
R(X, \xi )U, \xi
\bigr)
+ S
\bigl(
U,R(X, \xi )\xi
\bigr)
= 0.
Using (2.4) in the above equation, after straight forward computation we have \mu =
\beta
2
, provided
\alpha \not = \pm \beta . So, we have Theorem 4.1 for \alpha -Sasakian case \alpha = \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t} and \beta = 0. Thus we have
Corollary 4.1.
5. \bfitphi -Ricci symmetric three-dimensional trans-Sasakian manifold admitting \bfiteta -Ricci soliton.
The notion of \phi -Ricci symmetry was given by the first author in [10]. An almost contact manifold
is called \phi -Ricci symmetric if the Ricci operator Q satisfies
\phi 2(\nabla WQ)X = 0.
The manifold is called locally \phi -Ricci symmetric if X and W are orthogonal to \xi .
Theorem 5.1. A three-dimensional trans-Sasakian manifold admitting \eta -Ricci soliton is \phi -Ricci
symmetric if and only if \mu =
\beta
2
.
Proof. By virtue of (3.1) we obtain
QX = - 2\lambda + \beta
2
X +
2\mu - \beta
2
\eta (X)\xi .
Hence,
\phi 2(\nabla WQ)X = \alpha
\biggl(
\beta
2
- \mu
\biggr)
\eta (X)\phi 2(\phi W ) + \beta
\biggl(
\mu - \beta
2
\biggr)
\phi 2W.
The above equation proves Theorem 5.1.
6. Form of Riemann curvature tensors of trans-Sasakian manifolds of dimension greater
than three as Kagan subprojective spaces. Riemann curvature tensors for three-dimensional trans-
Sasakian manifolds have been deduced in the paper [11]. In this section, we like to deduce the form
of Riemann curvature tensors of trans-Sasakian manifolds of dimension greater than three as Kagan
subprojective spaces [19].
Theorem 6.1. The form of Riemann curvature tensor of a trans-Sasakian manifold of dimension
greater than three as Kagan subprojective space is given by
R(X,Y )Z =
\biggl(
r
2m(2m - 1)
- 2
2m - 1
\Bigl( r
2m
(\alpha 2 - \beta 2)
\Bigr) \biggr) \bigl(
g(Y, Z)X - g(X,Z)Y
\bigr)
+
+
1
2m - 1
\Bigl( r
2m
+ (2m+ 1)(\alpha 2 - \beta 2)
\Bigr) \bigl(
\eta (Y )\eta (Z)X - \eta (X)\eta (Z)Y
\bigr)
.
Theorem 6.2. A trans-Sasakian manifold of dimension greater than three as Kagan subprojec-
tive space is Einstein manifold. Hence, it does not admit Ricci soliton and \eta -Ricci soliton.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
SOME CHARACTERIZATIONS OF THREE-DIMENSIONAL TRANS-SASAKIAN MANIFOLDS . . . 431
Proof. T. Adati [1] have proved that Kagan subprojective spaces are conformally flat. We know
that the Weyl conformal curvature tensor C of a (2m + 1)-dimensional (m > 1) manifold is given
by
C(X,Y )Z = R(X,Y )Z - 1
2m - 1
\Bigl[
g(Y, Z)QX - g(X,Z)QY + S(Y,Z)X - S(X,Z)Y
\Bigr]
+
+
r
2m(2m - 1)
\bigl[
g(Y,Z)X - g(X,Z)Y
\bigr]
,
where S is Ricci tensor and Q is Ricci operator. Since Kagan subprojective spaces are conformally
flat, we get
R(X,Y )Z =
1
2m - 1
\Bigl[
g(Y,Z)QX - g(X,Z)QY + S(Y,Z)X - S(X,Z)Y
\Bigr]
-
- r
2m(2m - 1)
\Bigl[
g(Y,Z)X - g(X,Z)Y
\Bigr]
.
From the above equation, we find S and Q and obtain the results.
7. Some classification results of a trans-Sasakian manifold of dimension greater than three
as Kagan subprojective space.
Definition 7.1. A Riemannian manifold is called locally \phi -symmetric [26] if
\phi 2(\nabla WR)(X,Y )Z = 0,
for X,Y, Z orthogonal to \xi .
Theorem 7.1. A trans-Sasakian manifold of dimension greater than three as Kagan subprojec-
tive space is locally \phi -symmetric if and only if
dr
2m
= 4(\alpha d\alpha - \beta d\beta ).
Theorem 7.2. If the structure functions \alpha and \beta of a trans-Sasakian manifold of dimension
greater than three as Kagan subprojective space are same, then the manifold is locally \phi -symmetric
if and only if the scalar curvature of the manifold is constant.
Theorem 7.3. If the structure functions \alpha and \beta of a trans-Sasakian manifold of dimension
greater than three as Kagan subprojective space are constants, then the manifold is locally \phi -
symmetric.
