Introduction

Abstract

The theory of manifolds goes back to Riemann’s lecture “Ueber die Hypothesen, welche der Geometrie zu Grunde liegen” (“On the hypotheses which lie at the foundations of geometry”), which was delivered on June 10, 1854, to the faculty of the University of Göttingen. Since there were members of the faculty who knew little mathematics, Riemann chose a rather informal style of exposition to make his lecture intelligible. In part one of this lecture, he set himself the task of “constructing the concept of a multiply extended quantity from general notions of quantity”, a task he regards as being of a “philosophical nature, where difficulties lie more in the concepts than in the construction ....”