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PureMathematicsnØêÆ,2022,12(5),826-837
PublishedOnlineMay2022inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2022.125093
DŒÅdÄÈ¥dV6/WLÇ
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∗
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ÂvFϵ2022c418F¶¹^Fϵ2022c519F¶uÙFϵ2022c527F
Á‡
(M
1
,F
1
)Ú(M
2
,F
2
)´ü‡¥dV6/,DŒÅdÄÈ¥dVÝþ´3¦È6/M= M
1
×M
2
þDƒ¥dVÝþF=
p
f(K,H),Ù¥K= F
2
1
,H= F
2
2
,…f´È¼ê.©Ì‡ïÄDŒÅ
dÄÈ¥dV6/(M,F)WLÇÚ²þWLÇ,|^Üþ©Û{,DŒÅdÄÈ¥d
V6/(M,F)WLÇž”7‡^‡; 3(M
1
,F
1
)Ú(M
2
,F
2
)²þWLÇž”^‡
e,‰ÑDŒÅdÄÈ¥dV6/(M,F)²þWLÇž”¿©^‡,l‰Ñ˜«•x
äkAÏ5Ÿ¥dV6/k•{.
'…c
¥dV6/§DŒÅdÄȧWLǧ²þWLÇ
CartanTorsionofMinkowskian
ProductFinslerManifold
NaZhang,YongHe
∗
,HuiZhang,ShuwenLi
CollegeofMathematicalSciences,XinjiangNormalUniversity,UrumqiXinjiang
Received:Apr.18
th
,2022;accepted:May19
th
,2022;published:May27
th
,2022
∗ÏÕŠö"
©ÙÚ^:ÜA,Û].DŒÅdÄÈ¥dV6/WLÇ[J].nØêÆ,2022,12(5):826-837.
DOI:10.12677/pm.2022.125093
ÜA
Abstract
Let (M
1
,F
1
) and(M
2
,F
2
) be twoFinslermanifolds, Minkowskianproduct Finsler metric
istheFinslermetric F=
p
f(K,H)endowedontheproductmanifold M=M
1
×M
2
,
whereK=F
2
1
,H=F
2
2
,andfisproductfunction.Inthispaper,westudyCartan
torsionandmeanCartantorsion ofMinkowskianproduct Finslermanifold(M,F).By
usingtensoranalysis,weobtainthenecessaryconditionthatMinkowskianproduct
Finslermanifold(M,F)hasvanishingCartantorsion.Also,wegive thesufficient con-
ditionthatMinkowskianproductFinslermanifold(M,F)hasvanishingmeanCartan
torsionundertheconditionsofFinslermanifolds (M
1
,F
1
)and(M
2
,F
2
)havevanishing
meanCartantorsion.ThenaneffectivemethodforcharacteriseFinslermanifolds
withspecialpropertyisgiven.
Keywords
FinslerManifold, MinkowskianProduct,CartanTorsion,MeanCartanTorsion
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
¥dVAÛ3g,Ɖ+•A^2•,cÙ3ÔnÆ!)ÔÆ!ó§Eâ!&EAÛ!››Ø
Ú2ƒéØ•¡[1][2]. 1918c, Finsler3¦Ƭة¥Äg‰Ñ/¥dVAÛ0ù˜¶
¡,¿½ÂWLÇ[3]. 1934c, Cartan[4]Ú\Wéä¿y²WLÇCUïþ6/þ
¥dVÝþ†iùÝþ l§Ý.eC = 0,K¥dVÝþ´iùÝþ,ÏdWLÇ¡•¥
dVAÛ¥šiùAÛþ.
WLÇ÷Xÿ/‚CzÇ¡•Landsberg -Ç.1950c, EhresmannïÄAÏœ/e
Wéä[5].1972 c,Matsumoto |^WLÇ•x˜aAÏ¥dV6/,¡ƒ•C-Œ¥
dV6/[6],‘¦y²Wéä•˜5[7]. 1989 c,Fukui‰ÑäkWéäE¥dV
6/´iù6/7‡^‡[8]. 1998c,!§¬[9]y²äkÃ.WLÇ¥dV6/ØU
åi\?ÛDŒÅdĘm¥,¦„y²äkk.WL ÇR-g¥dV6/˜½
DOI:10.12677/pm.2022.125093827nØêÆ
ÜA
´Landsberg 6/[10].2010c,#•[11]í2©[10](J, =y²äkk.WLÇ
R-g¥dV6/˜½´6/.
