SYMPLECTIC GEOMETRY AND LAGRANGIAN SUBMANIFOLDS

This note provides background and exercises on Lagrangian submanifolds Λ ⊂ T ∗M in acotangent bundle of a manifold.

1. Symplectic linear algebra

We recall that a symplectic vector space (V, σ) is an even dimensional vector space with anon-degenerate skew-symmetric 2-form σ. That is, σ(v, w) = −σ(w, v) and v → σ(v, ·) is anisomorphism from V → V ∗.

Exercise Show that an odd-dimensional vector space cannot carry a non-degenerate skew-symmetric 2-form.

Exercise Show that there exists a symplectic basis ej, fk(j, k = 1, . . . , n) 2n = dimV for σ sothat σ(ej, ek) = 0 = σ(fj, fk), σ(ej, fk) = δjk.

A Lagrangian subspace of a symplectic vector space of dimension 2n is a subspace L ofdimension n so that σ|L = 0. That is σ(v, w) = 0 for all v, w ∈ L.

For instance, Re1, . . . , en, Rf1, . . . , fn are Lagrangian subspaces.A universal construction of symplectic vector spaces is as follows: Let W be a vector space

and W ∗ be its dual. Let V = W ⊕W ∗ with symplectic form

σ0(e⊕ f, e′ ⊕ f ′) = f ′(e)− f(e′).

Exercise Show that every symplectic vector space is equivalent to (V, σ0).

Consider the possible Lagrangian subspaces of (V, σ0) which are graphs over W , i.e. fj =∑nk=1 Ajkek. That is, let A : W → W ∗ be a linear map and let ΓA = (w,Aw) ⊂ W ⊕W ∗ be

its graph. Then ΓA is Lagrangian if and only if A is symmetric. Proof:

σ0((w,Aw), (w′, Aw′)) = (Aw′)(w)− (Aw)(w′).

2. Co-tangent bundles

The cotangent bundle is denoted by π : T ∗M →M .We recall that a cotangent bundle carries a canonical 1-form α = ξ · dx defined by

αx,ξ(v) = ξ(Dπv).

It may also be described as follows: A section of π : T ∗M → M is a co-vector field η : M →T ∗M . Then α is the unique element of Ω1(T ∗M) (i.e. unique 1-form on T ∗M) satisfyingη∗α = η.

Let x = (x1, . . . , xn) be any local coordinate system on M . Then dxj define local covectorfields giving a local trivialization of T ∗M → M . One defines the dual coordinates on T ∗Mof a covector η =

∑j ηjdxj by the coordinates relative to the frame field dx1, . . . , dxn, i.e.

ξj(η) = ηj.

Exercises1

2 SYMPLECTIC GEOMETRY AND LAGRANGIAN SUBMANIFOLDS

• Show that αx,ξ(v) := ξ(Dπv) implies that η∗α = η for all η.

• Show that in the (xj, ξk) coordinates, α =∑

j ξjdxj.

The canonical symplectic form ω of T ∗M is defined by: ω = dα. Thus, in the above localcoordinates,

ω =∑j

dξj ∧ dxj.

Such coordinates are called symplectic. In any coordinate system yj on T ∗M (j = 1, . . . , 2n)one could write

ω =n∑

j,k=1

ωijdyi ∧ dyj,

but in the (x, ξ) coordinates ω has constant coefficients

(ωij)

= J :=

0 I

−I 0

,

where we use the basis ∂∂xj, ∂∂ξk

for T (T ∗M). That is, we have

ω(∂

∂xj,∂

∂xk) = 0 = ω(

∂

∂ξj,∂

∂ξk), ω(

∂

∂xj,∂

∂ξk) = δjk.

Note that for fixed (x, ξ), T(x,ξ)T∗M is a symplectic vector space with the symplectic form

ωx,ξ.

2.1. Lagrangian submanifolds.

Definition 2.1. A submanifold Λ ⊂ T ∗M of dimension n = dimM is Lagrangian if ω|TΛ = 0.That is, ωλ(X, Y ) = 0 for all λ ∈ Λ, X, Y ∈ TλΛ.

A Lagrangian submanifold is called projectible if

π : Λ→M

is a diffeomorphism. If U ⊂ M is an open set we say that π : Λ|U → U is locally projectible ifthe projection is a local diffeomorphism.

A section of T ∗M is the same as a covector field η : M → T ∗M . The graph of a covectorfield is the submanifold Γη = (x, ηx) : x ∈M.

Exercise

• Show that Γη is a Lagrangian submanifold if and only if η is a closed 1-form.• Show that Λ ⊂ T ∗M is a globally projectible Lagrangian submanifold if and only if

Λ = Γη where η is a closed 1-form, i.e. dη = 0.• Show that if Λ is locally projectible over a contractible open set U , then Λ = ΓdS for

some smooth S : U → R, i.e. it is the graph of an exact form form dS. S is called alocal generating function of Λ. It is determined up to a constant.

In general, Lagrangian submanifolds are not projectible, even locally. For instance, everycurve in T ∗R defines a Lagrangian submanifold. The unit circle x2 +ξ2 = 1 is only projectibleaway from the “turning points” ξ = 0, x = ±1 and is a two-sheeted cover over (−1, 1). The

SYMPLECTIC GEOMETRY AND LAGRANGIAN SUBMANIFOLDS 3

image π(Λ) ⊂ M is called the classically allowed region for Λ. The singular set λ ∈ Λ :ker dπλ 6= 0 is called the Maslov singular cycle.

Since Lagrangian submanifolds are rarely projectible, they are often parametrized by La-grangian immersions,

ι : Λ→ T ∗M.

2.2. Hamilton vector fields. The Hamilton vector field XH of H : T ∗M → R is the symplec-tic gradient of H, i.e. ω(XH ·) = dH. Unlike the Riemannian metric gradient, XH is tangentto level sets of H.

Exercise Prove that XH is tangent to H = E.

A key fact is the following (Hamilton-Jacobi theorem)

Proposition 2.2. Suppose that Λ ⊂ T ∗M is a Lagrangian submanifold and tht Λ ⊂ H = E,Then XH is tangent to Λ.

Proof. By assumption ω|TΛ = 0 and since TΛ has dimension n, if X is a vector in Tλ(T∗M) such

that ωλ(X, Y ) = 0 for all Y ∈ TλΛ then X ∈ TλΛ. Hence it suffices to show that ωλ(XH , Y ) = 0or all Y ∈ TλΛ. But ωλ(XH , Y ) = dH(Y ) = 0 for all Y ∈ Tλ(H = E).

References