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PureMathematicsnØêÆ,2021,11(7),1389-1399
PublishedOnlineJuly2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.117156
fBerwaldVLÈFinslerÝþ
"""††††††§§§ÛÛÛ]]]
∗
§§§XXXjjj¦¦¦
#õ“‰ŒÆêƉÆÆ§#õ¿°7à
Email:dxxw523@126.com,
∗
heyong@xjnu.edu.cn,niqihui@126.com
ÂvFϵ2021c611F¶¹^Fϵ2021c713F¶uÙFϵ2021c721F
Á‡
©Ì‡ï ÄVLÈFinslerÝþ²þBerwald-ÇÚ••²þBerwald-ǧ‰ÑV
LÈFinslerÝþ´fBerwaldÝþ¿‡^‡§y²3˜½^‡eäk••²þBerwald
-ÇVLÈFinslerÝþ´fBerwaldÝþ"
'…c
FinslerÝþ§VLȧfBerwaldÝþ§••²þBerwald-Ç
WeaklyBerwaldDoubly-Twisted
ProductFinsler
Metrics
XiangxiangDeng,YongHe
∗
,QihuiNi
SchoolofMathematicsScience,XinjiangNormalUniversity,Urumqi Xinjiang
Email:dxxw523@126.com,
∗
heyong@xjnu.edu.cn,niqihui@126.com
Received:Jun.11
th
,2021;accepted:Jul.13
th
,2021;published:Jul.21
st
,2021
∗ÏÕŠö"
©ÙÚ^:"††,Û],Xj¦.fBerwaldVLÈFinslerÝþ[J].nØêÆ,2021,11(7):1389-1399.
DOI:10.12677/pm.2021.117156
"††
Abstract
ThispapermainlystudiesthemeanBerwaldcurvatureandisotropicmeanBerwald
curvaturedoubly-twistedproductofFinslermetrics.Thenecessaryandsufficient
conditionsforthedoubly-twisted productofFinslermetricsareweaklyBerwald met-
rics.Itisprovedthatundercertainconditionsthedoubly-twistedproductofFinsler
metricswithisotropicmeanBerwaldcurvatureisweaklyBerwaldmetrics.
Keywords
FinslerMetrics,DoublyTwistedProduct,WeaklyBerwaldMetrics,IsotropicMean
BerwaldCurvature
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
LÈ´˜«EAÏÝþk•{.1981c,Chen‰ÑRiemannÝþLÈVg[1].
2006c,Kozma,PeterÚShimadaòLÈí2FinslerAÛ¥,ïÄLÈFinslerÝþ
Cartanéä,ÿ/‚9Ù5[2].CAc,LÈFinslerÝþ˜Æö'5ÚïÄ[3][4].
3FinslerAÛ¥,fBerwaldÝþ´˜a-‡FinslerÝþ.BerwaldÄkJÑBerwald
-ÇVg[5][6].1986c,Matsumoto‰ÑBerwaldÝþ½Â[7].2001c,!§¬y²ä
kž”Berwald-ÇFinslerÝþ´BerwaldÝþ.Berwald-Ç,¡•²þBerwald-Ç,
…äk ž”²þBerwald-ÇFinslerÝþ•fBerwaldÝþ[8].2005c,§#Ú!§¬‰Ñ
••²þBerwald-Ç½Â[9],§´Berwald-Çí2.2013c,PeyghanÚTayebiy²
äk••²þBerwald-ÇLÈFinslerÝþ´fBerwaldÝþ[3].2020c,n!Û]ÚÜ
¡ y²VÛ-ÈFinsler6/äk••²þBerwald-Ç…=§´fBerwald6/[10].
©òLÈFinslerÝþí2•VLÈFinslerÝþ,̇ïÄVLÈFinslerÝþ²þ
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BerwaldÝþƒm'X.
