Ivancevic_Applied-Diff-Geom

532 Applied Differential Geometry: A Modern Introductionto these bases, the Lie bracket of sectionsξ = ξ α ∂ α + ξ p e p ,η = η µ ∂ µ + η q e qof the vector bundle T G P → Q reads[ξ, η] = (ξ µ ∂ µ η α − η µ ∂ µ ξ α )∂ α + (ξ α ∂ α η r − η α ∂ α ξ r + c r pqξ p η q )e r . (4.36)Putting ξ α = 0 and η µ = 0, we get the Lie bracket[ξ, η] = c r pqξ p η q e r (4.37)of sections of the vector bundle V G → P .A principal bundle P is also the general affine bundle modelled on theleft on the associated group bundle ˜P with the standard fibre G on which thestructure group G acts by the adjoint representation. The correspondingbundle map reads˜P × P −→ P,(˜p, p) ↦→ ˜pp.Note that the standard fibre of the group bundle ˜P is the group G, whilethat of the principal bundle P is the group space of G on which the structuregroup G acts on the left.A principal bundle P → Q with a structure Lie group G possesses thecanonical trivial vertical splittingα : V P → P × g l , π 2 ◦ α ◦ e m = J m ,where {J m } is a basis for the left Lie algebra g l and e m denotes the correspondingfundamental vector–fields on P . Given a principal bundle P → Q,the bundle T P → T Q is a principal bundleT P × T (Q × G) → T Pwith the structure group T G = G × g l where g l is the left Lie algebra ofleft–invariant vector–fields on the group G.If P → Q is a principal bundle with a structure group G, the exactsequence (4.13) can be reduced to the exact sequence0 → V G P ↩→ T G P → T Q → 0, (4.38)where T G P = T P/G, V G P = V P/Gare the quotients of the tangent bundle T P of P and the vertical tangentbundle V P of P respectively by the canonical action (4.31) of G on P on theright. The bundle V G P → Q is called the adjoint bundle. Its standard fibre