Shen's $L$-process on Berwald connection
The Shen connection cannot be obtained by using Matsumoto's processes from the other well-known connections.  Hence Tayebi–Najafi introduced two new processes called Shen's $C$ and $L$-processes and showed that the Shen connection is obtained from the Cher...
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2020
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512209583472640 |
|---|---|
| author | Faghfouri, M. Jazer , N. Faghfouri, Morteza Faghfouri, M. Jazer , N. |
| author_facet | Faghfouri, M. Jazer , N. Faghfouri, Morteza Faghfouri, M. Jazer , N. |
| author_sort | Faghfouri, M. |
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| datestamp_date | 2022-03-26T11:02:01Z |
| description | The Shen connection cannot be obtained by using Matsumoto's processes from the other well-known connections.  Hence Tayebi–Najafi introduced two new processes called Shen's $C$ and $L$-processes and showed that the Shen connection is obtained from the Chern connection by Shen's $C$-process.  In this paper, we  study the Shen's $C$- and $L$-process on Berwald connection and introduce two new torsion-free connections in Finsler geometry.  Then, we obtain all of Riemannian and non-Riemannian curvatures of these connections.  Using it, we find the explicit form of $hv$-curvatures of these connections and prove that $hv$-curvatures of these connections are vanishing if and only if the Finsler structures reduce to Berwaldian or Riemannian structures.  As an application, we consider compact Finsler manifolds and obtain ODEs.
|
| doi_str_mv | 10.37863/umzh.v72i8.6001 |
| first_indexed | 2026-03-24T03:25:09Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v72i8.6001
UDC 514.1
M. Faghfouri, N. Jazer (Univ. Tabriz, Iran)
SHEN’S \bfitL -PROCESS ON BERWALD CONNECTION
\bfitL -ПРОЦЕС ШЕНА НА ЗВ’ЯЗНОСТI БЕРВАЛЬДА
The Shen connection cannot be obtained by using Matsumoto’s processes from the other well-known connections. Hence
Tayebi – Najafi introduced two new processes called Shen’s C and L-processes and showed that the Shen connection is
obtained from the Chern connection by Shen’s C -process. In this paper, we study the Shen’s C - and L-process on Berwald
connection and introduce two new torsion-free connections in Finsler geometry. Then, we obtain all of Riemannian and
non-Riemannian curvatures of these connections. Using it, we find the explicit form of hv-curvatures of these connections
and prove that hv-curvatures of these connections are vanishing if and only if the Finsler structures reduce to Berwaldian
or Riemannian structures. As an application, we consider compact Finsler manifolds and obtain ODEs.
Зв’язнiсть Шена неможливо отримати за допомогою процесу Мацумото з iнших вiдомих процесiв. Тому Тайєбi
та Наджафi запропонували два нових процеси, названi C - та L-процесами Шена, i показали, що за допомогою
C -процесу Шена iз зв’язностi Черна можна отримати зв’язнiсть Шена. Ми вивчаємо C - та L-процеси Шена
на зв’язностi Бервальда i пропонуємо двi новi безторсiоннi зв’язностi у геометрiї Фiнслера. Далi отримуємо всi
рiмановi та нерiмановi кривини для цих зв’язностей. За допомогою цього знаходимо точну форму hv-кривини для
цих зв’язностей i доводимо, що hv-кривини для цих зв’язностей є нульовими тодi й тiльки тодi, коли структури
Фiнслера зводяться до структур Бервальда чи Рiмана. Як застосування розглядаємо компактнi фiнслеровi многовиди
та отримуємо звичайнi диференцiальнi рiвняння.
1. Introduction. In [8], Matsumoto introduced a satisfactory and truly aesthetical axiomatic de-
scription of Cartan’s connection in the sixties. After the Cartan connection has been constructed,
easy processes, baptized by Matsumoto “L-process” and “C -process” (or briefly “ML-process” and
“MC -process”), yield the Chern, the Hashiguchi and the Berwald connections. For other Finslerian
connections, see [3 – 8] and [14]. The space of all connections makes an affine space modeled on
the space of (1, 2)-tensors over pulled-back bundle \pi \ast TM. It means that adding a (1, 2)-tensor to
a connection makes a new connection. A Finsler metric F gives us two natural (1, 2)-tensors with
components Ci
jk and Li
jk. These two (1, 2)-tensors play key role in Matsumoto’s processes. The
C -processes use Cartan tensor, and the L-processes use Landsberg tensor:
Cartan connection
MC-process
- - - - \rightarrow Chern connection
| |
ML-\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{s} ML-\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{s}
\downarrow \downarrow
Hashiguchi connection
MC-process
- - - - \rightarrow Berwald connection.
It is well-known that vanishing hv-curvatures of Cartan and Berwald connections characterizes
Landsberg metrics and Berwald metrics, respectively.
In [11], Shen introduced a new connection in Finsler geometry, which vanishing hv-curvature
of this connection characterizes Riemannian metrics. In [9], Muzsnay and Nagy gave an invariant
treatment of Shen connection. The Shen connection can not be constructed by Matsumoto’s processes
from these known connections. Therefore, Tayebi and Najafi introduced two new processes on
c\bigcirc M. FAGHFOURI, N. JAZER, 2020
1134 ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 8
SHEN’S L-PROCESS ON BERWALD CONNECTION 1135
connections, called Shen’s C and L-processes [17]. For the sake of simplicity, we use “SC -process”
and “SL-process” instead of Shen’s C -process and Shen’s L-process, respectively. Let (M,F ) be a
Finsler manifold. Suppose that \nabla is a connection with connection forms \omega i
j . Define
\~\omega i
j := \omega i
j - Ci
jk\omega
k.
Then \~\omega i
j are connection forms of a connection \widetilde \nabla , that is called the connection obtained from \nabla by
Shen’s C -process. Similarly, one can define
\~\omega i
j := \omega i
j - Li
jk\omega
n+k.
Then \~\omega i
j are connection forms of a connection \widetilde \nabla , that is called the connection obtained from \nabla by
Shen’s L-process. Tayebi and Najafi showed that the Shen connection is obtained from the Chern
connection by Shen’s C -process.
