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.