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September 26th, 2007, 03:24 PM   #1
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Isometric surfaces - differential geometry

Hi everybody! I'm writing here because
I need to answer to this problem:
Let f:R^2--->R^3, (u,v)|-->(u,v,3u^3-4v^2) be a parametrization of the
Surface S and let g:R^2--->R^3, (u,v)|-->(u,v,3u^3-4(u+v)^2) be a
parametrization of the surface S'.
Tell if S and S' are (locally) isometric.

I find really difficult to understand what are 2 isometric surfaces! I
mean, I know about cilinder and plane, etc... But I really can't
understand how to prove that two general surfaces are isometric!

I tried to use the Gauss Egregium Theorem to prove that are not
isometric, but in this case it didn't work because the K are equal
for both surfaces! I even thought about finding a re-parametrization
for g : phi:R^2-->R^2, (u,v)|-->(u,v-u) but is this an isometry? Or
else, how can I find an isometry? How can I create such a map?
I really hope that someone can help me because this will be part of my
oral test....
Thank You so much
saturn4 is offline  
 
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