Authors’ abstract: “4-quasiplanar mappings of almost quaternionic spaces with affine connection without torsion are investigated. Geometrically motivated definitions of these mappings are presented. Based an these definitions, fundamental forms of these mappings are found, which are equivalent to the forms of 4-quasiplanar mappings introduced a priori by I. Kurbatova [Sov. Math. 30, 100-104 (1986; Zbl 0602.53029)]”.

A G-structure on a Riemannian manifold is said to be integrable if it is preserved by the Levi-Civita connection. In the presented paper, the following non-integrable G-structures are studied: SO(3)-structures in dimension 5; almost complex structures in dimension 6; G${}_2$-structures in dimension 7; Spin(7)-structures in dimension 8; Spin(9)-structures in dimension 16 and F${}_4$-structures in dimension 26. G-structures admitting an affine connection with totally skew-symmetric torsion are characterized....

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...

Summary: Starting from the physical point of view on the Miura transformation as reflectionless potential and its connection with supersymmetry we define a scaling $q$-deformation of this to obtain $q$-deformed supersymmetric quantum mechanics. An application to an inverse scattering problem is given.

From the text: The author reviews recent research on quantum deformations of the Poincaré supergroup and superalgebra. It is based on a series of papers (coauthored by P. Kosiński, J. Lukierski, P. Maślanka and A. Nowicki) and is motivated by both mathematics and physics. On the mathematical side, some new examples of noncommutative and noncocommutative Hopf superalgebras have been discovered. Moreover, it turns out that they have an interesting internal structure of graded bicrossproduct. As far...