When studying symplectic manifolds, we saw that symplectic manifolds are locally modeled on the standard symplectic space and therefore admit no local invariants. We now show that the cotangent bundle of a manifold \(Q\) gives a better local model for symplectic manifolds. In the cotangent bundle, \(Q\subset T^*Q\) is a half-dimensional submanifold on which the canonical symplectic form vanishes. Additionally, \(Q\) canonically determines the symplectic structure on \(T^*Q\). Lagrangian submanifolds are submanifolds of \(X\) which behave like the zero section \(Q\subset T^*Q\).

definition 0.0.1

Let \((X, \omega)\) be a symplectic manifold. A Lagrangian submanifold is a \(n\)-dimensional submanifold \(L\subset X\) such that \[\omega|_L=0.\] A Lagrangian manifold lies parallel to the symplectic form everywhere. The requirement that the dimension of \(L\) is half of \(\dim(X)\) is non-arbitrary --- this is the maximal dimension on which the symplectic form can vanish (due to the non-degeneracy of \(\omega\)).

1: examples of Lagrangian submanifolds

For symplectic surfaces, we have a complete classification of the Lagrangian submanifolds.

example 1.0.1

If \(n=1\), every curve is Lagrangian. Some Lagrangian submanifolds can be constructed easily from Lagrangian submanifolds in lower dimensions.

example 1.0.2

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 1.0.1. 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 1.0.3

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 1.0.4

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 \(Q\cap N^*V=V.\)

2: relations between Lagrangian submanifolds

In these notes, we will attempt to shed a little light on the construction and classification of Lagrangian submanifolds. Even in the simplest examples, this classification can become quite difficult. To give a meaningful answer to this question, we'll need a notion of equivalence between Lagrangian submanifolds. For submanifolds, a natural equivalence to consider is the isotopy class or the homotopy class. Let \(\li_t: L\times I\to X\) be a homotopy of Lagrangian submanifolds. We now describe the flux class of the homotopy, which is an element \(\Flux_{\li_t}\in H^1(L;\RR)\). To define this class we prescribe its values on chains of \(L\). Let \(c\in C_1(L)\) be a chain. Then the homotopy can be applied to \(c\) to give a 2-chain \(\li_t(c)\in C_2(X)\).

definition 2.0.1

The flux of a Lagrangian homotopy \(\li_t:L\times I\to X\) is the cohomology class defined by \[\Flux_{\li_t}(c):=\int_{\li_t(c)}\omega.\] To show that this is a cohomology class, we need to show that the flux homomorphism \(\Flux_{\li_t}\) vanishes on boundaries \(\partial b\in C_1(L_0)\) Without loss of generality, suppose that \(I=[0,1]\), and let \(b\in C_2(L)\) be a 2-chain. Then \(\li_t(b)\) similarly gives a 3-chain in \(X\) whose boundary components satisfy the relation \(\li_t(\partial b)=i_0(b)-i_1(b)+\partial(\li_t(b))\). Applying the definition of the flux homomorphism to this relation yields: \[ \Flux_{\li_t}(\partial b)=\int_{\li_t(\partial b)}\omega=\int_{i_0(b)}\omega-\int_{i_1(b)}\omega + \int_{\partial(\li_t(b))}\omega\] The first two terms vanish because \(i_0(b)\) and \(i_1(b)\) are subsets of a Lagrangian submanifold so \(\omega|_{\li_0(L)}=\omega|_{\li_1(L)}=0\). Because \(\omega\) is closed, an application of Stoke's theorem shows that the last term vanishes as well.

example 2.0.2

Let \(\CC^*\) be equipped with the symplectic form \(\omega=\frac{1}{2\pi} d(\log r) \wedge d\theta\). Let \(L_r\) be the Lagrangian \[L_r:=\{z \text{ such that } \log|z|=r\}.\] Let \(\li_r:L\times [r_0, r_1]\to X\) be the isotopy between \(L_{r_0}\) and \(L_{r_1}\). Let \(e\in H_1(L_r)\) be the fundamental class of \(L\). The amount of flux swept out between these two Lagrangian submanifolds is given by the difference of their \(r\)-values. \[\Flux_{i_r}(e)=\int_{S^1} \int_{r_0}^{r^1} \frac{1}{2\pi} d(\log |z|)\wedge d\theta = r_1-r_0.\] For this reason, the value \(r\) is sometimes called the ``flux coordinate'' of the fibration \(\CC^*\to \RR\). The flux class therefore provides a nice parameterization of the space of Lagrangian submanifolds in this example.

