Can you find a surface from a metric?

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In summary, the conversation discusses the relationship between a given metric and a corresponding surface in spherical coordinates. The question is raised about whether an arbitrary metric with two variables will always have a surface embedded in ℝ³ associated with it, and if it can be found using Python. The concept of a "surface" is further explored and the possibility of finding a geometrical demonstration for a given parametric function is discussed.
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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?

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

anuttarasammyak said:
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?

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1. Can a surface be uniquely determined by its metric?

Yes, a surface can be uniquely determined by its metric. This is because the metric contains information about the curvature and distances on the surface, which are fundamental properties of the surface.

2. What information does a metric provide about a surface?

A metric provides information about the curvature, distances, and angles on a surface. It also contains information about the local geometry and topology of the surface.

3. How is a metric used to find a surface?

A metric is used to find a surface by first defining a metric tensor, which describes the distances and angles on the surface. Then, using this metric tensor, the surface can be reconstructed using mathematical equations and techniques such as differential geometry.

4. Can a surface be found from any metric?

No, not all metrics can be used to find a surface. The metric must satisfy certain conditions, such as being positive definite and smooth, in order for a surface to be reconstructed from it.

5. What are some applications of finding a surface from a metric?

One application is in computer graphics and animation, where surfaces can be generated from metrics to create realistic 3D models. Another application is in physics, where metrics are used to describe the curvature of spacetime in general relativity.

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