Lectures on Alexandrov spaces with curvature bounded below

Vitali Kapovitch; Nina Lebedeva; Anton Petrunin

Lectures on Alexandrov spaces with curvature bounded below

Differential Geometry 2026-07-14 v1 General Topology Metric Geometry

Authors: Vitali Kapovitch , Nina Lebedeva , Anton Petrunin

Abstract

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.

Keywords

Alexandrov spaces curvature bounded below comparison geometry globalization theorem splitting theorem Gromov selection theorem

Cite

@article{arxiv.2607.12842,
  title  = {Lectures on Alexandrov spaces with curvature bounded below},
  author = {Vitali Kapovitch and Nina Lebedeva and Anton Petrunin},
  journal= {arXiv preprint arXiv:2607.12842},
  year   = {2026}
}

Comments

161 pages, 29 figures