Ivancevic_Applied-Diff-Geom

608 Applied Differential Geometry: A Modern IntroductionIn general, a vector–field τ = τ α ∂ α on a base X of a fibre bundle Y → Xinduces a vector–field on Y by means of a connection on this fibre bundle.Nevertheless, every natural fibre bundle Y → X admits the canonical lift ˜τonto Y of any vector–field τ on X. For example, if Y is the tensor bundle(4.11), the above canonical lift reads˜τ = τ µ ∂ µ + [∂ ν τ α1 ẋ να2···αmβ 1···β k+ . . . − ∂ β1 τ ν ẋ α1···αmνβ 2···β k− . . .]∂. (4.124)∂ẋ α1···αmβ 1···β kIn particular, we have the canonical lift onto the tangent bundle T X,and another one onto the cotangent bundle T ∗ X,˜τ = τ µ ∂ µ + ∂ ν τ α ẋ ν ∂∂ẋ α (4.125)˜τ = τ µ ∂ µ − ∂ β τ ν ∂ ẋ ν . (4.126)∂ẋ βA multivector–field ϑ of degree r (or simply a r-vector–field) on a manifoldM, by definition, is a global section of the bundle ∧ r T M → M. It isgiven by the coordinate expressionϑ = ϑ α1...αr ∂ α1 ∧ · · · ∧ ∂ αr , |ϑ| = r,where summation is over all ordered collections (λ 1 , ..., λ r ).Similarly, an exterior r−form on a manifold M with local coordinatesx α , by definition, is a global section of the skew–symmetric tensor bundle(exterior product) ∧ r T ∗ M → M,φ = 1 r! φ α 1...α rdx α1 ∧ · · · ∧ dx αr , |φ| = r.The 1–forms are also called the Pfaffian forms.The vector space V r (M) of r−vector–fields on a manifold M admits theSchouten–Nijenhuis bracket (or, SN bracket)[., .] SN : V r (M)×V s (M) → V r+s−1 (M)which generalizes the Lie bracket of vector–fields (4.123). The SN–brackethas the coordinate expression:ϑ = ϑ α1...αr ∂ α1 ∧ · · · ∧ ∂ αr , υ = υ α1...αs ∂ α1 ∧ · · · ∧ ∂ αs ,[ϑ, υ] SN = ϑ ⋆ υ + (−1) |ϑ||υ| υ ⋆ ϑ,whereϑ ⋆ υ = ϑ µα1...αr−1 ∂ µ υ α1...αs ∂ α1 ∧ · · · ∧ ∂ αr−1 ∧ ∂ α1 ∧ · · · ∧ ∂ αs .