TITLE: What is an "allowable surface patch"?
QUESTION [12 upvotes]: 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?

REPLY [5 votes]: 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 $u_1=0$.