D-homothetic deformation of normal almost contact metric manifolds
The object of the present paper is to study a transformation called the $D$-homothetic deformation of normal almost contact metric manifolds. In particular, it is shown that, in a $(2n + 1)$-dimensional normal almost contact metric manifold, the Ricci operator $Q$ commutes with the structure tenso...
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| author | De, U. C. Ghosh, S. Де, У. К. Хост, С. |
| author_facet | De, U. C. Ghosh, S. Де, У. К. Хост, С. |
| author_sort | De, U. C. |
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| datestamp_date | 2020-03-18T19:32:22Z |
| description | The object of the present paper is to study a transformation called the $D$-homothetic deformation of normal almost contact metric manifolds.
In particular, it is shown that, in a $(2n + 1)$-dimensional normal almost contact metric manifold, the Ricci operator $Q$ commutes with the structure
tensor $\phi$ under certain conditions, and the operator $Q\phi - \phi Q$ is invariant under a $D$-homothetic deformation.
We also discuss the invariance of $\eta$-Einstein manifolds, $\phi$-sectional curvature, and the local $\phi$-Ricci symmetry under a $D$-homothetic deformation.
Finally, we prove the existence of such manifolds by a concrete example. |
| first_indexed | 2026-03-24T02:27:51Z |
| format | Article |
| fulltext |
UDC 517.98
U. C. De (Univ. Calcutta, India),
S. Ghosh (Madanpur, Nadia, India)
D-HOMOTHETIC DEFORMATION
OF NORMAL ALMOST CONTACT METRIC MANIFOLDS
D-ГОМОТЕТИЧНА ДЕФОРМАЦIЯ
НОРМАЛЬНИХ МАЙЖЕ КОНТАКТНИХ МНОГОВИДIВ
The object of the present paper is to study a transformation called the D-homothetic deformation of normal almost contact
metric manifolds. In particular, it is shown that, in a (2n + 1)-dimensional normal almost contact metric manifold, the
Ricci operator Q commutes with the structure tensor φ under certain conditions, and the operator Qφ − φQ is invariant
under a D-homothetic deformation. We also discuss the invariance of η-Einstein manifolds, φ-sectional curvature, and the
local φ-Ricci symmetry under a D-homothetic deformation. Finally, we prove the existence of such manifolds by a concrete
example.
Метою цiєї статтi є вивчення перетворення, що називається D-гомотетичною деформацiєю нормальних майже
контактних многовидiв. Зокрема, показано, що у (2n + 1)-вимiрному нормальному майже контактному многовидi
оператор Рiччi Q комутує за певних умов iз структурним тензором φ, а оператор Qφ− φQ є iнварiантним щодо D-
гомотетичної деформацiї. Також розглянуто питання про iнварiантнiсть η-ейнштейнiвських многовидiв, φ-секцiйну
кривину та локальну φ-симетрiю Рiччi при D-гомотетичнiй деформацiї. Iснування таких многовидiв доведено на
конкретному прикладi.
1. Introduction. Let M be an almost contact metric manifold and (φ, ξ, η) its almost contact
structure. This means, M is an odd-dimensional differentiable manifold and φ, ξ, η are tensor fields
on M of types (1, 1), (1, 0) and (0, 1) respectively, such that
φ2 = −I + η ⊗ ξ, η(ξ) = 1, φξ = 0, η ◦ φ = 0. (1.1)
Let R be the real line and t a coordinate on R. Define an almost complex structure J on M × R
by
J
(
X,λ
d
dt
)
=
(
φX − λξ, η(X)
d
dt
)
, (1.2)
where the pair
(
X,λ
d
dt
)
denotes a tangent vector on M × R, X and λ
d
dt
being tangent to M and
R respectively.
M and (φ, ξ, η) are said to be normal if the structure J is integrable [1, 2]. The necessary and
sufficient condition for (φ, ξ, η) to be normal is
[φ, φ] + 2dη ⊗ ξ = 0, (1.3)
where the pair [φ, φ] is the Nijenhuis tensor of φ defined by
[φ, φ](X,Y ) = [φX, φY ] + φ2[X,Y ]− φ[φX, Y ]− φ[X,φY ] (1.4)
for any X,Y ∈ χ(M); χ(M) being the Lie algebra of vector fields on M.
We say that the contact form η has rank r = 2s if (dη)s 6= 0 and η ∧ (dη)s = 0 and has rank
r = 2s+ 1 if η ∧ (dη)s 6= 0 and (dη)s+1 = 0. We also say r is rank of the structure (φ, ξ, η).
c© U. C. DE, S. GHOSH, 2012
1330 ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10
D-HOMOTHETIC DEFORMATION OF NORMAL ALMOST CONTACT METRIC MANIFOLDS 1331
A Riemannian metric g on M satisfying the condition
g(φX, φY ) = g(X,Y )− η(X)η(Y ) (1.5)
for any X,Y ∈ χ(M), is said to be compatible with the structure (φ, ξ, η). If g is such a metric,
then the quadruple (φ, ξ, η, g) is called an almost contact metric structure on M and M is an almost
contact metric manifold. On such a manifold we also have
η(X) = g(X, ξ) (1.6)
for any X ∈ χ(M) and we can always define the 2-form Φ by
Φ(Y,Z) = g(Y, φZ), (1.7)
where Y,Z ∈ χ(M).
A normal almost contact metric structure (φ, ξ, η, g) satisfying additionally the condition dη = Φ
is called Sasakian. Of course, any such structure on M has rank 3. Also a normal almost contact
metric structure satisfying the condition dΦ = 0 is said to be quasi-Sasakian [3].
