Can you find a surface from a metric?

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  • Thread starter docnet
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  • #1
docnet
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Homework Statement:
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Relevant Equations:
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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?
 

Answers and Replies

  • #2
anuttarasammyak
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As for your example
[tex]w:=2\sqrt{u} u[/tex]
[tex]ds^2=dw^2+dv^2[/tex]
How are you defining "surface" for ##ds^2=dx^2+dy^2## ?
 
  • #3
docnet
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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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