Lagrangian submanifold

Let $(M,\omega)$ be a symplectic manifold of dimension $2n$ . An embedded submanifold $L\subset M$ is a Lagrangian submanifold if $\dim L=n$ and the pullback of $\omega$ along the inclusion $i:L\hookrightarrow M$ vanishes:

i^*\omega=0.

Equivalently, every tangent space $T_xL$ is a Lagrangian subspace of the symplectic vector space $T_xM$ . A Lagrangian embedding is an embedding $f:L\to M$ with $f^*\omega=0$ and $\dim L=\frac12\dim M$ . The vanishing condition is pointwise: every pair of tangent vectors to the submanifold has zero symplectic pairing. Both vanishing and the half-dimension condition are essential.

Equivalent formulations

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

(T_xL)^\omega=T_xL

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.

Standard local model

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 $T^*L$ .

Constructions and examples

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.

Conventions and scope

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.

References

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.
Dusa McDuff and Dietmar Salamon, Introduction to Symplectic Topology, 3rd ed., Oxford University Press, 2017. DOI record. Relevant: Chapter 3, Lagrangian submanifolds.