Characterization of weakly Berwald fourth root metrics
UDC 514.7 In recent studies, it is shown that the theory of fourth root metrics plays a very important role in physics, theory of space-time structures, gravitation, and general relativity. The class of weakly Berwald metrics contains the class of Berwald metrics as a special case. We establi...
Saved in:
| Date: | 2019 |
|---|---|
| Main Authors: | , , , |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2019
|
| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/1491 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507279480061952 |
|---|---|
| author | Abazari, N. Khoshdani, T. R. Абазарі, Н. Хошдані, Т. Р. |
| author_facet | Abazari, N. Khoshdani, T. R. Абазарі, Н. Хошдані, Т. Р. |
| author_sort | Abazari, N. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T08:57:08Z |
| description | UDC 514.7
In recent studies, it is shown that the theory of fourth root metrics plays a very important role in physics, theory of space-time structures, gravitation, and general relativity.
The class of weakly Berwald metrics contains the class of Berwald metrics as a special case.
We establish the necessary and sufficient condition under which the fourth root Finsler space with an $(\alpha, \beta)$-metric is a weakly Berwald space. |
| first_indexed | 2026-03-24T02:06:47Z |
| format | Article |
| fulltext |
UDC 514.7
T. R. Khoshdani, N. Abazari (Dep. Math., Univ. Mohaghegh Ardabili, Iran)
CHARACTERIZATION OF WEAKLY BERWALD FOURTH ROOT METRICS
ХАРАКТЕРИЗАЦIЯ СЛАБКИХ ЧОТИРИКОРЕНЕВИХ МЕТРИК БЕРВАЛЬДА
In recent studies, it is shown that the theory of fourth root metrics plays a very important role in physics, theory of
space-time structures, gravitation, and general relativity. The class of weakly Berwald metrics contains the class of Berwald
metrics as a special case. We establish the necessary and sufficient condition under which the fourth root Finsler space with
an (\alpha , \beta )-metric is a weakly Berwald space.
В останнiх дослiджeннях було встановлено, що теорiя чотирикореневих метрик вiдiграє важливу роль у фiзицi, теорiї
просторово-часових структур, гравiтацiї та загальнiй теорiї вiдносностi. Клас слабких метрик Бервальда мiстить
клас метрик Бервальда як частинний випадок. Встановлено необхiдну та достатню умову, за якої чотирикореневий
простiр Фiнслера з (\alpha , \beta )-метрикою є слабким простором Бервальда.
1. Introduction. In [15], Shimada developed the theory of m-th root Finsler metrics which applied
to biology as an ecological metric [1]. An m-th root metric is regarded as a direct generalization of
Riemannian metric in the sense that the second root metric is a Riemannian metric. The third and
fourth root metrics are called the cubic metric and quartic metric, respectively.
Recently studies show that the theory of m-th root Finsler metrics plays a very important role
in physics, theory of space-time structure, gravitation, general relativity and seismic ray theory
[9, 13, 14]. For quartic metrics, a study of the geodesics and of the related geometrical objects
are made by S. Lebedev [7]. Also, Einstein equations for some relativistic models relying on such
metrics are studied by V. Balan and N. Brinzei in [3].
In [18], Tayebi and Najafi characterized locally dually flat and Antonelli m-th root Finsler met-
rics. They showed that every m-th root Finsler metric of isotropic mean Berwald curvature reduces
to a weakly Berwald metric. In [19], they proved that every m-th root Finsler metric of isotropic
Landsberg metric reduces to a Landsberg metric. Then, they showed that every m-th root Finsler
metric with almost vanishing H-curvature satisfies \bfH = 0. Recently, Tayebi, Nankali and Peyghan
defined some non-Riemannian curvature properties for Cartan spaces and considered Cartan space
with the m-th root metric [20]. For other recent papers, see [18 – 22, 26, 27].
Let (M,F ) be a Finsler manifold of dimension n, TM its tangent bundle and (xi, yi) the
coordinates in a local chart on TM. Let F be a scalar function on TM defined by F = 4
\surd
A, where
A is given by
A := aijkl(x)y
iyjykyl, (1)
with aijkl symmetric in all its indices [15]. Then F is called an fourth root Finsler metric or an
quartic Finsler metric.
Let F be a Finsler metric on a manifold M. The geodesics of F are characterized locally by the
equation \"xi(t) + 2Gi(x, \.x(t)) = 0, where Gi are coefficients of a spray defined on M denoted by
\bfG (x, y) = yi
\partial
\partial xi
- 2Gi \partial
\partial yi
. A Finsler metric F is called a Berwald metric if Gi =
1
2
\Gamma i
jk(x)y
jyk
are quadratic in y \in TxM for any x \in M. Taking a trace of Berwald curvature yields mean Berwald
curvature \bfE . A Finsler metric with vanishing mean Berwald curvature is called weakly Berwald
c\bigcirc T. R. KHOSHDANI, N. ABAZARI, 2019
976 ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
CHARACTERIZATION OF WEAKLY BERWALD FOURTH ROOT METRICS 977
metric [23]. In [2], Bácsó and Yoshikawa studied some weakly Berwald metrics. Then I. Y. Lee
and M. H. Lee studied some weakly-Berwald spaces of special (\alpha , \beta )-metrics. In [23], Tayebi and
Peyghan studied the mean Berwald curvature of R-quadratic Finsler metrics. In [18], Tayebi and
Najafi showed that every m-th root metric of isotropic mean Berwald curvature is a weakly Berwald
metric. Recently, Najafi and Tayebi have found a condition on (\alpha , \beta )-metrics under which the
notions of isotropic S-curvature, weakly isotropic S-curvature and isotropic mean Berwald curvature
are equivalent [10]. Accordingly, it is necessary to study the weakly Berwald fourth root metrics.
Let L4 = c1\alpha
4 + c2\alpha
2\beta 2 + c3\beta
4 be a fourth root metric on a manifold M, where c1 \not = 0, c2 \not = 0,
c3 \not = 0 are real constants. Suppose that \alpha =
\sqrt{}
aij(x)yiyj be a Riemannian metric and \beta = bi(x)y
i
a 1-form on M such that b2 = bib
i \not = 0; then, we prove the following theorem.
Theorem 1.1. Let (M,Fn) be a Finsler manifold of dimension n \geq 3 equipped with quar-
tic metric L4 = c1\alpha
4 + c2\alpha
2\beta 2 + c3\beta
4, where \alpha =
\sqrt{}
aij(x)yiyj be a Riemannian metric and
\beta = bi(x)y
i a 1-form on M. Then Fn is a weakly-Berwald metric if and only if there exists a
homogeneous polynomial of degree 1, namely V = viy
i, such that the following hold:
si = 0, rij =
1
UW
\Bigl(
bivj + bjvi
\Bigr)
, (2)
where
U := - 2c81c2 -
1
2
b4c61c
3
2 - 2b2c71c
2
2 and W :=
3
2
b4c61c
3
2 - 12b2c81c3 + 3b2c71c
2
2 - 6b4c71c2c3
are real constants.
2. Preliminaries. Let M be an n-dimensional manifold C\infty . Denote by TxM the tangent
space at x \in M, by TM = \cup x\in MTxM the tangent bundle of M and by TM0 = TM \setminus \{ 0\} the slit
tangent bundle of M. A Finsler metric on a manifold M is a function F : TM \rightarrow [0,\infty ) with the
following properties: (i) F is C\infty on TM0; (ii) F (x, \lambda y) = \lambda F (x, y) \forall \lambda > 0, y \in TM ; (iii) for
each y \in TxM, the following quadratic form gy on TxM is positive definite:
gy(u, v) :=
1
2
\bigl[
F 2(y + su+ tv)
\bigr] \bigm| \bigm| \bigm|
s,t=0
, u, v \in TxM.
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 following:
Gi =
1
4
gil
\biggl[
\partial 2F 2
\partial xk\partial yl
yk - \partial F 2
\partial xl
\biggr]
. (3)
\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
.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
978 T. R. KHOSHDANI, N. ABAZARI
\bfB y(u, v, w) is symmetric in u, v and w. Based on the homogeneity of spray coefficients, we have
\bfB y(y, v, w) = 0. \bfB is called the Berwald curvature. A Finsler metric with vanishing Berwald
curvature is called a Berwald metric [11, 16].
Define the mean of Berwald curvature by \bfE y : TxM \otimes TxM \rightarrow \BbbR , where
\bfE y(u, v) :=
1
2
n\sum
i=1
gij(y)gy (\bfB y(u, v, ei), ej) . (4)
The family \bfE = \{ \bfE y\} y\in TM\setminus \{ 0\} is called the mean Berwald curvature or E-curvature. In local
coordinates, \bfE y(u, v) := Eij(y)u
ivj , where
Eij :=
1
2
Bm
mij .
By definition, \bfE y(u, v) is symmetric in u and v, and we have \bfE y(y, v) = 0. \bfE is called the mean
Berwald curvature. F is called a weakly Berwald metric if \bfE = \bfzero .
3. Proof of Theorem 1.1. A Finsler metric L(x, y) is called (\alpha , \beta )-metric if L is a positive-
homogeneous function of \alpha and \beta of degree one, where \alpha 2 = \alpha ij(x)y
iyj is a Riemannian metric
and \beta = \beta i(x)y
i is a one-form. The functions Gi of a Finsler space with an (\alpha , \beta )-metric are
given by 2Gi = \gamma i00 + 2Bi, where \gamma ijk stands for the Christoffel symbols in the space (M,\alpha )
(see [8, 17, 24, 25]). Let Gi
j =
\.\partial jG
i and Gi
jk = \.\partial kG
i
j . Then we have
Gi
j = \gamma i0j +Bi
j , Gi
jk = \gamma ijk +Bi
jkl,
where \.\partial jB
i = Bi
j and \.\partial kB
i
j = Bi
jkl.
Then a Finsler space with an (\alpha , \beta )-metric is a weakly-Berwald space if and only if Bm
m =
= \partial Bm/\partial ym is a one form.
Let us put
bi = airbr, b2 = arsbrbs, (5)
rij =
1
2
(bi/j + bj/i), sij =
1
2
(bi/j - bj/i), (6)
rij = airrrj , sij = airsrj , ri = brr
r
i , si = brs
r
i . (7)
Here, the symbol “/” denotes the h-covariant derivation with respect to the Riemannian connection
of (M,\alpha ). For Finslerian connections see [4, 5]. It is remarkable that if rij = 0 then \beta is a killing
1-form and if sij = 0 then \beta is a closed 1-from (see [12]).
