Suppose that $M$ is an embedded sub-manifold of $D$-dimensional Euclidean space $E^D$, with embedding $\phi:M \hookrightarrow E^D$; the embedding is not *explicitly* known.

And suppose that I know the induced Riemannian metric $g$ on $M$, which depends by construction on $\phi$ (which so happens that I don't explicitly know).

Can I express $\phi$ in terms of $g$ (closed form or as a PDE maybe?), supposing that I want $\phi$ to be the embedding into $M$ which has minimal normal curvature and contains a predetermined (fixed) set of points.