Riemannian 2-manifolds not realized by surfaces in $\mathbb{R}^3$? A smooth surface $S$ embedded in $\mathbb{R}^3$ whose metric is inherited from $\mathbb{R}^3$
(i.e., distance measured by shortest paths on $S$) is a Riemannian $2$-manifold: differentiable
because smooth and with the metric just described.  Two questions:


*

* Are such surfaces a subset of all Riemannian $2$-manifolds? Are there Riemannian 2-manifolds
that are not "realized" by any surface embedded in $\mathbb{R}^3$?  I assume _Yes_.

* If so, is there any characterization of which Riemannian 2-manifolds are realized by
such surfaces? In the absence of a characterization, examples would help.


Thanks!
Edit. In light of the useful responses, a sharpening of my question occurs to me:
3. Is the only impediment embedding vs. immersion? Is every Riemannian 2-manifold realized by
a surface immersed in $\mathbb{R}^3$?
 A: The hyperbolic plane cannot be smoothly isometrically embedded in $\mathbb{R}^3$. It can be so in $\mathbb{R}^5$. It is open (as far as I know) if it can be embedded in $\mathbb{R}^4$. I believe this is mentioned in Do Carmo's book on curves and surfaces. 
Edit: Not a complete characterization, but Amsler has shown (see below for reference) that any Riemannian surface with constant negative curvature, if attempted to be imbedded in $\mathbb{R}^3$, must have singularities. 

Amsler, M.H., Des surfaces a courbure negative constante dans l'espace a trois dimensions et de leurs singularites, Math. Ann. 130, 1955, 234-256
A: The Whitney embedding theorem says you can always embed a smooth $n$-manifold in $\mathbb{R}^{2n}$, and immerse it in $\mathbb{R}^{2n-1}$.  Nonorientable Riemann surfaces, for example, don't embed in $\mathbb{R}^3$, but there are some pretty good immersions (the typical picture of the Klein bottle is a good example).
For a Riemannian manifold, Nash and Kuiper proved that there's a $C^1$ globally isometric embedding into $\mathbb{R}^{2n+1}$ (and, in fact, that you can arbitrarily closely approximate any metric $C^\infty$ embedding into at least $\mathbb{R}^{n+1}$ by a global isometric $C^1$ embedding).  For a global isometric $C^\infty$ embedding, it looks like the current lower bound is max$(n(n+1)/2+2n,n(n+1)/2+n+5)$.  For a local one, you can do it into $n(n+1)/2+n$-space.  
This means that for a globally isometric and analytic embedding of a surface, you might have to go up to $\mathbb{R}^{10}$.  Ew.
A: There are pointers to a wealth of information on this question in the responses to the MathOverflow question, which you mentioned in your comment to Paul VanKoughnett's response.
In particular, Deane Yang's response gives a nice summary of the situation, and Bill Thurston's response seems to give a good perspective on the problem of trying to find a characterization of Riemannian manifolds that admit such an embedding.
Regarding the third question you mention in an edit. This is essentially a local problem. All this from the same MO question:
From BS's response:
there is not even a local isometric embedding in general:
https://doi.org/10.1007/s00526-007-0140-7.
From Will Jagy's response (and Deane Yang's comments on it):
If the metric is analytic, then you can construct a local isometric embedding.
Some recent progress on characterizing the requirements when the degree of smoothness is relaxed:
https://arxiv.org/abs/1009.6214.
The bibliography for that last one has no shortage of other relevant sounding titles.