According to [6], the necessary and sufficient condition for a Finsler space Fn with (\alpha , \beta )-metric
to be a weakly-Berwald space is that Gm
m = \gamma m0m + Bm
m and Bm
m is a homogeneous polynomial in
(ym) of degree one which is given by following:
Bm
m =
1
2\alpha L(\beta L\alpha )2\Omega 2
\bigl\{
2\Omega 2AC\ast + 2\alpha L\Omega 2Bs0 + \alpha 2LL\alpha L\alpha \alpha (Cr00 +Ds0 + Er0)
\bigr\}
, (8)
where
A = (n+ 1)\beta 2L\alpha (\beta L\alpha L\beta - \alpha LL\alpha \alpha ) + \alpha \gamma 2L
\bigl\{
\alpha (L\alpha \alpha )
2 - 2L\alpha L\alpha \alpha - \alpha L\alpha L\alpha \alpha \alpha
\bigr\}
, (9)
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
CHARACTERIZATION OF WEAKLY BERWALD FOURTH ROOT METRICS 979
B = \alpha 2LL\alpha \alpha , (10)
C = \beta \gamma 2
\bigl\{
- \beta 2(L\alpha )
2 + 2b2\alpha 3L\alpha L\alpha \alpha - \alpha 2\gamma 2(L\alpha \alpha )
2 + \alpha 2\gamma 2L\alpha L\alpha \alpha \alpha
\bigr\}
, (11)
D = 2\alpha
\bigl\{
\beta 3(\gamma 2 - \beta 2)L\alpha L\beta - \alpha 2\beta 2\gamma 2L\alpha L\alpha \alpha - 2\alpha \beta \gamma 2(\gamma 2 + 2\beta 2)L\beta L\alpha \alpha
\bigr\}
-
- \alpha 3\gamma 4(L\alpha \alpha )
2 - \alpha 2\beta \gamma 4L\beta L\alpha \alpha \alpha (12)
and
E = 2\alpha 2\beta 2L\alpha \Omega , (13)
\gamma 2 = b2\alpha 2 - \beta 2, (14)
C\ast =
\alpha \beta (r00L\alpha - 2\alpha s0L\beta )
2(\beta 2L\alpha + \alpha \gamma 2L\alpha \alpha )
, (15)
\Omega = \beta 2L\alpha + \alpha \gamma 2L\alpha \alpha (16)
provided that \Omega \not = 0.
In order to prove the Theorem 1.1, we need the following lemma.
Lemma 3.1. Let Fn be a Finsler space with an fourth root Finsler metric F (x, y). Then:
(1) In case of n = 2, F 4 is always written in the form
F 4 = d1\alpha
4 + d2\alpha
2\beta 2,
with constants d1, d2 by choosing suitable quadratic form \alpha 2 and one form \beta , where \alpha 2 may be
degenerate.
(2) In case of n \geq 3, if F is a function of a nondegenerate quadratic form \alpha 2 = \alpha ij(x)y
iyj and
a one-form \beta = bi(x)y
i which is homogeneous in \alpha and \beta of degree one, then F 4 is written in the
form
F 4 = c1\alpha
4 + c2\alpha
2\beta 2 + c3\beta
4
with constants c1, c2 and c3.
Proof. (1) First, we consider two-dimensional Finsler space with a quartic metric. Putting
t :=
y2
y1
, the quartic metric L is written as follows:
\biggl(
F
y1
\biggr) 4
= a1111 + 4a1112t+ 6a1122t
2 + 4a1222t
3 + a2222t
4(= f(t)). (17)
In case of a2222 \not = 0, the algebraic equation f(t) = 0 of the degree four with real coefficients has
always two real roots or four real roots or four complex roots.
Case A. If f(t) has two complex roots of z1 and z2, then it is clear that z1 and z2 are roots of
f(t). Hence, we obtain
f(t) = a2222
\biggl(
t2 - 1
2
\mathrm{R}\mathrm{e} z1t+ \gamma
\biggr) \biggl(
t2 - 1
2
\mathrm{R}\mathrm{e} z2t+ \delta
\biggr)
,
where \delta := z2z2 and \gamma := z1z1 are real numbers. Let us put
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
980 T. R. KHOSHDANI, N. ABAZARI
\alpha 2
1 := (y2)2 - 1
2
\mathrm{R}\mathrm{e} z1y
1y2 + \gamma (y1)2, \alpha 2
2 := a2222
\biggl\{
(y2)2 - 1
2
\mathrm{R}\mathrm{e} z2y
1y2 + \delta (y1)2
\biggr\}
.
Hence, we get F 4 = \alpha 4. In the case of a2222 = 0 and a1111 \not = 0, we have
f(t) = 4a1222t
3 + 6a1122t
2 + 4a1112t+ a1111.
Indeed, f(t) has always at least one real root \gamma . Then we obtain
f(t) = (t - \gamma )(4a1222t
2 + bt+ c),
where b and c are suitable real coefficients. Thus, we put
\alpha 2 = 4a1222(y
2)2 + by2y1 + c(y1)2 and \delta = y2 - \gamma y1,
to get F 4 = \alpha 2\beta 2.
If a2222 = 0 and a1111 = 0, we have f(t) = 4a1222t
3 + 6a1122t
2 + 4a1112t. By putting
\alpha 2 = 4a1222(y
2)2 + 6a1122y
2y1 + 4a1112(y
1)2 and \delta 1 = y1 and \delta 2 = y2,
one can obtain F 4 = \alpha 2\beta 2.
Case B. If f(t) has two real roots a1, a2 and two complex roots z, z, then we get
f(t) = a2222
\biggl(
t2 - 1
2
\mathrm{R}\mathrm{e} zt+ \gamma
\biggr)
(t - a1)(t - a2),
where \gamma = zz is a real number.
By putting \alpha 2 = (y2)2 - 1
2
\mathrm{R}\mathrm{e} z y1y2 + \gamma (y1)2 and \delta 1 = a2222\{ y2 - a1y
1\} and \delta 2 = y2 - a2y
1,
we obtain F 4 = \alpha 2\beta 2. In this case, if a2222 = 0, we have F 4 = \alpha 2\beta 2, again.
Case C. If f(t) has four real roots, then we have F 4 = \beta 4. This contradicts with our assumption
where \alpha 2 may be degenerate. Consequently, for n = 2, F 4 is always written in the form F 4 =
= a\alpha 4 + b\alpha 2\beta 2.
(2) Now, suppose that F be a homogeneous function of \alpha and \beta of degree one, where \alpha 2
is a nondegenerate quadratic form and \beta is an one-form. Then the Jacobian determinant
\partial (F, \alpha , \beta )/\partial (yi1 , yi2 , yi3) must be equal to zero for any different three indices of i1, i2, i3 = i, j, k, l.
For i1 = i, i2 = j and i3 = k, we have\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
ai aj ak
\alpha iay
a \alpha jay
a \alpha kay
a
bi bj bk
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| = 0
which is a homogeneous polynomial of degree four in yi. Equating every coefficients of yayhycyd
to zero, we get
\sum
(ijk)
\left\{ \sum
(ahcd)
\{ \alpha ia(bjakhcd - bkajhcd)\}
\right\} = 0, (18)
where the symbol
\sum
(ijk)\{ ...\} denotes the cyclic permutation of i, j, k and summation. Similarly,
for other indices, we have
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
CHARACTERIZATION OF WEAKLY BERWALD FOURTH ROOT METRICS 981
\sum
(jkl)
\left\{ \sum
(ahcd)
\{ \alpha ia(bjakhcd - bkajhcd)\}
\right\} = 0,
\sum
(kli)
\left\{ \sum
(ahcd)
\{ \alpha ia(bjakhcd - bkajhcd)\}
\right\} = 0,
\sum
(lij)
\left\{ \sum
(ahcd)
\{ \alpha ia(bjakhcd - bkajhcd)\}
\right\} = 0.
Here we continue with the equation (18). It is similar to other relationships.
Suppose that (\alpha ij) denotes the inverse matrix of (\alpha ij). Put
brarjkt = pjkt, bjpjkt = pkt, bkpkt = pt, btpt = p, \beta 2 = brbr,
\alpha rsaitrs = qit, biqit = qt, btqt = q = \alpha rsprs, piah\alpha
ah = qi.
Contracting (18) with \alpha jdbk implies that
(n+ 1)\beta 2aiahc + \alpha ia(phc - \beta 2qhc) + \alpha ih(pca - \beta 2qca) + \alpha ic(pah - \beta 2qah)+
+ nbi(baqhc + bhqca + bcqah) - (n+ 1)bipahc - n(bapihc + bbpica + bcpiah) = 0. (19)
Multiplying (19) with bc yields, we have
\beta 2piah+\alpha ia(ph - \beta 2qh)+\alpha ih(pa - \beta 2qa)+nbi(pah+baqh+bhqa) - (n+1)bipah - n(bapih+bhpia) = 0.
(20)
Now, by contracting (20) with bd, we get
\alpha ia(p - \beta 2q) + nbi(2pa + baq) - (n+ 1)bipa - npiba - (n - 1)\beta 2pia = 0. (21)
From (21), we obtain
pia =
1
(n - 1)\beta 2
\bigl\{
\alpha ia(p - \beta 2q) + (n - 1)bipa + nqbiba - npiba
\bigr\}
. (22)
Due to the symmetry property of pia and \alpha ia, from (21) one can get
pi = \gamma bi, (23)
where \gamma is scalar. Also, contraction of (20) by \alpha ah gives
qi = \eta bi, (24)
where \eta is scalar.
Substituting (23) in (22) yields
pia = e1\alpha ia + e2biba, (25)
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
982 T. R. KHOSHDANI, N. ABAZARI
where e1 and e2 are certain scalars.
By putting (23), (24) and (25) in (20), we obtain
piah = d1\alpha iabh + d2\alpha ihba + d3biba\beta h + e1bi\alpha ah. (26)
Now, substituting (23), (24), (25) and (26) in (19) yields
aiahc = \tau 1\alpha iabhbc + \tau 2\alpha ihbabc + \tau 3\alpha icbhba + \tau 4\alpha acbibh + \tau 5\alpha ahbcbi + \tau 6\alpha hcbiba+
+e1(\alpha ia\alpha hc + \alpha ih\alpha ca + \alpha ic\alpha ah) + \tau 7bibabhbc, (27)
where \tau i, i = 1, . . . , 7, are scalars.
By (27), we have F 4 = 3e1\alpha
4+\pi \alpha 2\beta 2+\tau 7\beta
4 where \pi is a linear combination of \tau i, i = 1, . . . , 6.
Since it was assumed that F is a function of \alpha and \beta alone, so it is obvious that e1 and \tau i,
i = 1, . . . , 7, and then \pi must be constant.
Proof of Theorem 1.1. The fourth root Finsler metric L(\alpha , \beta ) of Fn is given by
L4(\alpha , \beta ) = c1\alpha
4 + c2\alpha
2\beta 2 + c3\beta
4, (28)
where c1, c2 and c3 are constants. Let us put
X := c1\alpha
2 +
1
2
c2\beta
2, (29)
Y :=
1
2
c1c2\alpha
4 + 3\alpha 2\beta 2c1c3 -
1
4
\alpha 2\beta 2c22 +
1
2
c3c2\beta
4. (30)
Then we have
L3L\alpha = \alpha X, (31)
L3L\beta = c3\beta
3 +
1
2
c2\alpha
2\beta , (32)
L7L\alpha \alpha = Y \beta 2, (33)
L11L\alpha \alpha \alpha = - 3
2
\alpha 7\beta 2c21c2 - 15\alpha 5\beta 4c21c3 - 6\alpha 3\beta 6c1c3c2 +
3
8
\alpha 3\beta 6c32 -
9
4
\alpha \beta 8c22c3+
+6\beta 8\alpha c1c
2
3 +
3
2
\beta 4\alpha 5c1c
2
2. (34)
Substituting (31) – (34) in (9) – (16) and using (8), yield
\alpha 2\beta \Phi Bm
m + \alpha 2\beta 2\Psi r00 + \alpha 2\beta \Lambda r0 + \beta \Upsilon s0 = 0, (35)
where \Phi ,\Psi ,\Lambda and \Upsilon are listed in Appendix 1.