Proof. By using Theorem 6.1, we get
\phi 2(\nabla WR)(X,Y )Z =
=
\biggl(
dr
2m(2m - 1)
- 2
2m - 1
\biggl(
dr
2m
+ 2
\bigl(
\alpha d\alpha - \beta d\beta )
\biggr) \biggr) \Bigl(
g(X,Z)Y - g(Y, Z)X
\Bigr)
, (7.1)
for X, Y, Z orthogonal to \xi . Let us consider the following cases:
Case 1. Consider \alpha and \beta as arbitrary functions:
Subcase 1.1: Let \alpha \not = \beta . In that case we get Theorem 7.1 from (7.1).
Subcase 1.2: Let \alpha and \beta are equal functions. In that case \alpha d\alpha - \beta d\beta = 0. So, we obtain
Theorem 7.2 from (7.1).
Case 2. Let \alpha and \beta are constants. In that case, deducing S from R, we have Theorem 7.3.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
432 A. SARKAR, A. SIL, A. K. PAUL
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Received 08.06.17
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 3
|
| id | umjimathkievua-article-6047 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:25:50Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/52/229e440cd8f692dcacbd7d63f0ee9752.pdf |
| spelling | umjimathkievua-article-60472020-05-26T09:20:33Z Some characterizations of three-dimensional trans-Sasakian manifolds admitting η- Ricci solitons and trans-Sasakian manifolds as Kagan subprojective space Деякi характеристики тривимiрних транс-многовидiв Сасакяна, що допускають η -солiтони Рiччi, та транс-многовиди Сасакяна як субпроективнi простори Кагана Деякi характеристики тривимiрних транс-многовидiв Сасакяна, що допускають η -солiтони Рiччi, та транс-многовиди Сасакяна як субпроективнi простори Кагана Sarkar, A. Sil, A. Paul, A. K. Саркар, А. Сіл, А. Пауль, А. К. UDC 514.7 The object of the present paper is to study three-dimensional trans-Sasakian manifolds admitting $\eta$-Ricci soliton. Actually, we study such manifolds whose Ricci tensor satisfy some special conditions like cyclic parallelity, Ricci semisymmetry, $\phi$-Ricci semisymmetry, after reviewing the properties of second order parallel tensors on such manifolds. We determine the form of Riemann curvature tensor of trans-Sasakian manifolds of dimension greater than three as Kagan subprojective spaces. We also give some classification results of trans-Sasakian manifolds of dimension greater than three as Kagan subprojective spaces. УДК 514.7 Вивчаються тривимірні транс-многовиди Сасакяна, які допускають $\eta$-солітони Річчі.  Власне, після огляду властивостей паралельних тензорів другого порядку на таких многовидах ми вивчаємо многовиди, тензор Річчі яких задовольняє деякі спеціальні умови, такі як циклічна паралельність, напівсиметрія Річчі, $\phi$-напівсиметрія Річчі.  Визначено форму тензора кривини Рімана для транс-многовидів Сасакяна, розмірність яких більша ніж 3, як субпроективних просторів Кагана.  Також наведено деякі класифікаційні результати для транс-многовидів Сасакяна, розмірність яких більша ніж 3, як субпроективних просторів Кагана.  Institute of Mathematics, NAS of Ukraine 2020-03-28 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6047 10.37863/umzh.v72i3.6047 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 3 (2020); 427-432 Український математичний журнал; Том 72 № 3 (2020); 427-432 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6047/8685 |
| spellingShingle | Sarkar, A. Sil, A. Paul, A. K. Саркар, А. Сіл, А. Пауль, А. К. Some characterizations of three-dimensional trans-Sasakian manifolds admitting η- Ricci solitons and trans-Sasakian manifolds as Kagan subprojective space |
| title | Some characterizations of three-dimensional trans-Sasakian manifolds admitting η- Ricci solitons and trans-Sasakian manifolds as Kagan subprojective space |
| title_alt | Деякi характеристики тривимiрних транс-многовидiв Сасакяна, що допускають η -солiтони Рiччi, та транс-многовиди Сасакяна як субпроективнi простори Кагана Деякi характеристики тривимiрних транс-многовидiв Сасакяна, що допускають η -солiтони Рiччi, та транс-многовиди Сасакяна як субпроективнi простори Кагана |
| title_full | Some characterizations of three-dimensional trans-Sasakian manifolds admitting η- Ricci solitons and trans-Sasakian manifolds as Kagan subprojective space |
| title_fullStr | Some characterizations of three-dimensional trans-Sasakian manifolds admitting η- Ricci solitons and trans-Sasakian manifolds as Kagan subprojective space |
| title_full_unstemmed | Some characterizations of three-dimensional trans-Sasakian manifolds admitting η- Ricci solitons and trans-Sasakian manifolds as Kagan subprojective space |
| title_short | Some characterizations of three-dimensional trans-Sasakian manifolds admitting η- Ricci solitons and trans-Sasakian manifolds as Kagan subprojective space |
| title_sort | some characterizations of three-dimensional trans-sasakian manifolds admitting η- ricci solitons and trans-sasakian manifolds as kagan subprojective space |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6047 |
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