Critical point equation on almost Kenmotsu manifolds
We study the critical point equation $(CPE)$ conjecture on almost Kenmotsu manifolds. First, we prove that if a three-dimensional $(k,\mu)'$-almost Kenmotsu manifold satisfies the $CPE,$ then the manifold is either locally isometric to the product space $\mathbb H^2(-4)\times\mathbb R$ or t...
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| author | De, U. C. Mandal, K. De, U. C. Mandal, K. De, U. C. Mandal, K. |
| author_facet | De, U. C. Mandal, K. De, U. C. Mandal, K. De, U. C. Mandal, K. |
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| description | We study the critical point equation $(CPE)$ conjecture on almost Kenmotsu manifolds. First, we prove that if a three-dimensional $(k,\mu)'$-almost Kenmotsu manifold satisfies the $CPE,$ then the manifold is either locally isometric to the product space $\mathbb H^2(-4)\times\mathbb R$ or the manifold is Kenmotsu manifold. Further, we prove that if the metric of an almost Kenmotsu manifold with conformal Reeb foliation satisfies the $CPE$ conjecture, then the manifold is Einstein.
  |
| first_indexed | 2026-03-24T02:22:14Z |
| format | Article |
| fulltext |
UDC 514.7
U. C. De, K. Mandal (Univ. Calcutta, West Bengal, India)
CRITICAL POINT EQUATION ON ALMOST KENMOTSU MANIFOLDS
РIВНЯННЯ КРИТИЧНОЇ ТОЧКИ НА МНОГОВИДАХ,
ЩО Є МАЙЖЕ МНОГОВИДАМИ КЕНМОЦУ
We study the critical point equation (CPE) conjecture on almost Kenmotsu manifolds. First, we prove that if a three-
dimensional (k, \mu )\prime -almost Kenmotsu manifold satisfies the CPE, then the manifold is either locally isometric to the
product space \BbbH 2( - 4) \times \BbbR or the manifold is Kenmotsu manifold. Further, we prove that if the metric of an almost
Kenmotsu manifold with conformal Reeb foliation satisfies the CPE conjecture, then the manifold is Einstein.
Вивчається гiпотеза про рiвняння критичної точки (РКТ) на многовидах, що є майже многовидами Кенмоцу. Насам-
перед доведено, що у випадку, коли тривимiрний (k, \mu )\prime -майже многовид Кенмоцу задовольняє РКТ, цей многовид
є або локально iзометричним до добутку просторiв \BbbH 2( - 4) \times \BbbR , або многовидом Кенмоцу. Крiм того, доведено,
що у випадку, коли метрика многовиду, що є майже многовидом Кенмоцу з конформним розшаруванням Рiба,
задовольняє РКТ гiпотезу, цей многовид є многовидом Ейнштейна.
1. Introduction. By an almost contact metric manifold of odd dimensional we mean that a smooth
manifold together with an almost contact structure (\phi , \xi , \eta , g) given by a (1, 1) tensor field \phi , a
characteristic vector field \xi , a 1-form \eta and a compatible metric g satisfying the conditions [3, 4]
\phi 2 = - I + \eta \otimes \xi , \phi (\xi ) = 0, \eta (\xi ) = 1, \eta \circ \phi = 0,
g(\phi X, \phi Y ) = g(X,Y ) - \eta (X)\eta (Y ),
for any vector fields X and Y of TpM, where TpM denotes the tangent vector space of M at any
point p \in M. In 1972, Kenmotsu [13] introduced a new type of almost contact metric manifolds
named Kenmotsu manifolds nowadays. Later such manifolds were generalized to almost Kenmosu
manifolds by Janssens and Vanhecke [12]. Recently, Dileo and Pastore [9] introduced the notion of
(k, \mu )\prime -nullity distribution on an almost Kenmotsu manifold (M2n+1, \phi , \xi , \eta , g), which is defined for
any p \in M2n+1 and k, \mu \in \BbbR as follows:
Np(k, \mu )
\prime =
\bigl\{
Z \in TpM
2n+1 : R(X,Y )Z =
= k[g(Y, Z)X - g(X,Z)Y ] + \mu [g(Y, Z)h\prime X - g(X,Z)h\prime Y ]
\bigr\}
,
where two symmetric (1, 1)-type tensor fields defined by h\prime = h \circ \phi and 2h = \$\xi \phi . Since then
several authors such as Dileo and Pastore [8], De and Mandal [5 – 7], Wang and Liu [16 – 19] studied
almost Kenmotsu manifolds satisfying some nullity distributions.
