What is an "allowable surface patch"?

From the book Elementary Differential Geometry, Andrew Pressley, Second Edition, the author defined an allowable surface patch is follows:

If S is a surface, an allowable surface patch for S is a regular surface patch $\sigma:U \rightarrow R^3$ such that $\sigma$ is a homeomorphism from U to an open subset of S.

But the patch itself is actually a homeomorphism from U to an open subset of S, so what does the "such that..." part of the statement mean? Could it be that the author meant an allowable surface patch is a regular surface patch?

Solution 1:

This is a very good question, it isn't made terribly clear in the book (I only have a first edition copy of Pressley's book, but I'm assuming everything regarding these definitions is basically the same).

To use Pressley's terminology, a surface is a subset $S\subset\mathbb{R}^3$ with the property that for all $P\in S$, there are open sets $U\subset\mathbb{R}^2$, $W\subset\mathbb{R}^3$ such that $U$ and $S\cap W$ are homeomorphic, and $p\in S\cap W$. These homeomorphisms $\sigma:U\to S\cap W$ we've found are called surface patches, and given a surface $S$, the collection of them is called the atlas of $S$. Pressley then defines what it means for a surface patch to be regular, and then defines a smooth surface to be one whose atlas consists of regular surface patches.

The key issue with this definition is that given a smooth surface $S$, different people might find different atlases for it, and the way things have been set up, these would need to be considered as different surfaces, which is an undesirable situation. The usual solution is to have everyone expand the atlas they chose for $S$, to include every possible regular $\sigma:U\to S\cap W$, not just the ones you each found initially which verified $S$ was a smooth surface. Everyone will all agree on what this collection is. It is called the maximal atlas for $S$, and its members are called allowable surface patches.

So, yes, essentially the definition of "allowable surface patch" is "a regular homeomorphism $\sigma:U\to S\cap W$ where $U\subset\mathbb{R}^2$, $W\subset\mathbb{R}^3$ are open sets", and in that way allowable surface patches and regular surface patches have identical definitions. It's just that very often one wants to talk about regular surface patches which might be different than the ones you found initially, and "allowable" surface patch is a terminological distinction that indicates it may not be one from your original collection.

Solution 2:

If $\; U\subset \mathbb R^2$ is an open subset, a smooth map $\sigma: U\to S$ to the embedded smooth surface $S\subset \mathbb R^3$ is a coordinate patch if the map $U\to \sigma U$ is a homeomorphism and if $\sigma $ is an
immersion .
Immersion means the following: if you write $\sigma (u)=(\sigma_1 (u), \sigma_2 (u), \sigma_3 (u))$, then the jacobian matrix $Jac _{u_0}(\sigma)=(\frac{\partial \sigma_i}{\partial u_j}(u_0)) \in M_{3\times 2}(\mathbb R)$ has rank 2 (=maximal rank) at every $u_0\in U$.

The condition "immersion" does not follow from the condition "homeomorphism". Here is a counterexample:

Take $U=\mathbb R^2, S= \mathbb R^2 \times \lbrace 0\rbrace $ and $\sigma (u_1,u_2)=(u_1^3, u_2,0)$ .
The smooth map $\sigma$ is a homeomorphism $\sigma: \mathbb R^2\to S$ but is not an immersion (hence is not a coordinate patch) because $$Jac _{u}(\sigma) =\left(\begin{array}{rrr} 3u_1^2 & 0 \\ 0 & 1 \\ 0 & 0 \end{array}\right).$$
has rank $1$ instead of $2$ whenever $u_1=0$.