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Let $F$ be a $p$-dimensional foliation on an $n$-manifold $M$, and $T^{r}M$ the $r$-tangent bundle of $M$. The purpose of this paper is to present some reltionship between the foliation $F$ and a natural lifting of $F$ to the bundle $T^{r}M$. Let $L_q^r\left(F\right)$ $(q=0,1,\cdots ,r)$ be a foliation on $T^{r}M$ projectable onto $F$ and $L_q^r=\left\{L_q^r\left(F\right)\right\}$ a natural lifting of foliations to $T^{r}M$. The author proves the following theorem: Any natural lifting of foliations to the $r$-tangent bundle is equal to one of the liftings $L_0^r,L_1^r,\cdots ,L_n^r$. The exposition is clear and well organized.

[For the entire collection see Zbl 0699.00032.] Natural transformations of the Weil functor $T^A$ of A-velocities [, Commentat. Math. Univ. Carol. 27, 723-729 (1986; Zbl 0603.58001)] into an arbitrary bundle functor F are characterized. In the case where F is a linear bundle functor, the author deduces that the dimension of the vector space of all natural transformations of $T^A$ into F is finite and is less than or equal to $dim\left(F_0{ℛ}^k\right)$. The spaces of all natural transformations of Weil functors into linear functors...

One studies the flow prolongation of projectable vector fields with respect to a bundle functor of order $(r,s,q)$ on the category of fibered manifolds. As a result, one constructs an operator transforming connections on a fibered manifold $Y$ into connections on an arbitrary vertical bundle over $Y$. It is deduced that this operator is the only natural one of finite order and one presents a condition on vertical bundles over $Y$ under which every natural operator in question has finite order.

This paper is a contribution to the axiomatic approach to geometric objects. A collection of a manifold M, a topological space N, a group homomorphism E: Diff(M) → Homeo(N) and a function π: N → M is called a quasi-natural bundle if (1) π ∘ E(f) = f ∘ π for every f ∈ Diff(M) and (2) if f,g ∈ Diff(M) are two diffeomorphisms such that f|U = g|U for some open subset U of M, then E(f)|π^{-1}(U) = E(g)|π^{-1}(U). We give conditions which ensure that π: N → M is continuous. In particular, if (M,N,E,π)...

A complete description is given of all product preserving gauge bundle functors F on vector bundles in terms of pairs (A,V) consisting of a Weil algebra A and an A-module V with $di{m}_{ℝ}\left(V\right)<\infty$. Some applications of this result are presented.

Let $J^{r}T*M$ be the r-jet prolongation of the cotangent bundle of an n-dimensional manifold M and let $(J^{r}T*M)*$ be the dual vector bundle. For natural numbers r and n, a complete classification of all linear natural operators lifting 1-forms from M to 1-forms on $(J^{r}T*M)*$ is given.

We study the problem of how a map f:M → ℝ on an n-manifold M induces canonically an affinor $A\left(f\right):T{T}^{\left(r\right)}M\to T{T}^{\left(r\right)}M$ on the vector r-tangent bundle $T^{\left(r\right)}M=\left(J^r(M,ℝ)₀\right)*$ over M. This problem is reflected in the concept of natural operators $A:T_{|ℳfₙ}^{(0,0)}⇝T^{(1,1)}{T}^{\left(r\right)}$. For integers r ≥ 1 and n ≥ 2 we prove that the space of all such operators is a free (r+1)²-dimensional module over ${}^{\infty }\left(T^{\left(r\right)}ℝ\right)$ and we construct explicitly a basis of this module.

Let A be a Weil algebra and V be an A-module with dim V &lt; ∞. Let E → M be a vector bundle and let TE → TM be the vector bundle corresponding to (A,V). We construct canonically a linear semibasic tangent valued p-form Tφ : T E → ΛT*TM ⊗ TTE on TE → TM from a linear semibasic tangent valued p-form φ : E → ΛT*M ⊗­ TE on E → M. For the Frolicher-Nijenhuis bracket we prove that [[Tφ, Tψ]] = T ([[φ,ψ]]) for any linear semibasic tangent valued p- and q-forms φ and ψ on E → M. We apply these results...

Using a general connection Γ on a fibred manifold p:Y → M and a torsion free classical linear connection ∇ on M, we distinguish some “special” fibred coordinate systems on Y, and then we construct a general connection $\widetilde{ℱ}(\Gamma ,\nabla )$ on Fp:FY → FM for any vector bundle functor F: ℳ f → of finite order.

Let 𝓟𝓑 be the category of principal bundles and principal bundle homomorphisms. We describe completely the product preserving gauge bundle functors (ppgb-functors) on 𝓟𝓑 and their natural transformations in terms of the so-called admissible triples and their morphisms. Then we deduce that any ppgb-functor on 𝓟𝓑 admits a prolongation of principal connections to general ones. We also prove a "reduction" theorem for prolongations of principal connections into principal ones by means of Weil functors....

A classification of all $ℳf_m$-natural operators $A:G{r}_p⟿G{r}_{q}T*$ lifting p-dimensional distributions D ⊂ TM on m-manifolds M to q-dimensional distributions A(D) ⊂ TT*M on the cotangent bundle T*M is given.

We present a complete description of all product preserving bundle functors on the category ℱol of all foliated manifolds and their leaf respecting maps in terms of homomorphisms of Weil algebras.

We present a complete description of all fiber product preserving gauge bundle functors F on the category ${}_m$ of vector bundles with m-dimensional bases and vector bundle maps with local diffeomorphisms as base maps. Some corollaries of this result are presented.

All natural operators A transforming a linear vector field X on a vector bundle E into a vector field A(X) on the r-jet prolongation $J^{r}E$ of E are given. Similar results are deduced for the r-jet prolongations $J_v^{r}E$ and $J^{\left[r\right]}E$ in place of $J^{r}E$.

Let Y → M be a fibred manifold with m-dimensional base and n-dimensional fibres. Let r, m,n be positive integers. We present a construction $B^r$ of rth order holonomic connections $B^r(\Gamma ,\nabla ):Y\to J^{r}Y$ on Y → M from general connections Γ:Y → J¹Y on Y → M by means of torsion free classical linear connections ∇ on M. Then we prove that any construction B of rth order holonomic connections $B(\Gamma ,\nabla ):Y\to J^{r}Y$ on Y → M from general connections Γ:Y → J¹Y on Y → M by means of torsion free classical linear connections ∇ on M is equal to $B^r$. Applying...

A classification of natural liftings of foliations to the tangent bundle is given.

Let $F$ and $G$ be two natural bundles over $n$-manifolds. We prove that if $F$ is of type (I) and $G$ is of type (II), then any natural differential operator of $F$ into $G$ is of order 0. We give examples of natural bundles of type (I) or of type (II). As an application of the main theorem we determine all natural differential operators between some natural bundles.