Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 609The following relations hold for the SN–bracket:[ϑ, υ] SN = (−1) |ϑ||υ| [υ, ϑ] SN ,[ν, ϑ ∧ υ] SN = [ν, ϑ] SN ∧ υ + (−1) |ν||ϑ|+|ϑ| ϑ ∧ [ν, υ] SN ,(−1) |ν||ϑ|+|ν| [ν, ϑ ∧ υ] SN + (−1) |ϑ||ν|+|ϑ| [ϑ, υ ∧ ν] SN+ (−1) |υ||ϑ|+|υ| [υ, ν ∧ ϑ] SN = 0.In particular, let w = w µν ∂ µ ∧ ∂ ν be a bivector–field. We have[w, w] SN = w µα1 ∂ µ w α2α3 ∂ α1 ∧ ∂ α2 ∧ ∂ α3 . (4.127)Every bivector–field w on a manifold M induces the ‘sharp’ bundle mapw ♯ : T ∗ M → T M defined byw ♯ (p)⌋q := w(x)(p, q), w ♯ (p) = w µν (x)p µ ∂ ν , (p, q ∈ T ∗ x M).(4.128)A bivector–field w whose bracket (4.127) vanishes is called the Poissonbivector–field.Let ∧ r (M) denote the vector space of exterior r−forms on a manifoldM. By definition, ∧ 0 (M) = C ∞ (M) is the ring of smooth real functionson M. All exterior forms on M constitute the N−graded exterior algebra∧ ∗ (M) of global sections of the exterior bundle ∧T ∗ M with respect to theexterior product ∧. This algebra admits the exterior differentiald : ∧ r (M) → ∧ r+1 (M),dφ = dx µ ∧ ∂ µ φ = 1 r! ∂ µφ α1...α rdx µ ∧ dx α1 ∧ · · · dx αr ,which is nilpotent, i.e., d ◦ d = 0, and obeys the relationd(φ ∧ σ) = d(φ) ∧ σ + (−1) |φ| φ ∧ d(σ).The interior product (or, contraction) of a vector–field u = u µ ∂ µ and anexterior r−form φ on a manifold M is given by the coordinate expressionu⌋φ ==r∑ (−1) k−1u α kφr!α1...α k ...α rdx α1 ∧ · · · ∧ ̂dx α k∧ · · · ∧ dx(4.129)αrk=11(r − 1)! uµ φ µα2...α rdx α2 ∧ · · · ∧ dx αr ,