²þWLÇ•´¥dV AÛ¥˜a›©-‡šiùAÛþ. ²þWLÇ´WLÇ,,
§÷Xÿ/‚CzÇ¡•²þLandsberg-Ç. 1953c, Deicke[12]JÑͶDeicke ½
n,=˜‡½¥dV6/´iù6/…=T¥dV6/²þWLÇž”.d,ïÄAÏ
¥dV6/WLÇÚ²þWLǤ•NõÆö'59:¯K[13–18].
1982c,Okada[19]JÑDŒÅdÄÈ¥dV6/Vg,¿DŒÅdÄÈ¥dV6/þ
ÿ/‚†Ù©þ6/þÿ/‚ƒm'X.g,¯K ´XÛ•xDŒÅdÄ¥dV6/W
LÇÚ²þWLÇ, ±9&ÄDŒÅdÄ¥dV6/´Ääkž”WLÇÚ²þWLÇ.
éþã¯K, ©DŒÅdÄÈ¥dV6/WLÇž”7‡^‡, ¿3ü‡©þ6
/²þWLÇž”^‡e,‰ÑDŒÅdÄÈ¥dV6/²þWLÇž”¿©^‡.
2.ý•£
M´˜‡n‘1w6/,T
x
ML«x∈Mƒ˜m, TM=
S
x∈M
T
x
ML«Mƒm, 
MþÛÜ‹I•x
i
= (x
1
,...,x
n
),T
x
MþÛÜ‹I•(x
i
,y
i
) = (x
1
,...,x
n
,y
1
,...,y
n
).
½Â1.[20]6/Mþ¥dVÝþ´˜¼êF:TM→R
+
,÷v
1)G= F
2
3
˜
M= TM\{0}þ´1w¼ê;
2)éu?¿(x,y) ∈
˜
M,F(x,y) >0;
3)éu?¿(x,y) ∈
˜
M,λ∈R,kF(x,λy) = |λ|F(x,y);
4)çlÝ(G
αβ
) = (
∂
2
G
∂y
α
∂y
β
)3
˜
Mþ´½Ý.
(G
γα
)L«(G
αβ
)_Ý,=G
γα
G
αβ
= δ
γ
β
.
½Â2.[21](M,F)´˜‡¥dV6/,FWLÇC : T
x
M×T
x
M×T
x
M→R•
C= C
αβγ
dx
α
⊗dx
β
⊗dx
γ
,(2.1)
Ù¥
C
αβγ
=
1
4
G
y
α
y
β
y
γ
=
1
4
∂G
αβ
∂y
γ
.(2.2)
½Â3.[21](M,F)´˜‡¥dV6/,F²þWLÇI : T
x
M→R•
I= I
α
(x,y)dx
α
,(2.3)
Ù¥
I
α
=
1
2
G
βγ
∂G
αβ
∂y
γ
= 2G
βγ
C
αβγ
.(2.4)
DOI:10.12677/pm.2022.125093828nØêÆ
ÜA
(M
1
,F
1
)Ú(M
2
,F
2
)´ü‡¥dV6/, ‘ê©O•mÚn,KM=M
1
×M
2
´˜‡
m+n‘¦È6/.PM
1
,M
2
ÚMÛÜ‹I©O•(x
1
,...,x
m
),(x
m+1
,...,x
m+n
),
(x
1
,...,x
m
,x
m+1
,...,x
m+n
);M
1
,M
2
ÚMƒm©O´TM
1
,TM
2
ÚTM,§‚ÛÜ‹I
©O•(x
1
,...,x
m
,y
1
,...,y
m
),(x
m+1
,...,x
m+n
,y
m+1
,...,y
m+n
),(x
1
,...,x
m+n
,y
1
,...,y
m+n
).
TM
1
,TM
2
,TMƒm÷vTM
∼
=
TM
1
⊕TM
2
.
½Â4.[19]f: [0,+∞)×[0,+∞) →[0,+∞)´˜‡ëY¼ê,ef÷v
1)f(s,t) = 0…=(s,t) = (0,0);
2)é?¿λ∈[0,+∞),kf(λs,λt) = λf(s,t);
3)f3(0,+∞)×(0,+∞)þ´1w¼ê;
4)é?¿(s,t) ∈(0,+∞)×(0,+∞),k
∂f
∂s
6= 0,
∂f
∂t
6= 0;
5)é?¿(s,t) ∈(0,+∞)×(0,+∞),k
∂f
∂s
∂f
∂t
−2f
∂
2
f
∂s∂t
6= 0,
K¡f•È¼ê.
