Homotopy between Manifold
A homotopy from a smooth function
to a smooth function is a smooth function such that and for all . A homotopy is an isotopy if is an embedding for all .
Lagrangian Submanifold
A Lagrangian submanifold of a symplectic manifold
is a submanifold such that and . In other words, the restriction of the symplectic form to vanishes.
Moser’s Theorem
Moser Relative Theorem
Let
be a manifold, a compact submanifold of , the inclusion map, and symplectic forms on such that . Then there exists neighbourhoods , of and a diffeomorphism such that the following diagram commutes: and
.
Darboux Theorem
Let
be a symplectic manifold, and let be any point in . Then we can find a coordinate on containing such that
Proof Apply the Moser relative theorem to