一般計量空間の duality について

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(1)Title. 一般計量空間の duality について. Author(s). 叶, 長太郎. Citation. 北海道學藝大學紀要. 第二部, 6(1): 8-16. Issue Date. 1955-09. URL. http://s-ir.sap.hokkyodai.ac.jp/dspace/handle/123456789/5464. Rights. Hokkaido University of Education.

(2) Vol. 6, No. 1 Journal of Hokkaido Gakugei University Sep., 1955. ON THE DUALITY OF METRIC SPACES BASED ON A VECTOR DENSITY Chotaro KANO The Study of Mathematics, Hakodate Branch, Hokkaido Gakugei University.. ttf X*!IB : -®S:?+fi^FtflO Duality ^^>^-C. § 1. Introduction. The geometrical mentioned, associated with the names of Finsler and Cartan, differ essentially from Riemannian Geometry in that their fundamental quantities (metric and connection coefficients) are not only functions of position, but involve an element of support, which in Finsler geometry 1) is a line element, and in Cartan geometry 2) is a hypersnrface element. These two cases have been envisaged from a single point of view by SCHOTTTE.N and HAANTJBS, who took the element of support to be a vector density of arbitrary weight p. Finsler geometry then comes in as a special case in which the element of support is a contravariant vector density of weight p=0, while Cartan geometry has an element of support which is a covarianfc vector density p=—l. The conditions for the existence of such a geometry were studied in all their generality by SCHOUTEX and HAANTJES, and further were discussed by E. T. DAViK.-i.3) On the other hand the duality of the geometry of Finsler and Cartan were studied by Arthur MOOR. "1} In the present paper we shall treat the same problem as those treated in Arthur Moor's paper for the spaces whose element of support is a vector density,. §2. Fundamenta! tensors of metric space based on a vector density. We denote by /-<; a covariant vector density of weight —q, and by /fl a contravariant vector density +p. To covariant case we associate a scalar L (.r, /.() which is positive homogeneous of the first degree in the /M, and to contravariant case a scalar L* ^x, ju*) which is positive homogeneous of the first degree in the /-("''. These scalar is the "fundamental ftinction" of the geometry. We shall indicate by (a) any symbols or equations referring to the covariant case, and by (b) those referring to the contravariant case.. We have the following definitions for the metric coefficients : (2. 2) (a) ^-fta/("<l-l) a\t£^ (b) ^., =ffl-J'/(»"-l) yl2r: a//, a//, ~, -a/7i-a7?^, where. (a) a=det[3\~2L)\ (b) a^det[9\2L ^Ws''. \ya/?7. From (2. 1~), on carrying out the differentiation, we get.

(3) ON THE DUALITY OF METRIC SPACES BASED ON A VECTOR DENSITY .a^_r 32£ , 9^ 3£ ^^ ^-i^*_r* 92 r , ar 9r. (2.2) (a) gvgi3=L^^+-^-^, Cb) ^-^•=ra^?7+a^T a^r, and hence, using the homogeneity property of the L in the // and of the L* in the fi*, ,*' \ / ii*s. c...) (.)."(^.)(^)-., (b) A(p-^) (,.^)-., so that. =y_. L. ''I =. (2. 4) (a) ;; =. l*t. -^1. (b). 3£. 1. °r. y g. r/^'. ar 3,^. ^-. V g" 3^ ,. Next we introduce, for any function / (-x, /-() which is homogeneous in the /.(, the operator. (2. 5) (a) /FS=:7-. -I/ g.. a/. (b) fh=LV^P~. a/^ ,. ^L. ,*J a/^. then we define the tensors. (2. 6) (a) ^(? =- f gtk\'1. (b) ^ wi ° g glk ]1 A. and the vector. (b) ^ = ^ (log g%. (2. 7) (a) ^- ^ (log gy^\ By contraction with / we obtain C2. S) (a) A10" = Aotk - 9^,. (b). (2. 9) (a) A1 = -1— A001. (b). ». (b) (b). C2. S) (a) Al1io = 0 ,. (2. 22) <a) ^ ° = 0, Since for the determinant. g". !'u\. 9 log g. LloK ~. A*= i i to. ±_ -ooi A-^ —. ». 0,. A: - 0, 1 3g-. a/-<. pll^,. 'oOS -—. ^ ^g» ~^T,. g". g 9,«. we have. (b) ^ =. (2. 12-) (a) A,ml - A1 ,. (*))!. mi i. —^. so that the veector A is equal to one of the contractions of the torsion tensor. Passing now to consideration of the absolute differential of a vector, we write first, for anyvector Xt,. (2. 23) (a) DXi=dXt+X\r^dx'l+Ct^,,-), (b) DXt - rfA" + Jl'(ri.,A-'t + GW'^ If we put. (2.24) (a) r^r^+^Fra,,., (b) r^=n,.-^'^. the absolute differential of any vector may be written. (2.75) (a) DXt^dXI+r^XWl+A^^, (b) Mi^Zi+ F^XW+A^XW. Then we obtain F from the relations. 9 -.