In this paper, we are going to study the connections which obtain by Shen’s C - and L-process
on Berwald connection. In Section 3, we study the connection obtained by Shen’s L-process on the
Berwald connection, call it by D, and prove the existence and uniqueness of this connection. In
Section 4, we show that the hv-curvature of D vanishes if and only if F is a Berwald metric. Let
P j
i kl = P j
i kl(x, y) and P j
n kl = P j
i kly
i denote the hv-curvature and contracted hv-curvature of D,
respectively. In Section 5, we prove that on a compact Finsler manifold the contracted hv-curvature
of D is vanishing if and only if F is a Landsberg metric. In Section 6, we study the connection
obtained by Shen’s C -process on the Berwald connection, call it by \nabla , and prove the existence and
uniqueness of this connection. Finally, in Section 7, we show that the hv-curvature of \nabla vanishes if
and only if F reduce to a Riemannian metric.
2. Preliminaries. Let M be an n-dimensional C\infty manifold. Denote by TxM the tangent
space at x \in M, and by TM :=
\bigcup
x\in M TxM the tangent bundle of M. Each element of TM has
the form (x, y), where x \in M and y \in TxM. Let TM0 = TM \setminus \{ 0\} . The natural projection \pi :
TM \rightarrow M is given by \pi (x, y) := x.
The pull-back tangent bundle \pi \ast TM is a vector bundle over TM0 whose fiber \pi \ast
vTM at v \in
\in TM0 is TxM, where \pi (v) = x. Then
\pi \ast TM =
\Bigl\{
(x, y, v)
\bigm| \bigm| \bigm| y \in TxM0, v \in TxM
\Bigr\}
.
Some authors prefer to define connections in the pull-back tangent bundle \pi \ast TM. From geometrical
point of view, the construction of these connections on \pi \ast TM seems to be simple because here
the fibers are n-dimensional
\bigl(
i.e., \pi \ast (TM)u = T\pi (u)M \forall u \in TM
\bigr)
thus torsions and curvatures
are obtained quickly from the structure equations. When the construction is done on T (TM) many
geometrical objects appear twice and one needs to split T (TM) in the vertical and horizontal parts
where the latter is called horizontal distribution or nonlinear connection. Nevertheless we do not need
to split \pi \ast TM. Indeed the connection on \pi \ast (TM) is the most natural connection for physicists. In
order to define curvatures, it is more convenient to consider the pull-back tangent bundle than the
tangent bundle, because our geometric quantities depend on directions.
For the sake of simplicity, we denote by\Biggl\{
\partial i| v :=
\Biggl(
v,
\partial
\partial xi
\bigm| \bigm| \bigm| \bigm| \bigm|
x
\Biggr) \Biggr\} n
i=1
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 8
1136 M. FAGHFOURI, N. JAZER
the natural basis for \pi \ast
vTM. In Finsler geometry, we study connections and curvatures in (\pi \ast TM, g),
rather than in (TM,F ). The pull-back tangent bundle \pi \ast TM is very special tangent bundle.
A (globally defined) Finsler structure on a manifold M is a function F : TM \rightarrow [0,\infty ), with
the following properties:
(i) F is a differentiable function on the manifold TM0 and is continuous on the null section of
the projection \pi : TM \rightarrow M ;
(ii) F : TM \rightarrow [0,\infty ) is a positive scalar function;
(iii) F is positively 1-homogeneous on the fibers of tangent bundle TM ;
(iv) the Hessian of F 2 with elements
(gij) :=
\Biggl( \biggl[
1
2
F 2
\biggr]
yiyj
\Biggr)
is positively defined on TM0. Given a manifold M and a Finsler structure F on M, the pair (M,F )
is called a Finsler manifold. F is called Riemannian if gij(x, y) are independent of y \not = 0.
The Finsler structure F defines a fundamental tensor g : \pi \ast TM \otimes \pi \ast TM \rightarrow [0,\infty ) by the
formula g(\partial i| v, \partial j | v) = gij(x, y), where v = yi
\partial
\partial xi
\bigm| \bigm| \bigm|
x
. Let
gij(x, y) := FFyiyj + FyiFyj ,
where Fyi =
\partial F
\partial yi
. Then (\pi \ast TM, g) becomes a Riemannian vector bundle over TM0.
Put
Aijk(x, y) =
1
2
F (x, y)
\partial gij
\partial yk
(x, y) .
Clearly, Aijk is symmetric with respect to i, j, k. The Cartan tensor
A : \pi \ast TM \otimes \pi \ast TM \otimes \pi \ast TM \rightarrow R
is defined by
A(\partial i| v, \partial j | v, \partial k| v) = Aijk(x, y),
where v = yi
\partial
\partial xi
\bigm| \bigm| \bigm|
x
(see [15, 19]). In some literature Cijk =
Aijk
F
is called Cartan tensor. Rieman-
nian manifolds are characterized by A \equiv 0. The homogeneity condition (iii) holds in particular for
positive \lambda . Therefore, by Euler’s theorem we see that
yi
\partial gij
\partial yk
(x, y) = yj
\partial gij
\partial yk
(x, y) = yk
\partial gij
\partial yk
(x, y) = 0.
We recall that the canonical section \ell is defined by
\ell = \ell (x, y) =
yi
F (x, y)
\partial
\partial xi
=
yi
F
\partial
\partial xi
:= \ell i
\partial
\partial xi
.
Put \ell i := gij\ell
j = Fyi . Thus the canonical section \ell satisfies
g(\ell , \ell ) = gij
yi
F
yj
F
= 1
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 8
SHEN’S L-PROCESS ON BERWALD CONNECTION 1137
and
\ell iAijk = \ell jAijk = \ell kAijk = 0.
Thus, A(X,Y, \ell ) = 0.
Given an n-dimensional Finsler manifold (M,F ), then a global vector field \bfG is induced by F
on TM0, which in a standard coordinate (xi, yi) for TM0 is given by
\bfG = yi
\partial
\partial xi
- 2Gi(x, y)
\partial
\partial yi
,
where Gi = Gi(x, y) are called spray coefficients and given by the following:
Gi =
1
4
gil
\biggl[
\partial 2F 2
\partial xk\partial yl
yk - \partial F 2
\partial xl
\biggr]
.