In the paper [8], Olszak studied the curvature properties of normal almost contact manifold of
dimension three with several examples. Also in [4], U. C. De and A. K. Mondal studied three
dimensional normal almost contact metric manifolds satisfying certain curvature conditions.
An almost contact metric manifold is said to be η-Einstein if its Ricci tensor S is of the form
S = λg + µη ⊗ η (1.8)
where λ and µ are smooth functions on the manifold.
The notion of locally φ-symmetry first introduced by T. Takahashi [9] on a Sasakian manifold.
Again in a recent paper [5] U. C. De and Avijit Sarkar introduced the notion of locally φ-Ricci
symmetric Sasakian manifolds.
A three dimensional normal almost contact metric manifold is said to be locally φ-Ricci symmet-
ric if
φ2(∇XQ)(Y ) = 0,
where Q is the Ricci operator defined by g(QX,Y ) = S(X,Y ) and X,Y are orthogonal to ξ.
Let M (φ, ξ, η, g) be an almost contact metric manifold with dim M = m = 2n + 1. The
equation η = 0 defines an (m−1)-dimensional distribution D on M [12]. By an (m−1)-homothetic
deformation or D-homothetic deformation [10] we mean a change of structure tensors of the form
η̄ = aη, ξ̄ =
1
a
ξ, φ̄ = φ, ḡ = ag + a(a− 1)η ⊗ η,
where a is a positive constant. If M(φ, ξ, η, g) is an almost contact metric structure with contact form
η, then M(φ̄, ξ̄, η̄, ḡ) is also an almost contact metric structure [10]. Denoting by W i
jk the difference
Γ̄i
jk − Γi
jk of Christoffel symbols we have in an almost contact metric manifold [10]
W (X,Y ) = (1− a)[η(Y )φX + η(X)φY ] +
1
2
(
1− 1
a
)
[(∇Xη)(Y ) + (∇Y η)(X)]ξ (1.9)
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1332 U. C. DE, S. GHOSH
for all X,Y ∈ χ(M). If R and R̄ denote respectively the curvature tensor of the manifold
M(φ, ξ, η, g) and M(φ̄, ξ̄, η̄, ḡ), then we have [10]
R̄(X,Y )Z = R(X,Y )Z + (∇XW )(Z, Y )− (∇YW )(Z,X)+
+W (W (Z, Y ), X)−W (W (Z,X), Y ) (1.10)
for all X,Y, Z ∈ χ(M).
In [10, 13] the authors used D-homothetic deformation on a Sasakian and K-contact structures
to get results on the first Betti number, second Betti number and harmonic forms. Hence the D-
homothetic deformation can be used to get the results on the first Betti number, second Betti number
and harmonic forms of the normal almost contact structure. A plane section in the tangent space
Tp(M) is called a φ-section if there exists a unit vector X in Tp(M) orthogonal to ξ such that
{X,φX} is an orthonormal basis of the plane section. Then the sectional curvature
K(X,φX) = g(R(X,φX)X,φX)
is called a φ-sectional curvature. A contact metric manifold M(φ, ξ, η, g) is said to be of constant
φ-sectional curvature if at any point p ∈M, the sectional curvature K(X,φX) is independent of the
choice of non-zero X ∈ Dp, where D denotes the contact distribution of the contact metric manifold
defined by η = 0.
The model spaces of contact metric structure are complete and simply connected Sasakian mani-
folds of constant φ-sectional curvature H. These Sasakian manifolds admit the maximal dimensional
automorphism [14]. The Riemann curvature tensor R of Sasakian manifold of constant φ-sectional
curvature is determined by Ogiue [7]. The geometry of contact Riemannian manifold of constant
φ-sectional curvature is obtained by Tanno [15]. If the φ-sectional curvature H is constant on a K-
contact Riemannian manifold M(φ, ξ, η, g), then H can be deformed by a D-homothetic deformation
of the structure tensors [11]. If H > −3, then choosing a constant θ =
H + 3
4
, we get a K-contact
Riemannian manifold M
(
φ,
1
θ
ξ, θη, θg + (θ2 − θ)η ⊗ η
)
of constant φ-sectional curvature [11].
Hence Tanno posed a natural question that does there exist contact metric manifolds of constant φ-
sectional curvature which are not Sasakian [11]. Since the normal almost contact metric manifold
contains both the Sasakian and non-Sasakian structures, the existance of a non-Sasakian manifold of
both constant and non-constant φ-sectional curvature is ensured in our paper, which gives rise to the
answer of the question of Tanno [11] as affirmative.
In a Sasakian manifold, the Ricci operator Q commutes with the structure tensor φ, that is,
Qφ = φQ. But in (2n + 1)-dimensional normal almost contact metric manifold Qφ 6= φQ, in
general.
The present paper is organized as follows: After preliminaries in Section 3, we prove some
important lemmas. In Section 4, we study the properties of the expression Qφ − φQ in (2n + 1)-
dimensional normal almost contact metric manifolds and prove that Qφ = φQ in these manifolds,
provided α, β are constants. Beside this, in this section we also prove that the expression Qφ − φQ
of these manifolds is invariant under a D-homothetic deformation, provided α is constant. Section 5
deals with the study of (2n+ 1)-dimensional η-Einstein normal almost contact metric manifolds and
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10
D-HOMOTHETIC DEFORMATION OF NORMAL ALMOST CONTACT METRIC MANIFOLDS 1333
prove that these manifolds are invariant under a D-homothetic deformation, provided α = 0. Section
6 is devoted to study φ-sectional curvature tensor in a (2n + 1)-dimensional normal almost contact
metric manifold and we show that there exists a (2n+ 1)-dimensional normal almost contact metric
manifold (non-Sasakian) with non-zero and non-constant φ-sectional curvature. Section 7 deals with
locally φ-symmetric three dimensional normal almost contact metric manifold and we prove this
manifold is also invariant under a D-homothetic deformation, provided α = constant. Finally in
Section 8, we set an example of a three dimensional normal almost contact metric manifold which
verifies some theorems of Section 6.
