Lifting right-invariant vector fields and prolongation of connections

Abstract

Abstract. We describe all PBm(G)-gauge-natural operators A lifting right-invariant vector fields X on principal G-bundles P → M with m-dimensional bases into vector fields A(X) on the rth order principal prolongation W r P = P r M ×M J r P of P → M . In other words, we classify all PBm(G)-natural transformations J r LP ×M W r P → T W r P = LW r P ×M W r P covering the identity of W r P , where J r LP is the r-jet prolongation of the Lie algebroid LP = T P/G of P , i.e. we find all PBm(G)-natural transformations which are similar to the Kumpera-Spencer isomorphism J r LP = LW r P . We formulate axioms which characterize the flow operator of the gauge-bundle W r P → M . We apply the flow operator to prolongations of connections.0. Introduction. Let G be a Lie group with Lie algebra L(G) and e ∈ G be the unit element. Let PB m (G) denote the category of all principal G-bundles with m-dimensional bases and their (local) principal bundle isomorphisms over the identity group homomorphism.Let Φ, Ψ : P → Q be PB m (G)-maps. Let x ∈ M . The following conditions are equivalent: (i) j r p 0 Φ = j r p 0 Ψ for some p 0 ∈ P x ; (ii) j r p Φ = j r p Ψ for any p ∈ P x . We write j r x Φ = j r x Ψ if these conditions are satisfied (see [3]). Let P → M be a principal G-bundle with m-dimensional basis. Its rth principal prolongation W r P is defined as the space of all r-jets j r 0 ϕ of local trivializations ϕ : R m × G → P . By [3], W r P → M is a principal bundle with the structure group W r m G := J r 0 (R m × G, R m × G) 0 , and the fibred manifold W r P → M coincides with the fibred product P r M × M J r P , whereEvery PB m (G)-map Φ : P → Q can be extended (via composition of jets) to a principal bundle (local) isomorphism W r Φ :2000 Mathematics Subject Classification: 58A20, 58A32.