【摘要】 <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”;第更多还原

【Abstract】 In a previous paper of mine and Ku Chao-hao（1952）,we have consi-dered certain affinely connected spaces with given areal metric.Letxⁱ=xⁱ（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 du¹,du²,…,du^k and the func-tion F satisfies certain conditions of invariance.The connection coefficients Γ_jkⁱ there introduced are functions of（xⁱ）aswell as the K-ple supporting element（p_αⁱ）,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^*kdenotes 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,thatξⁱ 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（?）（C₁）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（xⁱ）as well as K-ple areal element（p_aⁱ）.（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（C₁）.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.更多还原