Differential Manifolds

Abstract

“ The general notion of manifold is quite difficult to define precisely. A surface gives the idea of a two-dimensional manifold. If we take for instance a sphere, or a torus, we can decompose this surface into a finite number of parts such that each of them can be bijectively mapped into a simply-connected region of the Euclidean plane.” This is the beginning of the third chapter of “ Leç ons sur la Gé omé trie des espaces de Riemann” by Elie Cartan (1928), that we strongly recommend to those who can read French. In fact, Cartan explains very neatly that these parts are what we call “open sets”. He explains also that if the domains of definition of two such maps (which are now called charts) overlap, one of them is gotten from the other by composition with a smooth map of the Euclidean space. This is just the formal definition of a differential (or smooth) manifold that we give in 1.A, and illustrate by the examples of the sphere, the torus, and also the projective spaces.