Differentiable Manifolds

Abstract

Let M be a set. An n-dimensional chart c on M is a pair (U,ϕ) , where U is a subset of M and ϕ a bijective mapping of U onto an open set of Rⁿ . U is called the domain of c,ϕ the coordinate mapping of c. For each pεU we can write (see Fig. 1)

$${\rm{f(p)}}\,{\rm{ = }}\,{\rm{(}}{{\rm{f}}_{{\rm{1}}\,}}{\rm{(p)}}\,{\rm{,}}...{\rm{,}}{{\rm{f}}_{\rm{n}}}\,{\rm{(p)}}\,{\rm{)}}\,{\rm{ = }}\,{\rm{(}}{{\rm{x}}_{\rm{1}}}\,{\rm{,}}...{\rm{,}}{{\rm{x}}_{\rm{n}}})$$

We call (x₁,...,x_n) the local coordinates of p in the chart (U,ϕ). The notation can be made more explicit by writing x₁,...,x_n in the form

$$({x_1}\,(p)\,,...,{x_{n\,}}\,(p)\,)$$

or, in short, x(p)

Lecture given at IX. Internationale Universitätswochen für Kernphysik, Schladming, February 23 March 7, 1970.