Contact $CR$-warped product of submanifolds of the generalized Sasakian space forms admitting the nearly trans-Sasakian structure
In the present paper, we apply Hopf’s lemma to the contact $CR$-warped product of submanifolds of the generalized Sasakian space forms admitting nearly trans-Sasakian structure and establish a characterization inequality for the existence of these types of warped products. This inequality generalize...
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2018
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507467297849344 |
|---|---|
| author | Meraj, Ali Khan Мерадж, Алі Хан |
| author_facet | Meraj, Ali Khan Мерадж, Алі Хан |
| author_sort | Meraj, Ali Khan |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:21:55Z |
| description | In the present paper, we apply Hopf’s lemma to the contact $CR$-warped product of submanifolds of the generalized Sasakian space forms admitting nearly trans-Sasakian structure and establish a characterization inequality for the existence of these
types of warped products. This inequality generalizes the inequalities obtained in [M. Atceken, Bull. Iran. Math. Soc. –
2013. – 39, № 3. – P. 415 – 429; M. Atceken, Collect. Math. – 2011. – 62, № 1. – P. 17 – 26, and Sibel Sular, Cihan O¨ zgu¨r,
Turkish J. Math. – 2012. – 36. – P. 485 – 497]. Moreover, we also compute another inequality for the squared norm of the
second fundamental form in terms of warping functions. This inequality is a generalization of the inequalities acquired in
[I. Mihai, Geom. Dedicata. – 2004. – 109. – P. 165 – 173 and K. Arslan, R. Ezentas, I. Mihai, C. Murathan, J. Korean Math.
Soc. – 2005. – 42, № 5. – P. 1101 – 1110]. The inequalities proved in the paper either generalize or improve all inequalities
available in the literature and related to the squared norm of the second fundamental form for contact CR-warped product
of submanifolds of any almost contact metric manifold. |
| first_indexed | 2026-03-24T02:09:46Z |
| format | Article |
| fulltext |
UDC 513.944
Meraj Ali Khan (Dep. Math., Univ. Tabuk, Kingdom of Saudi Arabia)
CONTACT CR-WARPED PRODUCT OF SUBMANIFOLDS
OF THE GENERALIZED SASAKIAN SPACE FORMS
ADMITTING THE NEARLY TRANS-SASAKIAN STRUCTURE
КОНТАКТНИЙ CR-ВИКРИВЛЕНИЙ ДОБУТОК ПIДМНОГОВИДIВ
УЗАГАЛЬНЕНИХ ПРОСТОРОВИХ ФОРМ САСАКI,
ЩО ДОПУСКАЮТЬ МАЙЖЕ ТРАНССТРУКТУРУ САСАКI
In the present paper, we apply Hopf’s lemma to the contact CR-warped product of submanifolds of the generalized Sasakian
space forms admitting nearly trans-Sasakian structure and establish a characterization inequality for the existence of these
types of warped products. This inequality generalizes the inequalities obtained in [M. Atceken, Bull. Iran. Math. Soc. –
2013. – 39, № 3. – P. 415 – 429; M. Atceken, Collect. Math. – 2011. – 62, № 1. – P. 17 – 26, and Sibel Sular, Cihan Özgür,
Turkish J. Math. – 2012. – 36. – P. 485 – 497]. Moreover, we also compute another inequality for the squared norm of the
second fundamental form in terms of warping functions. This inequality is a generalization of the inequalities acquired in
[I. Mihai, Geom. Dedicata. – 2004. – 109. – P. 165 – 173 and K. Arslan, R. Ezentas, I. Mihai, C. Murathan, J. Korean Math.
Soc. – 2005. – 42, № 5. – P. 1101 – 1110]. The inequalities proved in the paper either generalize or improve all inequalities
available in the literature and related to the squared norm of the second fundamental form for contact CR-warped product
of submanifolds of any almost contact metric manifold.
У роботi лему Хопфа застосовано до контактного CR-викривленого добутку пiдмноговидiв узагальнених просто-
рових форм Сасакi, що допускають майже трансструктуру Сасакi, та встановлено характеризацiйну нерiвнiсть
щодо iснування викривлених добуткiв такого типу. Ця нерiвнiсть узагальнює нерiвностi, що були встановленi в
[M. Atceken, Bull. Iran. Math. Soc. – 2013. – 39, № 3. – P. 415 – 429; M. Atceken, Collect. Math. – 2011. – 62, № 1. –
P. 17 – 26 та Sibel Sular, Cihan Özgür, Turkish J. Math. – 2012. – 36. – P. 485 – 497]. Крiм того, отримано iншу нерiвнiсть
для квадрата норми другої фундаментальної форми в термiнах викривляючих функцiй. Ця нерiвнiсть узагальнює
нерiвностi, що були встановленi в [I. Mihai, Geom. Dedicata. – 2004. – 109. – P. 165 – 173 та K. Arslan, R. Ezentas,
I. Mihai, C. Murathan, J. Korean Math. Soc. – 2005. – 42, № 5. – P. 1101 – 1110]. Нерiвностi, що доведенi в роботi,
узагальнюють або полiпшують усi нерiвностi, доступнi в лiтературi, що вiдносяться до квадрата норми другої
фундаментальної форми для контактного CR-викривленого добутку пiдмноговидiв будь-якого майже контактного
метричного многовиду.
