The paper deals with extensions of the finite operator calculus of G.-C. Rota called $\psi$-extensions which give a framework for corresponding quantum group investigations. This also covers the instance of the well-known $q$-analogue of umbral calculus. The article also contains glossaries of the most important terms and notations used by Ward, Viskov, Markowsky and Roman on one side and the Rota-oriented notations on the other side.

We completely classify Riemannian $g$-natural metrics of constant sectional curvature on the unit tangent sphere bundle $T_{1}M$ of a Riemannian manifold $(M,g)$. Since the base manifold $M$ turns out to be necessarily two-dimensional, weaker curvature conditions are also investigated for a Riemannian $g$-natural metric on the unit tangent sphere bundle of a Riemannian surface.

A Lie version of Turaev’s $\overline{G}$-Frobenius algebras from 2-dimensional homotopy quantum field theory is proposed. The foundation for this Lie version is a structure we call a $𝔤$-quasi-Frobenius Lie algebra for $𝔤$ a finite dimensional Lie algebra. The latter consists of a quasi-Frobenius Lie algebra $(𝔮,\beta )$ together with a left $𝔤$-module structure which acts on $𝔮$ via derivations and for which $\beta$ is $𝔤$-invariant. Geometrically, $𝔤$-quasi-Frobenius Lie algebras are the Lie algebra structures associated to symplectic...

There exist exactly four homomorphisms $\varphi$ from the pseudo-orthogonal group of index one $G=O(n,1,ℝ)$ into the group of real numbers ${ℝ}_0.$ Thus we have four $G$-spaces of $\varphi$-scalars $(ℝ,G,h_{\varphi })$ in the geometry of the group $G.$ The group $G$ operates also on the sphere $S^{n-2}$ forming a $G$-space of isotropic directions $(S^{n-2},G,*).$ In this note, we have solved the functional equation $F(A*q_1,A*q_2,\cdots ,A*q_m)=\varphi \left(A\right)·F(q_1,q_2,\cdots ,q_m)$ for given independent points $q_1,q_2,\cdots ,q_m\in S^{n-2}$ with $1\le m\le n$ and an arbitrary matrix $A\in G$ considering each of all four homomorphisms. Thereby we have determined all equivariant mappings $F:{\left(S^{n-2}\right)}^m\to ℝ.$

We introduce a generalization to the second order of the notion of the G1-structure, the so called generalized almost tangent structure. For this purpose, the concepts of the second order frame bundle H2(Vm), its structural group Lm2 and its associated tangent bundle of second order T2(Vm) of a differentiable manifold Vm are described from the point of view that is used. Then, a G1-structure of second order -called G12-structure- is constructed on Vm by an endorphism J acting on T2(Vm), satisfying...

In this article, we obtain a gap property of energy densities of harmonic maps from a closed Riemannian manifold to a Grassmannian and then, use it to Gaussian maps of some submanifolds to get a gap property of the second fundamental forms.

We study gauge transformations of Dirac structures and the relationship between gauge and Morita equivalences of Poisson manifolds. We describe how the symplectic structure of a symplectic groupoid is affected by a gauge transformation of the Poisson structure on its identity section, and prove that gauge-equivalent integrable Poisson structures are Morita equivalent. As an example, we study certain generic sets of Poisson structures on Riemann surfaces: we find complete gauge-equivalence invariants...

The classification of class VII surfaces is a very difficult classical problem in complex geometry. It is considered by experts to be the most important gap in the Enriques-Kodaira classification table for complex surfaces. The standard conjecture concerning this problem states that any minimal class VII surface with b₂ > 0 has b₂ curves. By the results of [Ka1]-[Ka3], [Na1]-[Na3], [DOT], [OT] this conjecture (if true) would solve the classification problem completely. We explain a new approach...