Symplectic and Poisson Manifolds Harry Smith, 30 November 2015. P

Symplectic and Poisson Manifolds Harry Smith, p 1/2. 30 November 2015.

1 Punchline of the talk

Every Poisson manifold is naturally partitioned into regularly immersed symplectic manifolds called symplectic leaves

2 Symplectic Manifolds 3 Poisson Manifolds

Defn 4. A Poisson manifold P is a differentiable man- ∞ Defn 1. A differential two-from ω on a differentiable ifold equipped with a Lie algebra structure on C (P ) manifold M, is non-degenerate at a point p ∈ M if defined by a Poisson Bracket {·, ·} : C∞(P ) × C∞(P ) −→ C∞(P ) ω(up, vp) = 0 for all vp implies up = 0 for all tangent satisfying the following conditions: vectors up, vp ∈ TpM. Furthermore, we say that a two-form is non-degenerate if it is non-degenerate at every point on the manifold. •{αf + βg, h} = α{f, h} + β{g, h} •{f, g} = −{g, f} •{f, {g, h}} + {g, {h, f}} + {h, {f, g}} = 0 •{fg, h} = f{g, h} + g{f, h} Defn 2. A Symplectic Manifold (M, ω) is a dif- for all f, g, h ∈ C∞(P ) and α, β ∈ R and with each ferentiable manifold M with a distinguished non- condition holding for both entries. degenerate two-form ω called the Symplectic Form. We can equip any differentiable manifold M with a trivial Poisson structure simply by setting {f, g} = 0 for all We find that all symplectic manifolds are necessarily of even f, g ∈ C∞(M). dimension.

We can equip any symplectic manifold (M, ω) with a Pois- Defn 3. A Symplectomorphism φ between two sym- son bracket {·, ·}ω by using the Hamiltonian vector ﬁelds ∞ plectic manifolds (X, ωX ) and (Y, ωY ) is a diﬀeomor- of functions in C (P ). ∗ phism φ : X → Y such that φ ωY = ωX . {f, g}ω = ω(Xf ,Xg)

∞ The most basic example of a symplectic manifold is the For any Poisson manifold P, by fixing some h ∈ C (P ) we plane R2 equipped with the two-form ω = dx ∧ dy. This can define a linear map can be generalized to any even dimensional real space R2n. ∞ ∞ Xh = {·, h} : C (P ) −→ C (P ) 2n If we take coordinates (x1, ..., xn, y1, ..., yn) ∈ R then we n Since our Poisson bracket satisfies Leibniz’s rule by defini- define our two-form to be ω = X dx ∧ dy i i tion, it follows that Xh defines a vector field on the Poisson i=1 manifold P . This is a generalized form of the Hamiltonian Vector Field for all Poisson manifolds and it agrees with We may also define a symplectic structure on the cotangent the definition for symplectic Poisson manifolds. bundle T ∗M of any differentiable manifold M.

Thm 1. (Darboux) Every 2n-dimensional symplectic manifold is locally symplectomorphic to (R2n, ω), with ω as above.

Any diﬀerentiable function h : M → R on a symplectic manifold (M, ω), uniquely determines a vector ﬁeld Xh on M. This is called the Hamiltonian Vector Field of h. Symplectic and Poisson Manifolds Harry Smith, p 2/2. 30 November 2015.

4 The Splitting Theorem Ex 4. The dual g∗ of a Lie algebra (g, [·, ·]) is a Pois- son manifold, where [·, ·] is the standard Lie bracket given by Defn 5. A Poisson mapping is a smooth mapping [X,Y ] = XY − YX φ : P → P between two Poisson manifolds P and 1 2 1 for X,Y ∈ g. As C∞(g∗) = (g∗)∗ = g, we can P with Poisson brackets {·, ·} and {·, ·} respec- 2 1 2 simply deﬁne a Poisson bracket {X,Y } = [X,Y ] for tively, such that {f ◦ φ, g ◦ φ} = {f, g} ◦ φ 1 2 any X,Y ∈ g. In this case the symplectic leaves are for all f, g ∈ C∞(P ). 2 realized as the coadjoint orbits of the Lie group G.

Note, the example 3 and example 4 may give us non- Defn 6. Given two Poisson manifolds P and P we 1 2 symplectic manifolds. define the product P1 × P2 to be the usual product of smooth manifolds equipped with a Poisson bracket 6 References {·, ·} such that each of the projections π : P × P → i 1 2 • A. Cannas da Silva, Symplectic Geometry, Hand- P are Poisson mappings and {f ◦ π , g ◦ π } = 0 for i 1 2 book of Differential Geometry, 2004 all f ∈ C∞(P ) and g ∈ C∞(P ). 1 2 • J. Watts, An Introduction to Poisson Manifolds, 2007 • A. Weinstein, The Local Structure of Poisson Thm 2. (Splitting Theorem) Let P be a Poisson man- Manifolds, J. Differential Geometry, 1983 ifold, and let p ∈ P . Then there exists an open neigh- • A. Zamorzaev, Introduction to Symplectic Topol- bourhood U ⊂ P containing p and a unique diffeo- ogy, Lecture Notes morphic Poisson mapping φ = φS × φN : U −→ S × N where S is a symplectic manifold (called a Symplectic Leaf) and N is a Poisson manifold.

In other words, every Poisson manifold is naturally partitioned into immersed symplectic manifolds.

5 Examples

Ex 1. On a diﬀerentiable manifold with the trivial Poisson structure, its symplectic leaves will be exactly the individual points x ∈ M.

Ex 2. If we take any symplectic manifold (M, ω) and equip it with the Poisson bracket {·, ·}ω then M will have a single symplectic leaf; itself.

Ex 3. Take any symplectic manifold M equipped with the symplectic Poisson structure, and any diﬀeren- tiable manifold N equipped with the trivial Poisson structure. Then M × N will be a Poisson manifold with its symplectic leaves being {M × {x}|x ∈ N}.