# Regular surface

A regular surface in $\R^3$ is a subset $S \subset \R^3$ satisfying the following equivalent conditions:
• It is a two-dimensional differential manifold, embedded inside $\R^3$
• There is an open subset $U \subset \R^3$ containing it, and a smooth map from $U$ to $\R$ 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 $\R^3$, and conversely, any compact regular surface in $\R^3$ is orientable.