Chart

Note: Sometimes called a coordinate chart

Note: see Transition map for moving between charts

Definition

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

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

Names

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

Shorthands

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

Comments

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

Centred chart

If [ilmath]\varphi(p)=0[/ilmath] then the chart [ilmath](U,\varphi)[/ilmath] is said to be centred at [ilmath]p[/ilmath]

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

See also

• Atlas
• Manifold
• Transition map