Canonical 1-forms on Higher Order Adapted Frame Bundles

Let (M,F) be a foliated m + n-dimensional manifold M with n-dimensional foliation F . Let V be a finite dimensional vector space over R. We describe all canonical (Folm,n-invariant) V -valued 1-forms Θ: TP r(M,F) → V on the r-th order adapted frame bundle P r(M,F) of (M,F). All manifolds and maps are assumed to be of class C∞. A definition of foliations can be found in [2]. Let Folm,n be the category of foliated m + n-dimensional manifolds with n-dimensional foliations and their foliation respecting local diffeomorphisms. Let (M,F) be a Folm,n-object. Then we have an adapted r-th order frame bundle P (M,F) = { j 0φ | φ : (Rm+n,Fm,n)→ (M,F) is a Folm,n-map } overM of (M,F) with the target projection, where F = { {a}×R } a∈Rm is the n-dimensional canonical foliation on Rm+n. We see that P (M,F) is a principal bundle with the standard Lie group Gm,n = P (R,F)0 (with the multiplication given by the composition of jets) acting on the right on P (M,F) by the composition of jets. Every Folm,n-map ψ : (M1,F1)→ (M2,F2) induces a local fibred diffeomorphism (even a principal bundle local isomorphism) P ψ : P (M1,F1)→ P (M2,F2) given by P ψ(j 0φ) = j 0(ψ ◦ φ). Definition 1. Let V be a finite dimensional vector space over R. We recall that a Folm,n-canonical V -valued 1-form Θ on P r is a family of Folm,n-invariant V -valued 1-forms Θ(M,F) : TP (M,F)→ V on P (M,F) for any Folm,n-object (M,F). The invariance means that the V -valued 1-forms Θ(M1,F1) and Θ(M2,F2) are P Φ-related (P ΦΘ(M2,F2) = Θ(M1,F1)) for any Folm,n-map Φ: (M1,F1)→ (M2,F2). It is rather-known the following Folm,n-canonical Rm+n-valued 1-form on P 1(M,F). Example 1. For every Folm,n-object (M,F) we define an Rm+n-valued 1-form θ(M,F) on P 1(M,F) as follows. Consider the target projection β : P 1(M,F)→M 2000 Mathematics Subject Classification: Primary: 58A20; Secondary: 58A32.