1. Introduction. Gray and Hervella classified the almost Hermitian manifolds [2], in this classifi-
cation there exists a class \scrW 4 of almost Hermitian manifolds, which is closely related to a locally
conformal Kaehler manifold. An almost contact metric structure on a manifold \=M is called a trans-
Sasakian structure if the product manifold \=M \times R belongs to class \scrW 4 [17]. The class \scrC 6 \oplus \scrC 5
coincides with the class of trans-Sasakian structure of the type (\alpha , \beta ). This trans-Sasakian structure
is cosymplectic or Sasakian or Kenmotsu if \alpha = 0, \beta = 0, or \beta = 0 or \alpha = 0. Later on, D. Chinea
and C. Gonzalez [11] generalized these structures actually they divided almost contact structure into
twelve different classes. An almost contact metric manifold is nearly trans-Sasakian manifold if it
is associated to the class \scrC 1 \oplus \scrC 5 \oplus \scrC 6. Recently, C. Gherghe [7] introduced a nearly trans-Sasakian
structure of the type (\alpha , \beta ) which is the generalization of the trans-Sasakian manifold of the type
(\alpha , \beta ). Moreover, if \beta = 0 or \alpha = 0 or \alpha = \beta = 0, then nearly trans-Sasakian structure of the type
(\alpha , \beta ) becomes nearly Sasakian [9] or nearly Kenmotsu [18] or nearly cosymplectic [8], respectively.
On the other hand the notion of CR-warped product submanifolds as a natural generalization of
CR-products was introduced by B. Y. Chen (see [3, 5]). Basically, Chen obtained some basic results
c\bigcirc MERAJ ALI KHAN, 2018
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 10 1417
1418 MERAJ ALI KHAN
for CR-warped product submanifolds of Kaehler manifolds and established a sharp relationship be-
tween the warping function f and squared norm of the second fundamental form. Later, I. Hesigawa
and I. Mihai proved a similar inequality for contact CR-warped product submanifolds of Sasakian
manifolds [14]. Moreover, I. Mihai in [15] improved same inequality for contact CR-warped product
submanifolds of Sasakian space form. Furthermore, in [19] K. Arslan et al. established a sharp esti-
mation for contact CR-warped product submanifolds in the setting of Kenmotsu space forms. Many
geometers obtained similar estimation for different setting of almost contact metric manifolds (see
references).
In the other direction, M. Ateceken [20, 21], Sibel Sular and Cihan Özgür [26] proved the
characterizing inequalities for existence the contact CR-warped product submanifolds of cosymplectic
space forms, Kenmotsu space forms and generalized Sasakian space forms admitting a trans-Sasakian
structure.
In the present study, we consider contact CR-warped product submanifolds of nearly trans-
Sasakian generalized Sasakian space forms and obtain a characterizing inequality for the squared
norm of the second fundamental form. Finally, we also establish a sharp inequality for squared norm
of the second fundamental form in terms of warping function. Our inequalities generalize or improve
all the inequalities for contact CR-warped products in any contact metric manifold.
2. Preliminaries. A (2n + 1)-dimensional C\infty -manifold \=M is said to have an almost contact
structure if there exist on \=M a tensor field \phi of type (1, 1) a vector field \xi and a 1-form \eta satisfying
[10]
\phi 2 = - I + \eta \otimes \xi , \phi \xi = 0, \eta \circ \phi = 0, \eta (\xi ) = 1.
There always exists a Riemannian metric g on an almost contact metric manifold \=M satisfying the
conditions
\eta (X) = g(X, \xi ), g(\phi X, \phi Y ) = g(X,Y ) - \eta (X)\eta (Y )
for all X,Y \in T \=M.
An almost contact structure (\phi , \xi , \eta ) is said to be normal if the almost complex structure J on
the product manifold \=M \times R given by
J
\biggl(
X, f
d
dt
\biggr)
=
\biggl(
\phi X - f\xi , \eta (X)
d
dt
\biggr)
,
where f is a C\infty -function on \=M \times R, has no torsion, that is J is integrable and the condition for
normality in terms of \phi , \xi and \eta is [\phi , \phi ] + 2d\eta \otimes \xi on \=M, where [\phi , \phi ] is the Nijenhuis tensor of
\phi . Finally, the fundamental 2-form \Phi is defined by \Phi (X,Y ) = g(X,\phi Y ).
An almost contact metric manifold is said to be trans-Sasakian manifold if [7]
( \=\nabla X\phi )Y = \alpha (g(X,Y )\xi - \eta (Y )X) + \beta (g(\phi X, Y )\xi - \eta (Y )\phi X) (2.1)
for all X,Y \in T \=M.
An almost contact metric manifold is said to be nearly trans-Sasakian manifold if
( \=\nabla X\phi )Y + ( \=\nabla Y \phi )X = \alpha (2g(X,Y ) - \eta (Y )X - \eta (X)Y ) - \beta (\eta (Y )\phi X + \eta (X)\phi Y (2.2)
for all X,Y \in T \=M.
