Exercises for Symplectic Geometry I
LMU M¨unchen, Summer term 09 Frederik Witt / Vincent Humili`ere Problem set 8, due for 02/07/09
Symplectic and Hamiltonian diffeomorphisms
1. The Hamiltonian group
Let (M, ω) be a symplectic manifold. We call(normalized) Hamiltonianany smooth functionH : [0,1]× M →Rsuch thatR1
0
R
MH(t, x)ωndt= 0 ifM is compact and which is compactly supported ifM is not compact. To any such Hamiltonian H(t, x) =Ht(x), one can associate the time dependent vector field XHt defined byιXHtω=dHt. We then denote byφtH the flow generated byXHt, and call itHamiltonian isotopy generated byH.
(a) InR2endowed with its standard symplectic form, write in coordinates the expression ofXHt. Does it reminds you of something ?
(b) Still inR2 in polar coordinates (r, θ) (thenω0=r dr∧dθ), suppose that H(t, r, θ) =h(r) for some functionh. Compute the Hamiltonian isotopy generated byH.
(c) Show thatφtH◦φtK=φtH]K and (φtH)−1=φtH, where
H]K(t, x) =H(t, x) +K(t,(φtH)−1(x)) and H(t, x) =−H(t, φtH(x)).
(d) Deduce that Ham(M, ω) = {φ1H|H Hamiltonian} is a subgroup of Symp(M, ω). The elements of Ham(M, ω) are calledHamiltonian diffeomorphisms.
(e) SupposeM is compact. We denote byAthe subset of Ham(M, ω) consisting of allφ1H forH time- independent hamiltonian (note that it is not a group). Prove that any Hamiltonian diffeomorphism can be written as the composition of elements inA.
Hint: consider the group generated byAand use Banyaga simplicity theorem.
2. Fixed points of symplectic diffeomorphisms Let (M, ω) be acompact symplectic manifold.
(a) With the notations of the previous exercise, show that any Hamiltonian diffeomorphism in Ahas at least two fixed points.
Hint: a smooth function on a compact manifold has always at least two critical points.
(b) In this question, we will prove that if H1(M,R) = 0, any symplectic diffeomorphism sufficiently close to the identity (in theC1 sens) admits at least two fixed points.
(i) We consider the symplectic manifold (M×M, ω⊕(−ω)). Show that there exists a neighbour- hood of the diagonal inM ×M symplectomorphic to a neighbourhood of the zero section in T∗M. Denote by Ψ such a symplectomorphism.
(ii) Let φbe a symplectic diffeomorphism ofM. Show that its graph Γφ is a Lagrangian subma- nifold of (M ×M, ω⊕(−ω)).
(iii) Suppose thatφisC1-close to the identity. Show that Ψ(Γφ) is the graph of a closed 1-form on M.
(iv) Conclude.
(c) Find a symplectic diffeomorphism of a compact manifold which does not have any fixed point.
3. A group homomorphism
Let (M, ω=dλ) be an exact symplectic manifold.
(a) Let φ be an Hamiltonian diffeomorphism. Show that there exists a unique compactly supported functionFφ such thatφ∗λ−λ=dFφ.
Hint: forHhamiltonian, considerFt=Rt
0(ιXHsλ−Hs)◦φsHdsand show thatφtH∗λ−λ=dFt
(b) Show that the formulaC(φ) =R
MFφωn defines a group homomorphismC: Ham(M, ω)→R (c) Deduce that for exact symplectic manifolds, Ham(M, ω) is not simple.
