Locally ϕ-Symmetric Generalized Sasakian-Space Forms
The aim of the present paper is to find necessary and sufficient conditions for locally ϕ-symmetric generalized Sasakian-space forms to have constant scalar curvature, η -parallel Ricci tensor, and cyclic parallel Ricci tensor. Illustrative examples are given.
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| author | Sarkar, A. Sen, M. Саркар, А. Сен, М. |
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UDC 517.5
A. Sarkar, M. Sen (Univ. Burdwan, India)
LOCALLY φ-SYMMETRIC GENERALIZED SASAKIAN-SPACE-FORMS
ЛОКАЛЬНО φ-СИМЕТРИЧНI УЗАГАЛЬНЕНI ФОРМИ
ПРОСТОРУ САСАКЯНА
The object of the present paper is to find necessary and sufficient conditions for locally φ-symmetric generalized Sasakian-
space-forms to have constant scalar curvature, η-parallel Ricci tensor and cyclic parallel Ricci tensor. Illustrative examples
are given.
Встановлено необхiднi та достатнi умови, при яких локально φ-симетричнi узагальненi форми простору Сасакяна
мають сталу скалярну кривизну, η-паралельний тензор Рiччi та циклiчний паралельний тензор Рiччi. Наведено
приклади.
1. Introduction. The nature of a Riemannian manifold mostly depends on the curvature tensor R of
the manifold. It is well known that the sectional curvatures of a manifold determine curvature tensor
completely. A Riemannian manifold with constant sectional curvature c is known as real-space-form
and its curvature tensor is given by
R(X,Y )Z = c{g(Y,Z)X − g(X,Z)Y }.
A Sasakian manifold with constant φ-sectional curvature is a Sasakian-space-form and it has a specific
form of its curvature tensor. Similar notion also holds for Kenmotsu and cosymplectic space-forms.
In order to generalize such space-forms in a common frame P. Alegre, D. E. Blair and A. Carriazo
introduced the notion of generalized Sasakian-space-forms in 2004 [1]. But, it is to be noted that
generalized Sasakian-space-forms are not merely generalization of such space-forms. It also contains
a large class of almost contact manifolds. For example, it is known that [2] any three-dimensional
(α, β)-trans Sasakian manifold with α, β depending on ξ is a generalized Sasakian-space-form. How-
ever, we can find generalized Sasakian-space-forms with non-constant functions and arbitrary dimen-
sions. In [1], the authors cited several examples of generalized Sasakian-space-forms in terms of
warped product spaces. In this connection, it should be mentioned that in 1989 Z. Olszak [12] stud-
ied generalized complex-space-forms and proved its existence. A generalized Sasakian-space-form is
defined as follows [1]:
Given an almost contact metric manifold M(φ, ξ, η, g), we say that M is generalized Sasakian-
space-form if there exist three functions f1, f2, f3 on M such that the curvature tensor R is given
by
R(X,Y )Z = f1{g(Y,Z)X − g(X,Z)Y }+
+f2{g(X,φZ)φY − g(Y, φZ)φX + 2g(X,φY )φZ}+
+f3{η(X)η(Z)Y − η(Y )η(Z)X + g(X,Z)η(Y )ξ − g(Y,Z)η(X)ξ},
for any vector fields X, Y, Z on M. In such a case we denote the manifold as M(f1, f2, f3). Here we
shall denote this manifold simply by M. In [1], the authors cited several examples of such manifolds.
c© A. SARKAR, M. SEN, 2013
1430 ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 10
LOCALLY φ-SYMMETRIC GENERALIZED SASAKIAN-SPACE-FORMS 1431
If f1 =
c+ 3
4
, f2 =
c− 1
4
and f3 =
c− 1
4
, then a generalized Sasakian-space-form with Sasakian
structure becomes Sasakian-space-form.