\bfG is called the spray associated to F.
Define \bfB y : TxM \otimes TxM \otimes TxM \rightarrow TxM by \bfB y(u, v, w) := Bi
jkl(y)u
jvkwl \partial
\partial xi
\bigm| \bigm| \bigm|
x
, where
Bi
jkl :=
\partial 3Gi
\partial yj\partial yk\partial yl
=
\partial 2N i
j
\partial yk\partial yl
.
\bfB y(u, v, w) is symmetric in u, v and w. From the homogeneity of spray coefficients, we have
\bfB y(y, v, w) = 0. \bfB is called the Berwald curvature. Indeed, L. Berwald first discovered that the
third order derivatives of spray coefficients give rise to an invariant for Finsler metrics. F is called a
Berwald metric if \bfB = \bfzero [16]. In this case, Gi are quadratic in y \in TxM for all x \in M, i.e., there
exists \Gamma i
jk = \Gamma i
jk(x) such that
Gi = \Gamma i
jky
jyk.
There is another equal definition for a Berwald metric as follows. A Finsler metric F is called a
Berwald metric if the Cartan torsion of F satisfies the following:
Aijk| l = 0,
where the ”| ” and ”,” denote the horizontal and vertical covariant derivatives with respect to the
Berwald connection.
For y \in TxM, define the Landsberg curvature \bfL y : TxM \otimes TxM \otimes TxM \rightarrow \BbbR by
\bfL y(u, v, w) := - 1
2
\bfg y (\bfB y(u, v, w), y) .
In local coordinates, \bfL y(u, v, w) := L ijk(y)u
ivjwk, where
Lijk := - 1
2
ylB
l
ijk.
\bfL y(u, v, w) is symmetric in u, v and w and \bfL y(y, v, w) = 0. \bfL is called the Landsberg curvature.
A Finsler metric F is called a Landsberg metric if \bfL y = 0 [12]. Equivalently, a Finsler metric F is
called a Landsberg metric if the Cartan torsion of F satisfies the following:
Aijk| mym = 0.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 8
1138 M. FAGHFOURI, N. JAZER
It is easy to see that, every Berwald metric is a Landsberg metric.
2.1. The bundle maps. In [1], Akbar-Zadeh developed the modern theory of global Finsler
geometry by establishing a global definition of Cartan connection. For this aim, he introduced two
bundle maps \rho and \mu . Here, we give a short introduction of these bundle maps. Let TTM be the
tangent bundle of TM and \rho the canonical linear mapping
\rho : TTM0 \rightarrow \pi \ast TM,
\^X \mapsto - \rightarrow
\Bigl(
z, \pi \ast ( \^X)
\Bigr)
,
where \^X \in TzTM0 and z \in TM0. The bundle map \rho satisfies
\rho
\biggl(
\partial
\partial xi
\biggr)
= \partial i, \rho
\biggl(
\partial
\partial yi
\biggr)
= 0.
Let VzTM be the set of vertical vectors at z, that is, the set of vectors tangent to the fiber through
z, or equivalently VzTM = \mathrm{k}\mathrm{e}\mathrm{r} \rho , called the vertical space.
By means of these considerations, one can see that the following sequence is exact:
0 \rightarrow V TM
i - \rightarrow TTM
\rho - \rightarrow \pi \ast TM - \rightarrow 0,
where i is the natural inclusion map.
Let \nabla be a linear connection on \pi \ast TM, that is \nabla : TzTM0 \times \pi \ast TM \rightarrow \pi \ast TM such that \nabla :
( \^X,Y ) \rightarrow \nabla \^XY. Let us define the linear mapping
\mu z : TzTM0 \rightarrow T\pi zM,
\^X \mapsto - \rightarrow \nabla \^XF\ell ,
where \^X \in TzTM0. For a torsion-free connection \nabla the bundle map \mu satisfies
\mu
\biggl(
\partial
\partial xi
\biggr)
= Nk
i \partial k, \mu
\biggl(
\partial
\partial yi
\biggr)
= \nabla \partial
\partial yi
F\ell = \rho
\biggl( \biggl[
\partial
\partial yi
, yk
\partial
\partial xk
\biggr] \biggr)
= \partial i,
where Nk
i = F\Gamma k
ij\ell
j and \Gamma k
ij are Christoffel symbols of \nabla .
Let us put
\delta
\delta xi
:=
\partial
\partial xi
- Nk
i
\partial
\partial yk
.
Then
\mu
\biggl(
\delta
\delta xi
\biggr)
= 0.
The connection \nabla is called a Finsler connection if for every z \in TM0, \mu z defines an isomorphism
of VzTM0 onto T\pi zM. Therefore, the tangent space TTM0 in z is decomposed as
TzTM0 = HzTM \oplus VzTM,
where HzTM = \mathrm{k}\mathrm{e}\mathrm{r}\mu z is called the horizontal space defined by \nabla . Indeed any tangent vector
\^X \in TzTM0 in z decomposes to
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 8
SHEN’S L-PROCESS ON BERWALD CONNECTION 1139
\^X = H \^X + V \^X,
where H \^X \in HzTM and V \^X \in VzTM. Thus \rho restricted to HTM is an isomorphism onto
\pi \ast TM, and \mu restricted to V TM is the bundle isomorphism onto \pi \ast TM.
The structural equations of the Finsler connection \nabla are
\scrT \nabla ( \^X, \^Y ) := \nabla \^XY - \nabla \^Y X - \rho [ \^X, \^Y ],
\Omega ( \^X, \^Y )Z := \nabla \^X\nabla \^Y Z - \nabla \^Y \nabla \^XZ - \nabla [ \^X, \^Y ]Z,
where X = \rho ( \^X), Y = \rho ( \^Y ) and Z = \rho ( \^Z). The tensors \scrT \nabla and \Omega are called, respectively, the
torsion and curvature tensors of \nabla . They determine two torsion tensors defined by
\scrS (X,Y ) := \scrT \nabla (H \^X,H \^Y ), \scrT ( \.X,Y ) := \scrT \nabla (V \^X,H \^Y )
and three curvature tensors defined by
R(X,Y ) := \Omega (H \^X,H \^Y ),
P (X, \.Y ) := \Omega (H \^X,V \^Y ),
Q( \.X, \.Y ) := \Omega (V \^X,V \^Y ),
where \.X = \mu ( \^X) and \.Y = \mu ( \^X).
