We determine all natural operators D transforming general connections Γ on fibred manifolds Y → M and torsion free classical linear connections ∇ on M into general connections D(Γ,∇) on the second order jet prolongation J2Y → M of Y → M
Any -dimensional subbundle of the cotangent bundle on an -dimensional manifold partitions into subsets ( being the minimum of and , the combinations of things taken 2 at a time). is the set on which the first derived systems of the subbundle has codimension .In this paper we prove the following:Theorem. Let and let be a generic -dimensional subbundle...
We classify all natural affinors on vertical fiber product preserving gauge bundle functors on vector bundles. We explain this result for some more known such . We present some applications. We remark a similar classification of all natural affinors on the gauge bundle functor dual to as above. We study also a similar problem for some (not all) not vertical fiber product preserving gauge bundle functors on vector bundles.
If (M,g) is a Riemannian manifold then there is the well-known base preserving vector bundle isomorphism TM → T* M given by v → g(v,−) between the tangent TM and the cotangent T* M bundles of M. In the present note first we generalize this isomorphism to the one JrTM → JrTM between the r-th order prolongation JrTM of tangent TM and the r-th order prolongation JrT M of cotangent TM bundles of M. Further we describe all base preserving vector bundle maps DM(g) : JrTM → JrT* M depending on a Riemannian...