Find explicit symplectomorphism

Question

How to solve the following:

Find explicit symplectomorphism between (2-sphere of radius r without one point) and (open disc of radius 2r)?

I don't have any idea how to solve this, so any help is welcome.

Thanks in advance.

Answer 1

Here is a proof strategy.

Fix $r > 0$. Let $S = \{ (x,y,z) \in \mathbb{R}^3 \, : \, x^2+y^2+z^2 = r^2 \}$, $n = (0,0,r)$ and $s = (0,0,-r)$. Equip $S$ with the (symplectic) 2-form $\omega$ induced by its inclusion into 3-space i.e. for $x \in S$ and $V, W \in T_xS$, set $\omega_x(V,W) = r \left( x \cdot (V \times W) \right) = r^2 \left( \frac{x}{\|x\|} \cdot (V \times W) \right)$ where $\cdot$ is the standard scalar product in 3-space and $\times$ is the standard vector product in 3-space .

Let $T = \{(x,y) \in \mathbb{R}^2 \, : \, x^2+y^2=r^2 \}$. Observe that the 'volume' 1-form on $T$ is $r d\theta$ where $\theta \in [0, 2\pi)$. Consider the cylinder $C = T \times (-r, r)$ equipped with the (symplectic) 2-form $\Omega = r d\theta \wedge dz$, where $z \in (-r, r)$.

1) Inspired by Archimedes, construct an explicit symplectomorphism between $S \setminus \{n,s\}$ equipped with $\omega$ and the cylinder $C$ equipped with $\Omega$. (Hint: Use cylindrical coordinates on $S$ and express $\omega$ in these coordinates.)

2) Construct an explicit symplectomorphism between $(C, \Omega)$ and the punctured disk $(D \setminus \{(0,0)\}, dx \wedge dy)$ where $D := \{(x,y) \in \mathbb{R}^2 \, : \, x^2+y^2 < (2r)^2 \}$. (Hint: Use polar coordinates on $D$ and express $dx \wedge dy$ in these coordinates.)

Hence you have obtained an explicit symplectic diffeomorphism from $D \setminus \{(0,0)\}$ to $S\setminus \{n, s \}$. It can be expressed in terms of euclidean coordinates on $D$ and on $S$. You only need to prove that the expressions for this map extend smoothly over $(0,0)$ (continuously is not enough) to a map $\Phi$ which covers $n$ or $s$ (the specific point depends on some choices you will have made). The symplectomorphism condition is automatically satisfied at $(0,0)$, as the 2-form $\Phi^*\omega - dx \wedge dy$ is continuous on $D$ and vanishes (by construction) outside $(0,0)$.