Invariant Einstein metrics on generalized Wallach spaces

Invariant Einstein metrics on generalized Wallach spaces have been classified except SO(k + l+ m)/SO(k) × SO(l) × SO(m). In this paper, we first give a survey on the study of invariant Einstein metrics on generalized Wallach spaces, and prove that there are infinitely many spaces of the type SO(k + l + m)/SO(k)× SO(l) × SO(m) admitting exactly two, three, or four invariant Einstein metrics up to a homothety.     

Correction to: Classification of generalized Wallach spaces

Geometriae Dedicata - A Correction to this paper has been published: https://doi.org/10.1007/s10711-015-0119-z     

Geodesics on weighted projective spaces

We study the inverse spectral problem for weighted projective spaces using wave-trace methods. We show that in many cases one can “hear” the weights of a weighted projective space.     

Geodesics in Randers spaces of constant curvature

Geodesics in Randers spaces of constant curvature are classified.

     

Geodesics on spaces of almost hermitian structures

A natural metric on the space of all almost hermitian structures on a given manifold is investigated.     

Automorphisms in generalized spaces

A survey of results on the theory of automorphisms in classical spaces with connections, Finsler spaces, and spaces of support elements are presented.Translated from Itogi Nauki i Tekhniki. Problemy Geometrii, Vol. 10, pp. 147–191, 1978.     

On the evolution of invariant Riemannian metrics on one class of generalized Wallach spaces under the influence of the normalized Ricci flow

The paper is devoted to the study of the evolution of invariant Riemannian metrics on the special class of generalized Wallach spaces corresponding to the case of a₁ = a₂ = a₃ = 1/4. We prove that the normalized Ricci flow evolves all generic invariant Riemannian metrics into metrics with positive Ricci curvature.     

Baire generalized topological spaces, generalized metric spaces and infinite games

Different generalizations of topological Baire spaces to the case of generalized topological spaces are considered and the properties of such spaces are examined. In particular, these considerations are focused on the relationship between Baire generalized topological spaces and semi-continuous real functions and infinite games. The notion of generalized metric spaces corresponding to generalized topological spaces is introduced as an important tool in this discussion.     

Completeness in generalized ultrametric spaces

We analyze the relationship between four notions of completeness for Γ-ultrametric spaces. The notions we consider include Cauchy completeness, strong Cauchy completeness, spherical completeness and injectivity. In the process we show that the category of Γ-ultrametric spaces is equivalent to the category of flabby separated presheaves on Γ ^op .     

Frames in generalized Fock spaces

Let A denote a real linear transformation on Cn which is symmetric and positive-definite relative to the real inner product Re〈z,w〉, z,w∈Cn. Let FA(Cn) denote the Fock space consisting of holomorphic functions on Cn which are square integrable with respect to the Gaussian measure . For w∈Cn, let , z∈Cn, where KA is the reproducing kernel for FA(Cn). The main aim of this paper is to show that there exist a,b>0 such that the set of functions forms a frame in FA.     

Geodesics as products of one-parameter subgroups in compact lie groups and homogeneous spaces

We study the geodesic equation for compact Lie groups G and homogeneous spaces $G/H$, and we prove that the geodesics are orbits of products $\exp (t{X}_1)\cdots \exp (t{X}_N)$ of one-parameter subgroups of G, provided that a simple algebraic condition for the Riemannian metric is satisfied. For the group $SO(3)$, we relate this type of geodesics to the free motion of a symmetric top. Moreover, by using series of Lie subgroups of G, we construct a wealth of metrics having the aforementioned type of geodesics.     

Elliptic problems in generalized Orlicz-Musielak spaces

We consider a strongly nonlinear monotone elliptic problem in generalized Orlicz-Musielak spaces. We assume neither a ??₂ nor ?₂-condition for an inhomogeneous and anisotropic N-function but assume it to be log-H?lder continuous with respect to x. We show the existence of weak solutions to the zero Dirichlet boundary value problem. Within the proof the L ^??-truncation method is coupled with a special version of the Minty-Browder trick for non-reflexive and non-separable Banach spaces.     

Regular sets in generalized topological spaces

We define and study regular and locally closed sets in generalized topological spaces.     

Newton's method in generalized banach spaces

A Kantorowitsch-analysis of Newton's method in generalized Banach spaces is given. The application of generalized norms – mappings from a linear space into a partially ordered Banach space - improves convergence results and error estimatescompared with the real norm theory.     

Adjacency in generalized projective Veronese spaces

A bijection of strong subspaces of a generalized Veronese space preserving the adjacency need not to be determined by an automorphism of the underlying space. We give conditions which assure that the adjacency preserving bijection of points (or lines) of a generalized Veronese space is determined by an automorphism of this space. The results are applied to Veronese spaces associated with projective structures.     

Wavelet bases in generalized Besov spaces

In this paper we obtain a wavelet representation in (inhomogeneous) Besov spaces of generalized smoothness via interpolation techniques. As consequence, we show that compactly supported wavelets of Daubechies type provide an unconditional Schauder basis in these spaces when the integrability parameters are finite.