So I read that the unique invariant of symplectic surfaces is the total area, i.e. two surfaces are symplectomorphic iff their area is the same. Consider $S^2$ with polar coordinates $(h,\theta)$ wherever these exist and the symplectic forms $\omega_1=dh\wedge d\theta$ and $\omega_2=-dh\wedge d\theta$. These should be non-symplectomorphic as the areas are not the same. But the reflection $\phi:(h,\theta)\rightarrow (-h,\theta)$ gives $\phi^*(\omega_1)=\omega_2$. What am I missing here?
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What are the areas you get using $\omega_1$ and $\omega_2$? – Michael Albanese Mar 17 '20 at 15:26
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If I take $h \in [0,1]$ and $\theta \in [0,2\pi)$ I get $2\pi$ and $-2\pi$. I think... But I am not used to working with two forms – confusedmathstudent Mar 17 '20 at 15:40
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Does it make sense for an area to be negative? – Michael Albanese Mar 17 '20 at 15:51
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Oh, I see. So "area" is not oriented... Thanks for your help! One last question - is this always true? I mean, for general symplectic manifolds are $\omega$ and $-\omega$ symplectomorphic? – confusedmathstudent Mar 17 '20 at 16:41
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Not in general no. See this question. – Michael Albanese Mar 17 '20 at 17:01