I have a question about the standard symplectic form on the 2-sphere:
$$(\sigma_{st})_x(v,u):= \langle x,v \times u\rangle$$
If I think about this geometrically, the cross product of two tangent vectors on $S^2$ in the same point $x$ would be parallel to the vector $x$, correct?
So $$\langle x,v \times u\rangle=\vert x \vert \vert v \times u \vert.$$
Is that correct?