Lagrangian submanifold

Summary

A Lagrangian submanifold of a symplectic manifold is a submanifold of half the dimension on which the symplectic form vanishes. Lagrangian submanifolds are the largest submanifolds on which the form vanishes identically. They include the zero section and the graphs of closed $1$-forms in cotangent bundles, and the graphs of symplectomorphisms. They are central objects of symplectic topology, Floer theory and mirror symmetry.