Riemannian Manifolds

Abstract

The metric properties of $\mathbb {R}^n$ (distances and angles) are determined by the canonical Cartesian coordinates. In a general differentiable manifold, however, there are no such preferred coordinates; to define distances and angles one must add more structure by choosing a special $2$-tensor field, called a Riemannian metric (much in the same way as a volume form must be selected to determine a notion of volume). This idea was introduced by Riemann in his 1854 habilitation lecture “On the hypotheses which underlie geometry”, following the discovery (around 1830) of non-Euclidean geometry by Gauss, Bolyai and Lobachevsky (in fact, it was Gauss who suggested the subject of Riemann’s lecture). It proved to be an extremely fruitful concept, having led, among other things, to the development of Einstein’s general theory of relativity. This chapter initiates the study of Riemannian geometry.