Introduction

Abstract

The theory of linear displacement is a result of an investigation into the foundations of differential geometry and also into the structure of geometry as a whole. Though based upon the analysis of space conception undertaken by Riemann in 1854¹, it received its impetus only with the advent of general relativity in 1916². Here, space-time is interpreted as a Riemannian manifold which is locally euclidean of the Minkowski type, so that the question arises how the comparison between the euclidean world at different points is being performed. This led to the discovery of parallelism in a Riemannian manifold³, then to the extension of this parallelism to manifolds of a more general type. The essential character of the local space changed, in these investigations, from euclidean to affine, the character of the general manifold from Riemannian to what is now called affine with or without torsion. This is the principal idea of the work done from 1917 to 1924 by Weyl, Schouten and Eddington ⁴. Closely connected with these investigations are the basic papers of Hessenberg and König ⁵, which deal with the purely mathematical aspect of the space problem only. From the beginning there has always been an intimate relation between the various attempts to improve or to generalize the theory of relativity and the systematical development of the mathematical theory. For evidence there is, for instance, the recent monograph of Veblen ⁶.