【摘要】 <正> 一、绪论本论文表面所讨论的空间,和前篇[1]一样地,是有两种结构的。第一种结构是:对于空间 S_N 的任何可导的 K 次元流形 V_K更多还原

【Abstract】 This note is a sequel to a previous one in which the geometry was as-sumed to be affine[1].Besides the notation newly introduced I shall usethe same notation.Let us assume that the space S_N is of affine connectionwhere H_αβⁱ is a homogeneous function-system symmetric in the indicesα,β[2].Further,we need to add the condition that the function-systemconstructed at each point of a differentiable variety V_K of K dimensionsxⁱ=xⁱ（u^α）should be tensor-invariant under both sorts of transformations[3]Especially,under the latter the function-system H_αβⁱ is transformed to（?） where we have placedConsequently,Γ_jkⁱ is transformed towith the abbreviationTherefore the function-system defined byis invariant under the parameter transformation.In the case ⊿=const.>0 we obtain the volumemary connection V_jkⁱ:Consider now the“volume integral”taken on a region R of V_K;the function F obeys the law Then follows the relationThis combined with the well-known identitysuffices to demonstrate thatIn attempting to obtain a tensor-invariant parallelism of any K-pleareal element（p_αⁱ）when the element itself is taken for the supporting ele-ment of the space we adopt Bortolotti’s differential[4]:whereand impose the condition that the metric function F（x,p）should remainunchanged when the areal element （?） of the variety V_K is sub-jected to the parallel transportIt is easily shown that the equations of connections are the same as in theformer note,namely,or in virtue of the above relationIutroducing the covariant derivatives of vectorsξⁱ（x,t）and（?）byusing the connection ccefficients V_jkⁱ,for example, （?）and adopting the notationwe are led to the second variation of the“volume integral”of an extremalvariety V_K:where B_ijk^h denotes the volumentary curvature tensor:更多还原