The Unapologetic Mathematician

Coordinate Patches

Let’s look back at yesterday’s example of a manifold. Not only did we cover the entire sphere by open neighborhoods with homeomorphisms to open regions of the plane, we did so in a “compatible” way, which I’ll explain soon. This notion of compatible coordinates is key to making a lot of differential topology and geometry work out right.

For the moment, though, let’s introduce a useful term. A “coordinate patch” on a manifold $M$ is an open subset $U\subseteq M$ together with a map $\phi:U\to\mathbb{R}^n$ that is a homeomorphism between $U$ and its image $\phi(U)$. Armed with this definition, we might say that a manifold is a topological space where every point can be contained in some coordinate patch. The only subtle point here is that this definition would put too much emphasis on the patches rather than on the local topology of the manifold itself.

The useful thing about a coordinate patch is that it lets us pull back coordinates from $\mathbb{R}^n$ to our manifold, or at least to the open set $U$. Let’s say $p\in U$ is sent to the point $\phi(p)\in\mathbb{R}^n$. We can now use the coordinate functions $x^i:\mathbb{R}^n\to\mathbb{R}$ to read off coordinates. In fact, when working in a particular coordinate patch, we will often abuse the notation and simply write

$\displaystyle x^i(p)=x^i(\phi(p))$

Of course, when we write $x^i(p)$ the actual number we get out for the $i$th component depends immensely on our coordinate homeomorphism $\phi$, and yet we’ve made no mention of it in our notation! This is one of the most confusing things about doing differential geometry and topology calculations involving coordinates, and it’s important to keep it in mind.

February 23, 2011 - Posted by | Differential Topology, Topology