figure 2.0.3:A basic computation of flux swept out between two submanifolds of \(\CC^*\)

example 2.0.4

Consider the symplectic manifold \(X=T^*L\). Consider a closed one form \(\eta \in \Omega^1(L, \RR)\). Use this to create a Lagrangian isotopy \begin{align*} \li_t: L\times [0, 1]\to& T^*L && (q,t) \mapsto& t\cdot \eta_p \end{align*} between the zero section and the graph of \(\eta\). To each loop \(\gamma\in L\), look at the cylinder \(\li_t\circ \gamma:S^1\times I\to X\). This can be explicitly parameterized by \begin{align*} c: S^1\times I \to& T^*L&& (\theta, t) \mapsto& (\gamma(\theta), t\cdot \eta_{\gamma(\theta)}). \end{align*} The cotangent bundle has an exact symplectic form, \(\omega=d\lambda\), so computing the flux can be simplified by using Stoke's theorem: \begin{align*} \Flux_{\li_t}(\gamma)=&\int_{c}\omega=\int_{c}d\lambda \\ =&\int_{S^1\times \{1\}}c^*\lambda - \int_{S^1\times\{0\}}c^*\lambda \\ =&\int_{S^1}\gamma^*\eta=\eta(\gamma) \end{align*} So \([\Flux_{\li_t}]=[\eta]\in \Omega(L)\). This example highlights two important properties of Lagrangian submanifolds, and Lagrangian homotopy. First, there homotopies a stronger notion of equivalence for Lagrangian submanifolds beyond homotopy by looking at those homotopies which sweep out zero flux.

definition 2.0.5

We call \(i_t: L\times I \to X\) an exact homotopy if for every subinterval \(J\subset I\), the flux of the restriction is exact, \[[\Flux_{i_{t}}|_J]=0\in H^1(L).\] Similar to the case of Hamiltonian isotopies, exact isotopies of Lagrangian submanifolds can be characterized in terms of the Lagrangian isotopy itself. Let \(\li_t: L\times I\to X\) be a Lagrangian homotopy. This is a exact homotopy if there exists a Hamiltonian function \(H_t:L\to \RR\) such that for all \(v\in TL\), \[\omega\left(\frac{d}{dt}\li_t,v\right)=dH_t(v).\] We show one direction here, which is that \(\omega\left(\frac{d}{dt}\li_t,v\right)=dH_t(v)\) implies \(\Flux_{\li_t}\) vanishes in cohomology. Let \(c:S^1\to L\) be any 1-cycle in \(L\). Parameterize a 2-chain \(\li_t\circ c: S^1\times I \to X\) with coordinates \((\theta, t)\). The flux class applied to \(c\) can be explicitly computed: \begin{align*} \Flux_{\li_t}(c)=&\int_{\li_t\circ c}\omega =\int_{I \times S^1} (\li_t\circ c )^* \omega\\ =&\int_I \int_{S^1} c^*\circ (\li_t)^* \omega =\int_I \int_{S^1} (c^* \iota_{\frac{d}{dt}\li_t}\omega ) dt\\ =&\int_I \left(\int_{S^1} (c^* dH_t) \right)dt =\int_I \left(\int_{S^1} d(c^*H_t)\right) dt \end{align*} By Stoke's theorem, the integral of an exact form over the circle is zero. For the reverse direction, fix a base point \(x_0\in L\). For every point \(x\in L\), pick a path \(\gamma_x: [0,1]\to L\) with \(\gamma_x(0)=x_0\) and \(\gamma_x(1)=x\). Define the function \(H_t: L\to \RR\) by \[dH_t(x_1):=\int_{\li_t\circ \gamma} \omega.\] Because the flux of the isotopy is zero, this integral does not depend on the choices of paths \(\gamma_x\) and gives a well defined function on \(L\). We now show that this function generates the Lagrangian isotopy. The vector field \(\frac{d}{dt}\li_t\) is determined by the form \(\iota_{\frac{d}{dt}\li_t}\omega\). Since \(\iota_{\frac{d}{dt}\li_t}\) is closed, \begin{align*} \int_{\gamma_x} \iota_{\frac{d}{dt}\li_t}\omega =& \end{align*} \todo{We now check that this these two things match up by computing the vector field.} This shows that when \(L\) is embedded, a time dependent Hamiltonian \(H_t: L\times (-\epsilon, \epsilon)\to \RR\) defines an isotopy of embeddings \(\li_t: L\times(-\epsilon, \epsilon)\to X\) for small values of \(t\). We say this is the isotopy generated by the Hamiltonian \(H\). Provided that \(L\) is compact, we immediately get the following statement on the non-displacibility of Lagrangian submanifolds by small exact Lagrangian homotopies.

