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?

  • 2
    $\begingroup$ We can even assume $\|x\|=1$, then it will be $\pm|v\times u|$. $\endgroup$ – Berci Feb 13 at 22:08
  • $\begingroup$ Is there a reason it is stated in this form? I mean of course you can but is there a reason why you don't just write $\sigma(u,v)=\vert v \times v \vert$? $\endgroup$ – User1 2 days ago
  • $\begingroup$ Do you mean the $\pm$ sign? Note that $v\times u=-u\times v$, so $\sigma(v,u)=-\sigma(u,v)$. $\endgroup$ – Berci 2 days ago

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