A Riemannian manifold (M, g) of dimension (2n + 1) \geq 3 with constant scalar curvature and
unit volume together with a non-constant smooth potential function \lambda satisfying\Bigl( r
2n
g - S
\Bigr)
\lambda - \mathrm{H}\mathrm{e}\mathrm{s}\mathrm{s}\lambda = S - r
2n+ 1
g, (1.1)
where S, r and \mathrm{H}\mathrm{e}\mathrm{s}\mathrm{s}\lambda are, respectively, Ricci tensor, scalar curvature and the Hess ian of the smooth
function \lambda on M is called a critical point equation (CPE ). Note that if \lambda = 0, then (1.1) becomes
c\bigcirc U. C. DE, K. MANDAL, 2020
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 1 61
62 U. C. DE, K. MANDAL
Einstein metric. Therefore, we consider only the non-trivial potential function \lambda . In [2], Besse
conjectured that the solution of the CPE is Einstein. Barros and Ribeiro [1] proved that the CPE
conjecture is true for half conformally flat. Recently, Hwang [11] proved that the CPE conjecture is
also true under certain condition on the bounds of the potential function \lambda . A necessary and sufficient
condition on the norm of the gradient of the potential function for a CPE metric is to be Einstein
obtained by Neto [14]. Ghosh and Patra [10] consider the CPE conjecture in the frame-work of
K -contact manifolds and (k, \mu )-contact manifolds.
Motivated by the above studies in this paper we study the CPE conjecture on almost Kenmotsu
manifolds. In Section 3, we prove that a three-dimensional (k, \mu )\prime -almost Kenmotsu manifold satis-
fying the CPE conjecture is either locally isometric to the product space \BbbH 2( - 4)\times \BbbR , or Kenmotsu.
In the final section, we prove that if the metric of an almost Knemotsu manifold with conformal Reeb
foliation satisfies the CPE conjecture, then the manifold is Einstein.
2. Almost Kenmotsu manifolds. Let us consider (M2n+1, \phi , \xi , \eta ) be an almost contact mani-
fold. The fundamental 2-form \Phi on an almost contact metric manifold is defined by \Phi (X,Y ) =
= g(X,\phi Y ) for any vector fields X, Y of TpM
2n+1. An almost Kenmotsu manifold is defined as an
almost contact metric manifold such that d\eta = 0 and d\Phi = 2\eta \wedge \Phi . An almost contact metric manifold
is said to be normal if the (1, 2)-type torsion tensor N\phi vanishes, where N\phi = [\phi , \phi ] + 2d\eta \otimes \xi ,
where [\phi , \phi ] is the Nijenhuis torsion of \phi [3]. A normal almost Kenmotsu manifold is a Kenmotsu
manifold. Also Kenmotsu manifolds can be characterized by (\nabla X\phi )Y = g(\phi X, Y )\xi - \eta (Y )\phi X, for
any vector fields X, Y. It is well known [13] that a Kenmotsu manifold M2n+1 is locally a warped
product I \times f N2n, where N2n is a Kähler manifold, I is an open interval with coordinate t and
the warping function f, defined by f = cet for some positive constant c. Let \scrD be the distribution
orthogonal to \xi and defined by \scrD = \mathrm{K}\mathrm{e}\mathrm{r} (\eta ) = \mathrm{I}\mathrm{m}(\phi ). In an almost Kenmotsu manifold \scrD is an
integrable distribution as \eta is closed. Let in an almost Kenmotsu manifold the two tensor fields h
and l are defined by h =
1
2
\$\xi \phi and l = R(\cdot , \xi )\xi . The tensor fields l and h are symmetric and
satisfy the following relations [8]:
h\xi = 0, l\xi = 0, tr(h) = 0, tr(h\phi ) = 0, h\phi + \phi h = 0, (2.1)
\nabla X\xi = - \phi 2X - \phi hX(\Rightarrow \nabla \xi \xi = 0), (2.2)
\phi l\phi - l = 2(h2 - \phi 2), (2.3)
R(X,Y )\xi = \eta (X)(Y - \phi hY ) - \eta (Y )(X - \phi hX) + (\nabla Y \phi h)X - (\nabla X\phi h)Y, (2.4)
for any vector fields X, Y.
