What is a Symplectic manifold

What is a Symplectic Manifold?

Peadar Coyle

University of Luxembourg - Fiber Bundles Seminar

April 8, 2013

Introduction

We want to introduce the notion of a Symplectic Manifold, butfirst we need to revise some Classical Mechanics, and DifferentialGeometry. In some sense, a Symplectic Manifold is simply thenatural generalization of Hamiltonian Mechanics.

Let us recall Hamilton’s equatons in Mechanics. Consider adynamical system characterized by a Hamiltonian. H = H(q, p),where q = (q1, · · · , qn) is position and p1, · · · , pn) is momentum

Introduction

We want to introduce the notion of a Symplectic Manifold, butfirst we need to revise some Classical Mechanics, and DifferentialGeometry. In some sense, a Symplectic Manifold is simply thenatural generalization of Hamiltonian Mechanics.Let us recall Hamilton’s equatons in Mechanics. Consider adynamical system characterized by a Hamiltonian. H = H(q, p),where q = (q1, · · · , qn) is position and p1, · · · , pn) is momentum

What is the Phase SpaceWe mentioned in the previous slide, a dynamical system. Below isa figure representation of the flow of a vector field

Figure: Flow Lines of a vector field

We can say that (q,p) denotes what we call the Phase Space. Forsimplicity let us imagine that this Phase space is R2n

Let q = ∂pH, p = −∂qH If we set x = (q, p) and denote by ω thesymplectic unit matrix

ω =

(0 1−1 0

)(1)

Vector fields and Phase Spaces

Let us recall

ω =

(0 1−1 0

)

the matrix product

ω (∂xH) = ω

(∂qH∂pH

)=

(∂pH−∂qH

)is a vector field fo the phase space.The motions are now given by

∂tX = Xx(t) (2)

i.e. by the flow of the vector field.

Vector fields and Phase Spaces

Let us recall

ω =

(0 1−1 0

)the matrix product

ω (∂xH) = ω

(∂qH∂pH

)=

(∂pH−∂qH

)is a vector field fo the phase space.The motions are now given by

∂tX = Xx(t) (2)

i.e. by the flow of the vector field.

Symplectic Manifolds

In usual particle mechanics, the phase space is given by theco-ordinates qi and their conjugate momenta, and its volume by

V =

∫ n∏i=1

dpidqi (3)

In some systems, however, the geometry of the phase space is notas simple. We can still consider R2n as a ’model’ of a symplecticmanifold, however it is worthwhile considering the general case.

Definition of a Symplectic Manifold

DefinitionA symplectic manifold M is a 2n− dimensional manifold that has asymplectic structure. Loosely speaking we can consider it theanti-symmetric counterpart to a pseudo-Riemannian manifold.

More specifically, it admits a two form ω =1

2ωijdx i ∧ dx j called

the symplectic form. The symplectic form has the followingproperties:

I Closed, namely dω = 0,

I Non-degenerate, namely the matrix ωij = −ωji is invertible.

Symplectic Manifolds: PropertiesDue to the invertibility property, detωij 6= 0, and hence

dV =1

n!6= 0

V =

∫dV =

∫1

n!ωn =

∫ (detωij

2n∏i=1

dx i

)(4)

Because the symplectic form is non-degenerate, we can define theinverse

ωijωjk = δik (5)

For the simple case above, the symplectic form is

ω =n∑

i=1

(dpi ∧ dqi

)(6)

and the symplectic matrix is

ω =

(0 1−1 0

)

Example of a two-sphere

Consider a non-trivial example of a compact phase space S2. Thesymplectic form is nothing but the surface area of the sphere

ω = J sin θdφ ∧ dθ (7)

It is trivially closed dω = 0 because there is no three-form on atwo-dimensional space, and non-degenerate because the symplecticmatrix

ωij =

(0 J sin θ

−J sin θ 0

)is invertible. The Poisson bracket is

A,B =−1

J sin θ

(∂A

∂φ

∂B

∂θ− ∂A

∂θ

∂B

∂φ

)(8)

Example of a 2-sphere: Continued

The symplectic form is locally exact

ω = dξ = d (J cos θdφ) (9)

and hence one can write the Lagrangian

L = J cos θφ− H (θ, φ) (10)

The consistency of the path integral requires∫S2

ω = 2π~N (11)

and hence J = j~ where j = N/2 is a half integer. This has deeplinks with spin in Quantum Mechanics, but we shall not delve intothat today!

On the Cohomology of Symplectic Manifolds

FactOne major restriction on symplectic manifolds, if compact, is thatthey need to have non-trivial second cohomology to allow for aclosed non-degenerate two-form. Namely there is a two-dimensonalsubsurface of the manifold that is closed (two-cycle C2) on whichthe symplectic form can be integrated. This puts an interestingrequirement on the normalization of the symplectic form as youwill see in the next section. Kahler manifolds (complex manifoldsof U(N) holonomy) are all symplectic.

Symplectic Connections

We introduced the notions of torsion and parallel transport in anadvanced Differential Geometry class. We want to consider aproposition describing symplectic connection

Proposition

Part A:On any Symplectic manifold (M, ω) there exists a torsionfree connection ∇ such that ∇ω = 0.PartB:The set of these connections is an affine space modelled onthe sections of the symmetric 3-forms denoted by:Sec

(S3T ∗M

)= Γ

(S3T ∗M

)



Proposition

Part A:On any Symplectic manifold (M, ω) there exists a torsionfree connection ∇ such that ∇ω = 0.

