Lagrangian submanifold

An $n$-dimensional submanifold of a $2n$-dimensional symplectic manifold is Lagrangian exactly when it is isotropic. Equivalently,

for every $x\in L$. The dimension hypothesis matters: lower-dimensional isotropic submanifolds are not Lagrangian, even though the symplectic form also restricts to zero on them.

The zero section of a cotangent bundle $T^*Q$, equipped with its canonical symplectic form, is Lagrangian. More generally, the graph of a one-form $\alpha$ on $Q$ is Lagrangian precisely when $d\alpha=0$; it is the graph of $df$ when $\alpha$ is exact. The Lagrangian neighborhood theorem says a neighborhood of any Lagrangian is symplectomorphic to a neighborhood of the zero section in $T^*L$ Cannas da Silva, §2.3.

The graph of a symplectomorphism $M\to N$ is Lagrangian in $M^{-}\times N$, where $M^{-}$ carries $-\omega_M$. This converts symplectic maps into Lagrangian correspondences. A smooth curve in a symplectic surface is automatically Lagrangian. By contrast, a symplectic submanifold of positive dimension cannot be Lagrangian because its restricted form is nondegenerate rather than zero.

Some literature permits immersed Lagrangians, defining them by a Lagrangian immersion rather than an embedded image. Unless explicitly qualified, this knowl uses embedded submanifolds. “Maximal isotropic” here means maximal by dimension in a symplectic vector space; maximality by set inclusion without regularity assumptions is not a substitute for the smooth half-dimensional condition.

1. Ana Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer, 2001. DOI record. Relevant: §2.3, Lagrangian submanifolds and the neighborhood theorem.
2. Dusa McDuff and Dietmar Salamon, Introduction to Symplectic Topology, 3rd ed., Oxford University Press, 2017. DOI record. Relevant: Chapter 3, Lagrangian submanifolds.