Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 629b.Let us consider automorphisms of Hamiltonian bundles [Lalonde andMcDuff (2002)]. As a guide note that in the trivial case when B = pt, aHamiltonian structure on P is an identification of P with M up to symplecticisotopy. Hence the group of automorphisms of this structure canbe identified with Symp 0 (M, ω). In general, if {a} is a Hamiltonian structureon (P, π) and Φ ∈ Symp 0 (P, π) then Φ ∗ ({a}) = {a} iff Φ ∗ (a) = a forsome a in the class {a}, because Φ induces the identity map on the baseand a − a ′ ∈ π ∗ (H 2 (B)) when a ∼ a ′ . We therefore make the followingdefinition.Let (P, π, {a}) be a Hamiltonian structure on the symplectic bundleP → B and let Φ ∈ Symp(P, π). Then Φ is an automorphism of theHamiltonian structure (P, π, {a}) if Φ ∈ Symp 0 (P, π) and Φ ∗ ({a}) = {a}.The group formed by these elements is denoted by Aut(P, π, {a}).The following result is not hard to prove, but is easiest to see in thecontext of a discussion of the action of Ham(M) on H ∗ (M).Let P → B be a Hamiltonian bundle and Φ ∈ Symp 0 (P, π). Then thefollowing statements are equivalent [Lalonde and McDuff (2002)]:(i) Φ is isotopic to an element of Ham(P, π);(ii) Φ ∗ ({a}) = {a} for some Hamiltonian structure {a} on P ;(iii) Φ ∗ ({a}) = {a} for all Hamiltonian structures {a} on P .For any Hamiltonian bundle P → B, the group Aut(P, π, {a}) does notdepend on the choice of the Hamiltonian structure {a} put on P . Moreover,it contains Ham(P, π) and each element of Aut(P, π, {a}) is isotopic to anelement in Ham(P, π).The following characterization is now obvious:Let P be the product B ×M and {a} any Hamiltonian structure. Then:(i) Ham(P, π) consists of all maps from B to Ham(M, ω).(ii) Aut(P, π, {a}) consists of all maps Φ : B → Symp 0 (M, ω) for whichthe following composition is trivialπ 1 (B) −→ Φ∗π 1 (Symp 0 (M)) F −→ luxωH 1 (M, R).The basic reason why the above proposition holds is that Hamiltonianautomorphisms of (P, π) act trivially on the set of extensions of the fibersymplectic class. This need not be true for symplectic automorphisms. Forexample, ifπ : P = S 1 × M → S 1