Let $S^2$ be the 2-sphere and $g$ some metric on it. Is it possible

- to construct embedding $\iota:S^2\rightarrow \mathrm{R}^3$ s.t. $g=\iota^* g_{\mathrm{R}^3}$ and
- decide when given $(S^2,g)$, there is such an embedding as above?

With $\iota: S^1\rightarrow \mathrm{R}^n$ ($n=2,3$) constant metrics generate arc-length parametrized curves so the above is trivial in that case but how much "less" are there embeddings satisfying 1 and 2.

constructan embedding, which I interpret to ask for a method that will 'produce' a solution for a specified $g$). Even for the positive curvature case, the question of how 'effective' the various existence proofs are for actually constructing a solution is an interesting one. $\endgroup$ – Robert Bryant Mar 2 '13 at 13:53