Submetric Class の面積空間における部分空間の誘導曲率テンゾルについて

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(1)Title. Submetric Class の面積空間における部分空間の誘導曲率テンゾルについて. 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, 19(1) : 28-32. Issue Date. 1968-09. URL. http://s-ir.sap.hokkyodai.ac.jp/dspace/handle/123456789/5890. Rights. Hokkaido University of Education.

(2) Vol. 19, No. 1 Journal of Hokkaido University of Education (Section II A) September, 1968. On the Induced Curvature Tensors of Subspaces in an Areal Space of the Submetric Class. Masao GAMA. Department of Mathematics, Asahigawa Branch. Hokkaido University of Education. ifjye : Submetric Class OSiK^fBl^.toH^gfS^-zgflfl o^^tt^T- '^•//iôi'-r. 0. Introduction In the previous paper El]*' the present author developed the theory of subspaces in the w-dimensional areal space A(-n,i' of the submetric class. In this paper it is the purpose to derive the generalized equations of Gauss and Codazzi in terms of the induced curvature tensors in an ^,-dimensional subspace A'-m) of Aw. We derive the induced curvature tensors in § 1 and the generalized equations of Gauss and Codazzi in § 2. Throughout the paper, Latin indices run from 1 to n, Greek a, (3, r, 8 from 1 to m ; \, p., v, a, r, t), p from 1 to Jt and S, '1, C from m+1 to n. § 1. Induced Curvature tensors An ^,-demensional subspace Aw of an ^-dimensional areal space ^°") can be represented parametrically by the equation xt=x\yx'), where we suppose that the variables yx form a coordinate system of A'-m\ The fundamental function F of ^<M). induced from F of Aw is given by W, P^=F(.«'W, B'^-), where B{ ='Qxi/'Qyx and we assume that the matrix |B^|| is of rank Ji throughout the paper.. The metric tensor g^ of Aw is defined by. g^-W^/m+p^g^ where L^ denotes the Legendre's form of the fundamental function F in Aw, namaly. L^F-l(F F; f; ^-F ; tF; ^+F; y, 0, ^det.Q^)'/2, pa^F-<F;^, *1 Numbers in brackets refer to the references at the end of the paper.. (28).

(3) Masao GAMA. and symbol ; ^ denoting 3/3^. Let us consider a vector such that. a. D X'=B^. We define the induced covariant differential 'DX^- by. (I. 2) /D^^B^DX!, where DX1 denotes the covariant differential defined by A. Kawaguchi and K, Tandai C2], namely (1. 3) DXi=DXt+Xfr:l"l,,dxk+X'C^Dpl and B)i =g,^Bj gl'lt, gx''' being determined by the relation g>"tg^ °= ^.. Then (1, 2) can be written in the form (1. 4) /DXX=dXX+r^X'Tdy'l+C^X'r/Dpv,, where F*^ - T^ - C\,s Tv^, / r\ _T>\rnt i ri s'Tti TTI nv i r*'' R^'R" an.=D'i'^Da,il +^jlTDffnllvP'a+1 JkDa°n^. » _r'. «R.<. w w. ^0-1 V = '-' Jll '->< u"(r-"V!. H'^r^B^+r^W), / Dp], =rK^+^r<!W)> v< •—f!'- ^1 n« yV_ KV _ r>V n'r n'"—F-ia/f'/an''. )';. =0',. —p'aP!, 1"r=°0'-r—P'tt P'r i lj~r °= r 'oi'l°lj?. and symbol, /' denotes 9/3.)"\. The induced curvature tensors for the covariant differential defined by (1. 4) are obtained in usual way as the coefficients of the following equations :. (/D/D - /D'D~)X^ = [K^rdy'dy7 + P'^CDp^ - /£»/W) 2112. 1. Z. 2112. +S-t^/DP^Dp^X'i I 1. where we put •••t-\ _ p*^ j_r'A'»/?*7 nfl. ^T=I\ ,^T-T îtio-"- evri-'a. )*,\ r'if\ P*,\ . ai r'f-a- ê i r*\ r'i'6. (1VT=Z /J.VIT —^ (IV > O-1 9Tl-'aTt ert liV. '*A , P*A . a p*<r r,s r<'\ r'*9. /AT, V "T~ A JU,T ? cr -t 0V^/rt — ^ 0VA fAT. *\ a n*\ . a; /-'^ os i /^\ p r'*T . ttr,9 liV la- — l itV > a-~ '-' Itlaft ~T u itlT ^ SV > al. Cl!i,^\v denoting the induced covarian tderivative of C^ with respect to yv, namely '\ a ^\ a /~'A » . P r'*T ^9 i /~ie a p*\ /"A tf r*9 /1?k nipirfl. lft>o:|v=°'L'i.i><i:!v—t-'f;;(r;^-< evpft ~\~'-''p.ia i- ev~'-'eia-1 ~ii.v~'-•y.ie1 a-VI. d S Â al.S i r\ p re a r'\ @, a, /~<\ ar'e p. p.xrlp = '^f.xnpT ^9 >p uft!o-— utt>E> ir— U0 >IT*^ ft'l> •. (25).

