Editing Symplectic manifold (section)

== Lagrangian and other submanifolds ==
There are several natural geometric notions of [[submanifold]] of a symplectic manifold <math> (M, \omega) </math>:

* '''Symplectic submanifolds''' of <math> M </math> (potentially of any even dimension) are submanifolds <math> S \subset M </math> such that <math> \omega|_S </math> is a symplectic form on <math> S </math>.
* '''Isotropic submanifolds''' are submanifolds where the symplectic form restricts to zero, i.e. each tangent space is an [[isotropic subspace]] of the ambient manifold's tangent space. Similarly, if each tangent subspace to a submanifold is co-isotropic (the dual of an isotropic subspace), the submanifold is called '''co-isotropic'''.
* '''Lagrangian submanifolds''' of a symplectic manifold <math>(M,\omega)</math> are submanifolds where the restriction of the symplectic form <math>\omega</math> to <math>L\subset M</math> is vanishing, i.e. <math>\omega|_L=0</math> and <math>\text{dim }L=\tfrac{1}{2}\dim M</math>. Lagrangian submanifolds are the maximal isotropic submanifolds.  

One major example is that the graph of a [[symplectomorphism]] in the product symplectic manifold {{nowrap|1=(''M'' × ''M'', ''ω'' × −''ω'')}} is Lagrangian. Their intersections display rigidity properties not possessed by smooth manifolds; the [[Arnold conjecture]] gives the sum of the submanifold's [[Betti number]]s as a lower bound for the number of self intersections of a smooth Lagrangian submanifold, rather than the [[Euler characteristic]] in the smooth case.

=== Examples ===
Let <math>\R^{2n}_{\textbf{x},\textbf{y}}</math> have global coordinates labelled <math>(x_1, \dotsc, x_n, y_1, \dotsc, y_n)</math>. Then, we can equip <math>\R_{\textbf{x},\textbf{y}}^{2n}</math> with the canonical symplectic form 

:<math>\omega =\mathrm{d}x_1\wedge \mathrm{d}y_1 + \dotsb + \mathrm{d}x_n\wedge \mathrm{d}y_n.</math> 

There is a standard Lagrangian submanifold given by <math>\R^n_{\mathbf{x}} \to \R^{2n}_{\mathbf{x},\mathbf{y}}</math>. The form <math>\omega</math> vanishes on <math>\R^n_{\mathbf{x}}</math> because given any pair of tangent vectors <math>X= f_i(\textbf{x}) \partial_{x_i}, Y=g_i(\textbf{x})\partial_{x_i},</math> we have that <math>\omega(X,Y) = 0.</math> To elucidate, consider the case <math>n=1</math>. Then, <math>X = f(x)\partial_x, Y=g(x)\partial_x,</math> and <math>\omega = \mathrm{d}x\wedge \mathrm{d}y</math>. Notice that when we expand this out

:<math>\omega(X,Y) = \omega(f(x)\partial_x,g(x)\partial_x) = \frac{1}{2}f(x)g(x)(\mathrm{d}x(\partial_x)\mathrm{d}y(\partial_x) - \mathrm{d}y(\partial_x)\mathrm{d}x(\partial_x))</math>

both terms we have a <math>\mathrm{d}y(\partial_x)</math> factor, which is 0, by definition.

====Example: Cotangent bundle====
The cotangent bundle of a manifold is locally modeled on a space similar to the first example. It can be shown that we can glue these affine symplectic forms hence this bundle forms a symplectic manifold. A less trivial example of a Lagrangian submanifold is the zero section of the cotangent bundle of a manifold. For example, let 

:<math>X = \{(x,y) \in \R^2 : y^2 - x = 0\}.</math>

Then, we can present <math>T^*X</math> as

:<math>T^*X = \{(x,y,\mathrm{d}x,\mathrm{d}y) \in \R^4 : y^2 - x = 0, 2y\mathrm{d}y - \mathrm{d}x = 0\}</math>

where we are treating the symbols <math>\mathrm{d}x,\mathrm{d}y</math> as coordinates of <math>\R^4 = T^*\R^2</math>. We can consider the subset where the coordinates <math>\mathrm{d}x=0</math> and <math>\mathrm{d}y=0</math>, giving us the zero section. This example can be repeated for any manifold defined by the vanishing locus of smooth functions <math>f_1,\dotsc,f_k</math> and their differentials <math>\mathrm{d}f_1,\dotsc,df_k</math>.