It should be noted that
4L4\beta 8\alpha 6X2(L12X3 + 3L8X2Y \alpha 2b2 + 3L4XY 2\alpha 4b4 + Y 3\alpha 6b6 - 3L8X2Y \beta 2 -
- 6L4XY 2\alpha 2b2\beta 2 - 3Y 3\alpha 4b4\beta 2 + 3L4XY 2\beta 4 + 3Y 3\alpha 2b2\beta 4 - Y 3\beta 6) \not = 0 (36)
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
CHARACTERIZATION OF WEAKLY BERWALD FOURTH ROOT METRICS 983
and
2\alpha \beta 2(XL4 + \alpha 2Y b2 - Y \beta 2) \not = 0 (37)
because it appears in the denominator of (8).
Thus, L4 \not = 0 and X \not = 0. Analogously to the above, this implies c1 \not = 0 and c2 \not = 0 and
c3 \not = 0. Suppose that Fn be a weakly-Berwald space, that is, Bm
m is hp(1). The term in (35) which
seemingly does not contain \alpha 2 is
\beta 23
\biggl(
- 6b2c1c
3
2c
5
3 +
27
8
c62c
3
3 +
9
4
b2c52c
4
3 + 24c21c
2
2c
5
3 - 18c1c
4
2c
4
3
\biggr)
s0
only, and we must have hp(22) V22 such that
\beta 23cs0 = \alpha 2V22,
where
c = - 6b2c1c
3
2c
5
3 +
27
8
c62c
3
3 +
9
4
b2c52c
4
3 + 24c21c
2
2c
5
3 - 18c1c
4
2c
4
3.
Analogously to the above, it implies that s0 = 0.
Substituting s0 = 0 into (35) implies that
\Phi Bm
m + \beta \Psi r00 + \Lambda r0 = 0. (38)
Let us put
U := - 2c81c2 -
1
2
b4c61c
3
2 - 2b2c71c
2
2,
T :=
1
2
b6c61c
3
2 + 3b4c71c
2
2 + 6b2c81c2 + 4c91.
Then only the term \alpha 20(Ur0 + TBm
m) of (38) seemingly does not contain \beta , and we must have
hp(20) V20 such that
\alpha 20(Ur0 + TBm
m) = \beta V20.
From \alpha 2 \not = 0 (\mathrm{m}\mathrm{o}\mathrm{d}\beta ) and b2 \not = 0, it follows that there exists a function k(x) such that
Ur0 + TBm
m = k(x)\beta . (39)
Multiplying (38) with U and using (39) one can obtain
\beta U\Psi r00 + \Lambda k(x)\beta + (U\Phi - \Lambda T )Bm
m = 0. (40)
Dividing (40) by \beta implies that
U\Psi r00 + \Lambda k(x) + \beta SBm
m = 0, (41)
where
S :=
U\Phi - \Lambda T
\beta 2
.
Let us put
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
984 T. R. KHOSHDANI, N. ABAZARI
W :=
3
2
b4c61c
3
2 - 12b2c81c3 + 3b2c71c
2
2 - 6b4c71c2c3.
Since only the term \alpha 18UWr00 +Uk(x)\alpha 20 of U\Psi r00 +\Lambda k(x) in (41) seemingly does not contain
\beta , thus there exists hp(19) V19 such that
\alpha 18(UWr00 + Uk(x)\alpha 2) = \beta V19.
Then there must exist hp(1) U1 satisfying
UWr00 + Uk(x)\alpha 2 = \beta U1. (42)
It is a contradiction and then k(x) = 0. Putting it into (41) implies that
U\Psi r00 + \beta SBm
m = 0. (43)
Then only the term \alpha 18UWr00 of U\Psi r00 in (43) seemingly does not contain \beta . Then we must have
hp(20) V20 such that
\alpha 18UWr00 = \beta V20.
According to \alpha 2 \not = 0 (\mathrm{m}\mathrm{o}\mathrm{d}\beta ) and b2 \not = 0, there must exists hp(1) V = viy
i satisfying
UWr00 = \beta V.
Hence, we have
r00 =
1
UW
\beta V. (44)
Conversely, let r00 =
1
UW
\beta V and s0 = 0 hold. Then we have
rij =
1
2UW
(bivj + bjvi), si = 0, (45)
where \beta = bi(x)y
i and V = viy
i. Multiplying (45) by biyj implies that
r0 =
1
2UW
(b2V + vb\beta ), rj =
1
2UW
(b2vj + vbbj), (46)
where vb = vib
i. According to assumptions and by putting (46) into (35), we obtain
\Phi Bm
m +
1
UW
\beta 2\Psi V +
1
2UW
\Lambda (b2V + vb\beta ) = 0. (47)
It is easy to see that only the term \alpha 20
\Bigl(
TBm
m+
1
2W
b2V
\Bigr)
of \Phi Bm
m+
1
2UW
\Lambda b2V in (47) seemingly
does not contain \beta . Then we must have hp(20) V20 such that
\alpha 20
\biggl(
TBm
m +
1
2W
b2V
\biggr)
= \beta V20.
The above shows the existence of a function g(x) satisfying
V20 = g(x)\alpha 20.
Then
Bm
m = - 1
2WT
b2V +
1
T
\beta g(x). (48)
Therefore, Bm
m is hp(1) and, hence, the Finsler space with (28) is a weakly-Berwald space.
Theorem 1.1 is proved.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
CHARACTERIZATION OF WEAKLY BERWALD FOURTH ROOT METRICS 985
4. Appendix 1.
\Phi =
27
64
\beta 20c82c3 + 4\alpha 20c91 + 4L4X2Y 3b6 - 3
2
b4\beta 20c1c
4
2c
4
3 + 6b2\beta 20c21c
3
2c
4
3 -
9
2
b2\beta 20c1c
5
2c
3
3 -
- 9
2
b2\beta 20c1c
5
2c
3
3 + 6\alpha 18b4\beta 2c61c
3
2 + 36\alpha 18b2\beta 2c81c3 + 21\alpha 18b2\beta 2c71c
2
2 + 108\alpha 16b4\beta 4c71c
2
3+
+
15
4
\alpha 16b4\beta 4c51c
4
2 +
69
2
\alpha 16b2\beta 4c61c
3
2 -
27
64
\alpha 4b4\beta 16c82c3 + 56\alpha 4\beta 16c41c
2
2c
3
3 -
15
2
\alpha 4\beta 16c31c
4
2c
2
3 -
- 27
2
\alpha 4\beta 16c21c
6
2c3 +
27
64
\alpha 2b2\beta 18c82c3 - 32\alpha 2\beta 18c41c2c
4
3 + 40\alpha 2\beta 18c31c
3
2c
3
3 -
27
2
\alpha 2\beta 18c21c
5
2c
2
3 -
- 3
4
\alpha 14b4\beta 6c41c
5
2 - 108\alpha 14b2\beta 6c71c
2
3 +
129
4
\alpha 14b2\beta 6c51c
4
2 - 80\alpha 14\beta 6c71c2c3 - 108\alpha 12b4\beta 8c61c
3
3 -
- 39
16
\alpha 12b4\beta 8c31c
6
2 +
123
8
\alpha 12b2\beta 8c41c
5
2 - 152\alpha 12\beta 8c61c
2
2c3 -
3
4
\alpha 10b4\beta 10c21c
7
2 +
3
16
\alpha 10b2\beta 10c31c
6
2+
+72\alpha 10\beta 10c61c2c
2
3 - 176\alpha 10\beta 10c51c
3
2c3 - 216\alpha 8b4\beta 12c51c
4
3 +
21
64
\alpha 8b4\beta 12c1c
8
2 -
141
32
\alpha 8b2\beta 12c21c
7
2+
+78\alpha 8\beta 12c51c
2
2c
2
3 -
259
2
\alpha 8\beta 12c41c
4
2c3 + 144\alpha 6b2\beta 14c51c
4
3 -
153
64
\alpha 6b2\beta 14c1c
8
2 + 32\alpha 6\beta 14c51c2c
3
3+
+36\alpha 6\beta 14c41c
3
2c
2
3 - 59\alpha 6\beta 14c31c
5
2c3 +36\alpha 18b4\beta 2c71c2c3+93\alpha 16b4\beta 4c61c
2
2c3+132\alpha 16b2\beta 4c71c2c3 -
- 132\alpha 4b4\beta 16c31c
2
2c
4
3 +
195
4
\alpha 4b4\beta 16c21c
4
2c
3
3 +
99
16
\alpha 4b4\beta 16c1c
6
2c
2
3 + 168\alpha 4b2\beta 16c41c2c
4
3 -
- 132\alpha 4b2\beta 16c31c
3
2c
3
3 -
9
8
\alpha 4b2\beta 16c21c
5
2c
2
3 +
189
16
\alpha 4b2\beta 16c1c
7
2c3 - 24\alpha 2b4\beta 18c21c
3
2c
4
3+
+
39
4
\alpha 2b4\beta 18c1c
5
2c
3
3 + 60\alpha 2b2\beta 18c31c
2
2c
4
3 - 48\alpha 2b2\beta 18c21c
4
2c
3
3 +
135
16
\alpha 2b2\beta 18c1c
6
2c
2
3+
+324\alpha 14b4\beta 6c61c2c
2
3 + 102\alpha 14b4\beta 6c51c
3
2c3 + 258\alpha 14b2\beta 6c61c
2
2c3 + 477\alpha 12b4\beta 8c51c
2
2c
2
3+
+
219
4
\alpha 12b4\beta 8c41c
4
2c3 - 342\alpha 12b2\beta 8c61c2c
2
3 + 339\alpha 12b2\beta 8c51c
3
2c3 - 324\alpha 10b4\beta 10c51c2c
3
3+
+486\alpha 10b4\beta 10c41c
3
2c
2
3 -
15
4
\alpha 10b4\beta 10c31c
5
2c3 - 459\alpha 10b2\beta 10c51c
2
2c
2
3 + 303\alpha 10b2\beta 10c41c
4
2c3 -
- 177\alpha 8b4\beta 12c41c
2
2c
3
