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 b s t r a c t T .K.P a n1) d e m o n s t r a t e dt h eg e n e r a l i z e df o r m u l a so fF r e n e tf o rav e c t o r五e l di nas u b s p a c eo f aR i e m a n n i a ns p a c e . T h i sp a p e r 巴x t e n d sh 1 Si n v e s t i g a t i o nt oah y p e r s u r f a c ea n das u b s p a c eo fa F i n s l e rs p a c e . 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, ・ ・ ・ , 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 t i a l has i sds2=めμ b t h efundamentalq u a d r a t i c formd s2ニ ' gαbdu"du . Letv ' beana r b i t r a r yb u t五xed 五ned a te v e r yp o i n to fF n l sucht h a t ザ ==v(~Ba). , ' g a b Vαが=1. u n i tv e c t o rf i e l d de 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 )
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