Under what conditions can an orientable Riemannian 3-manifold $\Sigma$ be defined implicitly?
What I mean by implicitly is that there exists a smooth function $f:\mathbb{R}^n\to \mathbb{R}^m$, such that $\Sigma$ is diffeomorphic to $f^{-1}(0)$, and the Euclidean metric on $\mathbb{R}^n$ pulled back to $f^{-1}(0)$ is equal to the metric on $\Sigma$.



Under what conditions can an orientable Riemannian 3-manifold be defined implicitly?
By the Nash embedding theorem every Riemannian manifold $M$ embeds isometrically into some ${\Bbb R}^n$. You may then take $f(x)=dist(x,M)$ for $x\in{\Bbb R}^n$.