An almost Kenmotsu manifold (M2n+1, \phi , \xi , \eta , g) with its characteristic vector field \xi belonging
to the (k, \mu )\prime -nullity distribution is known as (k, \mu )\prime -almost Kenmotsu manifolds and the curvature
tensor satisfies
R(X,Y )\xi = k[\eta (Y )X - \eta (X)Y ] + \mu [\eta (Y )h\prime X - \eta (X)h\prime Y ]. (2.5)
Now we provide some related results on almost Kenmotsu manifolds such that \xi belongs to some
nullity distributions. The (1, 1)-type symmetric tensor field h\prime = h \circ \phi is anticommuting with \phi
and h\prime \xi = 0. Also it is clear that
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CRITICAL POINT EQUATION ON ALMOST KENMOTSU MANIFOLDS 63
h = 0 \leftrightarrow h\prime = 0, h\prime 2 = (k + 1)\phi 2(\leftrightarrow h2 = (k + 1)\phi 2). (2.6)
Let X \in \scrD be the eigen vector of h\prime corresponding to the eigen value \lambda . It follows from (2.6) that
\lambda 2 = - (k + 1) is an constant. Therefore, k \leq - 1 and \lambda = \pm
\surd
- k - 1. We denote by [\lambda ]\prime and
[ - \lambda ]\prime the corresponding eigenspaces associated with h\prime corresponding to the non-zero eigen value \lambda
and - \lambda , respectively. We have the following lemmas.
Lemma 2.1 (Propositions 4.1 and 4.3 of [9]). Let (M2n+1, \phi , \xi , \eta , g) be an almost Kenmotsu
manifold such that \xi belongs to the (k, \mu )\prime -nullity distribution and h\prime \not = 0. Then k < - 1, \mu = - 2
and \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c} (h\prime ) = \{ 0, \lambda , - \lambda \} , with 0 as simple eigen value and \lambda =
\surd
- k - 1. The distributions
[\xi ] \oplus [\lambda ]\prime and [\xi ] \oplus [ - \lambda ]\prime are integrable with totally geodesic leaves. The distributions [\lambda ]\prime and
[ - \lambda ]\prime are integrable with totally umbilical leaves. Furthermore, the sectional curvature are given as
following:
(a) K(X, \xi ) = k - 2\lambda if X \in [\lambda ]\prime and K(X, \xi ) = k + 2\lambda if X \in [ - \lambda ]\prime ;
(b) K(X,Y ) = k - 2\lambda if X, Y \in [\lambda ]\prime ; K(X,Y ) = k + 2\lambda if X, Y \in [ - \lambda ]\prime and K(X,Y ) =
= - (k + 2) if X \in [\lambda ]\prime , Y \in [ - \lambda ]\prime ;
(c) M2n+1 has constant negative scalar curvature r = 2n(k - 2n).