2. Preliminaries. For a normal almost contact metric structure (φ, ξ, η, g) on M, we have [8]
(∇Xφ)(Y ) = g(φ∇Xξ, Y )− η(Y )φ∇Xξ, (2.1)
∇Xξ = α[X − η(X)ξ]− βφX, (2.2)
where 2α = div ξ and 2β = tr (φ∇ξ), div ξ is the divergent of ξ defined by div ξ = trace {X −→
−→ ∇Xξ} and tr (φ∇ξ) = trace {X −→ φ∇Xξ}. Using (2.2) in (2.1), we get
(∇Xφ)(Y ) = α[g(φX, Y )ξ − η(Y )φX] + β[g(X,Y )ξ − η(Y )X]. (2.3)
Also in this manifold the following relation holds:
R(X,Y )ξ = [Y α+ (α2 − β2)η(Y )]φ2X − [Xα+ (α2 − β2)η(X)]φ2Y+
+[Y β + 2αβη(Y )]φX − [Xβ + 2αβη(X)]φY, (2.4)
S(X, ξ) = −Xα− (φX)β − [ξα+ 2(α2 − β2)]η(X), (2.5)
ξβ + 2αβ = 0, (2.6)
where R denotes the curvature tensor and S is the Ricci tensor.
(∇Xη)(Y ) = αg(φX, φY )− βg(φX, Y ). (2.7)
On the other hand, the curvature tensor in a three dimensional Riemannian manifold always satisfies
R(X,Y )Z = S(Y,Z)X − S(X,Z)Y + g(Y, Z)QX − g(X,Z)QY−
−r
2
[g(Y, Z)X − g(X,Z)Y ], (2.8)
where r is the scalar curvature of the manifold.
By (2.4), (2.5) and (2.8) we can derive
S(Y,Z) =
(r
2
+ ξα+ α2 − β2
)
g(φY, φZ)−
−η(Y )(Zα+ (φZ)β)− η(Z)(Y α+ (φY )β)− 2(α2 − β2)η(Y )η(Z). (2.9)
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10
1334 U. C. DE, S. GHOSH
From (2.6) it follows that if α, β = constant, then the manifold is either β-Sasakian or α-Kenmotsu
[6] or cosymplectic [1]. Also we have a 3-dimensional normal almost contact metric manifold is
quasi-Sasakian if and only if α = 0 [8].
3. Some lemmas. In this section we shall state and prove some lemmas which will be needed to
prove the main results.
Lemma 3.1. In a normal almost contact metric manifold M the following relation holds:
g(R(X,Y )φZ,W ) + g(R(X,Y )Z, φW ) = (Xα)[g(φY,Z)η(W )−
−g(φY,W )η(Z)] + (Xβ)[g(Y,Z)η(W )−
−g(Y,W )η(Z)] + (Y α)[g(φX,W )η(Z)−
−g(φX,Z)η(W )] + (Y β)[g(X,W )η(Z)− g(X,Z)η(W )]+
+(α2 − β2)[g(φX,W )g(Y,Z) + g(φY,Z)g(X,W )−
−g(φY,W )g(X,Z)− g(φX,Z)g(Y,W )] + 2αβ[g(φY,W )g(φX,Z)−
−g(φX,W )g(φY,Z) + g(X,W )g(Y,Z)− g(Y,W )g(X,Z)]. (3.1)
Proof. Differentiating (1.7) covariantly with respect to X and using (2.3) and (2.7) we obtain
(∇XΦ)(Y,Z) = α[g(φX,Z)η(Y )− g(φX, Y )η(Z)]+
+β[g(X,Z)η(Y )− g(X,Y )η(Z)]. (3.2)
Again differentiating (3.2) covariantly and using (2.2), (2.3) and (2.7) yields
(∇X∇Y Φ)(Z,W ) = (Xα)[g(φY,W )η(Z)−
−g(φY,Z)η(W )] + (Xβ)[g(Y,W )η(Z)−
−g(Y,Z)η(W )] + α2[g(φY,W )g(φX, φZ)−
−g(φY,Z)g(φX, φW )− g(φX,W )η(Y )η(Z)+
+g(φX,Z)η(Y )η(W )] + β2[g(φX,W )g(Y, Z)−
−g(φX,Z)g(Y,W )] + αβ[g(φX,W )g(φY,Z)−
−g(φX,Z)g(φY,W ) + g(Y,W )g(φX, φZ)−
−g(Y, Z)g(φX, φW ) + g(X,Z)η(Y )η(W )−
−g(X,W )η(Y )η(Z)] + α[g(φ∇XY,W )η(Z)−
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D-HOMOTHETIC DEFORMATION OF NORMAL ALMOST CONTACT METRIC MANIFOLDS 1335
−g(φ∇XY,Z)η(W )] + β[g(∇XY,W )η(Z)− g(∇XY,Z)η(W )]. (3.3)
Using (3.2) and (3.3) we obtain
(∇X∇Y Φ)(Z,W )− (∇Y∇XΦ)(Z,W )− (∇[X,Y ]Φ)(Z,W ) =
= (Xα)[g(φY,W )η(Z)− g(φY,Z)η(W )]+
+(Xβ[g(Y,W )η(Z)− g(Y, Z)η(W )]−
−(Y α)[g(φX,W )η(Z)− g(φX,Z)η(W )]−
−(Y β)[g(X,W )η(Z)− g(X,Z)η(W )]+
+(α2 − β2)[g(φY,W )g(X,Z)− g(φX,W )g(Y,Z)−
−g(X,W )g(φY,Z) + g(Y,W )g(φX,Z)] + 2αβ[g(φX,W )g(φY,Z)−
−g(φX,Z)g(φY,W ) + g(X,Z)g(Y,W )− g(X,W )g(Y,Z)]. (3.4)
Then using (3.4) and by Ricci identity we easily obtain (3.1).
