Definition

Let be a -dimensional symplectic manifold. A submanifold is a Lagrangian submanifold if, at each point , is a Lagrangian subspace.

Equivalent definition: If is the inclusion map, then is Lagrangian if and only if and .

Lagrangian Submanifolds on Cotangent Bundles

For a smooth manifold , consider the cotangent bundle , with the tautological 1-form and the canonical symplectic form . Let be the section map given by a differential 1-form , and the image of . In this way, is a diffeomorphism. So, the following diagram commutes

Therefore,

The second to last line follows from the fact that pull backs along tautological 1-form preserves sections. So, Lagrangian submanifolds on are in 1-1 correspondence with the set of closed differential 1-forms on .

Let be a -dimensional submanifold, and the conormal bundle. Let be the inclusion map, and be the tautological 1-form. Then

which means that is a Lagrangian submanifold.

Let be a coordinate chart on adapted to , and be the associated cotangent coordinates. Thus, for , then

Therefore,