Ivancevic_Applied-Diff-Geom

612 Applied Differential Geometry: A Modern IntroductionWe shall deal with the following particular types of vector–fields anddifferential forms on a bundle Y −→ X [Sardanashvily (1993); Sardanashvily(1995); Giachetta et. al. (1997); Mangiarotti and Sardanashvily (2000a)]:• a projectable vector–field on Y ,u = u µ (x)∂ µ + u i (y)∂ i ,which covers a vector–field τ u = u µ (x)∂ µfollowing diagram commutes:on the base X such that theYπ❄Xuτ u✲ T YT π❄✲ T X• a vertical vector–field, u : Y → V Y, given by u = u i (y)∂ i , is aprojectable vector–field which covers τ u = 0;• an exterior horizontal form, φ : Y → ∧ r T ∗ X, given byφ = 1 r! φ α 1...α r(y)dx α1 ∧ · · · ∧ dx αr ;• a tangent–valued horizontal form, φ : Y → ∧ r T ∗ X ⊗ T Y, given byφ = 1 r! dxα1 ∧ · · · ∧ dx αr ⊗ [φ µ α 1...α r(y)∂ µ + φ i α 1...α r(y)∂ i ];• a vertical–valued horizontal form, φ : Y → ∧ r T ∗ X ⊗ V Y, given byφ = 1 r! φi α 1...α r(y)dx α1 ∧ · · · ∧ dx αr ⊗ ∂ i .• a vertical-valued soldering form, σ : Y → T ∗ X ⊗ V Y, given byσ = σ i α(y)dx α ⊗ ∂ i (4.138)and, in particular, the canonical soldering form on T X,θ X = dx α ⊗ ∂ α .