A classification of conformal vector fields on the tangent bundle
UDC 514.7 Let $(M,g)$ be a Riemannian manifold and $TM$ be its tangent bundle equipped with a Riemannian (or pseudo-Riemannian) lift metric derived from $g.$ We give a classification of infinitesimal fibre-preserving conformal transformations on the tangent bundle.
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2020
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| author | Raei, Zohre Latifi, Dariush Raei, Zohre Latifi, Dariush Raei, Zohre Latifi, Dariush |
| author_facet | Raei, Zohre Latifi, Dariush Raei, Zohre Latifi, Dariush Raei, Zohre Latifi, Dariush |
| author_sort | Raei, Zohre |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2022-03-26T11:01:43Z |
| description | UDC 514.7
Let $(M,g)$ be a Riemannian manifold and $TM$ be its tangent bundle equipped with a Riemannian (or pseudo-Riemannian) lift metric derived from $g.$ We give a classification of infinitesimal fibre-preserving conformal transformations on the tangent bundle. |
| doi_str_mv | 10.37863/umzh.v72i5.6013 |
| first_indexed | 2026-03-24T03:25:13Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v72i5.6013
UDC 514.7
Z. Raei, D. Latifi (Dep. Math., Univ. Mohaghegh Ardabili, Ardabil, Iran)
A CLASSIFICATION OF CONFORMAL VECTOR FIELDS
ON THE TANGENT BUNDLE
КЛАСИФIКАЦIЯ КОНФОРМНИХ ВЕКТОРНИХ ПОЛIВ
НА ДОТИЧНОМУ РОЗШАРУВАННI
Let (M, g) be a Riemannian manifold and TM be its tangent bundle equipped with a Riemannian (or pseudo-Riemannian)
lift metric derived from g. We give a classification of infinitesimal fibre-preserving conformal transformations on the
tangent bundle.
Нехай (M, g) — рiманiв многовид, TM — його дотичне розшарування з рiмановою (або псевдорiмановою) метрикою
пiдняття, яка породжується g. Наведено класифiкацiю нескiнченно малих конформних перетворень, що зберiгають
шари на дотичному розшаруваннi.
1. Introduction. Let M be a Riemannian manifold with a Riemannian metric g and X be a vector
field on M. Let us consider the local one-parameter group \{ \phi t\} of local transformations of M
generated by X. The vector field X is called an infinitesimal conformal transformation if each \phi t
is a local conformal transformation of M. As is well-known, the vector field X is an infinitesimal
conformal transformation or conformal vector field on M if and only if there exists a scalar function
\rho on M satisfying LXg = 2\rho g, where LX denotes the Lie derivation with respect to X. Especially,
the vector field X is called an infinitesimal homothetic one when \rho is constant and it is called an
isometry or Killing vector field when \rho vanishes.
Let TM be the tangent bundle over M and \Phi be a transformation of TM. If the transformation
\Phi preserves the fibres, it is called a fibre-preserving transformation. Consider a vector field \~X on
TM and the local one-parameter group \{ \Phi t\} of local transformations of TM generated by \~X. The
vector field \~X is called an infinitesimal fibre-preserving transformation if each \Phi t is a local fibre-
preserving transformation of TM. An infinitesimal fibre-preserving transformation \~X on TM is
called an infinitesimal fibre-preserving conformal transformation if each \Phi t is a local fibre-preserving
conformal transformation of TM. Let \~g be a Riemannian or pseudo-Riemannian metric on TM. \~X
is an infinitesimal conformal transformation of TM if and only if there exists a scalar function \Omega on
TM such that L \~X\~g = 2\Omega \~g, where L \~X denotes the Lie derivation with respect to \~X. An infinitesimal
conformal transformation \~X is called essential if \Omega depends only on (yi) with respect to the induced
coordinates (xi, yi) on TM, and is called inessential if \Omega depends only on (xi), that is, \Omega is a
constant on each fibre of TM. In this case, \Omega induces a function on M.