(4) On the Induced Curvature Tensors of Subspaces in an Areal Space of the Submetric Class. Let a vector Y' be transversal to J(m) :. Cl. 9) Yt=N'y, (. where Nl are n-\ linealy independent vectors transversal to A{m) and satisfy the relations. g,^lNJ=5^ N,B^Q, with N^g;,N'. S. i. '. S. '. S. -. '. i. We define the induced covariant differential of the vector Y{ by Cl. 10) /DYS=N,DY'. Then (1. 10) can be written in the form. (I. 11) 'DY^dy+^y^+C^Y^Dp^ where A^ = A^ - (7|,^ F9^ ^=N,^NI,,+r^,NJB',+C^N^H^ s. 'i. '. '. -v. '. 11. C^^N^NI;a,+C'^N-'Bfi. t. -V. 11. The induced curvature tensors for the covariant differential defined by (1. 11) are obtained as the coefficients of the following equations :. CD'D - /D/D-)^ = [K^wdy^df + P^CDp^dy^ - /Dp^dy^ 2112. 12. 21. I. '. 2. +S-WDp^/Dp^, 1 ' 2. where (1. 12) K'-t^=M^+C^R^,. a. is) M^ = A^ - A%; ? r^ + Âf^ . n> J-i-Va- nT _ /|*f /)*. CVlp. ~T IL (V > <r •£ w/^ ai — •" tjtl,'1 {V>. Ci. 14) p^,^^;^-c^+c^r^;^p,, a. 15) cf ,^ = c^,, - c^; g r^.pr, + c'i ,^ » A*v _ ("S <? r'*a. 'ipir"- Ci'- — uca •' all-1. a p _ rs o> • o . r'e s . « i r's P r"i a _ r's a'r'v e ('>A >;.t= ^( >A , il— '-' Oil , A+^;IH^(>,\ — °-;i>XU( >(l.. § 2. The generalized equations of Gauss and Codazzi. In the previous paper d] we defined the so-called D-symbolism in our space as follows : {ri. ~Vri Vi' . tf r'^v ^r i vrr'^f no; vi T>Ji'\7-i\c&. '\A'=A',x—A- ; f 1 '^pu+A'l •;.^, .Uî- = tS-^\. • |ft, '*'. _ ?r'-r" _L Nh. r.{". r\ = 1-'\± rlt -t- '* a\ °r!/i. ). ( 30 ).