Lemma 2.2 (Lemma 3 of [19]). Let (M2n+1, \phi , \xi , \eta , g) be an almost Kenmotsu manifold with
\xi belonging to the (k, \mu )\prime -nullity distribution. If h\prime \not = 0, then the Ricci operator Q of M2n+1 is
given by
Q = - 2nid+ 2n(k + 1)\eta \otimes \xi - 2nh\prime . (2.7)
Moreover, the scalar curvature of M2n+1 is 2n(k - 2n).
Lemma 2.3 (Lemma 4.1 of [9]). Let (M2n+1, \phi , \xi , \eta , g) be an almost Kenmotsu manifold with
h\prime \not = 0 and \xi belonging to the (k, - 2)\prime -nullity distribution. Then, for any X, Y \in TpM,
(\nabla Xh\prime )Y = - g(h\prime X + h\prime 2X,Y )\xi - \eta (Y )(h\prime X + h\prime 2X). (2.8)
Lemma 2.4 (Proposition 3 of [8]). An almost Kenmotsu manifold M3 such that \nabla \xi = - \phi 2 is
a Kenmotsu manifold.
3. (\bfitk , \bfitmu )\prime -Almost Kenmotsu manifolds satisfying the \bfitC \bfitP \bfitE conjecture. In this section, we
consider (k, \mu )\prime -almost Kenmotsu manifolds satisfying the critical point equation in dimension three.
Before proving our main result we recall the following result.
Lemma 3.1 [10]. Let (g, \lambda ) be a non-trivial solution of the CPE given by (1.1) on a (2n+1)-
dimensional Riemannian manifold M. Then the curvature tensor R can be expressed as
R(X,Y )D\lambda = (X\lambda )QY - (Y \lambda )QX + (\lambda + 1)(\nabla XQ)Y -
- (\lambda + 1)(\nabla Y Q)X + (Xf)Y - (Y f)X, (3.1)
where f = - r
\biggl(
\lambda
2n
+
1
2n+ 1
\biggr)
.
Now we prove the following theorem.
Theorem 3.1. If the metric of a three dimensional (k, \mu )\prime -almost Kenmotsu manifold satisfies
the critical point equation, then the manifold is either Kenmotsu manifold, or locally isometric to the
product space \BbbH 2( - 4)\times \BbbR .
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 1
64 U. C. DE, K. MANDAL
Proof. Replacing X by \xi in (3.1) and making use of (2.7), we have
R(\xi , Y )D\lambda = (\xi \lambda )QY - 2k(Y \lambda )\xi + (\lambda + 1)(\nabla \xi Q)Y -
- (\lambda + 1)(\nabla Y Q)\xi + (\xi f)Y - (Y f)\xi . (3.2)
Taking covariant differentiation of (1.1) along arbitrary vector field X and using (2.2), we obtain
(\nabla XQ)Y = 2(k + 1)\eta (Y )(X + h\prime X) - 2(\nabla Xh\prime )Y -
- 2(k + 1)\{ g(X,Y ) - 2\eta (X)\eta (Y ) + g(h\prime X,Y )\} \xi .