====Example: Parametric submanifold ====
Consider the canonical space <math>\R^{2n}</math> with coordinates <math>(q_1,\dotsc ,q_n,p_1,\dotsc ,p_n)</math>. A parametric submanifold <math>L</math> of <math>\R^{2n}</math> is one that is parameterized by coordinates <math>(u_1,\dotsc,u_n)</math> such that 
:<math>q_i=q_i(u_1,\dotsc,u_n) \quad p_i=p_i(u_1,\dotsc,u_n)</math>

This manifold is a Lagrangian submanifold if the [[Lagrange bracket]] <math>[u_i,u_j]</math> vanishes for all <math>i,j</math>. That is, it is Lagrangian if 

:<math>[u_i,u_j]=\sum_k \frac {\partial q_k}{\partial u_i}\frac {\partial p_k}{\partial u_j} 
- \frac {\partial p_k}{\partial u_i}\frac {\partial q_k}{\partial u_j} 
= 0</math>  

for all <math>i,j</math>.  This can be seen by expanding
:<math>
\frac {\partial }{\partial u_i}=
\frac {\partial q_k}{\partial u_i} \frac {\partial}{\partial q_k} 
+ \frac {\partial p_k}{\partial u_i} \frac {\partial}{\partial p_k} 
</math>
in the condition for a Lagrangian submanifold <math>L</math>. This is that the symplectic form must vanish on the [[tangent manifold]] <math>TL</math>; that is, it must vanish for all tangent vectors:
:<math>\omega\left( \frac {\partial}{\partial u_i}, \frac {\partial}{\partial u_j} \right)=0</math>
for all <math>i,j</math>. Simplify the result by making use of the canonical symplectic form on <math>\R^{2n}</math>:

:<math>
\omega\left( \frac {\partial }{\partial q_k}, \frac {\partial}{\partial p_k}\right) =
-\omega\left( \frac {\partial }{\partial p_k}, \frac {\partial}{\partial q_k}\right) = 1 
 </math>
and all others vanishing.

As [[Chart (topology)|local charts]] on a symplectic manifold take on the canonical form, this example suggests that Lagrangian submanifolds are relatively unconstrained. The classification of symplectic manifolds is done via [[Floer homology]]—this is an application of [[Morse theory]] to the [[Action (physics)|action functional]] for maps between Lagrangian submanifolds. In physics, the action describes the  time evolution of a physical system; here, it can be taken as the description of the dynamics of branes.

====Example: Morse theory====
Another useful class of Lagrangian submanifolds occur in [[Morse theory]]. Given a [[Morse function]] <math>f:M\to\R</math> and for a small enough <math>\varepsilon</math> one can construct a Lagrangian submanifold given by the vanishing locus <math>\mathbb{V}(\varepsilon\cdot \mathrm{d}f) \subset T^*M</math>. For a generic Morse function we have a Lagrangian intersection given by <math>M \cap \mathbb{V}(\varepsilon\cdot \mathrm{d}f) = \text{Crit}(f)</math>.

{{See also|symplectic category}}

=== Special Lagrangian submanifolds ===
In the case of [[Kähler manifold]]s (or [[Calabi–Yau manifolds]]) we can make a choice <math>\Omega=\Omega_1+\mathrm{i}\Omega_2</math> on <math>M</math> as a holomorphic n-form, where <math>\Omega_1</math> is the real part and <math>\Omega_2</math> imaginary. A Lagrangian submanifold <math>L</math> is called '''special''' if in addition to the above Lagrangian condition the restriction <math>\Omega_2</math> to <math>L</math> is vanishing. In other words, the real part <math>\Omega_1</math> restricted on <math>L</math> leads the volume form on <math>L</math>. The following examples are known as special Lagrangian submanifolds,
# complex Lagrangian submanifolds of [[hyperkähler manifold]]s,
# fixed points of a real structure of Calabi–Yau manifolds.
The [[SYZ conjecture]] deals with the study of special Lagrangian submanifolds in [[mirror symmetry (string theory)|mirror symmetry]]; see {{harv|Hitchin|1999}}. 

The [[Thomas–Yau conjecture]] predicts that the existence of a special Lagrangian submanifolds on Calabi–Yau manifolds in Hamiltonian isotopy classes of Lagrangians is equivalent to stability with respect to a [[Bridgeland stability condition|stability condition]] on the [[Fukaya category]] of the manifold.