\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\CritVal{\operatorname{CritVal}}
\def\FS{\operatorname{FS}}
\def\Sing{\operatorname{Sing}}
\def\Coh{\operatorname{Coh}}
\def\Vect{\operatorname{Vect}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
SympSnip: examples of Lagrangian submanifolds
examples of Lagrangian submanifolds
For symplectic surfaces, we have a complete classification of the Lagrangian submanifolds.
example 0.0.1
If \(n=1\), every curve is Lagrangian.
Some Lagrangian submanifolds can be constructed easily from Lagrangian submanifolds in lower dimensions.
example 0.0.2
The product torus is the Lagrangian torus in \(\CC^n\) given by
\[T^n=\{|z|\in \CC^n\;\text{ such that }\; |z_i|=1\}.\]
To prove that this is a Lagrangian submanifold we first observe that whenever \(L_1\subset (X_1, \omega_1)\) and \(L_2\subset (X_2, \omega_2)\) are Lagrangian submanifolds, then \(L_1\times L_2\subset (X_1\times X_2, \omega_1+\omega_2)\) is a Lagrangian submanifold as well.
The product torus comes from taking the product of circles \(S^1\subset \CC\), which are Lagrangian submanifolds from example 0.0.1.
We will later see that \(T^*Q\) gives a local model of a symplectic manifold. For this reason, Lagrangian submanifolds of \(T^*Q\) serve as local models for Lagrangian submanifolds in general \(X\).
example 0.0.3
Let \(Q\) be a smooth manifold, let \(\alpha\) be a \(1\)-form on \(X\).
Let \(L\) be the graph of \(\alpha\) in \(T^*Q\),
\[\{L:=(q, \alpha(q))\}\subset T^*Q.\]
\(L\) is a Lagrangian submanifold if and only if \(d\alpha=0\).
We parameterize the Lagrangian \(L\) by with the map
\begin{align*}
\li: Q\into& T^*Q &&
q\mapsto&(q, \alpha(q))
\end{align*}
We now wish to show that \(i^*\omega=0\).
Let \((q_1, \ldots q_n)\) be local coordinates on \(Q\), so that \(\alpha(q)=\sum_{k=1}^n \alpha_k(q)dq_{k}\).
Let \(\partial_{q_i}, \partial_{q_j}\) be two basis vectors for the tangent space of \(Q\).
Let \(\{(\partial_{q_i}, 0)\}\cup\{(0, \partial p_i)\}\) be a basis for the tangent space of \(T^*Q\).
\begin{align*}
\li^*\omega(\partial_{q_i}, \partial_{q_j})=& \omega \left(\left(\partial_{q_i}, \sum_{k=1}^n (\partial_{q_i}\alpha_k )\cdot \partial_{p_k}\right), \left(\partial_{q_j}, \sum_{k=1}^n (\partial_{q_j} \alpha_k)\cdot \partial_{p_k}\right)\right)\\
=&\left( \sum_{l=1}^n dp_l \wedge dq_l\right) \left(\left(\partial_i, \sum_{k=1}^n (\partial_{q_i}\alpha_k )\cdot \partial_{p_k}\right), \left(\partial_{q_j}, \sum_{k=1}^n (\partial_{q_j} \alpha_k )\cdot\partial_{p_k}\right)\right)\\
\end{align*}
Since \(dq_i\partial_{q_j}=\delta_{ij}\) and \(dp_i\partial_{p_j}=\delta_{ij}\)
\begin{align*}
=& \partial_{q_j} \alpha_i - \partial_{q_i} \alpha_j\\
=& d\alpha(\partial_{q_i}, \partial_{q_j})
\end{align*}
This vanishes for all \(i, j\) if and only if \(\alpha\) is closed.
In the setting of cotangent bundles we can also build local models for the intersections of Lagrangian submanifolds.
example 0.0.4
Let \(V\subset Q\) be a smooth submanifold. The conormal bundle \(N*V\subset T^*Q\) consists of all covectors \((q, p)\in T^*Q\) with \(q\in V\) and \(p(v)=0\) for all \(v\in TV\).
This is always an \(n\)-dimensional submanifold of \(T^*Q\).
We can choose local coordinates \(q_1, \ldots, q_n, p_1, \ldots p_k\) so that \(V=\{(0, \ldots, 0, q_{k+1}, \ldots, q_n)\;|\; q_i\in \RR\}\).
In these local coordinates, \(N^*V=\{(0, \ldots, q_{k+1}, \ldots, q_n, p_1, \ldots, p_k, 0, \ldots, 0)\}\); which is a Lagrangian linear subspace for the symplectic form \(\sum_{i=1}^n dq_i \wedge dp_i\).
The intersection of the zero section and a conormal bundle is \(Q\cap N^*V=V.\)