Riemann’s Blueprints for Architecture in Myriad Dimensions

Abstract

As we said in chapter 2, Riemann’s construction of the Riemannian manifold consisted first in building the foundation of the smooth manifold. He then established on that foundation the concept of a Riemannian metric. In the first two sections we will present smooth manifolds, and thereafter define Riemannian metrics. The notion of smooth manifold is at the same time extremely natural and quite hard to define correctly. This notion started with Riemann in 1854 and was widely used. Hermann Weyl was the first to lay down solid foundations for this notion in 1923. The definition became completely clear in the famous article Whitney 1936 [1259].