Submetric class の面積空間における共形対応について

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(1)Title. Submetric class の面積空間における共形対応について. Author(s). 蒲, 雅夫. Citation. 北海道教育大学紀要. 第二部. A, 数学・物理学・化学・工学編, 20(1) : 1-3. Issue Date. 1969-09. URL. http://s-ir.sap.hokkyodai.ac.jp/dspace/handle/123456789/5916. Rights. Hokkaido University of Education.

(2) Vol. 20, No. 1 Journal of Hokkaiclo University of Education (Section II A) September 1969. On Conformal Correspondence in ati Areal Space of the Submetric Class. Masao GAMA Department of Mathematics, Asahikawa Branch, Hokkaido University of Education. m ?^: : Submetric class OB^^rRl^^HÂ^Mj^ôi'-C. § 1. Introduction. Let us consider two w-dimehsional areal spaces Aw and A,(,'") of the submetric class of which the fundamental functions are given by F^x , ^)° and F<ixl, pla~) respectively referred to the same coordinates. Two areal spaces A»m and A1m) are called conformal, if the normalized metric tensors gij and gij of Aw and A,(m) respectively are connected by the relation (1.1) gtj=e'tvgi} where a is at most a point function by Theorem 3.1 in the paper [2J2). The main object of the present paper is to derive some invariants with respect to the conformal correspondence (1.1) and find the conditions for the factor of proportionality e2(r to be constant. Throughout the paper, we use the same notations as in the papers [I], [2]. § 2. Invariants. The normalized metric tensors gtj and gij satisfy the relations. (2.1) gw - &K - gi^ - 8^*? = 0, (2.2) gw - gwF^ - gi^l- gteT% = 0 respectively, the barred quantities referring to F, from which it follows that ,*?»_ (ft) ^rft^^-.*i»7-i*i i /-.*ai7-i*t r^*ar,*l. u = iyi — .§" \{-tr,tl if] + <^3r,ll at — ÎJ.U ' a.r ),. -ri*/i_ | li ( -^•hf7Vi-a,~~r,*l i 7»(i!n«( 7^*a7W. y= W —g^tr,ll'»]+l^jr~/l"?t—<^tj,ll'w).. From (1.1) and Theorem 3. 1 in the papeer [2], we have. (2.5) g^^Sw, (2.6) gw=ez'r(.2^gu+gu,^ from which it follows that. (2.7) \'u\ = Sftj + ^ + a,,8} - g"^,,.gu. Contracting (2.7) with 81,., we have _ 1 f ( A ( j ft. tf,.;=-^\ M - i"ji ^.. Also, from (2.3), (2.4) and (2.8), we have ~Fi*r , /^*w~jn*l- / T-I*)' > /^*rotn*!. °^= -^'^1 'r'J+ L,.; •;./It •»}[/ ';•'}rj+l^r, + L,.; "it '1=~H^.1 rj'+l^r. a}—V ll •^s)). aj!. Consequently from (2.3), (2.4), (2. 7) and (2.9), we have 1) Latain indices run from 1 to n and Greek indices from 1 to ttl. 2) Numbers in brackets refer to the references at the end of the paper.. Cl).

