Formulas of Frenet for a Vector Field in a Finsler Space

Muroran-IT Academic Resources Archive
Title
Author(s)
Citation
Issue Date
URL
Formulas of Frenet for a Vector Field in a Finsler Space
Nagata, Yukiyoshi
室蘭工業大学研究報告. Vol.3 No.3, pp.543-546, 1960
1960-06-15
http://hdl.handle.net/10258/3141
Rights
Type
Journal Article
See also Muroran-IT Academic Resources Archive Copyright Policy
Muroran Institute of Technology
FormulasofFrenetfora VectorFieldina FinslerSpace
YukiyoshiNagata
A
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.
1
. Formulas of Frenet for a Vector Field ina Hypersurface.
LetF
n
lbeah
y
p
e
r
s
u
r
f
a
c
eg
i
v
e
n by t
h
es
e
to
fe
q
u
a
t
i
o
n
s x'=x' (
u
"u2,
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・
・ ,
u
ニ1
,•.. ,
n
)i
n aF
i
n
s
l
e
rs
p
a
c
e,
Fπ thefundamentalquadraticform o
fwhich
(え
(
ι ど )dx'dx
ぺ Fn-lt
owhich t
h
ee
l
e
m
e
n
to
fs
u
p
p
o
r
ti
st
a
n
g
e
n
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l has
i
sds2=めμ
b
t
h
efundamentalq
u
a
d
r
a
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i
c formd
s2ニ '
gαbdu"du
. Letv
' beana
r
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i
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r
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r
yb
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五ned a
te
v
e
r
yp
o
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g
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u
n
i
tv
e
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t
o
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l
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Let C: ua=uα(
s
) (a=lγ ・. ,
n-1) bea c
u
r
v
eon,
F -1 andl
e
tN' bea u
n
i
tv
e
c
t
o
r
e
f
i
n
en v
e
c
t
o
r
sa
l
o
n
g C byt
h
ef
o
l
l
o
w
i
n
ge
q
u
a
t
i
o
n
s
:
normal t
oF
n
l・ W ed
n 1
-)
冗
tJ=ザフ ザ
可<
r+
1
/=D可γt,
'j
d
s
万
(
2
)
λ -
kN
',・・・,
(
J=2,
… ,n-1),
v
(
1
.1
)
wherevk=g'I'N'DvPjdsandDvμ2-3) d
e
n
o
t
e
st
h
ea
b
s
o
l
u
t
ed
i
旺e
r
e
n
t
i
a
la
l
o
n
gC o
ft
h
e
l
dげ a
t Po
fC
. When;
r
"
B
)λ (
s=l,
…,
n
)a
r
el
i
n
e
a
r
l
yi
n
d
e
p
e
n
d
e
n
t,t
h
e
v
e
c
t
o
r 五e
f
o
l
l
o
w
i
n
gn v
e
c
t
o
r
sσ(
p
/(p=l,・・・ ,n) whichareexpressed linearlywith the como
rJ=l,
•.• ,
p forma s
e
to
fm
u
t
u
a
l
l
yo
r
t
h
o
g
o
n
a
lv
e
c
t
o
r
s
:
ponents恥/ f
'
)-(町駅
(
J,ε=1,•.. ,
p
)
r
)
(
1
.2
)
where
f
oニ
1,
,
f = 1, f
p= I
f
r
'
l,
g,
p;
r
(
r
/
;
r
(
E
)
!
',
f
,
.
E =f~
f
,
.
EF
;=可.
P
u
t
t
i
n
g
Dι
ん
一一一干旦」ー
正
lS
(
p,q=1,・・・ ,n
),
σ(p)" - αqp
h
(
1
.3
)
fromσ
ω,
a(p)
ν=
d
gwehave
一 αqp= α
'^pq ,
(
1
5
7
)
(
1
.4
)
5
4
4
Y
u
k
i
y
o
s
h
iNagata
DσrnfhM
一一ーとζ ー = λαmσイ 川 r
q
ds
(
1
.5
)
P
"'
"
'
From (
1
.1
) and (
1
.2
)i
tf
o
l
l
o
w
st
h
a
t Dσ(
p
/
/
d
si
sa
t most a l
i
n
e
a
re
x
p
r
e
s
s
i
o
ni
n
d(
d(p+
1
/
' C
onsequently,α =O(ん+1<ん
)
.
Combiningt
h
i
sr
e
s
u
l
twith (
1
.4
),wehave
a♂,…, η
(肘,/andt
h
e
r
e
f
o
r
ei
n ♂ γ.. ,
帥
αp
-p
p
+
l一
- 一 αp
+
l
p一
-v
U.
.
.
