1: Geometry on Manifolds
In the broadest sense, differential geometry is the study of smooth manifolds equipped with additional data fixing some ``geometry'' on the space. Usually, this geometry is obtained by imposing some geometry of vector spaces onto the tangent space of a smooth manifold. An inner product on a vector space \(g: V\times V\to \RR\) determines a geometry on the vector space by specifying lengths and angles. A Riemannian manifold is a pair \((X, g)\) where \(X\) is a smooth manifold and \(g\) denotes a family of inner products \[g_p: T_pX\to T_pX\] which vary smoothly on the parameter in \(p\in X\). A Riemannian manifold has local notions of lengths and angles that determine geodesics, curvature, and volumes. Many constructions in Riemannian geometry are determined locally, as there are no global obstructions to locally modifying the Riemannian metric. However, Riemannian geometry posses some local invariants, which means that neighborhoods of points are distinguishable from each other using invariants such a curvature.definition 1.0.1
A complex space is a vector space \(V\) of dimension \(2n\), along with a map of vector bundles \(J: V\to V\) with \(J^2=-\operatorname{id}_V\).definition 1.0.2
An almost complex manifold is a pair \((X,J)\) where \(X\) is a \(2n\)-dimensional manifold equipped with a bundle morphism (called the almost complex structure) \[J: TX\to TX\] so at each fiber \((T_xX, J_x)\) is a complex vector space.definition 1.0.3
A symplectic vector space is a \(2n\)-dimensional vector space \(V\), along with a choice of symplectic form \(\omega: V\times V\to \RR\) which is an antisymmetric and satisfies any of the following equivalent non-degeneracy conditions:- \(v \in V\) is the zero vector if and only if \(\iota_v\omega=0\),
- The symplectic form gives an isomorphism between \(V\) and its dual: \begin{align*} \omega:V\to V^* && v\mapsto \iota_v\omega, \end{align*}
- The top form \(\omega^{n}\) is non-zero.
definition 1.0.4
A symplectic manifold is a pair \((X, \omega)\) where \(X\) is a smooth manifold of dimension \(2n\) equipped with a symplectic form \[\omega\in \Omega^2(X; \RR).\] which is closed (i.e. \(d\omega=0\)) and at each point \(x\in X\) makes the pair \((T_xX, \omega_x)\) a symplectic vector space.definition 1.0.5
Let \((X, \omega)\) be a symplectic manifold with almost complex structure \(J\). Then we say that \(J\) is a \(\omega\)-compatible almost complex structure if \[ g(v, w)=\omega(v, Jw)\] is a Riemannian metric on \(X\).proposition 1.0.6
Every symplectic manifold has a compatible almost-complex structure.2: examples of symplectic manifolds
example 2.0.1
The simplest example comes from \(\RR^{2n}\), which we give the coordinates \((q_i, p_i)\). In these local coordinates, we can define a symplectic form by \[\omega_{std}=\sum_{i=1}^n d p_i\wedge d q_i.\] Note that when \(n=1\), this gives the standard area form on \(\RR^2\). In these coordinates, it is easy to check that \(\frac{\omega_{std}^n}{n!}=\text{vol}_{\RR^{2n}}\), the standard volume form.example 2.0.2
Let \((X, g)\) be an oriented surface. Then \(g\) prescribes a volume form \(\omega:=\text{vol}_g\in \Omega^2(X;\RR)\), which is an example of a non-degenerate 2-form. Because \(\Omega^3(X;\RR)=0\), it trivially follows that \(\omega\) is closed. This example raises the possibility of the same space having many different symplectic forms, as an oriented surface can be equipped with several different metrics.example 2.0.3
An example that will be especially relevant later is \((\CC^*)^n\). We will equip this with a different symplectic form than the one inherited as a subset of \(\CC^n=\RR^{2n}\). Since \((\CC^*)^n\) is a group, it is natural to ask for a symplectic form on \((\CC^*)^n\) which is invariant under the group action. The symplectic form \[ \omega=\frac{1}{2\pi} d(\log |z|)\wedge d\theta \] gives an example of such a symplectic form. When \(n=1\), then this is the area form on \((\CC^*)\) which embeds into three dimensional space as an infinitely long cylinder, as drawn in figure 2.0.4.example 2.0.5
Let \(Q\) be a smooth \(n\)-dimensional manifold. We now describe a canonical symplectic form on the cotangent bundle, \(T^*Q\). At every point \(q\in Q\), there exists chart \(q\in U\subset Q\) which we can parameterize with coordinates \((q_1, \ldots, q_n)\). The cotangent bundle \(T^*U\) inherits coordinates \((q_1, p_1, q_2, p_2, \ldots, q_n, p_n)\), where the \(p_i\) linearly parameterize the fibers of the cotangent bundle in the direction of the basis element \(dq_i\). ^{0} In these coordinates, the canonical symplectic form on this chart is: \[\omega=\sum_{i=1}^n dq_i \wedge dp_i=-d(p dq).\]3: Symplectomorphisms and Hamiltonian isotopies
In previous definitions, we've presented- some diffeomorphic spaces (i.e. \(T^*T^n\) and \((\CC^*)^n\)) which should be considered the same as symplectic manifolds.