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CONTACT CR-WARPED PRODUCT OF SUBMANIFOLDS OF THE GENERALIZED SASAKIAN SPACE . . . 1419
Given an almost contact metric manifold \=M, it is said to be a generalized Sasakian space form
[23] if there exist three functions f1, f2 and f3 on \=M such that
\=R(X,Y )Z = f1\{ g(Y,Z)X - g(X,Z)Y \} + f2\{ g(X,\phi Z)\phi Y - g(Y, \phi Z)\phi X+
+2g(X,\phi Y )\phi Z\} + f3\{ \eta (X)\eta (Z)Y - \eta (Y )\eta (Z)X + g(X,Z)\eta (Y )\xi - g(Y, Z)\eta (X)\xi \} (2.3)
for any vector fields X,Y, Z on \=M, where \=R denotes the curvature tensor of \=M. If f1 =
c+ 3
4
,
f2 = f3 =
c - 1
4
, then \=M is Sasakian space form [10]; if f1 =
c - 3
4
, f2 = f3 =
c+ 1
4
, then \=M is
a Kenmotsu space form [18]; if f1 = f2 = f3 =
c
4
, then \=M is a cosymplectic space form [23].
Let M be a submanifold of an almost contact metric manifold \=M with induced metric g, and if
\nabla and \nabla \bot are the induced connection on the tangent bundle TM and the normal bundle T\bot M of
M, respectively, then the Gauss and Weingarten formulae are given by
\=\nabla XY = \nabla XY + h(X,Y ), (2.4)
\=\nabla XN = - ANX +\nabla \bot
XN, (2.5)
for each X,Y \in TM and N \in T\bot M, where h and AN are the second fundamental form and the
shape operator, respectively, for the immersion of M in \=M, they are related as
g(h(X,Y ), N) = g(ANX,Y ), (2.6)
where g denotes the Riemannian metric on \=M as well as on M.
The mean curvature vector H of M is given by
H =
1
n
n\sum
i=1
h(ei, ei),
where n is the dimension of M and \{ e1, e2, . . . , en\} is a local orthonormal frame of vector fields on
M. The squared norm of the second fundamental form is defined as
\| h\| 2 =
n\sum
i,j=1
g(h(ei, ej), h(ei, ej)). (2.7)
A submanifold M of \=M is said to be a totally geodesic submanifold if h(X,Y ) = 0, for each
X,Y \in TM, and totally umbilical submanifold if h(X,Y ) = g(X,Y )H.
For any X \in TM, we write
\phi X = PX + FX, (2.8)
where PX is the tangential component and FX is the normal component of \phi X.
Similarly, for N \in T\bot M, we can write
\phi N = tN + fN, (2.9)
where tN and fN are the tangential and normal components of \phi N, respectively.
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1420 MERAJ ALI KHAN
The covariant differentiation of the tensors \phi , P, F, t and f are defined as respectively
( \=\nabla X\phi )Y = \=\nabla X\phi Y - \phi \=\nabla XY, (2.10)
( \=\nabla XP )Y = \nabla XPY - P\nabla XY, (2.11)
( \=\nabla XF )Y = \nabla \bot
XFY - F\nabla XY, (2.12)
( \=\nabla Xt)N = \nabla XtN - t\nabla \bot
XN, (2.13)
( \=\nabla Xf)N = \nabla \bot
XfN - f\nabla \bot
XN. (2.14)
Furthermore, for any X,Y \in TM, the tangential and normal parts of ( \=\nabla X\phi )Y are denoted by \scrP XY
and \scrQ XY, i.e.,
( \=\nabla X\phi )Y = \scrP XY +\scrQ XY. (2.15)
On using equations (2.4) – (2.12) and (2.15), we may obtain
\scrP XY = ( \=\nabla XP )Y - AFYX - th(X,Y ),
\scrQ XY = ( \=\nabla XF )Y + h(X,TY ) - fh(X,Y ).
Similarly, for N \in T\bot M, denoting by \scrP XN and \scrQ XN respectively the tangential and normal parts
of ( \=\nabla X\phi )N, we find
\scrP XN = ( \=\nabla Xt)N + PANX - AfNX,
\scrQ XN = ( \=\nabla Xf)N + h(tN,X) + FANX.
On a submanifold M of a nearly trans-Sasakian manifold by (2.1) and (2.15)
\scrP XY + \scrP YX = \alpha (2g(X,Y )\xi - \eta (Y )X - \eta (X)Y ) - \beta (\eta (Y )PX + \eta (X)PY ), (2.16a)
\scrQ XY +\scrQ YX = - \beta (\eta (Y )FX + \eta (X)FY ) (2.16b)
for any X,Y \in TM.
An m-dimensional Riemannian submanifold M of an almost contact metric manifold \=M, where
\xi is tangent to M, is called contact CR-submanifold if it admits an invariant distribution D whose
orthogonal complementary distribution D\bot is anti invariant, that is
TM = D \oplus D\bot \oplus \langle \xi \rangle ,
where \phi D \subseteq D, \phi D\bot \subseteq T\bot M and \langle \xi \rangle denotes 1-dimensional distribution which is spanned by \xi .
If \mu is the invariant subspace of the normal bundle T\bot M, then in the case of contact CR-
submanifold, the normal bundle T\bot M can be decomposed as follows:
T\bot M = \mu \oplus \phi D\bot .
A contact CR-submanifold M is called the contact CR-product submanifold if the distributions
D and D\bot are parallel on M. In this case M is foliated by the leaves of these distributions. In
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 10
CONTACT CR-WARPED PRODUCT OF SUBMANIFOLDS OF THE GENERALIZED SASAKIAN SPACE . . . 1421
general, if N1 and N2 are Riemannian manifolds with Riemannian metrics g1 and g2 respectively,
then the product manifold (N1\times N2, g) is a Riemannian manifold with Riemannian metric g defined
as
g(X,Y ) = g1(d\pi 1X, d\pi 1Y ) + g2(d\pi 2X, d\pi 2Y ),
where \pi 1 and \pi 2 are the projection maps of M onto N1 and N2, respectively, and d\pi 1, d\pi 2 are
their differentials.
