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. Symplectic geometry, when taken outside of its historical context, seems a bit artificially constructed.

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. If Riemannian geometry yields a theory of lengths and complex geometry a theory of preferred right-angles, then symplectic geometry gives a theory of signed areas. There is a certain notion of compatibility between these three kinds of geometry. For example, notice that an inner product also identifies right angles and areas.

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 \(X\). Remarkably, having a symplectic structure is sufficient for the construction of a compatible almost complex structure.

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.

figure 2.0.4:The symplectic structure that we choose for \(\CC^*\) makes it an infinitely long cylinder.

The most important example of symplectic manifold comes from physics, which is the historical origin of symplectic geometry.

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: \[\omega=\sum_{i=1}^n dq_i \wedge dp_i=-d(p dq).\] In the physical literature, \(T^*Q\) is called the ``phase-space'' which encodes both the position and momentum of a particle. The canonical symplectic form describes the natural pairing of momentum and velocity. There is also a coordinate-free description of \(\omega\). To do so, we first define a canonical 1-form \(\eta\in \Omega^1(T^*Q)\). Let \((q,p)\in T^*Q\) be a point, where the coordinates take values \(q\in Q\) and \(p\in T^*_qQ\). The map \(\pi: T^*Q\to Q\) induces a map \(\pi_*: T(T^*Q)\to TQ\). The canonical 1-form is defined by its value on tangent vectors \(v\in T_{(q,p)}(T^*M)\) \[\eta_{(q,p)}(v):=p(\pi_*(v)).\] This describes the pairing between the \(Q\)-component (velocity in the base) of \(v\) and the momentum coordinate \(p\). From the canonical form, we obtain a coordinate-free definition of canonical symplectic form as \[\omega=-d\eta.\] Note that \(\omega\) is not simply a closed 2-form, but rather an exact symplectic form. One can also prove that the cotangent bundle serves as a general kind of local model for symplectic manifolds. We've already seen the cotangent bundle appear in the examples of symplectic structures on \(\RR^{2n}\) and \((\CC^*)^n\), which can be interpreted as the symplectic structures on the cotangent bundles \(T^*\RR^n\) and \(T^*T^n\) respectively. This example also gives an example of how an almost complex structure and symplectic structure can interact. Let \(Q\) be a manifold equipped with a connection. The tangent bundle of \(Q\) comes with an almost complex structure. With a choice of connection we obtain a splitting \[T_{(q,v)}TQ= T_v (T_q Q)\oplus T_q Q\] with an isomorphism \(A: T_v(T_q Q)\to T_qQ.\) One can then construct an almost complex structure by taking the matrix \[J:=\begin{pmatrix} 0 & A\\ -A^{-1} & 0\end{pmatrix}: T_{(q,v)}TQ\to T_{(q,v)}TQ.\] One can similarly (but not canonically) construct an almost complex structure for the cotangent bundle. Pick \(g\) a metric for \(Q\), which induces a bundle isomorphism between the tangent and cotangent bundle. Let \((q_1, \ldots, q_n, p_1, \ldots p_n)\) be local coordinates for \(T^*Q\) chosen so that \(\partial q_1, \ldots \partial q_n\) form an orthonormal basis at the origin and the coordinates \(p_1, \ldots, p_n\) parameterize the linear coordinates determined by the basis \(\{\iota_{\partial q_i}g\}\). Then an almost complex structure is specified by \begin{align*} J \partial q_i=\partial p_i && J\partial p_i = -\partial q_i. \end{align*} Note that the resulting almost complex structure depends on the choice of metric, which modifies the ``size'' of the cotangent fiber relative to the base. The metric \(\omega(J, -)\) is then the standard induced metric on the cotangent bundle.

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 \(\text{vol}_{2g}\)).

We now introduce an equivalence relation for symplectic structures on a manifold.

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 \[\psi^*\omega'=\omega\] One property of the canonical symplectic structure on the cotangent bundle \(T^*Q\) is that the symplectomorphism type of \(T^*Q\) is only dependent on the diffeomorphism type of \(Q\) [dS01]. Let \(f: Q_1\to Q_2\) be a diffeomorphism. Define the lift of \(f\) to be a map \(f_\#: T^*Q_1\to T^* Q_2\) defined by \[f_\#(q,p)=(f(q), ((df_q)^*)^{-1}p\] The lift plays well with the tautological 1-form, in the sense that if \(\lambda_i\) is the tautological 1-form for \(T^*Q_i\), then \((f_\#)^*\lambda_2=\lambda_1\). A particularly interesting type of symplectomorphism is one which arises as an isotopy. A symplectic isotopy is a smooth 1-parameter family of maps \(\phi_t: X\times I\to X\) which for fixed values of \(t\) give a symplectomorphism, and start at the identity (in the sense that \(\phi_0=\operatorname{id}_X\)). These are examples of smooth isotopies and so they can equivalently be described as the flow of a time-dependent vector field.

