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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...

Let $Y\to M$ be a fibred manifold with $m$-dimensional base and $n$-dimensional fibres and $E\to M$ be a vector bundle with the same base $M$ and with $n$-dimensional fibres (the same $n$). If $m\ge 2$ and $n\ge 3$, we classify all canonical constructions of a classical linear connection $A(\Gamma ,\Lambda ,\Phi ,\Delta )$ on $Y$ from a system $(\Gamma ,\Lambda ,\Phi ,\Delta )$ consisting of a general connection $\Gamma$ on $Y\to M$, a torsion free classical linear connection $\Lambda$ on $M$, a vertical parallelism $\Phi :Y{\times }_{M}E\to VY$ on $Y$ and a linear connection $\Delta$ on $E\to M$. An example of such $A(\Gamma ,\Lambda ,\Phi ,\Delta )$ is the connection $(\Gamma ,\Lambda ,\Phi ,\Delta )$ by I. Kolář.

The complete description of all product preserving bundle functors on fibered manifolds in terms of natural transformations between product preserving bundle functors on manifolds is given.