$ω_p(u,v):= \langle p, u × v\rangle$ defines symplectic form on $S^2\subset \mathbf{R}^3$ for all $u,v\in T_p S^2$.
How to show that it is in fact volume form on $S^2$?
Any help is welcome. Thanks in advance.
Symplectic form on $S^2$
1
$\begingroup$
analysis
differential-geometry
differential-forms
symplectic-geometry
1 Answers
4
If $J$ denotes the complex structure $Ju = p \times u$ induced by the cross product, $u$ is a unit tangent vector at $p$, and $v = Ju$, then $u \times v = p$, so $\omega_{p}(u, v) = \langle p, p\rangle = 1$.