·K1.[22]f´È¼ê,Ké?¿K6= 0,H6= 0, k
f
HK
K+f
HH
H= 0,(2.5)
f
K
K+f
H
H= f,(2.6)
f
KK
K+f
KH
H= 0,(2.7)
f
KKK
K+f
KKH
H= −f
KK
,(2.8)
f
KHK
K+f
KHH
H= −f
KH
,(2.9)
f
HHK
K+f
HHH
H= −f
HH
.(2.10)
Ù¥f
K
Úf
H
©OL«féKÚH ê,Xf
H
=
∂f
∂H
,f
KH
=
∂
2
f
∂K∂H
.
½Â5.[19] (M
1
,F
1
)Ú(M
2
,F
2
)´ü‡¥dV6/,-K= F
2
1
,H= F
2
2
,F
1
ÚF
2
'ufD
ŒÅdÄÈÝþ´3¦È6/M= M
1
×M
2
þDƒXe/ªÝþF: TM→R
+
F(x,y) =
p
f(K(x
i
,y
i
),H(x
i
0
,y
i
0
)),(2.11)
Ù¥f´È¼ê.w,(M,F)•´˜‡¥dV6/,¡(M,F)•(M
1
,F
1
)Ú(M
2
,F
2
) 'uf
DŒÅdÄÈ6/,~{¡(M,F)•DŒÅdÄÈ¥dV6/.
©¥, α,β,γF1i1Š‰Œ•1m+n; i,j,k.¶i1Š‰Œ
•1m;i
0
,j
0
,k
0
‘§.¶i1Š‰Œ•m+1m+n.'u(M
1
,F
1
)½(M
2
,F
2
)
AÛþ,·‚©O3Ùþ•V\•I1½2±««O, X
1
C
ijk
Ú
2
C
i
0
j
0
k
0
©OL«¥dV6/
(M
1
,F
1
)Ú(M
2
,F
2
)WLÇXê.
(M,F)´¥dV6/(M
1
,F
1
)Ú(M
2
,F
2
)DŒÅdÄÈ,KFÄÜþÝ•
G= (G
αβ
) = (
∂
2
G
∂y
α
∂y
β
) =
G
ij
G
ij
0
G
i
0
j
G
i
0
j
0
!
,
DOI:10.12677/pm.2022.125093829nØêÆ
ÜA
Ù¥

















G
ij
= f
K
K
ij
+f
KK
K
i
K
j
,(2.12)
G
ij
0
= f
KH
K
i
H
j
0
,(2.13)
G
i
0
j
= f
HK
H
i
0
K
j
= f
KH
H
i
0
K
j
,(2.14)
G
i
0
j
0
= f
H
H
i
0
j
0
+f
HH
H
i
0
H
j
0
.(2.15)
K
i
,H
i
0
©OL«K,Héy
i
,y
i
0
 ê,XK
i
=
∂K
∂y
i
,H
i
0
j
0
=
∂
2
H
∂y
i
0
∂y
j
0
.(G
αβ
)_Ý(G
βα
)•
(G
βα
) =
G
ji
G
ji
0
G
j
0
i
G
j
0
i
0
!
,
Ù¥





























G
ji
=
1
f
K
(K
ji
−
f
H
f
KK
∆
y
j
y
i
),(2.16)
G
ji
0
= −
1
∆
f
KH
y
j
y
i
0
,(2.17)
G
j
0
i
= −
1
∆
f
KH
y
j
0
y
i
,(2.18)
G
j
0
i
0
=
1
f
H
(H
j
0
i
0
−
f
K
f
HH
∆
y
j
0
y
i
0
),(2.19)
…
∆ = f
K
f
H
−2ff
KH
.(2.20)
3.DŒÅdÄÈ¥dV6/WLÇ
ŠâWLÇÚDŒÅdÄÈ¥dV6/½Â,!òíÑDŒÅdÄÈ¥dV6/
(M,F)WLÇX êLˆª, DŒÅdÄÈ¥dV6/(M,F)WLÇž”ž, ˜
‡ ‡©•§|,=‰ÑDŒÅdÄÈ¥dV6/(M,F) WLÇž”7‡^‡.
