Symplectic Manifold - Lagrangian Mapping

Lagrangian Mapping

Let L be a Lagrangian submanifold of a symplectic manifold (K,ω) given by an immersion i : L ↪ K (i is called a Lagrangian immersion). Let π : K ↠ B give a Lagrangian fibration of K. The composite (π ○ i) : L ↪ K ↠ B is a Lagrangian mapping. The critical value set of π ○ i is called a caustic.

Two Lagrangian maps (π₁ ○ i₁) : L₁ ↪ K₁ ↠ B₁ and (π₂ ○ i₂) : L₂ ↪ K₂ ↠ B₂ are called Lagrangian equivalent if there exist diffeomorphisms σ, τ and ν such that both sides of the diagram given on the right commute, and τ preserves the symplectic form. Symbolically:

where τ*ω₂ denotes the pull back of ω₂ by τ.