3. Shen’s \bfitL -process on Berwald connection. In this section, we are going to study the
connection obtained by Shen’s L-process on the Berwald connection. For this aim, we give a short
and exact definition of the Berwald connection.
In 1926, L. Berwald introduced a connection and two curvature tensors. The Berwald connection
is torsion-free, but is not necessarily metric-compatible [2]. It was Berwald who first successfully
extended the notion of Riemann curvature to Finsler spaces. He also introduced a notion of non-
Riemannian quantity called Berwald curvature.
The Berwald connection introduced by the following properties:
Berwald connection: Let (M,F ) be an n-dimensional Finsler manifold. Then the Berwald
connection \frakD is a linear connection in \pi \ast TM, which has the following properties:
(i) \frakD is torsion-free, i.e., for all \^X, \^Y \in C\infty (T (TM0)) ,
\scrT ( \^X, \^Y ) := \frakD \^X\rho ( \^Y ) - \frakD \^Y \rho (
\^X) - \rho
\Bigl(
[ \^X, \^Y ]
\Bigr)
= 0. (3.1)
(ii) \frakD is almost compatible with F in the following sence:
(\frakD \^Zg)(X,Y ) := \^Zg(X,Y ) - g(\frakD \^ZX,Y ) - g(X,\frakD \^ZY ) =
= 2F - 1A(\mu ( \^Z), X, Y ) - 2 \.A(\rho ( \^Z), X, Y ),
(3.2)
where X, Y \in C\infty (\pi \ast TM) and \^Z \in Tv(TM0).
In [17], Tayebi and Najafi did not consider the Shen’s C -process on Berwald connection. Here,
we apply Shen’s C -process on Berwald connection and find a new torsion-free connection. First, we
prove the existence of this linear Finslerian connection.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 8
1140 M. FAGHFOURI, N. JAZER
Theorem 3.1. Let (M,F ) be an n-dimensional Finsler manifold. Then there is a unique linear
connection D in \pi \ast TM, which has the following properties:
(i) D is torsion-free in the sense of (3.1);
(ii) D is almost compatible with the Finsler structure in the following sense: for all X,Y \in
\in C\infty (\pi \ast TM) and \^Z \in Tv(TM0),
(D \^Zg)(X,Y ) := - 2 \.A
\Bigl(
\rho ( \^Z), X, Y
\Bigr)
+ 2F - 1
\Bigl[
A
\Bigl(
\mu ( \^Z), X, Y
\Bigr)
+ \.A
\Bigl(
\mu ( \^Z), X, Y
\Bigr) \Bigr]
. (3.3)
Proof. In a standard local coordinate system (xi, yi) in TM0, we write
D \partial
\partial xi
\partial j = \Gamma k
ij\partial k, D \partial
\partial yi
\partial j = F k
ij\partial k.
Clearly, (3.1) and (3.3) are equivalent to the following:
\Gamma k
ij = \Gamma k
ji, (3.4)
F k
ij = 0, (3.5)
\partial (gij)
\partial xk
= \Gamma l
kiglj + \Gamma l
kjgil - 2 \.Aijk + 2\Gamma l
kmlm(Alij + \.Alij), (3.6)
\partial (gij)
\partial yk
= F s
ikgsj + F s
kjgis + 2F - 1(Aijk + \.Aijk) - 2F s
mkl
m \.Aijk. (3.7)
Note that (3.5) and (3.7) are just the definition of Aijk. We must compute \Gamma k
ij from (3.4) and (3.6).
Then making a permutation to i, j, k in (3.6), and, by using (3.4), we obtain
\Gamma k
ij = \gamma kij +
\.Ak
ij + gkl
\Bigl\{
\Gamma m
lb (Amij + \.Amij) - \Gamma m
ib (Amjl + \.Amlj) - \Gamma m
jb(Amil + \.Amil)
\Bigr\}
\ell b, (3.8)
where
\gamma kij :=
1
2
gkl
\biggl\{
\partial gjl
\partial xi
+
\partial gil
\partial xj
- \partial gij
\partial xl
\biggr\}
(3.9)
and Ak
ij = gklAijl. Multiplying (3.8) by \ell i implies that
\Gamma k
ib\ell
b = \gamma kib\ell
b - (Ak
im + \.Ak
im)\Gamma m
lb \ell
l\ell b. (3.10)
Contracting (3.10) with \ell j yields
\Gamma k
ab\ell
a\ell b = \gamma kab\ell
a\ell b. (3.11)
By putting (3.11) in (3.10), one can obtain
\Gamma k
ib\ell
b = \gamma kib\ell
b - \ell a\ell b\gamma mab(A
k
mi +
\.Ak
mi). (3.12)
Putting (3.12) in (3.8) gives us the following:
\Gamma k
ij = \gamma kij +
\.Ak
ij + gkl
\Bigl\{
\gamma mlb (Amij + \.Amij) - \gamma mib (Amlj + \.Amlj) - \gamma mjb(Amil + \.Amil)
\Bigr\}
\ell b+
+\gamma sab\ell
a\ell b
\Bigl\{
(Am
sj +
\.Am
sj)(A
k
mi +
\.Ak
mi) + (Am
si + \.Am
si)(A
k
mj + \.Ak
mj) -
- (Ak
sm + \.Ak
sm)(Am
ij + \.Am
ij )
\Bigr\}
.
This proves the uniqueness of D. The set \{ \Gamma k
ij , F
k
ij = 0\} , where \{ \Gamma k
ij\} are given by (3.1), defines a
linear connection D satisfying (3.1) and (3.3).