Let (M, g) be an n-dimensional Riemannian manifold. There are some lift metrics on TM =
=
\bigcup
x\in M TxM as follows: complete lift metric or g2, diagonal lift metric or g1 + g3, lift metric
g2 + g3 and lift metric g1 + g2, where
g1 := gijdx
idxj , g2 := 2gijdx
i\delta yj , g3 := gij\delta y
i\delta yj
are all bilinear differential forms defined globally on TM. Yamauchi [21] proved that every infinite-
simal fibre-preserving conformal transformation on TM with the metric g1+ g3 is homothetic and it
c\bigcirc Z. RAEI, D. LATIFI, 2020
694 ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
A CLASSIFICATION OF CONFORMAL VECTOR FIELDS ON THE TANGENT BUNDLE 695
induces an infinitesimal homothetic transformation on M. Also, in the case when M is a complete,
simply connected Riemannian manifold with a Riemannian metric, Hasegawa and Yamauchi [6]
showed that the Riemannian manifold M is isometric to the standard sphere when the tangent bundle
TM equipped with the metric g1 + g2 admits an essential infinitesimal conformal transformation.
In [3], Gezer has studied a similar problem in [20, 21] with respect to the synectic lift metric on
the tangent bundle. Also he represents the classification of infinitesimal fibre-preserving conformal
transformations on the tangent bundle, equipped with the Cheeger – Gromoll metric [4].
In [1], Abbassi and Sarih classified Killing vector fields on (TM, gCG); that is, they found
general forms of all Killing vector fields on (TM, gCG). Also, they showed that if (TM, gCG) is
the tangent bundle with the Cheeger – Gromoll metric gCG of a Riemannian, compact and orientable
manifold (M, g) with vanishing first and second Betti numbers, then the Lie algebras of Killing
vector fields on (M, g) and on (TM, gCG) are isomorphic. Finally, they showed that the sectional
curvature of the tangent bundle (TM, gCG) with the Cheeger – Gromoll metric gCG of a Riemannian
manifold (M, g) is never constant.
Peyghan, Tayebi and Zhong introduced a class of g-natural metrics Ga,b on the tangent bundle of
a Finsler manifold (M,F ) which generalizes the associated Sasaki - Matsumoto metric and Miron
metric and They investigated Killing vector fields associated to Ga,b in [11]. Two first authors
introduced two vector fields of horizontal Liouville type on a slit tangent bundle endowed with a
Riemannian metric of Sasaki – Finsler type and proved that these vector fields are Killing if and only
if the base Finsler manifold is of positive constant curvature. In the special case of one of them, they
showed that if it is Killing vector field then the base manifold is the Einstein - Finsler manifold [14].
For the other progress, see [5, 7 – 10, 12, 13, 15 – 18].
In [2], Bidabad introduced a new Riemannian (or pseudo-Riemannian) lift metrics on TM , \~g =
= \alpha g1 + \beta g2 + \mu g3, where \alpha , \beta and \mu are certain constant real numbers. That is a combination of
diagonal lift, and complete lift metrics. He had proved that if (M, g) is an n-dimensional Riemannian
manifold and TM is its tangent bundle with metric \~g, Then every complete lift conformal vector
field on TM is homothetic.
The purpose of the present paper is to characterize infinitesimal fibre-preserving conformal trans-
formations with respect to the lift metric \~g.
2. Preliminaries. Let M be a real n-dimensional manifold of class C\infty . We denote by
TM \rightarrow M the bundle of tangent vectors and by \pi : TM\setminus \{ 0\} \rightarrow M the fiber bundle of non-
zero tangent vectors. Let \scrV vTM = \mathrm{k}\mathrm{e}\mathrm{r}\pi v
\ast be the set of the vectors tangent to the fiber through
v \in TM\setminus \{ 0\} . Then a vertical vector bundle on M is defined by \scrV TM :=
\bigcup
v\in TM\setminus \{ 0\} \scrV vTM. A
nonlinear connection or a horizontal distribution on TM\setminus \{ 0\} is a complementary distribution \scrH TM
for \scrV TM on T (TM\setminus \{ 0\} ). Therefore, we have the decomposition
T (TM\setminus \{ 0\} ) = \scrV TM \oplus \scrH TM.
