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Local isometry

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Poirot

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Feb 15, 2012
250
Let S={(x,y,z):$a^{2}x +a^{2}y=z^2$, a>0}. Let f:$R^2/{(0,0)}$->S be given by

f(rcosx,rsinx)=(brcos2x,brsin2x,abr) where b>0. Show f is a local isometry iff $a=3^{0.5}$ and $b=0.5$.

My thoughts: I can use the fact that f is a local isometry iff for each p in the domain, there is a basis $e_{1}$, $e_{2}$ for the tangent space at p such that $df_{p}(e_{i}).df_{p}(e_{j})=e_{i}.e_{j}$ fo i,j=1,2, where $df_{p}$ is the differential of f at p. I thought of getting the basis vectors from a co-ordinate chart.

Thanks
 
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Poirot

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Feb 15, 2012
250
I should be clearer; I want to know what co-ordinate chart I should use to yield the basis vectors for the tangent space to the domain at (r,x).