Ivancevic_Applied-Diff-Geom

610 Applied Differential Geometry: A Modern Introductionwhere the caret ̂· denotes omission. The following relations hold:φ(u 1 , . . . , u r ) = u r ⌋ · · · u 1 ⌋φ, (4.130)u⌋(φ ∧ σ) = u⌋φ ∧ σ + (−1) |φ| φ ∧ u⌋σ, (4.131)[u, u ′ ]⌋φ = u⌋d(u ′ ⌋φ) − u ′ ⌋d(u⌋φ) − u ′ ⌋u⌋dφ, (φ ∈ ∧ 1 (M)). (4.132)Recall from section 3.7 above, that the Lie derivative L u σ of an exteriorform σ along a vector–field u is defined by the Cartan relationIt satisfies the relationIn particular, if f is a function, thenL u σ = u⌋dσ + d(u⌋σ).L u (φ ∧ σ) = L u φ ∧ σ + φ ∧ L u σ.L u f = u(f) = u⌋df.It is important for dynamical applications that an exterior form φ is invariantunder a local 1–parameter group of diffeomorphisms G t of M (i.e.,G ∗ t φ = φ) iff its Lie derivative L u φ along the vector–field u, generating G t ,vanishes.Let Ω be a two–form on M. It defines the ‘flat’ bundle map Ω ♭ , asΩ ♭ : T M → T ∗ M, Ω ♭ (v) = −v⌋Ω(x), (v ∈ T x M). (4.133)In coordinates, if Ω = Ω µν dx µ ∧ dx ν and v = v µ ∂ µ , thenΩ ♭ (v) = −Ω µν v µ dx ν .One says that Ω is of constant rank k if the corresponding map (4.133) isof constant rank k (i.e., k is the greatest integer n such that Ω n is not thezero form). The rank of a nondegenerate two–form is equal to dim M. Anondegenerate closed two–form is called the symplectic form.Given a manifold map f : M → M ′ , any exterior k-form φ on M ′induces the pull–back exterior form f ∗ φ on M by the conditionf ∗ φ(v 1 , . . . , v k )(x) = φ(T f(v 1 ), . . . , T f(v k ))(f(x))for an arbitrary collection of tangent vectors v 1 , · · · , v k ∈ T x M. The followingrelations hold:f ∗ (φ ∧ σ) = f ∗ φ ∧ f ∗ σ,df ∗ φ = f ∗ (dφ).