# When does a 2D metric have a 3D-surface representation?

Certain non-flat, 2D metrics can be visualized as a 3D surface. The metric for the surface of the unit sphere, $$ds^2 = d\theta^2+\sin^2\theta\,d\phi^2,$$ would be the most familiar example. Others are more esoteric: in Martin's General Relativity: A Guide ..., he visualizes the following metric as a infinitely long trumpet-shaped surface: $$ds^2=\frac{1}{r^2}dr^2 + r^2d\phi^2,$$

What determines whether or not a 2D metric can be described by a 3D surface?

• Would Mathematics be a better home for this question? See also Phys.SE chat discussion. – Qmechanic May 9 '17 at 2:40
• @NeuroFuzzy, your flat torus example doesn't seem correct: m.pnas.org/content/109/19/7218 – cutculus May 15 '17 at 2:04
• @Qmechanic I should think it would. – E.P. May 15 '17 at 12:44
• @NeuroFuzzy, you may be right on that; the embedding is "only" $C^1$. The authors say that "the normal vector exhibits a fractal behaviour" which would perhaps be undesirable for physics applications. – cutculus May 15 '17 at 12:46
• It depends on the degree of smoothness you require: If it is $C^1$ then there are no restrictions on the metric (Nash-Kuiper's theorem). Otherwise ($C^2$-smooth), there is no known complete answer, only partial results, e.g. every positively curved metric embeds isometrically in $R^3$. – Moishe Kohan May 24 '17 at 2:36