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.

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...

Summary: The AHS-structures on manifolds are the simplest cases of the so called parabolic geometries which are modeled on homogeneous spaces corresponding to a parabolic subgroup in a semisimple Lie group. It covers the cases where the negative parts of the graded Lie algebras in question are abelian. In the series the authors developed a consistent frame bundle approach to the subject. Here we give explicit descriptions of the obstructions against the flatness of such structures based on the latter...

This paper has two parts. Part one is mainly intended as a general introduction to the problem of sectioning vector bundles (in particular tangent bundles of smooth manifolds) by everywhere linearly independent sections, giving a survey of some ideas, methods and results.Part two then records some recent progress in sectioning tangent bundles of several families of specific manifolds.

The main result of this brief note asserts, incorrectly, that there exists a rational fibration $S^2\to E\to ℂP^3$ whose total space admits nonzero Massey products. The methods used would be appropriate for showing results of this kind, if the circumstances were to allow for it. Unfortunately the author makes a simple, but nonetheless fatal, computational error in his calculation that ostensibly shows the existence of a nonzero Massey product (p. 249, 1.13: $ab\ne D\left(x^{2}y\right))$. In fact, for any rational fibration $S^2\to E\to ℂP^3$ the total space...

[For the entire collection see Zbl 0699.00032.] A connection structure (M,H) and a path structure (M,S) on the manifold M are called compatible, if $S\left(v\right)=H(v,v),\forall v\in TM,$ locally $G^i(x,y)=y^j{\Gamma }_j^i(x,y),$ where $G^i$ and ${\Gamma }_j^i$ express the semi-spray S and the connection map H, resp. In the linear case of H its geodesic spray S is quadratic: $G^i(x,y)={\Gamma }_{jk}^i\left(k\right)y^j{y}^k.$ On the contrary, the homogeneity condition of S induces the relation for the compatible connection H, $y^j({\Gamma }_j^i\circ {\mu }_t)=t{y}^j{\Gamma }_j^i,$ whence it follows not that H is linear, i.e. if a connection structure is compatible with a spray, then...

[For the entire collection see Zbl 0699.00032.] The author considers the conformal relation between twistors and spinors on a Riemannian spin manifold of dimension $n\ge 3$. A first integral is constructed for a twistor spinor and various geometric properties of the spin manifold are deduced. The notions of a conformal deformation and a Killing spinor are considered and such a deformation of a twistor spinor into a Killing spinor and conditions for the equivalence of these quantities is indicated.

[For the entire collection see Zbl 0699.00032.] A new cohomology theory suitable for understanding of nonlinear partial differential equations is presented. This paper is a continuation of the following paper of the author [Differ. geometry and its appl., Proc. Conf., Brno/Czech. 1986, Commun., 235-244 (1987; Zbl 0629.58033)].

The opportunity for verifying the basic principles of quantum theory and possible $q$-deformation appears in quantum cryptography (QC) – a new discipline of physics and information theory.The author, member of the group of cryptology of Praha, presents in this paper the possibility to verify the $q$-deformation of Heisenberg uncertainty relation $q$-deformed QM and possible discretization on the base of a model presented in the fourth section.In the seven sections, the author discusses these problems....