(4) Chotaro Kano (2. 16') (a) ^ i^/i, = Tljlt + îj'Tmoli + Ayml „„,; — Aiiml moj , <*i (*•> m (*) '* _v* _ A* r1 in _ A*S- rlm _i_ A* r'm. l.Sli,~lljh ^^ljmj- on ^•ij1i.m'- at I •l± ItlW- o.f. (*). (*-!. (*). (2. 17~) (a) îo;t°=)';o7t+9^t^"'^tnof—^;ti^tn»o> (b) 1 ojh='1ojh~-^Jhm^'oo~\~pllf'^m-^1ol) ,. (2. 18) (a) r^=r^+qAnT,^-ql,A"T^, (b) rofo=?^-^X>,^+p<rs,. »l- r.. - 1 (Hr + ^ - 3^). Following Cartan, we may consider the elementary vector increment undergone by any vector X1 when the unit vector I is transported by parallelism from the point x to the near point x+dx. In that case the absolute differential can be written in the form X1 \ „ dx'\ where. (,2.19) (a) xt\,^3,xl+xTr^+rv", (b) Jfi,,,=a,ji-z'|,,,r:?»+r^,,jM>, which are the "covariant derivatives" of X1 in the two cases. For the curvature tensors we have the following. (2.20) (a) R{u=R{u-A3:"R,,^,., (b) ^=^+^>X"^, where 'v-i- r*-< i!'" r"" -i- r*j. p?'" —. -'; Kls=:0;l •t~fe-r^ u..|! 1 ,uol~t~1 ml1 tk —"-](,.. (*•) (*1 (*-) (*•] (.*). Wj _^ P*J p*.) |] r'*m _L P*J /-'*ni_;. Li'itf.=0;l ji, —/ .("i.|;,>, ^ oi.~ "r 1 mll IK ~tt\ I' •. § 3. A correspondence of the "fundamental elements". We shall define a correspodence between (.r, ,u) and G-^. /<* ) by the following equation :. (3- ^ //<= 77^^,-^) ^ ^'' ^ ^l' (3. 2) ffi-/g^(x^yglk (.-, /.) /A., From (3. 7) and (3. 2) it follows that (3. 3) g^x, ^/g^" (z-, !^gkjt^~) gtJCx, p~)=8}. By contraction with gji (^x, //) we obtain from (3. 3). (3. 4) gU Cv, /^) -gn: (.x, 1^. Further, by means of the contraction with g'", we obtain (3.5-) gtl\ :v, f^=g'"^, /-<). and then we have the following equation. (3. 6) g\x, ^=g<^, i^. From these equation we have the identity of the "fundamental function" in the one-to- 10 -.

(5) ON THE DUALITY OF METRIC SPACES BASED ON A VECTOR DENSITY one correspondence (X, /-() <-—-)• (^X, /^ ). By means of (2. 2) we obtain, using the homogeneity property of the £ in the /i, L£ Qx, //) sa=g'l^}^^f. Further, by virture of (3. 2), (3. 4) and (3. 5), the above equation becomes 2 r^. ,,~\ 8 ^x' f1) r*2^,- .;^ _ r*2r,^ ,,*-. .x, sj-)= g"l^v, f7JlJ~^x' ^^=1J'^X' t1)'. (3. 7) L = L\ Thus we have proved the identity of the "fundamental function".. § 4. The torsion tensor ^ and A* . In this section we shall give the identity of the vectors /*f and /(, and then, if p^q, we shall obtain Afi=A' =0.. By virture of (2. 4) the equation (3. 2) become. 4AA i, = -£= ^-,. z- = -£1&A /;. 77^77) il = ^HF gilc l~ = ^?%^T (i' From (3. 7) and (3. 6) we have ^ =^ . Further we have, from (5. 5), /*( =ll. Next,. differentiating (3. 7) by /jt, we get. /. =—=L= 9r (^ /"') 9^. v^C^T -—a/^- 3^7,. where //k a ^/jv^ „. ^ ,_ 'Qe"1'. (4. 2) ^ = --^ g-' ^, + V^T ^ ^, 4- <— ^< to On the other hand, from (2. 7a) and (2. 6a), we have. ^!=L -J^ ^- ôk =- ^ ¥ ^= ^ ^. 2 ^g^ a/A, " - 2 3^ /'8-71' " • Using these we have. _ .P^L (4. 3) ^— =y -p—^V^+a y.(^+ Vg^+'T g.<V. ^pj. Substituting (4. 3) in (3. 7) and using ln=lt, we find (6-g) ^' =0. Thus, if p^q we have A' =0 and A'°& =6>. By means of (3. 6), since A''=0, we have therefore A*,=0. Finally, we have the identity of the torsion tensor A\^ and AJK in the one-to-one correspondence (A', //* ) <-—> <^X, /J.'). Differentiating (3. 5) by AS and using (4. 3) with ,M Q^.t. A1 =0, we have -g— = ^- Vg^ g"-8. On modifying by (2. 6a) we get (4. 4) ^'8 = .- -^- ^ ^". Hence, by virture of (4. 4), (3. 5), (3. 7) and (2. 6a), we have At'ts=A*vts.. § 5. Osculating Riemannian geometry- Let us consider a set of the element of support, - 11.