Using the local coordinates (xi, yi) on TM we have the local field of frames
\biggl\{
\partial
\partial xi
,
\partial
\partial yi
\biggr\}
on
TTM. It is well-known that we can choose a local field of frames
\biggl\{
\delta
\delta xi
,
\partial
\partial yi
\biggr\}
adapted to the above
decomposition, i.e.,
\delta
\delta xi
\in \Gamma (\scrH TM) and
\partial
\partial yi
\in \Gamma (\scrV TM) set of vector fields on \scrH TM and \scrV TM,
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
696 Z. RAEI, D. LATIFI
where
\delta
\delta xi
=
\partial
\partial xi
- N j
i
\partial
\partial yj
and N j
i (x, y) are the nonlinear differentiable functions on TM, called coefficients of the nonlinear
connection.
Let X = Xi \partial
\partial xi
be the local expression in U \subset TM of a vector field X on M. Then the
vertical lift Xv, the horizontal lift Xh and the complete lift XC of X are given, with respect to the
induced coordinates, by
Xv = Xi \partial
\partial yi
,
Xh = Xi \partial
\partial xi
- ys\Gamma i
skX
k \partial
\partial yi
,
XC = Xi \partial
\partial xi
+ ys\partial sX
i \partial
\partial yi
= Xi \delta
\delta xi
+ ys\nabla sX
i \partial
\partial yi
,
where \Gamma i
jk are the coefficients of the Levi – Civita connection \nabla of g.
Suppose that we are given a tensor field S \in \Im p
q(M), q > 1, where \Im p
q(M) is the set of all
tensor fields of type (p, q) on M. We define a tensor field \gamma S \in \Im p
q(TM) on \pi - 1(U) by
\gamma S =
\bigl(
yeS
j1...jp
ei2...iq
\bigr) \partial
\partial yj1
\otimes . . .\otimes \partial
\partial yjp
\otimes dxi2 \otimes . . .\otimes dxiq
with respect to the induced coordinates (xi, yi). We easily see that \gamma A has components, with respect
to the induced coordinates (xi, yi),
(\gamma A) =
\bigl(
0, yiAj
i
\bigr)
for any A \in \Im 1
1(M) and (\gamma A)(fv) = 0, f \in \Im 0
0(M), i.e., \gamma A is a vertical vector field on TM.
The bracket operation of vertical and horizontal vector fields is given by the following formulae:\biggl[
\delta
\delta xi
,
\delta
\delta xj
\biggr]
= - ysRh
sij
\partial
\partial yh
,
\biggl[
\delta
\delta xi
,
\partial
\partial yj
\biggr]
= \Gamma h
ij
\partial
\partial yh
,
\biggl[
\partial
\partial yi
,
\partial
\partial yj
\biggr]
= 0,
where Rh
sij denotes the components of the Riemannian curvature tensor of g defined by
R(X,Y ) =
\bigl[
\nabla X ,\nabla Y
\bigr]
- \nabla [X,Y ].
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
A CLASSIFICATION OF CONFORMAL VECTOR FIELDS ON THE TANGENT BUNDLE 697
3. A lift metric on tangent bundle. Let (M, g) be a Riemannian manifold. In this section
we introduce a new Riemannian or pseudo-Riemannian metric on TM derived from g. This metric
is in some sense more general than the other lift metrics defined previously on TM. By mean of
the dual basis \{ dxi, \delta yi\} analogously to the Riemannian geometry the tensors; g1 := gijdx
idxj ,
g2 := 2gijdx
i\delta yj and g3 := gij\delta y
i\delta yj are all quadratic differential tensors defined globally on TM,
see [3]. Now let’s consider the Riemannian metric tensor g with the components gij(x, y). The
tensor field \~g = \alpha g1 + \beta g2 + \mu g3 on TM, where the coefficient \alpha , \beta and \gamma are real numbers, has
the components \Biggl(
\alpha g \beta g
\beta g \mu g
\Biggr)
with respect to the dual basis of TM. From the linear algebra we have
\mathrm{d}\mathrm{e}\mathrm{t} \~g = (\alpha \mu - \beta 2)n \mathrm{d}\mathrm{e}\mathrm{t} g2.
Therefore, \~g is nonsingular if \alpha \mu - \beta 2 \not = 0 and it is positive definite if \alpha , \mu are positive and
\alpha \mu - \beta 2 > 0. Indeed \~g defines respectively a pseudo-Riemannian or a Riemannian lift metric on
TM.