Generalized Sasakian-space-forms have been studied by several authors, viz., [1, 2, 9]. As a
weaker notion of locally symmetric manifolds T. Takahashi [13] introduced and studied locally φ-
symmetric Sasakian manifolds. Locally φ-symmetric manifolds have also been studied in the papers
[5, 6]. Symmetry of a manifold primarily depends on curvature tensor and Ricci tensor of the mani-
fold. In the paper [4], locally φ-symmetric generalized Sasakian-space-forms have been studied and
determined the condition for the manifold to be locally φ-symmetric with the additional condition
that the manifold is conformally flat. In the present paper, we study locally φ-symmetric generalized
Sasakian-space-forms and show that every locally φ-symmetric generalized Sasakian-space-form is
conformally flat. So, the present paper improves the result of the paper [4]. The present paper is
organized as follows:
Section 2 of this paper contains some preliminary results. In Section 3, we study locally φ-
symmetric generalized Sasakian-space-forms, and prove that every generalized Sasakian-space-form
which is locally φ-symmetric is conformally flat. In this section, we also find the conditions for a
locally φ-symmetric generalized Sasakian-space-form to have constant scalar curvature, η-parallel
Ricci tensor and cyclic parallel Ricci tensor. Interestingly, we show that in a locally φ-symmetric
generalized Sasakian-space-form all these properties hold if and only if f3 is constant. The last
section contains illustrative examples.
2. Preliminaries. This section contains some basic results and formulas which we will use in
need for.
A (2n+1)-dimensional Riemannian manifold (M, g) is called an almost contact metric manifold
if the following results hold [3]:
φ2X = −X + η(X)ξ, φξ = 0, g(X, ξ) = η(X). (2.1)
Here X is any vector field on the manifold, φ is a (1, 1) tensor, ξ is a unit vector field, η is an
1-form and g is a Riemannian metric. This metric induces an inner product on the tangent space of
the manifold. An almost contact metric manifold is called contact metric manifold if
dη(X,Y ) = Φ(X,Y ) = g(X,φY ),
for any vector fields X,Y on the manifold. Φ is called the fundamental two form of the manifold.
An almost contact metric structure is said to be normal if the induced almost complex structure J on
the product manifold M × R defined by
J
(
X, f
d
dt
)
=
(
φX − fξ, η(X)
d
dt
)
is integrable, where X is tangent to M, t is the coordinate of R, and f is a smooth function on
M × R [3]. A normal contact metric manifold is known as Sasakian manifold. An almost contact
metric manifold is Sasakian if and only if
(∇Xφ)Y = g(X,Y )ξ − η(Y )X,
for any vector fieldsX,Y on the manifold [3]. Here∇ is the Levi – Civita connection on the manifold.
It is also called operator of covariant differentiation.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 10
1432 A. SARKAR, M. SEN
For a (2n+ 1)-dimensional generalized Sasakian-space-form we have [1]
R(X,Y )Z = f1{g(Y,Z)X − g(X,Z)Y }+
+f2{g(X,φZ)φY − g(Y, φZ)φX + 2g(X,φY )φZ}+
+f3{η(X)η(Z)Y − η(Y )η(Z)X + g(X,Z)η(Y )ξ − g(Y,Z)η(X)ξ}, (2.2)
S(X,Y ) = (2nf1 + 3f2 − f3)g(X,Y )− (3f2 + (2n− 1)f3)η(X)η(Y ), (2.3)
r = 2n(2n+ 1)f1 + 6nf2 − 4nf3. (2.4)
Here S is the Ricci tensor and r is the scalar curvature of the space-form.
A generalized Sasakian-space-form of dimension greater than three is said to be conformally flat
if its Weyl conformal curvature tensor vanishes. It is known that [9] a (2n+ 1)-dimensional (n > 1)
generalized Sasakian-space-form M(f1, f2, f3) is conformally flat if and only if f2 = 0.
3. Locally φ-symmetric generalized Sasakian space-forms.
Definition 3.1. A generalized Sasakian space form is said to be locally φ-symmetric if
φ2(∇WR)(X,Y )Z = 0,
for all vector fields X,Y, Z orthogonal to ξ.
This notion was introduced by T. Takahashi for Sasakian manifolds [13].
Definition 3.2. The Ricci tensor S of a generalized Sasakian-space-form is called η-parallel if
it satisfies
(∇WS)(φX, φY ) = 0,
for any vector fields X,Y,W.
The notion of η-parallel Ricci tensor was introduced by M. Kon in the context of Sasakian
geometry [11].
If X,Y, Z are orthogonal to ξ, then (2.2) takes the form
R(X,Y )Z = f1{g(Y, Z)X − g(X,Z)Y }+
+f2{g(X,φZ)φY − g(Y, φZ)φX + 2g(X,φY )φZ}.
