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

  1. to construct embedding $\iota:S^2\rightarrow \mathrm{R}^3$ s.t. $g=\iota^* g_{\mathrm{R}^3}$ and
  2. 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.

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    $\begingroup$ There are a few closely related questions here mathoverflow.net/questions/37708/… mathoverflow.net/questions/76955/… $\endgroup$ – j.c. Mar 1 '13 at 11:09
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    $\begingroup$ I would say that the answers so far have concentrated on Part 2 (the existence question) rather than Part 1 (can one construct an 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
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Although, as jc says, this question was addressed in earlier questions, the answer is scattered among the accepted answer as well as the comments. So here is a summary:

  1. By the Nash-Kuiper theorem (which I believe is one of the original uses of the so-called h-principle), there always exists a global $C^1$ isometric embedding of a surface with a $C^1$ or better Riemannian metric. This holds whether the surface is closed, open, or has boundary and whether the metric is complete or not. This is both amazing and somewhat unsatisfactory. If the embedding is only $C^1$, there is no second fundamental form, which is in some sense the most important geometric invariant of a surface in $R^3$

  2. If the surface is closed and has a metric with strictly positive curvature, then the question is known as the Weyl problem. The existence of a global isometric embedding, if the metric is sufficiently smooth, was proved by Nirenberg in a 1953 CPAM paper. Its uniqueness was proved much earlier by Cohn-Vossen. Nirenberg's paper is a landmark paper, because it was one of the first to develop and use a priori estimates for nonlinear elliptic PDE's to prove a theorem in global differential geometry. This approach to differential geometry led eventually to the important and exciting work by, among many others, Yau, Schoen, Uhlenbeck, Taubes, Donaldson, Hamilton. But the solution of the Weyl problem is also credited to Pogorelov and maybe also Alexandrov. Unfortunately, their work was not well understood back in the 50's and 60's and did not get the attention it deserved until much later.

  3. If the curvature is strictly negative, it's easy to prove that no global isometric embedding can exist.

  4. If the curvature changes sign, then no global theorem is known. ADDED: This is not quite right. I just found a paper by Lin and Han that gives sufficient conditions for a 2-torus to be isometrically embedded in $R^3$. But the conditions are quite restrictive.

  5. With suitable assumptions on curvature, the existence of local isometric embeddings in the neighborhood of a point is known. But that's another long story.

As Alexandre Eremenko says, you can find all the details in the book by Han and Hong. (Names of authors corrected)

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In general, the answer is no. This is a difficult problem and there was a lot of research on it. For example, for a metric with positive curvature there is also an embedding. The answer also depends on the degree of the smoothness of the metric. Here is a recent survey: MR2261749 Han, Qing; Hong, Jia-Xing Isometric embedding of Riemannian manifolds in Euclidean spaces.

See also MR0133090 Pogorelov, A. V. On the isometric immersion of a two-dimensional Riemannian manifold homeomorphic to a sphere into a three-dimensional Riemannian space, (Russian) Dokl. Akad. Nauk SSSR 139 1961 818–820.

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