Lemma 3.2. Let M(φ, ξ, η, g) be a normal almost contact metric manifold of dimension (2n+
+ 1). Then for any X, Y, Z and W on M, the following relation holds:
g(R(X,Y )φZ, φW ) = g(R(X,Y )Z,W ) + (Xα)[g(Y,Z)η(W )−
−g(Y,W )η(Z)]− (Xβ)[g(φY,Z)η(W )−
−g(φY,W )η(Z)] + (Y α)[g(X,W )η(Z)−
−g(X,Z)η(W )] + (Y β)[g(φX,Z)η(W )−
−g(φX,W )η(Z)] + (α2 − β2)[g(X,W )g(Y,Z)−
−g(X,Z)g(Y,W ) + g(φX,Z)g(φY,W )−
−g(φX,W )g(φY,Z)] + 2αβ[g(Y,W )g(φX,Z)−
−g(X,W )g(φY,Z) + g(X,Z)g(φY,W )− g(Y,Z)g(φX,W )]. (3.5)
Proof. Replacing W by φW in (3.1) and using (1.1), (1.6) and (2.4) we easily obtain (3.5).
Lemma 3.3. Let M(φ, ξ, η, g) be a normal almost contact metric manifold of dimension (2n+
+ 1). Then for any X, Y, Z and W on M, the following relation holds:
g(R(φX, φY )φZ, φW ) = g(R(X,Y )Z,W ) + (α2 − β2)[g(Y,Z)η(X)η(W )−
−g(X,Z)η(Y )η(W ) + g(X,W )η(Y )η(Z)− g(Y,W )η(X)η(Z)]+
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10
1336 U. C. DE, S. GHOSH
+2αβ[2g(φX,W )g(Y,Z)− 2g(φY,W )g(X,Z)+
+2g(φY,Z)g(X,W )− 2g(φX,Z)g(Y,W )+
+g(φY,W )η(X)η(Z)− g(φX,W )η(Y )η(Z)+
+g(φX,Z)η(Y )η(W )− g(φY,Z)η(X)η(W )]+
+(Zα)[g(X,W )η(Y )− g(Y,W )η(X)]−
−(Zβ)[g(φY,W )η(X)− g(φX,W )η(Y )]+
+(Wα)[g(Y,Z)η(X)− g(X,Z)η(Y )]+
+(Wβ)[g(φY,Z)η(X)− g(φX,Z)η(Y )]+
+(φXα)[g(φY,Z)η(W )− g(φY,W )η(Z)]−
−(φXβ)[g(Y,W )η(Z)− g(Y, Z)η(W )]+
+(φY α)[g(φX,W )η(Z)− g(φX,Z)η(W )]+
+(φY β)[g(X,W )η(Z)− g(X,Z)η(W )]. (3.6)
Proof. Putting φX and φY instead of X and Y respectively in (3.5) and using (1.1), (1.6) and
(3.5) we easily obtain (3.6).
Proposition 3.1. In a (2n+ 1)-dimensional η-Einstein normal almost contact metric manifold
M(φ, ξ, η, g), the Ricci tensor is expressed as
S(X,Y ) =
[ r
2n
+ ξα+ (α2 − β2)
]
g(X,Y )−
−
[ r
2n
+ (2n+ 1)ξα+ (2n+ 1)(α2 − β2)
]
η(X)η(Y ). (3.7)
Proof. From (1.8) we have by contraction
r = (2n+ 1)λ+ µ, (3.8)
where r is the scalar curvature of the manifold. Again putting X = ξ in (2.5), we obtain
λ+ µ = −2nξα− 2n(α2 − β2). (3.9)
Solving above two equations we get
λ =
r
2n
+ ξα+ (α2 − β2), (3.10)
and
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D-HOMOTHETIC DEFORMATION OF NORMAL ALMOST CONTACT METRIC MANIFOLDS 1337
µ = − r
2n
− (2n+ 1)ξα− (2n+ 1)(α2 − β2). (3.11)
Putting the values of λ and µ in (1.8) we get (3.7).
Proposition 3.1 is proved.
4. Properties of the expression Qφ − φQ. In this section we investigate the properties of the
expression Qφ− φQ in a (2n+ 1)-dimensional normal almost contact metric manifold M.