PartB:The set of these connections is an affine space modelled onthe sections of the symmetric 3-forms denoted by:Sec

(S3T ∗M

)= Γ

(S3T ∗M

)



Proposition

Part A:On any Symplectic manifold (M, ω) there exists a torsionfree connection ∇ such that ∇ω = 0.PartB:The set of these connections is an affine space modelled onthe sections of the symmetric 3-forms denoted by:Sec

(S3T ∗M

)= Γ

(S3T ∗M

)

Proposition: The Strategy of the Proof

Let us break down the strategy of the proof into some steps:

I Part A1: Firstly we choose a local trivialization of TMinduced by Darboux charts.

I Part A2: Transport the trivial connection ∇0 fromψ(U)× R2n to TU

I Part A3:Use partititon of unity type arguments to imply thatthe connection is torsion free and parallel

We also have to prove the second part of the proposition

I Part B1: We shall prove the fact that symplectic connectionsform an affine space modelled on the sections of thesymmetric 3-forms, by comparing two connections on M.

I Part B2: We shall verify the using the defintions of parallelconnections and torsion free connections.

Part A1: Local trivialization

We now proceed with the proof

Proof.Firstly we choose a local trivialization of TM induced by Darbouxcharts (

U, ψ =(p1, · · · , pn, q

1, · · · qn))

Let us recall that the symplectic form in a chart reads

ω|U =∑i

dpi ∧ dqi

i.e. thatω|U = ψ∗ω0

where ω0 is the canonical symplectic form of R2n

Part A2:Transportation of the Connection

Proof.We need to transform the trivial connection ∇0 from ψ(U)× R2N

to TU.

Indeed, (ψ∗, ψ) where φ∗ is the tangent map of ψ, is anisomorphism between the tangent bundle TU and ψ(U)× R2n.Hence by the equivalence relation for connections, we have thatconnection transport is defined for any X,Y ∈ V(U) by

ψ∗ (∇XY ) = ∇0ψ∗Xψ∗Y (12)

Here ψ∗Y = ψ∗ Y ψ−1 is the pushforward of Y by thediffeomorphism ψ.

Part A2:Transportation of the Connection

Proof.We need to transform the trivial connection ∇0 from ψ(U)× R2N

to TU. Indeed, (ψ∗, ψ) where φ∗ is the tangent map of ψ, is anisomorphism between the tangent bundle TU and ψ(U)× R2n.Hence by the equivalence relation for connections, we have thatconnection transport is defined for any X,Y ∈ V(U) by

ψ∗ (∇XY ) = ∇0ψ∗Xψ∗Y (12)

Here ψ∗Y = ψ∗ Y ψ−1 is the pushforward of Y by thediffeomorphism ψ.

Part A3:Partition of Unity arguments

Proof.Up until now we have constructed a local connection ∇ in U orbetter, since (U, ψ) runs through an atlas, ∇α in Uα such thatthere is T∇α = 0 and∇αω|Uα = 0 As in other proofs involvingconnections, we use the gluing by partition of unity arguments. Letfα - be the partition of unity.

It is easy to see that the global connection, defined for anyX ,Y ∈ V(M)

∇XY =∑α

fα∇αX |UalphaY |Uα

is also torsion and parallel free.

Part A3:Partition of Unity arguments

Proof.Up until now we have constructed a local connection ∇ in U orbetter, since (U, ψ) runs through an atlas, ∇α in Uα such thatthere is T∇α = 0 and∇αω|Uα = 0 As in other proofs involvingconnections, we use the gluing by partition of unity arguments. Letfα - be the partition of unity.It is easy to see that the global connection, defined for anyX ,Y ∈ V(M)

∇XY =∑α

fα∇αX |UalphaY |Uα

is also torsion and parallel free.

Part B1:Comparison of Connections

Let us prove that the symplectic connection forms an affine spacemodelled on the field of Symmetric-3-forms

Proof.If ∇1 is another connection on M, we know that

Ω = ∇1 −∇ ∈ Γ(⊗12TM) (13)

Since

∇1Z (ω) = LZ (ω(X ,Y ))− ω(∇1

ZX ,Y )− ω(X ,∇1ZY )

and similarly for ∇, we gets

∇1Z (ω) = ∇Z (ω(X ,Y ))− ω(Ω(Z ,X ),Y )− ω(X ,Ω(Z ,Y ))

for any X,Y,Z ∈ V(M).

Part B2:Parallelness and Torsionness

Proof.Hence we can say that the connection ∇1 is parallel iff

ω (Ω(Z ,X ),Y ) = ω(X ,Ω(Z ,Y )) (14)

for any X,Y,Z ∈ V(M).

In other words, the tensor field

Ω ∈ Γ(⊗3T ∗M)

,defined byΩ(X ,Y ,Z ) = ω(Ω(X ,Y ),Z )

has to be symmetric w.r.t the last two arguments.On the other hand, it follows from an earlier argument that ∇1 istorsion free iff Ω is symmetric, i.e. if Ω is symmetric w.r.t the firsttwo arguments. We have thus proved the proposition.

Thank you for listening.



ω (Ω(Z ,X ),Y ) = ω(X ,Ω(Z ,Y )) (14)

for any X,Y,Z ∈ V(M).In other words, the tensor field

Ω ∈ Γ(⊗3T ∗M)


has to be symmetric w.r.t the last two arguments.

On the other hand, it follows from an earlier argument that ∇1 istorsion free iff Ω is symmetric, i.e. if Ω is symmetric w.r.t the firsttwo arguments. We have thus proved the proposition.




ω (Ω(Z ,X ),Y ) = ω(X ,Ω(Z ,Y )) (14)

for any X,Y,Z ∈ V(M).In other words, the tensor field

Ω ∈ Γ(⊗3T ∗M)


has to be symmetric w.r.t the last two arguments.On the other hand, it follows from an earlier argument that ∇1 istorsion free iff Ω is symmetric, i.e. if Ω is symmetric w.r.t the firsttwo arguments. We have thus proved the proposition.


What is a Symplectic manifold

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