L込
(
1
.6
)
(
qキ p+1),
α別 ニ O
whereのKpi
sd
e
f
i
n
e
dbyt
h
ef
i
r
s
to
ft
h
e
s
ee
q
u
a
t
i
o
n
s
. Hence e
1
.5
)a
r
e
q
u
a
t
i
o
n
s(
r
e
d
u
c
e
dt
o
竺
p
{
_==- vKp内 一 什 vKp
,
σ(PHJE
(ρ=2γ ・.,
n-1),
(
1
.7
)
正
IS
where v
Kpf
o
rp=l,
・
・,
n-1 a
r
ec
a
l
l
e
d,r
e
s
p
e
c
t
i
v
e
l
y,t
h
ea
s
s
o
c
i
a
t
ec
u
r
v
a
t
u
r
e
so
f
o
r
d
e
r1
γ ・. ,
n-1 o
ft
h
ev
e
c
t
o
r五e
l
dv f
o
rt
h
ec
u
r
v
eC
. (
1
.7
) maybec
o
n
s
i
d
e
r
e
d
a
sa g
e
n
e
r
a
l
i
z
a
t
i
o
no
ft
h
eF
r
e
n
e
tf
o
r
m
u
l
a
sf
o
rac
u
r
v
e andh
o
l
de
x
c
e
p
tt
h
ec
a
s
e
p =1
. And (
1
.7
)a
p
p
l
yt
ot
h
ec
a
s
e
ρ =nw
i
t
ht
h
eu
n
d
e
r
s
t
a
n
d
i
n
gt
h
a
tvK二 O
. We
renetoft
h
esecondkindforv along C i
nF
n
.
c
a
l
lt
h
e
s
et
h
eformulasofF
Int
h
ef
o
l
l
o
w
i
n
g,wes
h
a
l
ld
e
r
i
v
et
h
ef
o
r
m
u
l
a
so
fF
r
e
n
e
to
ft
h
ef
i
r
s
tk
i
n
df
o
r
va
l
o
n
gC i
nFn・ Wep
u
t
η
=ザ , ~(2)Æ =Dく
ご1
/
/
d
s="
K
'
切にー・,
f
、
r
+1
ゾ=D~(r)Æ/ds ,
~(1)Æ
,
(
1
.8
)
where vKi
st
h
ea
b
s
o
l
u
t
ec
u
r
v
a
t
u
r
eo
fv a
tP withr
e
s
p
e
c
tt
o C andt
h
es
e
n
s
eo
f
ft
h
e
s
ev
e
c
t
o
r
s ~(æ)Æ (α=1,
・
ー,
n
)
ω i
s choseni
nsuchawaya
st
omakevK>O. I
a
r
eassumedt
o bel
i
n
e
a
r
l
yi
n
d
e
p
e
n
d
e
n
t,t
h
ef
o
l
l
o
w
i
n
gl
i
n
e
a
rc
o
m
b
i
n
a
t
i
o
n
so
f them
'・
.,
n formas
e
to
fn m
u
t
u
a
l
l
yo
r
t
h
o
g
o
n
a
lv
e
c
t
o
r
s:
f
o
rP=l,
λ=(ι)
い
(
r
,E=1
γ .,
p
)
μ(p)
(
1
.9
)
where
y
;
Y
O=1, Y1! =I I
,
y
c=y~ =gÆP~(r)ÆごくE/ ,
Y
;Y~
=o
;.
Andwehave f
1
(
l
)
'=v
,
"f
1
(
2
/
=
W
J
.
.
P
u
t
t
i
n
g(D
f
1
(
h
)j
d
s
)μων=んk(
h,k=l,… ,n
),from,
1
fh),
f
1(
k
)
'
=ほ weh
ave
(
1
.1
0
)
μ
a
(
1
5
8
)
Dμa
z
h
一
一
虹ゐ
s
k
1
' =ー
んk
(
1
.1
1
)
F
o
r
m
u
l
a
so
fF
r
e
n
e
tf
o
raV
e
c
t
o
rF
i
e
l
di
naF
i
n
s
l
e
rS
p
a
c
e
545
(
1
.1
1
)a
r
ereducedt
o
全正 =
正IS
-vLk
J
約 一 什 ムμ山
ん
(=2,
・
・
・,
n-1)
(
1
.1
2
)
where vLk=skk+l=ーん +lk・
(
1
.1
2
)a
p
p
l
yt
ot
h
ec
a
s
e んニ 1 w
i
t
ht
h
eu
n
d
e
r
s
t
a
n
d
i
n
gt
h
a
t Lo=O andVL'=VK.
Also,wehave(
1
.1
2
)f
o
r ん=n w
i
t
ht
h
eu
n
d
e
r
s
t
a
n
d
i
n
gt
h
a
tvLn=O. vLkん
(=1,
一,
n-1) a
r
ec
a
l
l
e
d,r
e
s
p
e
c
t
i
v
e
l
y,t
h
ea
s
s
o
c
i
a
t
ec
u
r
v
a
t
u
r
e
so
fo
r
d
e
r 1,••• ,
n-1 o
f
t
h
ev
e
c
t
o
r五e
l
dv f
o
rt
h
ec
u
r
v
e C. We c
a
l
l(
1
.1
2
)t
h
eformulas ofFrenet of
t
h
ef
i
r
s
tkindforv α
:
l
o
n
gC i
n Fn.