·K2.(M,F)´¥dV6/(M
1
,F
1
)Ú(M
2
,F
2
)DŒÅdÄÈ,K(M,F)WLÇXê
C
αβγ
•
C
ijk
= f
K
1
C
ijk
+
1
4
(f
KKK
K
i
K
j
K
k
+f
KK
K
ij
K
k
+f
KK
K
ik
K
j
+f
KK
K
jk
K
i
),(3.1)
C
i
0
jk
=
1
4
(f
HKK
H
i
0
K
j
K
k
+f
HK
H
i
0
K
jk
),(3.2)
DOI:10.12677/pm.2022.125093830nØêÆ
ÜA
C
ij
0
k
=
1
4
(f
KHK
K
i
H
j
0
K
k
+f
KH
K
ik
H
j
0
),(3.3)
C
ijk
0
=
1
4
(f
KKH
K
i
K
j
H
k
0
+f
KH
K
ij
H
k
0
),(3.4)
C
i
0
j
0
k
=
1
4
(f
HHK
H
i
0
H
j
0
K
k
+f
HK
H
i
0
j
0
K
k
),(3.5)
C
i
0
jk
0
=
1
4
(f
HKH
H
i
0
K
j
H
k
0
+f
HK
H
i
0
k
0
K
j
),(3.6)
C
ij
0
k
0
=
1
4
(f
KHH
K
i
H
j
0
H
k
0
+f
KH
K
i
H
j
0
k
0
),(3.7)
C
i
0
j
0
k
0
= f
H
2
C
i
0
j
0
k
0
+
1
4
(f
HHH
H
i
0
H
j
0
H
k
0
+f
HH
H
i
0
j
0
H
k
0
+f
HH
H
i
0
k
0
H
j
0
+f
HH
H
j
0
k
0
H
i
0
).(3.8)
y².-(2.2)¥α= i,β= j,γ= k, ¿ò(2.12) “\,²†OŽŒ
C
ijk
=
1
4
∂G
ij
∂y
k
=
1
4
(f
KKK
K
i
K
j
K
k
+f
KK
K
ik
K
j
+f
KK
K
i
K
jk
+f
KK
K
k
K
ij
+f
K
K
ijk
).(3.9)
36/(M
1
,F
1
)þA^½Â2,¿5¿K= F
2
1
,Kk
K
ijk
= 4
1
C
ijk
.(3.10)
ò(3.10)“\(3.9),Œ
C
ijk
= f
K
1
C
ijk
+
1
4
(f
KKK
K
i
K
j
K
k
+f
KK
K
ij
K
k
+f
KK
K
ik
K
j
+f
KK
K
jk
K
i
).
=(3.1)¤á.ÓnŒy(3.2)-(3.8)¤á.y..
½n1.(M,F)´¥dV6/(M
1
,F
1
)Ú(M
2
,F
2
)DŒÅdÄÈ. e(M,F)WLÇC ž
”,Ke•§|¤á









2f
KKK
K+3f
KK
= 0,(3.11)
2f
HHH
H+3f
HH
= 0,(3.12)
f
KHK
K−f
KHH
H= 0.(3.13)
y².d(2.1)Œ•,(M,F) WLÇž”…=
C
ijk
= C
i
0
jk
= C
ij
0
k
= C
ijk
0
= C
i
0
j
0
k
= C
i
0
jk
0
= C
ij
0
k
0
= C
i
0
j
0
k
0
= 0.(3.14)
DOI:10.12677/pm.2022.125093831nØêÆ
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Šâ(3.1),(3.14)¥C
ijk
= 0du
f
K
1
C
ijk
+
1
4
(f
KKK
K
i
K
j
K
k
+f
KK
K
ij
K
k
+f
KK
K
ik
K
j
+f
KK
K
jk
K
i
) = 0.(3.15)
(3.15)ü>Óž†y
i
y
j
y
k
¿,¿5¿y
k
1
C
ijk
= 0,Œ
y
i
y
j
y
k
(f
KKK
K
i
K
j
K
k
+f
KK
K
ij
K
k
+f
KK
K
ik
K
j
+f
KK
K
jk
K
i
) = 0.(3.16)
duK= F
2
1
'uyäkgàg5,Šâî.½nk
K
ij
y
i
= K
j
,(3.17)
K
i
y
i
= 2K.(3.18)
ò(3.17)Ú(3.18)“\(3.16),¿5¿K6= 0, Kk
2f
KKK
K+3f
KK
= 0.
ÓnŠâ(3.8)Ú(3.14)¥C
i
0
j
0
k
0
= 0,Œ±í
2f
HHH
H+3f
HH
= 0.
