2 - Differential and Analytic Manifolds

Abstract

The notion of a differential manifold proposed by Riemann in his inaugural lecture in 1854 is a generalization of curves in the plane, or curves or surfaces in everyday space; these curves and surfaces (extensively studied by Gauss, who supervised Riemann’s thesis), and more generally manifolds, are assumed to be regular. For a curve, being regular means having a tangent at every point; for a surface, it means having a tangent plane at every point. Locally, each curve “looks like” its tangent and each surface “looks like” its tangent plane. Manifolds can be studied locally using charts in the same way that we study regions of our planet geographically. It was Gauss who suggested charts to study curves and surfaces locally. We cannot represent the whole Earth with a single chart unless we are willing to accept significant deformations of some of its regions, for example the poles in the classical Mercator projection. Hence, we require more than one chart; these charts are assembled into an atlas.