As a generalization of the product manifold and in particular of contact CR-product submanifold,
one can consider the warped product of manifolds which are defined as follows.
Definition 2.1. Let (B, gB) and (C, gC) be two Riemannian manifolds with Riemannian metric
gB and gC , respectively, and f be a positive differentiable function on B. The warped product of B
and C is the Riemannian manifold (B \times C, g), where
g = gB + f2gC .
For a warped product manifold N1 \times f N2, we denote by D1 and D2 the distributions defined
by the vectors tangent to the leaves and fibers, respectively. In other words, D1 is obtained by the
tangent vectors of N1 via the horizontal lift and D2 is obtained by the tangent vectors of N2 via
vertical lift. In case of contact CR-warped product submanifolds D1 and D2 are replaced by D and
D\bot , respectively.
The warped product manifold (B \times C, g) is denoted by B \times f C. If X is the tangent vector field
to M = B \times f C at (p, q), then
\| X\| 2 = \| d\pi 1X\| 2 + f2(p)\| d\pi 2X\| 2.
R. L. Bishop and B. O’Neill [24] proved the following theorem.
Theorem 2.1. Let M = B\times f C be warped product manifolds. If X,Y \in TB and V,W \in TC,
then:
(i) \nabla XY \in TB,
(ii) \nabla XV = \nabla VX =
\biggl(
Xf
f
\biggr)
V,
(iii) \nabla VW = \nabla C
VW - g(V,W )\nabla \mathrm{l}\mathrm{n} f.
From above theorem, for the warped product M = B \times f C it is easy to conclude that
\nabla XV = \nabla VX = (X\mathrm{l}\mathrm{n} f)V (2.17)
for any X \in TB and V \in TC.
\nabla f is the gradient of f and is defined as
g(\nabla f,X) = Xf (2.18)
for all X \in TM.
Corollary 2.1. On a warped product manifold M = N1 \times f N2, the following statements hold:
(i) N1 is totally geodesic in M,
(ii) N2 is totally umbilical in M.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 10
1422 MERAJ ALI KHAN
In what follows, N\bot and NT will denote a anti-invariant and invariant submanifold, respectively,
of an almost contact metric manifold \=M.
A warped product manifold is said to be trivial if its warping function f is constant. More
generally, a trivial warped product manifold M = N1 \times N2 is a Riemannian product N1 \times Nf
2 ,
where Nf
2 is the manifold with the Riemannian metric f2g2 which is homothetic to the original
metric g2 of N2. For example, a trivial contact CR-warped product is contact CR-product.
Let M be a m-dimensional Riemannian manifold with Riemannian metric g and let \{ e1, . . . , em\}
be an orthogonal basis of TM. As a consequence of (2.18), we have
\| \nabla f\| 2 =
m\sum
i=1
(ei(f))
2.
The Laplacian of f is defined by
\Delta f =
m\sum
i=1
\{ (\nabla eiei)f - eieif\} . (2.19)
Now, we state the Hopf’s lemma.
Hopf’s lemma [5]. Let M be a n-dimensional connected compact Riemannian manifold. If \psi is
differentiable function on M such that \Delta \psi \geq 0 everywhere on M (or \Delta \psi \leq 0 everywhere on M),
then \psi is a constant function.
3. Contact CR-warped product submanifolds. In this section we consider contact CR-warped
product of the type NT \times f N\bot of the nearly trans-Sasakian manifolds \=M, where NT and N\bot are the
invariant and anti-invariant submanifolds respectively of \=M. Throughout, this section, we consider
\xi tangent to NT .
Now we have some fundamental results in the following lemma for later use.
Lemma 3.1 [1]. Let M = NT \times f N\bot be a contact CR-warped product submanifold of a nearly
trans-Sasakian manifold \=M such that NT and N\bot are invariant and anti-invariant submanifolds of
\=M, respectively. Then we have:
(i) \xi \mathrm{l}\mathrm{n} f = \beta ,
(ii) g(h(X,Y ), \phi Z) = 0,
(iii) g(h(X,Z), \phi Z) = - \{ (\phi X \mathrm{l}\mathrm{n} f) + \alpha \eta (X)\} \| Z\| 2,
(iv) g(h(\xi , Z), \phi Z) = - \alpha \| Z\| 2,
(v) g(h(\phi X,Z), \phi Z) = (X \mathrm{l}\mathrm{n} f - \beta \eta (X))\| Z\| 2
for any X \in TNT and Z \in TN\bot .
Now, we have the following lemma.
Lemma 3.2. Let M = NT \times f N\bot be a contact CR-warped product submanifold of a nearly
trans-Sasakian manifold \=M. Then
g(h(\phi X,Z), \phi h(X,Z)) = \| h\mu (X,Z)\| 2 - g(\phi h(X,Z),\scrQ XZ)
for any X \in TNT and Z \in TN\bot .
Proof. By (2.4) and (2.10)
h(\phi X,Z) = ( \=\nabla Z\phi )X + \phi \nabla ZX + \phi h(X,Z) - \nabla Z\phi X.
