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 $\mathbb{R}^3$?