Question : How to show that $(\mathbb{R}^2,\delta)$ is conformal related with $(\mathbb{S}^2,\sigma)$ ? Here we have
1.1) $\delta$ is just the standard euclidian metric in spherical coordinates;
1.2) $\sigma$ is the standard metric of $\mathbb{S}^2$.
Example I know how to do the compactification of $\mathbb{R}^3$ into $\mathbb{S}^3$. We have
2.1) $\sigma=d\psi\otimes d\psi +\sin^2(\psi)d\theta \otimes d\theta + \sin^2(\psi)\sin^2(\theta)d\phi \otimes d\phi$
2.2) $\delta= dr \otimes dr+r^2 d\theta \otimes d\theta + r^2 \sin^2(\theta)d\phi \otimes d\phi$
Thus we set $\sigma=\omega^2 \delta$ where $\omega$ is the conformal factor and solve to $\omega,r(\psi)$
Problem I tried this same idea to solve my question, but it did not work. I appreciate any help, thanks!