高次空間における共形微分について
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(2) Vol. 21, No. 1 Journal of HokkaiBo University of Education (Section II A) September 1970. On the Conformal Differential in a Higher Order Space. Chotaro KANO Department of Mathematics, Hakodate Branch, Hokkaido University of Education. f^ ^*?: i^^S^rfc^S^^^^-.^-^^-C. Introduction In the previous paper [4], by use of a non-intrinsic conformal derivative, we have given an intrinsic conformal connection in the space with the metric s= | \Ai(,x,X')x"t+B^x,x'~)y"'dt. In this paper, by use of a intrinsic conformal derivative, we shall give the same conformal connection.. In the first place we shall give an intrinsic confprmal derivative and the effect of this operation for relative conformal tensors. In the second place, making use of this intrinsic conformal derivative we shall have the intrinsic conformal differential. At last we shall give the base connection and the conformal covariant derivative which is intrinsic for a change of parameter.. ,. ,. ,. ,. § 1. Intrinsic conformal derivative In the previous paper [3], we have defined a conformal derivative for any relative conformal tensor of weight k, as follows : (1.1) ^(V) = V^ - 2V^Hw- kF-^A^x "i "V. Let V go into fhV by transformation of parameter t. Then, for a change of parameter, we have. ,. (1.2) ^(V)(7)=r't-'^(y)-r't-^(y(^-M/yF-Q, {Vw-kAsVF-^(f)=rh-\Vw-kA)VF-^. On. the. other. hand. we. have. ,rfr (1.3) /<o=r^<-2r-1(1.3) /<0=r^-2r-1-^-. where. r=-%, Kx, x', x")=^MuwX^xW'CXMwX^lr\ [3]. From (1.2) and (1.3) we obtain the intrinsic conformal derivative :. (1.4) ^V)=v>^V)-^Vw-kF-^VY Thus we have. (8). ..
(3) ^21^ ^1^ .. , , , Affiiia^.^^SiS (^2^iA). : ..,! Bg^45^9J! ^(y)=e«-^(V), ^(y)(7)=rh-'0XTQ. ' I. 3. When V and V are relative conformal tensors of the same type, we obtain. (1.5) <l,^aV+bV)=aMV^+b^V), where a and b are constant. Further, when both V and W are any relative conformal tensor and. P-tensor. [5],. we. obtain. i. ,. i.;. .. ,. .. .;,. •. ,. ,. ,,.. (1.6) <f>^V-W)=</>^V)-W+V-</>^W). . , .:• ,, , ' After short calculation we have. (1.7) {^(.V^wW='f'^V^-^^Vw-^-kF-'V^A^F-'+/zw-)Af. Specially it can be verified that . ^(^)=3?, 0yCF'°')=0, 0/3^=0, , ,,,, (1-8). 1. _. .. ^(A)=^(A)+^-/<F-1A,A^, 0/(Ai)^=0, 0,(AOi;"=-A^.. Applying (1.1) to the equation AiXwt=AiXi=F, we have. (1.9) 0/A,)^t=-A^(X()=0, where. .... !. ,1.1;. Xt=x^t+^x", W)=^W-^x": On the other hand we have. (1.10) 0/A()^=^(A()A;'2'/+-|-/<4(=AT. Next, differentiating (1.3) with respect to Xwp we have j"ft>i>(0='r'-j^(2)p-' ' ' ..! . — ,. Further, after short calculation, we have (1.11) {HMX"-=-2, mwXP=Q. Thus, by virtue, of ,(1.4), (1.6), (1.7) and (1.11), we have , , . , , (1.12) T?(y^-20/A()+2^(Aj)^*:AiF-1, , , , , , , , ,i, where H= -2^(A,)^'2":= -2AT+/^Ai. Then, from (1.8), (1.9) and (1.11), we have the following relations ; (1.13) T^x!i=-2At,. TwjXlt=2Ai.; , ' , , ;. Further,' applying (1.1) to (1.8) and using (1.8), we .have i (1.14) ^^Ai')x'^=-^At~), <f>^(Ai')x":=-<f>^At'), 0^(Ai)A:"=-^(A)-0.(Aj).. §2. Intrinsic conformal differential ; i In the first place we shall give (2.1) S*"-(A»(AO = -^Skn+ ^An)X»x"CF-^. • . . Applying (1.1) to S*"CTiw]=Sj and contracting the resultant equations with X'3, we have, by virtue of (1.6), (1.13),. (1.8) and (1.10), . .. (9).
