I was playing with piece of paper which has the form of semisphere, to be more precise we may assume that it satisfies $x^2+y^2+z^2=1$ for nonnegative $z$. I tried to make it flat without stretching and cutting, just by bending. I couldn't perform this action, so I decide to write this problem mathematically:
Let $S \subset R^3$ be a space of all points satisfying previously defined equation. There is a natural metric $d'$ on this set, which is not induced by euclidian metric on $R^3$, but it is the length of the geodesic curve between 2 points. I 'm searching for continuous injection $f$ from $S$ to $R^2$ preserving metrics(i.e. $d'(a,b)=d(f(a), f(b))$.
Intuitively there is no such map. But I'm wondering about related more general theorems. I haven't taken differential geometry yet, and I don't know which tags would be more appropriate, so feel free to edit. Even if there is no related theorem, I'm interested in the proof of this specific problem. But again, feel free to answer anything related.