特殊河口空間における計量テンソルについて
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(2) Vol. 19, No. 1 Journal of Hokkaido University of Education (Secation II A) September, 1968. On Metric Tensors in Special Kawaguchi Spaces. Chotaro KANO. Department of Mathematics, Hakodate Branch,. Hokkaido University of Education. ti- X*? : 'NF%NosrBlr*i^-%j-l-a7"^y^^^»-'-c. Introduction The conformal theory of the special Kawagachi space with the metric s= j" F~pdf, where F=AiX"'+B, was studied by the author C33. Further the conformal theory of the special Kawaguchi space with the metric s= 5 F^vdi, where Fi=aiajX"ix"i+. 2bajX"i+c, was developed by T. Nakagawa and the author [41 In the present paper we shall introduce the intrinsic metric tensors g-^ in these spaces. Then gij==ez°'gij and gij==ep<rgij provide a characterization of a conformal transformation in our spaces. We shall prefix our equations by (a) or (b) according to whether they refer to the space with the metric s= f F~pdt or to the space with the metric s=Fipdi.. § 1. Conformal derivative In the previous paper we have introduced the conformal base vector xW, where xW=x"l+2H'i. Using this vector, we have (1. 1) (a) F-'A^x^^F-^A^x^+paw, (b) 7-1«p(k)-rizlp=/-'«p(k)-K? +|-tf(io, (1. 2) (a) 2''-'Ap^A(?=F-iAp(^Px?+Mio.vl2lk, (b) f-}a^x^xW^f-}a^x^x^ +-^awXWlt, where (a) F=AiX"i+B, (b) /=fliA-//I+6+{e(c-A2)K By use of (1. 1) we shall introduce the conformal derivative. Lot V be a relative conformal tensor of weight &, that is, (1. 3) (a) V=e^vV. (b) V=e^"V.. Differentiating (1. 3) partially with respect to xfi and eliminating (TQ) by means of. ( 3 ).
(3) On Metric Tensors in Special Kawaguchi Spaces. (1. 1), we have (a) V(j, - /'.F-'AP(,)^IP V= e^-i V^ - kF-'A^x^ V}, (b) Y(J) - kf-^d^x^ V= e^v { V<j) - &/-'ffp(j)^lp V\. In the above equations the second term is a tensor, but the first term is not. Then, making use of the base connection H'^x, X1'), we have for the coordinate transformation A-i=.1;t(A') a. 17(j)-2TW/('(j) =^j-( Vw-2V^H\^, where (a) H[=ri-Kx'; (b) H'=A'-Kx'i. Since 2l/(2)qffqo) is the conformal quantities, we can define the conformal derivative. by (1. 4) (a) v^V)=V^-2V,^H\^-kF-\A^x^V, (b) v^v^=v^- 2 Vu^ff\^ - kf-^a^x^ V, and then, for the conformal transformation, we have. (a) p,(V)=ek^(y), (b) ^(y)=e1"p^(V). Applying this to the vectors A\ and a;, we have (1. 5) (a) v^Ai)=A^x^'A^ASS-^ (b) v^-)=a^-f-^a^x^>a,=a*^ and hence these quantities are the relative conformal tensors of weight k. Fnrther it. can be verified easily that the following relations hold : (1. 6) (a) v^x'^0, v^AQx'^-A,, V^A,-)x^=Q, (b) ?'j(ffi);t/j=0, x^a0x'l=-as, <p^ai)xw=0. § 2. Conformal metric tensor In this paragraph we shall define the conformal metric tensor. In the first place, from the Craig vectors (a) Ti= -2G^xW and (b) Si=2G^xW, we have the relative conformal vectors of weight /;, that is, (2. 1) (a) T,*=-2A^xW+2F-^A,wx?'x^A;=-2Alls^w,. (b) ^=2aw)XW-2f-}a^x^x^ai=W^ xW, and hence we have (2. 2) (a) TiX''=2F, (b) ^*;r/i= -2/. Now, on account of (2. 2) and (1. 6), we define the metric tensors in onr spaces. by (2. 3) (a) g^x, x', x")=F^Aa';,+A-t-^+T-*T'f,,. (b) g^x, x' x")=f(.a^,+^+S'f^,. From (1. 6) and (2. 2), we have the following relations :. ( 4 ).