Definition 3.1 [2]. Let (M, g) be a Riemannian manifold. Consider tensor field \~g = \alpha g1 +
+ \beta g2 + \mu g3 on TM, where the coefficient \alpha , \mu and \beta are real numbers. If \alpha \mu - \beta 2 \not = 0, then
\~g is nonsingular and it can be regarded as a pseudo-Riemannian metric on TM. If \alpha and \mu are
positive such that \alpha \mu - \beta 2 > 0, then \~g is positive definite and it can consequently be regarded as a
Riemannian metric on TM ; \~g is called the lift metric of g on TM.
4. Main results. Let \~X be a vector field on TM with components (vh, wh) with respect to
the adapted frame
\biggl\{
\delta
\delta xi
,
\partial
\partial yi
\biggr\}
. Then \~X is a fibre-preserving vector field on TM if and only if vh
depend only on the variables (xh). Therefore, every fibre-preserving vector field \~X on TM induces
a vector field X = vh
\partial
\partial xh
on M.
Let M be an n-dimensional manifold, X a vector field on M and \{ \phi t\} a 1-parameter local
group of local transformations of M generated by X. Take any tensor field S on M, and denote by
\phi \ast
t (S) the pulled back of S by \phi t. Then the Lie derivation of S with respect to X is a tensor field
LXS on M defined by
LXS =
\partial
\partial t
\phi \ast
t (S)
\bigm| \bigm| \bigm| \bigm|
t=0
= \mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow 0
\phi \ast
t (S) - (S)
t
,
on the domain of t. The mapping LX which maps S to LX(S) is called the Lie derivation with
respect to X. Then we obtain the following.
Lemma 4.1 [20, 21]. The Lie derivations of the adapted frame and its dual basis with respect
to \~X = vh
\delta
\delta xh
+ wh \partial
\partial yh
are given as follows:
1) L \~X
\delta
\delta xh
= - \partial hv
a \delta
\delta xa
-
\biggl(
vbRa
bh + wb\Gamma a
bh +
\delta wa
\delta xh
\biggr)
\partial
\partial ya
,
2) L \~X
\partial
\partial yh
=
\biggl(
vb\Gamma a
bh -
\partial wa
\partial yh
\biggr)
\partial
\partial ya
,
3) L \~Xdxh = \partial mvhdxm,
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
698 Z. RAEI, D. LATIFI
4) L \~X\delta yh =
\biggl(
vbRh
bm + wb\Gamma h
bm +
\delta wa
\delta xm
\biggr)
dxm -
\biggl(
vb\Gamma h
bm - \partial wh
\partial ym
\biggr)
\delta ym.
Lemma 4.2. The Lie derivative L \~X\~g with respect to the fibre-preserving vector field \~X are
given as follows:
L \~X\~g =
\biggl(
\alpha LXgij + 2\beta gai
\biggl(
vpRa
pj + wp\Gamma a
pj +
\delta wa
\delta xj
\biggr) \biggr)
dxidxj+
+2
\biggl(
\beta
\biggl(
LXgij - gia\nabla jv
a + gai
\partial wa
\partial yj
\biggr)
+
+\mu gaj
\biggl(
vpRa
pi + wp\Gamma a
pi +
\delta wa
\delta xi
\biggr) \biggr)
dxi\delta yj+
+\mu
\biggl(
gaj
\partial wa
\partial yi
+ gai
\partial wa
\partial yj
\biggr)
\delta yi\delta yj .
Proof. The statement is a direct consequence of Lemma 4.1.
Let TM be the tangent bundle over M with the lift metric \~g, and let \~X be an infinitesimal
fibre-preserving conformal transformation on (TM, \~g) such that
L \~X\~g = 2\Omega \~g. (4.1)
By means of Lemma 4.2, we have
2\alpha \Omega gijdx
idxj + 4\beta \Omega gijdx
i\delta yj + 2\mu \Omega gij\delta y
i\delta yj =
=
\biggl(
\alpha LXgij + 2\beta gai
\biggl(
vpRa
pj + wp\Gamma a
pj +
\delta wa
\delta xj
\biggr) \biggr)
dxidxj+
+2
\biggl(
\beta
\biggl(
LXgij - gia\nabla jv
a + gai
\partial
\partial yj
\biggr)
+
+\mu gaj
\biggl(
vpRa
pi + wp\Gamma a
pi +
\delta wa
\delta xi
\biggr) \biggr)
dxi\delta yj+
+\mu
\biggl(
gaj
\partial wa
\partial yi
+ gai
\partial wa
\partial yj
\biggr)
\delta yi\delta yj .
