The main result of this paper determines a system of linear partial differential equations of Cauchy type whose solutions correspond exactly to holomorphically projective mappings of a given equiaffine space onto a Kählerian space. The special case of constant holomorphic curvature is also studied.

The author obtains sufficient conditions of the finite independence and the commutativity for local as well as non-local homogeneous symmetries of a large class of $(1+1)$-dimensional evolution systems.

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...

In this paper a Weil approach to quasijets is discussed. For given manifolds $M$ and $N$, a quasijet with source $x\in M$ and target $y\in N$ is a mapping $T_x^{r}M\to T_y^{r}N$ which is a vector homomorphism for each one of the $r$ vector bundle structures of the iterated tangent bundle $T^r$ [A. Dekrét, Casopis Pest. Mat. 111, No. 4, 345-352 (1986; Zbl 0611.58004)]. Let us denote by $Q{J}^r(M,N)$ the bundle of quasijets from $M$ to $N$; the space ${\tilde{J}}^r(M,N)$ of non-holonomic $r$-jets from $M$ to $N$ is embeded into $Q{J}^r(M,N)$. On the other hand, the bundle $Q{T}_m^{r}N$ of $(m,r)$-quasivelocities...

Let $G_{n,k}$ (${\tilde{G}}_{n,k}$) denote the Grassmann manifold of linear $k$-spaces (resp. oriented $k$-spaces) in ${ℝ}^n$, $d_{n,k}=k(n-k)=\text{dim}{G}_{n,k}$ and suppose $n\ge 2k$. As an easy consequence of the Steenrod obstruction theory, one sees that $(d_{n,k}+1)$-fold Whitney sum $(d_{n,k}+1){\xi }_{n,k}$ of the nontrivial line bundle ${\xi }_{n,k}$ over $G_{n,k}$ always has a nowhere vanishing section. The author deals with the following question: What is the least $s$ ($=s_{n,k}$) such that the vector bundle $s{\xi }_{n,k}$ admits a nowhere vanishing section ? Obviously, $s_{n,k}\le d_{n,k}+1$, and for the special case in which $k=1$, it is known that $s_{n,1}=d_{n,1}+1$. Using results...

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...

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....