(3) Masao GAMA. ~n*}t. J- r'rV^r , /=;>*WFi*i ^ ^t- ± i'~n*r , T^rct, 7^*1 \ ^t. n's - -^ (T^ + C"r,'°ir'a))8"i - — ^F;'t + C7,Val )8]. +^g"sg^+cy.xn\~) loiTW i T^rtTW —ft'h/^* of~n*l l,~t1 a^+u ],l1 »t—g ^tj,ll a'r. (2.io) = r%- ^-(r^ + c^r^)^" - ^(r?< + c^rM +^g"sg^w+cr.W) ^!ic&n*l t r>*hd>T^^ _r}i/-^*(6ri*!. 1,'i^ '»)+ ^ J.'if at — g L'W^ we.. The relation (2.10) shows that the quantities G[] defined by -?t _ j-i*h ^ 1 f j-,*r i ^*»-a r,*;. ^ ;,'t. Tij= 1 'U ~~n'^1 >-^+ L- '•,"'•' ai)°l. -^(.rrt+cr,xrw. +^W^+c^r:'0 <*/itf n*l , /^*/ttf T^*^ ^'/l-/^* a T^*I t,~~l.l'»}+ 0 ),'ll 'st — g {-'l),t1 »•'•. are invariant with respect to the conformal correspondence (1.1). In particular, when C?^=0, since it follows from (2.3) that Fij = \i'}\ , (2.11) reduces to. G"y= {u} —^{^}8'l—^{n}8']+^gMgiA^} 'A. f !i' i •j- i r i ^ J- f »' i ^ i •L ^hs _ / v. which represents that the quantities Gy are the conformal connection parameters of J. M. Thomas in Riemannian space." We call Gij the conformal connection parameters of our space. From the definition (2.11) of the conformal connection parameters, we have Gij=Gji, G!i.j=0, Gij; ip^=0 by means of a straight-forward calculation. From (2.8) and (2.9), we have -ri»>- i '^*ra~r,*l r >• i _ r.»)' , ^.*>'a;r.*( ( '•. ,•/+<-,•,(/ »J—\rji=l f]+^ r, it aj—{rj).. With respect to the conformal correspondence (1.1), we have (2.13) P(A-',^)=e"?7<'(;v',^) from which by differentiation with respect to x3 it follows that Tiâ? ri*f6\. (2.14) ",3=^r7}-ra°°s) i*tf rî n*)t 1 7^*t>i r^l ~n*!<. where ra3=(%.nj and F<»'5=/%r"u. From (2.8) and (2.14), we have 1 -r,*ce 1 ( ;7( _ 1 )i*a 1 (h. (2.15) y^-Yz W =m/^-t !/:''s • In virtue of the relations )! r,^h _ al (/i ( of rW' _ a( (ft.. </, / '(.;• = PA ii.;i, pill IJ = p* iv. which can be derived from (2.3) and (2.4) respectively, the relation (2.15) reduces to 3) See [4].. (2).

(4) On Conformal Correspondence in an Areal Space of the Submetric Class. c2.16) ^^-^-^^-^ which can also be obtained by means of a straight-forward calculation from (2,7).. From (2.9) and (2.14) it follows that J_p*tf _^_f'~T*r , ,^*'r»~r*l ^ _J_7-i*tf J^/'r'*)' i ^~'*fa> n*; ~ml «J~^^1 rJ+l^r<11 u-<J=m1 w~~n^1 ^+0 '•-'•' as^. The relations (2.12), (2.15), (2.16) and (2.17) represent that all of the quantities i*y i ^*ra j-i*l | r \ 1 r.*(t 1 f ft )•}• -t- <-- r,"ll aj- ir.,j , ^ «'J~~M. in. i^-(. 1 ( '• 1 a» 1 I ft I 1 T-i*a! 1 / n*r , i^*r» n*l. s:l\Pr--^\hj<i m'" "" n '"" ', -^,1 m' »j-~^\l "' n »-.?-t-L- r,ll.»l.. are invariant with respect to the conformal correspondence (1.1). § 3. Conditions for vanishing' of a,j. The Euler vectors with respect to F and F are. given by. (3.1) Et = L?/( ^,p + i»i ^) + i«S ; ro?^, (3.2) E^L?W,,+ [M ? + W; WPS» respectively0. From(2.7), (3.1) and (3.2), we have (3.3) EÊi-m.a,rt\ from which it follows that the Euler vector is invariant with respect to the conformal correspondence (1.1), if and only if (3.4) arv;=Q, where a1 = glra,r. The condition (3.4) implies that a' may be represented in the form 0.5) ff=r^. If our space is positively regular,55 ^? in (3.5) vanish by means of Theorem 3.2 in the. paper [2]. We proceed to find conditions that â> vanish, i.e. a is constant. From (2.14), we have Theorem. In order that a be constant, it is necessary and sufficient that. (3.6) r^=r^. We can prove without difficulty that the equations (3.6) are equivalent to the equations ri*/' i /^*}'c6~n*l T^*)' > ^rcd ri*f. rj -1- ^ r, 1.1 0..!=^ rj-\-^r,ll a.l.. Moreover, from (2.7) it can be proved that the equations (3.6) are equivalent to the equations " I _ (1. 5.8) h"; i = ;(}i .. REFERENCES [1] Gama, M. (1965), On areal spaces of the submetric class. Tensor, N.S., Vol. 16, p. 262-268. [2] Kikuchi, S. (1966), Some remarks on areal spaces of the submetric class. Tensor, N.S., Vol. 17, p. 44-48. [3] Tanclai, K, (1963), On general connection in an areal space 1. General connection on a fibre of the tangent m-frame bundle. Tensor, N.S., Vol. 13, p. 277-291. [4] Yano, K. (1942), Shots Riman-Kikagaku. Kangaekata Kenkyusha. Tokyo,. 4) See (2.11) in [1]. 5) See (5.2) in [3] or [2].. (3).

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