= Symplectic topology
<Symplectic topology> studies global properties of <symplectic manifolds> and maps preserving their <symplectic forms>. Local coordinates put every <symplectic form> in the same standard form, by the <Symplectic Darboux theorem>; global embedding and deformation problems can nevertheless have rigid obstructions.
A striking example is the <Gromov non-squeezing theorem>. In $\mathbb R^{2n}$ with its standard <symplectic form>, the ball $B^{2n}(R)$ embeds symplectically into the cylinder $B^2(r)\times\mathbb R^{2n-2}$ only if $R\le r$. For $n\ge2$ the cylinder has infinite <volume>, so ordinary volume comparison cannot explain the obstruction. The selected two-dimensional plane is a conjugate coordinate plane. Tools such as <J-holomorphic curves> connect this global rigidity to <topology> and <Hamiltonian mechanics>.<Pseudo holomorphic curves in symplectic manifolds by Mikhail Gromov>[section 0.3.A, Corollary, printed page 310]{ref}
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