Let {ei, φei, ξ}, i = 1, 2, . . . , n, be a local φ-basis at any point of the manifold. Then putting
Y = Z = ei in (3.6) and taking summation over i = 1 to n, we obtain by virtue of η(ei) = 0,
−
n∑
i=1
φR(φX, φei)φei =
n∑
i=1
R(X, ei)ei + n(α2 − β2)η(X)ξ+
+[(n− 1)gradα− (φ gradβ)]η(X)+
+4(n− 2)αβ(φX) + (Xα)ξ + (n− 1)(φXβ)ξ. (4.1)
Again putting Y = Z = φei in (3.6) and taking summation over i = 1 to n then using (1.1) and
η(ei) = 0, we obtain
−
n∑
i=1
φR(φX, ei)ei =
n∑
i=1
R(X,φei)φei+
+n(α2 − β2)η(X)ξ + [(n− 1)gradα− (φ gradβ)]η(X)+
+4(n− 2)αβ(φX) + (Xα)ξ + (n− 1)(φXβ)ξ. (4.2)
Adding (4.1) and (4.2) and using the definition of Ricci operator, we obtain
−φQ(φX) + φR(φX, ξ)ξ = QX −R(X, ξ)ξ+
+2n(α2 − β2)η(X)ξ + 8(n− 2)αβ(φX)+
+2[(n− 1)gradα− φ(gradβ)]η(X) + 2(Xα)ξ + 2(n− 1)(φXβ)ξ. (4.3)
From (2.4) by virtue of (2.6), it follows that
R(φX, ξ)ξ = −[ξα+ (α2 − β2)](φX). (4.4)
In view of (2.4), (2.6) and (4.4), the relation (4.3) takes the form
−φQ(φX) = QX + 2n(α2 − β2)η(X)ξ + 8(n− 2)αβ(φX)+
+2[(n− 1)gradα− φ(gradβ)]η(X) + 2(Xα)ξ + 2(n− 1)(φXβ)ξ. (4.5)
Operating φ on both sides of (4.5) and using (1.1) we get
QφX − φQX = S(φX, ξ)ξ + 8(n− 2)αβ(φ2X)+
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1338 U. C. DE, S. GHOSH
+2[(n− 1)φ(gradα)− φ2(gradβ)]η(X). (4.6)
From (2.5) we have
S(φX, ξ) = −(φX)α− (φ2X)β. (4.7)
By virtue of (4.7) and (2.6), (4.6) reduces to
[Qφ− φQ]X = (Xβ)ξ − (n− 2)(4ξβ)X − (φXα)ξ+
+(4n− 7)(ξβ)η(X)ξ + 2[(n− 1)φ(gradα)− φ2(gradβ)]η(X). (4.8)
Hence we state the following theorem.
Theorem 4.1. In a (2n + 1)-dimensional normal almost contact metric manifold Qφ = φQ,
provided α, β are constants.
By virtue of (2.7), the relation (1.10) reduces to
W (X,Y ) = (1− a)[η(Y )φX + η(X)φY ] +
(
1− 1
a
)
α[g(X,Y )− η(X)η(Y )]ξ. (4.9)
In view of (2.2), (2.3) and (2.7), the relation (4.9) yields
(∇XW )(Y,Z) = (1− a)[α{g(φX, Y )η(Z)ξ+
+g(φX,Z)η(Y )ξ + g(X,Z)φY + g(X,Y )φZ−
−η(X)η(Y )φZ − η(X)η(Z)φY − 2η(Y )η(Z)φX}+ β{g(X,Y )η(Z)ξ+
+g(X,Z)η(Y )ξ − g(φX,Z)φY − g(φX, Y )φZ − 2η(Y )η(Z)X}]+
+
a− 1
a
(Xα)[g(Y, Z)− η(Y )η(Z)]ξ − a− 1
a
α[α{g(X,Y )η(Z)ξ+
+g(X,Z)η(Y )ξ + g(Y,Z)η(X)ξ − g(Y,Z)X + η(Y )η(Z)X−
−3η(X)η(Y )η(Z)ξ}+ β{g(Y,Z)φX − g(φX,Z)η(Y )ξ−
−g(φX, Y )η(Z)ξ − η(Y )η(Z)φX}]. (4.10)
Using (4.9) and (4.10) into (1.11), we obtain by virtue of (2.4) and (2.7) that
R̄(X,Y )Z = R(X,Y )Z + (1− a)[α{g(φX,Z)η(Y )ξ−
−g(φY,Z)η(X)ξ + 2g(φX, Y )η(Z)ξ + g(X,Z)φY − g(Y,Z)φX+
+η(X)η(Z)φY − η(Y )η(Z)φX}+ β{g(X,Z)η(Y )ξ − g(Y, Z)η(X)ξ−
−2g(φX, Y )φZ − g(φX,Z)φY + g(φY,Z)φX − 2η(Y )η(Z)X+
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D-HOMOTHETIC DEFORMATION OF NORMAL ALMOST CONTACT METRIC MANIFOLDS 1339
+2η(X)η(Z)Y }] +
a− 1
a
(Xα)[g(Y, Z)−
−η(Y )η(Z)]ξ − a− 1
a
(Y α)[g(X,Z)− η(X)η(Z)]ξ +
a− 1
a
α[α{g(Y,Z)X−
−g(X,Z)Y + η(X)η(Z)Y − η(Y )η(Z)X}+
+β{g(X,Z)φY − g(Y,Z)φX + 2g(φX, Y )η(Z)ξ + g(φX,Z)η(Y )ξ−
−g(φY,Z)η(X)ξ + η(Y )η(Z)φX − η(X)η(Z)φY }]+
+(1− a)2[η(X)η(Z)φ2Y − η(Y )η(Z)φ2X]−
−(1− a)2
a
[α{g(φZ,X)η(Y )ξ − 2g(φX, Y )η(Z)ξ+
+g(Y,Z)φX − g(X,Z)φY + η(X)η(Z)φY − η(Y )η(Z)φX + g(φY,Z)η(X)ξ}]. (4.11)
Putting Y = Z = ξ in (4.11) and using (1.1) we obtain
R̄(X, ξ)ξ = R(X, ξ)ξ + 2(1− a)[β(φ2X)− α(φX)]− (1− a)2φ2X. (4.12)
Let {ei, φei, ξ}, i = 1, 2, . . . , n, be a local φ-basis at any point of the manifold. Then putting
Y = Z = ei in (4.11) and taking summation over i = 1 to n we obtain by virtue of η(ei) = 0,
n∑
i=1
R̄(X, ei)ei =
n∑
i=1
