If you are given the line element (or, for that matter, the metric or area element) on a two-dimensional surface, is it possible to determine an equation for that surface in three-dimensions?
For example, if I were to give you the line element on a sphere (namely, $(ds)^2=R^2(d\phi)^2+R^2\sin^2(\phi)(d\theta)^2$), how would you deduce that the equation for that surface is either $$(x-h)^2+(y-k)^2+(z-l)^2=R^2 $$ or $$x=R\sin(\phi)\cos(\theta)+h\\y=R\sin(\phi)\sin(\theta)+k\\z=R\cos(\phi)+l $$ for any $h,\ k,\text{ and }l$ (the sphere's center), $0\leq \phi\leq \pi$ (the azimuthal angle), and $0\leq \theta\leq 2\pi$ (the polar angle)?