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SHEN’S L-PROCESS ON BERWALD CONNECTION 1141
4. Curvatures of the connection \bfitD . The curvature tensor \Omega of D is defined by
\Omega ( \^X, \^Y )Z = D \^XD \^Y Z - D \^Y D \^XZ - D[ \^X, \^Y ]Z,
where \^X, \^Y \in C\infty (T (TM0)) and Z \in C\infty (\pi \ast TM). Let \{ ei\} ni=1 be a local orthonormal (with
respect to g) frame field for the vector bundle \pi \ast TM such that g(ei, en) = 0, i = 1, . . . , n - 1 and
en :=
y
F
=
yi
F (x, y)
\partial
\partial xi
= \ell .
Let \{ \omega i\} ni=1 be its dual co-frame field. These are local sections of dual bundle \pi \ast TM. One readily
finds that
\omega n :=
\partial F
\partial xi
= \ell idx
i = \omega ,
which is the Hilbert form. It is obvious that \omega (\ell ) = 0.
Now, let us put
\rho = \omega i \otimes ei, Dei = \omega j
i \otimes ej , \Omega ei = 2\Omega j
i \otimes ej .
\{ \Omega j
i \} and \{ \omega j
i \} are called the curvature forms and connection forms of D with respect to \{ ei\} .
We have \mu := DF\ell = F\{ \omega i
n + d(\mathrm{l}\mathrm{o}\mathrm{g}F )\delta in\} \otimes ei. Put \omega n+i := \omega i
n + d(\mathrm{l}\mathrm{o}\mathrm{g}F )\delta in. It is easy to
see that \{ \omega i, \omega n+i\} ni=1 is a local basis for T \ast (TM0). By definition \rho = \omega i \otimes ei, \mu = F\omega n+i \otimes ei.
Using the above formula for Theorem 3.1, it then re-expresses the structure equation of the new
connection D as follows:
d\omega i = \omega j \wedge \omega i
j , (4.1)
dgij = gkj\omega
k
i + gki\omega
k
j - 2 \.Aijk\omega
k + 2(Aijk + \.Aijk)\omega
n+k. (4.2)
Define gij.k and gij| k by
dgij - gkj\omega
k
i - gik\omega
k
j = gij| k\omega
k + gij.k\omega
n+k,
where gij.k and gij| k are, respectively, the vertical and horizontal covariant derivative of gij with
respect to the connection D. This gives
gij| k = - 2 \.Aijk,
gij.k = 2(Aijk + \.Aijk).
It can be shown that \delta ij| s = 0 and \delta ij.s = 0, thus (gijgjk)| s = 0 and (gijgjk).s = 0. So,
gij| s = 2 \.Aij
s , gij.s = 2( \.Aij
s +Aij
s ).
Moreover, torsion freeness is equivalent to the absent of dyk in \{ \omega i
j \} namely
\omega i
j = \Gamma i
jk(x, y)dx
k,
which is equivalent to
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1142 M. FAGHFOURI, N. JAZER
d\omega j
i - \omega k
i \wedge \omega j
k = \Omega j
i . (4.3)
Since the \Omega i
j are 2-forms on the manifold TM0, they can be generally expanded as
\Omega j
i =
1
2
R j
i kl\omega
k \wedge \omega l + P j
i kl\omega
k \wedge \omega n+l +
1
2
Q j
i kl\omega
n+k \wedge \omega n+l. (4.4)
The objects R, P and Q are respectively the hh-, hv- and vv-curvature tensors of the connection D.
Let \{ \=ei, \.ei\} ni=1 be the local basis for T (TM0), which is dual to \{ \omega i, \omega n+i\} ni=1, i.e., \=ei \in HTM, \.ei \in
\in V TM such that \rho (\=ei) = ei, \mu ( \.ei) = Fei. Let us put
R(\=ek, \=el)ei = R j
i klej , P (\=ek, \.el)ei = P j
i klej , Q( \.ek, \.el)ei = Q j
i klej .
The connection defined in Theorem 3.1 is torsion-free. Then we have Q = 0. First Bianchi identity
for R is given by
R j
i kl +R j
k li +R j
l ik = 0
and
P j
i kl = P j
k il. (4.5)
Exterior differentiation of (4.3) gives the second Bianchi identity
d\Omega j
i - \omega k
i \wedge \Omega j
k + \omega j
k \wedge \Omega k
i = 0. (4.6)
We decompose the covariant derivative of the Cartan tensor on TM
dAijk - Aljk\omega
l
i - Ailk\omega
l
j - Aijl\omega
l
k = Aijk| l\omega
l +Aijk.l\omega
n+l. (4.7)
Similarly, for \.Aijk, we get
d \.Aijk - \.Aljk\omega
l
i - \.Ailk\omega
l
j - \.Aijl\omega
l
k = \.Aijk| l\omega
l + \.Aijk.l\omega
n+l. (4.8)
It is easy to see that, Aijk| l, Aijk.l, \.Aijk| l and \.Aijk.l are symmetric with respect to indices i, j and
k.
Put \.Aijk = \.A(ei, ej , ek). Then
Aijk| n = \.Aijk.
By (4.7) and (4.8), we get
Anjk| l = 0, and Anjk.l = - Ajkl,
\.Anjk| l = 0, and \.Anjk.l = - \.Ajkl.
Theorem 4.1. Let (M,F ) be a Finsler manifold. Suppose that D is the linear torsion-free
connection obtained by Shen’s L-process on Berwald’s connection. Then the hv-curvature of D
vanishes if and only if F is a Berwald metric.
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SHEN’S L-PROCESS ON BERWALD CONNECTION 1143
Proof. Let (M,F ) be a Finsler manifold. Differentiating (4.2), and using (4.1), (4.2), (4.3), (4.7)
and (4.8) leads to
d\omega i = \omega j \wedge \omega i
j , (4.9)
dgij = gkj\omega
k
i + gki\omega
k
j - 2 \.Aijk\omega
k + 2(Aijk + \.Aijk)\omega
n+k. (4.10)
By differentiating of (4.10), we get
0 = dgik\omega
k
j + gikd\omega
k
j + dgjk\omega
k
i + gjkd\omega
k
i + 2(dAijk + d \.Aijk)\omega
n+k+
+2(Aijk + \.Aijk)d\omega
n+k - 2d \.Aijk\omega
k - 2 \.Aijkd\omega
k.