R(X, ei)ei−
−(1− a)[α(n− 1)(φX) + β{nη(X)ξ − 3X}] +
a− 1
a
(n− 1)(Xα)ξ+
+
a− 1
a
α2(n− 1)X − a− 1
a
αβ(n− 1)φX − (1− a)2
a
α(n− 1)φX. (4.13)
Again, putting Y = Z = φei in (4.11) and taking summation over i = 1 to n then using (1.1)
and η(ei) = 0, we obtain
n∑
i=1
R̄(X,φei)φei =
n∑
i=1
R(X,φei)φei−
−(1− a)[α(n− 1)(φX) + β{nη(X)ξ − 3X}] +
a− 1
a
(n− 1)(Xα)ξ+
+
a− 1
a
α2(n− 1)X − a− 1
a
αβ(n− 1)φX − (1− a)2
a
α(n− 1)φX. (4.14)
Adding (4.13) and (4.14) and using the definition of Ricci operator we have
Q̄X − R̄(X, ξ)ξ = QX −R(X, ξ)ξ − 2(1− a)[α{(n− 1)φX}+
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1340 U. C. DE, S. GHOSH
+β{nη(X)ξ − 3X}] +
2(a− 1)
a
(n− 1)(Xα)ξ +
2(a− 1)
a
α2(n− 1)X−
−2(a− 1)
a
αβ(n− 1)φX − 2(1− a)2
a
α(n− 1)φX. (4.15)
In view of (4.12) we get from (4.15)
S̄(X,Y ) = S(X,Y )− 2(1− a)[αng(φX, Y )−
−β{g(φ2X,Y ) + nη(X)η(Y )− 3g(X,Y )}]+
+
2(a− 1)
a
(n− 1)[(Xα)η(Y ) + α2g(X,Y )−
−αβg(φX, Y )− (a− 1)αg(φX, Y )], (4.16)
which implies that
Q̄X = QX − 2(1− a)[αnφX − β{φ2X + nη(X)ξ − 3X}]+
+
2(a− 1)
a
(n− 1)[(Xα)ξ + α2X − αβ(φX)− (a− 1)α(φX)]. (4.17)
Operating φ̄ = φ on both sides of (4.17) from the left we have
φ̄Q̄X = φQX − 2(1− a)[αn(φ2X) + 4β(φX)]+
+
2(a− 1)
a
(n− 1)[α2(φX)− αβ(φ2X)− (a− 1)α(φ2X)]. (4.18)
Again, putting φ̄X = φX in (4.17) we have
Q̄φ̄X = QφX − 2(1− a)[αn(φ2X) + 4β(φX)]+
+
2(a− 1)
a
(n− 1)[(φXα)ξ + α2(φX)− αβ(φ2X)− (a− 1)α(φ2X)]. (4.19)
Subtracting (4.18) and (4.19) we get
(φ̄Q̄− Q̄φ̄)X = (φQ−Qφ)X − 2(a− 1)
a
(n− 1)(φXα)ξ. (4.20)
Therefore we can state the following theorem.
Theorem 4.2. Under a D-homothetic deformation, the expression Qφ − φQ of a (2n + 1)-
dimensional normal almost contact metric manifold is invariant, provided α is constant.
In view of (4.20) we state the following corollary.
Corollary 4.1. Under a D-homothetic deformation, the expression Qφ−φQ of a 3-dimensional
normal almost contact metric manifold is invariant.
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D-HOMOTHETIC DEFORMATION OF NORMAL ALMOST CONTACT METRIC MANIFOLDS 1341
5. η-Einstein normal almost contact metric manifolds. Let M(φ, ξ, η, g) be a (2n + 1)-
dimensional η-Einstein normal almost contact metric manifold which reduces to M(φ̄, ξ̄, η̄, ḡ) under
a D-homothetic deformation. Then from (4.16) it follows by virtue of (3.7) that
S̄(X,Y ) = λ̄ḡ(X,Y ) + µ̄η̄(X)η̄(Y ) +
2(a− 1)
a2
(n− 1)(Xα)η̄(Y )−
−
[
2(1− a)
a
αn+
2(a− 1)
a2
αβ(n− 1) +
2(a− 1)2
a2
(n− 1)α
]
ḡ(φ̄X, Y ), (5.1)
where λ̄, µ̄ are smooth functions given by
λ̄ =
1
a
[ r
2n
+ ξα+ (α2 − β2)
]
− 8
(1− a)
a
β +
2(a− 1)
a2
(n− 1)α2 (5.2)
and
µ̄ = −a− 1
a
[ r
2n
+ ξα+ (α2 − β2)
]
− 1
a2
{ r
2n
+ (2n+ 1)(ξα+ α2 − β2)
}
+
+2β(n+ 1)
1− a
a2
− 8β
(a− 1)2
a
− 2α2(n− 1)
(a− 1)2
a2
. (5.3)
In view of the relation (5.1) we state the following theorem.
Theorem 5.1. Under a D-homothetic deformation, a (2n+ 1)-dimensional η-Einstein normal
almost contact metric manifold is invariant, provided α = 0.
6. φ-Sectional curvature of normal almost contact metric manifolds. In this section we con-
sider the φ-sectional curvature on a (2n+ 1)-dimensional normal almost contact metric manifold.
From (4.11) it can be easily seen that
K̄(X,φX)−K(X,φX) =
a− 1
a
[3aβ − α2] (6.1)
and hence we state the following theorem.
Theorem 6.1. Under a D-homothetic deformation, the φ-sectional curvature of a (2n + 1)-
dimensional normal almost contact metric manifold is invariant.
