Coordinate Systems on Manifolds

Definition: Coordinate System

Let $U$ be an open subset of an $n$-manifold $M$.

A coordinate system or coordinate map on $U$ is a homeomorphism $\varphi :U\to {\mathbb{R}}^n$ from the subspace $U$ to an open subset of the Euclidean space ${\mathbb{R}}^n$.

Definition: Coordinates on a Subset

The component functions ${\varphi }^1,\dots ,{\varphi }^n:U\to \mathbb{R}$ of $\varphi$ are known as local coordinates or simply coordinates on $U$.

NOTATION

The component functions of a coordinate system $\varphi$ are usually denoted using superscripts instead of subscripts, i.e. ${\varphi }^1,\dots ,{\varphi }^n$ instead of ${\varphi }_1,\dots ,{\varphi }_n$.

Definition: Coordinates of a Point

Given a point $p\in U$, we call ${\varphi }^k(p)$ the $k$-th coordinate of $p$.

NOTATION

The coordinates of a point $p$ are usually written together as an $n$-tuple:

$$({\varphi }^1(p),\dots ,{\varphi }^n(p))$$

Oftentimes, we denote the coordinates of $p$ using superscripts in order to make it clear that they are coordinates of some point:

$$(p^1,\dots ,p^n),$$

where $p^k={\varphi }_k(p)$.

INTUITION

In its essence, a coordinate system is just a way to identify each point of $U$ with a point (vector) of ${\mathbb{R}}^n$. More importantly, it is a way to uniquely do so, since $\varphi$ is a homeomorphism and therefore bijective. No two points of $U$ can correspond to the same vector of ${\mathbb{R}}^n$ and no two vectors in ${\mathbb{R}}^n$ can correspond to the same point in $U$. This means that each point $u\in U$ can be uniquely assigned coordinates

$$u^1={\varphi }_1(u),\dots ,u^n={\varphi }_n(u)$$

Another useful property of coordinate systems is the fact that they are continuous (since they are homeomorphism by definition). Intuitively, this means that if two points $p$ and $q$ of $U$ are “close”, then the Euclidean distance between the vectors they correspond to will be small.

Definition: Origin of a Coordinate System

We say that a point $p\in U$ is the origin of $\varphi$ or that $\varphi$ is centered at $p$ iff $\phi (p)=\vec{0}$.

Definition: Global Coordinate System

A coordinate system on an $n$-manifold $M$ is global iff its domain is the entirety of $M$.

INTUITION

A global coordinate system extends to the entire manifold - each point $p\in M$ can be uniquely assigned $n$ coordinates.

WARNING

Most manifolds do not admit global coordinate systems because this would require that they are homeomorphic to ${\mathbb{R}}^n$.

Charts on Manifolds

Definition: Chart

Let $M$ be an $n$-manifold.

A chart $(U,\varphi )$ for $M$ is an open subset $U\subseteq M$ equipped with a coordinate system $\varphi :U\to {\mathbb{R}}^n$ on it.

Definition: Coordinate Domain

We call $U$ the coordinate domain of $(U,\varphi )$.

Definition: Chart Compatibility

Let $(U_{\alpha },{\varphi }_{\alpha })$ and $(U_{\beta },{\varphi }_{\beta })$ be two charts on an $n$-manifold $M$.

We say that $(U_{\alpha },{\varphi }_{\alpha })$ and $(U_{\beta },{\varphi }_{\beta })$ are $C^k$-compatible (where $k\in {\mathbb{N}}_0\cup \{\infty \}$) iff the transition maps between them are $k$-times continuously partially differentiable or $U_{\alpha }\cap U_{\beta }=\varnothing$.