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 \[(V_t)_p:=\frac{d}{ds}\phi_s(\phi^{-1}_t(p))|_{s=t}.\]

We say that \( V_t\) is the infinitesimal generator associated to \(\phi_t\). There is an equivalence between vector fields and flows.

theorem 3.0.3

On compact manifolds, every time-dependent vector field has a well defined flow. One can check that a smooth isotopy \(\phi_t\) is a symplectic isotopy by verifying that the symplectic form is preserved by the infinitesimal generator \(V_t\). This yields an easy to check criterion.

proposition 3.0.4

\(\phi_t\) is a symplectic isotopy if and only if its infinitesimal generator \(V_t\) satisfies \[d(\iota_{V_t}\omega)=0.\] We prove the forward direction, using that \(\phi^*_t\omega=\omega\) for all \(t\). By taking the derivative with respect to \(t\) of both sides, we obtain: \begin{align*} 0=&\frac{d}{dt}\phi^*\omega =\phi^*_t\mathcal L_{V_t}\omega \end{align*} Since \(\phi\) is a diffeomorphism, this is equivalent to the vanishing of \(\mathcal L_{V_t}\omega\), \begin{align*} 0=& \mathcal L_{V_t}\omega \end{align*} Applying Cartan's formula, and using that \(\omega\) is closed, \begin{align*} 0=&d(\iota_{V_t}\omega)+ \iota_{V_t}(d\omega)=d(\iota_{V_t}\omega). \end{align*} This means that \(\iota_{V_t}\omega\) is closed. One could ask for the stronger condition of exactness for the 1-form \(\iota_{V_t}\omega\). In this case, we can describe vector field \(V_t\) by a function on \(X\).

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 \(dH=\iota_{V_H}\omega\). The uniqueness of this vector field arises from the non-degeneracy of the symplectic form \(\omega\). This additionally means that to every exact symplectic isotopy we can associate a generating Hamiltonian function. The Hamiltonian isotopies give a large set of easy-to-describe symplectic isotopies, and the relation between Hamiltonian isotopies and all symplectic isotopies has a nice interpretation in terms of the topology of \(X\). If \(H^1(X, \RR)=0\), then every symplectic isotopy is Hamiltonian.

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*} A notable feature of Hamiltonian flow \(V_H\) is that it preserves the level sets of \(H\), as \[V_H(H)=dH(V_H)=\omega(V_H, V_H)=0.\]

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, \(V_H=\partial_\theta\).

figure 3.0.8:The Hamiltonian flow of the standard height function rotates the sphere counterclockwise relative to the north pole

The Hamiltonian flow is sometimes called the symplectic gradient. In the setting where we have a compatible triple \((X, \omega,g, J)\), the Hamiltonian flow and gradient are related by the almost complex structure.

lemma 3.0.9

Let \((X, \omega, g, J)\) be a compatible triple. Let \(H:X\to \RR\) be a Hamiltonian. \[\grad H = J V_H\] This is a direct computation. On any test vector \(v\), \begin{align*} g(J V_H, v)=\omega(V_H, v)=dH(v)= g(\grad(H), v) \end{align*} Because \(g\) is nondegenerate, \(\grad(H)=JV_H\).

