On bundles of covelocities

Abstract

For a Weil algebra A = \( \mathbb{D}\) r k /I = ℝ ⊕ N A and a manifold M satisfying dimM = m ≥ k, the coincidence of the space T A *M of A-covelocities T x A f: T x A M → T 0 A ℝ with the bundle of the r-th order covelocities T r *M is proved. For a Lie subgroup G A ⊆ G m r of I-preserving \( \mathbb{D}\) m r -automorphisms and a Lie group homomorphism p: G m r → G A it is proved that the space T V,p A *M of T x A f restricted to individual regular p(G m r )-orbits on T m r → M together with the extensions to other regular p(G m r )-orbits coincides with the natural bundle P r M[N A , l] with the standard fiber N A and the left action l: G m r × N A → N A induced by p.