Thus by using (2.15) and (2.17)
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CONTACT CR-WARPED PRODUCT OF SUBMANIFOLDS OF THE GENERALIZED SASAKIAN SPACE . . . 1423
h(\phi X,Z) = \scrP ZX +\scrQ ZX +X \mathrm{l}\mathrm{n} f\phi Z + \phi h(X,Z) - \phi X \mathrm{l}\mathrm{n} fZ.
Comparing normal parts
h(\phi X,Z) = \scrQ ZX +X \mathrm{l}\mathrm{n} f\phi Z + \phi h\mu (X,Z)
or
g(h(\phi X,Z), \phi h(X,Z)) = g(\scrQ ZX,\phi h(X,Z)) + \| h\mu (X,Z)\| 2.
By using (2.16b), we get
g(h(\phi X,Z), \phi h(X,Z)) = \| h\mu (X,Z)\| 2 - g(\phi h(X,Z),\scrQ XZ).
Next we prove the following characterization theorem.
Theorem 3.1. Let M = NT \times fN\bot be a contact CR-warped product submanifold of generalized
Sasakian space form \=M(f1, f2, f3) admitting the nearly trans-Sasakian structure such that NT is
connected and compact. Then M is contact CR-product submanifold if either one of the following
inequality holds:
(i)
\sum p
i=1
\sum q
j=1
\| h\mu (ei, ej)\| 2 \geq f2pq +
\sum p
i=1
\sum q
j=1
\| \scrQ eie
j\| 2,
(ii)
\sum p
i=1
\sum q
j=1
\| h\mu (ei, ej)\| 2 \leq f2pq,
where h\mu denotes the component of h in \mu , p+ 1 and q are the dimensions of NT and N\bot .
Proof. For any unit vector fields X tangent to NT and orthogonal to \xi and Z tangent to N\bot .
Then from (2.3) we have
\=R(X,\phi X,Z, \phi Z) = - 2f2g(X,X)g(Z,Z). (3.1)
On the other hand by Coddazi equation
\=R(X,\phi X,Z, \phi Z) = g(\nabla \bot
Xh(\phi X,Z), \phi Z) - g(h(\nabla X\phi X,Z), \phi Z) -
- g(h(\phi X,\nabla XZ), \phi Z) - g(\nabla \bot
\phi Xh(X,Z), \phi Z)+
+g(h(\nabla \phi XX,Z), \phi Z) + g(h(X,\nabla \phi XZ), \phi Z). (3.2)
By using part (iii) of Lemma 3.1, (2.10), (2.4) and (2.15), we get
g(\nabla \bot
Xh(\phi X,Z), \phi Z) = Xg(h(\phi X,Z), \phi Z) - g(h(\phi X,Z), \=\nabla X\phi Z) =
= X(X \mathrm{l}\mathrm{n} fg(Z,Z)) - g(h(\phi X,Z), ( \=\nabla X\phi )Z + \phi \=\nabla XZ).
On further simplification above equation yields
g(\nabla \bot
Xh(\phi X,Z), \phi Z) = X2 \mathrm{l}\mathrm{n} fg(Z,Z) + 2(X \mathrm{l}\mathrm{n} f)2g(Z,Z) - g(h(\phi X,Z),\scrQ XZ) -
- g(h(\phi X,Z), \phi h(X,Z)) - X \mathrm{l}\mathrm{n} fg(h(\phi X,Z), \phi Z).
Utilizing the part (v) of Lemma 3.1 and Lemma 3.2, we have
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 10
1424 MERAJ ALI KHAN
g(\nabla \bot
Xh(\phi X,Z), \phi Z) = X2 \mathrm{l}\mathrm{n} fg(Z,Z) + (X \mathrm{l}\mathrm{n} f)2g(Z,Z) - \| h\mu (X,Z)\| 2 -
- g(\phi h(X,Z) - h(\phi X,Z),\scrQ XZ).
Further, using (2.4), (2.15), (2.16b) and (2.17) in the last term of above equation, we obtain
g(\nabla \bot
Xh(\phi X,Z), \phi Z) = X2 \mathrm{l}\mathrm{n} fg(Z,Z) + (X \mathrm{l}\mathrm{n} f)2g(Z,Z) - \| h\mu (X,Z)\| 2 + \| \scrQ XZ\| 2. (3.3)
In the same way, we can calculate
- g(\nabla \bot
\phi Xh(X,Z), \phi Z) = (\phi X)2 \mathrm{l}\mathrm{n} fg(Z,Z) + (\phi X \mathrm{l}\mathrm{n} f)2g(Z,Z) -
- \| h\mu (\phi X,Z)\| 2 + \| \scrQ \phi XZ\| 2. (3.4)
From part (iii) of Lemma 3.1, we get
g(A\phi ZZ, \phi X) = X \mathrm{l}\mathrm{n} f,
replacing X by \nabla XX
g(A\phi ZZ, \phi \nabla XX) = \nabla XX \mathrm{l}\mathrm{n} f.
By using the Gauss formula in preceding equation, we have
g(A\phi ZZ, \phi ( \=\nabla XX - h(X,X)) = \nabla XX \mathrm{l}\mathrm{n} f. (3.5)
By use of (2.4), (2.10), (2.2) and (2.17), it is straightforward to see that h(X,X) \in \mu , applying this
fact in (3.5), we obtain
g(A\phi ZZ, \=\nabla X\phi X - ( \=\nabla X\phi )X) = \nabla XX \mathrm{l}\mathrm{n} f.