By using the above equation, we get
(\nabla XQ)Y - (\nabla Y Q)X =
= - 2(k + 1)\{ \eta (X)(Y + h\prime Y ) - \eta (Y )(X + h\prime X)\} -
- 2\{ (\nabla Xh\prime )Y - (\nabla Y h
\prime )X\} (3.3)
for any vector fields X, Y. Putting X = \xi in (3.3), we have
(\nabla \xi Q)Y - (\nabla Y Q)\xi = - 2(k + 1)\{ Y + h\prime Y - \eta (Y )\xi \} -
- 2\{ (\nabla \xi h
\prime )Y - (\nabla Y h
\prime )\xi \} . (3.4)
From (2.8), we obtain
(\nabla \xi h
\prime )Y - (\nabla Y h
\prime )\xi = h\prime Y + h\prime 2Y. (3.5)
Taking inner product of (3.4) and using (3.5), we get
g((\nabla \xi Q)Y - (\nabla Y Q)\xi , \xi ) = 0. (3.6)
It follows from (3.2) and (3.6) that
g(R(\xi , Y )D\lambda , \xi ) = 2nk\xi (\lambda )\eta (Y ) - 2nkY (\lambda ) + \xi (f)\eta (Y ) - Y (f). (3.7)
On the other hand, from (2.5) and Lemma 2.1 we have
g(R(\xi , Y )D\lambda , \xi ) = - g(R(\xi , Y )\xi ,D\lambda ) =
= k[g(D\lambda , Y ) - \xi (\lambda )\eta (Y )] - 2g(h\prime D\lambda , Y ). (3.8)
Making use of (3.7) and (3.8), we get
3kD\lambda - k\xi (\lambda )\xi - 2h\prime D\lambda = 2k\xi (\lambda )\xi + \xi (f)\xi - Df. (3.9)
Now, from Lemma 3.1, we have f = - r
\biggl(
\lambda
2
+
1
3
\biggr)
. Differentiating this equation, we get
\xi (f) = - (k - 2)\xi (\lambda ) and Df = - (k - 2)D\lambda , (3.10)
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 1
CRITICAL POINT EQUATION ON ALMOST KENMOTSU MANIFOLDS 65
where we have used Lemma 2.1. Relations (3.9) and (3.10) both gives
h\prime D\lambda = (k + 1)(D\lambda - \xi (\lambda )\xi ). (3.11)
Applying h\prime on both sides of (3.11) and using (2.6), we have
- (k + 1)(D\lambda - \xi (\lambda )\xi ) = (k + 1)h\prime D\lambda . (3.12)
Substituting the value of h\prime D\lambda from (3.11) in (3.12) yields
(k + 1)(k + 2)(D\lambda - \xi (\lambda )\xi ) = 0.
We consider three cases:
Case 1. Let k = - 1, then from (2.6) we have h\prime = 0. Using this in (2.2) gives \nabla \xi = - \phi 2.
Hence from Lemma 2.4 the manifold becomes Kenmotsu manifold.
Case 2. Let k = - 2. Then, from Remark 5.1 of [9], we can state that the manifold is locally
isometric to the Riemannian product \BbbH 2( - 4)\times \BbbR .
Case 3. Let D\lambda = \xi (\lambda )\xi . Taking trace of the equation (1.1), we have \bigtriangleup g\lambda = - r\lambda
2n
. Using this
and Lemma 2.2 in (1.1) gives
\nabla XD\lambda = (\lambda + 1)QX + fX. (3.13)
Putting D\lambda = \xi (\lambda )\xi in (3.13) yields
(\lambda + 1)QX = \{ X\xi (\lambda ) - \xi (\lambda )\eta (X)\} \xi + \xi (\lambda )h\prime X+
+
\biggl\{
\xi (\lambda ) + 2(k - 2)
\biggl(
\lambda
2
+
1
3
\biggr) \biggr\}
X.
Comparing this relation with (2.7), we have
X\xi (\lambda ) - \xi (\lambda )\eta (X) = 2(k + 1)(\lambda + 1)\eta (X), (3.14)
\xi \lambda + 2(k - 2)
\biggl(
\lambda
2
+
1
3
\biggr)
= - 2(\lambda + 1), (3.15)
\xi (\lambda ) = - 2(\lambda + 1), (3.16)
for any vector field X. By using (3.16) in (3.15), we have \lambda is a constant, which is a contradiction.
Theorem 3.1 is proved.
