Submetric Classの面積空間におけるリー微分についての注意

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(1)Title. Submetric Classの面積空間におけるリー微分についての注意. 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, 30(2) : 123-126. Issue Date. 1980-03. URL. http://s-ir.sap.hokkyodai.ac.jp/dspace/handle/123456789/6045. Rights. Hokkaido University of Education.

(2) Journal of Hokkaido University of Education (Section II A) Vol. 30, No. 2 March, 1980. ^m~^H±^&m (US 2 $[SA) ^ 30 ^ ^ 2^- Hg% 55 ^ 3 ^. Remarks on Lie Derivatives in Areal Spaces. of the Submetric Class. Masao GAMA Mathematics Laboratory, Asahikawa College, Hokkaido University of Education, Asahikawa 070. ^ : Submetric Class ^m^FV^W ^ 'J -%^^^^T^^ «Mir^™jii^M''Nsfci. § 0. Introduction. Let A(n be an areal space of the submetric class whose metric tensor gij, connection parameters F^' and CJ"al) were introduced by A. Kawaguchi and K.. Tandai [2]2). In the previous paper [1] we defined Lie derivatives in Aw and exhibited some of their applications. The main purpose of this paper is to exhibit a necessary and sufficient condition that the Lie derivatives of the connection parameters vanish. We employ some notations and results. used in the previous paper [1] without explanations. § 1. Some preliminaries. Consider an infinitesimal transformation (1. 1) xi=xi+^{xJ)dT where ^l{xj) is a contravariant vector field and dr an infinitesimal constant. By this transformation, pa undergoes the transformation. (1. 2) Pa=pla+^ Pi dr. In the previous paper [1] the Lie derivative of a geometric object £2 {xi,pa} with respect to ^ was defined as follows: y m. (1. 3) £ Q.={dQ.-dQ.)ldr,. 1) Latin indices run from 1 to n and Greek indices from 1 to m. 2) Numbers in brackets refer to the references at the end of the paper.. (123).

(3) Masao GAMA. where dQ.=Q(xi,pa)-Q.{xi,p^, m dQ.=Q{xi,1)la)-Q.{xi,pla).. By the definition (1. 3), we obtained the following results : (1.. 4). £ Pa=Q,. (1.. 5). £ 9ti=gr, ^r[i+gir SrU+9i^r ^rlsp!,. (1.. 6). £ P?=^;az ^s\r=r^pjga13 £ grs,. (1.. 7). £ F=F /9J ^-= ^ g^ pl ps,£ grs,. (1.. 8). (1.. 9). s. •. -. -. •. ^. -. -. —s. * ;_ jp 2 £:r_i_ -p *z.-l ^r. ^ p4-fit,.,. Jh—njhr S- "Ti Jh,r S- |s P\~r? \j\h,. ?. i a —_^^,ihfl „ .ff or ^6n/' 2>m^s s>m-^s\ _ o s-r ^s a. Cj,a=~^~g"\\gmn','a t^'b gu'\ GFTh — ff'h T] ) ~~ ^ Q'h Cj ,~a] A 5rrs. ^.^--.._-. -. ^. +(^^J+&r /3S^(£ grs);aa} Also we defined a motion in A(n and proved that a necessary and sufficient condition that the transformation (1.1) be a motion was given by £ F = 0. s. Moreover we found that if the Lie derivative of the normalized metric tensor vanishes, by virtue of (1.6), (1.7) and (1.9) £ p?=0, £ F=0, £ c], 2=0. ?. '. '. ?. '. ?. Take a general tensor T^. we obtain the following formulas: Tij)\k~ £^ (Tij\k) =~ Tij ^ T-rft+ Trj £ F ik . -. ^ ^. ^. h -f.' T-'* r_i_ rrh.a ^ T^*a. ir ^ 1 j-A-t- 1 tj;a^ i ah,. e. e. (1. 11) {£ TWr-£{T^=-T?,£ C^r +T^ £ C?^JrTla£ C?,/r, s. e. s. (1. 12) {£ T^r-£(n-/r)=0. ?. f. For p", and cja,a, we obtain. (1. 13) {£ paa}\t-£{p^=rW£ csrf^ e. ". '. ?. ". '. '. -. ?. (124). f. '. e.

(4) Remarks on Lie Derivatives in Areal Spaces of the Submetric Class. ^. id \\ft _ _c( ^{a \p\ — _^ra _c ^{P. -L î a -C ^ r^-L î.a -C ^rf_ _ îe _c ^"ft ',al\'b —aC\^j,a\bi— —^j,aoC Cr,& ~T C-r,a oC Cj,t,~T Cj,r •^- G-a,6—C.j,a°C C.e,'&.. -—. -. ^. -•. .. --. ^. •-. •. ^. -.-. -•. ^. -. -•. ^. For F*{?, Om P. Singh [3] gave the following relation: d. 15) (^T^)i.-(o6T?*')i,=^^^+r?;;^rLr-r^;Aro6TLr. c. s. s. s. ?. § 2. Conditions for the vanishing of £ F ?J1. Applying the formula (1. 10) to the normalized metric tensor gu, we obtain. (2. 1) (£ 9i,)^=(grj^+gir^+gij;Arp!)£rU from which. (2. 2) ^<7^(^'7^)i.+(â)i.-(^(7^] S. f. S. '*.h-\-^- nha( n _.-i1 -r r'*.n-t- /!._•'" -^ r'*." —^...^ ^ T-i*n'. ij~T-9~ y \Uaj,n aC 1 tH~ryia,n »p l A? ~ y ij','n <Z- l\ua;.. ?. ^. ?. ?. '. ?. From (2.2), we obtain (2. 3) 4- ^/'a[(^^)i.+(°f^^)i.—(^5r")^^=w? o^n." S. S. S. f. where. (2. 4) W^S^=Shn^^+^gha(gaj-/n^+g,a/n^-gij-/n^)pi.. In A\m) we may assume that the determinant [TTÎ with respect to the system of indices (ns p.) and (hj A) in (2.4) does not, in general, vanish. Under that assumption, by virtue of (2.1), (2.2) and (2.3), we have Theorem. In order that the Lie derivative of the connection parameter F *ijh with respect to $l vanish, it is necessary and sufficient that (2. 5) (£gi,)\H=0. s. From this theorem and (1.10), we have Corollary 1. // the normalized metric tensor satisfies (2.5), then the operations of the Lie derivative with respect to ^ and the covariant derivative with respect to x are commutative. From this theorem and (1.15), we have Corollary 2. If the normalized metric tensor satisfies (2.5), then the Lie derivative of the curvature tensor R^-/, with respect to ^ vanishes.. (125).

(5) Masao GAMA. By virtue of (1.9), we have Corollary 3. If the Lie derivative of the normalized metric tensor with respect to ^ vanishes, then the Lie derivative of the connection parameter cjjaa with respect to ^l vanishes.. From Corollary 3 and (1.11), we have Corollary 4. If the Lie derivative of the normalized metric tensor zuith respect to ^f vanishes, then the operations of the Lie derivative with respect to ^ and the covariant derivative with respect to pa are commutative.. REFERENCES [1] Gama, M. (1971), On motions in an areal space. Journal of Hokkaido University of Education, Section II A, Vol. 22. No. 1, p. 1-3.. [2] Kawaguchi, A. and Tandai, K. (1952), On areal spaces V. Normalized metric tensor and connection parameters in a space of the submetric class. Tensor, N. S. Vol. 2, p. 47-58. [3] Singh, Om P. (1974), On the curvature collineation in an areal space of submetric class. Tensor, N. S. Vol. 28, p. 263-268.. (126).

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