By covariant differentiation of R(X,Y )Z with respect to W, we obtain from the above equation
(∇WR)(X,Y )Z = ∇WR(X,Y )Z −R(∇WX,Y )Z −R(X,∇WY )Z −R(X,Y )∇WZ =
= df1(W ){g(Y,Z)X − g(X,Z)Y }+
+f1{∇W g(Y,Z)X + g(Y, Z)∇WX −∇W g(X,Z)Y − g(X,Z)∇WY }+
+df2(W ){g(X,φZ)φY − g(Y, φZ)φX + 2g(X,φY )φZ}+
+f2{∇W g(X,φZ)φY + g(X,φZ)∇W (φY )−
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LOCALLY φ-SYMMETRIC GENERALIZED SASAKIAN-SPACE-FORMS 1433
−∇W g(Y, φZ)φX − g(Y, φZ)∇W (φX)+
+2∇W g(X,φY )φZ + 2g(X,φY )∇W (φZ)}−
−f1{g(Y,Z)∇WX − g(∇WX,Z)Y }−
−f2{g(∇WX,φZ)φY − g(Y, φZ)φ∇WX + 2g(∇WX,φY )φZ}−
−f1{g(∇WY,Z)X − g(X,Z)∇WY }−
−f2{g(X,φZ)φ∇WY − g(∇WY, φZ)φX + 2g(X,φ∇WY )φZ}−
−f1{g(Y,∇WZ)X − g(X,∇WZ)Y }−
−f2{g(X,φ∇WZ)φY − g(Y, φ∇WZ)φX + 2g(X,φY )φ∇WZ}. (3.1)
Arranging the terms of the above equation, we have
(∇WR)(X,Y )Z = df1(W ){g(Y,Z)X − g(X,Z)Y }+
+df2(W ){g(X,φZ)φY − g(Y, φZ)φX + 2g(X,φY )φZ}+
+f1{∇W g(Y,Z)X + g(Y,Z)∇WX −∇W g(X,Z)Y − g(X,Z)∇WY−
−g(Y, Z)∇WX + g(∇WX,Z)Y − g(∇WY,Z)X + g(X,Z)∇WY−
−g(Y,∇WZ)X + g(X,∇WZ)Y }+
+f2{∇W g(X,φZ)φY + g(X,φZ)∇W (φY )−
−∇W g(Y, φZ)φX − g(Y, φZ)∇W (φX)+
+2∇W g(X,φY )φZ + 2g(X,φY )∇W (φZ)−
−g(∇WX,φZ)φY + g(Y, φZ)φ∇WX − 2g(∇WX,φY )φZ−
−g(X,φZ)φ∇WY + g(∇WY, φZ)φX − 2g(X,φ∇WY )φZ−
−g(X,φ∇WZ)φY + g(Y, φ∇WZ)φX − 2g(X,φY )φ∇WZ}.
After canceling some terms in the coefficient of f1 in the above equation, using the result
(∇Wφ)X = ∇W (φX)− φ∇WX and arranging the terms, we get from the above equation
(∇WR)(X,Y )Z = df1(W ){g(Y,Z)X − g(X,Z)Y }+
+df2(W ){g(X,φZ)φY − g(Y, φZ)φX + 2g(X,φY )φZ}+
+f1{(∇W g(Y, Z)− g(∇WY, Z)− g(Y,∇WZ))X−
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 10
1434 A. SARKAR, M. SEN
−(∇W g(X,Z)− g(∇WX,Z)− g(X,∇WZ))Y }+
+f2{(∇W g(X,φZ)− g(∇WX,φZ)− g(X,∇W (φZ)))φY+
+g(X, (∇Wφ)Z)φY − (∇W g(Y, φZ)− g(∇WY, φZ)−
−g(Y,∇W (φZ)))φX − g(Y, (∇Wφ)Z)φX+
+2(∇W g(X,φY )− g(∇WX,φY )−
−g(X,∇W (φY )))φZ + 2g(X, (∇Wφ)Y )φZ+
+g(X,φZ)(∇Wφ)Y − g(Y, φZ)(∇Wφ)X + 2g(X,φY )(∇Wφ)Z}.