If a (2n + 1)-dimensional normal almost contact metric manifold M(φ̄, ξ̄, η̄, ḡ) satisfies
R(X,Y )ξ = 0 for all X,Y (for example the tangent sphere bundle of a flat Riemannian manifold
admits a contact metric structure with R(X,Y )ξ = 0), then it can be easily seen that K(X,φX) = 0
and hence from (6.1) it follows that
K̄(X,φX) =
a− 1
a2
[3aβ − α2] 6= 0
for a 6= 1 and α2 6= 3aβ, where X is a unit vector field orthogonal to ξ and K(X,φX) is the
φ-sectional curvature. This implies that the φ-sectional curvature K̄(X,φX) is non-vanishing and
non-constant for a 6= 1 and α2 6= 3aβ. Therefore, we state the following theorem.
Theorem 6.2. There exists (2n+ 1)-dimensional normal almost contact metric manifold (non-
Sasakian) with non-zero and non-constant φ-sectional curvature.
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1342 U. C. DE, S. GHOSH
7. Locally φ-Ricci symmetric three dimensional normal almost contact metric manifolds. In
this section we study locally φ-Ricci symmetry on a three dimensional normal almost contact metric
manifold.
Differentiating (4.17) covariantly with respect to W and using (2.3) we obtain
(∇W Q̄)(X) = (∇WQ)(X)− 2(1− a)(Wα)φX−
−2(1− a)α[α{g(φW,X)ξ − η(X)φW}+ β{g(W,X)ξ − η(X)W}]−
−(1− a)2(∇W η)(X)ξ − (1− a)2η(X)∇W ξ. (7.1)
Operating φ2 on both sides of (7.1) and taking X as an orthonormal vector to ξ we obtain
φ̄2(∇W Q̄)(X) = φ2(∇WQ)(X) + 2(1− a)(Wα)(φX). (7.2)
In view of the relation (7.2) we state the following theorem.
Theorem 7.1. Under a D-homothetic deformation a locally φ-Ricci symmetry on a three di-
mensional normal almost contact metric manifold is invariant, provided α = constant.
8. Example. We consider the three dimensional manifold M = {(x, y, z) ∈ R3, z 6= 0}, where
(x, y, z) are standard coordinate of R3. The vector fields
e1 = z
(
∂
∂x
+ y
∂
∂z
)
, e2 = z
∂
∂y
, e3 =
∂
∂z
are linearly independent at each point of M.
Let g be a Riemannian metric defined by
g(e1, e3) = g(e1, e2) = g(e2, e3) = 0,
g(e1, e1) = g(e2, e2) = g(e3, e3) = 1.
Let η be the 1-form defined by η(Z) = g(Z, e3) for any Z ∈ χ(M). Let φ be the (1, 1) tensor
field defined by
φ(e1) = e2, φ(e2) = −e1, φ(e3) = 0.
Then using the identity of φ and g, we have
η(e3) = 1,
φ2Z = −Z + η(Z)e3,
g(φZ, φW ) = g(Z,W )− η(Z)η(W )
for any Z,W ∈ χ(M). Then for e3 = ξ, the structure (φ, ξ, η, g) defines an almost contact metric
structure on M.
Let ∇ be the Levi – Civita connection with respect to the metric g. Then we have
[e1, e3] = ye2 − z2e3, [e1, e3] = −1
z
e1 and [e2, e3] = −1
z
e2.
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D-HOMOTHETIC DEFORMATION OF NORMAL ALMOST CONTACT METRIC MANIFOLDS 1343
The Riemannian connection ∇ of the metric g is given by
2g(∇XY,Z) = Xg(Y,Z) + Y g(Z,X)− Zg(X,Y )−
−g(X, [Y,Z])− g(Y, [X,Z]) + g(Z, [X,Y ]), (8.1)
which is known as Koszul’s formula. Using (8.1) we can easily calculate the following:
∇e1e3 = −1
z
e1 +
z2
2
e2, ∇e1e2 = −1
2
z2e3, ∇e1e1 =
1
z
e3,
∇e2e3 = −1
z
e2 −
1
2
z2e1, ∇e2e2 = ye1 +
1
z
e3, ∇e2e1 =
1
2
z2e3 − ye2, (8.2)
∇e3e3 = 0, ∇e3e2 = −1
2
z2e1, ∇e3e1 =
1
2
z2e2.
From (8.2) it can be easily seen that (φ, ξ, η, g) is a normal almost contact metric manifold with
α = −1
z
6= 0 and β = −1
2
z2 6= 0.
It is known that
R(X,Y )Z = ∇X∇Y Z −∇Y∇XZ −∇[X,Y ]Z. (8.3)
With the help of (8.3) and using (8.2) we can easily calculate
R(e1, e2)e1 =
(
3z4
4
+
1
z2
+ y2
)
e2 + (yz2)e3, R(e2, e1)e2 =
(
3z4
4
+
1
z2
+ y2
)
e1 +
y
z
e3,
R(e1, e3)e3 =
(
z4
4
− 2
z2
)
e1, R(e2, e3)e3 =
(
z4
4
− 2
z2
)
e2,
R(e3, e1)e1 =
(
z4
4
− 2
z2
)
e3 − (yz2)e2, R(e3, e2)e2 =
(
z4
4
− 2
z2
)
e3 −
y
z
e1.
From the above expressions of the curvature tensor we obtain
S(e1, e1) = g(R(e1, e2)e2, e1) + g(R(e1, e3)e3, e1) = −z
4
2
− 3
z2
− y2.
Similarly we have
S(e2, e2) = −z
4
2
− 3
z2
− y2 and S(e3, e3) =
z4
2
− 4
z2
.
Therefore
r = S(e1, e1) + S(e2, e2) + S(e3, e3) = −z
4
2
− 10
z2
− 2y2.