DOI:10.12677/pm.2021.1171561390nØêÆ
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DOI:10.12677/pm.2021.1171561391nØêÆ
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2
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kh
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i
g
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kh
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j
g
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kh
∂y
l
g
ij
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kh
∂g
ij
∂y
l
)
∂f
2
2
∂x
h
−
1
4f
2
2
∂
3
g
kh
∂y
i
∂y
j
∂y
l
(
∂f
2
2
∂x
h
F
2
1
+
∂f
2
1
∂x
h
F
2
2
),(3.1)
B
k
iβl
= B
k
ilβ
= B
k
βil
= −
1
4f
2
2
∂
2
g
kh
∂y
i
∂y
l
∂f
2
1
∂x
h
∂F
2
2
∂v
β
,(3.2)
DOI:10.12677/pm.2021.1171561392nØêÆ
"††
B
k
αβl
= B
k
αlβ
= B
k
lαβ
= −
1
2f
2
2
∂g
kh
∂y
l
∂f
2
1
∂x
h
g
αβ
,(3.3)
B
k
αβλ
= −
1
f
2
2
g
kh
∂f
2
1
∂x
h
C
αβλ
,(3.4)
B
γ
ijl
= −
1
f
2
1
g
γµ
∂f
2
2
∂u
µ
C
ijl
,(3.5)
B
γ
iβl
= B
γ
βil
= B
γ
ilβ
= −
1
2f
2
1
∂g
γµ
∂v
β
∂f
2
2
∂u
µ
g
il
,(3.6)
B
γ
αβl
= B
γ
αlβ
= B
γ
lαβ
= −
1
4f
2
1
∂
2
g
γµ
∂v
α
∂v
β
∂f
2
2
∂u
µ
∂F
2
1
∂y
l
,(3.7)
B
γ
αβλ
=
2
B
γ
αβλ
−
1
4f
2
1
(
∂
2
g
γµ
∂v
β
∂v
λ
∂F
2
2
∂v
α
+
∂
2
g
γµ
∂v
α
∂v
λ
∂F
2
2
∂v
β
+
∂
2
g
γµ
∂v
α
∂v
β
∂F
2
2
∂v
λ
)
∂f
2
1
∂u
µ
−
1
2f
2
1
(
∂g
γµ
∂v
α
g
βλ
+
∂g
γµ
∂v
β
g
αλ
+
∂g
γµ
∂v
λ
g
αβ
+g
γµ
∂g
αβ
∂v
λ
)
∂f
2
1
∂u
µ
−
1
4f
2
1
∂
3
g
γµ
∂v
α
∂v
β
∂v
λ
(
∂f
2
2
∂u
µ
F
2
1
+
∂f
2
1
∂u
µ
F
2
2
).(3.8)
y².Šâ(2.1)
G
A
=
1
4
G
AB
(
∂
2
F
2
∂Y
B
∂X
C
Y
C
−
∂F
2
∂X
B
),(3.9)
-(3.9)¥A= i,K
G
i
=
1
4
G
ih
(
∂
2
F
2
∂y
h
∂x
j
y
j
+
∂
2
F
2
∂y
h
∂u
α
v
α
−
∂F
2
∂x
h
)+
1
4
G
iµ
(
∂
2
F
2
∂v
µ
∂x
j
y
j
+
∂
2
F
2
∂v
µ
∂u
α
v
α
−
∂F
2
∂u
µ
),(3.10)
d(2.6)†OŽk
∂F
2
∂y
h
= f
2
2
∂F
2
1
∂y
h
,
∂F
2
∂x
h
=
∂f
2
2
∂x
h
F
2
1
+f
2
2
∂F
2
1
∂x
h
+
∂f
2
1
∂x
h
F
2
2
,(3.11)
∂
2
F
2
∂y
h
∂u
α
=
∂f
2
2
∂u
α
∂F
2
1
∂y
h
,
∂
2
F
2
∂y
h
∂x
j
=
∂f
2
2
∂x
j
∂F
2
1
∂y
h
+f
2
2
∂
2
F
2
1
∂y
h
∂x
j
,(3.12)
r(2.9),(3.11)Ú(3.12)“\(3.10),Œ
G
i
=
1
G
i
+
1
4f
2
2
g
ih
[(
∂f
2
2
∂x
j
y
j
+
∂f
2
2
∂u
α
v
α
)
∂F
2
1
∂y
h
−
∂f
2
2
∂x
h
F
2
1
−
∂f
2
1
∂x
h
F
2
2
],(3.13)
(3.13)ü>Óž'uy
j
‡©Œ
∂G
i
∂y
j
=
∂
1
G
i
∂y
j
+
1
4f
2
2
∂g
ih
∂y
j
[(
∂f
2
2
∂x
k
y
k
+
∂f
2
2
∂u
α
v
α
)
∂F
2
1
∂y
h
−
∂f
2
2
∂x
h
F
2
1
−
∂f
2
1
∂x
h
F
2
2
]
+
1
4f
2
2
g
ih
[
∂f
2
2
∂x
j
∂F
2
1
∂y
h
+(
∂f
2
2
∂x
k
y
k
+
∂f
2
2
∂u
α
v
α
)
∂
2
F
2
1
∂y
h
∂y
j
−
∂f
2
2
∂x
h
∂F
2
1
∂y
j
],(3.14)
DOI:10.12677/pm.2021.1171561393nØêÆ
"††
d(2.7)(i)
∂F
2
1
∂y
h
= 2g
hk
y
k
.Ïd,
∂g
ih
∂y
j
∂F
2
1
∂y
h
= 2
∂g
ih
∂y
j
g
hk
y
k
= −2g
ih
∂g
hk
∂y
j
y
k
= 0,(3.15)
¿5¿g
ih
g
hj
= δ
i
j
,¤±
1
8f
2
2
g
ih
(
∂f
2
2
∂x
k
y
k
+
∂f
2
2
∂u
α
v
α
)
∂
2
F
2
1
∂y
h
∂y
j
=
1
4f
2
2
(
∂f
2
2
∂x
k
y
k
+
∂f
2
2
∂u
α
v
α
)δ
i
j
,(3.16)
r(3.15)Ú(3.16)“\(3.14),Œ
∂G
i
∂y
j
=
∂
1
G
i
∂y
j
+
1
4f
2
2
g
ih
(
∂f
2
2
∂x
j
∂F
2
1
∂y
h
−
∂f
2
2
∂x
h
∂F
2
1
∂y
j
)+
1
2f
2
2
(
∂f
2
2
∂x
k
y
k
+
∂f
2
2
∂u
α
v
α
)δ
i
j
−
1
4f
2
2
∂g
ih
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1
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h
F
2
2
),(3.17)
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j
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F
2
1
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h
F
2
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),(3.18)
(3.18)ü>Óž'uy
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i
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µ
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,(3.20)
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2
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2
F
2
2
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α
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.(3.21)
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ij
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k
ijk
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γ
ijγ
,(3.22)
DOI:10.12677/pm.2021.1171561394nØêÆ
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2
1
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,(3.23)
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jk
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jk
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i
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h
j
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i
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k
F
2
1
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2
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h
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2
1
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γ
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F
2
2
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β
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2
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F
2
2
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1
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k
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2
1
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h
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2
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µ
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kh
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i
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1
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h
= 0,(4.5)
DOI:10.12677/pm.2021.1171561395nØêÆ
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j
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2
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kh
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2
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µ
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1
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2
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2
1
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2
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2
F
2
1
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DOI:10.12677/pm.2021.1171561396nØêÆ
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k
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2
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h
F
2
1
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kh
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2
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k
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2
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h
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∂v
γ
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2
1
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µ
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DOI:10.12677/pm.2021.1171561397nØêÆ
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k
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1
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h
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2
1
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2
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1
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k
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2
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1
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i
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i
v
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kh
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k
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2
1
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h
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f
2
1
f
2
2
F
3
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2
1
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i
,(4.26)
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DOI:10.12677/pm.2021.1171561398nØêÆ
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DOI:10.12677/pm.2021.1171561399nØêÆ

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