Abstract
We describe all natural operators \(A\) transforming general connections \(\Gamma\) on fibred manifolds \(Y \rightarrow M\) and torsion-free classical linear connections \(\Lambda\) on \(M\) into general connections \(A(\Gamma,\Lambda)\) on the fibred product \(J^{&lt;q&gt;}Y \rightarrow M\) of \(q\) copies of the first jet prolongation \(J^{1}Y \rightarrow M\).<br /><br />
Highlights
All manifolds are smooth, Hausdorff, finite dimensional and without boundaries
Let us recall that an r-th order connection on a fibred manifold p : Y → M is a section Θ : Y → JrY of the r-jet prolongation β : JrY → Y of p : Y → M
If F : F Mm,n → F M is a bundle functor on F Mm,n of order r = 1 and Γ is a general connection on an F Mm,n-object p : Y → M and ∇ is a torsion-free classical linear connection on M, one can obtain the general connection F(Γ, ∇) as in [2]
Summary
Hausdorff, finite dimensional and without boundaries. Maps are assumed to be smooth, i.e. of class C∞. If F : F Mm,n → F M is a bundle functor on F Mm,n of order r = 1 and Γ is a general connection on an F Mm,n-object p : Y → M and ∇ is a torsion-free classical linear connection on M , one can obtain the general connection F(Γ, ∇) as in [2]. An F Mm,n-natural operator D : J1 × Qτ (B) J1(F → B) transforming general connections Γ on fibred manifolds Y → M and torsion-free classical linear connections ∇ on M into general connections D(Γ, ∇) : F Y → J1F Y on F Y → M is a system of regular operators DY : Con(Y → M )×Qτ (M ) → Con(F Y → M ), (p : Y → M ) ∈ Obj(F Mm,n) satisfying the F Mm,ninvariance condition. |α|+|β|≤r−1 j=1 k=1 holds for any multiindex β ∈ (N ∪ {0})n such that |β| ≤ r − 1, where j0r−1 m dxi
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