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I am rather confused by the idea of "geodesic polar coordinates", so I hope someone would kindly explain it to me.

As far as my understanding goes, given a Riemannian metric $ds^2=E\,dx^2+2F\,dx\,dy+G\,dy^2$ we somehow have to find other variables $u,v$ so that $ds^2=du^2+G(u,v)\,dv^2$.

  1. How can I convert $ds^2={dr^2+r^2\,d\theta^2\over f(r)^2}$ where $f(r)$ is some function of $r$ into that form?
  2. Is there an "algorithm" to achieve this change of form?
  3. Is there some significant benefit in doing so?

Thank you.

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Geodesic polar coordinates $(\rho,\theta)$ are easy to define with exponencial map basic properties, only difference with normal coordinates $(u,v)$ is that you use polar coordinates in your domain. They are a simple change of variables like any others that you can define.

An algorithm is like any change of variables studied in multivariable calculus.

This change of variable to polar coordinates is very meaningful and give us some powerful theorems like the Gauss lemma (which say that $F(\rho,\theta)=0$).