Natural liftings $D:I\to IT{T}^{*}$ are classified for $n\ge 2$. It is proved that they form a 5-parameter family of operators.

We clarify how the natural transformations of fiber product preserving bundle functors on $ℱ{ℳ}_m$ can be constructed by using reductions of the rth order frame bundle of the base, $ℱ{ℳ}_m$ being the category of fibered manifolds with m-dimensional bases and fiber preserving maps with local diffeomorphisms as base maps. The iteration of two general r-jet functors is discussed in detail.

The authors study some geometrical constructions on the cotangent bundle $T^{*}M$ from the viewpoint of natural operations. First they deduce that all natural operators transforming functions on $T^{*}M$ into vector fields on $T^{*}M$ are linearly generated by the Hamiltonian vector field with respect to the canonical symplectic structure of $T^{*}M$ and by the Liouville vector field of $T^{*}M$. Then they determine all natural operators transforming pairs of functions on $T^{*}M$ into functions on $T^{*}M$. In this case, the main generator is...

We prove that the only natural differential operations between holomorphic forms on a complex manifold are those obtained using linear combinations, the exterior product and the exterior differential. In order to accomplish this task we first develop the basics of the theory of natural holomorphic bundles over a fixed manifold, making explicit its Galoisian structure by proving a categorical equivalence à la Galois.

[For the entire collection see Zbl 0742.00067.]This paper is devoted to a method permitting to determine explicitly all multilinear natural operators between vector-valued differential forms and between sections of several other natural vector bundles.

We prove, that $r$-th order gauge natural operators on the bundle of Cartan connections with a target in the gauge natural bundles of the order $(1,0)$ (“tensor bundles”) factorize through the curvature and its invariant derivatives up to order $r-1$. On the course to this result we also prove that the invariant derivations (a generalization of the covariant derivation for Cartan geometries) of the curvature function of a Cartan connection have the tensor character. A modification of the theorem is given for...

For natural numbers n ≥ 3 and r ≥ 1 all natural operators $A:T_{|ℳfₙ}^{(0,0)}⟿T^{(1,1)}{T}^{r*}$ transforming functions from n-manifolds into affinors (i.e. tensor fields of type (1,1)) on the r-cotangent bundle are classified.

The complete description of all natural operators lifting real valued functions to bundle functors on fibered manifolds is given. The full collection of all natural operators lifting projectable real valued functions to bundle functors on fibered manifolds is 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 $A$ be a Weil algebra. The bijection between all natural operators lifting vector fields from $m$-manifolds to the bundle functor $K^A$ of Weil contact elements and the subalgebra of fixed elements $SA$ of the Weil algebra $A$ is determined and the bijection between all natural affinors on $K^A$ and $SA$ is deduced. Furthermore, the rigidity of the functor $K^A$ is proved. Requisite results about the structure of $SA$ are obtained by a purely algebraic approach, namely the existence of nontrivial $SA$ is discussed.

Let $F_1$ be a natural bundle of order $r_1$; a basis of the $s$-th order differential operators of $F_1$ with values in $r_2$-th order bundles is an operator $D$ of that type such that any other one is obtained by composing $D$ with a suitable zero-order operator. In this article a basis is found in the following two cases: for $F_1=\text{semi}{F}^{r_1}$ (semi-holonomic $r_1$-th order frame bundle), $s=0$, $r_2<r_1$ and $F_1=F^1$ ($1$-st order frame bundle), $r_2\le s$. The author uses here the so-called method of orbit reduction which provides one with a criterion for checking...

Let $Y\to M$ be a fibered manifold over a manifold $M$ and $\mu :A\to B$ be a homomorphism between Weil algebras $A$ and $B$. Using the results of Mikulski and others, which classify product preserving bundle functors on the category of fibered manifolds, the author classifies all natural operators $T_{\text{proj}}Y\to T^{\mu }Y$, where $T_{\text{proj}}Y$ denotes the space of projective vector fields on $Y$ and $T^{\mu }$ the bundle functors associated with $\mu$.

Let us consider two closed surfaces $ℳ$, $𝒩$ of class $C^1$ and two functions $\varphi :ℳ\to ℝ$, $\psi :𝒩\to ℝ$ of class $C^1$, called measuring functions. The natural pseudodistance $d$ between the pairs $(ℳ,)$, $(𝒩,\psi )$ is defined as the infimum of $\Theta \left(f\right):={max}_{P\in ℳ}\left|\varphi \left(P\right)-\psi \left(f\left(P\right)\right)\right|$ as $f$ varies in the set of all homeomorphisms from $ℳ$ onto $𝒩$. In this paper we prove that the natural pseudodistance equals either $|c_1-c_2|$, $\frac{1}{2}\left|c_1-c_2\right|$, or $\frac{1}{3}\left|c_1-c_2\right|$, where $c_1$ and $c_2$ are two suitable critical values of the measuring functions. This shows that a previous relation between the natural pseudodistance and critical values...

A natural $T$-function on a natural bundle $F$ is a natural operator transforming vector fields on a manifold $M$ into functions on $FM$. For any Weil algebra $A$ satisfying $dimM\ge \mathrm{w}idth\left(A\right)+1$ we determine all natural $T$-functions on $T^{*}{T}^{A}M$, the cotangent bundle to a Weil bundle $T^{A}M$.

We determine all natural transformations T²₁T*→ T*T²₁ where $T_k^{r}M=J_0^r({ℝ}^k,M)$. We also give a geometric characterization of the canonical isomorphism ψ₂ defined by Cantrijn et al.