The reason that I am posting this is because I'm having a hard time figuring out how to pose the question in a manner that makes any sense. But I thought I'd post it here and see if anyone can help guide me.

Suppose we have a smooth surface defined in 3-space that is given by $\displaystyle X^3 = X^3 ( X^1 , ~X^2 )$. (The superscripts indicate a contravariant index, not a power.)

Suppose we have a parametrization of the surface, $\displaystyle X^1 = v^1$, $\displaystyle X^2 = v^2$, and thus $\displaystyle X^3 = X^3(v^1,~ v^2 )$. My question is: "Is there a transformation of coordinates of this surface such that the transformed coordinates are not allowable, but still have the same domain as the original parametrization?"

I haven't been able to do much in the way of a proof but geometrically speaking I don't think such a transformation can be done. The surface is, of course, independent of the parametrization so it has the same number of critical points, the same curvature at a given point, etc. But I can't see how to make a proof out of it.

I'm probably making a mountain out of a molehill but it bothers me that I can't come up with a way to write this thing down in clear terms.

Thanks!

-Dan