### Natural transformations of foliations into foliations on the cotangent bundle

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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 $dim\left({F}_{0}{\mathcal{R}}^{k}\right)$. 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}_{\mathbb{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 ${T}^{\left(r\right)}M=\left({J}^{r}(M,\mathbb{R})\u2080\right)*$ over M. This problem is reflected in the concept of natural operators $A:{T}_{|\mathcal{M}f\u2099}^{(0,0)}\u21dd{T}^{(1,1)}{T}^{\left(r\right)}$. 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 ${}^{\infty}\left({T}^{\left(r\right)}\mathbb{R}\right)$ and we construct explicitly a basis of this module.

Let A be a Weil algebra and V be an A-module with dim V < ∞. 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 $\tilde{\mathcal{F}}(\Gamma ,\nabla )$ 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 $\mathcal{M}{f}_{m}$-natural operators $A:G{r}_{p}\u27ffG{r}_{q}T*$ 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 $F={F}^{(A,H,t)}$ and ${F}^{1}={F}^{({A}^{1},{H}^{1},{t}^{1})}$ be fiber product preserving bundle functors on the category ${\mathrm{\mathcal{F}\mathcal{M}}}_{m}$ of fibred manifolds $Y$ with $m$-dimensional bases and fibred maps covering local diffeomorphisms. We define a quasi-morphism $(A,H,t)\to ({A}^{1},{H}^{1},{t}^{1})$ to be a $GL\left(m\right)$-invariant algebra homomorphism $\nu :A\to {A}^{1}$ with ${t}^{1}=\nu \circ t$. The main result is that there exists an ${\mathrm{\mathcal{F}\mathcal{M}}}_{m}$-natural transformation $FY\to {F}^{1}Y$ depending on a classical linear connection on the base of $Y$ if and only if there exists a quasi-morphism $(A,H,t)\to ({A}^{1},{H}^{1},{t}^{1})$. As applications, we study existence problems of symmetrization (holonomization)...

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.

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