J. Kurek

The natural affinors on the r-th order frame bundle

We describe all Mfm-natural affinors on the r-th order frame bundle UM = invJ5(Rm, M) over M. Manifolds and maps are assumed to be of class C°°. Manifolds are assumed to be finite dimensional and without boundaries. Let Mfm denote the category of m-dimensional manifolds and their embeddings (i.e. diffeomorphisms onto open subsets) and TM. denote the category of fibred manifolds and their fibred maps. For any m-manifold M we have the r-th order frame bundle LrM = mvJo(Rm, M) of M. Every Mfm-map ijj : M N induces Lrtp : LrM LrN by = O M is a A^/m-map. The correspondence Lr : M.fm ~* FM. is a bundle functor in the sense of [3]. A .M/m-natural affinor A on Lr is a family of Ai /m-invariant affinors (tensor fields of type (1,1)) A : TV M -> TLrM on UM for any Al/m-object M. The invariance means that A o TLrij) = TLrip o A for any M /m-map t/j : M —> N. In this note we describe all ,M/m-natural affinors A on Lr. We have the following examples of _A/f/m-natural affinors on Lr. EXAMPLE 1. We have the identity -M/m-natural affinor Id on Lr such that Id : TLrM —> TLrM is the identity map for any A^/m-object M. EXAMPLE 2. Let B : J ^ _ 1 T R M -»• ( J ^ T R M ) 0 be a linear map, where J J ~ T R M = {J^X I X G ; T ( R M ) } and ( J ^ T R M ) 0 = {jR0X X € * ( R M ) with XQ = 0} are the vector spaces. Given a vVf/m-object M we define a

Lagrangians and Euler morphisms from connections on the frame bundle

We classify all natural operators transforming torsion free classical linear connections ∇ on m‐dimensional manifolds M into r‐th order Lagrangians λ(∇) and Euler morphisms E(∇) on the linear frame bundle P1M. We also briefly write how this classification result can be generalized on higher order frame bundles PkM instead of P1M.

DISTRIBUTIONS ON THE COTANGENT BUNDLE FROM TORSION-FREE CONNECTIONS

of M such that dim(A(∇)ω) = q for any ω ∈ T ∗M . We have the following simple examples: (a) The zero distribution A[1](∇) such that A(∇)ω = {0} for any ω ∈ T ∗M ; (b) The vertical distribution A[2](∇) = V T ∗M ; (c) The full distribution A[3](∇) = TT ∗M ; (d) The ∇∗-horizontal distribution A[0](∇) = H∇ on T ∗M (i.e. the horizontal distribution of the linear general connection ∇∗ on T ∗M →M dual to ∇). We have the decomposition TT ∗M = V T ∗M ⊕H∇ . We recall that given v ∈ TxM and ω ∈ (T ∗M)x, x ∈ M , we have the ∇∗-horizontal lift v∇ ∗ ω of v at ω, i.e. the unique vector from H ∇∗ ω over v with respect to the cotangent bundle projection. We have the following family of modifications of H∇ ∗ ; (e) Given a canonically dependent on ∇ fibred map B(∇) : T ∗M → T ∗M⊗T ∗M covering idM , we have theB(∇)-modification A[B](∇) ofH∇ such that A(∇)ω := {v∇ ω + d dt |t=0(ω+t < B(∇)(ω), v >) | v ∈ TxM} for any ω ∈ (T ∗M)x, x ∈M . Then A[B](∇) is a smooth distribution on T ∗M of dimension m at any point. Clearly, if B(∇) = 0 then A[0](∇) = H∇ .