Comparing both sides of the above equation, we obtain the following three relations:
\alpha LXgij + 2\beta gai
\biggl(
vpRa
pj + wp\Gamma a
pj +
\delta wa
\delta xj
\biggr)
= 2\alpha \Omega gij , (4.2)
\beta
\biggl(
LXgij - gia\nabla jv
a + gai
\partial
\partial yj
\biggr)
+ \mu gaj
\biggl(
vpRa
pi + wp\Gamma a
pi +
\delta wa
\delta xi
\biggr)
= 2\beta \Omega gij , (4.3)
\mu
\biggl(
gaj
\partial wa
\partial yi
+ gai
\partial wa
\partial yj
\biggr)
= 2\mu \Omega gij . (4.4)
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
A CLASSIFICATION OF CONFORMAL VECTOR FIELDS ON THE TANGENT BUNDLE 699
Theorem 4.1. Let (M, g) be a Riemannian manifold and TM be its tangent bundle equipped
with the lift metric \~g. A vector field \~X on TM is a fiber-preserving conformal vector field with
respect to \~g if and only if
\~X = XC + \gamma A+Bv,
where B,X \in \Im 1
0(M) and A = (Aa
h) \in \Im 1
1(M) such that:
1) if \mu = 0, then
\beta LBgij + \alpha LXgij = 2\alpha \Omega gij , (4.5)
vpRispj + gai\nabla jP
a
s = 0, (4.6)
gaj\nabla iv
a + gaiP
a
j = 2\Omega gij ; (4.7)
2) if \mu \not = 0, then
\beta LBgij + \alpha LXgij = 2\alpha \Omega gij , (4.8)
vpRispj + gai\nabla jP
a
s = 0, (4.9)
\beta
\bigl(
LXgij - gai\nabla jv
a + gaiP
a
j
\bigr)
+ \mu gaj\nabla iB
a = 2\beta \Omega gij , (4.10)
gaiP
a
j + gajP
a
i = 2\Omega gij , (4.11)
where Aa
h = P a
h - \nabla hX
a.
Proof. We consider the 0-section (yi = 0) in the coordinate neighborhood \pi - 1(U) in TM and
its neighborhood W. For a vector field \~X = vh
\delta
\delta xh
+ wh \partial
\partial yh
on TM, and (x, y) = (xi, yi) in W,
we can write, by Taylor’s theorem,
vh(x, y) = vh(x, 0) + (\partial .
rv
h)(x, 0)yr +
1
2
(\partial .
r\partial
.
sv
i)(x, 0)yrys + . . .+ [\ast ]hm,
wh(x, y) = wh(x, 0) + (\partial .
rw
h)(x, 0)yr +
1
2
(\partial .
r\partial
.
sw
i)(x, 0)yrys + . . .+ [\ast ]hm,
where [\ast ]hm, h = 1, 2, . . . , 2n, is of the form
[\ast ]hm =
1
m!
\biggl(
\partial mvh
\partial yi1\partial yi2 . . . \partial yim
\biggr)
(xa, \theta (x, y)yb)yi1yi2 . . . yim ,
where 1 \leq i1, . . . , im \leq n. Tanno [19] Proved that in this situation the following:
X =
\bigl(
Xi(x)
\bigr)
=
\bigl(
vi(x, 0)
\bigr)
, Y =
\bigl(
Y i(x)
\bigr)
=
\bigl(
wi(x, 0)
\bigr)
,
K =
\bigl(
Ki
r(x)
\bigr)
=
\bigl(
(\partial .
r\partial
.
sv
i)(x, 0)
\bigr)
, E =
\bigl(
Ei
rs(x)
\bigr)
=
\bigl(
(\partial .
r\partial
.
sv
i)(x, 0)
\bigr)
,
P =
\bigl(
P i
r(x)
\bigr)
=
\bigl(
(\partial .
rw
i)(x, 0) - (\partial .