Now using (2.9) in (2.8) we get
g(R(X,Y )Z,W ) =
[r
2
+ ξα+ (α2 − β2)
]
[g(φY, φZ)g(X,W )−
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1344 U. C. DE, S. GHOSH
−g(φX, φZ)g(Y,W ) + g(φX, φW )g(Y,Z)− g(φY, φW )g(X,Z)]−
−{Xα+ (φX)β}[g(Y,Z)η(W )− g(Y,W )η(Z)−
−{Y α+ (φY )β}[g(X,W )η(Z)− g(X,Z)η(W )]−
−{Wα+ (φW )β}[g(Y, Z)η(X)− g(X,Z)η(Y )]−
−2(α2 − β2)[g(X,W )η(Y )η(Z)− g(Y,W )η(X)η(Z)+
+g(Y,Z)η(X)η(W )− g(X,Z)η(Y )η(W )]−
−r
2
[g(Y,Z)g(X,W )− g(X,Z)g(Y,W )].
In view of the above relation we get
K(e1, φe1) = K(e2, φe2) = 2(β2 − α2)− 2(ξα)− r
2
.
Now, in this example we have
K(e1, φe1) = g(R(e1, φe1)e1, φe1) = g(R(e1, e2)e1, e2) =
=
3z4
4
+
1
z2
+ y2 = 2(β2 − α2)− 2(ξα)− r
2
.
Similarly we have
K(e2, φe2) =
3z4
4
+
1
z2
+ y2 = 2(β2 − α2)− 2(ξα)− r
2
.
Again from (4.11) it can be easily shown that
K̄(e1, φe1) =
3z4
4
+
1
z2
+ y2 +
a− 1
a
(3αβ − α2) =
= K(e1, φe1) +
a− 1
a
(
−3az2
2
−
(
−1
z
)2
)
,
which implies that
K̄(e1, φe1)−K(e1, φe1) =
a− 1
a
(3aβ − α2).
Similarly, we have
K̄(e2, φe2)−K(e2, φe2) =
a− 1
a
(3aβ − α2).
Therefore such a normal almost contact metric manifold satisfies the relation (6.1) and hence Theorem
6.1 is verified.
1. Blair D. E. Contact manifolds in Riemannian geometry // Lect. Notes Math. – 1976. – 509.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10
D-HOMOTHETIC DEFORMATION OF NORMAL ALMOST CONTACT METRIC MANIFOLDS 1345
2. Blair D. E. Riemannian geometry of contact and symplectic manifolds // Progr. Math. – 2002. – 203.
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P. 47 – 52.
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9. Takahashi T. Sasakian φ-symmetric spaces // Tohoku Math. J. – 1977. – 29. – P. 91 – 113.
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Received 23.07.11,
after revision — 19.02.12
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 10
|
| id | umjimathkievua-article-2662 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:27:51Z |
| publishDate | 2012 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/01/e3c9a4f5a7fda65afac75472062ccc01.pdf |
| spelling | umjimathkievua-article-26622020-03-18T19:32:22Z D-homothetic deformation of normal almost contact metric manifolds $D$-гомотетична деформацiя нормальних майже контактних многовидiв De, U. C. Ghosh, S. Де, У. К. Хост, С. The object of the present paper is to study a transformation called the $D$-homothetic deformation of normal almost contact metric manifolds. In particular, it is shown that, in a $(2n + 1)$-dimensional normal almost contact metric manifold, the Ricci operator $Q$ commutes with the structure tensor $\phi$ under certain conditions, and the operator $Q\phi - \phi Q$ is invariant under a $D$-homothetic deformation. We also discuss the invariance of $\eta$-Einstein manifolds, $\phi$-sectional curvature, and the local $\phi$-Ricci symmetry under a $D$-homothetic deformation. Finally, we prove the existence of such manifolds by a concrete example. Метою цiєї статтi є вивчення перетворення, що називається D-гомотетичною деформацiєю нормальних майже контактних многовидiв. Зокрема, показано, що у $(2n + 1)$-вимiрному нормальному майже контактному многовидi оператор Рiччi $Q$ комутує за певних умов iз структурним тензором $\phi$, а оператор $Q\phi - \phi Q$ є iнварiантним щодо $D$-гомотетичної деформацiї. Також розглянуто питання про iнварiантнiсть $\eta$-ейнштейнiвських многовидiв, $\phi$-секцiйну кривину та локальну $\phi$-симетрiю Рiччi при $D$-гомотетичнiй деформацiї. Iснування таких многовидiв доведено на конкретному прикладi. Institute of Mathematics, NAS of Ukraine 2012-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2662 Ukrains’kyi Matematychnyi Zhurnal; Vol. 64 No. 10 (2012); 1330-1329 Український математичний журнал; Том 64 № 10 (2012); 1330-1329 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2662/2083 https://umj.imath.kiev.ua/index.php/umj/article/view/2662/2084 Copyright (c) 2012 De U. C.; Ghosh S. |
| spellingShingle | De, U. C. Ghosh, S. Де, У. К. Хост, С. D-homothetic deformation of normal almost contact metric manifolds |
| title | D-homothetic deformation of normal almost contact metric manifolds |
| title_alt | $D$-гомотетична деформацiя нормальних майже контактних многовидiв |
| title_full | D-homothetic deformation of normal almost contact metric manifolds |
| title_fullStr | D-homothetic deformation of normal almost contact metric manifolds |
| title_full_unstemmed | D-homothetic deformation of normal almost contact metric manifolds |
| title_short | D-homothetic deformation of normal almost contact metric manifolds |
| title_sort | d-homothetic deformation of normal almost contact metric manifolds |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2662 |
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