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(2) •22^ Hl-^- . ^^^aW^^BS (^ II ^ A) RS^O 46^ 9^. On Motions in an Areal Space. M:asao GAMA Department of Mathematics, Asahigawa Branch Hokkaido University of Education. %^: ^ ^ ^' H %.» ^ ^ ^ -r. § 0. Introduction We consider an areal space All of the submetric class with the fundamental function. F^l,p^, where. F^det\guP^Pj,\,. gu=(^f+prp^g^, L^=F-l{FF;a^-F;fF;^+F^F;^, pf=F-iF;f, ;f=919p'^ The purpose of this paper is to define a motion in our space by the use of Lie derivatives and show some results concerning it. Throughout the paper Latin indices run from 1 to n and Greek from 1 to m.. 1. Lie derivatives in an areal space Consider an infinitesimal transformation (1.1) xi=xi+^i(xj)dT, where ^l{xj) is a contravariant vector field of class C2 and dr an infinitesimal, which carries a point C-v() of a surface A(,n) : -v =A:i(Mal) to a point (xf) of another surface A(,n) : xi=xt(iia').. By this transformation p^ undergoes this transformation. (1.2) P'^=P^+^,jP^dT, where, j = 9]Qxj. According to [2]1:> and [3] the Lie derivative of a geometric object £S{xi,pi^) (e.g. a tensor, connection parameters etc.) with respect to the vector field ^ may be defined thus : 1) TO. (1.3) iQ={dQ-dQ)idr, where. ^=^(x(,%)-^(^,^), m. ^. —.,. ..... d^=Q(%t,p'^-Q^xi,p^,. 1) Numbers in brackets refer to the references at the end of the paper.. (1).
(3) Vol. 22, No. 1 Journal of Hokkaido University of Education (Section II A) September 1971 and ~Q (x(, pta,') denotes the component of iQ(,xi,pa') in the coordinate system (xt), (1.1) being interpreted as a coodrinate transformation from (;K<) to (x). By definition the Lie derivative of gij with respect to Si is. (1.4) £^,=2{^i,)+C(,i,),^^, where ^=gi^j,. C^=g^f Cl^=lgl'c{g^+2grs-W^}3\ r?=^-^, ^=Pr.Pt symbol \j denoting the covariant derivative with respect to xji:> and <5^ the Kronecker delta. Lie derivatives of p^, pf and the fundamental function F with respect to ^i are as follows :. (1.5) £p^=0,. (1.6) y=^f,U= W+Wf)^, (1.7) £F=F^ respectively. From (1.6) and (1.7) we have. (1.8) ^FY,?=pf£F+Fip? on taking account of the relation ^y-a>==i ^]± Me'i —*-'•. As to Lie derivatives of the connection parameters -T"^ and C^, ? with respect to ^i, we. find. (1.9) iT^R^+n ;^W+^n. where a —os r<*i _ r~'*i .01 p*{ j_ r'*; r'*i. L^ifc—^1.J KH,^~i jOl, '|;|A |al|A:3T-1 XA-1 1?W' '*; _ r'*i ^,1aifc—-* rk^ct,'. (1.10) £C^ = ign{ (<^fc + 2r^) (£^);? + ACw^PzW - 2cyg^} respectively.. From (1.4) and (1.9) we get. d.ii) (£^j)i.= (^ ;!^+<3^+^)^r^ on taking account of the relation Ri^+ R,^ + gv ;fR^= O55.. 2) The round brackets denote the symmetric part, e.g. 2S(i\j)=^iU+§jn3) The square brackets denote the alternating part, e.g. 2r^i=r^-r^t.. 4) See [1]. 5) See [1].. (2).
(4) ; 22 ^ tt 1 ^ »M»;W5^ (^ II ^ A) TO 46 4^ 9 H 2. Motions in an areal space When the transformation (1.1) does not change the fundamental function {Fxl, p1^) of an areal space Anm), we call this transformation a motion in the areal space.. By this definition we have Theorem 1. In order that an infinitesimal transformation (1.1) be a motion in an areal space, it is necessary and sufficient that the Lie derivative of the fundamental function with respect to ^l vanish, that is, (2.1) £^=0 from which, taking account of (1.7) we have. Corollary. In order that an infinitesimal transformation (1.1) be a motion in an areal space, it is necessary and sufficient that (2.2) ^f-0. By virtue of (1.4), (2.2) is equivalent to. (2.3) ga^g^=0. Corollary. If the Lie derivative of gzj ivith respect to ^ vanishes, then the transformation (1.1) is a motion. The Lie derivative of L^f with respect to ^' is expressed as follows :. (2.4) £L^=p^p?+p^+ (£^%P Consequently by virtue of Theorem 1, (1.8) and (2.4) we have Theorem 2. If the transformation (1.1) ;'s a motion, then ive have. (2.5) £pf=0, £Lff=0. ^. By virtue of corollary, Theorem 2 and (1.10), we have. Theorem 3. // the Lie derivative of gzj with respect to ^l vanishes, then zue have (2.5) and (2.6) £%=0. From (1.11) we get. (2.7) (^£^)i.=2^£Fr,. Thus by virtue of Corollary and (2.7) we have. Theorem 4. If the transformation (1.1) is a motion, then we have. (2.8) ^F^=0. REFERENCES [1] Gama, M. (1965), On areal spaces of the submetric class. Tensor, N. S., Vol. 16, p. 262-268. [2] Rund, H. (1959), The Differential Geometry of Finsler spaces. Springer-Verlag.. [3] Yano, K. (1955), The theory of Lie derivatives and its applications. North-Holland.. (3).
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