(6) Chotaro Kano. by 0. VABGA,. (5. 2a) ^^-( 0). (5. 2b) ^-p' (0. We suppose that an Extremal curves Xt (s) of the variation problem \ L* (A'(r), /.(* W)dv pass through each of the elements of the set (5. 2). Hence we have an one-parameter family of extremals, covering a certain domain B of the n-dimensional space Xi. For each point xl of B we associate a direction, namely the direction of the tangent vectors. (5..) xi-rt^=-7y^^'^ of the extremal curves. Further we suppose that. y. 3) ,•(»-,.. (,(,))- ^^^^,.(.,(,)). Substituting (5. 2~) in gu(.x, /<*), we find. (5. 4) rL(--v) = g:,(^, ^ (.r)), fi^-) = îfc(^, ^ (.z-)). Hence I*IK and T*tk are a fundamental tensor of Riemannian space in B. The auto-parallel curves, which the tangent vector is autoparallel (with respect to the connection defined in the space) coincide with the extremal curves which the distance between any two points is extremal, under Aft=0.. Hence we have along the curve (5. 1~) r; (p). (5.5) -^- + T^rô, where (p) (pl. (5.5a) n,. = fts ^,. ^ _A ^3rl , 9rl 3Fk'. (5. 5b) ^ =Y ^^- + -^ - -a^. Now we have a set of the element of support of the covariant case in the correspondence <ix, /-<) <-—)• <ix, /u*);. (5. 6a) ^ =.'vi (0,. (5. 6b) /.. = /.. co = TT^-TilTTW ^c' (<)> ^(<)) ^co • Further we have in B, by (5. 2), (5. 2) and (5. 3), (5. 7) /., - ^ (0 = ^-^ ?1. C^-) )-'• C^-), - 12 -.

(7) ON THE DUALITY OF METRIC SPACES BASED ON A VECTOR DENSITY where. f^ ^det OU-v)). Then, substituting (5. 7) in gn- (.X, /<), we have in B. (5. Sa) 1M W -gtK (^ ^ M) = ^<i: (^, -^S^ ^ ^) (5. Sb) ?-» W -git. (.x, p W) ° g-" (x, -^w ry'W ). Now we find from (3. 4), (5. 8) and (5. 4) that (5. 9) ftj (^) = ri} W, ,-:, (^ = r,, ^). Finally we get easily from (5. 5) (5.20) -|^- - r\ ,-, = 0, where, ,-,, - ^ ra .. 5. The identity of the Curvature tensor. In this section we shall prove the following : (p).. Theorem I. The curvature tensor Riw of the Riemannian space, which osculate along the curve (5. la), coincide with the cnrvaiure tensor RVw of the contravariani case,. if following relations. ays a. Qx' -Qx1 ~ ~ '9xt. .(r ^r^ )+g-L(^ ^/^^y* v7i>-^,. are satisfied along the curve (5. la). (p).. Theorem II. The cnrvati^re tensor Riw. of the Riemannian space, mhichosculate along the curve (5. 6a), coincide mith the curvature tensor R-iw. of the covariant case, if. the relations. y^ a ^ p r...} \ ,+ ^ ( L ^ r,.. \ _L_ )-£_ (-I= (-^^ -£soi f i- ^ -^ I /—"T" x sol I / —"n"" ''- r. tot. '9xr-9xl Qxl c1 \ V gq ~ ""' / ' 8//t \V g-" ~ '"* / V g'". are satisfied along the curve (5. 6a). Since the proofs of these Theorem are essentially the same, we give only the argument for the Theorem I. (p) (*). Proof. First we prove the following relations f^;(») =Ikt • Substituting (5. 4) in (5. 5b) we have. c^.^ _ 1^ ^ _S^\^(^^\ J^,^ _9^ 3/^1 w^'J = ^\^^~^~^^)^^\^~W^~^~^~^^~Q^)On modifying the equations by (2. 6b) we get - 13 -.

