[For the entire collection see Zbl 0699.00032.] It was previously known that for every principal fibre bundle P there is some corresponding transitive Lie algebroid A(P) - a vector bundle equipped with some structure like the structure of a Lie algebra in the module of sections. The author of this article shows that the Chern-Weil homomorphism of P is a notion of the Lie algebroid of P, i.e. knowing only A(P) of P one can uniquely reproduce the ring of invariant polynomials ${\left(V{g}^{*}\right)}_I$ and the Chern-Weil...

[For the entire collection see Zbl 0699.00032.] In this interesting paper the authors show that all natural operators transforming every projectable vector field on a fibered manifold Y into a vector field on its r-th prolongation $J^{r}Y$ are the constant multiples of the flow operator. Then they deduce an analogous result for the natural operators transforming every vector field on a manifold M into a vector field on any bundle of contact elements over M.

A product preserving functor is a covariant functor $ℱ$ from the category of all manifolds and smooth mappings into the category of fibered manifolds satisfying a list of axioms the main of which is product preserving: $ℱ(M_1\times M_2)=ℱ\left(M_1\right)\times ℱ\left(M_2\right)$. It is known that any product preserving functor $ℱ$ is equivalent to a Weil functor $T^A$. Here $T^A\left(M\right)$ is the set of equivalence classes of smooth maps $\varphi :{ℝ}^n\to M$ and $\varphi ,{\varphi }^{\text{'}}$ are equivalent if and only if for every smooth function $f:M\to ℝ$ the formal Taylor series at 0 of $f\circ \varphi$ and $f\circ {\varphi }^{\text{'}}$ are equal in $A=ℝ\left[[x_1,\cdots ,x_n]\right]/𝔞$. In this paper all...