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...

The paper generalizes results of H. H. Hacisalihoglu and A. Kh. Amirov [Dokl. Akad. Nauk, Ross. Akad. Nauk 351, No. 3, 295-296 (1996; Zbl 0895.53038) and Sib. Mat. Zh. 39, No. 4, 1005-1012 (1998; Zbl 0913.53019)] on the existence and uniqueness of a Riemannian metric on a domain in ${ℝ}^n$ given prescribed values for some of the components of the Riemann curvature tensor and initial values of the metric and its partial derivatives. The authors establish the construction (existence and uniqueness) of a...

[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...

The article contains a list of 7 problems related to operads and configuration spaces. Problems 1-2 are about the compactification of configuration spaces (homology and Koszulness, geometric decompositions). Problems 3-4 are about configuration spaces related to knot invariants, their geometry and Koszulness. Problems 5 to 7 are related to (operadically defined) traces and cyclic homology.

[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.

Some special linear connection introduced in the Finsler space by Ichijyō has the property that the curvature tensors under conformal changes remain invariant. Two Ichijyō manifolds $(M,E,\nabla )$ and $(M,\overline{E},\overline{\nabla })$ are said to be conformally equivalent if $\overline{E}=(exp{\sigma }^v)E$, $\sigma \in C^{\infty }\left(M\right)$.It is proved, that in this case, the following assertions are equivalent: 1. $\sigma$ is constant, 2. $h_{\nabla }=h_{\overline{\nabla }}$, 3. $S_{\nabla }=S_{\overline{\nabla }}$, 4. $t_{\nabla }=t_{\overline{\nabla }}$.It is also proved (when the above conditions are satisfied) that1. If $(M,E,\nabla )$ is a generalized Berwald manifold, then $(M,\overline{E},\overline{\nabla })$ is also a generalized Berwald manifold.2....

Author’s abstract: “We introduce the concept of the flux homomorphism for regular Poisson manifolds. First we establish a one-to-one correspondence between Poisson diffeomorphisms close to $id$ and closed foliated 1-forms close to 0. This allows to show that the group of Poisson automorphisms is locally contractible and to define the flux locally. Then, by means of the foliated cohomology, we extend this local homomorphism to a global one”.

[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)].

Summary: The $r$-th order variational sequence is the quotient sequence of the De Rham sequence on the $r$th jet prolongation of a fibered manifold, factored through its contact subsequence.In this paper, the first order variational sequence on a fibered manifold with one-dimensional base is considered. A new representation of all quotient spaces as some spaces of (global) forms is given. The factorization procedure is based on a modification of the interior Euler operator, used in the theory of (infinite)...

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....

In this paper the authors compare two different approaches to the second order absolute differentiation of a fibered manifold (one of them was studied by the authors [Arch. Math., Brno 33, 23-35 (1997; Zbl 0910.53014)]. The main goal is the extension of one approach to connections on functional bundles of all smooth maps between the fibers of two fibered manifolds over the same base (we refer to the book “Natural Operations in Differential Geometry” [Springer, Berlin (1993; Zbl 0782.53013)] I. Kolar,...

The author considers the problem to give explicit descriptions for several types of bundles on smooth manifolds, naturally related with the bundle of $k$-dimensional velocities, or $k$-jets. In fact, this kind of bundles are very natural objects in differential geometry, mechanics and Lagrangian dynamics. For this the author considers Weil bundles that arose from Weil algebras. If a suitable combinatorial data is provided by a simplicial coloured structure, then the author describes the corresponding...

[For the entire collection see Zbl 0742.00067.]In the first part some general results on Hecke algebras are recalled; the structure constants corresponding to the standard basis are defined; in the following the example of the commuting algebra of the Gelfand- Graev representation of the general linear group $GL(2,F)$ is examined; here $F$ is a finite field of $q$ elements; the structure constants are explicitly determined first for the standard basis and then for a new basis obtained via a Mellin-transformation....