Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 627B Ham(M)˜g1 ✒❄B ✲g B Symp(M)Hamiltonian structures are in bijective correspondence with homotopyclasses of such lifts. There are two stages to choosing the lift ˜g: one firstlifts g to a map ĝ into B Symp 0 (M, ω), where Symp 0 is the identity componentof Symp, and then to a map ˜g into B Ham(M, ω). As we will showbelow, choosing ĝ is equivalent to fixing the isotopy class of an identificationof (M, ω) with the fiber (M b0 , ω b0 ) over the base point b 0 . If B issimply connected, in particular if B is a single point, there is then a uniqueHamiltonian structure on P , i.e., a unique choice of lift ˜g. Before describingwhat happens in the general case, we discuss properties of the extensionsτ.Let τ ∈ Ω 2 (P ) be a closed extension of the symplectic forms on thefibers. Given a loop, γ : S 1 → B, based at b 0 , and a symplectic trivializationT γ : γ ∗ (P ) → S 1 × (M, ω) that extends the given identification of M b0 withM, push forward τ to a form (T γ ) ∗ τ on S 1 × (M, ω). Its characteristic flowround S 1 is transverse to the fibers and defines a symplectic isotopy φ t of(M, ω) = (M b0 , ω b0 ) whose flux, as a map from H 1 (M) → R, is equal to(T γ ) ∗ [τ]([S 1 ] ⊗ ·). This flux depends only on the cohomology class a of τ.Moreover, as we mentioned above, any extension a of the fiber class [ω] canbe represented by a form τ that extends the ω b . Thus, given T γ and anextension a = [τ] ∈ H 2 (P ) of the fiber symplectic class [ω], it makes senseto define the flux class f(T γ , a) ∈ H 1 (M, R) byf(T γ , a)(δ) = (T γ ) ∗ (a)(γ ⊗ δ)for all δ ∈ H 1 (M).The equivalence class [f(T γ , a)] ∈ H 1 (M, R)/Γ ω does not depend on thechoice of T γ : indeed two such choices differ by a loop φ in Symp 0 (M, ω)and so the differencef(T γ , a) − f(T ′ γ, a) = f(T γ , a) ◦ Tr φ = ω ◦ Tr φbelongs to Γ ω . The following lemma is elementary: If π : P → B is asymplectic bundle satisfying the above conditions, there is an extension a