Regular surface

Definition

A regular surface in ${\mathbb{R}}^3$ is a subset $S\subset {\mathbb{R}}^3$ satisfying the following equivalent conditions:

• It is a two-dimensional differential manifold, embedded inside ${\mathbb{R}}^3$
• There is an open subset $U\subset {\mathbb{R}}^3$ containing it, and a smooth map from $U$ to $\mathbb{R}$ under which $S$ is the inverse image of a regular value

It is a theorem that any 2-dimensional compact connected orientable differential manifold can be realized as a regular surface inside ${\mathbb{R}}^3$, and conversely, any compact regular surface in ${\mathbb{R}}^3$ is orientable.