For 300 years or more, all sorts of ways have been invented to construct surfaces in Euclidean 3-space. Thus to be interesting, a surface ought to have reasonable Gaussian curvature. For anyone interested in surfaces, these pages may be a good introduction to Mathematica or Maple. Most welcome of all are reports of errors.
Surfaces are very intuitive objects.
Typically the coordinates of R 2 will be u,v. It guarantees that, locally at least, the image of x really is a surface. D R n into any Euclidean space. To get a surface in R 4 a natural choice is a complex function C C . Proof of the lemma.