Regular surface

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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.