Symplectic Rigidity of Real Bidisc

Let $\mathbb{D}$ be the unit disc in $\mathbb{C}$, then ${\mathbb{D}}^n(r)$ is the complex or symplectic $n$-discs of radius $r$. Let $z_j = x_j+iy_j\in$\mathbb{C}$, j=1,2$and$$\mathbb{D}$_{$\mathbb{R}$}^2={(z_1,z_2) : |x_1|^2+|x_2|^2<1,|y_1|^2+|y_2|^2<1}$ be the real bidisc. In this paper we will prove the following two theorems: 1) If $T\in O(4)$ is an orthogonal transformation on ${\mathbb{R}}^4$, then $T({\mathbb{D}}^2)$ is symplectomorphic to ${\mathbb{D}}^2$ w.r.t. the standard symplectic form on ${\mathbb{R}}^4$ if and only if $T$ is unitary or conjugate to unitary. 2) For $r\ge 1$ and $n\ge 2$, ${\mathbb{D}}_{\mathbb{R}}^2\times {\mathbb{D}}^{n-2}(r)$ and $$\mathbb{D}$^2\times $\mathbb{D}$^{n-2}®$ are not symplectomorphic w.r.t. the standard symplectic form on ${\mathbb{C}}^n$.