具有面积测度的一些仿射联络空间

<正> 为了要把 K 展空间和具有 K 重面积测度的空间结合起来,笔者和谷超豪讨论过具有两种结构的一些空间,第一种结构是:空间具有 K 维面积测度,就是说:对于空间的任何 K 维可微分流形 V_K 的一部分给定了一个 K 重积分,作为这部分的“面积”;第

CERTAIN AFFINELY CONNECTED SPACES WITH AREAL METRIC

In a previous paper of mine and Ku Chao-hao(1952),we have consi-dered certain affinely connected spaces with given areal metric.Letx~i=x~i(u~α)(i=1,…,N;α=1,…,K)be the equations of a differentiable K-dimensional variety V_k in an N-di-mensional space S_N,and let the'area'of a certain portion R of the varietygiven by a K-ple integral(?)where(?)is an abbreviation for du~1,du~2,…,du~k and the func-tion F satisfies certain conditions of invariance.The connection coefficients Γ_(jk)~i there introduced are functions of(x~i)aswell as the K-ple supporting element(p_α~i),and are supposed to satisfy aset of conditions which suffice to insure that(?)(*)These Γ's are related to the metric function F by the equations of connec-tion(?)(**)where we have placed(?)In Riemannian spaces these conditions(*)and(**)are satisfied by theChristoffel symbols of the second species(?)and the metric function(?)of a K-dimensional differentiable variety VK in the space S_N,where gλ_udenotes the induced metric tensor of V_k,so that the general formula for thesecond variation of the'area'gives immediately the one due to E.T.Daviesas its special case.It is natural to inquire whether or not our theory contains the corres-ponding theories for Finsler and Cartan spaces.In the present paper,we demonstrate that the equations of connectionstill hold good in the geometries of Finsler space and a regular Cartanspace as a necessary consequence of the generalized Ricci Lemmas in thesespaces. On the contrary,the conditions(*)are by no means valid in Finsler orCartan spaces.For the purpose of finding more extensive conditions inorder to include both Finsler and Cartan geometries,we have to investigateEulerian vector E_i in each of these spaces.In the former,it is readily shown that(*)should be replaced by thefollowing ones:(?)where Γ_(jh)~(*k)denotes the connection coefficient of Cartan as well as that ofBerwald and therefore that E_i is equal to the covariant curvature vector ofthe curve in consideration.Denoting the integrand of the second variation of the are under theinfinitesimal transformation(?)by F"and assuming,in particular,thatξ~i is independent of t,we obtain(?)(F2)where R_(jikh) denotes the curvature tensor of the space.In a regular Cartan space we have to put K=N-l and obtain that(?)(C_1)These relations suggest us to consider a further generalization of af-finely connected spaces with areal metric in the following manner:(Ⅰ)The coefficients of affine connections,Γ_(jh)~(*k),are functions of position(x~i)as well as K-ple areal element(p_a~i).(Ⅱ)The metric function F(x,p)is related to these Γ's by the conditionthat the metric of any K-ple areal element should be invariant with respectto the parallel transport of the connection when the element itself is takenfor the supporting element.This naturally leads to the equations of con-nection.(Ⅲ)The Eulerian vector E_i is given by(?)(E)where we have placed(?)There is no difficulty in showing that(E)is equivalent to(?)(E')which implies(C_1).Thus we have extended the spaces to such ones which may be seen ascontaining Riemannian,Finslerian and Cartannian geometries.The formulafor the second variation of the'area'as established in the previous paperremains valid.