The operator ∇ of the covariant differentiation is called metric connection if (∇W g)(X,Y ) = 0,
i.e., ∇W g(X,Y ) − g(∇WX,Y ) − g(X,∇WY ) = 0. Here we take ∇ as metric connection. Then,
we also have (∇W g)(X,φY ) = 0. Thus, the above equation gives
(∇WR)(X,Y )Z = df1(W ){g(Y,Z)X − g(X,Z)Y }+
+df2(W ){g(X,φZ)φY − g(Y, φZ)φX + 2g(X,φY )φZ}+
+f2{g(X,φZ)(∇Wφ)Y − g(Y, φZ)(∇Wφ)X+
+2g(X,φY )(∇Wφ)Z + g(X, (∇Wφ)Z)φY−
−g(Y, (∇Wφ)Z)φX + 2g(X, (∇Wφ)Y )φZ}. (3.2)
Applying φ2 on both sides of (3.2) and using (2.1), we get
φ2(∇WR)(X,Y )Z = df1(W ){g(X,Z)Y − g(Y,Z)X}+
+df2(W ){g(Y, φZ)φX − 2g(X,φY )φZ − g(X,φZ)φY }+
+f2{g(X,φZ)φ2((∇Wφ)Y )− g(Y, φZ)φ2((∇Wφ)X)+
+2g(X,φY )φ2((∇Wφ)Z)− g(X, (∇Wφ)Z)φY+
+g(Y, (∇Wφ)Z)φX − 2g(X, (∇Wφ)Y )φZ}. (3.3)
Suppose that the manifold is locally φ-symmetric. Then (3.3) yields
df1(W ){g(X,Z)Y − g(Y, Z)X}+
+df2(W ){g(Y, φZ)φX − 2g(X,φY )φZ − g(X,φZ)φY }+
+f2{g(X,φZ)φ2((∇Wφ)Y )− g(Y, φZ)φ2((∇Wφ)X)+
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 10
LOCALLY φ-SYMMETRIC GENERALIZED SASAKIAN-SPACE-FORMS 1435
+2g(X,φY )φ2((∇Wφ)Z)− g(X, (∇Wφ)Z)φY + g(Y, (∇Wφ)Z)φX−
−2g(X, (∇Wφ)Y )φZ} = 0. (3.4)
Taking the inner product g in both sides of the above equation with W we have
df1(W ){g(X,Z)g(Y,W )− g(Y,Z)g(X,W )}+
+df2(W ){g(Y, φZ)g(φX,W )− 2g(X,φY )g(φZ,W )− g(X,φZ)g(φY,W )}+
+f2{g(X,φZ)g(φ2((∇Wφ)Y ),W )− g(Y, φZ)g(φ2((∇Wφ)X),W )+
+2g(X,φY )g(φ2((∇Wφ)Z),W )− g(X, (∇Wφ)Z)g(φY,W ) + g(Y, (∇Wφ)Z)g(φX,W )−
−2g(X, (∇Wφ)Y )g(φZ,W )} = 0. (3.5)
In (3.5) putting X = W = ei, where {ei} is an orthonormal basis of the tangent space at each
point of the manifold and taking summation over i, i = 1, 2, 3, . . . , 2n+ 1, we get
2ndf1(W )g(Y, Z) + 3df2(W )g(Y,Z)− f2{g(φZ, φ2(∇eiφ)Y )−
−
∑
i
g(Y, φZ)g(φ2(∇eiφ)ei, ei) + 2g(φY, φ2(∇eiφ)Z)−
−g((∇Wφ)Z, φY )− 2g((∇Wφ)Y, φZ)} = 0. (3.6)
Putting Z = φY, we have from the above equation
f2{g(φ2Y, φ2(∇eiφY ))−
∑
i
g(Y, φ2Y )g(φ2(∇eiφ)ei, ei)+
+2g(φY, (∇eiφ)φY )− g((∇Wφ)φY, φY )− 2g((∇Wφ)Y, φ2Y )} = 0. (3.7)
The above equation is true for any arbitrary Y orthogonal to ξ. We observe from (3.7) that for Y 6= ξ
g(φ2Y, φ2(∇eiφY ))−
∑
i
g(Y, φ2Y )g(φ2(∇eiφ)ei, ei)+
+2g(φY, (∇eiφ)φY )− g((∇Wφ)φY, φY )− 2g((∇Wφ)Y, φ2Y ) 6= 0.