(8) Chotaro Kano ,*t. (^ , i f ^ a//*' , „ 3f^ ,, •9^t\. (6. 2a~) I M = n.M + £!i-7^r ^ ^9^' + A "'^ ~ ^ KltQ^. (^ ^ ^ 1_(^\-^. ^_ ^ a^. (6.2b) ^t(=nl+£^7^r^&°(a7+^''"a^-^;t8-^/ where. ,_lâ^ ++^-^_- -9^ ^' n,.=yr"^3^ On contracting with ]'1C and using the fact that A*p \ ffi=0 (since A*=0~) we have w. .. .. .. 1. ..... 3u*(. (6. 4) r... ^ rL rK + y-^r A^ ^ f. On the other, from (2. 17\j), we have (*1. f«1. (6. 5) I oM =* Tokt ^l-y> ^ oo = ToKl ^<;u> ?"m> •. By virtur of (5. 5), (5. 2) and the fact that A* =0 we get. ^ ^) ^-f= - ^7^( ^^' 1,. ( 77^) rt r' +^ ra r;). Substituting (6. 6) in (6. 4) and further from (6. 5) we have (6. 7) 7P1.,, rl: = ?-,„ /& --^:» FL ;-t ?•'. (p) \ (*'). =L [ Tost ~ ^ sii. 1 00 )=L ÔSl >. <p».. (6. S) T^f=VF\,. Further, using (6. 2a), (6. S), (2. 76 b) and the equations. 3S - ^ '{^W "•• ^^ --^'^•(.7T')''"+L'"r'-), we have (p). (*). (p). (*). (6.9) r^W=r^, rl,,(x)=m. - 14 -.

(9) ON THE DUALITY OP METRIC SPACES BASED ON A VECTOR DENSITY Then, under conditions (^4), we have along the element (5. la) (p). _. (*).. (*>. _ar,^ OQ _ ê^^, , en-n 3,^. 'a^— = "av^ + "a/7i- ~o^,. (pt. f*-). (•*). II1_ _ _ 3^ 3^ _._ _._ 3^_ ^Zl_ 3^_ 9^ ;8— = -^^S— -t- -^7,*(-. ^r- " —Qy^— T- ^7<- av'-.. Hence, substituting (6. II) in the equation (p) fp; <p'>. ?)Fi,. 7)rj. <P) <?_>. (-P) <^) l UJ~ tK _ _^t_(_L _i_ r 'm ps _ fm Vs. in ~ 3^.'. — g^-a ~T~ i •i» •' "" -~ •t ''• •l "t'6. and using (6. II), we have <e\ ?^_ •i kl =-"-("it;. •. Next we shall prove the following relations. m m '!";• ?iF*m. m (*'• ?*'. <*i.. tin l-/xoft ^•lo_a__ i T-itin || 'T-I tp _ pk»» n r'*i». o ».k= ^yK ~ —gî— -T -1 oftllu-' o'A—-t o 7i lln ± ok="J-. Using the fact that /(*^ (*)^ '9L _ /(*!. <*1. -i* _ n V*m. ^?-. 001 — f ± ml. jp~. ._. _. (*)^_^. f*>. (*). 81_ = n ,/^n>' r'*"1 r«. =' :j"— = /J V g" I 'm I. 1 IttK = 1 ioh •. -Sx. we have (*). a1 f ,,*s __ /"r)r*s. (*} <*}. ^9z-fc 3x. ~* j~r?y~ { u i o" _~r'tsA\ r*v.. =-^ v ^" V^^T- -'' °-t"1'' ('''' /. '9'tus. a2u*s. -. .. =..... Hence, from ^j%^ - ^r^..i sssso, we have Rtom? 3xt '9xK '9xK Qx' "' ••""-•" -oh*; -0. "•. Thus, from (2. 20~) and the Theorem I, we obtain in B (p). •*l _ V*l P (. J »• = ^LJ »•( = ll.; /tf.. In like manner we have (p). ' — Rt — T?i. J Kl — •"-J W. — ±'-j Kl •. Thus we have the identity of the curvature tensor R'j ki and K j r;, in B. - 15 -.

(10) Chotaro Kano. REFERENCES 1) E. CAHTAN : Les espaces de Finsler, Actualites ScientHigues, 79 Cl934). 2) E. CARTAN : Les espaces metriques tondes sur la notion daire, Actzsalitss Scientifigzies, 72(1933). 3) E. T. DAVIES : On the metric spaces based on a vector density, Pro. London Math. Soc., 49 (1947) 241~259. 4) Arthur. MOOB : Uber die Diialititt von Finslerschen und Cartanschen Riiumen, Ada. Math. (1952). 5) L. BBP.WALD : Uber die n - dimensionalen Cartansche Rsiume und eine Normalform der zweiten Variation eines (w-1) -fachen Oberfiiichen integrals, Acta. Math, 71 (1939).. - 16 -.

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