proposition 2.0.6

Let \(H: L\to \RR\) be a non-time dependent Hamiltonian . Let \(\li_t: L\times[0, c]\to X\) be the isotopy generated by \(H\). There exists \(\epsilon>0\) such that for all \(t_0, t_1\in [0, \epsilon]\), \begin{align*} |\li_{t_0}(L)\cap \li_{t_1}(L)|\geq \sum_{i=1}^n b_i(L). \end{align*} Let \(\Crit(H)\) be the set of critical points of \(H\). From , whenever \(x\in \Crit(H)\), \(\frac{d\li}{dt}(x)=0\) and so \(\li_{t_0}(x)=\li_{t_1}(x)\). Therefore, \(|\li_{t_0}(L)\cap \li_{t_1}(L)|\subset\Crit(H)\). The proposition follows from the Morse inequalities. In fact, this can extends to the case of time-dependent Hamiltonians for sufficiently small \(t\). Let \(H_t: L\to \RR\) generate a Hamiltonian isotopy. Then there exists \(\epsilon>0\) such that for all \(t_0, t_1\in [0,\epsilon)\), \[|\li_{t_0}(L)\cap \li_{t_1}(L)|\geq \sum_{i=1}^n b_i(L).\] Secondly, every cohomological class is realizable as a flux class in the cotangent bundle. This observation motivates the following theorem, which shows that this behavior occurs more generally.

theorem 2.0.7 [Wei71]

Let \(\li: L\to X\) be a compact Lagrangian submanifold of \((X, \omega)\). Then there exists a neighborhood of the zero section \(T^*_\epsilon L\subset TL\) and a symplectic embedding \(\phi: T^*_\epsilon L\to X\) which agrees on the zero section, in the sense that \[\phi|_L=\li.\]In other words, a symplectic neighborhood of a Lagrangian submanifold is determined by the diffeomorphism class of the Lagrangian. This local model will be useful later, as many of our constructions involving Lagrangian submanifolds will involve restricting to a Weinstein neighborhood, performing the construction there, and then implanting our construction into the original symplectic manifold. An example of such a construction is the following.

