Under what conditions can an orientable Riemannian 3-manifold $\Sigma$ be defined implicitly? What I mean by implicitly is that there exists a 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 $\Sigma$ 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 a Riemannian 3-manifold $\Sigma$ be defined implicitly? What I mean by implicitly is that there exists a 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 $\Sigma$ be defined implicitly? What I mean by implicitly is that there exists a 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 $\Sigma$.