3 Curvature and the Notion of Space

a lecture that nearly did not occur, Georg Friedrich Bernhard Riemann (1826–1866) proposed a visionary concept for the study of space [223, pp. 132–133]. To obtain the position of an unsalaried lecturer (Privatdozent) in the German university system, Riemann was required to submit an inaugural paper (Habilitationsschrift) as well as to present an inaugural lecture (Habilitationsvortrag). The topic of the lecture was selected from a list of three provided by the candidate, with tradition suggesting that the first would be chosen. The most prominent member of the faculty at Göttingen and arguably the preeminent mathematician of his time, Carl Friedrich Gauss (1777–1855) passed over Riemann's first two topics (concerning his recent investigations into complex functions and trigonometric series), and chose the third as the subject of the lecture: ¨ Uber die Hypothe-sen, welche der Geometrie zu Grunde liegen (On the Hypotheses That Lie at the Foundations of Geometry) [173, p. 22]. Gauss's decision, undoubtedly motivated by his own unpublished work on non-Euclidean geometry, elicited a lecture that changed the course of differential geometry. Some years prior to Riemann's lecture, Gauss had developed a consistent system of geometry in which the Euclidean parallel postulate (see below) does not hold, but wishing to avoid controversy, he did not publish these results [101]. Riemann, however, did not present a lecture tied to the tenets of a particular geometry (Euclidean, hyperbolic, or otherwise), but offered a new paradigm for the study of mathematical space with his notion of an n-dimensional manifold. His ideas remain the standard for the classification of space today. Although many modern textbooks on geometry and topology offer a rather technical definition of a manifold, the ultimate goal of this chapter is to present Rie-mann's own lucid description of what space ought to be. What developments in mathematics helped to precipitate Riemann's lecture? What mathematical concepts are needed for an appreciation of the ideas therein? Why have his thoughts endured the test of time?