Ivancevic_Applied-Diff-Geom

806 Applied Differential Geometry: A Modern Introduction• a tangent–valued projectable horizontal formφ = dx α1 ∧ · · · ∧ dx αr ⊗ (φ µ α 1...α r∂ µ + φ i α 1...α r∂ i )= dx α1 ∧ · · · ∧ dx αr ⊗ [φ µ α 1...α r(∂ µ + y i µ∂ i ) + (φ i α 1...α r− φ µ α 1...α ry i µ)∂ i ]and, e.g., the canonical 1–formθ Y = dx α ⊗ ∂ α + dy i ⊗ ∂ i = α + θ = dx α ⊗ ̂∂ α + ̂dy i ⊗ ∂ i= dx α ⊗ (∂ α + y i α∂ i ) + (dy i − y i αdx α ) ⊗ ∂ i . (5.15)The splitting (5.15) implies the canonical horizontal splitting of theexterior differentiald = d θY = d H + d V = d α + d θ . (5.16)Its components d H and d Vact on the pull–backsφ α1...α r(y)dx α1 ∧ · · · ∧ dx αrof horizontal exterior forms on a bundle Y → X onto J 1 (X, Y ) by π 01 . Inthis case, d H makes the sense of the total differentiald H φ α1...α r(y)dx α1 ∧· · ·∧dx αr = (∂ µ +y i µ∂ i )φ α1...α r(y)dx µ ∧dx α1 ∧· · ·∧dx αr ,whereas d Vis the vertical differentiald V φ α1...α r(y)dx α1 ∧· · ·∧dx αr = ∂ i φ α1...α r(y)(dy i −y i µdx µ )∧dx α1 ∧· · ·∧dx αr .If φ = ˜φω is an exterior horizontal density on Y → X, we havedφ = d V φ = ∂ i˜φdy i ∧ ω.5.3 Connections as Jet FieldsRecall that one can introduce the notion of connections on a general fibrebundle Y −→ X in several equivalent ways. In this section, following [Giachettaet. al. (1997); Kolar et al. (1993); Mangiarotti and Sardanashvily(2000a); Saunders (1989)], we start from the traditional definition of a connectionas a horizontal splitting of the tangent space to Y at every pointy ∈ Y .A connection on a fibre bundle Y → X is usually defined as a linearbundle monomorphismΓ : Y × T X → T Y, Γ : ẋ α ∂ α ↦→ ẋ α (∂ α + Γ i α(y)∂ i ), (5.17)