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We determine all natural transformations of the rth order cotangent bundle functor $T^{r*}$ into $T^{s*}$ in the following cases: r = s, r < s, r > s. We deduce that all natural transformations of $T^{r*}$ into itself form an r-parameter family linearly generated by the pth power transformations with p =1,...,r.

All natural affinors on the $r$-th order cotangent bundle $T^{r*}M$ are determined. Basic affinors of this type are the identity affinor id of $T{T}^{r*}M$ and the $s$-th power affinors $Q_M^s:T{T}^{r*}M\to V{T}^{r*}M$ with $s=1,\cdots ,r$ defined by the $s$-th power transformations $A_s^{r,r}$ of $T^{r*}M$. An arbitrary natural affinor is a linear combination of the basic ones.

For natural numbers r,s,q,m,n with s≥r≤q we determine all natural functions g: T *(J (r,s,q)(Y, R 1,1)0)*→R for any fibered manifold Y with m-dimensional base and n-dimensional fibers. For natural numbers r,s,m,n with s≥r we determine all natural functions g: T *(J (r,s)(Y, R)0)*→R for any Y as above.

All natural operators T ↝ T(T ⊗ T*) lifting vector fields X from n-dimensional manifolds M to vector fields B(X) on the bundle of affinors ™ ⊗ T*M are described.

Let P be a principal fiber bundle with the basis M and with the structural group G. A trivialization of P is a section of P. It is proved that there exists only one gauge natural operator transforming trivializations of P into principal connections in P. All gauge natural operators transforming trivializations of P and torsion free classical linear connections on M into classical linear connections on P are completely described.

We study the so-called inverse problem. Namely, given a prescribed skew-symmetric Ricci tensor we find (locally) a respective linear connection.

If $(M,g)$ is a Riemannian manifold, we have the well-known base preserving   vector bundle isomorphism $TM\widetilde{=}{T}^{*}M$ given by $v\to g(v,-)$ between the tangent $TM$ and the cotangent $T^{*}M$ bundles of $M$. In the present note, we generalize this isomorphism to the one $T^{\left(r\right)}M\widetilde{=}{T}^{r*}M$ between the $r$-th order vector tangent $T^{\left(r\right)}M={\left(J^r{(M,R)}_0\right)}^{*}$ and the $r$-th order cotangent $T^{r*}M=J^r{(M,R)}_0$ bundles of $M$. Next, we describe all base preserving  vector bundle maps $C_M\left(g\right):T^{\left(r\right)}M\to T^{r*}M$ depending on a Riemannian metric $g$ in terms of natural (in $g$) tensor fields on $M$.

We study  how a projectable general connection $\Gamma$ in a 2-fibred manifold $Y^2\to Y^1\to Y^0$  and a general vertical connection $\Theta$ in $Y^2\to Y^1\to Y^0$ induce a general connection $A(\Gamma ,\Theta )$ in $Y^2\to Y^1$.

By a torsion of a general connection $\Gamma$ on a fibered manifold $Y\to M$ we understand the Frölicher-Nijenhuis bracket of $\Gamma$ and some canonical tangent valued one-form (affinor) on $Y$. Using all natural affinors on higher order cotangent bundles, we determine all torsions of general connections on such bundles. We present the geometrical interpretation and study some properties of the torsions.

If (M,g) is a Riemannian manifold, we have the well-known base preserving vector bundle isomorphism TM ≅ T∗ M given by υ → g(υ,−) between the tangent TM and the cotangent T∗ M bundles of M. In the present note, we generalize this isomorphism to the one T(r)M ≅ Tr∗ M between the r-th order vector tangent T(r)M = (Jr(M,R)0)∗ and the r-th order cotangent Tr∗ M = Jr(M,R)0 bundles of M. Next, we describe all base preserving vector bundle maps CM(g) : T(r)M → Tr∗ M depending on a Riemannian metric g in...

