The phenomenon of determining a geometric structure on a manifold by the group of its automorphisms is a modern analogue of the basic ideas of the Erlangen Program of F. Klein. The author calls such diffeomorphism groups admissible and he describes them by imposing some axioms. The main result is the followingTheorem. Let $(M_i,{\alpha }_i)$, $i=1,2$, be a geometric structure such that its group of automorphisms $G(M_i,{\alpha }_i)$ satisfies either axioms 1, 2, 3 and 4, or axioms 1, 2, 3’, 4, 5, 6 and 7, and $M_i$ is compact, or axioms 1, 2,...

The author studies relations between the following two types of natural operators: 1. Natural operators transforming vector fields on manifolds into vector fields on a natural bundle $F$; 2. Natural operators transforming vector fields on manifolds into functions on the cotangent bundle of $F$. It is deduced that under certain assumptions on $F$, all natural operators of the second type can be constructed through those of the first one.

The author develops a $q$-analogue of Rota’s finite operator calculus in enumerative combinatorics.

Contrary to a statement of Borsuk the author proves that every locally plane Peano continuum is embeddable into a 2-manifold.

Let $M$ be a manifold with all structures smooth which admits a metric $g$. Let $\Gamma$ be a linear connection on $M$ such that the associated covariant derivative $\nabla$ satisfies $\nabla g=g\otimes w$ for some 1-form $w$ on $M$. Then one refers to the above setup as a Weyl structure on $M$ and says that the pair $(g,w)$ fits $\Gamma$. If $\sigma :M\to ℝ$ and if $(g,w)$ fits $\Gamma$, then $(e^{\sigma g},w+d\sigma )$ fits $\Gamma$. Thus if one thinks of this as a map $g\mapsto w$, then $e^{\sigma g}\mapsto w+d\sigma$.In this paper, the author attempts to apply Weyl’s idea above to Finsler spaces. A Finsler fundamental function $L:TM\to ℝ$ satisfies (i) $L\left(u\right)>0$ for...

[For the entire collection see Zbl 0742.00067.]Let ${𝔤}_k$ be the Lie algebra $𝔤l(k,𝒞)$, and let $U_k$ be the universal enveloping algebra for ${𝔤}_k$. Let $Z_k$ be the center of $U_k$. The authors consider the chain of Lie algebras ${𝔤}_n\supset {𝔤}_{n-1}\supset \cdots \supset {𝔤}_1$. Then $Z=\langle Z_k\mid k=1,2,\cdots n\rangle$ is an associative algebra which is called the Gel’fand-Zetlin subalgebra of $U_n$. A ${𝔤}_n$ module $V$ is called a $GZ$-module if $V={\sum }_x\oplus V\left(x\right)$, where the summation is over the space of characters of $Z$ and $V\left(x\right)=\{v\in V\mid {(a-x\left(a\right))}^{m}v=0$, $m\in {𝒵}_{+}$, $a\in 𝒵\}$. The authors describe several properties of $GZ$- modules. For example, they prove that if $V\left(x\right)=0$ for some $x$...

The author previously studied with F. Ilosvay and B. Kis [Publ. Math. 42, 139-144 (1993; Zbl 0796.53022)] the diffeomorphisms between two Finsler spaces $F^n=(M^n,L)$ and ${\overline{F}}^n=(M^n,\overline{L})$ which map the geodesics of $F^n$ to geodesics of ${\overline{F}}^n$ (geodesic mappings).Now, he investigates the geodesic mappings between a Finsler space $F^n$ and a Riemannian space ${\overline{ℝ}}^n$. The main result of this paper is as follows: if $F^n$ is of constant curvature $K$ and the mapping $F^n\to {\overline{ℝ}}^n$ is a strongly geodesic mapping then $K=0$ or $K\ne 0$ and $\overline{L}=e^{\varphi \left(x\right)}L$.