4. Almost Kenmotsu manifolds with conformal Reeb foliation satisfying the \bfitC \bfitP \bfitE conjec-
ture. This section is devoted to study almost Kenmotsu manifolds with conformal Reeb foliation
satisfying the CPE conjecture of dimension \geq 5. An almost contact Riemannian manifold M is
said to be an \eta -Einstein manifold if the Ricci tensor S satisfies the condition
S(X,Y ) = \gamma g(X,Y ) + \delta \eta (X)\eta (Y ),
where \gamma , \delta are smooth functions and X, Y are vector fields on the manifold. In particular, if \delta = 0,
then M is an Einstein manifold. Pastore and Saltarelli [15] prove that on an almost Kenmotsu
manifold the Reeb foliation is conformal if and only if h = 0. We present the following result.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 1
66 U. C. DE, K. MANDAL
Lemma 4.1 [15]. Let (M2n+1, \phi , \xi , \eta , g), n > 1, be an \eta -Einstein almost Kenmotsu manifold
with conformal Reeb foliation, then either the manifold is Einstein, or \delta is not constant, X(\delta ) = 0
for any vector field X \bot \xi , \xi (\delta ) = - 2\delta and in this case the Ricci operator is given by QX =
= - (2n+ \delta )X + \delta \eta (X)\xi , where \delta is locally given by \delta = ce - 2t for some non-zero constant c.
Now we prove the following theorem.
Theorem 4.1. Let (M2n+1, \phi , \xi , \eta , g), n > 1, be an almost Kenmotsu manifold with conformal
Reeb foliation. If M satisfies the critical point equation, then the manifold is an Einstein manifold
provided the scalar curvature r \not = - 2n(2n+ 1).
Proof. Since h = 0, we have, from (2.4),
R(X,Y )\xi = \eta (X)Y - \eta (Y )X (4.1)
for any vector fields X, Y. From this we obtain
Q\xi = - 2n\xi . (4.2)
Since
f = - r
\biggl(
\lambda
2n
+
1
2n+ 1
\biggr)
,
then we get
(\xi f) = - r
2n
(\xi \lambda ) and (Y f) = - r
2n
(Y \lambda ). (4.3)
Replacing \xi instead of X in (3.1) and using (4.2) yields
R(\xi , Y )D\lambda = (\xi \lambda )QY + 2n(Y \lambda )\xi + (\lambda + 1)(\nabla \xi Q)Y -
- (\lambda + 1)(\nabla Y Q)\xi + (\xi f)Y - (Y f)\xi . (4.4)
Taking inner product of (4.4) with \xi and making use of (4.3) gives
g(R(\xi , Y )D\lambda , \xi ) = -
\Bigl(
2n+
r
2n
\Bigr)
(\xi \lambda )\eta (Y ) +
\Bigl(
2n+
r
2n
\Bigr)
(Y \lambda ).
Also, from (4.1), we obtain
g(R(\xi , Y )D\lambda , \xi ) = - g(R(\xi , Y )\xi ,D\lambda ) = (\xi \lambda )\eta (Y ) - (Y \lambda ).
Comparing the above two equations, we have\Bigl(
2n+ 1 +
r
2n
\Bigr)
\{ (Y \lambda ) - (\xi \lambda )\eta (Y )\} = 0,
from which it follows that \Bigl(
2n+ 1 +
r
2n
\Bigr)
\{ D\lambda - (\xi \lambda )\xi \} = 0. (4.5)
Let us assume that the scalar curvature r \not = - 2n(2n+1), then we have from (4.5) that D\lambda = (\xi \lambda )\xi .
By using this in (3.13), we get
(\lambda + 1)QX =
\biggl(
(\xi \lambda ) +
r\lambda
2n
+
r
2n+ 1
\biggr)
X + (X(\xi \lambda ) - (\xi \lambda )\eta (X))\xi . (4.6)
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 1
CRITICAL POINT EQUATION ON ALMOST KENMOTSU MANIFOLDS 67
This shows the manifold is an \eta -Einstein manifold. Without loss of any generality we may assume
that the manifold is not Einstein. From Lemma 4.1 we see that the second case is true, that is,
QX = - (2n+ \delta )X + \delta \eta (X)\xi , (4.7)
where \delta is locally given by \delta = ce - 2t for some non-zero constant c. Now comparing the relations
(4.6) and (4.7), we have
(\xi \lambda ) +
r\lambda
2n
+
r
2n+ 1
= - (\lambda + 1)(2n+ \delta ), (4.8)
X(\xi \lambda ) - (\xi \lambda )\eta (X) = \delta (\lambda + 1)\eta (X) (4.9)
for any vector field X. With the help of (4.8) and (4.9) we get
\xi (\xi \lambda ) = - r\lambda
2n
- r
2n+ 1
- (\lambda + 1)2n. (4.10)
Tracing (4.6) gives
\xi (\xi \lambda ) = - 2n(\xi \lambda ) - r\lambda
2n
. (4.11)
In view of (4.10) and (4.11), we obtain
\xi \lambda =
r
2n(2n+ 1)
+ \lambda + 1. (4.12)
Making use of (4.12) in (4.10) yields
(\lambda + 1)
\Bigl( r
2n
+ 1 + 2n
\Bigr)
= 0.
