Summary: Let $𝔤$ be a real semisimple $\left|k\right|$-graded Lie algebra such that the Lie algebra cohomology group $H^1({𝔤}_{-},𝔤)$ is contained in negative homogeneous degrees. We show that if we choose $G=Aut\left(𝔤\right)$ and denote by $P$ the parabolic subgroup determined by the grading, there is an equivalence between regular, normal parabolic geometries of type $(G,P)$ and filtrations of the tangent bundle, such that each symbol algebra $\text{gr}\left(T_{x}M\right)$ is isomorphic to the graded Lie algebra ${𝔤}_{-}$. Examples of parabolic geometries determined by filtrations of the...

Let $\Omega \subset {ℂ}^n$ be a domain with smooth boundary and $p\in \partial \Omega$. A holomorphic function $f$ on $\Omega$ is called a $C^k$ ($k=0,1,2,\cdots$) peak function at $p$ if $f\in C^k\left(\overline{\Omega }\right)$, $f\left(p\right)=1$, and $\left|f\right(q\left)\right|<1$ for all $q\in \overline{\Omega }\setminus \left\{p\right\}$. If $\Omega$ is strongly pseudoconvex, then $C^{\infty }$ peak functions exist. On the other hand, J. E. Fornaess constructed an example in ${ℂ}^2$ to show that this result fails, even for $C^1$ functions, on a weakly pseudoconvex domain [Math. Ann. 227, 173-175 (1977; Zbl 0346.32026)]. Subsequently, E. Bedford and J. E. Fornaess showed that there is always a continuous peak function on a...

[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...