Using (4.4), (4.6) and (4.7), one can obtain
Rijkl +Rjikl = - 2AijsR
s
n kl - 2 \.AijsR
s
n kl - 4 \.Aijk| l, (4.11)
Pijkl + Pjikl = - 2 \.Aijk,l - 2(Aijl| k + \.Aijl| k) - 2(Aijs + \.Aijs)P
s
n kl, (4.12)
Aijk.m + \.Aijk.m = 0. (4.13)
Permuting i, j, k in (4.12) yields
Pjkil + Pkjil = - 2 \.Ajki.l - 2(Ajkl| i + \.Ajkl| i) - 2(Ajks + \.Ajks)P
s
n il, (4.14)
Pkijl + Pikjl = - 2 \.Akij.l - 2(Akil| j + \.Akil| j) - 2(Akis + \.Akis)P
s
n jl. (4.15)
From (4.12), (4.14) and (4.15), we get
Pijkl = - \.Aijk,l -
\Bigl[
(Aijl| k + \.Aijl| k) + (Ajkl| i + \.Ajkl| i) - (Akil| j + \.Akil| j)
\Bigr]
-
-
\Bigl[
(Aijs + \.Aijs)P
s
n kl + (Ajks + \.Ajks)P
s
n il - (Akis + \.Akis)P
s
n jl
\Bigr]
. (4.16)
Taking a vertical derivation of \.Aijky
i = 0 with respect to yl implies that
\.Aijk,ly
i = - \.Ajkl. (4.17)
Multiplying (4.17) with yj yields
\.Aijk,ly
iyj = 0. (4.18)
By contracting (4.16) with yi and considering Aijl| ky
i = 0, (4.18) and (4.17), we get
Pnjkl = - \"Ajkl - (Ajks + \.Ajks)P
s
n nl. (4.19)
On the other hand, multiplying (4.12) with yiyj implies that Pnnkl = 0. Thus, by (4.5), we have
Pknnl = 0. (4.20)
Contracting (4.12) with yjyk yields
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1144 M. FAGHFOURI, N. JAZER
Pinnl + Pninl = 0. (4.21)
By (4.20) and (4.21) it follows that
Pninl = 0. (4.22)
Putting (4.22) in (4.19) implies that
Pnjkl = - \"Ajkl. (4.23)
Let F be a Berwald metric. Thus from (4.23), we get Pnjkl = 0 or, equivalently, P j
n kl = 0. By
putting it and Aijk| l = 0 in (4.16), we get P = 0.
Conversely let P = 0. By (4.23), it follows that \"Ajkl = 0. By assumption, (4.16) reduces to
following:
\.Aijk,l = - (Aijl| k + \.Aijl| k) - (Ajkl| i + \.Ajkl| i) + (Akil| j + \.Akil| j). (4.24)
Permuting i, j, k in the above identity leads to
\.Ajki,l = - (Ajkl| i + \.Ajkl| i) - (Akil| j + \.Akil| j) + (Aijl| k + \.Aijl| k). (4.25)
(4.24), (4.25) yields
Aijl| k + \.Aijl| k = Akil| j + \.Akil| j . (4.26)
Contracting (4.26) with yk implies that
\.Aijl = - \"Aijl. (4.27)
Since \"Ajkl = 0, then (4.27) reduces to \.Aijk = 0. Putting it and P = 0 in (4.12) imply Aijk| l = 0.
This means that F is a Berwald metric.
5. Compact Finsler manifolds. Let \=\ell denote the unique vector field in HTM such that
\rho (\=\ell ) = \ell . We call \=\ell the geodesic field on TM0, because it determines all geodesics and it is called a
spray.
Let c : [a, b] \rightarrow (M,F ) be a unit speed C\infty curve. The canonical lift of c to TM0 is defined by
\^c :=
dc
dt
\in TM0. It is easy to see that \rho
\biggl(
d\^c
dt
\biggr)
= \ell \^c. The curve c is called a geodesic if its canonical
lift \^c satisfies
d\^c
dt
= \ell \^c, where \=\ell is the geodesic field on TM0, i.e., \ell \in HTM, \rho (\=\ell ) = \ell .
Let IxM = \{ v \in TxM,F (v) = 1\} and IM =
\bigcup
p\in M IxM. The IxM is called indicatrix, and
it is a compact set. We can show that the projection of integral curve \varphi (t) of \=\ell with \varphi (0) \in IM is
a unit speed geodesics c whose canonical lift is \^c(t) = \varphi (t). A Finsler manifold (M,F ) is called
complete if any unit speed geodesic c : [a, b] \rightarrow M can be extended to a geodesic defined on R. This
is equivalent to requiring that the geodesic field \=\ell restricted to IM is complete.
Let (M,F ) be a Finsler manifold and c be a unit speed geodesic in M. A section X = X(t) of
\pi \ast TM along \^c is said to be parallel if D d\^c
dt
X = 0. For v \in TM0, let us define
\| A\| v = \mathrm{s}\mathrm{u}\mathrm{p}A(X,Y, Z),
where the supremum is taken over all unit vectors of \pi \ast
vTM. Put \| A\| v = \mathrm{s}\mathrm{u}\mathrm{p}v\in IM \| A\| v. Then we
have the following theorem.
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SHEN’S L-PROCESS ON BERWALD CONNECTION 1145
Theorem 5.1. Let (M,F ) be a compact Finsler manifold. Then F is a Landsberg metric if and
only if
\"A = 0. (5.1)
This means that, on compact manifolds Pnjkl = 0 if and only if F is a Landsberg metric.