(4) Vol. 21, No. 1 Journal of Hokkaido University of Education (Section II A) September 1970. (2.2) Ai0n(S*a) =-|-0p(A,.)WF-1. On the other hand, applying (1.1) to AiS*<k= --^-X"°, we obtain. S*ik0,.(A,) = -^ - A<^(SW), Substituting (2.2) for the above equations, we have (2.1).. Thus, applying (1.1) to (2.1), we have (2.3) (&»(Sitm00.(A»)=-S't»^,>^(A,,0-^^p(A,)X^'^='—|-0^At)X<F-'. If we put £>yn=0»(S*">Q^(A«), it follows that (2.4) DVnX"t=Q, D^x'n=S*"t^<iAm), and. (2.5) Dt^Dli, Dl{(t)=D^. Further, on account of (2.4) if we put. Dl =23^ + S*m^t(A,»)0XA»)Z"F - ', then we have D'^x"c=0, D'iin(t)=D'^, and D'^=D'^n, D'^x'n=0.. Thus, with the help of the conformal tensor D'i'n, an absolute differential can be defined. by 8vi = dvl + Hw mvW + D'^v"8*x", which is invariant for a change of parameter t, and this defferential coincides with the intrinsic conformal differential D"vt in the previous paper [4].. §4. Base connection Let vl be a relative conformal vector, which is v'=ekvffvt for the conformal transformation, and let vl go into rhvl by transformation of parameter t. In § 2 we have obtained the intrinsic conformal differential, as follows ;. (3.1) Svt=dvi+At^+Avi, where (3.2) Aii=H^wdxlc+ D'jt8*x"c, ACh, k) = {h- ^p-2~)k}v-kQjX'F-\ v=^wS*xlle. From (3.1) the expressions 1(1). .. 1(2). (3.3) 8xi=dx'i+AijX'J+A<il,0')x't, Sxf = dXl + A^ +y((2,0)^1 (.10).
(5) 1¥21^ ^1-f ^)l@;il»^*^^^ (^2SRA) gg^45^9^ are the components of a contravariant vector.. Hence the expressions (3.3) give the base connections in KS, a special Kawaguchi space of order 2 and dimension n. Along the curve, the equations (3.3) become id).. Ka). •( AVt. '^=x"i+2Hi+^x't=x^i+^x't^Xi, u^=^-+Ht^Xi+^Xt. From vi(t')=fhvt, we have vwsx"=0, vwx'-l+vw^=hvi.. Thus, using these equations and (3.3), we have a covariant derivative, that is, I. i. I. a. Ka). a. 8v{ = dx*f,cvt + 8x"cf^itVi + Sx{FkVi, where. ^v{=^+Hw^-v^H^-kF-^^vi-vw^Hi+Hw^-2H^Hw +^H1^), 7kVi=vtw +D'^v'+R^-Vwt(^Hm +D'^X^tJtwXl) -^p.v\w =W)+D'}^-vwD's.X'-^nwVi{h-(p-2~)k}-VwXluiw, :, ^vi=vlw,. .. REFERENCES [1] Kawaguchi, A, (1933), Geometry in an n.dimeiisional space with the arc length s= | (Aw"t+B)l/PA, Trans, Math. Soc., 44. p. 153-167.. [2] Kano, C, (1956), Conformal geometry in an n-dimensional space with the arc length s= | (A<x"(+B)'/rrf<, Tensor, New Series, 5, p. 187-196. [3] Kano, C. (1969), Some remarks on the conformal differential in a special Kawaguchi space. Tensor, New Series, 20, p. 75-78. [4] Kano, C. (1969), On Conformal differentials in a special Kawaguchi space. Tensor, New Series, 20, p' 79-82.. [5] Kawaguchi, A. (1941), Die Differentialgeometrie h6here Ordnung 111, Jour. Fac. Sci. Hokkaido. Univ., Ser. I, 10, p. 77-156.. (^).
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