Riemann’s Habilitation Texts

The two posthumously published works discussed in this chapter came from Riemann's Habilitation: his postdoctoral thesis and the text he read for his final qualifying lecture. Although unrelated mathematically, they were nevertheless linked by virtue of this important event in Riemann's life. His thesis topic may well have been suggested to him by Dirichlet, who also discussed it with him in 1852, when Dirichlet made a trip to Göttingen. In order to extend the class of real functions that could be represented by a trigonometric series, Riemann needed to extend the class of integrable functions. This was the original motivation behind his definition of what today is called the Riemann integral. Even more famous, though, was Riemann's lecture: On the Hypotheses underlying the Foundations of Geometry. In it, he clearly distinguished between the topological and metrical properties of a manifold, an original concept with no significant precursors. Gauss's theory of surface curvature provided a special case, however, which Riemann duly acknowledged while extending this to higher dimensions. The text only alludes to the machinery required to do this, though Riemann later gave some details in his Paris prize paper (see Chapter 8). In the third and closing section of his Habilitation lecture, Riemann discussed the relevance of his theory for the physical world, both in the large as well as in the small. However, it took fourteen years before this text found its way into print. This circumstance had great import for its immediate reception, which coincided with works by two other authors, namely, Helmholtz and Beltrami. After reading Riemann, Helmholtz considered that his arguments supported Helmholtz's own quite different reflections on the epistemic roots of spatial intuition. At this same time, Beltrami published an important paper on 3-dimensional Lobachevskian geometry, in which he explicitly pointed to the relevance of Riemann's text.