包
2
. Extension.
Wec
o
n
s
i
d
e
ra s
u
b
s
p
a
c
eF
. (m<n)givenbyX'=ど (
z
人...,
u叫
)(
え
ニ 1,"
'
,n
)
i
n aF
i
n
s
l
e
rs
p
a
c
eF
n
. Theelement o
fs
u
p
p
o
r
ti
st
a
n
g
e
n
t
i
a
lt
oF
. LetN;(pニ
m十 1,.・.,
n
)ben-mmutuallyorthogonalu
n
i
tv
e
c
t
o
r
snormal t
o Fmwithr
e
s
p
e
c
t
t
ot
h
em
e
t
r
i
co
fF
n
. Letザ
vλ bean 但
a
r
七
b
i
t
仕r
a
r
可y b
u
t五xedu
n
i
tv
e
c
t
o
r五e
l
d白
d
e
白
五ne
吋da
討t
e
v
e
r
守y p
o
i
n
to
fFm such出
t
h抗
a
tザ
v 1 ニ UαB~p ラ'gαb山
v(
汽
r
b
民ea 叩
c
urveonF
W ed
e
n
o
t
et
h
ea
b
s
o
l
u
t
ed
i
旺e
r
e
n
t
i
a
lo
fv
' with r
e
s
p
e
c
tt
oC a
tP byDザ and
de
五nethefollowingvectors:
泊
汎・
叫
NL-・
布
。)λ ニザ,
ザ
玖 2)λ ニ ム
D;
r
l
r
)
'
一
一
一
λ
(
,
十 1ヲ
9
(
rニ 2,…, η-1),
ー
(
2
.
1
)
正
IS
whereλ= め ,(N~Dv l'/ds.
If可ゆえ (s=l
,・
.,
n
)a
r
el
i
n
e
a
r
l
yi
n
d
e
p
e
n
d
e
n
t,t
h
ef
o
l
l
o
w
i
n
gl
i
n
e
a
rc
o
m
b
i
n
a
t
i
o
n
so
f
ー ,
n forma s
e
to
fn m
u
t
u
a
l
l
yo
r
t
h
o
g
o
n
a
lv
e
c
t
o
r
s
:
them f
o
rpニ 1,・
_
h
_
_¥
!_
,
'F!
σ Aー(
γ
(,ε=1γ .
.,
p
)
T
!
,
¥jirl/'MP
(
2
.
2
)
where
fo=l, ,
1 = 1,
五 =I
f
r
"
l,
f
;=f
l=g
'
p
;
r
く
J守的 μ,
ξ
frEF~ 士。;.
Thereforep
u
t
t
i
n
g (Da(q,
j
d
s
)σων=αqp,wehave
11,
. μ
句
一よ
K
ι
p一 内 一 円幻K凶
三与E2__
α
正
IS
where"K
,
,=
α
p
士
~pp 寸 1-
ψ
一
1)
αp1
11
'.
v¥.
十
(
15
9
)
(
pニ 2
,"
'
, n-1),
(
2
.
3
)
546
Y
u
k
i
y
o
s
h
iN
a
g
a
t
a
2
.
3
)t
h
eformul
ω
, ofF
renetザ t
h
esecond的 dforv alongC i
nF
n
.
We c
a
l
l(
vKp(ρ=1,
"
'
, n-1)arec
a
l
l
e
d,r
e
s
p
e
c
t
i
v
e
l
y
,thea
s
s
o
c
i
a
t
ec
u
r
v
a
t
u
r
e
so
fo
r
d
e
r1,
"
'
,
n-1 ofthevector五eldvforthe curve C
. (2.3) holdexceptthecasep=1and
h
eu
n
d
e
r
s
t
a
n
d
i
n
gt
h
a
tvK
η=0.
a
p
p
l
yt
ot
h
ec
a
s
ep=nwitht
While
,the formulas ofFrenet ofthe 五
回tkindf
o
rv a
l
o
n
gC i
nFnmay be
.
d
e
r
i
v
e
di
nt
h
esame waya
si
smentionedi
n1
(
R
e
c
e
i
v
e
d]
a
n
.26,1960)
R
e
f
e
r
e
n
c
e
s
P
a
n,T. K.: P
r
o
c
.A
m
e
r
.M
a
t
h
.S
o
c
.8,294 (1957)
D
a
v
i
e
s,E
.T
.
: P
r
o
c
.L
o
n
d
o
nM
a
t
h
.S
o
c
.49,19 (1945)
a
r
t
a
n,E
.
: L
e
se
s
p
a
c
e
sd
eF
i
n
s
l
e
rP
.40(
P
a
r
i
sHermann1934)
3
) C
1
)
2
)
(
16
0
)