特殊的仿射聯絡空間

<正> §1.在n維的仿射聯絡空間中,一對稱張量b_(ij)及二個仿射聯絡G_(ik)~i,Γ_(ik)~i如滿足依據诺爾勤的說法,聯絡偶G_(ik)~i,Γ_(ik)~i关於張量b_(ij)是共軛的.作者曾經擴充這個思想而定義m個聯絡關於一個m階的對稱共變張量是共軛的場合,當時曾提出由m-1個聯絡如何决定第m個聯絡的問題,從而得到一系列的

SOME SPECIAL AFFINELY CONNECTED SPACES

Norgen has defined a pair of affine connections G_(ik)~k, Γ_(ik)~i as conjugate connections with respect to a symmetric tensor b_(ij), when they satisfy the condition In a recent paper the author has generalized this notion to the case of a symmetric tensor of order m, and has solved the problem of determing the mth connection from the given m-1 connections. We have system of equations and for the consistency and uniqueness of the solution of the system of equations we obtained some restrictions which must be added to the tensor or the m-1 connections.In the present note we consider n conjugate connections with respect to an antisymmetric tensor ei_1 i_2 ... i_n and prove the following theorem:If n connections are conjugate with respect to an antisymmetric tensor ei_1 i_2 ... i_n, then(i)one of them for example, can be choosen arbitrarily,(ii)the remaining connections are common-pseudo with(hence mutually common-pseudo),(iii)the common-pseudo vectors must satisfy the following condition:As a geometrical interpretation of these connections we have: If there exist a volume in an affinely connected space formed by n vectors such that it remains unaltered when these vectors undergo parallel displacements respectively with regard to the connections then these connections must satisfy the conditions (i), (ii), (iii).If G_(ik)~i is such an affine connection in our space, then we can always find n connections common-pseudo with G_(ik)~i such that the volume formed by certain n vectors remains unaltered when the forming vectors undergo parallel displacements with regard to respectively.If the connection G_(ik)~i is equi-affine and any one of the n connections is commonpseudo with it, then the directions of the vectors remain the same when they are parallelly displaced under respectively.Especially, when the n connections are all symmetric, then they become one and the same.