Smooth chart (coordinate chart)

Definition

Let $M$ be an $n$-dimensional topological manifold (in particular, any smooth manifold has such an underlying space). A chart (or coordinate chart) on $M$ is a pair $(U,\varphi)$ where

• $U\subset M$ is open, and
• $\varphi:U\to \varphi(U)\subset \mathbb{R}^n$ is a homeomorphism onto an open subset of $\mathbb{R}^n$.

The component functions of $\varphi$ are the local coordinates on $U$: writing $\varphi(p)=(x^1(p),\dots,x^n(p))$ defines coordinate functions $x^i:U\to\mathbb{R}$.

Given two charts $(U,\varphi)$ and $(V,\psi)$ with $U\cap V\neq\varnothing$, the transition map is the map $\psi\circ\varphi^{-1}$ from $\varphi(U\cap V)$ to $\psi(U\cap V)$. A collection of charts forms a smooth atlas precisely when all such transition maps are smooth in the usual multivariable sense.

Examples

1. On $\mathbb{R}^n$, the pair $(\mathbb{R}^n,\mathrm{id})$ is a global chart; restricting $\mathrm{id}$ to any open set $U\subset\mathbb{R}^n$ gives a chart $(U,\mathrm{id}|_U)$.
2. On the sphere $S^n$, stereographic projection from the north pole gives a chart $(S^n\setminus\{N\},\sigma_N)$ with image $\mathbb{R}^n$; stereographic projection from the south pole gives a second chart that overlaps smoothly with the first, generating a smooth structure (see smooth manifold ).
3. On the circle $S^1\subset\mathbb{R}^2$, stereographic projection from a point $p\in S^1$ gives a chart $(S^1\setminus\{p\},\sigma_p)$ with image $\mathbb{R}$, exhibiting $S^1$ as a $1$-dimensional smooth manifold.