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May 11th, 2013, 11:33 AM   #1
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metric space Isometry

Notes: $R^2_{+}={(x,y):y>0=$ and the metric riemannian g at any point (x,y) is given by $g(u,v)=\frac{u\cdot v}{y^2}$
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 May 11th, 2013, 11:42 AM #2 Senior Member   Joined: Feb 2012 Posts: 628 Thanks: 1 Re: metric space Isometry Does the map $(x, y) \rightarrow \left(\frac{9x}{x^2 + y^2}, \frac{9y}{x^2 + y^2}\right)$ work?
 May 11th, 2013, 11:58 AM #3 Newbie   Joined: Dec 2012 Posts: 13 Thanks: 0 Re: metric space Isometry How did you find that map?
 May 11th, 2013, 12:04 PM #4 Senior Member   Joined: Feb 2012 Posts: 628 Thanks: 1 Re: metric space Isometry It's an inversion about the circle $x^2 + y^2= 9$. It's the only map that I know of that preserves the points on the circle but is not the identity. Basically, an inversion takes points outside the circle and puts them inside the circle, and takes points inside the circle and puts them outside the circle. It also preserves angles from the origin. The distance (calculated the normal way) from any point to the origin before the transformation is r, and after the transformation is $9/r$.

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