Hence, in view of (3.7) we must have
f2 = 0. (3.8)
It is known that [9] a generalized Sasakian-space-form is conformally flat if and only if f2 = 0.
Thus, we have the following theorem.
Theorem 3.1. A locally φ-symmetric generalized Sasakian-space-form is conformally flat.
The above theorem gives a new result regarding the relation between locally φ-symmetric gener-
alized Sasakian-space-forms and conformally flat generalized Sasakian-space-forms.
By virtue of (3.8), (3.5) takes the form
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 10
1436 A. SARKAR, M. SEN
df1(W ) = 0.
The above equation yields f1 is a constant. Hence, for a locally φ-symmetric generalized Sasakian-
space-form f2 = 0 and f1 is constant. Therefore, from (2.4), it follows that
r = 2n(2n+ 1)f1 − 4nf3.
The above quation yields
dr(W ) = −4ndf3(W ). (3.9)
In view of the above equation we obtain the following theorem.
Theorem 3.2. The scalar curvature of a locally φ-symmetric generalized Sasakian-space-form
is constant if and only if f3 is constant.
From (2.3) we have
(∇WS)(φX, φY ) = d(2nf1 + 3f2 − f3)(W )g(φX, φY ), (3.10)
where X,Y are orthogonal to ξ. If the manifold is locally φ-symmetric, then the above equation
takes the form
(∇WS)(φX, φY ) = −d(f3)(W )g(X,Y ).
The above equation leads us to state the following theorem.
Theorem 3.3. A locally φ-symmetric generalized Sasakian-space-form has η-parallel Ricci ten-
sor if and only if f3 is constant.
A. Gray [8] introduced two classes of Riemannian manifolds determined by the covariant deriva-
tive of the Ricci tensor. The first one is the class A consisting of all Riemannian manifolds whose
Ricci tensor S is a Codazzi tensor, that is,
(∇XS)(Y,Z) = (∇Y S)(X,Z).
The second one is the class B consisting of all Riemannian manifolds whose Ricci tensor is cyclic
parallel, that is,
(∇XS)(Y, Z) + (∇Y S)(X,Z) + (∇ZS)(X,Y ) = 0.
It is known that [10] the Ricci tensor of Cartan hypersurface is cyclic parallel. Now, we like to find
under what condition a locally φ-symmetric generalized Sasakian space-form has cyclic parallel Ricci
tensor. In view of (2.3), and for X,Y, Z orthogonal to ξ, we get
(∇XS)(Y, Z) + (∇Y S)(X,Z) + (∇ZS)(X,Y ) = d(2nf1 + 3f2 − f3)(X)g(Y,Z)+
+d(2nf1 + 3f2 − f3)(Y )g(X,Z) + d(2nf1 + 3f2 − f3)(Z)g(X,Y ). (3.11)
For a locally φ-symmetric generalized Sasakian-space-form f2 = 0 and f1 is consant. Hence, the
above equation yields
(∇XS)(Y,Z) + (∇Y S)(X,Z) + (∇ZS)(X,Y ) =
= −d(f3)(X)g(Y, Z)− d(f3)(Y )g(X,Z)− d(f3)(Z)g(X,Y ). (3.12)
The above equation enables us to state the following theorem.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 10
LOCALLY φ-SYMMETRIC GENERALIZED SASAKIAN-SPACE-FORMS 1437
Theorem 3.4. A locally φ-symmetric generalized Sasakian-space-form has cyclic parallel Ricci
tensor if and only if f3 is constant.
By virtue of Theorems 3.2, 3.3, 3.4, we obtain the following corollary.
Corollary 3.1. For a locally φ-symmetric generalized Sasakian-space-form the following condi-
tions are equivalent:
(i) the manifold has constant scalar curvature,
(ii) the manifold has η-parallel Ricci tensor,
(iii) the manifold has cyclic parallel Ricci tensor.
The above corollary gives a new result.
Remark 3.1. The notion of quarter-symmetric metric connection was introduced by S. Golab
[7]. The torsion tensor of the quarter-symmetric metric connection is given by
T (X,Y ) = η(Y )X − η(X)Y.