rv
i)(x, 0)
\bigr)
are tensor fields on M. So for a fibre-preserving vector field \~X = vh
\delta
\delta xh
+wh \partial
\partial yh
on TM, we can
write
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
700 Z. RAEI, D. LATIFI
vh(x, y) = Xh, (4.12)
wh(x, y) = Bh + P h
r y
r +
1
2
Qh
rsy
rys + . . .+ [\ast ]hm, (4.13)
where P h
r and Qh
rs are given by P h
r = (\partial .
rw
h)(x, 0) and Qh
rs = \partial .
r\partial
.
sw
h)(x, 0). By putting (4.12)
and (4.13) into (4.2) – (4.4), we have
\alpha LXgij + \beta
\bigl(
gai\nabla jB
a + gaj\nabla iB
a
\bigr)
+ 2\beta gaiy
s
\bigl(
vpRa
spj +\nabla jP
a
s - N t
j\partial
.
tP
a
s
\bigr)
+
+\beta gaiy
syr
\bigl(
\nabla jQ
a
rs +N t
j\partial
.
tQ
a
rs
\bigr)
= 2\alpha \Omega gij , (4.14)
\beta
\bigl(
LXgij - gai\nabla jv
a + gaiP
a
j
\bigr)
+ \beta ysgai
\bigl(
\partial .
jP
a
s +Qa
js
\bigr)
+ \mu ysgaj
\bigl(
vpRa
spi +\nabla iP
a
s - N t
i \partial
.
tP
a
r
\bigr)
+
+\mu gai\nabla iB
a +
\beta
2
gaiy
rys\partial .
jQ
a
rs +
\mu
2
gajy
rys
\bigl(
\nabla iQ
a
rs - N t
i \partial
.
tQ
a
rs
\bigr)
= 2\beta \Omega gij , (4.15)
\mu yr
\bigl(
gaj(Q
a
ir + \partial .
iP
a
r ) + gai(Q
a
jr + \partial .
jP
a
r )
\bigr)
+
+\mu yrys(gaj\partial
.
iQ
a
rs + gai\partial
.
jQ
a
rs) + \mu
\bigl(
P a
i gaj + P a
j gai
\bigr)
= 2\mu \Omega gij . (4.16)
Case 1. If \mu = 0, since \alpha \mu - \beta 2 \not = 0, we have \beta \not = 0, so from (4.15) we get
\bigl(
LXgij - gai\nabla jv
a + gaiP
a
j
\bigr)
+ ysgai
\bigl(
\partial .
jP
a
s +Qa
js
\bigr)
+
1
2
gaiy
rys\partial .
jQ
a
rs = 2\Omega gij . (4.17)
Putting (4.17) into (4.14), and taking into account the part which does not contain yr, we obtain
\beta LBgij + \alpha gai\nabla jv
a = \alpha gaiP
a
j . (4.18)
Therefore, P a
j depends only on the variables (xh).
Taking into account the parts which contain yr and yrys, we get
vpRa
spj +\nabla jP
a
s = 0,
Qa
js = 0.
Since \partial .
jP
a
s = Qa
js = 0. So, (4.17) turns to
gaj\nabla iv
a + gaiP
a
j = 2\Omega gij . (4.19)
From (4.18) and (4.19), we have
\beta LBgij + \alpha LXgij = 2\alpha \Omega gij .
Case 2. If \mu \not = 0, then from (4.16) we have
(P a
i gaj + P a
j gai) + 2yr(gajQ
a
ir + gaiQ
a
jr)+
+yrys(gaj\partial
.
iQ
a
rs + gai\partial
.
jQ
a
rs) = 2\Omega gij . (4.20)
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
A CLASSIFICATION OF CONFORMAL VECTOR FIELDS ON THE TANGENT BUNDLE 701
Putting (4.20) into (4.14) and taking into account the part which does not contain yr, we obtain
\alpha LXgij + \beta LBgij = \alpha
\bigl(
gaiP
a
j + gajP
a
i
\bigr)
,
therefore, P a
j depends only on the variables (xh). We get
\partial .
jP
a
s = Qa
js = 0. (4.21)
Thus (4.20) turns to
gaiP
a
j + gajP
a
i = 2\Omega gij . (4.22)
Putting (4.20) into (4.15), and taking into account the part which does not contain yr and (4.22), we
obtain
\beta
\bigl(
LXgijgai\nabla jv
a + gaiP
a
j
\bigr)
+ \mu gaj\nabla iB
a = \beta
\bigl(
gaiP
a
j + gajP
a
i
\bigr)
= 2\beta \Omega gij .
