Lobachevsky Geometry and Stellar Parallaxes

Siberian Mathematical Journal - Using Beltrami–Poincaré models in the Euclidean semiplane and semispace for two-dimensional and three-dimensional Lobachevsky spaces, we give...     

Solution of a Busemann problem

We give a solution to the problem posed by Busemann which is as follows: Determine the noncompact Busemann G-spaces such that for every two geodesics there exists a motion taking one to the other. We prove that each of these spaces is isometric to the Euclidean space or to one of the noncompact symmetric spaces of rank 1 (of negative sectional curvature).     

Uniform universal covers of uniform spaces

We develop a generalized covering space theory for a class of uniform spaces called coverable spaces. Coverable spaces include all geodesic metric spaces, connected and locally pathwise connected compact topological spaces, in particular Peano continua, as well as more pathological spaces like the topologist's sine curve. The uniform universal cover of a coverable space is a kind of generalized cover with universal and lifting properties in the category of uniform spaces and uniformly continuous mappings. Associated with the uniform universal cover is a functorial uniform space invariant called the deck group, which is related to the classical fundamental group by a natural homomorphism. We obtain some specific results for one-dimensional spaces.     

On Urysohn’s ℝ-Tree

In the short note of 1927, Urysohn constructed the metric space R that is nowhere locally separable. There is no publication with indications that R is a (noncomplete) ?-tree that has valency c at each point. The author in 1989, as well as Polterovich and Shnirelman in 1997, constructed ?-trees isometric to R unaware of the paper by Urysohn. In this paper the author considers various constructions of the ?-tree R and of the minimal complete ?-tree of valency c including R, as well as the characterizations of ?-trees, their properties, and connections with ultrametric spaces.

     

Abnormal Extremals of Left-Invariant Sub-Finsler Quasimetrics on Four-Dimensional Lie Groups with Three-Dimensional Generating Distributions

Siberian Mathematical Journal - We find the three-dimensional subspaces of four-dimensional Lie algebras which generate the algebras, as well as abnormal extremals on the connected Lie groups...     

Abnormal Extremals of Left-Invariant Sub-Finsler Quasimetrics on Four-Dimensional Lie Groups

Siberian Mathematical Journal - We find the abnormal extremals on four-dimensional connected Lie groups with left-invariant sub-Finsler quasimetric defined by a seminorm on a...     

A Metric Characterization of Riemannian Submersions

A map between metric spaces is called a submetry if it mapsballs of radius R around a point onto balls of the sameradius around the image point. We show that when the domain and targetspaces are complete Riemannian manifolds, submetries correspond toC ^1,1 Riemannian submersions. We also study someconsequences of this fact, and introduce the notion of submetries with asoul.