On the existence of prolongations of connections by bundle functors

Abstract

We construct canonically a general connection Aᶠ (Ґ,∇) on Fₚ: FY → FM from a general connection Ґ on a fibred manifold p: Y → M by means of a projectable classical linear connection ∇ on Y , where 𝐹 : M𝑓 → Vβ is a vector bundle functor. In the case of a not necessarily vector bundle functor F : M𝑓 →FM we find some simple equivalent condition on the existence of a general connection A(Ґ,∇) on Fₚ : FY →FM from a general connection Ґ on Y → M by means of a projectable classical linear connection. ∇ on Y . We present a construction of a classical linear connection Aᶠ (∇) on FY from a projectable classical linear connection ∇ on Y for any fiber product preserving bundle functor 𝐹: FMₘ → FM. We characterize bundle functors 𝐹: Fₘ,ₙ→ FM which admit a construction of a classical linear connection A(∇) on FY from a projectable classical linear connection ∇ on Y . We characterize gauge bundle functors 𝐹: VBₘ,ₙ → FM which admit a construction of a classical linear connection A (D, ∇) on FE from a linear general connection D on E → M by means of a classical linear connection ∇ on M.