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The 2-sphere is given as example of symplectic manifolds, with a symplectic form [tex]\Omega = \sin{\varphi} d \varphi \wedge d \theta[/tex]. Here the parametrization is given by [tex](x,y,z) = (\cos{\theta}\sin{\varphi}, \sin{\theta}\sin{\varphi}, \cos{\varphi})[/tex] with [tex] \varphi \in [0,\pi],\ \theta \in [0, 2\pi) [/tex].

Now my question is, at the points [tex]\varphi = 0, \pi[/tex], which are the north and the south pole, is the one-form [tex]d \theta[/tex] well-defined? If yes, how? If not then how does one make [tex]\Omega[/tex] globally well defined?

Thanks in advance :)

Ram.

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# Symplectic form on a 2-sphere

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