Symplectic cut

In mathematics, specifically in symplectic geometry, the symplectic cut is a geometric modification on symplectic manifolds. Its effect is to decompose a given manifold into two pieces. There is an inverse operation, the symplectic sum, that glues two manifolds together into one. The symplectic cut can also be viewed as a generalization of symplectic blow up. The cut was introduced in 1995 by Eugene Lerman, who used it to study the symplectic quotient and other operations on manifolds.

Let ${\displaystyle (X,\omega )}$ be any symplectic manifold and

a Hamiltonian on ${\displaystyle X}$. Let ${\displaystyle \epsilon }$ be any regular value of ${\displaystyle \mu }$, so that the level set ${\displaystyle \mu ^{-1}(\epsilon )}$ is a smooth manifold. Assume furthermore that ${\displaystyle \mu ^{-1}(\epsilon )}$ is fibered in circles, each of which is an integral curve of the induced Hamiltonian vector field.

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