Symplectic Manifold - Lagrangian Fibration

A Lagrangian fibration of a symplectic manifold M is a fibration where all of the fibres are Lagrangian submanifolds. Since M is even dimensional we can take local coordinates (p₁,…,p_n,q₁,…,q_n), and by Darboux's theorem the symplectic form ω can be, at least locally, written as ω = ∑ dp_k ∧ dq_k, where d denotes the exterior derivative and ∧ denotes the exterior product. Using this set-up we can locally think of M as being the cotangent bundle T*Rn, and the Lagrangian fibration as the trivial fibration π : T*Rn ↠ Rn. This is the canonical picture.