We describe all canonical 2-forms Λ(ω) on the r-th order tangent bundle TM = J (;M) of a symplectic manifold (M, ω). As a corollary we deduce that all canonical symplectic structures Λ(ω) on TM over a symplectic manifold (M, ω) are of the form Λ(ω) = Σ αω for all real numbers α with α ≠ 0, where ω is the (k)-lift (in the sense of A. Morimoto) of ω to TM.

We extend the concept of r-order connections on fibred manifolds to the one of (r,s,q)-order projectable connections on fibred-fibred manifolds, where r,s,q are arbitrary non-negative integers with s ≥ r ≤ q. Similarly to the fibred manifold case, given a bundle functor F of order r on (m₁,m₂,n₁,n₂)-dimensional fibred-fibred manifolds Y → M, we construct a general connection ℱ(Γ,Λ):FY → J¹FY on FY → M from a projectable general (i.e. (1,1,1)-order) connection $\Gamma :Y\to J^{1,1,1}Y$ on Y → M by means of an (r,r,r)-order...

We prove that the problem of finding all $ℳf_m$-natural operators $B:Q⟿Q{T}^A$ lifting classical linear connections ∇ on m-manifolds M to classical linear connections $B_M\left(\nabla \right)$ on the Weil bundle $T^{A}M$ corresponding to a p-dimensional (over ℝ) Weil algebra A is equivalent to the one of finding all $ℳf_m$-natural operators $C:Q⟿(T{¹}_{p-1},T*\otimes T*\otimes T)$ transforming classical linear connections ∇ on m-manifolds M into base-preserving fibred maps $C_M\left(\nabla \right):T{¹}_{p-1}M={⨁}_M^{p-1}TM\to T*M\otimes T*M\otimes TM$.

Let F:ℱol → ℱℳ be a product preserving bundle functor on the category ℱol of foliated manifolds (M,ℱ) without singularities and leaf respecting maps. We describe all natural operators C transforming infinitesimal automorphisms X ∈ 𝒳(M,ℱ) of foliated manifolds (M,ℱ) into vector fields C(X)∈ 𝒳(F(M,ℱ)) on F(M,ℱ).

We reduce the problem of describing all $ℳf_m$-natural operators  transforming general affine connections on $m$-manifolds into general affine ones to the known description of all $GL\left({𝐑}^m\right)$-invariant maps ${𝐑}^{m*}\otimes {𝐑}^m\to {\otimes }^k{𝐑}^{m*}\otimes {\otimes }^k{𝐑}^m$ for $k=1,3$.

Let $ℳf_m$ be the category of $m$-dimensional manifolds and local diffeomorphisms and  let $T$ be the tangent functor on $ℳf_m$. Let $𝒱$ be the category of real vector spaces and linear maps and let ${𝒱}_m$ be the category of $m$-dimensional real vector spaces and linear isomorphisms. We characterize all regular covariant functors $F:{𝒱}_m\to 𝒱$ admitting $ℳf_m$-natural operators $\tilde{J}$ transforming classical linear connections $\nabla$ on $m$-dimensional manifolds $M$ into almost complex structures $\tilde{J}\left(\nabla \right)$ on $F\left(T\right)M={\bigcup }_{x\in M}F\left(T_{x}M\right)$.

If $m\ge p+1\ge 2$ (or $m=p\ge 3$), all  natural bilinear  operators $A$ transforming pairs of couples of vector fields and $p$-forms on $m$-manifolds $M$ into couples of vector fields and $p$-forms on $M$ are described. It is observed that  any natural skew-symmetric bilinear operator $A$ as above coincides with the generalized Courant bracket up to three (two, respectively) real constants.

We classify all natural affinors on vertical fiber product preserving gauge bundle functors $F$ on vector bundles. We explain this result for some more known such $F$. We present some applications. We remark a similar classification of all natural affinors on the gauge bundle functor $F^{*}$ dual to $F$ as above. We study also a similar problem for some (not all) not vertical fiber product preserving gauge bundle functors on vector bundles.