3 +
1125
4
\alpha 8b4\beta 12c31c
4
2c
2
3 -
285
16
\alpha 8b4\beta 12c21c
6
2c3 + 24\alpha 8b2\beta 12c51c2c
3
3 -
- 711
2
\alpha 8b2\beta 12c41c
3
2c
2
3 +
735
4
\alpha 8b2\beta 12c31c
5
2c3 - 288\alpha 6b4\beta 14c41c2c
4
3 + 42\alpha 6b4\beta 14c31c
3
2c
3
3+
+
297
4
\alpha 6b4\beta 14c21c
5
2c
2
3 - 6\alpha 6b4\beta 14c1c
7
2c3 - 96\alpha 6b2\beta 14c41c
2
2c
3
3 -
513
4
\alpha 6b2\beta 14c31c
4
2c
2
3+
+
543
8
\alpha 6b2\beta 14c21c
6
2c3 - 20\alpha 16\beta 4c81c3 + 50\alpha 16\beta 4c71c
2
2 -
27
64
\alpha 4b2\beta 16c92 - 32\alpha 4\beta 16c51c
4
3+
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
986 T. R. KHOSHDANI, N. ABAZARI
+
243
64
\alpha 4\beta 16c1c
8
2 + 3\alpha 20b4c71c
2
2 + 6\alpha 20b2c81c2 + 80\alpha 14\beta 6c61c
3
2 + 24\alpha 12\beta 8c71c
2
3 + 89\alpha 12\beta 8c51c
4
2+
+71\alpha 10\beta 10c41c
5
2 + 16\alpha 8\beta 12c61c
3
3 +
323
8
\alpha 8\beta 12c31c
6
2 +
9
64
\alpha 6b4\beta 14c92 +
63
4
\alpha 6\beta 14c21c
7
2+
+
9
16
b4\beta 20c62c
3
3 +
27
32
b2\beta 20c72c
2
3 - 8\beta 20c31c
2
2c
4
3 + 9\beta 20c21c
4
2c
3
3 -
- 27
8
\beta 20c1c
6
2c
2
3 + 20\alpha 18\beta 2c81c2 +
27
64
\alpha 2\beta 18c92,
\Psi = - L4X3Y 2b4c2n - 3\alpha 18c81c2+189\alpha 8b6\beta 10c41c2c
4
3 - 18\alpha 8b6\beta 10c31c
3
2c
3
3 +
3
4
\alpha 8b6\beta 10c21c
5
2c
2
3 -
- 1
16
\alpha 8b6\beta 10c1c
7
2c3 + 36\alpha 8b4\beta 10c51c
4
3n+
1
128
\alpha 8b4\beta 10c1c
8
2n - 249
2
\alpha 8b4\beta 10c41c
2
2c
3
3+
+
1167
8
\alpha 8b4\beta 10c31c
4
2c
2
3 -
131
16
\alpha 8b4\beta 10c21c
6
2c3 +
5
16
\alpha 8b2\beta 10c21c
7
2n+ 54\alpha 8b2\beta 10c51c2c
3
3 -
- 1233
4
\alpha 8b2\beta 10c41c
3
2c
2
3 +
787
16
\alpha 8b2\beta 10c31c
5
2c3 + 6\alpha 8\beta 10c61c
2
3c3n - 255
2
\alpha 8\beta 10c51c
2
2c
2
3n+
+
71
2
\alpha 8\beta 10c41c
4
2c3n+
9
2
\alpha 8\beta 10c41c
4
2c3n - 30\alpha 6X2Y 2b6c21c3 + 3\alpha 6X2Y 2b6c1c
2
2+
+81\alpha 6b6\beta 12c31c
2
2c
4
3 - 13\alpha 6b6\beta 12c21c
4
2c
3
3 +
15
16
\alpha 6b6\beta 12c1c
6
2c
2
3 + 132\alpha 6b4\beta 12c41c2c
4
3 -
- 98\alpha 6b4\beta 12c31c
3
2c
3
3 +
885
16
\alpha 6b4\beta 12c21c
5
2c
2
3 -
109
32
\alpha 6b4\beta 12c1c
7
2c3 - 48\alpha 6b2\beta 12c51c
4
3n+
+
13
32
\alpha 6b2\beta 12c1c
8
2n+ 15\alpha 6b2\beta 12c41c
2
2c
3
3 -
273
2
\alpha 6b2\beta 12c31c
4
2c
2
3 +
713
32
\alpha 6b2\beta 12c21c
6
2c3 -
- 21\alpha 6\beta 12c51c2c
3
3n - 435
4
\alpha 6\beta 12c41c
3
2c
2
3n+ 22\alpha 6\beta 12c31c
5
2c3n+ 9\alpha 6X2Y 2b4c21c2+
+
33
2
\alpha 4b6\beta 14c21c
3
2c
4
3 -
9
4
\alpha 4b6\beta 14c1c
5
2c
3
3 +
3
128
\alpha 4b4\beta 14c82c3n+ 52\alpha 4b4\beta 14c31c
2
2c
4
3 -
- 251
8
\alpha 4b4\beta 14c21c
4
2c
3
3 +
351
32
\alpha 4b4\beta 14c1c
6
2c
2
3 + 76\alpha 4b2\beta 14c41c2c
4
3 - 37\alpha 4b2\beta 14c31c
3
2c
3
3 -
- 339
16
\alpha 4b2\beta 14c21c
5
2c
2
3 +
225
64
\alpha 4b2\beta 14c1c
7
2c3 - 6\alpha 4\beta 14c51c
3
3c3n - 52\alpha 4\beta 14c41c
2
2c
3
3n -
- 195
4
\alpha 4\beta 14c31c
4
2c
2
3n+
243
32
\alpha 4\beta 14c21c
6
2c3n - 39
32
\alpha 4\beta 14c21c
6
2c3n+
3
4
\alpha 4X2Y 2b6\beta 2c32+
+3\alpha 2X4c1c
2
2b
2L8 - 30\alpha 2X4c21c3b
2L8 +
13
8
\alpha 2\beta 16b6c1c
4
3c
4
2 + 7\alpha 2\beta 16b4c21c
4
3c
3
2 -
- 49
16
\alpha 2\beta 16b4c1c
5
2c
3
3 -
3
32
\alpha 2\beta 16b2nc82c3 + 22\alpha 2\beta 16c31c
4
3c
2
2b
2 - 125
8
\alpha 2\beta 16c21c
4
2c
3
3b
2+
+
15
16
\alpha 2\beta 16c1c
6
2c
2
3b
2 + 23\alpha 2\beta 16nc41c
4
3c2 -
95
2
\alpha 2\beta 16nc31c
3
2c
3
3 -
75
8
\alpha 2\beta 16nc21c
5
2c
2
3+
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
CHARACTERIZATION OF WEAKLY BERWALD FOURTH ROOT METRICS 987
+
59
64
\alpha 2\beta 16nc1c
7
2c3 -
35
64
\alpha 2\beta 16c3nc
7
2c1 -
9
4
\alpha 2X2Y 2b4\beta 4c32 + 3\alpha 2X2Y 2\beta 4c21c2+
+
111
8
\beta 18c3nc
2
1c
4
2c
2
3 +
57
32
\beta 18c3nc1c
6
2c3 - 36X2Y 2b4\beta 6c1c
2
3 +
27
2
X2Y b4\beta 6c22c3 -
- 3L4X3Y b2\beta 4c32 - 60L4X3Y \beta 4c21c3 + 6L4X3Y \beta 4c1c
2
2 +
3
2
X2Y 2b2\beta 4c32\gamma
2+
+30X2Y 2\beta 4c21c3\gamma
2 - 3X2Y 2\beta 4c1c
2
2\gamma
2 + 3\alpha 16b6\beta 2c51c
3
2c3 +
1
8
\alpha 16b4\beta 2c51c
4
2n+
+16\alpha 16b4\beta 2c61c
2
2c3 - \alpha 16b2\beta 2c61c
3
2n+ 37\alpha 16b2\beta 2c71c2c3 +
65
2
\alpha 14b2\beta 4c61c
2
2c3+
+14\alpha 14\beta 4c71c2c3n+ 2\alpha 14\beta 4c71c2c3n+ 27\alpha 14b6\beta 4c51c
2
2c
2
3 -
5
2
\alpha 14b6\beta 4c41c
4
2c3 -
- 1
4
\alpha 14b4\beta 4c41c
5
2n+ 141\alpha 14b4\beta 4c61c2c
2
3 +
53
2
\alpha 14b4\beta 4c51c
3
2c3 + 24\alpha 14b2\beta 4c71c
2
3n -
- 5
2
\alpha 14b2\beta 4c51c
4
2n+ 108\alpha 12b6\beta 6c51c2c
3
3 -
9
2
\alpha 12b6\beta 6c41c
3
2c
2
3 -
1
2
\alpha 12b6\beta 6c31c
5
2c3+
+36\alpha 12b4\beta 6c61c
3
3n - 1
16
\alpha 12b4\beta 6c31c
6
2n+
471
2
\alpha 12b4\beta 6c51c
2
2c
2
3 +
73
4
\alpha 12b4\beta 6c41c
4
2c3 -
- 5
2
\alpha 12b2\beta 6c41c
5
2n - 273\alpha 12b2\beta 6c61c2c
2
3 +
175
4
\alpha 12b2\beta 6c51c
3
2c3 + 12\alpha 12\beta 6c71c3c3n+
+
55
2
\alpha 12\beta 6c61c
2
2c3n+
13
2
\alpha 12\beta 6c61c
2
2c3n+ 54\alpha 10b6\beta 8c41c
2
2c
3
3 -
21
2
\alpha 10b6\beta 8c31c
4
2c
2
3+
+
3
4
\alpha 10b6\beta 8c21c
6
2c3 +
5
64
\alpha 10b4\beta 8c21c
7
2n - 87\alpha 10b4\beta 8c51c2c
3
3 +
945
4
\alpha 10b4\beta 8c41c
3
2c
2
3 -
- 41
16
\alpha 10b4\beta 8c31c
5
2c3 - 24\alpha 10b2\beta 8c61c
3
3n - \alpha 10b2\beta 8c31c
6
2n - 375\alpha 10b2\beta 8c51c
2
2c
2
3+
+
229
4
\alpha 10b2\beta 8c41c
4
2c3 - 84\alpha 10\beta 8c61c2c
2
3n+
151
4
\alpha 10\beta 8c51c
3
2c3n+
33
4
\alpha 10\beta 8c51c
3
2c3n+
+
1
4
\beta 18b4c43nc1c
4
2 - 2\beta 18b2nc21c
3
2c
4
3 +
5
4
\beta 18b2nc1c
5
2c
3
3 - \beta 18c3nc
3
1c
3
3c
2
2 - 14\alpha 16\beta 2c81c3 -
- 23
2
\alpha 16\beta 2c71c
2
2 - 22\alpha 14\beta 4c61c
3
2 + 30\alpha 12\beta 6c71c
2
3 -
53
2
\alpha 12\beta 6c51c
4
2 -
337
16
\alpha 10\beta 8c41c
5
2+
+
1
256
\alpha 8b6\beta 10c92 + 12\alpha 8\beta 10c61c
3
3 -
353
32
\alpha 8\beta 10c31c
6
2 +
1
16
\alpha 6b4\beta 12c92 -
55
16
\alpha 6\beta 12c21c
7
2 -
- 57
256
\alpha 4b2\beta 14c92 -
3
64
\beta 18b2c72c
2
3 - 32\alpha 4\beta 14c51c
4
3 -
15
32
\alpha 4\beta 14c1c
8
2 -
9
64
\alpha 2\beta 16nc92+
+
1
16
\beta 18b6c52c
4
3 -
1
32
\beta 18b4c62c
3
3 -
1
16
\beta 18nc82c3 -
5
64
\beta 18c3nc
8
2 - 8\beta 18c31c
