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3 Lagrangian Submanifolds
Homework 3:
Tautological Form and Symplectomorphisms
This set of problems is from [53].
1. Let (M, ω ) be a symplectic manifold, and let α be a 1-form such that
ω = −d α .
Show that there exists a unique vector field v such that its interior product with
ω is α , i.e., ıv ω = −α .
Prove that, if g is a symplectomorphism which preserves α (that is, g∗ α = α ),
then g commutes with the one-parameter group of diffeomorphisms generated
by v, i.e.,
(exptv) ◦ g = g ◦ (exp tv) .
Hint: Recall that, for p ∈ M, (exptv)(p) is the unique curve in M solving the ordinary
differential equation
! d
dt (exptv(p)) = v(exptv(p))
(exptv)(p)|t=0 = p
for t in some neighborhood of 0. Show that g ◦ (exptv) ◦ g−1 is the one-parameter
group of diffeomorphisms generated by g∗ v. (The push-forward of v by g is defined
by (g∗ v)g(p) = dg p (v p ).) Finally check that g preserves v (that is, g∗ v = v).
2. Let X be an arbitrary n-dimensional manifold, and let M = T ∗ X. Let
(U, x1 , . . . , xn ) be a coordinate system on X, and let x1 , . . . , xn , ξ1 , . . . , ξn be
the corresponding coordinates on T ∗ U.
Show that, when α is the tautological 1-form on M (which, in these coordinates, is ∑ ξi dxi ), the vector field v in the previous exercise is just the vector field
∑ ξi ∂∂ξi .
Let exptv, −∞ < t < ∞, be the one-parameter group of diffeomorphisms generated by v.
Show that, for every point p = (x, ξ ) in M,
(exptv)(p) = pt
where
pt = (x, et ξ ) .
3. Let M be as in exercise 2.
Show that, if g is a symplectomorphism of M which preserves α , then
g(x, ξ ) = (y, η )
=⇒
g(x, λ ξ ) = (y, λ η )
for all (x, ξ ) ∈ M and λ ∈ R.
Conclude that g has to preserve the cotangent fibration, i.e., show that there exists
a diffeomorphism f : X → X such that π ◦ g = f ◦ π , where π : M → X is the
projection map π (x, ξ ) = x.
Finally prove that g = f# , the map f# being the symplectomorphism of M lifting f .
3.4 Application to Symplectomorphisms
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Hint: Suppose that g(p) = q where p = (x, ξ ) and q = (y, η ).
Combine the identity
(dg p )∗ αq = α p
with the identity
d πq ◦ dg p = d fx ◦ d π p .
(The first identity expresses the fact that g∗ α = α , and the second identity is obtained
by differentiating both sides of the equation π ◦ g = f ◦ π at p.)
4. Let M be as in exercise 2, and let h be a smooth function on X. Define τh : M → M
by setting
τh (x, ξ ) = (x, ξ + dhx ) .
Prove that
τh∗ α = α + π ∗ dh
where π is the projection map
M (x, ξ )
↓
↓π
X
x
Deduce that
τh∗ ω = ω ,
i.e., that τh is a symplectomorphism.