Jiří M. Tomáš

On bundles of covelocities

AbstractFor a Weil algebra A =

The general rigidity result for bundles of A-covelocities and A-jets

\mathbb{D}

Product preserving bundle functors on fibered fibered manifolds

rk/I = ℝ ⊕ NA and a manifold M satisfying dimM = m ≥ k, the coincidence of the space TA*M of A-covelocities TxAf: TxAM → T0Aℝ with the bundle of the r-th order covelocities Tr*M is proved. For a Lie subgroup GA ⊆ Gmr of I-preserving

T

\mathbb{D}

Gauge-natural transformations of some cotangent bundles

mr-automorphisms and a Lie group homomorphism p: Gmr → GA it is proved that the space TV,pA*M of TxAf restricted to individual regular p(Gmr)-orbits on Tmr → M together with the extensions to other regular p(Gmr)-orbits coincides with the natural bundle PrM[NA, ℓ] with the standard fiber NA and the left action ℓ: Gmr × NA → NA induced by p.

Erratum to: “On Bundles of Covelocities”

For any product-preserving bundle functor F defined on the category TbA of fibered-fibered manifolds, we determine all natural operators transforming projectableprojectable vector fields on Y 6 Ob(J-^SA) to vector fields on FY. We also determine all natural afRnors on FY and prove a composition property analogous to that concerning Weil bundles. 0. Preliminaries The classical results by Kainz and Michor, [2], Eck, [1] and Luciano, [5] read that the product preserving bundle functors on the category Mf of manifolds are just Weil bundles, [4], Let us remind the result by Kolar, [3]. For a bundle functor F : Mf —> FM, denote by T the flow operator associated to F which is defined by TX = F(Fl?) for any vector field X on M. Further, consider an element c of a Weil algebra A and let L(c)M • TTM TTM denote the natural affinor by Koszul, [3] and [4]. Then we have a natural operator L(C)M°T : TM TTM lifting vector fields on a manifold M to a Weil bundle TM. As for the absolute natural operators T —• TT, i.e. independent on a vector field X, they are of the form AD for a derivation D € Der A. They are defined as follows. The Lie algebra Aut(A) associated to the Lie group of all algebra automorphisms of A is identified with the algebra of derivations Der A of A. For 1991 Mathematics Subject Classification: 58A20.