In view of (2.2) the previous equation abridged to
g(h(\nabla X\phi X,Z), \phi Z) = \nabla XX \mathrm{l}\mathrm{n} fg(Z,Z). (3.6)
Correspondingly,
g(h(\nabla \phi XX,Z), \phi Z) = - \nabla \phi X\phi X \mathrm{l}\mathrm{n} fg(Z,Z). (3.7)
By use of (2.17) and part (iii) of Lemma 3.1, it is simple to see the following:
g(h(\phi X,\nabla XZ), \phi Z) = (X \mathrm{l}\mathrm{n} f)2g(Z,Z) (3.8)
and
g(h(X,\nabla \phi XZ), \phi Z) = - (\phi X \mathrm{l}\mathrm{n} f)2g(Z,Z). (3.9)
Substituting (3.3), (3.4), (3.6), (3.7), (3.8) and (3.9) in (3.2), we find
\=R(X,\phi X,Z, \phi Z) = X2 \mathrm{l}\mathrm{n} fg(Z,Z) + (\phi X)2 \mathrm{l}\mathrm{n} fg(Z,Z) - \nabla XX \mathrm{l}\mathrm{n} fg(Z,Z) -
- \nabla \phi X\phi Xg(Z,Z) - \| h\mu (X,Z)\| 2 - \| h(\phi X,Z)\| 2 + \| \scrQ XZ\| 2 + \| \scrQ \phi XZ\| 2. (3.10)
Let \{ e0 = \xi , e1, e2, . . . , ep/2, \phi e1, \phi e2, . . . , ep = \phi ep/2,e
1, e2, . . . , eq\} be an orthonormal frame of
TM such that \{ e0, e1, . . . , ep/2, \phi e1, \phi e2, . . . , \phi ep/2\} are tangent to TNT and \{ e1, e2, . . . , eq\} are
tangent to TN\bot .
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CONTACT CR-WARPED PRODUCT OF SUBMANIFOLDS OF THE GENERALIZED SASAKIAN SPACE . . . 1425
By using (3.1) and (2.19) in (3.10) and summing over i = 1, 2, . . . , p and j = 1, 2, . . . , q, we get
q\Delta \mathrm{l}\mathrm{n} f = f2pq -
p\sum
i=1
q\sum
j=1
\| h\mu (ei, ej)\| 2 +
p\sum
i=1
q\sum
j=1
\| \scrQ eie
j\| 2. (3.11)
From Hopf’s lemma and (3.11), if
p\sum
i=1
q\sum
j=1
\| h\mu (ei, ej)\| 2 \geq f2pq +
p\sum
i=1
\| \scrQ eie
j\| 2
or
p\sum
i=1
q\sum
j=1
\| h\mu (ei, ej)\| 2 \leq f2pq,
then the warping function f is constant on M, i.e., M is simply a contact CR-product submanifold,
which proves the theorem completely.
Now we have the following corollary, which can be confirmed straightforwardly.
Corollary 3.1. Let M = NT \times f N\bot be a contact CR-warped product submanifolds of a genera-
lized Sasakian space form \=M(f1, f2, f3) admitting the nearly trans-Sasakian structure such that NT
is connected and compact. Then M is contact CR-product if and only if
p\sum
i=1
q\sum
j=1
\| h\mu (ei, ej)\| 2 = f2pq +
p\sum
i=1
q\sum
j=1
\| \scrQ eie
j\| 2,
where h\mu denotes the component of h in \mu , p+ 1 and q are the dimensions of NT and N\bot .
Furthermore, if the ambient manifold \=M is a generalized Sasakian manifolds with trans-Sasakian
structure, then from above findings we have the following corollary.
Corollary 3.2. Let M = NT \times f N\bot be a contact CR-warped product submanifold of a genera-
lized Sasakian space form \=M(f1, f2, f3) admitting the trans-Sasakian structure such that NT is
connected and compact. Then M is contact CR-product submanifold if either one of the inequality
p\sum
i=1
q\sum
j=1
\| h\mu (ei, ej)\| 2 \geq f2pq,
or
p\sum
i=1
q\sum
j=1
\| h\mu (ei, ej)\| 2 \leq f2pq
holds, where h\mu denotes the component of h in \mu , p+ 1 and q are the dimensions of NT and N\bot .
Remark 3.1. In the above corollary, the first characterizing inequality was also proved by Sibel
Sular and Cihan Özgür in [26]. In particualr the above inequalities also generalize the results obtained
in [20, 21].
Corollary 3.3. Let M = NT \times f N\bot be a contact CR-warped product submanifolds of a genera-
lized Sasakian space form \=M(f1, f2, f3) admitting the trans-Sasakian structure such that NT is
compact. Then M is contact CR-product if and only if
p\sum
i=1
q\sum
j=1
\| h\mu (ei, ej)\| 2 = f2pq,
where h\mu denotes the component of h in \mu , p+ 1 and q are the dimensions of NT and N\bot .
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 10
1426 MERAJ ALI KHAN
Remark 3.2. Similar consequence in Corollary 3.3 was also proved in [26].
4. Another inequality. In the present section, we estimate the squared norm of the second
fundamental form in terms of warping function.
Theorem 4.1. Let \=M(f1, f2, f3) be a (2n + 1)-dimensional generalized Sasakian space form
admitting the nearly trans-Sasakian structure and M = NT\times fN\bot be an m-dimensional contact CR-
warped product submanifold, such that N1 is (p+1)-dimensional invariant submanifold tangent to \xi ,
where p is an even number and N\bot be a q-dimensional anti-invariant submanifold of \=M(f1, f2, f3).
