Can you find a surface from a metric?

[docnet]

Homework Statement

.

Relevant Equations

.

if a metric like $ds^2=dr^2+r^2d\theta^2+r^2\sin^2\theta d\phi^2$ is given, we know it corresponds to a sphere in spherical coordinates .

if you are given an arbitrary metric with two variables for example $ds^2=\frac{du^2}{u}+dv^2$ is ther guarenteed to be a surface embedded in $R^3$ associated with the metric? could a surface be found using python?

 

[anuttarasammyak]

As for your example
$$w:=2\sqrt{u} u$$
$$ds^2=dw^2+dv^2$$
How are you defining "surface" for $ds^2=dx^2+dy^2$ ?

 

[docnet]

How are you defining "surface" for $ds^2=dx^2+dy^2$ ?

ummm I'm not sure if i understand your question.. just a regular plane with the standard basis $(1,0)$ and $(0,1)$? actually I'm not sure if I understand the question i was trying to ask.

$x=2\sqrt{u}u, y=v$ is like a parametric function that corresponds to a map from $R^2$ to $R^2$, right? like i was wondering if there is a geometrical demonstration for it. like could you scrunch a paper with uniform grid spacing and make another shape with non-standard grid spacing that somehow describes the image geometrically?

like when you wrap a plane around a sphere, you end up deforming the standard grid spacing in a way that reflects the new metric on the sphere.. does every induced metric of this kind have a corresponding surface that can be drawn on a paper?