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Nina Lebedeva

Publications and source records attributed to Nina Lebedeva.

At least 19 recordsLinked to original sources
arXiv2026

Lectures on Alexandrov spaces with curvature bounded below

An introduction to Alexandrov spaces with curvature bounded below. Topics include various comparison conditions, the globalization theorem, tangent spaces and spaces of directions, gradient flows, the splitting theorem, dimension and volume, Gromov's selection theorem, the boundary and the doubling theorem, and quotient spaces. We also give a brief overview of the two-dimensional theory, the main precursor to modern Alexandrov geometry.

math.DG↗
arXiv2025

Quadratic metric comparisons

We study the effects on length spaces imposed by quadratic inequalities on the six distances between the points in every quadruple.

math.DG↗
arXiv2023

All-set-homogeneous spaces

A metric space is said to be all-set-homogeneous if any of its partial isometries can be extended to a genuine isometry. We give a classification of a certain subclass of all-set-homogeneous length spaces.

math.MG↗
arXiv2023

Graph comparison meets Alexandrov

Graph comparison is a certain type of condition on metric space encoded by a finite graph. We show that any nontrivial graph comparison implies one of Alexandrov's comparisons. The proof gives a complete description of graphs with trivial graph comparisons.

math.MG↗
arXiv2023

Five-point Toponogov theorem

We give an if-and-only-if condition on a five-point metric spaces that admit embeddings into nonnegatively curved Riemannian manifolds.

math.DG↗
arXiv2019

Self-contracted curves in spaces with weak lower curvature bound

We show that bounded self-contracted curves are rectifiable in metric spaces with weak lower curvature bound in a sense we introduce in this article. This class of spaces is wide and includes, for example, finite-dimensional Alexandrov spaces of curvature bounded below and Berwald spaces of nonnegative flag curvature. (To be more precise, our condition is regarded as a strengthened doubling condition and holds also for a certain class of metric spaces with upper curvature bound.) We also provide the non-embeddability of large snowflakes into (balls in) metric spaces in the same class. We follow the strategy of the last author's previous paper based on the small rough angle condition, where spaces with upper curvature bound are considered. The results in this article show that such a strategy applies to spaces with lower curvature bound as well.

math.MG↗
arXiv2019

Bipolar comparison

We define a new type of metric comparison similar to the comparison of Alexandrov. We show that it has strong connections to continuity of optimal transport between regular measures on a Riemannian manifold, in particular to the so called MTW condition introduced by Xi-Nan Ma, Neil Trudinger and Xu-Jia Wang.

math.DG↗
arXiv2018

On open flat sets in spaces with bipolar comparison

We show that if a Riemannian manifold satisfies (3,3)-bipolar comparisons and has an open flat subset then it is flat. The same holds for a version of MTW where the perpendicularity is dropped. In particular we get that the (3,3)-bipolar comparison is strictly stronger than the Alexandrov comparison.

math.DG↗
arXiv2015

Smoothing 3-dimensional polyhedral spaces

We show that 3-dimensional polyhedral manifolds with nonnegative curvature in the sense of Alexandrov can be approximated by nonnegatively curved 3-dimensional Riemannian manifolds.

math.DG↗
arXiv2013

Alexandrov spaces with maximal number of extremal points

We show that any n-dimensional nonnegatively curved Alexandrov space with the maximal possible number of extremal points is isometric to a quotient space of Euclidean n -space by an action of a crystallographic group. We describe all such actions.

math.MG↗