Since \lambda being a non-constant smooth function, then from the above equation we have
r = - 2n(2n + 1), which is a contradiction. Hence, using Lemma 4.1, we complete the proof
of theorem.
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Received 03.12.16
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 1
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| id | umjimathkievua-article-2330 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:22:14Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/76/89173b2ff9ca908ffe6d0d4698e4c776.pdf |
| spelling | umjimathkievua-article-23302020-01-27T12:30:33Z Critical point equation on almost Kenmotsu manifolds Рiвняння критичної точки на многовидах, що є майже многовидами Кенмоцу De, U. C. Mandal, K. De, U. C. Mandal, K. De, U. C. Mandal, K. Многовиди Ейнштейна многовид Кенмоцу Almost Kenmotsu manifold nullity distribution critical point equation Einstein manifold We study the critical point equation $(CPE)$ conjecture on almost Kenmotsu manifolds. First, we prove that if a three-dimensional $(k,\mu)'$-almost Kenmotsu manifold satisfies the $CPE,$ then the manifold is either locally isometric to the product space $\mathbb H^2(-4)\times\mathbb R$ or the manifold is Kenmotsu manifold. Further, we prove that if the metric of an almost Kenmotsu manifold with conformal Reeb foliation satisfies the $CPE$ conjecture, then the manifold is Einstein. &nbsp; Вивчається гіпотеза про рівняння критичної точки (РКТ) на многовидах, що є майже многовидами Кенмоцу. Насамперед доведено, що у випадку, коли тривимірний $(k,\mu)'$-майже многовид Кенмоцу задовольняє РКТ, цей многовид є або локально ізометричним до добутку просторів $\mathbb H^2(-4)\times\mathbb R,$ або многовидом Кенмоцу. Крім того, доведено, що у випадку, коли метрика многовиду, що є майже многовидом Кенмоцу з конформним розшаруванням Ріба, задовольняє РКТ гіпотезу, цей многовид є многовидом Ейнштейна. Institute of Mathematics, NAS of Ukraine 2020-01-15 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2330 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 1 (2020); 61-68 Український математичний журнал; Том 72 № 1 (2020); 61-68 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2330/1548 Copyright (c) 2020 Elina Dichter (Менеджер журналу) |
| spellingShingle | De, U. C. Mandal, K. De, U. C. Mandal, K. De, U. C. Mandal, K. Critical point equation on almost Kenmotsu manifolds |
| title | Critical point equation on almost Kenmotsu manifolds |
| title_alt | Рiвняння критичної точки на многовидах, що є майже многовидами Кенмоцу |
| title_full | Critical point equation on almost Kenmotsu manifolds |
| title_fullStr | Critical point equation on almost Kenmotsu manifolds |
| title_full_unstemmed | Critical point equation on almost Kenmotsu manifolds |
| title_short | Critical point equation on almost Kenmotsu manifolds |
| title_sort | critical point equation on almost kenmotsu manifolds |
| topic_facet | Многовиди Ейнштейна многовид Кенмоцу Almost Kenmotsu manifold nullity distribution critical point equation Einstein manifold |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2330 |
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