lemma 2.0.8

Let \(L\subset T^*Q\) be a Lagrangian submanifold. Suppose that the intersections between \(L\) and \(Q\) are transverse. Then there exists a Hamiltonian isotopy \(\phi\) of \(L\) so that \[\phi(L)\cap B^*_\epsilon Q=\bigcup_{p\in L\cap Q} T^*_pQ.\] At each \(q\in L\cap Q\), consider the Lagrangian submanifold \(T^*qQ\). Take local coordinates \((q_1, \ldots, q_n, p_1, \ldots, p_n)\) identifying \(q\) with the origin so that \(T^*qQ\) is the linear subspace in the \(p_i\) directions. We can take a Weinstein neighborhood \(T^*_\epsilon(T^*_qQ)\) of \(T^*_qQ\), whose cotangent bundle structure is \begin{align*} T^*_\epsilon(T^*_qQ)\to T^*qQ && (q_1, \ldots, q_n, p_1, \ldots, p_n)\mapsto (p_1, \ldots, p_n). \end{align*} Since the intersection \(L\cap Q\) is transverse, the projection \(T_qL\to T_q(T^*qQ)\) is surjective. Therefore when restricted to a small enough neighborhood of \(\in U\subset T^*_qQ\) the Lagrangian \(L|_{T^*_\epsilon U}\) presents itself as a section of \(T^*_\epsilon U\to U\). Therefore, there exists a one form \(\eta\in \Omega^1(U)\) so that \(L|_{T^*U}\) is parameterized by \((p, \eta_p)\). By taking an even smaller \(U\), we may assume that \(U\) is a contractible neighborhood, and \(\eta=dH\) is an exact one-form. Pick \(\rho\) a function which vanishes in a neighborhood of \(q\in U\), and takes the value \(1\) in a neighborhood of \(\partial U\). Consider the Lagrangian section of \(T^*U\) parameterized by \(d(\rho \cdot H)\). This section is Hamiltonian isotopic to \(L|_{T^*_\epsilon U}\) relative boundary. Additionally, \(d(\rho\cdot H)\) agrees with \(U=T^*_qQ\) in a small neighborhood of \(q\). The Lagrangian submanifold \(L\setminus (L|_{T^*U})\cup (d(\rho\cdot H))\) is Hamiltonian isotopic to \(L\).

3: exercises on Lagrangian submanifolds

exercise 3.0.1

Prove example 1.0.3 without using coordinates. Try showing that \(\omega|_L=\pm d\alpha\).

exercise 3.0.2

Show that there are no Lagrangian spheres inside of \(\CP^2\).

exercise 3.0.3

Prove the converse of , showing that every Hamiltonian Lagrangian isotopy is generated by the flow of a function \(H_t: L\times \RR\to \RR\).

exercise 3.0.4

Let \(M\) be an oriented manifold, and \(H\subset M\) be an oriented hypersurface. Let \(X=T^*M\) be the cotangent bundle equipped with the canonical symplectic form.

Show that the conormal bundle \(N^*H\subset T^*M\) is a Lagrangian submanifold.
Let \(B_\epsilon(H)\subset X\) be a small neighborhood of \(H\subset M\subset T^*M\) . Find an embedded Lagrangian submanifold \(L\subset T^*M\) which, outside of a small neighborhood \(B_\epsilon(H)\subset X\), agrees with \(N^*H\cup M\), \[ L\setminus (B_\epsilon(H))=(N^*H\cup M)\setminus H.\]

exercise 3.0.5

Let \(\Sigma\) be a symplectic manifold with \(\dim_\RR(\Sigma)=2\). Let \(L_0, L_1\subset \Sigma\) be two Lagrangian submanifolds which intersect transversely at points \(p_i\). Show that there exists a Lagrangian submanifold \(L'\subset X\) which agrees with \(L_0, L_1\) outside a small neighborhood of the \(p_i\): \[L'\setminus \left(\bigcup_{i} B_\epsilon(p_i)\right)=(L_0\cup L_1) \setminus \left(\bigcup_{i} B_\epsilon(p_i)\right).\]

exercise 3.0.6

Suppose that \(\alpha\) is a closed one form. Find a Hamiltonian isotopy of \(T^*N\) for which sends the zero section to \(\alpha\).

exercise 3.0.7

Let \(H_t: L\times [0, c]\to \RR\) be a time-dependent Hamiltonian. Suppose for each \(t_0\) the function \(H_{t_0}: L\to \RR\) is Morse. Let \(\li_t: L\to X\) be the induced isotopy. Show that for each critical point \(x\in \Crit(H_0)\), there is a sequence of points and times \(x_i\in L\) , \(t_i\in [0, c]\) so that \(\lim_{i\to\infty} t_i=0\), \(\li_{t_i}(x_i)\subset\li_{t_i}(L)\cap \li_{0}(L)\) and \(\lim_{i\to\infty}\li_{t_i}(x_i)=x\).

exercise 3.0.8

Which circles on \(S^2\) can be displaced by Hamiltonian isotopy?

References

[Wei71]

Alan Weinstein. Symplectic manifolds and their Lagrangian submanifolds. Advances in Mathematics, 6(3):329--346, 1971.