Šâ(3.2),(3.14)¥C
i
0
jk
= 0du
f
HKK
H
i
0
K
j
K
k
+f
HK
H
i
0
K
jk
= 0.(3.19)
(3.19)ü>Óž†y
i
0
y
j
y
k
¿,¿A^(3.17)Ú(3.18),Œ
2f
HKK
K+f
KH
= 0.(3.20)
r(2.9)“\(3.20),k
f
KHK
K−f
KHH
H= 0.
Ónd(3.14)¥Ù§ª½ŒíÑ(3.13)¤á.y..
4.DŒÅdÄÈ¥dV6/²þWLÇ
Šâ²þWLÇÚDŒÅdÄÈ¥dV6/½Â,!ò íÑDŒÅdÄÈ¥dV6/
(M,F)²þWLÇXêLˆª, 3˜½^‡e‰ÑDŒÅdÄÈ¥dV6/(M,F)²þW
LÇž”¿©^‡,=•xäkAÏLÇ5Ÿ¥dV6/.
·K3.(M,F)´¥dV6/(M
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DOI:10.12677/pm.2022.125093832nØêÆ
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XêI
α
•
I
i
=
1
I
i
−
K
i
2∆
[f
H
(f
KKK
K+3f
KKH
H)−f
K
(f
KHH
H−2f
KHK
K)],(4.1)
I
i
0
=
2
I
i
0
−
H
i
0
2∆
[f
K
(f
HHH
H+3f
HHK
K)−f
H
(f
HKK
K−2f
HKH
H)].(4.2)
y².-(2.4)¥α= i,k
I
i
= 2G
βγ
C
iβγ
= 2(G
jk
C
ijk
+G
j
0
k
C
ij
0
k
+G
jk
0
C
ijk
0
+G
j
0
k
0
C
ij
0
k
0
).(4.3)
Šâ(2.16)Ú(3.1),k
G
jk
C
ijk
=
1
f
K
(K
jk
−
f
H
f
KK
∆
y
j
y
k
)[f
K
1
C
ijk
+
1
4
(f
KKK
K
i
K
j
K
k
+f
KK
K
ij
K
k
+f
KK
K
ik
K
j
+f
KK
K
jk
K
i
)]
=K
jk
1
C
ijk
+
1
4f
K
(f
KK
K
i
+f
KK
δ
k
i
K
k
+f
KK
δ
j
i
K
j
)
+
1
4f
K
K
jk
f
KKK
K
i
K
j
K
k
−
1
4f
K
f
H
f
KK
∆
y
j
y
k
(f
KKK
K
i
K
j
K
k
+f
KK
K
ij
K
k
+f
KK
K
ik
K
j
+f
KK
K
jk
K
i
)−
2f
H
f
KK
∆
y
j
y
k
1
C
ijk
=K
jk
1
C
ijk
+
3
4f
K
f
KK
K
i
+
1
4f
K
K
jk
f
KKK
K
i
K
j
K
k
−
1
4f
K
f
H
f
KK
∆
y
j
y
k
(f
KKK
K
i
K
j
K
k
+f
KK
K
ij
K
k
+f
KK
K
ik
K
j
+f
KK
K
jk
K
i
),(4.4)
Ù¥1n‡ªí¥A^y
k
1
C
ijk
= 0.(3.17)ü>Óž†K
jk
¿,Œ
K
jk
K
j
= y
k
.(4.5)
r(3.18)Ú(4.5)“\(4.4),²†OŽŒ
G
jk
C
ijk
=K
jk
1
C
ijk
+
K
i
4f
K
(2f
KKK
K+3f
KK
)−
K
i
2∆
f
H
f
KK
K
f
K
(2f
KKK
K+3f
KK
)
=K
jk
1
C
ijk
+
K
i
2∆
∆
2f
K
(2f
KKK
K+3f
KK
)−
K
i
2∆
Kf
H
f
KK
f
K
(2f
KKK
K
+3f
KK
).(4.6)
DOI:10.12677/pm.2022.125093833nØêÆ
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5¿∆ = f
K
f
H
−2ff
KH
,·‚k
K
i
2∆
∆
2f
K
(2f
KKK
K+3f
KK
) =
K
i
2∆
[
f
K
f
H
−2ff
KH
2f
K
(2f
KKK
K+3f
KK
)]
=
K
i
2∆
(
1
2
f
H
−
ff
KH
f
K
)(2f
KKK
K+3f
KK
).(4.7)
r(4.7)“\(4.6),Œ
G
jk
C
ijk
=K
jk
1
C
ijk
+
K
i
2∆
(
1
2
f
H
−
ff
KH
f
K
)(2f
KKK
K+3f
KK
)