(4) Chotaro KANO. (2. 4) (a) g^x'l=-FAi+2FTV,, g,:,x'5= -FA.+2FT*, g^x'^^F\ (b) g;sxli=-fa,-2f^,. g.,x'^-fa,-2f^, g^x''=^f\ Furthermore we have, under the conformal transformation, (2. 5) (a) ^.i=e?p°^j. (b) g^e^g,,. Thus the conformal transformation F=evlrF implies g,j == e21:>°gij. Conversely on account of (2. 4), (2. 5) implies F=eplrF. On the other hand Fi=ef"rFi implies g\, I. ~. 1. ~. ~. 1. =epvgw and conversely (2. 5) implies f=e^paf. Hence we have ai=e^pa'ai, A+(e^)2' 1. 1. 1. =e^p°'(A+(e^)-2-). If we put u=6+(e^)~2-, we obtain u-26u+6-e^=0. Hence we 1. _. have t(i+Us=l>, and consequency b=etpab. Further we obtain c=epvc and then. from 1. ~. Fi=f2-2^a[X"'+l/)t/i~2, we have Fi=ePVFi. Thus (2. 5) provides a characterization of a conformal transformation in our space. But unfortunately the vector (2. 3) is not invariant for a change of parameter t.. § 3. Intrinsic metric tensor In this paragraphe we shall introduce the relative conformal metric tensor, which is invariant for a change of parameter t. Both the scalars (a) GWAp^ and (b) Gpci<?p(q) are the conformal scalars, homogeneous in x' of degree zero, and hence JV[^)X'3=Q, where (a) M-^G^A^w, (b) M=GP-flp(,), Hence M is invariant for a change of parameter i. Further we can see that M(J) is the conformal covariant vector and consequently M^x^j, the conformal scalar. Then, for a change of parameter, we have (3. 1) /-<Tt=r^-2r-'^, I.. where /.^x, x', :v//)=(M(,i)(,.).?l2lk)(M(o;^')-' :5]. On the othet hand, we have (a) (^,.,)T= rr-^i.i +rp-5-^ 44i ^-1, !T. (b) C^!,j)T=r^-1ff"<,i + r?-5^-fli<?,/-', 1;. and (a) (T*,)T=rP-iT"<,-2rP-3^-4, <^T. (b) a!pi)T=r?-'?!ti+2rf-3-^Lfli. Elimiuating r~1—3L- from these equations and (3. 1), we obtain. T. (a) (Z,,)T==rp-34,, (£^=rP-'£^, (b) (4,)T= r~l-3l;h (£!",)T=^-'£!t-i, where. ( 5 ).
(5) On Metric Tensors in Special Kawaguchi Spaces (3. 2) (a) Jij=^,,+V2MAF'-1, £-*,=Tsi!i-/-<4,. (b) 7,,=^,j+V^,ffj/-', 2?*,=^1!,+^,. Since /-< is the conformal scalar, we have (a) Z,j=eP'rJ,j, E*i=e'"rE*;, p - p (b) Ji,,=^°'J,,, Ji,,=e^j,,, E:v;=e^EW,.. Further it can be verified easily that the following relations hold : (3. 3) (a) I;sx'i=-A,, ly.x'^Q, Esf-iXn=2F, (b) I^x\=-a,, I;,x'^0, E:t;x'i=-2f.. Mow we shall define a metric tensor g\\ by the following : (3. 4) (a) g-^x, x', x"^WF(l^l^+E*-^^F-^\ (b) g-^x, x', x")=Nf^+I^+E*,EV-,, where ^ = F P, A/= M(j).t;121-i.. From (3. 2), (3. 3) and (3. 4), we have easily (a) g^'i=3i-^-t\-NFA,+2FEf), (b) g-^=-Nfa,-2fE:*,. and hence the determinant of the matrix (^ij) is not vanished identically. Furthermore we have (3. 5) (a) gi,'vr'x"=y, (b) gnx'ix'^=^f\ Next, by means of (4. 2), we have for a change of parameter. (3. 6) (a) gisW=g.,W, (b) ^,i(0=rp-Wt), and under the conformal transformation (3. 7) (a) g;,=e^g;s, (b) g-^e^g;;.. Thus by concerning (3. 5) and (3, 6), (3. 7) provides a characterization of a con formal transformation in our space.. REFERENCES [1] Hokari H. (1936), Die Geometrie des Integrals s= \ (.aw:,x"'x"l+'ibas»"tJrC~)vdt, Proc. Jmp. Acad. Jap., 12. p, 209-212. C2] Kawaguchi, A. (1933), Geometry in an n-dimeasional space with the arc length s= \ (^A\"'l+ B~)vdt, Trans. Amer. Math. Soc., 44 p. 153-167. [3] Kano, C. (1956), Coaformal geometry in an n-dimensional space with the arc length s= \ 1 (A,A-/"+S)Prf;, Tensor, New Series, 5, p. 187-196. C 4 ] Kano, C. (1968), Some remarks on the conformal differential in a special Kawaguchi space. Tensor, New Series, in press. C 5 ] Nakagawa, T. and Kano, C. (1965), Conformal geometry in an n-dimensional space with the arc length s= \ iiaiaiX'"x"5+2bajX":i+C')~pdt, Jour. Hokkiado. Gakugei Univ., nC, 16, p. 1-3.. ( 6 ).
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