On symmetrization of jets

Abstract

Let F = F ^(A,H,t) and ${F^1} = {F^{({A^1},{H^1},{t^1})}}$ be fiber product preserving bundle functors on the category FM _m of fibred manifolds Y with m-dimensional bases and fibred maps covering local diffeomorphisms. We define a quasi-morphism (A, H, t) → (A ¹, H ¹, t ¹) to be a GL(m)-invariant algebra homomorphism ν: A → A ¹ with t ¹ = ν ∘ t. The main result is that there exists an FM _m -natural transformation FY → F ¹ Y depending on a classical linear connection on the base of Y if and only if there exists a quasi-morphism (A, H, t) → (A ¹, H ¹, t ¹). As applications, we study existence problems of symmetrization (holonomization) of higher order jets and of holonomic prolongation of general connections.