An axiom of unique reflection
We characterize neutral planes by three metric conditions: existence of geodesics, side-side-side congruence, and uniqueness of reflection.
math.MG↗
arXiv subjects
Nina Lebedeva
Publications and source records attributed to Nina Lebedeva.
Metric foundations of geometry
A metric space is called all-set-homogeneous if every isometry between two of its subsets extends to an isometry of the whole space. We classify all-set-homogeneous length spaces: besides the classical examples, they include the universal metric trees of finite valence.
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Alexandrov Regions
We introduce the notion of Alexandrov region. In contrast to Alexandrov spaces these are intrinsic metric spaces, which satisfy a lower curvature bound and have finite Hausdorff dimension, might however, be non-complete. We show a quantitative version of the statement that such regions are locally isometric to an Alexandrov space.
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Quadratic metric comparisons
We study the effects on length spaces imposed by quadratic inequalities on the six distances between the points in every quadruple.
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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.
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Trees meet octahedron comparison
We show that trees and their products meet octahedron comparison.
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Curvature tensor of smoothable Alexandrov spaces
We prove weak convergence of curvature tensors of Riemannian manifolds for converging noncollapsing sequences with a lower bound on sectional curvature.
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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.
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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.
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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.
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Alexandrov's embedding theorem
Written for the book "Mathematicians from Saint Petersburg and their theorems".
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5-point CAT(0) spaces after Tetsu Toyoda
We give another proof of Toyoda's theorem that describes 5-point subpaces in CAT(0) length spaces
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Locally homogeneous $C^0$-Riemannian manifolds
We show that locally homogeneous $\mathcal{C}^0$-Riemannian manifolds are smooth.
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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.
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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.
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On the total curvature of minimizing geodesics on convex surfaces
We give a universal upper bound for the total curvature of minimizing geodesic on a convex surface in the Euclidean space.
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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.
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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.
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