2
2c
4
3 + 5\beta 18c21c
4
2c
3
3 -
- 3
4
\beta 18c1c
6
2c
2
3 +
1
8
\alpha 18b6c51c
4
2 -
1
2
\alpha 18b4c61c
3
2 + 18\alpha 18b2c81c3 -
9
2
\alpha 18b2c71c
2
2 + 108\alpha 8b4\beta 10c51c
4
3+
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
988 T. R. KHOSHDANI, N. ABAZARI
+
11
64
\alpha 8b4\beta 10c1c
8
2 -
103
64
\alpha 8b2\beta 10c21c
7
2 - 6\alpha 8\beta 10c61c
3
3n - 115
16
\alpha 8\beta 10c31c
6
2n+ 84\alpha 8\beta 10c51c
2
2c
2
3 -
- 199
8
\alpha 8\beta 10c41c
4
2c3 -
1
32
\alpha 6b6\beta 12c82c3 -
1
128
\alpha 6b4\beta 12c92n+ 72\alpha 6b2\beta 12c51c
4
3 -
- 137
128
\alpha 6b2\beta 12c1c
8
2 -
29
8
\alpha 6\beta 12c21c
7
2n - 6\alpha 6\beta 12c51c2c
3
3 + 60\alpha 6\beta 12c41c
3
2c
2
3 - 14\alpha 6\beta 12c31c
5
2c3+
+
3
32
\alpha 4b6\beta 14c72c
2
3 -
15
32
\alpha 4b4\beta 14c82c3 +
3
32
\alpha 4b2\beta 14c92n+ 22\alpha 4\beta 14c51c
4
3n - 69
64
\alpha 4\beta 14c1c
8
2n+
+5\alpha 4\beta 14c41c
2
2c
3
3 +
201
18
\alpha 4\beta 14c31c
4
2c
2
3 -
495
32
\alpha 4\beta 14c21c
6
2c3 +
147
16
\alpha 4\beta 14c21c
6
2c3 + 3\alpha 2X4c21c2L
8 -
- 1
8
\alpha 2\beta 16b6c62c
3
3 +
45
64
\alpha 2\beta 16b4c72c
2
3 -
21
128
\alpha 2\beta 16c82c3b
2 - 32\alpha 2\beta 16c43c
4
1c2 +
31
2
\alpha 2\beta 16c33c
3
1c
3
2+
+
39
16
\alpha 2\beta 16c23c
2
1c
5
2 -
39
32
\alpha 2\beta 16c3c
7
2c1 - 2XY 4b6 + 48L4X3Y b2\beta 4c1c2c3 -
- 24X2Y 2b2\beta 4c1c2c3\gamma
2 - 12\alpha 4X2Y 2b6\beta 2c1c2c3 - 3\alpha 2L4X3Y \gamma 2c1c
2
2b
2+30\alpha 2L4X3Y \gamma 2c21c3b
2+
+36\alpha 2X2Y 2b4\beta 4c1c2c3 + 120\alpha 2L4X3Y b2\beta 2c21c3 + 9\alpha 18b4c71c2c3 -
3
16
\alpha 16b6\beta 2c41c
5
2+
+54\alpha 16b4\beta 2c71c
2
3 +
1
2
\alpha 16b4\beta 2c51c
4
2 -
37
4
\alpha 16b2\beta 2c61c
3
2 + 4\alpha 16\beta 2c81c3n - \alpha 16\beta 2c71c
2
2n -
- 72\alpha 14b2\beta 4c71c
2
3 -
47
8
\alpha 14b2\beta 4c51c
4
2 - 4\alpha 14\beta 4c61c
3
2n - 38\alpha 14\beta 4c71c2c3 +
1
16
\alpha 14b6\beta 4c31c
6
2+
+
1
8
\alpha 14b4\beta 4c41c
5
2 -
97
2
\alpha 12\beta 6c61c
2
2c3 +
1
32
\alpha 12b6\beta 6c21c
7
2 - 54\alpha 12b4\beta 6c61c
3
3 -
- 13
16
\alpha 12b4\beta 6c31c
6
2 -
19
16
\alpha 12b2\beta 6c41c
5
2 - 24\alpha 12\beta 6c71c
2
3n - 31
4
\alpha 12\beta 6c51c
4
2n+
+162\alpha 10b6\beta 8c51c
4
3 -
3
128
\alpha 10b6\beta 8c1c
8
2 -
1
4
\alpha 10b4\beta 8c21c
7
2 -
- 18\alpha 10b2\beta 8c61c
3
3 -
23
32
\alpha 10b2\beta 8c31c
6
2 -
37
4
\alpha 10\beta 8c41c
5
2n+ 81\alpha 10\beta 8c61c2c
2
3 - 41\alpha 10\beta 8c51c
3
2c3 -
- 1
32
\beta 18b4nc62c
3
3 +
1
4
\beta 18b4c1c
4
2c
4
3 -
3
16
\beta 18b2nc72c
2
3 + \beta 18b2c21c
3
2c
4
3 + 22\alpha 4b4\beta 14c31c
2
2c
4
3n -
- 60\alpha 4L4X3Y b4c21c3 -
5
8
\beta 18b2c1c
5
2c
3
3 + 5\beta 18nc31c
4
3c
2
2 -
143
8
\beta 18nc21c
4
2c
3
3 -
15
32
\beta 18nc1c
6
2c
2
3+
+
3
4
\beta 2X4c32b
2L8c21c2 + 30\alpha 4X2Y 2b4c21c3\gamma
2 - 6\alpha 4X2Y 2b2c21c2\gamma
2 - 3\alpha 4X2Y 2b4c1c
2
2\gamma
2+
+12\alpha 4L4X3Y b2 +
315
4
\alpha 6\beta 12c41c
3
2c3c3n+ 6\alpha 4L4X3Y b4c1c
2
2+
+
99
2
\alpha 6b2\beta 12c31c
4
2c
2
3n - 25
2
\alpha 6b2\beta 12c21c
6
2c3n+ 21\alpha 6\beta 12c51c2c
3
3n+
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
CHARACTERIZATION OF WEAKLY BERWALD FOURTH ROOT METRICS 989
+
19
64
\alpha 6b4\beta 12c1c
7
2c3n - 12\alpha 6b2\beta 12c41c
2
2c
3
3n+
165
2
\alpha 8\beta 10c51c
2
2c3c3n+ 48\alpha 6b4\beta 12c41c2c
4
3n+
+
47
2
\alpha 6b4\beta 12c31c
3
2c
3
3n - 75
16
\alpha 6b4\beta 12c21c
5
2c
2
3n+
1
16
\alpha 8b4\beta 10c21c
6
2c3n - 60\alpha 8b2\beta 10c51c2c
3
3n+
+105\alpha 8b2\beta 10c41c
3
2c
2
3n - 95
4
\alpha 8b2\beta 10c31c
5
2c3n - 19
8
\alpha 10b4\beta 8c31c
5
2c3n+ 108\alpha 10b2\beta 8c51c
2
2c
2
3n -
- 43
2
\alpha 10b2\beta 8c41c
4
2c3n+ 48\alpha 10\beta 8c61c2c3c3n+
171
2
\alpha 8b4\beta 10c41c
2
2c
3
3n - 33
4
\alpha 8b4\beta 10c31c
4
2c
2
3n -
- 3X2Y 2\beta 6c1c
2
2 +
1
4
\alpha 18b4nc61c
3
2 - L4X3Y 2b4c2 + 30\beta 2X4c21c3L
8 - 3\beta 2X4c1c
2
2L
8+
+
9
4
X2Y 2b2\beta 6c32 + 30X2Y 2\beta 6c21c3 +
1
2
\alpha 4b4\beta 14c21c
4
2c
3
3n -
- 15
32
\alpha 4b4\beta 14c1c
6
2c
2
3n - 56\alpha 4b2\beta 14c41c2c
4
3n+ 26\alpha 4b2\beta 14c31c
3
2c
3
3n+ 6\alpha 4b2\beta 14c21c
5
2c
2
3n -
- 21
8
\alpha 4b2\beta 14c1c
7
2c3n+ 36\alpha 4\beta 14c41c
2
2c
3
3n+
87
2
\alpha 4\beta 14c31c
4
2c3c3n+
+90\alpha 4X2Y 2b4\beta 2c21c3 - 9\alpha 4X2Y 2b4\beta 2c1c
2
2 - 9\alpha 4X2Y 2b2\beta 2c21c2 -
- 3\alpha 2L4X3Y \gamma 2c21c2 + 4\alpha 2\beta 16b4c43nc
2
1c
3
2 -
7
16
\alpha 2\beta 16b4c33nc1c
5
2 - 20\alpha 2\beta 16b2nc31c
4
3c
2
2+
+
25
2
\alpha 2\beta 16b2nc21c
4
2c
3
3 -
3
2
\alpha 2\beta 16b2nc1c
6
2c
2
3 - 7\alpha 2\beta 16c3nc
4
1c
3
3c2 +
63
2
\alpha 2\beta 16c3nc
3
1c
3
2c
2
3+
+
105
8
\alpha 2\beta 16c3nc
2
1c
5
2c3 - 90\alpha 2X2Y 2b2\beta 4c21c3 + 9\alpha 2X2Y 2b2\beta 4c1c
2
2 + 12\alpha 2X2Y 2b6\beta 4c1c
2
3 -
- 9
2
\alpha 2X2Y 2b6\beta 4c22c3 +
3
2
\alpha 2L4X3Y b4\beta 2c32 -
3
4
\alpha 2X2Y 2b4\beta 2c32\gamma
2 - 6\alpha 2L4X3Y \beta 2c21c2+
+3\alpha 2X2Y 2\beta 2c21c2\gamma
2 - 12\beta 2X4c1c3c2b
2L8 - 30\beta 2L4X3Y \gamma 2c21c3 -
3
4
\beta 2L4X3Y \gamma 2c32b
2+
+3\beta 2L4X3Y \gamma 2c1c
2
2 - 36X2Y 2b2\beta 6c1c2c3 + 24L4X3Y b4\beta 4c1c
2
3 - 9L4X3Y b4\beta 4c22c3 -
- 12X2Y 2b4\beta 4c1c
2
3\gamma
2 +
9
2
X2Y 2b4\beta 4c22c3\gamma
2 + 4\alpha 16b4\beta 2c61c
2
2c3n+ 4\alpha 16b2\beta 2c71c2c3n+
+21\alpha 14b4\beta 4c61c2c
2
3n+
19
4
\alpha 14b4\beta 4c51c
3
2c3n+ 4\alpha 14b2\beta 4c61c
2
2c3n+
81
2
\alpha 12b4\beta 6c51c
2
2c
2
3n -
- 13
8
\alpha 12b4\beta 6c41c
4
2c3n+ 72\alpha 12b2\beta 6c61c2c
2
3n - 8\alpha 12b2\beta 6c51c
3
2c3n+ 105\alpha 10b4\beta 8c51c2c
3
3n+
+
27
2
\alpha 10b4\beta 8c41c
3
2c
2
3n - 12\alpha 2L4X3Y b2\beta 2c1c
2
2 - 60\alpha 2X2Y 2b2\beta 2c21c3\gamma
2+
+6\alpha 2X2Y 2b2\beta 2c1c
2
2\gamma
2 - 24\alpha 2L4X3Y b4\beta 2c1c2c3+
+12\beta 2L4X3Y \gamma 2c1c3c2b
2 + 12\alpha 2X2Y 2b4\beta 2c1c2c3\gamma
2,
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
990 T. R. KHOSHDANI, N. ABAZARI
\Lambda = - 375
2
\alpha 8b2\beta 12c31c
4
2c
2
3 +
95
8
\alpha 8b2\beta 12c21c
6
2c3 + 192\alpha 6b2\beta 14c41c2c
4
3 -
- 28\alpha 6b2\beta 14c31c
3
2c
3
3 -
99
2
\alpha 6b2\beta 14c21c
5
2c
2
3 + 4\alpha 6b2\beta 14c1c