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 \[\phi_1^*\Omega_1=\Omega_0.\] Without loss of generality, let \(V_0=V_1\) and let \(\phi_0\) be the identity. Let \(\Omega_t=(1-t)\Omega_0+t\Omega_1\). We want to find \(\{\phi_t\}\) with \(\phi_t^*\Omega_t=\Omega_0\). We could equivalently describe such an isotopy \(\phi_t\) by the vector field \[V_t:=\left(\frac{d}{dt}\phi_t\right)\circ\phi_t.\] Taking Lie derivatives, we have the pullback condition is equivalent to \[\frac{d}{dt}(\phi_t^*\Omega_t)=\phi_t^*\left(\mathcal L_{V_t} \Omega_t+\frac{d}{dt}\Omega_t\right)\] So we are looking for \(V_t\) with \(\mathcal L_{V_t}\Omega_t+\frac{d}{dt}\Omega_t=0\). Simplifying further gives us \[ d\iota_{V_t}\Omega_t=-\frac{d}{dt}\Omega_t=\Omega_0-\Omega_1\] By Stoke's theorem, the right term is exact as \(\int_X\Omega_0=\int_X \Omega_1\). Therefore there exists some \(n-1\) form \(\eta\) so that \(d\iota_{V_t}\Omega_t=d\eta\). So now we have to solve \(\iota_{V_t}\Omega_t=\eta\). Since \(\Omega_t\) is a volume form, then there exists a \(V_t\) satisfying this equation. This tells us that symplectomorphisms of surfaces are boring because any diffeomorphism can be made into a symplectomorphism. In fact, every isotopy of symplectic forms in the same cohomology can be realized by a family of symplectomorphisms.

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 \(\phi_t^*\omega_t=\omega_0\). Instead of finding \(\phi_t\), we instead search for the vector \(V_t\) generating the isotopy. Because the symplectic form should be invariant under this isotopy, we obtain the following condition on \(V_t\): \begin{align*} 0=&\frac{d}{dt}(\phi_t^*\omega_t)=\phi_t^*\left(\mathcal L_{V_t}\omega_t+\frac{d}{dt}\omega_t\right)\\ =&\mathcal L_{V_t}\omega_t + \frac{d}{dt}\omega_t\\ =&d\iota_{V_t}\omega+\frac{d}{dt}\omega_t \end{align*} Since the cohomology class \([\omega_t]\) is constant, the time derivative \(\frac{d}{dt}\omega_t\) is exact. Therefore there exists ⁰ \(\eta_t\) with \(d\eta_t=\frac{d}{dt}\omega_t\). This reduces our previous computation to \begin{align*} =&d\iota_{V_t}\omega_t+d\eta_t \end{align*} Since \(\omega_t\) is nondegenerate, there exists unique \(V_t\) with \(\iota_{V_t}\omega_t+\eta_t=0\). We also have a relative version of the Moser theorem.

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 \(p\in X\). Let \(\omega_t=(1-t)\omega_0+t\omega_1\) for \(t\in [0, 1)\). Then \(\omega_t\) is symplectic for all \(t\) in a sufficiently small tubular neighborhood \(N\) of \(X\). We want to find \(\phi_t\) with \(\phi_t^*\omega_t=\omega_0\) for all \(t\). Let \(V_t=\frac{d}{dt}\phi_t\circ \phi_t\). Recall that \begin{align*} \frac{d}{dt}(\phi_t^*\omega_t)=&\phi^*_t(\mathcal L_{V_t}\omega_t+(\omega_1-\omega_0))\\ =&\phi^*_t(di_{V_t}\omega_t+(\omega_1-\omega_0)) \end{align*} Since \(i^*\omega_0=i^*\omega_1\), we have that \([\omega_0]=[\omega_1]\in H^2(N)\) . Then there exists \(\eta\) such that \(d\eta=\omega_0-\omega_1\) and \(\eta(p)=0\) for all \(p\in X\). \begin{align*} =&\phi^*d(i_{V_t}\omega-\eta) \end{align*} Since \(\omega_t\) is nondegenerate, there exists unique vector field \(V_t\) with \(i_{V_t}\omega_t=\eta\). Since \(\eta(p)=0\) for all \(p\in X\), it follows that \(V_t(p)=0\) for all \(p\in X\). For this \(V_t\) we have that \(\frac{d}{dt}(\phi^*_t\omega_t)=0\). Therefore \(V_t\) generates a family of diffeomorphism \(\{\phi_t\}\) which are the identity on \(X\) and defined on a neighborhood of \(X\). The application of this theorem to symplectic geometry shows that there are no local invariants of symplectic manifolds. If \((X, \omega_0)\) is symplectic then for all \(p\in X\) there exists neighborhoods \(U_0\supset \{p\}\) and \(U_1\subset \RR^{2n}\) and a symplectomorphism \((U_0, \omega_0)\to (U_1, \omega_{std})\).

References

[dS01]

Ana Cannas da Silva. Lectures on symplectic geometry. Lecture Notes in Mathematics, 1764, 2001.