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.

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....

For natural numbers $r$ and $n\ge 2$ all natural operators $T_{|ℳf_n}⇝T^{*}\left(J^r{T}^{*}\right)$ transforming vector fields from $n$-manifolds $M$ into $1$-forms on $J^r{T}^{*}M=\{j_x^r\left(\omega \right)\mid \omega \in {\Omega }^1\left(M\right),x\in M\}$ are classified. A similar problem with fibered manifolds instead of manifolds is discussed.

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...

In this paper we study the geometry of direct connections in smooth vector bundles (see N. Teleman [Tn.3]); we show that the infinitesimal part, ${\nabla }^{\tau }$, of a direct connection τ is a linear connection. We determine the curvature tensor of the associated linear connection ${\nabla }^{\tau }.$ As an application of these results, we present a direct proof of N. Teleman’s Theorem 6.2 [Tn.3], which shows that it is possible to represent the Chern character of smooth vector bundles as the periodic cyclic homology class of a...

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...