Riemann’s Result and Consequences for Physics and Philosophy

Abstract

Riemann commented on his main result as follows: “The common character of those manifolds whose curvature is constant may also be expressed thus: that figures may be viewed in them without stretching. For clearly figures could not be arbitrarily shifted and turned around in them if the curvature at each point were not the same in all directions at one point as at another, and consequently the same constructions can be made from it; whence it follows that in aggregates with constant curvature, figures may have any arbitrary position given them. The measure-relations of these manifolds depend only on the value of the curvature, and in relation to the analytic expression it may be remarked that if this value is denoted by K, the expression for the line-element may be written as in (41.17).”