Taking into account the part which contains yr and (4.21), we have
vpRa
spj +\nabla jP
a
s = 0.
In both situation, we set Aa
j := P a
j - \nabla jv
a and obtain
\~X = vh
\delta
\delta xh
+
\biggl(
Bh + P h
s y
s +
1
2
Qa
rsy
rys
\biggr)
\partial
\partial yh
=
= vh
\delta
\delta xh
+ ys\nabla sv
h \partial
\partial yh
+ ysAh
s
\partial
\partial yh
+Bh \partial
\partial yh
=
= XC + \gamma A+Bv.
Conversely, if \~X = XC + \gamma A + Bv is given such that X, B and A satisfy in (4.5) – (4.7) or
(4.8) – (4.11), with a simple calculation we see that
L \~X\~g = 2\Omega \~g,
thus \~X is a fiber-preserving conformal vector field with respect to \~g.
Theorem 4.1 is proved.
Corollary 4.1. Let (M, g) be a C\infty Riemannian manifold, TM its tangent bundle and \~g =
= \alpha g + 2\beta g + \mu g the Riemannian (or pseudo-Riemannian) metric on TM derived from g. Every
infinitesimal fibre-preserving conformal transformation on (TM, \~g) is inessential.
Proof. By taking into account proof of Theorem 4.1, we deduce that \Omega depends only on the
variables (xh). Thus \~X is inessential with respect to the induced coordinates (xi, yi) on TM.
Corollary 4.2. Let (M, g) be a C\infty connected Riemannian manifold, TM its tangent bundle
and \~g = \alpha g + 2\beta g + \mu g the Riemannian (or pseudo-Riemannian) metric on TM derived from g.
Every infinitesimal fibre-preserving conformal transformation on (TM, \~g) is homothetic if \mu \not = 0.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
702 Z. RAEI, D. LATIFI
Proof. By using Corollary 4.1, we deduce that \Omega depends on the variables (xh). Applying the
covariant derivative \nabla s to the both sides of part (4.11) from Theorem 4.1, we obtain
\nabla sP
a
i gaj +\nabla sP
a
j gai = 2gij\nabla s\Omega .
By using (4.9), we get
gai\nabla sP
a
j = - vpRijps.
Therefore,
2gij\nabla s\Omega = - vpRijps - vpRjips = 0.
We get \nabla s\Omega =
\partial \Omega
\partial xs
= 0. Since M is connected, the scalar function \Omega is constant.
Corollary 4.2 is proved.
Corollary 4.3. Let (M, g) be a Riemannian manifold and TM be its tangent bundle equipped
with the Riemannian (or pseudo-Riemannian) lift metric \~g. A vector field \~X on TM is a fiber-
preserving Killing vector field with respect to \~g if and only if
\~X = XC + \gamma A+Bv,
where B,X \in \Im 1
0(M) and A = (Aa
h) \in \Im 1
1(M) such that:
1) if \mu = 0, then
A = (Aa
h) = - giaLXgih,
\alpha LXgij + \beta LBgij = 0,
LX\Gamma a
ij = 0;
2) if \mu \not = 0, then
\alpha LXgij + \beta LBgij = 0,
LX\Gamma a
ij = 0,
\beta
\bigl(
gaj\nabla iv
a + gaiP
a
j
\bigr)
+ \mu gaj\nabla iB
a = 0,
gaiP
a
j + gajP
a
i = 0,
where Aa
j := P a
j - \nabla jv
a.
Proof. A vector field \~X is a Killing vector field on TM with respect to \~g if and only if
L \~X\~g = 0. By Theorem 4.1, we say that \~X = vh
\delta
\delta xh
+ wh \partial
\partial yh
is a fiber-preserving Killing vector
field on TM with respect to \~g if and only if the following relations hold:
\~X = XC + \gamma A+Bv,
where B,X \in \Im 1
0(M) and A = (Aa
h) \in \Im 1
1(M).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
A CLASSIFICATION OF CONFORMAL VECTOR FIELDS ON THE TANGENT BUNDLE 703
Case 1. If \mu = 0, then
\beta LBgij + \alpha LXgij = 0, (4.23)
vpRispj + gai\nabla jP
a
s = 0, (4.24)
gaj\nabla iv
a + gaiP
a
j = 0. (4.25)
Since \alpha \mu - \beta 2 \not = 0, then we get \beta \not = 0. Thus, by (4.25) we have
P a
j = \nabla jv
a - gaiLXgij , (4.26)
Thus we set Aa
j := - gaiLXgij .
