Theorem. Characterization of k-dimensional smooth submanifolds of cartesian space [0093]

For a subset of the following are equivalent

1. is a k-dimensional smooth submanifold of cartesian space.

2. "Locally M is the zero set of a regular height function."

   For all points in there is an open neighbourhood of and a smooth map such that , is surjective and .

3. "Locally M is the graph of a smooth function."

   For all points in there is a diffeomorphism permuting coordinates, and open neighbourhood of and a smooth map such that .