−
K
i
2∆
Kf
H
f
KK
f
K
(2f
KKK
K+3f
KK
).(4.8)
r(2.6)-(2.8)“\(4.8),²{üOŽŒ±
G
jk
C
ijk
= K
jk
1
C
ijk
−
K
i
2∆
(
1
2
f
H
−f
KH
K)(f
KKK
K+3f
KKH
H)
=
1
2
1
I
i
−
K
i
2∆
(
1
2
f
H
−f
KH
K)(f
KKK
K+3f
KKH
H).(4.9)
ÓnŒ
G
j
0
k
C
ij
0
k
= −
K
i
2∆
f
KH
H(f
KHK
K−f
HKH
H),(4.10)
G
jk
0
C
ijk
0
= −
K
i
2∆
f
KH
H(f
KHK
K−f
HKH
H),(4.11)
G
j
0
k
0
C
ij
0
k
0
=
K
i
2∆
(
1
2
f
K
−f
KH
H)(f
KHH
H−f
KHK
K).(4.12)
r(4.9)-(4.12)“\(4.3),²nŒ
I
i
=2(G
jk
C
ijk
+G
j
0
k
C
ij
0
k
+G
jk
0
C
ijk
0
+G
j
0
k
0
C
ij
0
k
0
)
=
1
I
i
−
K
i
∆
(
1
2
f
H
−f
KH
K)(f
KKK
K+3f
KKH
H)−
2K
i
∆
f
KH
H(f
KHK
K−f
HKH
H)
+
K
i
∆
(
1
2
f
K
−f
KH
H)(f
KHH
H−f
KHK
K)
=
1
I
i
−
K
i
∆
[
1
2
f
H
(f
KKK
K+3f
KKH
H)−
1
2
f
K
(f
KHH
H−f
KHK
K)]−
K
i
∆
f
KH
[−K
(f
KKK
K+f
KHK
H)−H(3f
KKH
K+2f
HKH
H)+H(2f
KHK
K+f
KHH
H)].(4.13)
r(2.7)-(2.9)“\(4.13),Œ
I
i
=
1
I
i
−
K
i
2∆
[f
H
(f
KKK
K+3f
KKH
H)−f
K
(f
KHH
H−2f
KHK
K)].
=(4.1)¤á.ÓnŒy(4.2)¤á.y..
½n2.(M,F)´¥dV6/(M
1
,F
1
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2
,F
2
)DŒÅdÄÈ, …(M
1
,F
1
)Ú(M
2
,F
2
)
DOI:10.12677/pm.2022.125093834nØêÆ
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²þWLÇž”,eF÷ve•§|







f
KKK
K+3f
KKH
H= 0,(4.14)
f
HHH
H+3f
KHH
K= 0,(4.15)
f
KHK
K= f
KHH
H= 0,(4.16)
K(M,F) ²þWLÇž”.
y².Šâ(2.3)Œ•,(M
1
,F
1
)²þWLÇž”du
1
I
i
= 0.(4.17)
ò(4.17)“\(4.1),Œ
I
i
= −
K
i
2∆
[f
H
(f
KKK
K+3f
KKH
H)−f
K
(f
KHH
H−2f
KHK
K)].(4.18)
®•(4.14)-(4.16)¤á, ò(4.14) Ú(4.16) “\(4.18), ŒI
i
=0. Ónd(M
2
,F
2
) ²þWL
Çž”,¿ò(4.15)Ú(4.16)“\(4.2),ŒI
i
0
= 0.
Ïd
I= I
i
dx
i
+I
i
0
dx
i
0
= 0.
y..
5.(Ø
©DŒÅdÄÈ¥dV6/WLÇž”7‡^ ‡, =3DŒÅdÄÈ¥dV6/
WLÇž”^‡e, ‰Ñ‡©•§•x. …3DŒÅdÄÈ¥dV 6/ü‡©þ6/
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äkAÏ5Ÿ¥dV6/. Ï·‚òÏL¦)þãüa ‡©•§|, ‰Ñäkž”WL
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ë•©z
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[2]Asanov,G.S.(2012)FinslerGeometry,RelativityandGaugeTheories.SpringerScienceand
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