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symplectic structure of tangent bundle of $\mathbb{S}^{n-1}$

Here is a formula for an explicit symplectomorphism $F$ from $T^*S^{n-1}$ to the affine quadric $\{\sum z_{j}^{2}=1\}$ in $\mathbb{C}^n$: $$ F(p,q) = \left(\frac{1+\sqrt{1+4|p|^2}}{2}\right)^{1/2} …
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