Charts#
Definition: Coordinate System
Let \(U\) be an open set of an \(n\)-manifold.
A coordinate system on \(U\) is a homeomorphism \(\phi: U \to \mathbb{R}^n\) between the subspace \(U\) and the Euclidean space \(\mathbb{R}^n\).
Definition: Local Coordinates
The component functions of \(\phi\) are known as (local) coordinates on \(U\). When evaluated at some \(p \in U\), we call them (local) coordinates of \(p\).
Notation
Local coordinates of \(\phi\) are usually denoted via superscripts:
\[\phi^1 \qquad \cdots \qquad \phi^n\]
\[p^k = \phi^k(p)\]
Definition: Chart
A chart \((U, \phi)\) on an \(n\)-manifold \(M\) is an open subset \(U \subseteq M\) equipped with a coordinate system \(\phi: U \to \mathbb{R}^n\).
Notation
We also write \((U, \phi^1, \dotsc, \phi^n)\) instead of \((U,\phi)\).