7
2c3 -
- 24\alpha 18b2\beta 2c71c2c3 - 2\alpha 20c81c2 +
9
64
\alpha 4\beta 16c92 -
9
32
\beta 20c72c
2
3 +
1
2
\alpha 14b2\beta 6c41c
5
2 -
- 86\alpha 14\beta 6c61c
2
2c3 + 72\alpha 12b2\beta 8c61c
3
3 +
13
8
\alpha 12b2\beta 8c31c
6
2 + 114\alpha 12\beta 8c61c2c
2
3 -
- 113\alpha 12\beta 8c51c
3
2c3 +
1
2
\alpha 10b2\beta 10c21c
7
2 + 153\alpha 10\beta 10c51c
2
2c
2
3 - 101\alpha 10\beta 10c41c
4
2c3+
+144\alpha 8b2\beta 12c51c
4
3 -
7
32
\alpha 8b2\beta 12c1c
8
2 - 8\alpha 8\beta 12c51c2c
3
3 +
237
2
\alpha 8\beta 12c41c
3
2c
2
3 -
- 245
4
\alpha 8\beta 12c31c
5
2c3 + 32\alpha 6\beta 14c41c
2
2c
3
3 +
171
4
\alpha 6\beta 14c31c
4
2c
2
3 -
181
8
\alpha 6\beta 14c21c
6
2c3 -
- 62\alpha 16b2\beta 4c61c
2
2c3 + 88\alpha 4b2\beta 16c31c
2
2c
4
3 -
65
2
\alpha 4b2\beta 16c21c
4
2c
3
3 -
- 33
8
\alpha 4b2\beta 16c1c
6
2c
2
3 + 16\alpha 2b2\beta 18c21c
3
2c
4
3 -
13
2
\alpha 2b2\beta 18c1c
5
2c
3
3 - 216\alpha 14b2\beta 6c61c2c
2
3 -
- 3
32
\alpha 6b2\beta 14c92 - 48\alpha 6\beta 14c51c
4
3 +
51
64
\alpha 6\beta 14c1c
8
2 -
41
8
\alpha 12\beta 8c41c
5
2 -
1
16
\alpha 10\beta 10c31c
6
2+
+
47
32
\alpha 8\beta 12c21c
7
2 -
9
64
\alpha 2\beta 18c82c3 - 2\alpha 20b2c71c
2
2 + 36\alpha 14\beta 6c71c
2
3 -
43
4
\alpha 14\beta 6c51c
4
2 -
- 3
8
b2\beta 20c62c
3
3 - 2\beta 20c21c
3
2c
4
3 +
3
2
\beta 20c1c
5
2c
3
3 - 12\alpha 18\beta 2c81c3 - 7\alpha 18\beta 2c71c
2
2 -
- 23
2
\alpha 16\beta 4c61c
3
2 - 4L4X2Y 3b4 - 20\alpha 2\beta 18c31c
2
2c
4
3+
+16\alpha 2\beta 18c21c
4
2c
3
3 -
45
16
\alpha 2\beta 18c1c
6
2c
2
3 +
3
8
\alpha 4\beta 16c21c
5
2c
2
3 -
63
16
\alpha 4\beta 16c1c
7
2c3 - 44\alpha 16\beta 4c71c2c3+
+
9
32
\alpha 4b2\beta 16c82c3 - 56\alpha 4\beta 16c41c2c
4
3 + 44\alpha 4\beta 16c31c
3
2c
3
3 + b2\beta 20c1c
4
2c
4
3 -
- 4\alpha 18b2\beta 2c61c
3
2 - 72\alpha 16b2\beta 4c71c
2
3 -
5
2
\alpha 16b2\beta 4c51c
4
2 + 118\alpha 8b2\beta 12c41c
2
2c
3
3+
+
5
2
\alpha 10b2\beta 10c31c
5
2c3 + 216\alpha 10b2\beta 10c51c2c
3
3 - 324\alpha 10b2\beta 10c41c
3
2c
2
3 -
- 73
2
\alpha 12b2\beta 8c41c
4
2c3 - 318\alpha 12b2\beta 8c51c
2
2c
2
3 - 68\alpha 14b2\beta 6c51c
3
2c3,
\Upsilon = - \beta 2b2nc32c
6
1\alpha
20 + \beta 4nc32c
6
1\alpha
18 - 30\beta 2b4c61c
2
2c3\alpha
20 - 48\beta 2b2c3c
7
1c2\alpha
20 - 4\beta 4c3nc
7
1c2\alpha
18 -
- 3
2
\beta 4b2nc42c
5
1\alpha
18 - 67
2
\beta 4b4c3c
5
1c
3
2\alpha
18 - 198\beta 4b4c61c2c
2
3\alpha
18 - 126\beta 4b2c3c
6
1c
2
2\alpha
18 -
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
CHARACTERIZATION OF WEAKLY BERWALD FOURTH ROOT METRICS 991
- 10\beta 4b2nc3c
6
1c
2
2\alpha
18 - 1
2
\beta 4b4nc3c
5
1c
3
2\alpha
18 - 32\beta 6b2c23nc
6
1c2\alpha
16 -
- 18\beta 6b2nc3c
5
1c
3
2\alpha
16 - 8\beta 6b4nc23c
5
1c
2
2\alpha
16 - 25
2
\beta 2b2c61c
3
2\alpha
20 - 3
4
\beta 2b4c51c
4
2\alpha
20+
+
3
4
\beta 4b4c41c
5
2\alpha
18 - 108\beta 4b2c23c
7
1\alpha
18 - 29
2
\beta 4b2c51c
4
2\alpha
18 -
- 24\beta 4c3c
7
1c2\alpha
18 - 8\beta 6c23nc
7
1\alpha
16 +
7
2
\beta 6nc42c
5
1\alpha
16 - 23\beta 6c3c
6
1c
2
2\alpha
16 - 9\beta 6b2c41c
5
2\alpha
16 -
- 5b2c71c
2
2\alpha
22 - 3
2
b4c61c
3
2\alpha
22 - 12\beta 2c81c3\alpha
20 - 4\beta 2c71c
2
2\alpha
20 - 3
2
\beta 4c61c
3
2\alpha
18 - 2c81c2\alpha
22 -
- 170\beta 6b2c3c
5
1c
3
2\alpha
16 - 12\beta 6c3nc
6
1c
2
2\alpha
16 - 15
2
\beta 6b4c3c
4
1c
4
2\alpha
16 - 218\beta 6b2c23c
6
1c2\alpha
16 -
- 1
2
\beta 6b2nc52c
4
1\alpha
16 - 25
2
\beta 8b2nc3c
4
1c
4
2\alpha
14 - 58\beta 8b2c23nc
5
1c
2
2\alpha
14 -
- 11
2
\beta 8b4nc23c
4
1c
3
2\alpha
14 - 42\beta 8b4nc33c
5
1c2\alpha
14 +
1
2
\beta 8b4nc3c
3
1c
5
2\alpha
14 -
- 45\beta 10b2c23nc
4
1c
3
2\alpha
12 - 96\beta 10b2c33nc
5
1c2\alpha
12 - 5
2
\beta 10b2nc3c
3
1c
5
2\alpha
12 -
- 432\beta 6b4c61c
3
3\alpha
16 +
3
4
\beta 6b4c31c
6
2\alpha
16 + 146\beta 8c23c
6
1c2\alpha
14 - 10\beta 8c3c
5
1c
3
2\alpha
14 +
3
32
\beta 8b4c21c
7
2\alpha
14+
+216\beta 8b2c33c
6
1\alpha
14 - 23
16
\beta 8b2c31c
6
2\alpha
14 + 6\beta 8nc52c
4
1\alpha
14 +
11
2
\beta 6c51c
4
2\alpha
16 + 64\beta 6c23c
7
1\alpha
16+
+
85
8
\beta 8c41c
5
2\alpha
14 - 326\beta 6b4c23c
5
1c
2
2\alpha
16 +
1
2
\beta 8b2nc62c
3
1\alpha
14 - 798\beta 8b4c33c
5
1c2\alpha
14 -
- 268\beta 8b4c23c
4
1c
3
2\alpha
14 +
53
4
\beta 8b4c3c
3
1c
5
2\alpha
14 - 48\beta 8b2c33nc
6
1\alpha
14 -
- 214\beta 8b2c23c
5
1c
2
2\alpha
14 - 154\beta 8b2c3c
4
1c
4
2\alpha
14 - 16\beta 8c23nc
6
1c2\alpha
14 - 20\beta 8c3nc
5
1c
3
2\alpha
14 -
- 1
8
\beta 10b4nc3c
2
1c
6
2\alpha
12 + 6\beta 10b4nc23c
3
1c
4
2\alpha
12 - 60\beta 10b4nc33c
4
1c
2
2\alpha
12 -
- 18\beta 2XY 2b4c21c
2
2\alpha
10 - 6\beta 2XY 2b6c1c
3
2\alpha
10 - 174\beta 12b4nc43c
4
1c2\alpha
10 + 3\beta 12b4nc33c
3
1c
3
2\alpha
10+
+
7
4
\beta 12b4nc23c
2
1c
5
2\alpha
10 - 3
32
\beta 12b4nc3c1c
7
2\alpha
10 - 96\beta 12b2c33nc
4
1c
2
2\alpha
10 + 684\beta 10b4c43c
5
1\alpha
12+
+153\beta 10c23c
5
1c
2
2\alpha
12 + 24\beta 10c33nc
6
1\alpha
12 +
45
4
\beta 10c3c
4
1c
4
2\alpha
12b4c21c
2
2L
4\alpha 10 - 84\beta 14b4nc43c
3
1c
2
2\alpha
8+
+15\beta 14b4nc33c
2
1c
4
2\alpha
8 + 144\beta 14b2c43nc
4
1c2\alpha
8 - 130\beta 14b2c33nc
3
1c
3
2\alpha
8 + 11\beta 14b2c23nc
2
1c
5
2\alpha
8+
+
7
4
\beta 14b2nc3c1c
7
2\alpha
8 + 18\beta 4XY 2b2c21c
2
2\alpha
8 + 18\beta 4XY 2b4c1c
3
2\alpha
8 + 120\beta 4XY 2b6c21c
2
3\alpha
8+
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
992 T. R. KHOSHDANI, N. ABAZARI
+468\beta 12b2c43c
5
1\alpha
10 +
43
32
\beta 12b2c1c
8
2\alpha
10 +
65
16
\beta 12nc72c
2
1\alpha
10 -
- 104\beta 12c33c
5
1c2\alpha
10 +
263
2
\beta 12c23c
4
1c
3
2\alpha
10 + 16\beta 12c3c
3
1c
5
2\alpha
10 -
- 216\beta 4XY 2b4c21c2c3\alpha
8 + 12\beta 4XY 2b6c1c
2
2c3\alpha
8 - 18\beta 2L4X2Y b2c21c
2
2\alpha
8 -
- 9\beta 2L4X2Y b4c1c
3
2\alpha
8 +
9
64
\beta 12b4c92\alpha
10 +
135
32
\beta 12c21c
7
2\alpha
10 + 1068\beta 12b4c43c
4
1c2\alpha
10 -
- 993\beta 12b4c33c
3
1c
3
2\alpha
10 +
701
8
\beta 12b4c23c
2
1c
5
2\alpha
10 - 135
32
\beta 12b4c3c1c
7
2\alpha
10 + 48\beta 12b2c43nc
5
1\alpha
10 -
- 1
32
\beta 12b2nc82c1\alpha
10 + 346\beta 12b2c33c
4
1c
2
2\alpha
10 - 205
2
\beta 12b2c23c
3
1c
4
2\alpha
10 - 143
4
\beta 12b2c3c
2
1c
6
2\alpha
10+
+60\beta 12c33nc
5
1c2\alpha