If X, Y are orthogonal to ξ, then the torsion tensor vanishes and the quarter-symmetric metric
connection reduces to Levi-Civita connection. Therefore, all the results of the present paper are of
the same form with respect to quarter-symmetric metric connection and Levi-Civita connection.
4. Examples. Let us now give an example of a generalized Sasakian-space-form which is locally
φ-symmetric.
Example 4.1. In [1], it is shown that R×f Cm is a generalized Sasakian-space-form with
f1 = −(f ′)2
f2
, f2 = 0, f3 = −(f ′)2
f2
+
f ′′
f
,
where f = f(t), t ∈ R and f ′ denotes derivative of f with respect to t. If we choose m = 4, and
f(t) = et, then M is a 5-dimensional conformally flat generalized Sasakian-space-form, because
f2 = 0. We also see that f3 = 0, which is a constant. Therefore, by the results obtained in the present
paper M is locally φ-symmetric and has constant scalar curvature, η-parallel Ricci tensor and cyclic
parallel Ricci tensor.
Example 4.2. Let N(a, b) be a generalized complex space-form of dimension 4, then by [1],
M = R ×f N, endowed with the almost contact metric structure (φ, ξ, η, gf ) is a generalized
Sasakian-space-form M(f1, f2, f3) of dimension 5 with
f1 =
a− f ′2
f2
, f2 =
b
f2
, f3 =
a− f ′2
f2
+
f ′′
f
where f is a function of t ∈ R and f ′ denotes differentiation of f with respect to t. Let us choose f
and a as constants and b = 0. Then f2 = 0 and f3 is a constant. Therefore, by theorems obtained in
the present paper M locally φ-symmetric and has constant scalar curvature, η-parallel Ricci tensor
and cyclic parallel Ricci tensor.
Example 4.3. For a Sasakian-space-form of dimension greater than three and of constant φ-
sectional curvature 1, f1 = 1, f2 = f3 = 0. Therefore, by theorems obtained in the present paper M
is locally φ-symmetric and has constant scalar curvature, η-parallel Ricci tensor and cyclic parallel
Ricci tensor.
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1438 A. SARKAR, M. SEN
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Received 01.06.11,
after revision — 14.02.12
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 10
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| id | umjimathkievua-article-2521 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:25:05Z |
| publishDate | 2013 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/b1/0d0ce14c3821a8925674a4f6ecb860b1.pdf |
| spelling | umjimathkievua-article-25212020-03-18T19:17:19Z Locally ϕ-Symmetric Generalized Sasakian-Space Forms Локально ϕ-симетричні узагальнені форми простору Сасакяна Sarkar, A. Sen, M. Саркар, А. Сен, М. The aim of the present paper is to find necessary and sufficient conditions for locally ϕ-symmetric generalized Sasakian-space forms to have constant scalar curvature, η -parallel Ricci tensor, and cyclic parallel Ricci tensor. Illustrative examples are given. Встановлено необхідні та достатні умови, при яких локально ϕ-симетричні узагальнені форми простору Сасакяна мають сталу скалярну кривизну, η-паралельний тензор Річчі та циклічний паралельний тензор Річчі. Наведено приклади. Institute of Mathematics, NAS of Ukraine 2013-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2521 Ukrains’kyi Matematychnyi Zhurnal; Vol. 65 No. 10 (2013); 1430–1438 Український математичний журнал; Том 65 № 10 (2013); 1430–1438 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2521/1806 https://umj.imath.kiev.ua/index.php/umj/article/view/2521/1807 Copyright (c) 2013 Sarkar A.; Sen M. |
| spellingShingle | Sarkar, A. Sen, M. Саркар, А. Сен, М. Locally ϕ-Symmetric Generalized Sasakian-Space Forms |
| title | Locally ϕ-Symmetric Generalized Sasakian-Space Forms |
| title_alt | Локально ϕ-симетричні узагальнені форми простору Сасакяна |
| title_full | Locally ϕ-Symmetric Generalized Sasakian-Space Forms |
| title_fullStr | Locally ϕ-Symmetric Generalized Sasakian-Space Forms |
| title_full_unstemmed | Locally ϕ-Symmetric Generalized Sasakian-Space Forms |
| title_short | Locally ϕ-Symmetric Generalized Sasakian-Space Forms |
| title_sort | locally ϕ-symmetric generalized sasakian-space forms |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2521 |
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