Complementary areas obstruct symplectic equivalence of separating curves
= Complementary areas obstruct symplectic equivalence of separating curves
A <symplectomorphism> of a closed <symplectic surface> carries the complementary components of a separating simple closed curve to those of its image, preserving their unordered <symplectic areas>. On the unit <sphere> a latitude at height $h$ has complementary areas $2\pi(1-h)$ and $2\pi(1+h)$. Thus the equator cannot be sent to any nonequatorial latitude.