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(2) Vol. 13 No. 1 Journal of Hokkaido Gakugel University Dec 1962. on a special Kawaguchi space Goro KITAMURA The Study of Mathematics, Asahigawa Branch, Hokkaido Gakugei University. ;fWEH& ; ^AM n ^^^-3^-1: The fundamental theory of n-dimensional spaces such that the arc length of a curve. x!-==x'(t) is given by the integral s = J { A' (x, x')x//t + B (x, x/) } " dt has been established by Prof. Kawaguchi" and thereafter studied by many others.. In the present paper we study a generalization of an absolute derivative with p^ 1,3 when the components of a vector vl are homogeneous of degree h with regard to the. xf (§1) and lead a necessary and sufficient condition that the special Kawaguchi space of even dimensions with | Hi j \ =/- 0 be locally flat by another method from Watanabe's proof2''(§ 2). The notations are the same as those of Kawaguchi's paper. § I. A generalization of an absolute derivative.. The definitions of the absolute derivative 8 vl = dv' + F'^^ v> dx1' have no geometrical meaning if v' are homogeneous of degree /; with regard to the x. For we obtain ( 8 v!),- = a"- 8 vl + hv' a"-1 do ( ff == ^ ) by transformation of the parameter t : /==/(?).. When we put 5* y;== 5 v' - A.yi /(f), we get ( 8* vl),. = a'1 { 8 v' - hv'(f)i} + hv' ff"-1 da by transformation of t. Therefore / must has the following form by transformation t. </>.-/+<°. On the other hand (8 A, xWt )r. = a" 8 A, x^i +(p-2)F -r"-1 da- a"-2 -^- A, 8 x", ( T< 8 x" }r -=- a" Tt 8 x'1 + pF ff"-1 d a - (2p - 3) a"-2 -d^r- Ai 8 x" .. Accodingly we get. AA^IA = A/k^L ^ A-l _^tf_ „ ^-2 _^ A 3 .,// 7^-).. = ~7^- + ^r ~^~ ~ ^ff-' ~w A''x"' -12-.
(3) On a special Kawaguchi space. (^Tt8xii^-\ = .Ttsx/l_ + 1 -^^ - -4- ^ -^ A, 5 .v/i. \(2p-3)~pFh = ~(2p-S)pF ~t~ Yp^3 ~a~ ~ "pF " - ~W "'i " A ' / (2p -3) 8 At x'-2't -T,8 x't \ (2p- 3) 8 Ai x'»t - T< 8 x/i. \ (2p-3)pF !,. (2p-3) pF. P-2 _1_\ A". p 2p-3/ <r •. Consequently we can put (2p-3) SAix'^-Tt 8x/l. 2(^+1) F. Using the relations : x[2'-l:=-Ti Gu-, G«,= 2 Aiw - A^I) ,. we put. 2p-3. /i==^TT^ ^v<A,, /'( = (p^Y)F XWK { (P-2) AW) + Aw } , Then we obtain as the required absolute differential 8* vl == 8 vt - hvl (/?., f/A-^ + /(., 8 x/-i) ,. accompanied by the covariant derivatives 2p - 3. yy vl == -Vj Vt - -2^4. ^~YF- lwi x i~w v-f Ak'. VJ/ V1 = V/ y' - -^^-p- hvi xwk { (p-2)A^ + A,w } . F ~ P 3#vl are indepent of a change of the parameter t. § 2. Another proof of the theorem. Theorem : A necessary and sufficient condition that the special Kawaguchi space of. even dimensions with \Hij \ +- 0 be locally flat is that the following relations are satisfied : ^0)m=0' ^wi=o> ^i=0> ^<u)=0Proof. Necessity. If the space be locally flat, the relations Aao,) = const., B (y, y') = 0 are satisfied in the suitable chosen y coordinate system. Hereafter we use the indexes /', j, k,...; a, b, c,... in regard to x and y coordinate system respectively. We can lead easily the relation AH^) = 0 from Aa,w = const. . Differentiating Aaw /" = (p-2)Aa with respect to y'e, we get p==3. Consequently we have the following relations : Aaw y'b = Aa., A,,w y'" = - Aa, :. A,,^-) = - Aaw .. Besides Hah == 2 AaW , Gal, = 3 Aa(,,i == const.,. 2rc=(2Aa,,y/l>- Bw) Gae /«. ;ffl6 •I. „/& _9 ( —-'——- I „/(>. la <-r ;;> J' —•. 6. =0.. -13-.
(4) Goro Kitamui'a. Hence we have R'j^ = 0, B'^1 == 0 from Fc = 0 and 8 Aaw = 0 from Aaw =^ const... Sufficiency. From B'^t( = 0, then I^ij^i are functions only of the x' s. If we consider now the. simultaneous partial differential equations 3V _ pa -a^-'1 >a. ( _ pa r-m. 3^ = ' '» ' (w) ' these equations satisfy the integrability conditions from R'y == 0, B'^ =0. Hence we choose the above solution ya = yct (x} as the coordinate transformation.. Differentiating Pf Q{ = 8{ (^=-^'r) by ^' behave Qia =-QlaQi Q"i PC^ -- - Q^ Q'S ri,w .. Differentiating Ay^ == Q^ Q{ A,^ by yc and using the above relations, we can see. A^ =(Q^ Qi +Ql» %) A^+ Qi (3^ 0^' A,^, = Q'. Qi Q's v. A^ =0,. B (y, y')= B( x, x') -A,, Q'^ P^ x/t x's =5(.v,.r/)-A,,, F^ A-/<A-/J =0.. Hence this space be locally flat.. Refferences 1. A. Kawaguchi : Geometry in an n-dimensional space with the arc length s = | (A; x"1 +B) (It, Trails, of the Amer. Mafli. Soc., 44, 0938;. 2. S. Watanabe ; On special Kawaguchi spaces, Tonsor, 7, (1957).. 14 -.
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