Then:
(i) The squared norm of the second fundamental form h satisfies
\| h\| 2 \geq q[\| \nabla \mathrm{l}\mathrm{n} f\| 2 - \Delta \mathrm{l}\mathrm{n} f - \alpha - \beta ] + f2pq + \| \scrQ DD
\bot \| 2, (4.1)
where \Delta denotes the Laplace operator on NT .
(ii) The equality sign of (4.1) holds identically if and only if we have
(a) NT is totally geodesic invariant submanifold of \=M(f1, f2, f3); hence, NT is a generalized
Sasakian space form admitting the nearly trans-Sasakian structure,
(b) N\bot is a totally umbilical anti-invariant submanifold of \=M(f1, f2, f3).
Proof. For any X \in TNT - \langle \xi \rangle and Z \in TN\bot , from Lemma 3.1 we have
g(h(\xi , Z), \phi Z) = - \alpha \| Z\| 2
and
g(h(\phi X,Z), \phi Z) = X \mathrm{l}\mathrm{n} f\| Z\| 2.
Since \xi \mathrm{l}\mathrm{n} f = \beta , then combining this with above two equations, we get
p\sum
i=0
q\sum
j=1
\| h\phi D\bot (ei, e
j)\| 2 = q[\| \nabla \mathrm{l}\mathrm{n} f\| 2 - \alpha - \beta ]. (4.2)
Once more from (3.11)
p\sum
i=1
q\sum
j=1
\| h\mu (ei, ej)\| 2 = f2pq - q\Delta \mathrm{l}\mathrm{n} f +
p\sum
i=1
q\sum
j=1
\| \scrQ eie
j\| 2. (4.3)
We use the following notation:
p\sum
i=1
q\sum
j=1
\| \scrQ eie
j\| 2 = \| \scrQ DD
\bot \| 2.
Substituting above notation in (4.3) and combining it with (4.2), we acquire the inequality (4.1).
Let h\prime \prime be the second fundamental form of N\bot in M. Then we have
g(h\prime \prime (Z,W ), X) = g(\nabla ZW,X) = - X \mathrm{l}\mathrm{n} fg(Z,W ),
on using (2.18), we get
h\prime \prime (Z,W ) = - g(Z,W )\nabla \mathrm{l}\mathrm{n} f. (4.4)
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 10
CONTACT CR-WARPED PRODUCT OF SUBMANIFOLDS OF THE GENERALIZED SASAKIAN SPACE . . . 1427
If the equality sign of (4.1) holds identically, then we achieve
h(D,D) = 0, h(D\bot , D\bot ) = 0. (4.5)
The first condition of (4.5) implies that NT is totally geodesic in M. On the other hand, one has
g(h(X,\phi Y ), \phi Z) = g( \=\nabla X\phi Y, \phi Z) = - g(\phi Y, ( \=\nabla X\phi )Z). (4.6)
By use of (2.10) and (2.4) we get the following equation:
g(\phi Y, ( \=\nabla Z\phi )X) = g(\phi Y,\nabla Z\phi X) - g(Y,\nabla ZX),
in view of (2.17) the above equation reduced to
g(\phi Y, ( \=\nabla Z\phi )X) = 0. (4.7)
From (4.6), (4.7) and (2.16a) we have
g(h(X,\phi Y ), \phi Z) = - g(\phi Y, ( \=\nabla X\phi )Z + ( \=\nabla Z\phi )X) = 0. (4.8)
From (4.8), it is evident that NT is totally geodesic in \=M(f1, f2, f3) and hence is a generalized
Sasakian space form admitting the nearly trans-Sasakian structure.
The second condition of (4.5) and (4.4) imply that N\bot is totally umbilical in \=M(f1, f2, f3).
In the last we have the following corollary which can be deduced from inequality (4.1).
Corollary 4.1. Let M = NT \times f N\bot be a contact CR-warped product submanifold of a genera-
lized Sasakian space form \=M(f1, f2, f3) admitting the trans-Sasakian structure, then squared norm
of the second fundamental form satisfies
\| h\| 2 \geq q[\| \nabla \mathrm{l}\mathrm{n} f\| 2 - \Delta \mathrm{l}\mathrm{n} f - \alpha - \beta ] + f2pq,
where \Delta is the Laplace operator on NT , and p + 1 and q are the dimensions of NT and N\bot ,
respectively.
Remark 4.1. For the contact CR-warped product submanifolds of generalized Sasakian space
forms \=M(f1, f2, f3), admitting nearly trans-Sasakian structure if we consider the tensorial equation
as follows:
( \=\nabla X\phi )Y + ( \=\nabla Y \phi )X = \alpha (2g(X,Y ) - \eta (Y )X - \eta (X)Y ) - \beta (\eta (Y )\phi X + \eta (X)\phi Y
and dimension of invariant submanifold NT as 2p + 1. Then by similar calculations the inequality
(4.1) will be change as follows:
\| h\| 2 \geq q[\| \nabla \mathrm{l}\mathrm{n} f\| 2 - \Delta \mathrm{l}\mathrm{n} f - \alpha - \beta ] + 2f2pq + \| \scrQ DD
\bot \| 2. (4.9)
Now we have the following conclusions:
(i) If the ambient manifold \=M(f1, f2, f3) is Kenmotsu space form, i.e., \alpha = 0, \beta = 1, f2 =
=
c+ 1
4
and \scrQ DD
\bot = 0, then the inequality (4.9) reduced to the inequality obtained in Theorem
4.1 of [19].
