Lagrangian Submanifolds

Lagrangian submanifolds are the natural "half-dimensional" objects in symplectic geometry. They play a central role in classical mechanics (as configuration spaces), mirror symmetry (as objects of the Fukaya category), and microlocal analysis.

Basic Definitions

Definition6.1Lagrangian Submanifold

A submanifold $L \subseteq (M^{2n}, \omega)$ is Lagrangian if $\dim L = n$ and $\omega|_L = 0$ (the restriction of $\omega$ to $L$ vanishes). Equivalently, $T_x L$ is a Lagrangian subspace of $(T_x M, \omega_x)$ for every $x \in L$ : $T_x L = (T_x L)^\omega$ , where $(T_x L)^\omega = \{v : \omega(v, w) = 0 \, \forall w \in T_x L\}$ .

Definition6.2Special Lagrangian

On a Calabi-Yau manifold $(M, \omega, \Omega)$ with holomorphic volume form $\Omega$ , a Lagrangian $L$ is special Lagrangian if $\mathrm{Im}(\Omega)|_L = 0$ . Special Lagrangians are volume-minimizing in their homology class and play a central role in the SYZ conjecture of mirror symmetry.

ExampleZero Section of $T^*Q$

The zero section $Q \hookrightarrow T^*Q$ is Lagrangian in the cotangent bundle with its canonical symplectic form $\omega = -d\theta = \sum dp_i \wedge dq_i$ . More generally, the graph of any closed 1-form $\alpha$ on $Q$ is Lagrangian: $\mathrm{Graph}(\alpha) = \{(q, \alpha_q) : q \in Q\}$ .

Nearby Lagrangians

RemarkWeinstein Neighborhood Theorem

The Weinstein Lagrangian neighborhood theorem states that a neighborhood of any Lagrangian $L$ in $(M, \omega)$ is symplectomorphic to a neighborhood of the zero section in $(T^*L, \omega_{\text{can}})$ . This means the local symplectic geometry near a Lagrangian is completely determined by the Lagrangian itself.

Definition6.3Exact and Monotone Lagrangians

In an exact symplectic manifold $(M, d\lambda)$ , a Lagrangian $L$ is exact if $\lambda|_L = df$ for some $f \in C^\infty(L)$ . A Lagrangian in a symplectic manifold is monotone if $[\omega]|_{\pi_2(M,L)} = c \cdot \mu|_{\pi_2(M,L)}$ for a positive constant $c$ , where $\mu$ is the Maslov class.