Symplectic surface

A symplectic surface is a two-dimensional symplectic manifold. Every embedded curve in it is a Lagrangian submanifold.

The symplectic area of a compact symplectic surface $(\Sigma ,\omega )$ is ${\int }_{\Sigma }\omega$, using the orientation determined by the symplectic form.

Two compact connected symplectic surfaces are symplectomorphic exactly when their genus of a surface and symplectic area agree. The sufficiency follows from Moser theorem applied to a positive interpolation of their symplectic forms after an orientation-preserving diffeomorphism.

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.