- some spaces which are diffeomorphic (i.e. \(S^2\) and itself) which can be equipped with clearly different symplectic structures (i.e. \(\text{vol}_g\) and \(\text{vol}_{2g}\)).
definition 3.0.1
Let \((X, \omega)\) and \((X', \omega')\) be symplectic manifolds, and \(\phi: X\to X' \) be a diffeomorphism. We call \(\phi\) a symplectomorphism if \[\phi^*\omega'=\omega.\] Given an embedding \(\psi: X\to X'\), we say that \(\psi\) is a symplectic embedding if \[\psi^*\omega'=\omega\]definition 3.0.2
Given a time-dependent vector field \(V_t\), the flow associated to \(V\) is the function \(\phi_t: X\times \RR\to X\) which- for all \(t\), \(\phi_t:X\to X\) is a diffeomorphisms and;
- is the identity at time 0 so that \(\phi_0=\operatorname{id}_M\) and;
- generates the vector field \(V_t\) in the sense that \[(V_t)_p:=\frac{d}{ds}\phi_s(\phi^{-1}_t(p))|_{s=t}.\]
theorem 3.0.3
On compact manifolds, every time-dependent vector field has a well defined flow.proposition 3.0.4
\(\phi_t\) is a symplectic isotopy if and only if its infinitesimal generator \(V_t\) satisfies \[d(\iota_{V_t}\omega)=0.\]definition 3.0.5
Let \((X, \omega)\) be a symplectic manifold, and \(H: X\to \RR\) is a smooth function. The Hamiltonian vector field of \(H\) is the unique vector field \(V_H\) such that \(dH=\iota_{V_H}\omega\).example 3.0.6
Suppose that we are working in \(\RR^{2n}=(q_1, \ldots , q_n,p_1, \ldots p_n) \), equipped with the standard symplectic form \(\omega= \sum_{i=1}^n dp_i \wedge dq_i\). We can compute the Hamiltonian vector field for \(H: \RR^{2n}\to \RR\), in local coordinates as \[V_H=\sum_{k=1}^n \left(a_k \partial_{q_k}+b_k \partial_{p_k}\right)\] where the functions \(a_k\) and \(b_k\) are given by the formulas \begin{align*} \frac{\partial p_k}{\partial H}=-b_k & & \frac{\partial q_k}{\partial H}=a_k. \end{align*}example 3.0.7
Consider \(S^2=\{(x_0, x_1, x_2)\;|\;x_0^2+x_1^2+x_2^2=1\}\) equipped with the symplectic form agreeing with the standard metric induced from \(\RR^3\). Take the Hamiltonian \begin{align*} H:S^2\to&\RR\\ (x_0,x_1,x_2)\mapsto& x_2 \end{align*} as drawn in figure 3.0.8. Since Hamiltonian flow preserves the level sets of \(H\), we know that the latitudinal slices are orbits under the action of the Hamiltonian flow. To show that the Hamiltonian flow uniformly rotates the sphere, consider the map \(\phi:S^2\setminus\{(0,0,1), (0,0,-1)\}\into S^1\times \RR\subset \RR^3\), where \(S^1\times \RR=\{(x_0, x_1, x_2)\;|\; x_0^2+x_1^2=1\}\), and the embedding is given by the latitudinal projection. This projection (the Gall-Peters map projection) is area-preserving, and so \(\phi\) is a symplectic embedding. In these new coordinates, \(\omega=d\theta\wedge dx_2\) and \(H=x_2\). In the Gall-Peters' coordinates, \(V_H=\partial_\theta\).lemma 3.0.9
Let \((X, \omega, g, J)\) be a compatible triple. Let \(H:X\to \RR\) be a Hamiltonian. \[\grad H = J V_H\]4: local models for symplectic manifolds
In contrast to Riemannian geometry, symplectic manifolds are ``locally symplectomorphic.'' This means that one cannot distinguish two symplectic manifolds simply based on the symplectic geometry of a small neighborhood of a point. In this section, we sketch a proof of this fact.theorem 4.0.1
Let \(V_0\) and \(V_1\) be closed connected \(n\) manifolds with volume forms \(\Omega_0\) and \(\Omega_1\). Suppose that they have the same total volume, that is \[\int_{V_0} \Omega_0 = \int_{V_1} \Omega_1.\] Let \(\phi_0: V_0\to V_1\) be a diffeomorphism. Then \(\phi_0\) is isotopic to a volume preserving diffeomorphism \(\phi_1\); that is \[\phi_1^*\Omega_1=\Omega_0.\]theorem 4.0.2
Let \(X\) be a closed \((2n)\)-manifold. Let \(\{\omega_t\}\) be a smooth family of symplectic forms in the same cohomology class. Then there exists a smooth family of diffeomorphism \(\{\phi_t\}\) with \(\phi_0=\operatorname{id}_X\) and \(\phi_t^*\omega_t=\omega_0\).theorem 4.0.3
Let \(X\) be a manifold. Let \(Y\) be a compact submanifold of \(X\). Let \(\omega_0, \omega_1\) be symplectic forms on \(Y\) such that \(\omega_0(p)=\omega_1(p)\) for all \(p\in X\). Then there exists neighborhoods \(U_0, U_1\supset X\) and a symplectomorphism \(\phi:(U_0, \omega_0)\to (U_1, \omega_1)\) such that \(\phi(p)=p\) for all \(p\in X\).References
[dS01] | Ana Cannas da Silva. Lectures on symplectic geometry. Lecture Notes in Mathematics, 1764, 2001. |