10 - 35\beta 12c23nc
4
1c
3
2\alpha
10 - 45
4
\beta 12c3nc
3
1c
5
2\alpha
10 -
- \beta 14b4nc23c1c
6
2\alpha
8 + 144\beta 14b4c53c
4
1\alpha
8 - 35
32
\beta 14b4c3c
8
2\alpha
8 -
- 3
32
\beta 14b2nc92\alpha
8 +
51
32
\beta 14nc82c1\alpha
8 + 44\beta 14c33c
4
1c
2
2\alpha
8 + 68\beta 14c23c
3
1c
4
2\alpha
8+
+
179
16
\beta 14c3c
2
1c
6
2\alpha
8 + 108\beta 2L4X2Y b4c21c2c3\alpha
8 - 3
2
\beta 4XY 2b6c42\alpha
8 - 72\beta 14b4nc53c
4
1\alpha
8+
+
1
32
\beta 14b4nc3c
8
2\alpha
8 + 336\beta 14b4c43c
3
1c
2
2\alpha
8 - 330\beta 14b4c33c
2
1c
4
2\alpha
8+
+
259
8
\beta 14b4c23c1c
6
2\alpha
8 + 318\beta 14b2c43c
4
1c2\alpha
8 + 178\beta 14b2c33c
3
1c
3
2\alpha
8 -
- 217
4
\beta 14b2c23c
2
1c
5
2\alpha
8 - 115
16
\beta 14b2c3c1c
7
2\alpha
8 + 60\beta 14c33nc
4
1c
2
2\alpha
8 -
- 65
2
\beta 14c23nc
3
1c
4
2\alpha
8 - 2\beta 14c3nc
2
1c
6
2\alpha
8 + 3X3c21c
2
2b
2L8\alpha 8 +
3
8
\beta 14b2c92\alpha
8 - 216\beta 14c43c
5
1\alpha
8+
+
27
32
\beta 14c1c
8
2\alpha
8 - 36\beta 6XY 2b4c1c
2
2c3\alpha
6 + 216\beta 6XY 2b2c21c2c3\alpha
6 + 18\beta 4L4X2Y b2c1c
3
2\alpha
6+
+180\beta 4L4X2Y b4c21c
2
3\alpha
6 + 36\beta 2X3c21c2b
2c3L
8\alpha 6 + 24\beta 6XY 2b6c1c2c
2
3\alpha
6 - 9
4
\beta 4L4X2Y b4c42\alpha
6 -
- 3\beta 2X3c1c
3
2b
2L8\alpha 6 - 60\beta 16b4nc53c
3
1c2\alpha
6 - 5\beta 16b4nc43c
2
1c
3
2\alpha
6+
+
15
8
\beta 16b4nc33c1c
5
2\alpha
6 + 8\beta 16b2c43nc
3
1c
2
2\alpha
6 - 48\beta 16b2c33nc
2
1c
4
2\alpha
6+
+
81
8
\beta 16b2c23nc1c
6
2\alpha
6 + 6\beta 6XY 2b6c32c3\alpha
6 - 360\beta 6XY 2b4c21c
2
3\alpha
6 - 18\beta 6XY 2b2c1c
3
2\alpha
6+
+9\beta 4L4X2Y c21c
2
2\alpha
6 +
39
16
\beta 16b4c23c
7
2\alpha
6 - 1152\beta 16b2c53c
4
1\alpha
6 +
9
32
\beta 16b2c3c
8
2\alpha
6 -
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
CHARACTERIZATION OF WEAKLY BERWALD FOURTH ROOT METRICS 993
- 336\beta 16c43c
4
1c2\alpha
6 + 72\beta 16c33c
3
1c
3
2\alpha
6 +
351
8
\beta 16c23c
2
1c
5
2\alpha
6 +
3
2
\beta 16c3c1c
7
2\alpha
6+
+18\beta 4L4X2Y b4cc31c2\alpha
6 - 216\beta 4L4X2Y b2c21c2c3\alpha
6 - 6\beta 6XY 2c21c
2
2\alpha
6 - 3\beta 2X3c21c
2
2L
8\alpha 6 -
- 3
32
\beta 16b4nc23c
7
2\alpha
6 + 120\beta 16b4c53c
3
1c2\alpha
6 - 11\beta 16b4c43c
2
1c
3
2\alpha
6 - 303
8
\beta 16b4c33c1c
5
2\alpha
6+
+96\beta 16b2c53nc
4
1\alpha
6 - 9
32
\beta 16b2nc3c
8
2\alpha
6 + 897\beta 16b2c43c
3
1c
2
2\alpha
6 -
- 203
2
\beta 16b2c33c
2
1c
4
2\alpha
6 - 177
16
\beta 16b2c23c1c
6
2\alpha
6 - 16\beta 16c43nc
4
1c2\alpha
6 + 46\beta 16c33nc
3
1c
3
2\alpha
6 -
- 45
2
\beta 16c23nc
2
1c
5
2\alpha
6+
15
8
\beta 16c3nc1c
7
2\alpha
6+
9
2
\beta 6XY 2b4c42\alpha
6+
9
32
\beta 16nc92\alpha
6 + 9\beta 6L4X2Y b4c32c3\alpha
4+
+108\beta 6L4X2Y c21c2c3\alpha
4 - 360\beta 6L4X2Y b2c21c
2
3\alpha
4+ 6\beta 4X3c1c
2
2b
2c3L
8\alpha 4+ \beta 2L4X2Y 2b4c22n\alpha
4+
+36\beta 8XY 2b2c1c
2
2c3\alpha
4 - 72\beta 8XY 2b4c1c2c
2
3\alpha
4 + 60\beta 4X3c21c
2
3b
2L8\alpha 4 + \beta 2L4X2Y 2b4c22\alpha
4 -
- 14\beta 18b4nc53c
2
1c
2
2\alpha
4 +
3
2
\beta 18b4nc43c1c
4
2\alpha
4 + 64\beta 18b2c53nc
3
1c2\alpha
4 -
- 36\beta 18b2c43nc
2
1c
3
2\alpha
4 +
5
2
\beta 18b2c33nc1c
5
2\alpha
4 - 18\beta 8XY 2b4c32c3\alpha
4 -
- 72\beta 8XY 2c21c2c3\alpha
4 + 18\beta 8XY 2b6c22c
2
3\alpha
4 + 360\beta 8XY 2b2c21c
2
3\alpha
4 - 48\beta 8XY 2b6c1c
3
3\alpha
4 -
- 9\beta 6L4X2Y c1c
3
2\alpha
4 +
9
2
\beta 6L4X2Y b2c42\alpha
4 - 36\beta 4X3c21c2c3L
8\alpha 4 - 9
8
\beta 18b4c33c
6
2\alpha
4+
+
135
32
\beta 18b2c23c
7
2\alpha
4 - 32\beta 18c53nc
4
1\alpha
4 +
27
32
\beta 18c3nc
8
2\alpha
4 - 446\beta 18c43c
3
1c
2
2\alpha
4 +
381
4
\beta 18c33c
2
1c
4
2\alpha
4+
+
219
16
\beta 18c23c1c
6
2\alpha
4 + 2\beta 2Y 4b6c2\alpha
4 + 36\beta 6L4X2Y b4c1c2c
2
3\alpha
4 -
- 36\beta 6L4X2Y b2c1c
2
2c3\alpha
4 - 3
4
\beta 4X3c42b
2L8\alpha 4 + 3\beta 4X3c1c
3
2L
8\alpha 4+
+
1139
2
\beta 18b2c43c
2
1c
3
2\alpha
4 - 403
4
\beta 18b2c33c
5
2c1\alpha
4 + 24\beta 18c43nc
3
1c
2
2\alpha
4+
+
7
2
\beta 18c33nc
2
1c
4
2\alpha
4 - 21
4
\beta 18c23nc1c
6
2\alpha
4 - 9
2
\beta 8XY 2b2c42\alpha
4 + 6\beta 8XY 2c1c
3
2\alpha
4 + 28\beta 18b4c53c
2
1c
2
2\alpha
4 -
- 51
4
\beta 18b4c43c1c
4
2\alpha
4 +
9
16
\beta 18b2c23nc
7
2\alpha
4 - 824\beta 18b2c53c
3
1c2\alpha
4 + 352\beta 18c53c
4
1\alpha
4 - 27
32
\beta 18c3c
8
2\alpha
4+
+4XY 4b6\alpha 4 + 12\beta 6X3c1c
2
3b
2c2L
8\alpha 2 + 4\beta 4L4X2Y 2b4c2c3\alpha
2 -
- 2\beta 4L4X2Y 2b2c22n\alpha
2 + 2\beta 2X3Y b2c22nL
8\alpha 2 - 2\beta 2Y 3b4Xnc2L
4\alpha 2+
+72\beta 10XY 2b2c1c2c
2
3\alpha
2 + 18\beta 8X2Y c1c3c
2
2L
4\alpha 2 - 72\beta 8X2Y b4c1c
3
3L
4\alpha 2 -
- 18\beta 8X2Y b2c32c3L
4\alpha 2 + 27\beta 8X2Y b4c22c
2
3L
4\alpha 2 + 2\beta 2X3Y b2c22L
8\alpha 2+
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
994 T. R. KHOSHDANI, N. ABAZARI
+10\beta 2XY 3b4c2L
4\alpha 2 + 144\beta 10XY 2b4c1c
3
3\alpha
2 - 12\beta 10XY 2c1c
2
2c3\alpha
2 - 54\beta 10XY 2b4c22c
2
3\alpha
2+
+18\beta 10XY 2b2c32c3\alpha
2 + 180\beta 8X2Y c21c
2
3L
4\alpha 2 + 3\beta 6X3c32b
2c3L
8\alpha 2 -
- 6\beta 6X3c1c
2
2c3L
8\alpha 2 - 2\beta 4L4X2Y 2b2c22\alpha
2 - \beta 20b4nc53c1c
3
2\alpha
2 + 8\beta 20b2c53nc
2
1c
2
2\alpha
2 -
- 5\beta 20b2c43nc1c
4
2\alpha
2 + 8L4X2Y 3b4\alpha 2 - \beta 20b4c43c
5
2\alpha
2 -
- 69
8
\beta 20b2c33c
6
2\alpha
2 +
9
16
\beta 20c23nc
7
2\alpha
2 + 224\beta 20c53c
3
1c2\alpha
2 - 202\beta 20c43c
2
1c
3
2\alpha
2 +
93
2
\beta 20c33c1c
5
2\alpha
2+
+
3
2
\beta 10XY 2c42\alpha
2 +
3
4
\beta 6X3c42L
8\alpha 2 - 6\beta 4Y 4b4c2\alpha
2 + 4\beta 4Y 4b6c3\alpha
2 - 12\beta 2XY 4b4\alpha 2 -
- 72\beta 8X2Y b2c1c
2
3c2L
4\alpha 2 + 4\beta 4L4X2Y 2b4c2c3n\alpha
2 - 120\beta 10XY 2c21c
2
3\alpha
2 -
- 9
4
\beta 8X2Y c42L
4\alpha 2 - 60\beta 6X3c21c
2
3L
8\alpha 2 - 6L8X2Y 2b4c2\alpha
2 +
1
8
\beta 20b4nc43c
5
2\alpha
2+
+2\beta 20b4c53c1c
3
2\alpha
2 +
3
4
\beta 20b2nc33c
6
2\alpha
2 - 136\beta 20b2c53c
2
1c
2
2\alpha
2 +
151
2
\beta 20b2c43c
4
2c1\alpha
2 -
- 16\beta 20c53nc
3
1c2\alpha
2 + 16\beta 20c43nc
2
1c
3
2\alpha
2 - 21
4
\beta 20c33nc1c
5
2\alpha
2 - 27
32
\beta 20c23c
7
2\alpha
2 - 4L4Y 4b6\alpha 2 -
- 6b2\beta 22c1c
3
2c
5
3 +
27
8
\beta 22c62c
3
3 +
9
4
b2\beta 22c52c
4
3 + 24\beta 22c21c
2
2c
5
3 - 18\beta 22c1c
4
2c
4
3.
