I have been given a the Riemannian metric of a sphere of radius R in stereographic coordinates: $$G=4R^4\frac{du^2+dv^2}{(R^2+u^2+v^2)^2}.$$ I have shown that this metric is preserved under rotation, but now I must give an example of a non-linear transformation of coordinates (u,v) so that the metric G is preserved.
Any help would be much appreciated!
In stereographic coordinates, the isometries of a sphere become Möbius transformations. The simplest nonlinear Möbius transformation is inversion, reflection about a circle. But one has to get the radius of the circle right. Instead of guessing, look for the longest circle centered at the origin: since $u^2+v^2=r^2$ has length $\dfrac{4R^2 }{(R^2+r^2)^2} (2\pi r)^2$, the maximum is attained at $r=R$. So, $$(u,v)\mapsto \frac{R^2}{u^2+v^2}(u,v)$$ is the transformation you want. (You still have to check it'san isometry, of course.)