Manifolds

Let n be a positive integer. Given a set M, an n-dimensional chart (coordinate system) on M is a bijection ϕ from a subset U of M to an open subset φ(U) of the n-dimensional real space. An atlas for M is a collection of charts {(Ua, ϕa)} such that

• The union of all Ua is the set M
• If the intersection of two charts is not null, then the map$F={\varphi }_a◦{\varphi }_b^{-1}:{\mathbb{R}}^n\to {\mathbb{R}}^n({\tilde{x}}^{\mu }(x^{\nu }))$ is differentiable where it is defined. A maximal atlas is one containing all the charts satisfying the above definition. An n-dimensional manifold is the set M along with a maximal atlas. Examples:
• n-dimensional real space - just take ϕ to be the identity map. A maximal atlas includes the identity map and all differentiable coordinates.
• Two sphere - needs at least 2 charts to cover it.
• Circle - need 2 charts {(0 < θ < 2π), (-π < 0 < π)}

Mathematically, the utility of a manifold is that it defines a differentiable structure on the underlying set (smoothness). Given any function $f:M\to \mathbb{R},f:p\to f(p)$ we can define functions$f◦{\varphi }^{-1}:\mathbb{R}\to \mathbb{R},f:x^{\mu }\to f(p(x^{\mu }))$ We say that f is differentiable if these functions are differentiable where they are defined.