Chart

Note: Sometimes called a coordinate chart

Note: see Transition map for moving between charts, and Smoothly compatible charts for the smooth form.

Definition

A coordinate chart - or just chart on a topological manifold of dimension $n$ is a pair $(U,\varphi)$ where:

• $U\subseteq M$ that is open
• $\varphi:U\rightarrow\hat{U}$ is a homeomorphism from $U$ to an open subset $\hat{U}=\varphi(U)\subseteq\mathbb{R}^n$

Names

• $U$ is called the coordinate domain or coordinate neighbourhood of each of its points
• If $\varphi(U)$ is an open ball then $U$ may be called a coordinate ball, or cube or whatever is applicable.
• $\varphi$ is called a local coordinate map or just coordinate map
• The component functions $$(x^1,\cdots,x^n)=\varphi$$ are defined by $$\varphi(p)=(x^1(p),\cdots,x^n(p))$$ and are called local coordinates on U

Shorthands

• To emphasise coordinate functions over coordinate map, we may denote the chart by $$(U,(x^1,\cdots,x^n))$$ or $$(U,(x^i))$$
• $(U,\varphi)$ is a chart containing $p$ is shorthand for "$(U,\varphi)$ is a chart whose domain, $U$, contains $p$"

Comments

• By definition each point of the manifold is contained in some chart
• If $\varphi(p)=0$ the chart is said to be centred at $p$ (see below)

Centred chart

If $\varphi(p)=0$ then the chart $(U,\varphi)$ is said to be centred at $p$

• Given any chart whose domain contains $p$ it is easy to obtain a chart centred at $p$ simply by subtracting the constant vector $\varphi(p)$

See also

• Atlas
• Manifold
• Transition map
• Smoothly compatible charts