How to prove Sphere $S^3$ minus point is conformal to submanifold with boundary of $\mathbb R\times (M,g)$ with $M$ is 2 dimensional compact simple manifold ?
Definition: A manifold $(M,g)$ with boundary is simple if $\partial M $ is strictly convex and for any point $x\in M$ the exponential map $exp_x$ is diffeomorphism form closed neibourhood of $0$ in $T_xM$ onto $M$.
I was reading one notes in that it is given that $S^3\setminus\{S\}$ where $S$ is South Pole is conformal to submanifold with boundary of $\mathbb R\times (M,g)$ with $M$ is 2 dimensional compact simple manifold ?
I thought that $S^3\setminus\{S\}$ and $R^3$ are conformal using stereographic projection. But How to write $R^3= R\times M$ where $M$ is 2 dimensional compact manifold (simple).
Is there any other conformal map exists? Any help will be appreciated.