(ii) Moreover, if the ambient manifold \=M(f1, f2, f3) is Sasakian space form, i.e., \alpha = 0, \beta =
= 1, f2 =
c+ 1
4
and \scrQ DD
\bot = 0, then the inequality (4.9) will be very much similar to the inequality
(3.1) of Theorem 3.1 in [15].
Remark 4.2. The inequality (4.1) of Theorem 4.1 is also an improved version of the inequality
4.1 proved in [1].
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 10
1428 MERAJ ALI KHAN
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Received 25.05.16
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 10
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| id | umjimathkievua-article-1645 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:09:46Z |
| publishDate | 2018 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/a7/0c89f9ba5dfa6b2fb4a37b2b5a2bbca7.pdf |
| spelling | umjimathkievua-article-16452019-12-05T09:21:55Z Contact $CR$-warped product of submanifolds of the generalized Sasakian space forms admitting the nearly trans-Sasakian structure Контактний $CR$-викривлений добуток пiдмноговидiв узагальнених просторових форм Сасакi, що допускають майже трансструктуру Сасакi Meraj, Ali Khan Мерадж, Алі Хан In the present paper, we apply Hopf’s lemma to the contact $CR$-warped product of submanifolds of the generalized Sasakian space forms admitting nearly trans-Sasakian structure and establish a characterization inequality for the existence of these types of warped products. This inequality generalizes the inequalities obtained in [M. Atceken, Bull. Iran. Math. Soc. – 2013. – 39, № 3. – P. 415 – 429; M. Atceken, Collect. Math. – 2011. – 62, № 1. – P. 17 – 26, and Sibel Sular, Cihan O¨ zgu¨r, Turkish J. Math. – 2012. – 36. – P. 485 – 497]. Moreover, we also compute another inequality for the squared norm of the second fundamental form in terms of warping functions. This inequality is a generalization of the inequalities acquired in [I. Mihai, Geom. Dedicata. – 2004. – 109. – P. 165 – 173 and K. Arslan, R. Ezentas, I. Mihai, C. Murathan, J. Korean Math. Soc. – 2005. – 42, № 5. – P. 1101 – 1110]. The inequalities proved in the paper either generalize or improve all inequalities available in the literature and related to the squared norm of the second fundamental form for contact CR-warped product of submanifolds of any almost contact metric manifold. У роботi лему Хопфа застосовано до контактного $CR$-викривленого добутку пiдмноговидiв узагальнених просто рових форм Сасакi, що допускають майже трансструктуру Сасакi, та встановлено характеризацiйну нерiвнiсть щодо iснування викривлених добуткiв такого типу. Ця нерiвнiсть узагальнює нерiвностi, що були встановленi в [M. Atceken, Bull. Iran. Math. Soc. – 2013. – 39, № 3. – P. 415 – 429; M. Atceken, Collect. Math. – 2011. – 62, № 1. –P. 17 – 26 та Sibel Sular, Cihan Ozgur, Turkish J. Math. – 2012. – 36. – P. 485 – 497]. Крiм того, отримано iншу нерiвнiсть для квадрата норми другої фундаментальної форми в термiнах викривляючих функцiй. Ця нерiвнiсть узагальнює нерiвностi, що були встановленi в [I. Mihai, Geom. Dedicata. – 2004. – 109. – P. 165 – 173 та K. Arslan, R. Ezentas, I. Mihai, C. Murathan, J. Korean Math. Soc. – 2005. – 42, № 5. – P. 1101 – 1110]. Нерiвностi, що доведенi в роботi, узагальнюють або полiпшують усi нерiвностi, доступнi в лiтературi, що вiдносяться до квадрата норми другої фундаментальної форми для контактного CR-викривленого добутку пiдмноговидiв будь-якого майже контактного метричного многовиду. Institute of Mathematics, NAS of Ukraine 2018-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1645 Ukrains’kyi Matematychnyi Zhurnal; Vol. 70 No. 10 (2018); 1417-1428 Український математичний журнал; Том 70 № 10 (2018); 1417-1428 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1645/627 Copyright (c) 2018 Meraj Ali Khan |
| spellingShingle | Meraj, Ali Khan Мерадж, Алі Хан Contact $CR$-warped product of submanifolds of the generalized Sasakian space forms admitting the nearly trans-Sasakian structure |
| title | Contact $CR$-warped product of submanifolds of the generalized Sasakian space forms admitting the nearly trans-Sasakian structure |
| title_alt | Контактний $CR$-викривлений добуток пiдмноговидiв
узагальнених просторових форм Сасакi,
що допускають майже трансструктуру Сасакi |
| title_full | Contact $CR$-warped product of submanifolds of the generalized Sasakian space forms admitting the nearly trans-Sasakian structure |
| title_fullStr | Contact $CR$-warped product of submanifolds of the generalized Sasakian space forms admitting the nearly trans-Sasakian structure |
| title_full_unstemmed | Contact $CR$-warped product of submanifolds of the generalized Sasakian space forms admitting the nearly trans-Sasakian structure |
| title_short | Contact $CR$-warped product of submanifolds of the generalized Sasakian space forms admitting the nearly trans-Sasakian structure |
| title_sort | contact $cr$-warped product of submanifolds of the generalized sasakian space forms admitting the nearly trans-sasakian structure |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1645 |
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