Presentation of the Text

Abstract

In his work, Riemann analyzed the mathematical structure of space in a conceptually novel way. Through Riemann’s work, physical space gets firstly empirically determinable characteristics and secondly loses its uniqueness as a mathematical space. For this purpose, Riemann first introduces the concept of a multiply extended variety or manifold. A manifold is characterized in that its sufficiently small parts can be fully and non-redundantly described by n coordinates. That number n is then the dimension of the manifold. It is fundamental that this manifold structure determines in modern terminology only the topology, i.e. the qualitative aspects of position, but does not yet provide for any measurements. Riemann thus recognizes that in order to measure lengths and angles, an additional structure is required which is of a quantitative nature. This additional structure is arbitrary (obeying certain natural constraints). This structure can then be restricted on the one hand by conditions of simplicity and on the other hand by empirical testing if it is supposed to describe the actual physical space. Riemann then describes the quantitative structure by a so-called metric tensor, which for simplicity is chosen as quadratic (this will be explained later). Using this metric tensor, one can then determine curve lengths and distances between points and sizes of angles, that is the usual metric quantities. But since a manifold can be described locally by coordinates in different ways, it becomes the central task of geometrical investigations to identify quantities that do not depend on the choice of coordinates. This then are the invariants of the manifold provided with a metric. Riemann thus goes on to identify a complete set of invariants under his conditions. This set of invariants is represented by the curvature tensor. This represents a far-reaching generalization of the Gaussian theory of surfaces. Through additional requirements on the geometric properties, the curvature tensor can be more narrowly constrained. In particular, it follows from the requirement of the free mobility of rigid bodies that the curvature of the space has to be constant, a result which Helmholtz will later put at the center of his considerations. The Riemannian spaces of constant negative curvature turn out to be models of the non-Euclidean geometry of Bolyai and Lobatchevsky, as subsequently emphasized by Beltrami. Riemann therefore had found a new and much more general approach to non-Euclidean geometry, of which, incidentally, he had apparently not even been aware of when composing his work. For Riemann, this generality is particularly important from natural philosophical reasons because he already hints at the relationship, fundamental for Einstein’s general theory of relativity, between the geometry of space and the forces caused by the objects contained in it. This extends far beyond the class of spaces of constant curvature, since then bodies moving in space affect the latter’s geometry and then, conversely, the geometry can determine the motion of bodies.