# Regular surface

From Diffgeom

## Definition

A **regular surface** in is a subset satisfying the following equivalent conditions:

- It is a two-dimensional differential manifold, embedded inside
- There is an open subset containing it, and a smooth map from to under which 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 , and conversely, any compact regular surface in is orientable.