(5) Masao GAMA 'ft _A7''>n"^R' i /^ P R'-H's nT i /-I*1 7?rRs a\=sl'i'iP»^.CSV!\+{-'n's S'vtl\rP'p+ I 'rsts'Vt!'kji •7t_;A R7t RT Yi|a_yi.n) i yr<ri|- a ,'=IT—DTDI '• I A ]ft =A' ;A -t-A'(.';.!?;.. For a vector X' = B[X1', we define the operators D^, D^ by ? _ Yit n« YV' _ Rl* Ri'-Y' 1<» 'AAr-=AI~|M ^'XAr'=o'(-°,(A I/;, 'v- — Tl1 _ YI'- • " r*v nr _i_ y°-/-"I!^ |,\=^ >,\—/1 > v z TX/â 1-yt ^ aA.. For a vector Yi=N'YS, we define the operators D^, D^ by. s. . ai p-i-v^ nT _i_y»/l*f. r)?V'=n7.R"^V|»>'. 'X'1 =.-t >A—'t >vl T\/ya)-t--t "• v\i ^'\J- ~= 1-1 ILJ\^ \hJ.. (. In the previous paper HI], we defined the generalized Eulerian curvature tensors and generalized equations of Gauss and Codazzi making use of the -D-symbolism. The generalized first and second Eulerian curvature tensors are defined by. (2. 2) N[^D,B^B^+BT,r^-B'^^, (2. 3) L[^D,NI^N!,,-N'^r^pl+N'T^-NIA\ ?. f. ?. ?. 1). respectively. Applying the 2?A-operator to N' y and then subtracting the result obtained by interchanging /., p., we have '' _ D A?"' = R'tMi . — R( /?*0' . j-Ni f r"'f? _ r'*0'-. '^v |U,V—'"^'t'\V=UV'I"ft.ft,<—u<TX'-VfX\-rl'<rV^Z A(l—-t ^^^>. where we define the curvature tensor : "' _ ^r r"f-i p*» • » r"t-a- r,r \ r*i r"r-r. /,,l,\=a^^-t ,,tii.,\1—-t n.cn . \v\l |T]\]/•*«-1-' r(\l \n\n.'}j-. Similarly, we have from (2. 3) (2. 6) D^-D^=N"M',^-N'M^. -?;^^-^(r^-r^). We stated in Cl] that N{ and L{,; can be expressed in the from. (2. 7) N{,=N"N^, (2. 8) L,^B,L^ respectively. From (2. 7) and (2. 8), it follows that. (2. 9) D^=BI^N^+NID,N^. ?. Accordingly from (2. 4) and (2. 9), we have. (2. 10) BW^-L^N^+N'^D^-D.N^ = BW^ - ^ R^. + N'^r'^ - r^). Multiplying (2. 10) with BTi, we have. (31).

(6) On the Induced Curvature Tensors of Subspaces in an Areal Space of the Submetfic Class ~.T. W? _ T.T . Nf = R"-RT A/r( . _ P*T '\fi1ll.V~ ^^'XV = UVUi luU!t\— •*'- V(l\. which represent the generalized equations of Gauss.. Noting that >*A _ r\e C2. 12) ^=C^%\, ev<r. where C^ = S^8e, - C^ ,?p^, (2. 11) can be written by. C2. iiy L\^ - L^ N^ = B'WM',^ - W-^ . From (2. 4) and (2. 11), we have. (2. 13) L^N^-L^Nl^B^D,N^-D,N^. Multiplying (2. 10), with Ni, we have V. '" — D ?V? =» W'N.IVl'- . J- AT" CF*? — F'*".. 'hll li,v~ ^'ltll \vs=''-'Vi'lilvl!il'-\~fi'1 a-v^ \/i.~ '- y.\J. 1). From (2. 7) and (2. 8), we have. (2. 15) D^=N!N^L^+B^D^. 7?. Accordingly from (2. 6) and (2. 15), it follows that. (2. 16) NW^L^-N^L^+B^D^-D^-) T). r" M1 . _ ?Vi M" . _ A/'1 • "' ??*" . nT — T.l .c r*". _ r'*?' t. 7l^\—ly trlfll.^~11 ) V -"- Til.\lja ~l-'a'S\.1 ftX—'1 Aft^. ' ' 1 " t. Multiplying (2. 16) with Be, we have C2. 17) W, - 5,J.t. = BflAT" M;,,,- B8Ari; ^!II^p; n«o. p*o. •a-f^ )U—^ \ll.J-. Equations (2. 14) and (2. 17) represent the generalization of the equations of Codazzi.. Making use of (2. 12), (2. 17) can be written by (2. 17)' Ox ^ - D, Le,, = fifl?- M;,,,, - BeNt; WK^p, "So- r«a. •<rf\.M it\~ ^ \ii.J-. References C 1 ] Gama, M. (1967), Theory of subspaces in areal spaces of the submetric class, Tensor, N. S.,18, p. 168-180. C2] Kawaguchi, A. and Tanda, K, (1952), On areal spaces V. Normalized metric tensor and connection parameters in a space of the snbmetric class, Tensor, N. S., 2, p. 47-58.. ( 32 ).

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