Now, by putting (4.26) into (4.24), we get
0 = vpRispj + gia\nabla j\nabla sv
a - \nabla jLXgis =
= vpRispj + gia(LX\Gamma a
js - vpRispj) + gja(LX\Gamma a
is - vpRjspi) =
= - gasLX\Gamma a
ij
from which it follows that LX\Gamma a
ij = 0, i.e., X is an infinitesimal affine transformation on M.
Case 2. If \mu \not = 0, then
\beta LBgij + \alpha LXgij = 0,
vpRispj + gai\nabla jP
a
s = 0,
\beta
\bigl(
LXgij - gai\nabla jv
a + gaiP
a
j
\bigr)
+ \mu gaj\nabla iB
a = 0,
gaiP
a
j + gajP
a
i = 0,
where Aa
h = P a
h - \nabla hX
a. By the same method used in case 1, we have
\alpha LXgij + \beta LBgij = 0,
LX\Gamma a
ij = 0,
\beta (gaj\nabla iv
a + gaiP
a
j ) + \mu gaj\nabla iB
a = 0,
gaiP
a
j + gajP
a
i = 0,
where Aa
j := P a
j - \nabla jv
a.
Corollary 4.3 is proved.
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ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 5
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| id | umjimathkievua-article-6013 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:25:13Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/c5/6b1914560f46f9628dd5c158de91a4c5.pdf |
| spelling | umjimathkievua-article-60132022-03-26T11:01:43Z A classification of conformal vector fields on the tangent bundle A classification of conformal vector fields on the tangent bundle Класифiкацiя конформних векторних полiв на дотичному розшаруваннi Raei, Zohre Latifi, Dariush Raei, Zohre Latifi, Dariush Raei, Zohre Latifi, Dariush Fibre-preserving vector field Infinitesimal conformal transformation Lift metric Mathematics subject Classification: 53A45, 53B20, 53C07 UDC 514.7 Let $(M,g)$ be a Riemannian manifold and $TM$ be its tangent bundle equipped with a Riemannian (or pseudo-Riemannian) lift metric derived from $g.$ We give a classification of infinitesimal fibre-preserving conformal transformations on the tangent bundle. Нехай $(M,g)$  --- ріманів многовид, $TM$ --- його дотичне розшарування з рімановою (або псевдорімановою) метрикою підняття, яка породжується $g.$ Наведено класифікацію нескінченно малих  конформних перетворень, що зберігають шари на дотичному розшаруванні. Нехай $(M,g)$  — ріманів многовид, $TM$ — його дотичне розшарування з рімановою (або псевдорімановою) метрикою підняття, яка породжується $g.$ Наведено класифікацію нескінченно малих  конформних перетворень, що зберігають шари на дотичному розшаруванні. Institute of Mathematics, NAS of Ukraine 2020-04-29 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6013 10.37863/umzh.v72i5.6013 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 5 (2020); 694–704 Український математичний журнал; Том 72 № 5 (2020); 694–704 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6013/8697 |
| spellingShingle | Raei, Zohre Latifi, Dariush Raei, Zohre Latifi, Dariush Raei, Zohre Latifi, Dariush A classification of conformal vector fields on the tangent bundle |
| title | A classification of conformal vector fields on the tangent bundle |
| title_alt | A classification of conformal vector fields on the tangent bundle Класифiкацiя конформних векторних полiв на дотичному розшаруваннi |
| title_full | A classification of conformal vector fields on the tangent bundle |
| title_fullStr | A classification of conformal vector fields on the tangent bundle |
| title_full_unstemmed | A classification of conformal vector fields on the tangent bundle |
| title_short | A classification of conformal vector fields on the tangent bundle |
| title_sort | classification of conformal vector fields on the tangent bundle |
| topic_facet | Fibre-preserving vector field Infinitesimal conformal transformation Lift metric Mathematics subject Classification: 53A45 53B20 53C07 |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6013 |
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