Proof. Let us fix X, Y, Z \in \pi \ast TM at v \in IxM. Suppose that c : M \rightarrow R is the unit speed
geodesic with
dc
dt
(0) = v. Let X(t), Y (t) and Z(t) denote the parallel sections along \^c with
X(0) = X, Y (0) = Y and Z(0) = Z. Put
A(t) = A (X(t), Y (t), Z(t)) ,
\.\bfA (t) = \.A (X(t), Y (t), Z(t)) ,
\"\bfA (t) = \"A (X(t), Y (t), Z(t)) .
By definition, (5.1) implies that
\.\bfA =
d\bfA
dt
\mathrm{a}\mathrm{n}\mathrm{d} \"\bfA =
d \.\bfA
dt
. (5.2)
Therefore, from (5.1) and (5.2), we have
d \.\bfA
dt
= 0.
Then
A(t) = t \.\bfA (0) + A(0).
Since M is compact then it is complete and \| \.A\| < \infty . Then by letting t \rightarrow - \infty or t \rightarrow \infty , we get
\.A(0) = \.A(X,Y, Z) = 0.
Thus, F is a Landsberg metric.
Remark 5.1. Suppose that F satisfies (5.1). This equation is equivalent to that for any linearly
parallel vector fields u, v, w along a geodesic c, the following holds:
d
dt
\Bigl[
\.\bfA \.c(u, v, w)
\Bigr]
= 0.
The geometric meaning of this is that the rate of change of the Landsberg curvature is constant along
any Finslerian geodesic [22].
An (\alpha , \beta )-metric is a Finsler metric defined by F := \alpha \phi (s), s = \beta /\alpha , where \phi is a smooth
function on a symmetric interval ( - b0, b0) with certain regularity, \alpha is a Riemannian metric and
\beta is a 1-form on the base manifold (see [13, 20, 21]). There is a special class of (\alpha , \beta )-metric,
namely Randers metrics. A Randers metric F = \alpha +\beta on a manifold M is just a Riemannian metric
\alpha perturbated by a one form \beta on M such that the Riemanninan length of \beta \sharp is less than 1 (see
[10, 18]).
In the proof of the main theorem in [22], the authors used the condition gijLijk| sy
s = 0 and
proved that every Randers metric with closed one form \beta is a stretch metric if and only if it is
Berwaldian. Then, we get the following corollary.
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1146 M. FAGHFOURI, N. JAZER
Corollary 5.1. Let (M,F ) be a Finsler – Randers manifold equipped with the Finsler connection
D. Then F is a Berwald metric if and only if Pnjkl = 0.
The Corollary 5.1 can be considered as an extension of Theorem 5.1. We delete the condition
“compact” and replace the Randers manifold instead of arbitrary manifold.
6. Shen’s \bfitC -process on Berwald connection. In this section, we are going to study the
connection obtained by Shen’s C -process on the Berwald connection.
Theorem 6.1. Let (M,F ) be an n-dimensional Finsler manifold. Then there is a unique linear
connection \nabla in \pi \ast TM, which has the following properties:
(i) \nabla is torsion-free in the sense of (3.1);
(ii) \nabla is almost compatible with the Finsler structure in the following sense:
(\nabla \^Zg)(X,Y ) := 2
\Bigl[
A(\rho ( \^Z), X, Y ) - \.A(\rho ( \^Z), X, Y )
\Bigr]
+ 2F - 1A(\mu ( \^Z), X, Y ), (6.1)
where X,Y \in C\infty (\pi \ast TM) and \^Z \in Tv(TM0).
Proof. In a standard local coordinate system (xi, yi) in TM0, we write
\nabla \partial
\partial xi
\partial j = \Gamma k
ij\partial k, \nabla \partial
\partial yi
\partial j = F k
ij\partial k.
The equations (3.1) and (6.1) are equivalent to
\Gamma k
ij = \Gamma k
ji, (6.2)
F k
ij = 0, (6.3)
\partial (gij)
\partial xk
= \Gamma l
kigjl + \Gamma l
kjgli + 2(Aijk - \.Aijk) + 2\Gamma l
kmlmAijl, (6.4)
\partial (gij)
\partial yk
= F l
ikglj + F l
kjgli + 2F - 1Aijk + 2F l
mkl
mAijl. (6.5)
Then making a permutation to i, j, k in (6.4), and by using (6.2), we obtain
\Gamma k
ij = \gamma kij - (Ak
ij - \.Ak
ij) + gkl
\bigl\{
Aijm\Gamma m
lb - Aljm\Gamma m
ib - \Gamma m
jbAilm
\bigr\}
\ell b,
where \gamma kij defined by (3.9). By the same argument used in Theorem 3.1, we get
\Gamma k
ij = \gamma kij - (Ak
ij - \.Ak
ij) + gkl
\bigl\{
Aijm\gamma mlb - Ajlm\gamma mib - Alim\gamma mjb
\bigr\}
\ell b+
+
\Bigl\{
Ak
jmAm
is +Ak
imAm
js - Ak
smAm
ij
\Bigr\}
\gamma sab\ell
a\ell b. (6.6)
This proves the uniqueness of \nabla . The set \{ \Gamma k
ij , F
k
ij = 0\} , where \{ \Gamma k
ij\} are given by (6.6), defines
a linear torsion-free and almost compatible connection \nabla satisfying (3.1) and (6.1).
Here, we remark that the connection \nabla can be expressed by the following equations
d\omega i = \omega j \wedge \omega i
j , (6.7)
dgij = gki\omega
k
j + gjk\omega
k
i + 2(Aijk - \.Aijk)\omega
k + 2Aijk\omega
n+k. (6.8)
Thus,
gij| k = 2(Aijk - \.Aijk), gij.k = 2Aijk.
By a simple calculation, we get the following theorem.
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SHEN’S L-PROCESS ON BERWALD CONNECTION 1147
Theorem 6.2. The new connection \nabla can be obtained from the Shen connection by Matsumoto’s
L-process.
Thus we get the following diagram:
Chern connection
SC-process
- - - - \rightarrow Shen connection
| |
ML-\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{s} ML-\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{c}\mathrm{e}\mathrm{s}\mathrm{s}
\downarrow \downarrow
Berwald connection
SC-process
- - - - \rightarrow the connection \nabla .