References
1. Antonelli P. L., Ingarden R., Matsumoto M. The theory of sprays and Finsler spaces with applications in physics and
biology. – Netherlands: Kluwer Acad. Publ., 1993.
2. Bácsó S., Yoshikawa R. Weakly-Berwald spaces // Publ. Math. Debrecen. – 2002. – 61. – P. 219 – 231.
3. Balan V., Brinzei N. Berwald – Moor-type (h, v)-metric physical models // Hypercomplex Numbers Geom. and
Phys. – 2005. – 2, № 2(4). – P. 114 – 122.
4. Bidabad B., Tayebi A. A classification of some Finsler connection and their applications // Publ. Math. Debrecen. –
2007. – 71. – P. 253 – 266.
5. Bidabad B., Tayebi A. Properties of generalized Berwald connections // Bull. Iran. Math. Soc. – 2009. – 35. –
P. 237 – 254.
6. Lee I. Y., Lee M. H. On weakly-Berwald spaces of special (\alpha , \beta )-metrics // Bull. Korean Math. Soc. – 2006. – 43,
№ 2. – P. 425 – 441.
7. Lebedev S. V. The generalized Finslerian metric tensors // Hypercomplex Numbers Geom. and Phys. – 2005. – 4,
№ 2. – P. 44 – 50.
8. Matsumoto M. The Berwald connection of a Finsler space with an (\alpha , \beta )-metric // Tensor (N.S.). – 1991. – 50,
№ 1. – P. 18 – 21.
9. Matsumoto M., Shimada H. On Finsler spaces with 1-form metric. II. Berwald – Moór’s metric L =
\bigl(
y1y2...yn
\bigr) 1/n
//
Tensor (N.S.). – 1978. – 32. – P. 275 – 278.
10. Najafi B., Tayebi A. On a class of isotropic mean Berwald metrics // Acta. Math. Acad. Paedagog. Nyiregyhaziensis. –
2016. – 32. – P. 113 – 123.
11. Peyghan E., Tayebi A. Generalized Berwald metrics // Turkish J. Math. – 2012. – 36. – P. 475 – 484.
12. Peyghan E., Tayebi A. Killing vector fields of horizontal Liouville type // C. R. Acad. Sci. Paris. Ser. I. – 2011. –
349. – P. 205 – 208.
13. Pavlov D. G. (Ed.) Space-time structure. Algebra and Geometry // Collect. Papers. – Moscow: TETRU, 2006.
14. Pavlov D. G. Four-dimensional time // Hypercomplex Numbers Geom. and Phys. – 2004. – 1, № 1. – P. 31 – 39.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
CHARACTERIZATION OF WEAKLY BERWALD FOURTH ROOT METRICS 995
15. Shimada H. On Finsler spaces with metric L = m
\sqrt{}
ai1i2...imyi1yi2 ...yim // Tensor (N.S.). – 1979. – 33. – P. 365 – 372.
16. Tayebi A. On the class of generalized Landsberg manifolds // Period. Math. Hung. – 2016. – 72. – P. 29 – 36.
17. Tayebi A., Barzagari M. Generalized Berwald spaces with (\alpha , \beta )-metrics // Indag. Math. – 2016. – 27. – P. 670 – 683.
18. Tayebi A., Najafi B. On m-th root Finsler metrics // J. Geom. and Phys. – 2011. – 61. – P. 1479 – 1484.
19. Tayebi A., Najafi B. On m-th root metrics with special curvature properties // C. R. Acad. Sci. Paris. Ser. I. – 2011. –
349. – P. 691 – 693.
20. Tayebi A., Nankali A., Peyghan E. Some curvature properties of Cartan spaces with m-th root metrics // Lith.
Math. J. – 2014. – 54, № 1. – P. 106 – 114.
21. Tayebi A., Nankali A., Peyghan E. Some properties of m-th root Finsler metrics // J. Contemp. Math. Anal. – 2014. –
49, № 4. – P. 157 – 166.
22. Tayebi A., Peyghan E., Shahbazi Nia M. On generalized m-th root Finsler metrics // Linear Algebra and Appl. –
2012. – 437. – P. 675 – 683.
23. Tayebi A., Peyghan E. On \bfitE -curvature of R-quadratic Finsler metrics // Acta Math. Acad. Paedagog. Nyiregyhazi-
ensis. – 2012. – 28. – P. 83 – 89.
24. Tayebi A., Sadeghi H. Generalized P -reducible (\alpha , \beta )-metrics with vanishing S -curvature // Ann. Polon. Math. –
2015. – 114. – P. 67 – 79.
25. Tayebi A., Sadeghi H. On generalized Douglas – Weyl (\alpha , \beta )-metrics // Acta Math. Sin. Engl. Ser. – 2015. – 31,
№ 10. – P. 1611 – 1620.
26. Tayebi A., Shahbazi Nia M. A new class of projectively flat Finsler metrics with constant flag curvature K = 1 //
Different. Geom. and Appl. – 2015. – 41. – P. 123 – 133.
27. Tayebi A., Tabatabaeifar T., Peyghan E. On Kropina-change of m-th root Finsler metrics // Ukr. Math. J. – 2014. –
66, № 1. – P. 160 – 164.
Received 14.07.16
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 7
|
| id | umjimathkievua-article-1491 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:06:47Z |
| publishDate | 2019 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/89/bb437971aeec671f9ce7d0d9d2874e89.pdf |
| spelling | umjimathkievua-article-14912019-12-05T08:57:08Z Characterization of weakly Berwald fourth root metrics Характеризацiя слабких чотирикореневих метрик Бервальда Abazari, N. Khoshdani, T. R. Абазарі, Н. Хошдані, Т. Р. UDC 514.7 In recent studies, it is shown that the theory of fourth root metrics plays a very important role in physics, theory of space-time structures, gravitation, and general relativity. The class of weakly Berwald metrics contains the class of Berwald metrics as a special case. We establish the necessary and sufficient condition under which the fourth root Finsler space with an $(\alpha, \beta)$-metric is a weakly Berwald space. УДК 514.7 В останніх досліджeннях було встановлено, що теорія чотирикореневих метрик відіграє важливу роль у фізиці, теорії просторово-часових структур, гравітації та загальній теорії відносності. Клас слабких метрик Бервальда містить клас метрик Бервальда як частинний випадок. Встановлено необхідну та достатню умову, за якої чотирикореневий простір Фінслера з $(\alpha, \beta)$-метрикою є слабким простором Бервальда. Institute of Mathematics, NAS of Ukraine 2019-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1491 Ukrains’kyi Matematychnyi Zhurnal; Vol. 71 No. 7 (2019); 976-995 Український математичний журнал; Том 71 № 7 (2019); 976-995 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1491/475 Copyright (c) 2019 Abazari N.; Khoshdani T. R. |
| spellingShingle | Abazari, N. Khoshdani, T. R. Абазарі, Н. Хошдані, Т. Р. Characterization of weakly Berwald fourth root metrics |
| title | Characterization of weakly Berwald fourth root metrics |
| title_alt | Характеризацiя слабких чотирикореневих метрик Бервальда |
| title_full | Characterization of weakly Berwald fourth root metrics |
| title_fullStr | Characterization of weakly Berwald fourth root metrics |
| title_full_unstemmed | Characterization of weakly Berwald fourth root metrics |
| title_short | Characterization of weakly Berwald fourth root metrics |
| title_sort | characterization of weakly berwald fourth root metrics |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1491 |
| work_keys_str_mv | AT abazarin characterizationofweaklyberwaldfourthrootmetrics AT khoshdanitr characterizationofweaklyberwaldfourthrootmetrics AT abazarín characterizationofweaklyberwaldfourthrootmetrics AT hošdanítr characterizationofweaklyberwaldfourthrootmetrics AT abazarin harakterizaciâslabkihčotirikorenevihmetrikbervalʹda AT khoshdanitr harakterizaciâslabkihčotirikorenevihmetrikbervalʹda AT abazarín harakterizaciâslabkihčotirikorenevihmetrikbervalʹda AT hošdanítr harakterizaciâslabkihčotirikorenevihmetrikbervalʹda |