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 describe all ${}_m\left(G\right)$-gauge-natural operators lifting right-invariant vector fields X on principal G-bundles P → M with m-dimensional bases into vector fields (X) on the rth order principal prolongation $W^{r}P=P^{r}M{\times }_M{J}^{r}P$ of P → M. In other words, we classify all ${}_m\left(G\right)$-natural transformations $J^{r}LP{\times }_M{W}^{r}P\to T{W}^{r}P=L{W}^{r}P{\times }_M{W}^{r}P$ covering the identity of $W^{r}P$, where $J^{r}LP$ is the r-jet prolongation of the Lie algebroid LP=TP/G of P, i.e. we find all ${}_m\left(G\right)$-natural transformations which are similar to the Kumpera-Spencer isomorphism $J^{r}LP=L{W}^{r}P$. We formulate axioms which characterize...

In a previous paper we have given a complete description of linear liftings of p-forms on n-dimensional manifolds M to q-forms on $T^{A}M$, where $T^A$ is a Weil functor, for all non-negative integers n, p and q, except the case p = n and q = 0. We now establish formulas connecting such liftings and the exterior derivative of forms. These formulas contain a boundary operator, which enables us to define a homology of the Weil algebra A. We next study the case p = n and q = 0 under the condition that A is acyclic....

The second order transverse bundle $T_{}^{2}M$ of a foliated manifold $M$ carries a natural structure of a smooth manifold over the algebra ${𝔻}^2$ of truncated polynomials of degree two in one variable. Prolongations of foliated mappings to second order transverse bundles are a partial case of more general ${𝔻}^2$-smooth foliated mappings between second order transverse bundles. We establish necessary and sufficient conditions under which a ${𝔻}^2$-smooth foliated diffeomorphism between two second order transverse bundles maps...

We give a classification of all linear natural operators transforming affinors on each n-dimensional manifold M into affinors on $T^{A}M$, where $T^A$ is the product preserving bundle functor given by a Weil algebra A, under the condition that n ≥ 2.

The paper contains a classification of linear liftings of skew symmetric tensor fields of type $(1,2)$ on $n$-dimensional manifolds to tensor fields of type $(1,2)$ on Weil bundles under the condition that $n\ge 3.$ It complements author’s paper “Linear liftings of symmetric tensor fields of type $(1,2)$ to Weil bundles” (Ann. Polon. Math. 92, 2007, pp. 13–27), where similar liftings of symmetric tensor fields were studied. We apply this result to generalize that of author’s paper “Affine liftings of torsion-free connections...

We define equivariant tensors for every non-negative integer $p$ and every Weil algebra $A$ and establish a one-to-one correspondence between the equivariant tensors and linear natural operators lifting skew-symmetric tensor fields of type $(p,0)$ on an $n$-dimensional manifold $M$ to tensor fields of type $(p,0)$ on $T^{A}M$ if $1\le p\le n$. Moreover, we determine explicitly the equivariant tensors for the Weil algebras ${𝔻}_k^r$, where $k$ and $r$ are non-negative integers.

This paper contains a classification of all linear liftings of symmetric tensor fields of type (1,2) on n-dimensional manifolds to any tensor fields of type (1,2) on Weil bundles under the condition that n ≥ 3.

We give a classification of all linear natural operators transforming $p$-vectors (i.e., skew-symmetric tensor fields of type $(p,0)$) on $n$-dimensional manifolds $M$ to tensor fields of type $(q,0)$ on $T^{A}M$, where $T^A$ is a Weil bundle, under the condition that $p\ge 1$, $n\ge p$ and $n\ge q$. The main result of the paper states that, roughly speaking, each linear natural operator lifting $p$-vectors to tensor fields of type $(q,0)$ on $T^A$ is a sum of operators obtained by permuting the indices of the tensor products of linear natural operators lifting...

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.

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.

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.

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 consider a vector bundle $E\to M$ and the principal bundle $PE$ of frames of $E$. We determine all natural transformations of the connection bundle of the first order principal prolongation of principal bundle $PE$ into itself.

We study geometrical properties of natural transformations $T^{A}T*\to T*T^A$ depending on a linear function defined on the Weil algebra A. We show that for many particular cases of A, all natural transformations $T^{A}T*\to T*T^A$ can be described in a uniform way by means of a simple geometrical construction.

Let $M$ be a differentiable manifold with a pseudo-Riemannian metric $g$ and a linear symmetric connection $K$. We classify all natural (in the sense of [KMS]) 0-order vector fields and 2-vector fields on $TM$ generated by $g$ and $K$. We get that all natural vector fields are of the form $$E\left(u\right)=\alpha \left(h\left(u\right)\right)u^H+\beta \left(h\left(u\right)\right)u^V,$$ where $u^V$ is the vertical lift of $u\in T_{x}M$, $u^H$ is the horizontal lift of $u$ with respect to $K$, $h\left(u\right)=1/2g(u,u)$ and $\alpha ,\beta$ are smooth real functions defined on $R$. All natural 2-vector fields are of the form $$\Lambda \left(u\right)={\gamma }_1\left(h\left(u\right)\right)\Lambda (g,K)+{\gamma }_2\left(h\left(u\right)\right)u^H\wedge u^V,$$ where ${\gamma }_1$, ${\gamma }_2$ are smooth real functions defined...

New versions of Slovák’s formulas expressing the covariant derivative and curvature of the linear connection ${}_A\Gamma$ are presented.