7. Curvatures of the connection \nabla .
Theorem 7.1. Let (M,F ) be an n-dimensional Finsler manifold. Suppose that \nabla is obtained
from the Berwald connection by Shen’s C -process. Then the hv-curvature of \nabla vanishes if and only
if F is Riemannian.
Proof. Let (M,F ) be a Finsler manifold. Differentiating (6.8), and using (6.7), (6.8), (4.3), (4.7)
and (4.8) leads to
d\omega i = \omega j \wedge \omega i
j , (7.1)
dgij = gkj\omega
k
i + gki\omega
k
j + 2(Aijk - \.Aijk)\omega
k + 2Aijk\omega
n+k. (7.2)
By differentiating of (7.2), we get
dgik\omega
k
j + gikd\omega
k
j + dgjk\omega
k
i + gjkd\omega
k
i + 2(dAijk - d \.Aijk)\omega
k+
+2(Aijk - \.Aijk)d\omega
k + 2dAijk\omega
n+k + 2Aijkd\omega
n+k = 0. (7.3)
Putting (4.4) in (7.3) implies that
Rijkl +Rjikl = - 2AijsR
s
n kl, (7.4)
Pijkl + Pjikl = - 2 \.Aijk.l + 2(Aijk.l - Aijl| k) - 2AijsP
s
nkl, (7.5)
Aijk.l = Aijl.k. (7.6)
Permuting i, j, k in (7.5) yields
Pjkil + Pkjil = - 2 \.Ajki.l + 2(Ajki.l - Ajkl| i) - 2AjksP
s
n il, (7.7)
Pkijl + Pikjl = - 2 \.Akij.l + 2(Akij.l - Akil| j) - 2AkisP
s
n jl. (7.8)
From (7.5), (7.7) and (7.8), we have
Pijkl = - \.Aijk.l +Aijk.l - (Aijl| k +Ajkl| i - Akil| j) -
- AijsP
s
n kl - AjksP
s
n il +AkisP
s
n jl. (7.9)
Therefore,
Pnjkl = - Ajkl. (7.10)
By (7.9) and (7.10), it follows that the hv-curvature of F is vanishing if and only if F reduces
to a Riemannian metric.
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1148 M. FAGHFOURI, N. JAZER
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Received 24.06.17
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| institution | Ukrains’kyi Matematychnyi Zhurnal |
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| resource_txt_mv | umjimathkievua/9a/04c0173c3c247278dd21c297439e329a.pdf |
| spelling | umjimathkievua-article-60012022-03-26T11:02:01Z Shen's $L$-process on Berwald connection Shen's $L$-process on Berwald connection Faghfouri, M. Jazer , N. Faghfouri, Morteza Faghfouri, M. Jazer , N. Shen’s C and L-Processes Landsberg metric Shen connection Shen’s C and L-Processes Landsberg metric Shen connection The Shen connection cannot be obtained by using Matsumoto's processes from the other well-known connections.&nbsp;&nbsp;Hence Tayebi–Najafi introduced two new processes called Shen's $C$ and $L$-processes and showed that the Shen connection is obtained from the Chern connection by Shen's $C$-process.&nbsp;&nbsp;In this paper, we&nbsp; study the Shen's $C$- and $L$-process on Berwald connection and introduce two new torsion-free connections in Finsler geometry.&nbsp;&nbsp;Then, we obtain all of Riemannian and non-Riemannian curvatures of these connections.&nbsp;&nbsp;Using it, we find the explicit form of $hv$-curvatures of these connections and prove that $hv$-curvatures of these connections are vanishing if and only if the Finsler structures reduce to Berwaldian or Riemannian structures.&nbsp;&nbsp;As an application, we consider compact Finsler manifolds and obtain ODEs. Зв'язність Шена неможливо отримати за допомогою процесу Мацумото з інших відомих процесів.&nbsp;&nbsp;Тому Тайєбі та Наджафі запропонували два нових процеси,&nbsp; названі&nbsp; $C$- та $L$-процесами Шена, і показали, що за допомогою $C$-процесу Шена із зв'язності Черна можна отримати зв'язність Шена.&nbsp;&nbsp;Ми вивчаємо $C$- та $L$-процеси Шена на зв'язності Бервальда і пропонуємо дві нові безторсіонні зв'язності у геометрії Фінслера.&nbsp;Далі&nbsp; отримуємо всі ріманові та неріманові кривини для цих зв'язностей.&nbsp;За допомогою цього&nbsp; знаходимо точну форму $hv$-кривини для цих зв'язностей і доводимо, що $hv$-кривини для цих зв'язностей є нульовими тоді й тільки тоді, коли структури Фінслера зводяться до структур Бервальда чи Рімана.&nbsp;Як застосування&nbsp; розглядаємо компактні фінслерові многовиди та отримуємо звичайні диференціальні рівняння. Institute of Mathematics, NAS of Ukraine 2020-08-18 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6001 10.37863/umzh.v72i8.6001 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 8 (2020); 1134-1148 Український математичний журнал; Том 72 № 8 (2020); 1134-1148 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6001/8744 |
| spellingShingle | Faghfouri, M. Jazer , N. Faghfouri, Morteza Faghfouri, M. Jazer , N. Shen's $L$-process on Berwald connection |
| title | Shen's $L$-process on Berwald connection |
| title_alt | Shen's $L$-process on Berwald connection |
| title_full | Shen's $L$-process on Berwald connection |
| title_fullStr | Shen's $L$-process on Berwald connection |
| title_full_unstemmed | Shen's $L$-process on Berwald connection |
| title_short | Shen's $L$-process on Berwald connection |
| title_sort | shen's $l$-process on berwald connection |
| topic_facet | Shen’s C and L-Processes Landsberg metric Shen connection Shen’s C and L-Processes Landsberg metric Shen connection |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6001 |
| work_keys_str_mv | AT faghfourim shen039slprocessonberwaldconnection AT jazern shen039slprocessonberwaldconnection AT faghfourimorteza shen039slprocessonberwaldconnection AT faghfourim shen039slprocessonberwaldconnection AT jazern shen039slprocessonberwaldconnection |